Cohen-Tannoudji, Diu and Laloë - Quantum Mechanics (vol. I, II and III, 2nd ed.).pdf

QUANTUMMECHANICS
olume I
Basic Concepts, Tools, and Applications
Claude Cohen-Tannoudji, Bernard Diu,
and Franck Laloë
Translated from the French bySusan Reid Hemley,
Nicole Ostrowsky, and Dan Ostrowsky
V

Authors
Prof. Dr. Claude Cohen-Tannoudji
Laboratoire Kastler Brossel (ENS)
24 rue Lhomond
75231 Paris Cedex 05
France
Prof. Dr. Bernard Diu
4 rue du Docteur Roux
91440 Boures-sur-Yvette
France
Prof. Dr. Frank Laloë
Laboratoire Kastler Brossel (ENS)
24 rue Lhomond
75231 Paris Cedex 05
France
Cover Image
© antishock/Getty Images
Second Edition
All books published by Wiley-VCH are
carefully produced. Nevertheless, authors,
editors, and publisher do not warrant the
information contained in these books,
including this book, to be free of errors.
Readers are advised to keep in mind that
statements, data, illustrations, procedural
details or other items may inadvertently be
inaccurate.
Library of Congress Card No.:
applied for
British Library Cataloguing-in-Publication
Data:
A catalogue record for this book is available
from the British Library.
Bibliographic information published by the
Deutsche Nationalbibliothek
The Deutsche Nationalbibliothek lists this
publication in the Deutsche Nationalbiblio
graﬁe; detailed bibliographic data are avail
able on the Internet at http://dnb.d-nb.de.
© 2020 WILEY-VCH Verlag GmbH &
Co. KGaA, Boschstr. 12, 69469 Weinheim,
Germany
All rights reserved (including those of trans
lation into other languages). No part of
this book may be reproduced in any form –
by photoprinting, microﬁlm, or any other
means – nor transmitted or translated into
a machine language without written per
mission from the publishers. Registered
names, trademarks, etc. used in this book,
even when not speciﬁcally marked as such,
are not to be considered unprotected by
law.
Print ISBN978-3-527-34553-3
ePDF ISBN978-3-527-82270-6
ePub ISBN978-3-527-82271-3
Cover DesignTata Consulting Services
Printing and BindingCPI Ebner & Spiegel
Printed on acid-free paper.

Directions for Use
This book is composed of chapters and their complements:
The chapterscontain the fundamental concepts. Except for a few
additions and variations, they correspond to a course given in the last
year of a typical undergraduate physics program (Volume I) or of a
graduate program (Volumes II and III). The 21 chapters arecomplete in
themselvesand can be studied independently of the complements.
The complementsfollow the corresponding chapter. Each is labelled
by a letter followed by a subscript, which gives the number of the chapter
(for example, the complements of Chapter V are, in order, AV, BV, CV,
etc.). They can be recognized immediately by the symbolthat appears
at the top of each of their pages.
The complements vary in character. Some are intended to expand the
treatment of the corresponding chapter or to provide a more detailed
discussion of certain points. Others describe concrete examples or in-
troduce various physical concepts. One of the complements (usually the
last one) is a collection of exercises.
Thedicultyof the complements varies. Some are very simple examples
or extensions of the chapter. Others are more dicult and at the grad-
uate level or close to current research. In any case, the reader should
have studied the material in the chapter before using the complements.
The complements are generally independent of one another.The student
should not try to study all the complements of a chapter at once. In
accordance with his/her aims and interests, he/she should choose a small
number of them (two or three, for example), plus a few exercises. The
other complements can be left for later study. To help with the choise,
the complements are listed at the end of each chapter in a reader's
guide, which discusses the diculty and importance of each.
Some passages within the book have been set in small type, and these
can be omitted on a rst reading.

Foreword
Foreword
Quantum mechanics is a branch of physics whose importance has continually in-
creased over the last decades. It is essential for understanding the structure and dynamics
of microscopic objects such as atoms, molecules and their interactions with electromag-
netic radiation. It is also the basis for understanding the functioning of numerous new
systems with countless practical applications. This includes lasers (in communications,
medicine, milling, etc.), atomic clocks (essential in particular for the GPS), transistors
(communications, computers), magnetic resonance imaging, energy production (solar
panels, nuclear reactors), etc. Quantum mechanics also permits understanding surpris-
ing physical properties such as superuidity or supraconductivity. There is currently a
great interest in entangled quantum states whose non-intuitive properties of nonlocality
and nonseparability permit conceiving remarkable applications in the emerging eld of
quantum information. Our civilization is increasingly impacted by technological appli-
cations based on quantum concepts. This why a particular eort should be made in the
teaching of quantum mechanics, which is the object of these three volumes.
The rst contact with quantum mechanics can be disconcerting. Our work grew
out of the authors' experiences while teaching quantum mechanics for many years. It
was conceived with the objective of easing a rst approach, and then aiding the reader
to progress to a more advance level of quantum mechanics. The rst two volumes, rst
published more than forty years ago, have been used throughout the world. They remain
however at an intermediate level. They have now been completed with a third volume
treating more advanced subjects. Throughout we have used a progressive approach to
problems, where no diculty goes untreated and each aspect of the diverse questions is
discussed in detail (often starting with a classical review).
This willingness to go further without cheating or taking shortcuts is built into
the book structure, using two distinct linked texts:chaptersandcomplements. As we
just outlined in the Directions for use, the chapters present the general ideas and
basic concepts, whereas the complements illustrate both the methods and concepts just
exposed.
Volume I presents a general introduction of the subject, followed by a second
chapter describing the basic mathematical tools used in quantum mechanics. While
this chapter can appear long and dense, the teaching experience of the authors has
shown that such a presentation is the most ecient. In the third chapter the postulates
are announced and illustrated in many of the complements. We then go on to certain
important applications of quantum mechanics, such as the harmonic oscillator, which
lead to numerous applications (molecular vibrations, phonons, etc.). Many of these are
the object of specic complements.
Volume II pursues this development, while expanding its scope at a slightly higher
level. It treats collision theory, spin, addition of angular momenta, and both time-
dependent and time-independent perturbation theory. It also presents a rst approach
to the study of identical particles. In this volume as in the previous one, each theoretical
concept is immediately illustrated by diverse applications presented in the complements.
Both volumes I and II have beneted from several recent corrections, but there have also
been additions. Chapter
perturbations, and a complement concerning relaxation has been added.
iv

Foreword
Volume III extends the two volumes at a slightly higher level. It is based on the
use of the creation and annihilation operator formalism (second quantization), which is
commonly used in quantum eld theory. We start with a study of systems of identical
particles, fermions or bosons. The properties of ideal gases in thermal equilibrium are
presented. For fermions, the Hartree-Fock method is developed in detail. It is the base
of many studies in chemistry, atomic physics and solid state physics, etc. For bosons, the
Gross-Pitaevskii equation and the Bogolubov theory are discussed. An original presen-
tation that treats the pairing eect of both fermions and bosons permits obtaining the
BCS (Bardeen-Cooper-Schrieer) and Bogolubov theories in a unied framework. The
second part of volume III treats quantum electrodynamics, its general introduction, the
study of interactions between atoms and photons, and various applications (spontaneous
emission, multiphoton transitions, optical pumping, etc.). The dressed atom method is
presented and illustrated for concrete cases. A nal chapter discusses the notion of quan-
tum entanglement and certain fundamental aspects of quantum mechanics, in particular
the Bell inequalities and their violations.
Finally note that we have not treated either the philosophical implications of quan-
tum mechanics, or the diverse interpretations of this theory, despite the great interest
of these subjects. We have in fact limited ourselves to presenting what is commonly
called the orthodox point of view. It is only in Chapter
questions concerning the foundations of quantum mechanics (nonlocality, etc.). We have
made this choice because we feel that one can address such questions more eciently
after mastering the manipulation of the quantum mechanical formalism as well as its nu-
merous applications. These subjects are addressed in the bookDo we really understand
quantum mechanics?(F. Laloë, Cambridge University Press, 2019); see also section 5 of
the bibliography of volumes I and II.
v

Foreword
Acknowledgments:
Volumes I and II:
The teaching experience out of which this text grew were group eorts, pursued
over several years. We wish to thank all the members of the various groups and partic-
ularly Jacques Dupont-Roc and Serge Haroche, for their friendly collaboration, for the
fruitful discussions we have had in our weekly meetings and for the ideas for problems
and exercises that they have suggested. Without their enthusiasm and valuable help, we
would never have been able to undertake and carry out the writing of this book.
Nor can we forget what we owe to the physicists who introduced us to research,
Alfred Kastler and Jean Brossel for two of us and Maurice Levy for the third. It was in
the context of their laboratories that we discovered the beauty and power of quantum
mechanics. Neither have we forgotten the importance to us of the modern physics taught
at the C.E.A. by Albert Messiah, Claude Bloch and Anatole Abragam, at a time when
graduate studies were not yet incorporated into French university programs.
We wish to express our gratitude to Ms. Aucher, Baudrit, Boy, Brodschi, Emo,
Heywaerts, Lemirre, Touzeau for preparation of the mansucript.
Volume III:
We are very grateful to Nicole and Daniel Ostrowsky, who, as they translated this
Volume from French into English, proposed numerous improvements and clarications.
More recently, Carsten Henkel also made many useful suggestions during his transla-
tion of the text into German; we are very grateful for the improvements of the text
that resulted from this exchange. There are actually many colleagues and friends who
greatly contributed, each in his own way, to nalizing this book. All their complementary
remarks and suggestions have been very helpful and we are in particular thankful to:
Pierre-François Cohadon
Jean Dalibard
Sébastien Gleyzes
Markus Holzmann
Thibaut Jacqmin
Philippe Jacquier
Amaury Mouchet
Jean-Michel Raimond
Félix Werner
Some delicate aspects of Latex typography have been resolved thanks to Marco
Picco, Pierre Cladé and Jean Hare. Roger Balian, Edouard Brézin and William Mullin
have oered useful advice and suggestions. Finally, our sincere thanks go to Geneviève
Tastevin, Pierre-François Cohadon and Samuel Deléglise for their help with a number of
gures.
vi

Table of contents
Volume I
Table of contents
I WAVES AND PARTICLES. INTRODUCTION TO THE BASIC
IDEAS OF QUANTUM MECHANICS
A Electromagnetic waves and photons
B Material particles and matter waves
C Quantum description of a particle. Wave packets
D Particle in a time-independent scalar potential
READER'S GUIDE FOR COMPLEMENTS 33
AIOrder of magnitude of the wavelengths associated with material
particles
BIConstraints imposed by the uncertainty relations
1 Macroscopic system
2 Microscopic system
CIHeisenberg relation and atomic parameters
DIAn experiment illustrating the Heisenberg relations
EIA simple treatment of a two-dimensional wave packet
1 Introduction
2 Angular dispersion and lateral dimensions
3 Discussion
FIThe relationship between one- and three-dimensional problems
1 Three-dimensional wave packet
2 Justication of one-dimensional models
GIOne-dimensional Gaussian wave packet: spreading of the wave packet
1 Denition of a Gaussian wave packet
2 Calculation of and; uncertainty relation
3 Evolution of the wave packet
HIStationary states of a particle in one-dimensional square potentials
1 Behavior of a stationary wave function(). . . . . . . . . . . . . . . . . . .
2 Some simple cases
JIBehavior of a wave packet at a potential step
1 Total reection: 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Partial reection: 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
KIExercises
***********
vii

Table of contents
II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
A Space of the one-particle wave function
B State space. Dirac notation
C Representations in state space
D Eigenvalue equations. Observables
E Two important examples of representations and observables
F Tensor product of state spaces
READER'S GUIDE FOR COMPLEMENTS 159
AIIThe Schwarz inequality
BIIReview of some useful properties of linear operators
1 Trace of an operator
2 Commutator algebra
3 Restriction of an operator to a subspace
4 Functions of operators
5 Derivative of an operator
CIIUnitary operators
1 General properties of unitary operators
2 Unitary transformations of operators
3 The innitesimal unitary operator
DIIA more detailed study of the r and p representations
1 The rrepresentation
2 The prepresentation
EIISome general properties of two observables,and, whose commu-
tator is equal to~ 187
1 The operator (): denition, properties
2 Eigenvalues and eigenvectors of. . . . . . . . . . . . . . . . . . . . . . . .
3 The qrepresentation
4 The prepresentation. The symmetric nature of thePandQobservables
FIIThe parity operator
1 The parity operator
2 Even and odd operators
3 Eigenstates of an even observableB+. . . . . . . . . . . . . . . . . . . . . .
4 Application to an important special case
GIIAn application of the properties of the tensor product: the two-
dimensional innite well
1 Denition; eigenstates
2 Study of the energy levels
HIIExercises
viii

Table of contents
III THE POSTULATES OF QUANTUM MECHANICS
A Introduction
B Statement of the postulates
C The physical interpretation of the postulates concerning observables and their
measurement
D The physical implications of the Schrödinger equation
E The superposition principle and physical predictions
READER'S GUIDE FOR COMPLEMENTS 267
AIIIParticle in an innite one-dimensional potential well
1 Distribution of the momentum values in a stationary state
2 Evolution of the particle's wave function
3 Perturbation created by a position measurement
BIIIStudy of the probability current in some special cases
1 Expression for the current in constant potential regions
2 Application to potential step problems
3 Probability current of incident and evanescent waves, in the case of reection
from a two-dimensional potential step
CIIIRoot mean square deviations of two conjugate observables
1 The Heisenberg relation forand. . . . . . . . . . . . . . . . . . . . . . .
2 The minimum wave packet
DIIIMeasurements bearing on only one part of a physical system
1 Calculation of the physical predictions
2 Physical meaning of a tensor product state
3 Physical meaning of a state that is not a tensor product
EIIIThe density operator
1 Outline of the problem
2 The concept of a statistical mixture of states
3 The pure case. Introduction of the density operator
4 A statistical mixture of states (non-pure case)
5 Use of the density operator: some applications
FIIIThe evolution operator
1 General properties
2 Case of conservative systems
GIIIThe Schrödinger and Heisenberg pictures
HIIIGauge invariance
1 Outline of the problem: scalar and vector potentials associated with an elec-
tromagnetic eld; concept of a gauge
2 Gauge invariance in classical mechanics
3 Gauge invariance in quantum mechanics
ix

Table of contents
JIIIPropagator for the Schrödinger equation
1 Introduction
2 Existence and properties of a propagator(21). . . . . . . . . . . . . . . .
3 Lagrangian formulation of quantum mechanics
KIIIUnstable states. Lifetime
1 Introduction
2 Denition of the lifetime
3 Phenomenological description of the instability of a state
LIIIExercises
MIIIBound states in a potential well of arbitrary shape
1 Quantization of the bound state energies
2 Minimum value of the ground state energy
NIIIUnbound states of a particle in the presence of a potential well or
barrier
1 Transmission matrix (). . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Transmission and reection coecients
3 Example
OIIIQuantum properties of a particle in a one-dimensional periodic struc-
ture 375
1 Passage through several successive identical potential barriers
2 Discussion: the concept of an allowed or forbidden energy band
3 Quantization of energy levels in a periodic potential; eect of boundary con-
ditions
***********
IV APPLICATIONS OF THE POSTULATES TO SIMPLE CASES:
SPIN 1/2 AND TWO-LEVEL SYSTEMS
A Spin 1/2 particle: quantization of the angular momentum
B Illustration of the postulates in the case of a spin 1/2
C General study of two-level systems
READER'S GUIDE FOR COMPLEMENTS 423
AIVThe Pauli matrices
1 Denition; eigenvalues and eigenvectors
2 Simple properties
3 A convenient basis of the22matrix space
BIVDiagonalization of a22Hermitian matrix
1 Introduction
2 Changing the eigenvalue origin
3 Calculation of the eigenvalues and eigenvectors
x

Table of contents
CIVFictitious spin 1/2 associated with a two-level system
1 Introduction
2 Interpretation of the Hamiltonian in terms of ctitious spin
3 Geometrical interpretation
DIVSystem of two spin 1/2 particles
1 Quantum mechanical description
2 Prediction of the measurement results
EIVSpin12density matrix
1 Introduction
2 Density matrix of a perfectly polarized spin (pure case)
3 Example of a statistical mixture: unpolarized spin
4 Spin 1/2 at thermodynamic equilibrium in a static eld
5 Expansion of the density matrix in terms of the Pauli matrices
FIVSpin 1/2 particle in a static and a rotating magnetic elds: magnetic
resonance
1 Classical treatment; rotating reference frame
2 Quantum mechanical treatment
3 Relation between the classical treatment and the quantum mechanical treat-
ment: evolution ofM. . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Bloch equations
GIVA simple model of the ammonia molecule
1 Description of the model
2 Eigenfunctions and eigenvalues of the Hamiltonian
3 The ammonia molecule considered as a two-level system
HIVEects of a coupling between a stable state and an unstable state
1 Introduction. Notation
2 Inuence of a weak coupling on states of dierent energies
3 Inuence of an arbitrary coupling on states of the same energy
JIVExercises
***********
V THE ONE-DIMENSIONAL HARMONIC OSCILLATOR
A Introduction
B Eigenvalues of the Hamiltonian
C Eigenstates of the Hamiltonian
D Discussion
READER'S GUIDE FOR COMPLEMENTS 525
xi

Table of contents
AVSome examples of harmonic oscillators
1 Vibration of the nuclei of a diatomic molecule
2 Vibration of the nuclei in a crystal
3 Torsional oscillations of a molecule: ethylene
4 Heavy muonic atoms
BVStudy of the stationary states in the x representation. Hermite poly-
nomials
1 Hermite polynomials
2 The eigenfunctions of the harmonic oscillator Hamiltonian
CVSolving the eigenvalue equation of the harmonic oscillator by the
polynomial method
1 Changing the function and the variable
2 The polynomial method
DVStudy of the stationary states in the momentum representation
1 Wave functions in momentum space
2 Discussion
EVThe isotropic three-dimensional harmonic oscillator
1 The Hamiltonian operator
2 Separation of the variables in Cartesian coordinates
3 Degeneracy of the energy levels
FVA charged harmonic oscillator in a uniform electric eld
1 Eigenvalue equation of(E)in the representation
2 Discussion
3 Use of the translation operator
GVCoherent quasi-classical states of the harmonic oscillator
1 Quasi-classical states
2 Properties of the states
3 Time evolution of a quasi-classical state
4 Example: quantum mechanical treatment of a macroscopic oscillator
HVNormal vibrational modes of two coupled harmonic oscillators
1 Vibration of the two coupled in classical mechanics
2 Vibrational states of the system in quantum mechanics
JVVibrational modes of an innite linear chain of coupled harmonic
oscillators; phonons
1 Classical treatment
2 Quantum mechanical treatment
3 Application to the study of crystal vibrations: phonons
xii

Table of contents
KV Vibrational modes of a continuous physical system. Photons
1 Outline of the problem
2 Vibrational modes of a continuous mechanical system: example of a vibrating
string
3 Vibrational modes of radiation: photons
LVOne-dimensional harmonic oscillator in thermodynamic equilibrium
at a temperature 647
1 Mean value of the energy
2 Discussion
3 Applications
4 Probability distribution of the observable. . . . . . . . . . . . . . . . . . .
MVExercises
***********
VI GENERAL PROPERTIES OF ANGULAR MOMENTUM IN QUAN-
TUM MECHANICS
A Introduction: the importance of angular momentum
B Commutation relations characteristic of angular momentum
C General theory of angular momentum
D Application to orbital angular momentum
READER'S GUIDE FOR COMPLEMENTS 703
AVISpherical harmonics
1 Calculation of spherical harmonics
2 Properties of spherical harmonics
BVIAngular momentum and rotations
1 Introduction
2 Brief study of geometrical rotationsR. . . . . . . . . . . . . . . . . . . . . .
3 Rotation operators in state space. Example: a spinless particle
4 Rotation operators in the state space of an arbitrary system
5 Rotation of observables
6 Rotation invariance
CVIRotation of diatomic molecules
1 Introduction
2 Rigid rotator. Classical study
3 Quantization of the rigid rotator
4 Experimental evidence for the rotation of molecules
DVIAngular momentum of stationary states of a two-dimensional har-
monic oscillator
1 Introduction
2 Classication of the stationary states by the quantum numbersand . .
3 Classication of the stationary states in terms of their angular momenta
xiii

Table of contents
4 Quasi-classical states
EVIA charged particle in a magnetic eld: Landau levels
1 Review of the classical problem
2 General quantum mechanical properties of a particle in a magnetic eld
3 Case of a uniform magnetic eld
FVIExercises
***********
VII PARTICLE IN A CENTRAL POTENTIAL, HYDROGEN ATOM
A Stationary states of a particle in a central potential
B Motion of the center of mass and relative motion for a system of two inter-
acting particles
C The hydrogen atom
READER'S GUIDE FOR COMPLEMENTS 831
AVIIHydrogen-like systems
1 Hydrogen-like systems with one electron
2 Hydrogen-like systems without an electron
BVIIA soluble example of a central potential: The isotropic three-dimensional
harmonic oscillator
1 Solving the radial equation
2 Energy levels and stationary wave functions
CVIIProbability currents associated with the stationary states of the hy-
drogen atom
1 General expression for the probability current
2 Application to the stationary states of the hydrogen atom
DVIIThe hydrogen atom placed in a uniform magnetic eld. Paramag-
netism and diamagnetism. The Zeeman eect
1 The Hamiltonian of the problem. The paramagnetic term and the diamagnetic
term
2 The Zeeman eect
EVIISome atomic orbitals. Hybrid orbitals
1 Introduction
2 Atomic orbitals associated with real wave functions
3 hybridization
4
2
hybridization
5
3
hybridization
xiv

Table of contents
FVIIVibrational-rotational levels of diatomic molecules
1 Introduction
2 Approximate solution of the radial equation
3 Evaluation of some corrections
GVIIExercises
1 Particle in a cylindrically symmetric potential
2 Three-dimensional harmonic oscillator in a uniform magnetic eld
INDEX 901
***********
xv

Table of contents
Volume II
VOLUME II
Table of contents
VIII AN ELEMENTARY APPROACH TO THE QUANTUM THEORY
OF SCATTERING BY A POTENTIAL
READER'S GUIDE FOR COMPLEMENTS 957
AVIIIThe free particle: stationary states
with well-dened angular momentum
BVIIIPhenomenological description of collisions with absorption
CVIIISome simple applications of scattering theory
***********
IX ELECTRON SPIN
READER'S GUIDE FOR COMPLEMENTS 999
AIX Rotation operators for a spin 1/2 particle
BIX Exercises
***********
X ADDITION OF ANGULAR MOMENTA
READER'S GUIDE FOR COMPLEMENTS 1041
AX Examples of addition of angular momenta
BX Clebsch-Gordan coecients
CX Addition of spherical harmonics
DX Vector operators: the Wigner-Eckart theorem
EX Electric multipole moments
FX Two angular momenta J1andJ2coupled by
an interactionJ1J2 1091
GX Exercises
xvi

Table of contents
***********
XI STATIONARY PERTURBATION THEORY
READER'S GUIDE FOR COMPLEMENTS 1129
AXIA one-dimensional harmonic oscillator subjected to a perturbing
potential in,
2
,
3
1131
BXIInteraction between the magnetic dipoles of two spin 1/2
particles
CXIVan der Waals forces
DXIThe volume eect: the inuence of the spatial extension of the nu-
cleus on the atomic levels
EXIThe variational method
FXIEnergy bands of electrons in solids: a simple model
GXIA simple example of the chemical bond: the H
+
2
ion
HXIExercises
***********
XII AN APPLICATION OF PERTURBATION THEORY: THE FINE
AND HYPERFINE STRUCTURE OF THE HYDROGEN ATOM
READER'S GUIDE FOR COMPLEMENTS 1265
AXIIThe magnetic hyperne Hamiltonian
BXIICalculation of the average values of the ne-structure Hamiltonian
in the1,2and2states
CXIIThe hyperne structure and the Zeeman eect for muonium and
positronium
DXIIThe inuence of the electronic spin on the Zeeman eect of the
hydrogen resonance line
EXIIThe Stark eect for the hydrogen atom
***********
xvii

Table of contents
XIII APPROXIMATION METHODS FOR TIME-DEPENDENT
PROBLEMS
READER'S GUIDE FOR COMPLEMENTS 1337
AXIIIInteraction of an atom with an electromagnetic wave
BXIIILinear and non-linear responses of a two-level system subject to a
sinusoidal perturbation
CXIIIOscillations of a system between two discrete states under the
eect of a sinusoidal resonant perturbation
DXIIIDecay of a discrete state resonantly coupled to a continuum of nal
states
EXIIITime-dependent random perturbation, relaxation
FXIIIExercises
***********
XIV SYSTEMS OF IDENTICAL PARTICLES
READER'S GUIDE FOR COMPLEMENTS 1457
AXIVMany-electron atoms. Electronic congurations
BXIVEnergy levels of the helium atom. Congurations, terms, multi-
plets
CXIVPhysical properties of an electron gas. Application to solids
DXIVExercises
***********
APPENDICES
I Fourier series and Fourier transforms
II The Dirac -function
III Lagrangian and Hamiltonian in classical mechanics
BIBLIOGRAPHY OF VOLUMES I AND II
INDEX
***********
xviii

Table of contents
Volume III
VOLUME III
Table of contents
XV CREATION AND ANNIHILATION OPERATORS FOR IDENTI-
CAL PARTICLES
READER'S GUIDE FOR COMPLEMENTS 1617
AXVParticles and holes
BXVIdeal gas in thermal equilibrium; quantum distribution functions
CXVCondensed boson system, Gross-Pitaevskii equation
DXVTime-dependent Gross-Pitaevskii equation
EXVFermion system, Hartree-Fock approximation
FXVFermions, time-dependent Hartree-Fock approximation
GXVFermions or Bosons: Mean eld thermal equilibrium
HXVApplications of the mean eld method for non-zero temperature
***********
XVI FIELD OPERATOR
READER'S GUIDE FOR COMPLEMENTS 1767
AXVISpatial correlations in an ideal gas of bosons or fermions
BXVISpatio-temporal correlation functions, Green's functions
CXVIWick's theorem
***********
XVII PAIRED STATES OF IDENTICAL PARTICLES
READER'S GUIDE FOR COMPLEMENTS 1843
xix

Table of contents
AXVIIPair eld operator for identical particles
BXVIIAverage energy in a paired state
CXVIIFermion pairing, BCS theory
DXVIICooper pairs
EXVIICondensed repulsive bosons
***********
XVIII REVIEW OF CLASSICAL ELECTRODYNAMICS
READER'S GUIDE FOR COMPLEMENTS 1977
AXVIIILagrangian formulation of electrodynamics
***********
XIX QUANTIZATION OF ELECTROMAGNETIC RADIATION
READER'S GUIDE FOR COMPLEMENTS 2017
AXIXMomentum exchange between atoms and photons
BXIXAngular momentum of radiation
CXIXAngular momentum exchange between atoms and photons
***********
XX ABSORPTION, EMISSION AND SCATTERING OF PHOTONS
BY ATOMS
READER'S GUIDE FOR COMPLEMENTS 2095
AXXA multiphoton process: two-photon absorption
BXXPhotoionization
CXXTwo-level atom in a monochromatic eld. Dressed-atom method
DXXLight shifts: a tool for manipulating atoms and elds
EXXDetection of one- or two-photon wave packets, interference
***********
XXI QUANTUM ENTANGLEMENT, MEASUREMENTS, BELL'S IN-
EQUALITIES
xx

Table of contents
READER'S GUIDE FOR COMPLEMENTS 2215
AXXIDensity operator and correlations; separability
BXXIGHZ states, entanglement swapping
CXXIMeasurement induced relative phase between two condensates
DXXIEmergence of a relative phase with spin condensates; macroscopic
non-locality and the EPR argument
***********
APPENDICES
IV Feynman path integral
V Lagrange multipliers
VI Brief review of Quantum Statistical Mechanics
VII Wigner transform
BIBLIOGRAPHY OF VOLUME III
INDEX
xxi

Chapter I
Waves and particles.
Introduction to the
fundamental ideas of quantum
mechanics
A Electromagnetic waves and photons
A-1 Light quanta and the Planck-Einstein relations
A-2 Wave-particle duality
A-3 The principle of spectral decomposition
B Material particles and matter waves
B-1 The de Broglie relations
B-2 Wave functions. Schrödinger equation
C Quantum description of a particle. Wave packets
C-1 Free particle
C-2 Form of the wave packet at a given time
C-3 Heisenberg relations
C-4 Time evolution of a free wave packet
D Particle in a time-independent scalar potential
D-1 Separation of variables. Stationary states
D-2 One-dimensional square potentials. Qualitative study
In the present state of scientic knowledge, quantum mechanics plays a fundamen-
tal role in the description and understanding of natural phenomena. In fact, phenomena
that occur on a very small (atomic or subatomic) scale cannot be explained outside the
framework of quantum physics. For example, the existence and the properties of atoms,
the chemical bond and the propagation of an electron in a crystal cannot be understood
Quantum Mechanics, Volume I, Second Edition. C. Cohen-Tannoudji, B. Diu, and F. Laloë.
© 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

CHAPTER I INTRODUCTION TO THE BASIC IDEAS OF QUANTUM MECHANICS
in terms of classical mechanics. Even when we are concerned only with macroscopic
physical objects (that is, whose dimensions are comparable to those encountered in ev-
eryday life), it is necessary, in principle, to begin by studying the behavior of their various
constituent atoms, ions, electrons, in order to arrive at a complete scientic description.
Actually, there are many phenomena that reveal, on a macroscopic scale, the quantum
behaviour of nature. It is in this sense that it can be said that quantum mechanics is the
basis of our present understanding of all natural phenomena, including those traditionally
treated in chemistry, biology, etc...
From a historical point of view, quantum ideas contributed to a remarkable uni-
cation of the concepts of fundamental physics by treating material particles and radiation
on the same footing. At the end of the nineteenth century, people distinguished between
two entities in physical phenomena: matter and radiation. Completely dierent laws
were used for each one. To predict the motion of material bodies, the laws ofNewtonian
mechanics(cf. Appendix) were utilized. Their success, though of long standing,
was none the less impressive. With regard to radiation, thetheory of electromagnetism,
thanks to the introduction of Maxwell's equations, had produced a unied interpretation
of a set of phenomena which had previously been considered as belonging to dierent
domains: electricity, magnetism and optics. In particular, the electromagnetic theory of
radiation had been spectacularly conrmed experimentally by the discovery of Hertzian
waves. Finally,interactions between radiation and matterwere well explained by the
Lorentz force. This set of laws had brought physics to a point which could be considered
satisfactory, in view of the experimental data at the time.
However, at the beginning of the twentieth century, physics was to be marked by
the profound upheaval that led to the introduction of relativistic mechanics and quantum
mechanics. The relativistic revolution and the quantum revolution were, to a large
extent, independent, since they challenged classical physics on dierent points. Classical
laws cease to be valid for material bodies travelling at very high speeds, comparable to
that of light (relativistic domain). In addition, they are also found to be wanting on
an atomic or subatomic scale (quantum domain). However, it is important to note that
classical physics, in both cases, can be seen as an approximation of the new theories,
an approximation which is valid for most phenomena on an everyday scale. For exam-
ple, Newtonian mechanics enables us to predict correctly the motion of a solid body,
providing it is non-relativistic (speeds much smaller than that of light) and macroscopic
(dimensions much greater than atomic ones). Nevertheless, from a fundamental point of
view, quantum theory remains indispensable. It is the only theory which enables us to
understand the very existence of a solid body and the values of the macroscopic parame-
ters (density, specic heat, elasticity, etc...) associated with it. It is possible to develop a
theory that is at the same time quantum and relativistic, but such a theory is relatively
complex. However, most atomic and molecular phenomena are well explained by the
non-relativistic quantum mechanicsthat we intend to examine here.
This chapter is an introduction to quantum ideas and vocabulary.No attempt
is made here to be rigorous or complete. The essential goal is to awaken the curiosity
of the reader. Phenomena will be described which unsettle ideas as rmly anchored in
our intuition as the concept of a trajectory. We want to render the quantum theory
plausible for the reader by showing simply and qualitatively how it enables us to solve
the problems which are encountered on an atomic scale. We shall later return to the
various ideas introduced in this chapter and go into further detail, either from the point
2

A. ELECTROMAGNETIC WAVES AND PHOTONS
of view of the mathematical formalism (Chap.) or from the physical point of view
(Chap.).
In the rst section (), we introduce the basic quantum ideas (wave-particle
duality, the measurement process), relying on well-known optical experiments. Then
we show () how these ideas can be extended to material particles (wave function,
Schrödinger equation). We next study in more detail the characteristics of the wave
packet associated with a particle, and we introduce the Heisenberg relations ().
Finally, we discuss some simple examples of typical quantum eects ().
A. Electromagnetic waves and photons
A-1. Light quanta and the Planck-Einstein relations
Newton considered light to be a beam of particles, able, for example, to bounce
back upon reection from a mirror. During the rst half of the nineteenth century, the
wavelike nature of light was demonstrated (interference, diraction). This later enabled
optics to be integrated into electromagnetic theory. In this framework, the speed of light,
, is related to electric and magnetic constants and light polarization phenomena can be
interpreted as manifestations of the vectorial character of the electric eld.
However, the study ofblackbodyradiation, which electromagnetic theory could not
explain, led Planck to suggest the hypothesis of thequantization of energy(1900): for an
electromagnetic wave of frequency, the only possible energies are integral multiples of
the quantum, whereis a new fundamental constant. Generalizing this hypothesis,
Einstein proposed a return to the particle theory (1905): light consists of a beam of
photons, each possessing an energy. Einstein showed how the introduction of photons
made it possible to understand, in a very simple way, certain as yet unexplained char-
acteristics of the photoelectric eect. Twenty years had to elapse before the photon was
actually shown to exist, as a distinct entity, by the Compton eect (1924).
These results lead to the following conclusion: the interaction of an electromagnetic
wave with matter occurs by means ofelementary indivisible processes, in which the
radiation appears to be composed of particles, the photons. Particle parameters (the
energyand the momentumpof a photon) and wave parameters (the angular frequency
= 2and the wave vectork, wherek= 2, withthe frequency andthe
wavelength) are linked by the fundamental relations:
= =~
p=~k
(Planck-Einstein relations) (A-1)
where~=2is dened in terms of the Planck constant:
662 10
34
Joulesecond (A-2)
During each elementary process, energy and total momentum must be conserved.
A-2. Wave-particle duality
Thus we have returned to a particle conception of light. Does this mean that
we must abandon the wave theory? Certainly not. We shall see that typical wave
3

CHAPTER I INTRODUCTION TO THE BASIC IDEAS OF QUANTUM MECHANICS
phenomena such as interference and diraction could not be explained in a purely particle
framework. Analyzing Young's well-known double-slit experiment will lead us to the
following conclusion: a complete interpretation of the phenomena can be obtained only
by conservingboththe wave aspect and the particle aspect of light (although they seem
a prioriirreconcilable). We shall then show how this paradox can be resolved by the
introduction of the fundamental quantum concepts.
A-2-a. Analysis of Young's double-slit experiment
The device used in this experiment is shown schematically in Figure. The
monochromatic light emitted by the sourceSfalls on an opaque screenPpierced by
two narrow slits1and2, which illuminate the observation screenE(a photographic
plate, for example). If we block2, we obtain onEa light intensity distribution1()
which is the diraction pattern of1. In the same way, when1is obstructed, the
diraction pattern of2is described by2(). When the two slits1and2are open at
the same time, we observe a system of interference fringes on the screen. In particular,
we note that the corresponding intensity()is not the sum of the intensities produced
by1and2separately:
()=1() +2() (A-3)
How could one conceive of explaining, in terms of a particle theory (seen, in the
preceding section, to be necessary), the experimental results just described? The exis-
tence of a diraction pattern when only one of the two slits is open could, for example, be
explained as being due to photon collisions with the edges of the slit. Such an explanation
would, of course, have to be developed more precisely, and a more detailed study would
show it to be insucient. Instead, let us concentrate on the interference phenomenon.
We could attempt to explain it by an interaction between the photons which pass through
the slit1and those which pass through the slit2. Such an explanation would lead to
the following prediction: if the intensity of the sourceS(the number of photons emitted
per second) is diminished until the photons strike the screen practically one by one, the
interaction between the photons must diminish and, eventually, vanish. The interference
fringes should therefore disappear.
Before we indicate the answer given by experiment, recall that the wave theory
provides a completely natural interpretation of the fringes. The light intensity at a point
of the screenEis proportional to the square of the amplitude of the electric eld at this
point. If1()and2()represent, in complex notation, the electric elds produced
atby slits1and2respectively (the slits behave like secondary sources), the total
resultant eld at this point when1and2are both open is
1
:
() =1() +2() (A-4)
Using complex notation, we then have:
() ()
2
= 1() +2()
2
(A-5)
Since the intensities1()and2()are proportional, respectively, to1()
2
and
2()
2
, formula (A-5) shows that()diers from1() +2()by an interference
1
Since the experiment studied here is performed with unpolarized light, the vectorial character of
the electric eld does not play an essential role. For the sake of simplicity, we ignore it in this paragraph.
4

A. ELECTROMAGNETIC WAVES AND PHOTONS
Figure 1: Diagram of Young's double-slit light interference experiment (g. a). Each of
the slits1and2produces a diraction pattern on the screen. The corresponding
intensities are1()and2()(solid lines in gure b). When the two slits1and
2are open simultaneously, the intensity()observed on the screen is not the sum
1() +2()(dashed lines in gure b), but shows oscillations due to the interference
between the electric elds radiated by1and2(solid line in gure c).
term which depends on the phase dierence between1and2and whose presence
explains the fringes. The wave theory thus predicts that diminishing the intensity of the
sourceSwill simply cause the fringes to diminish in intensity but not vanish.
What actually happens whenSemits photons practically one by one?Neither
the predictions of the wave theory nor those of the particle theory are veried. In fact:
(i) If we cover the screenEwith a photographic plate and increase the exposure
time so as to capture a large number of photons on each photograph, we observe when
we develop them that thefringes have not disappeared. Therefore, the purely corpuscular
interpretation, according to which the fringes are due to an interaction between photons,
must be rejected.
()On the other hand, we can expose the photographic plate during a time so
short that it can only receive a few photons. We then observe that each photon produces
alocalized impactonEand not a very weak interference pattern.Therefore, the purely
wave interpretation must also be rejected.
In reality, as more and more photons strike the photographic plate, the following
phenomenon occurs. Their individual impacts seem to be distributed in arandom man-
ner, and only when a great number of them have reachedEdoes the distribution of the
impacts begin to have a continuous aspect. The density of the impacts at each point ofE
corresponds to the interference fringes: maximum on a bright fringe and zero on a dark
fringe. It can thus be said that the photons, as they arrive, build up the interference
pattern.
The result of this experiment therefore leads, apparently, to a paradox. Within
the framework of the particle theory, for example, it can be expressed in the following
way. Since photon-photon interactions are excluded, each photon must be considered
separately. But then it is not clear why the phenomena should change drastically ac-
cording to whether only one slit or both slits are open. For a photon passing through
one of the slits, why should the fact that the other is open or closed have such a critical
5

CHAPTER I INTRODUCTION TO THE BASIC IDEAS OF QUANTUM MECHANICS
importance?
Before we discuss this problem, note that in the preceding experiment we did not
seek to determine through which slit each photon passed before it reached the screen.
In order to obtain this information, we can imagine placing detectors (photomultipliers)
behind1and2. It will then be observed that, if the photons arrive one by one, each
one passes through a well-determined slit (a signal is recorded either by the detector
placed behind1or by the one covering2but not by both at once). But, obviously,
the photons detected in this way are absorbed and do not reach the screen. Remove the
photomultiplier which blocks1, for example. The one which remains behind2tells us
that, out of a large number of photons, about half pass through2. We conclude that
the others (which can continue as far as the screen) pass through1. But the pattern
that they gradually construct on the screen is not an interference pattern, since2is
blocked. It is only the diraction pattern of1.
A-2-b. Quantum unication of the two aspects of light
The preceding analysis shows that it is impossible to explain all the phenomena
observed if only one of the two aspects of light, wave or particle, is considered. Now
these two aspects seem to be mutually exclusive. To overcome this diculty, it thus
becomes indispensable to reconsider in a critical way the concepts of classical physics.
We must accept the possibility that these concepts, although our everyday experience
leads us to consider them well-founded, may not be valid in the new (microscopic)
domain which we are entering. For example, an essential characteristic of this new
domain appeared when we placed counters behind Young's slits:when one performs a
measurement on a microscopic system, one disturbs it in a fundamental fashion. This
is a new property since, in the macroscopic domain, we always have the possibility of
conceiving measurement devices whose inuence on the system is practically as weak as
one might wish. This critical revision of classical physics is imposed by experiment and
must of course be guided by experiment.
Let us reconsider the paradox stated above concerning the photon which passes
through one slit but behaves dierently depending on whether the other slit is open or
closed. We saw that if we try to detect the photons when they cross the slits, we prevent
them from reaching the screen. More generally, a detailed experimental analysis shows
thatit is impossible to observe the interference pattern and to know at the same time
through which slit each photon has passed(cf. ComplementI). Thus it is necessary, in
order to resolve the paradox, to give up the idea that a photon inevitably passes through
a particular slit. We are then led to question the concept, which is a fundamental one of
classical physics, of a particle's trajectory.
Moreover, as the photons arrive one by one, their impacts on the screen gradually
build up the interference pattern. This implies that, for a particular photon, we are not
certain in advance where it will strike the screen. Now these photons are all emitted
under the same conditions. Thus another classical idea has been destroyed: that the
initial conditions completely determine the subsequent motion of a particle. We can
only say, when a photon is emitted, that the probability of its striking the screen atis
proportional to the intensity()calculated using wave theory, that is, to()
2
.
After many tentative eorts that we shall not describe here, the concept ofwave-
particle dualitywas formulated. We can summarize it schematically as follows
2
:
2
It is worth noting that this interpretation of physical phenomena, generally considered to be ortho-
6

A. ELECTROMAGNETIC WAVES AND PHOTONS
()The particle and wave aspects of light are inseparable.Light behaves simul-
taneously like a wave and like a ux of particles, the wave enabling us to calculate the
probability of the manifestation of a particle.
()Predictions about the behavior of a photon can only be probabilistic.
()The information about a photon at timeis given by the wave(r), which
is a solution of Maxwell's equations. We say that this wave characterizes the state of
the photons at time;(r)is interpreted as theprobability amplitudeof a photon
appearing, at time, at the pointr. This means that the corresponding probability is
proportional to(r)
2
.
Comments:
(i) Since Maxwell's equations are linear and homogeneous, we can use asu-
perposition principle: if1and2are two solutions of these equations, then
=11+22, where1and2are constants, is also a solution. It is
this superposition principle which explains wave phenomena in classical optics
(interference, diraction). In quantum physics, the interpretation of(r)as a
probability amplitude is thus essential to the persistence of such phenomena.
()The theory merely allows one to calculate the probability of the occurence of a
given event. Experimental verications must thus be founded on the repetition of
a large number of identical experiments. In the above experiment, a large number
of photons, all produced in the same way, are emitted successively and build up
the interference pattern, according to the calculated probabilities.
()We are talking here about the photon state so as to be able to develop in
(r)and the wave function(r)that characterizes
the quantum state of a material particle. This optical analogy is very fruitful.
In particular, as we shall see in , it allows us to understand, simply and without
recourse to calculation, various quantum properties of material particles. However,
we should not push it too far and let it lead us to believe that it is rigorously correct
to consider(r)as characterizing the quantum state of a photon.
Furthermore, we shall see that the fact that(r)is complex is essential in quan-
tum mechanics, while the complex notation(r)is used in optics purely for convenience
(only its real part has a physical meaning). The precise denition of the (complex) quan-
tum state of radiation can only be given in the framework of quantum electrodynamics,
a theory which is simultaneously quantum mechanical and relativistic. We shall not
consider these problems here (they will briey be discussed in ComplementV).
A-3. The principle of spectral decomposition
Armed with the ideas introduced in , we are now going to discuss another
simple optical experiment, whose subject is the polarization of light. This will permit
us to introduce the fundamental concepts which concern the measurement of physical
quantities.
dox, is not unique; other interpretations have been proposed, which are still discussed among physicists.
7

CHAPTER I INTRODUCTION TO THE BASIC IDEAS OF QUANTUM MECHANICSe
y
e
x
e
p
y
z
A
E
P
x
O
θ
Figure 2: A simple measurement experiment relating to the polarization of a light wave.
A beam of light propagates along the directionand crosses successively the polarizer
and the analyzer; is the angle betweenand the electric eld of the wave
transmitted by. The vibrations transmitted byare parallel to.
The experiment consists of directing a polarized monochromatic plane light wave
onto an analyzer. designates the direction of propagation of this wave ande,
the unit vector describing its polarization (cf.Fig. 2). The analyzertransmits light
polarized parallel toand absorbs light polarized parallel to.
The classical description of this experiment (a description which is valid for a
suciently intense light beam) is the following. The polarized plane wave is characterized
by an electric eld of the form:
E(r) =0ee
( )
(A-6)
where0is a constant. The light intensityis proportional to0
2
. After its passage
through the analyzer, the plane wave is polarized along:
E(r) =
0ee
( )
(A-7)
and its intensity, proportional to
0
2
, is given byMalus' law:
=cos
2
(A-8)
[eis the unit vector of theaxis andis the angle betweeneande].
What will happen on the quantum level, that is, whenis weak enough for the
photons to reach the analyzer one by one? (We then place a photon detector behind
this analyser.) First of all, the detector never registers a fraction of a photon. Either
the photon crosses the analyzer or it is entirely absorbed by it. Next (except in special
cases that we shall examine in a moment), we cannot predict with certainty whether a
given incident photon will pass or be absorbed. We can only know the corresponding
8

A. ELECTROMAGNETIC WAVES AND PHOTONS
probabilities. Finally, if we send out a large numberof photons one after the other,
the result will correspond to the classical law, in the sense that aboutcos
2
photons
will be detected after the analyzer.
We shall retain the following ideas from this description:
() The measurement device (the analyzer, in this case) can give only certain
privileged results, which we shall calleigen(or proper)results
3
. In the above experiment,
there are only two possible results: the photon crosses the analyzer or it is stopped.
One says that there is quantization of the result of the measurement, in contrast to the
classical case [cf.formula (A-8)] where the transmitted intensitycan vary continuously,
according to the value of, between0and.
() To each of these eigen results corresponds aneigenstate. Here, the two eigen-
states are characterized by:
e=e
ore=e
(A-9)
(eis the unit vector of theaxis). Ife=e, we know with certainty that the photon
will traverse the analyzer; ife=e, it will, on the contrary, denitely be stopped.
The correspondence between eigen results and eigenstates is therefore the following. If
the particle is, before the measurement, in one of the eigenstates, the result of this
measurement is certain: it can only be the associated eigen result.
() When the state before the measurement is arbitrary, only the probabilities of
obtaining the dierent eigen results can be predicted. To nd these probabilities, one
decomposes the state of the particles into a linear combination of the various eigenstates.
Here, for an arbitrarye, we write:
e=ecos+esin (A-10)
The probability of obtaining a given eigen result is then proportional to the square of
the absolute value of the coecient of the corresponding eigenstate.
The proportionality factor is determined by the condition that the sum of all these
probabilities must be equal to 1. We thus deduce from (A-10) that each photon has a
probabilitycos
2
of traversing the analyzer and a probabilitysin
2
of being absorbed
by it (we know thatcos
2
+ sin
2
= 1). This is indeed what was stated above. This
rule is called in quantum mechanics theprinciple of spectral decomposition. Note that
the decomposition to be performed depends on the type of measurement device being
considered, since one must use the eigenstates which correspond to it: in formula (A-10),
the choice of the axesand is xed by the analyzer.
() After passing through the analyzer, the light is completely polarized alonge.
If we place, after the rst analyzer, a second analyzer, having the same axis, all
the photons which traversedwill also traverse. According to what we have just
seen in point (), this means that, after they have crossed, the state of the photons
is the eigenstate characterized bye. There has therefore been an abrupt change in the
state of the particles. Before the measurement, this state was dened by a vectorE(r)
which was collinear withe. After the measurement, we possess an additional piece of
information (the photon has passed) which is incorporated by describing the state by a
dierent vector, which is now collinear withe. This expresses the fact, already pointed
3
The reason for this name will appear in Chapter.
9

CHAPTER I INTRODUCTION TO THE BASIC IDEAS OF QUANTUM MECHANICS
out in , thatthe measurement disturbs the microscopic system(here, the photon)
ina fundamental fashion.
Comment:
The certain prediction of the result whene=eore=eis only a special case.
The probability of one of the possible events is then indeed equal to 1. But, in order to
verify this prediction, one must perform a large number of experiments. One must be
sure thatallthe photons pass (or are stopped), since the fact that a particular photon
crosses the analyzer (or is absorbed) is not characteristic ofe=e(ore=e).
B. Material particles and matter waves
B-1. The de Broglie relations
Parallel to the discovery of photons, the study of atomic emission and absorption
spectra uncovered a fundamental fact, which classical physics was unable to explain:
these spectra are composed ofnarrow lines. In other words, a given atom emits or
absorbs only photons having well-determined frequencies (that is, energies). This fact
can be interpreted very easily if one accepts thatthe energy of the atom is quantized,
that is, it can take on only certain discrete values(= 12 ): the emission or
absorption of a photon is then accompanied by a jump in the energy of the atom from
one permitted valueto another. Conservation of energy implies that the photon
has a frequencysuch that:
= (B-1)
Only frequencies which obey (B-1) can therefore be emitted or absorbed by the atom.
The existence of such discrete energy levels was conrmed independently by the
Franck-Hertz experiment. Bohr interpreted this in terms of privileged electronic orbits
and stated, with Sommerfeld, an empirical rule which permitted the calculation of these
orbits for the case of the hydrogen atom. But the fundamental origin of these quantization
rules remained mysterious.
In 1923, however, de Broglie put forth the following hypothesis:material particles,
just like photons, can have a wavelike aspect. He then derived the Bohr-Sommerfeld
quantization rules as a consequence of this hypothesis, the various permitted energy levels
appearing as analogues of the normal modes of a vibrating string. Electron diraction
experiments (Davisson and Germer, 1927) strikingly conrmed the existence of a wavelike
aspect of matter by showing thatinterference patterns could be obtained with material
particles such as electrons.
One therefore associates with a material particle of energyand momentump, a
wave whose angular frequency= 2and wave vectorkare given by the same relations
as for photons (cf. ):
= =~
p=~k
(B-2)
10

B. MATERIAL PARTICLES AND MATTER WAVES
In other words, the corresponding wavelength is:
=
2
k
=
p
(de Broglie relation) (B-3)
Comment:
The very small value of the Planck constantexplains why the wavelike nature
of matter is very dicult to demonstrate on a macroscopic scale. ComplementIof this
chapter discusses the orders of magnitude of the de Broglie wavelengths associated with
various material particles.
B-2. Wave functions. Schrödinger equation
In accordance with de Broglie's hypothesis, we shall apply the ideas introduced in

paragraph, we are led to the following formulation:
(i) For the classical concept of a position and momentum, we must substitute the
concept of a time-varyingstate. The quantum state of a particle such as the electron
4
is
characterized by awave function(r), which contains all the information it is possible
to obtain about the particle.
(ii)(r)is interpreted as aprobability amplitude of the particle's presence. Since
the possible positions of the particle form a continuum, the probabilitydP(r)of the
particle being, at time, in a volume elementd
3
= dddsituated at the pointr
must be proportional tod
3
. It is therefore innitesimal, and(r)
2
is interpreted as
the correspondingprobability density, with:
dP(r) = (r)
2
d
3
(B-4)
whereis a normalization constant [see comment (i) at the end of ].
(iii)The principle of spectral decompositionapplies to the measurement of an
arbitrary physical quantity:
The result found must belong to a set of eigen results {}.
With each eigenvalueis associated an eigenstate, that is, an eigenfunction
(r). This function is such that, if(r0) =(r)(where0is the time at which the
measurement is performed), the measurement will always yield.
For any(r), the probabilityPof nding the eigenvaluefor a measurement
at time0is found by decomposing(r0)in terms of the functions(r):
(r0) = (r) (B-5)
Then:
P=
2
2
(B-6)
4
We shall not take into account here the existence of electron spin (cf.Chap.).
11

CHAPTER I INTRODUCTION TO THE BASIC IDEAS OF QUANTUM MECHANICS
(the presence of the denominator ensures that the total probability is equal to unity:
P= 1).
If the measurement indeed yields, the wave function of the particle immediately
after the measurement is:
(r0) =(r) (B-7)
(iv) The equation describing the evolution of the function(r)remains to be
written. It is possible to introduce it in a very natural way, using the Planck and
de Broglie relations. Nevertheless, we have no intention of proving this fundamental
equation, which is called theSchrödinger equation. We shall simply assume it. Later,
we shall discuss some of its consequences (whose experimental verication will prove its
validity). Besides, we shall consider this equation in much more detail in Chapter.
When the particle (of mass) is subjected to the inuence of a potential
5
(r),
the Schrödinger equation takes on the form:
~
(r) =
~
2
2
(r) +(r)(r)
(B-8)
whereis the Laplacian operator
2 2
+
2 2
+
2 2
.
We notice immediately that this equation is linear and homogeneous in. Consequently,
for material particles, there exists a superposition principle which, combined with the
interpretation ofas a probability amplitude, is the source of wavelike eects. Note,
moreover, that the dierential equation (B-8) is rst-order with respect to time. This
condition is necessary if the state of the particle at a time0, characterized by(r0),
is to determine its subsequent state.
Thus there exists a fundamental analogy between matter and radiation: in both
cases, a correct description of the phenomena necessitates the introduction of quantum
concepts, and, in particular, the idea of wave-particle duality.
Comments:
(i) For a system composed of only one particle, the total probability of nding the
particle anywhere in space, at time, is equal to 1:
dP(r) = 1 (B-9)
SincedP(r)is given by formula (B-4), we conclude that thewave function
(r)must be square-integrable:
(r)
2
d
3
is nite (B-10)
5
(r)designates a potential energy here. For example, it may be the product of an electric
potential and the particle's charge. In quantum mechanics,(r)is commonly called a potential.
12

C. QUANTUM DESCRIPTION OF A PARTICLE. WAVE PACKETS
The normalization constantthat appears in (B-4) is then given by the relation:
1
= (r)
2
d
3
(B-11)
(we shall later see that the form of the Schrödinger equation implies thatis
time-independent). One often uses wave functions which are normalized, such
that:
(r)
2
d
3
= 1 (B-12)
The constantis then equal to 1.
(ii) Note the important dierence between the concepts of classical states and
quantum states. The classical state of a particle is determined at timeby the
specication of six parameters characterizing its position and its velocity at time:
,,;,,. The quantum state of a particle is determined by aninnite
numberof parameters: the values at the various points in space of the wave
function(r)which is associated with it. For the classical idea of a trajectory
(the succession in time of the various states of the classical particle), we must
substitute the idea of the propagation of the wave associated with the particle.
Consider, for example, Young's double-slit experiment, previously described for
the case of photons, but which in principle can also be performed with material
particles such as electrons. When the interference pattern is observed, it makes no
sense to ask through which slit each particle has passed, since the wave associated
with it passed through both.
(iii) It is worth noting that, unlike photons, which can be emitted or absorbed
during an experiment, material particles can neither be created nor destroyed. The
electrons emitted by a heated lament for example already existed in the lament.
In the same way, an electron absorbed by a counter does not disappear; it becomes
part of an atom or an electric current. Actually, the theory of relativity shows that
it is possible to create and annihilate material particles: for example, a photon
having sucient energy, passing near an atom, can materialize into an electron-
positron pair. Inversely, the positron, when it collides with an electron, annihilates
with it, emitting photons. However, we pointed out in the beginning of this chapter
that we would limit ourselves here to the non-relativistic quantum domain, and we
have indeed treated time and space coordinates asymmetrically. In the framework
of non-relativistic quantum mechanics, material particles can neither be created
nor annihilated. This conservation law, as we shall see, plays a role of primary
importance. The need to abandon it is one of the important diculties encountered
when one tries to construct a relativistic quantum mechanics.
C. Quantum description of a particle. Wave packets
In the preceding paragraph, we introduced the fundamental concepts necessary for the
quantum description of a particle. In this paragraph, we are going to familiarize ourselves
with these concepts and deduce from them several very important properties. We start
with the very simple case of a free particle.
13

CHAPTER I INTRODUCTION TO THE BASIC IDEAS OF QUANTUM MECHANICS
C-1. Free particle
Consider a particle whose potential energy is zero (or has a constant value) at
every point in space. The particle is thus not subjected to any force; it is said to be free.
When(r) = 0, the Schrödinger equation becomes:
~
(r) =
~
2
2
(r) (C-1)
This dierential equation is obviously satised by solutions of the form:
(r) =e
(kr )
(C-2)
(whereis a constant), on the condition thatkandsatisfy the relation:
=
~k
2
2
(C-3)
Observe that, according to the de Broglie relations [see (B-2)], condition (C-3) expresses
the fact that the energyand the momentumpof a free particle satisfy the equation,
which is well-known in classical mechanics:
=
p
2
2
(C-4)
We shall come back later () to the physical interpretation of a state of the form
(C-2). We already see that, since
(r)
2
=
2
(C-5)
a plane wave of this type represents a particle whose probability of presence is uniform
throughout all space (see comment below).
The principle of superposition tells us that every linear combination of plane waves
satisfying (C-3) will also be a solution of equation (C-1). Such a superposition can be
written:
(r) =
1
(2)
32
(k) e
[kr()]
d
3
(C-6)
(d
3
represents, by denition, the innitesimal volume element ink-space:ddd).
(k), which can be complex, must be suciently regular to allow dierentiation inside the
integral. It can be shown, moreover, that any square-integrable solution can be written
in the form (C-6).
A wave function such as (C-6), a superposition of plane waves, is called a three-
dimensional wave packet. For the sake of simplicity, we shall often be led to study the
case of a one-dimensional wave packet
6
of plane waves all propagating parallel to.
The wave function then depends only onand:
() =
1
2
+
() e
[ ()]
d (C-7)
6
A simple model of a two-dimensional wave packet is presented in ComplementI. Some general
properties of three-dimensional wave packets are studied in ComplementI, which also shows how, in
certain cases, a three-dimensional problem can be reduced to several one-dimensional problems.
14

C. QUANTUM DESCRIPTION OF A PARTICLE. WAVE PACKETS
In the following paragraph, we shall be interested in the form of the wave packet at a
given instant. If we choose this instant as the time origin, the wave function is written:
(0) =
1
2
() ed (C-8)
We see that()is simply the Fourier transform (cf.Appendix) of(0):
() =
1
2
(0) ed (C-9)
Consequently, the validity of formula (C-8) is not limited to the case of the free particle:
whatever the potential,(0)can always be written in this form. The consequences
that we shall derive from this in
not until
Comment:
A plane wave of type (C-2), whose modulus is constant throughout all space [cf.
(C-5)], is not square-integrable. Therefore, rigorously, it cannot represent a phys-
ical state of the particle (in the same way as, in optics, a monochromatic plane
wave is not physically realizable). On the other hand, a superposition of plane
waves like (C-7) can be square-integrable.
C-2. Form of the wave packet at a given time
The form of the wave packet is given by the-dependence of(0)dened by
equation (C-8). Imagine that()has the shape depicted in Figure, with a pronounced
peak situated at=0and a width (dened, for example, at half its maximum value)
of.0
k
0
k
Δk
g(k)
Figure 3: Shape of the function(), mod-
ulus of the Fourier transform of(0): we
assume that it is centered at=0, where it
reaches a maximum, and has a width of.
Let us begin by trying to understand qualitatively the behavior of(0)through
the study of a very simple special case. Let(0), instead of being the superposition
of an innite number of plane waveseas in formula (C-8), be the sum of only three
plane waves. The wave vectors of these plane waves are0,0
2
,0+
2
, and their
15

CHAPTER I INTRODUCTION TO THE BASIC IDEAS OF QUANTUM MECHANICS
amplitudes are proportional, respectively, to 1, 1/2 and 1/2. We then have:
() =
(0)
2
e
0
+
1
2
e
(0
2)
+
1
2
e
(0+
2)
=
(0)
2
e
0
1 + cos
2
(C-10)
We see that()is maximum when= 0. This result is due to the fact that, when
takes on this value, the three waves are in phase and interfere constructively, as shown in
Figure. As one moves away from the value= 0, the waves become more and more out
of phase, and()decreases. The interference becomes completely destructive when
the phase shift betweene
0
ande
(0 2)
is equal to:()goes to zero when
= 2,being given by:
= 4 (C-11)
This formula shows that the smaller the widthof the function(), the larger the
width of the function()(the distance between two zeros of()).
Comment:
Formula (C-10) shows that()is periodic inand therefore has a series of maxima
and minima. This arises from the fact that()is the superposition of a nite number
of waves (here, three). For a continuous superposition of an innite number of waves,
as in formula (C-8), such a phenomenon does not occur, and(0)can have only one
maximum.
Let us now return to the general wave packet of formula (C-8). Its form also results
from an interference phenomenon:(0)is maximum when the dierent plane waves
interfere constructively.
Let()be the argument of the function():
() =()e
()
(C-12)
Assume that()varies suciently smoothly within the interval0
2
0+
2
where()is appreciable; then, whenis suciently small, one can expand()in
the neighborhood of=0:
() (0) + ( 0)
d
d
=0
(C-13)
which enables us to rewrite (C-8) in the form:
(0)
e
[0+(0)]
2
+
()e
( 0)( 0)
d (C-14)
16

C. QUANTUM DESCRIPTION OF A PARTICLE. WAVE PACKETS0
k
0
k
0
k
0
+
Δk
2
–
Δk
2
–
Δx
2
+
Δx
2
x
Re { ψ(x) }
Figure 4: The real parts of the three waves whose sum gives the function()of (C-10).
At= 0, the three waves are in phase and interfere constructively. As one moves away
from= 0, they go out of phase and interfere destructively for=2.
In the lower part of the gure, Re()is shown. The dashed-line curve corresponds
to the function1 + cos

2
, which, according to (C-10), gives()(and therefore,
the form of the wave packet).
with:
0=
d
d
=0
(C-15)
The form (C-14) is useful for studying the variations of(0)in terms of.
When 0is large, the function ofwhich is to be integrated oscillates a very large
number of times within the interval. We then see (cf.Fig.-a, in which the real part
of this function is depicted) that the contributions of the successive oscillations cancel
each other out, and the integral overbecomes negligible. In other words, whenis
xed at a value far from0, the phases of the various waves which make up(0)vary
very rapidly in the domain, and these waves destroy each other by interference. On
the other hand, if0, the function to be integrated overoscillates hardly at all (cf.
Fig.-b), and(0)is maximum.
The position(0)of the center of the wave packet is therefore:
(0) =0=
d
d
=0
(C-16)
Actually the result (C-16) can be obtained very easily. An integral such as the
one appearing in (C-8) will be maximum (in absolute value) when the waves having the
17

CHAPTER I INTRODUCTION TO THE BASIC IDEAS OF QUANTUM MECHANICS0 0
k k
k
0
k
0
g(k)
x – x
0
>
g(k)
Re { g(k)e
i(k – k
0
)(x – x
0
)
} Re { g(k)e
i(k – k
0
)(x – x
0
)
}
1
k
x – x
0
<
1
k
a b
Figure 5: Variations with respect toof the function to be integrated overin order to
obtain(0). In gure (a),is xed at a value such that 01, and the
function to be integrated oscillates several times within the interval. In gure (b),
is xed such that 01, and the function to be integrated hardly oscillates, so
that its integral overtakes on a relatively large value. Consequently, the center of the
wave packet [point where(0)is maximum] is situated at=0.
largest amplitude (those withclose to0) interfere constructively. This occurs when
the-dependent phases of these waves vary only slightly around=0. To obtain
the center of the wave packet, one then imposes (stationary phasecondition) that the
derivative with respect toof the phase is zero for=0. In the particular case which
we are studying, the phase of the wave corresponding tois+(). Therefore,(0)
is that value offor which the derivative+ ddis zero at=0.
Whenmoves away from the value0,(0)decreases. This decrease becomes
appreciable ife
( 0)( 0)
oscillates approximately once whentraverses the domain
, that is, when:
( 0)1 (C-17)
Ifis the approximate width of the wave packet, we therefore have:
&1 (C-18)
We are thus brought back to a classical relation between the widths of two functions
which are Fourier transforms of each other. The important fact is that the product
has a lower bound; the exact value of this bound clearly depends on the precise
denition of the widthsand.
A wave packet such as (C-7) thus represents the state of a particle whose probabil-
ity of presence, at the time= 0, is practically zero outside an interval of approximate
width centered at the value0.
18

C. QUANTUM DESCRIPTION OF A PARTICLE. WAVE PACKETS
Comment:
The preceding argument could lead one to believe that the product is
always of the order of 1 [cf.(C-17)]. Let us stress the fact that this is a lower
limit. Although it is impossible to construct wave packets for which the product
is negligible compared to 1, it is perfectly possible to construct packets
for which this product is as large as desired [see, for example, ComplementI,
especially comment (ii) of C-18) is written in the form of an
inequality.
C-3. Heisenberg relations
In quantum mechanics, inequality (C-18) has extremely important physical con-
sequences. We intend to discuss these now (we shall stay, for simplicity, within the
framework of a one-dimensional model).
We have seen that a plane wavee
(0 0)
corresponds to a constant probability
density for the particle's presence along theaxis, for all values of. This result can be
roughly expressed by saying that the corresponding value ofis innite. On the other
hand, only one angular frequency0and one wave vector0are involved. According to
the de Broglie relations, this means that the energy and momentum of the particle are
well-dened:=~0and=~0. Such a plane wave can, moreover, be considered to
be a special case of (C-7), for which()is a delta function (cf.Appendix):
() =( 0) (C-19)
The corresponding value ofis then zero.
This property can also be interpreted in the following manner, using the principle
of spectral decomposition (cf.). To say that a particle, described at= 0
by the wave function(0) =e, has a well-determined momentum, is to say that
a measurement of the momentum at this time will denitely yield=~. From this we
deduce thatecharacterizes the eigenstate corresponding to=~. Since there exists
a plane wave for every real value of, the eigenvalues which one can expect to nd in
a measurement of the momentum on an arbitrary state include all real values. In this
case, there is no quantization of the possible results: as in classical mechanics, all values
of the momentum are allowed.
Now consider formula (C-8 (0)appears as a linear superpo-
sition of the momentum eigenfunctions in which the coecient ofeis(). We are
thus led to interpret()
2
(to within a constant factor) as the probability of nding
=~if one measures, at= 0, the momentum of a particle whose state is described
by(). In reality, the possible values of, like those of, form a continuous set,
and()
2
is proportional to aprobability density: the probability
dP()of obtaining a
value between~and~(+d)is, to within a constant factor,()
2
d. More precisely,
if we rewrite formula (C-8) in the form:
(0) =
1
2~() e
~
d (C-20)
we know that
()and(0)satisfy the Parseval-Plancherel relation (cf.Appendix):
19

CHAPTER I INTRODUCTION TO THE BASIC IDEAS OF QUANTUM MECHANICS
+
(0)
2
d=
+
()
2
d (C-21)
If the common value of these integrals is,dP() =
1
(0)
2
dis the probability
of the particle being found, at= 0, betweenand+ d. In the same way:
dP() =
1
()
2
d (C-22)
is the probability that the measurement of the momentum will yield a result included
betweenand+ d[relation (C-21) then insures that the total probability of nding
any value is indeed equal to 1].
Now let us go back to the inequality (C-18). We can write it as:
&~ (C-23)
(=~is the width of the curve representing
()). Consider a particle whose state
is dened by the wave packet (C-20). We know that its position probability at= 0, is
appreciable only within a region of widthabout0: its position is known within an
uncertainty. If one measures the momentum of this particle at the same time, one
will nd a value between0+
2
and0
2
, since
()
2
is practically zero outside
this interval: the uncertainty in the momentum is therefore. The interpretation of
relation (C-23) is then as follows: it is impossible to dene at a given time both the
position of the particle and its momentum to an arbitrary degree of accuracy. When
the lower limit imposed by (C-23) is reached, increasing the accuracy in the position
(decreasing) implies that the accuracy in the momentum diminishes (increases),
and vice versa. This relation is called theHeisenberg relation(or sometimesuncertainty
relation, for historical reasons).
We know of nothing like this in classical mechanics. The limitation expressed
by (C-23) arises from the fact thatis not zero. It is the very small value ofon the
macroscopic scale which renders this limitation totally negligible in classical mechanics
(an example is discussed in detail in ComplementI).
Comment:
The inequality (C-18) with which we started is not an inherently quantum mechanical
principle. It merely expresses a general property of Fourier transforms, numerous appli-
cations of which can be found in classical physics. For example, it is well known from
electromagnetic theory that there exists no packet of electromagnetic waves for which
one can dene the position and the wavelength with innite accuracy at the same time.
Quantum mechanics enters when one associates a wave with a material particle and
requires that the wavelength and the momentum satisfy de Broglie's relation.
C-4. Time evolution of a free wave packet
Until now, we have been concerned only with the form of a wave packet at a given
instant; in this paragraph, we are going to study its time evolution. Let us return,
20

C. QUANTUM DESCRIPTION OF A PARTICLE. WAVE PACKETS
therefore, to the case of a free particle whose state is described by the one-dimensional
wave packet (C-7).
A given plane wavee
( )
propagates along theaxis with the velocity:
() =
(C-24)
since it depends onandonly through
;()is called thephase velocityof
the plane wave.
We know that in the case of an electromagnetic wave propagating in a vacuum,
is independent ofand equal to the speed of light. All the waves making up a
wave packet move at the same velocity, so that the packet as a whole also moves with
the same velocity, without changing its shape. On the other hand, we know that this is
not true in a dispersive medium, where the phase velocity is given by:
() =
()
(C-25)
()being the index of the medium, which varies with the wavelength.
The case that we are considering here corresponds to a dispersive medium, since
the phase velocity is equal to [cf.equation (C-3)]:
() =
~
2
(C-26)
We shall see that when the dierent waves have unequal phase velocities, the velocity
of the maximum of the wave packet is not the average phase velocity
0
0
=
~0
2
,
contrary to what one might expect.
As we did before, we shall begin by trying to understand qualitatively what hap-
pens, before taking a more general point of view. Therefore, let us return to the super-
position of three waves considered in . For arbitrary,()is given by:
() =
(0)
2
e
[0 0]
+
1
2
e
[(0
2)(0
2)]
+
1
2
e
[(0+
2)(0+
2)]
=
(0)
2
e
[0 0]
1 + cos
22
(C-27)
We see that the maximum of(), which was at= 0at= 0, is now at point:
() =
(C-28)
and not at point=
0
0
. The physical origin of this result appears in Figure.
Parta) of this gure represents the position at time= 0of three adjacent maxima
(1), (2), (3), for the real parts of each of the three waves. Since the maxima denoted
by the index (2) coincide at= 0, there is constructive interference at this point,
21

CHAPTER I INTRODUCTION TO THE BASIC IDEAS OF QUANTUM MECHANICS(1)
2
(2) (3)
(1) (2) (3)
(1) (2) (3)
(1) (2) (3)
(1) (2) (3)
(1)
00
xx
x
M
(t)x
M
(0)
a b
k
0
Δk
k
0
k
0
–
(2) (3)
2
Δk
Figure 6: Positions of the maxima of the three waves of Figure = 0(g. a)
and at a subsequent time(g. b). At time= 0, it is the maxima (2), situated at= 0,
which interfere constructively: the position of the center of the wave packet is(0) = 0.
At time, the three waves have advanced with dierent phase velocities. It is then the
maxima (3) which interfere constructively and the center of the wave packet is situated at
= (). We thus see that the velocity of the center of the wave packet (group velocity)
is dierent from the phase velocities of the three waves.
which thus corresponds to the position of the maximum of(0). Since the phase
velocity increases with[formula (C-26)], the maximum (3) of the wave0+
2
will gradually catch up with that of the wave(0), which will in turn catch up with that
of the wave0
2
. After a certain time, we shall thus have the situation shown in
Figure-b: it will be the maxima (3) which coincide and thus determine the position of
the maximum ()of(). We clearly see in the gure that()is not equal to
0
0
, and a simple calculation again yields (C-28).
The shift of the center of the wave packet (C-7) can be found in an analogous
fashion, by applying the stationary phase method. It can be seen from the form (C-7)
of the free wave packet that, in order to go from(0)to(), all we need to do is
change()to() e
()
. The reasoning of
that we replace the argument()of()by:
()() (C-29)
The condition (C-16) then gives:
() =
d
d
=0
d
d
=0
(C-30)
We are thus brought back to result (C-28): the velocity of the maximum of the
wave packet is:
(0) =
d
d
=0
(C-31)
22

D. PARTICLE IN A TIME-INDEPENDENT SCALAR POTENTIAL
(0)is called thegroup velocityindexGroup velocity of the wave packet. With the
dispersion relation given in (C-3), we obtain:
(0) =
~0
= 2(0) (C-32)
This result is important, for it enables us to retrieve the classical description of the
free particle, for the cases where this description is valid. For example, when one is
dealing with a macroscopic particle (and the example of the dust particle discussed in
ComplementIshows how small it can be), the uncertainty relation does not introduce
an observable limit on the accuracy with which its position and momentum are known.
This means that we can construct, in order to describe such a particle in a quantum
mechanical way, a wave packet whose characteristic widthsand are negligible.
We would then speak, in classical terms, of the position()and the momentum 0
of the particle. But then its velocity must be=
0
. This is indeed what is implied
by formula (C-32), obtained in the quantum description: in the cases whereand
can both be made negligible, the maximum of the wave packet moves like a particle that
obeys the laws of classical mechanics.
Comment:
We have stressed here the motion of the center of the free wave packet. It is also
possible to study the way in which its form evolves in time. It is then easy to
show that, if the widthis a constant of the motion,varies over time and,
for suciently long times, increases without limit (spreading of the wave packet).
The discussion of this phenomenon is given in ComplementI, where the special
case of a Gaussian wave packet is treated.
D. Particle in a time-independent scalar potential
We have seen, in , how the quantum mechanical description of a particle reduces
to the classical description when Planck's constantcan be considered to be negligible.
In the classical approximation, the wavelike character does not appear because the wave-
length=
associated with the particle is much smaller than the characteristic lengths
of its motion. This situation is analogous to the one encountered in optics. Geometrical
optics, which ignores the wavelike properties of light, constitutes a good approximation
when the corresponding wavelength can be neglected compared to the lengths with which
one is concerned. Classical mechanics thus plays, with respect to quantum mechanics,
the same role played by geometrical optics with respect to wave optics.
In this paragraph, we are going to be concerned with a particle in a time-independent
potential. What we have just said implies that typically quantum eects (that is, those
of wave origin) should arise when the potential varies appreciably over distances shorter
than the wavelength, which cannot then be neglected. This is why we are going to study
the behavior of a quantum particle placed in various square potentials, that is, step
potentials, as shown in Figure-a. Such a potential, which is discontinuous, clearly
varies considerably over intervals of the order of the wavelength, however small it is:
quantum eects must therefore always appear. Before beginning this investigation, we
23

CHAPTER I INTRODUCTION TO THE BASIC IDEAS OF QUANTUM MECHANICS
shall discuss some important properties of the Schrödinger equation when the potential
is not time-dependent.
D-1. Separation of variables. Stationary states
The wave function of a particle whose polential energy(r)is not time-dependent
must satisfy the Schrödinger equation:
~
(r) =
~
2
2
(r) +(r)(r) (D-1)
D-1-a. Existence of stationary states
Let us see if there exist solutions of this equation of the form:
(r) =(r)() (D-2)
Substituting (D-2) into (D-1), we obtain:
~(r)
d()
d
=()
~
2
2
(r)+()(r)(r) (D-3)
If we divide both sides by the product(r)(), we nd:
~
()
d()
d
=
1
(r)
~
2
2
(r)+(r) (D-4)
This equation equates a function ofonly (left-hand side) and a function ofronly (right-
hand side). This equality is only possible if each of these functions is in fact a constant,
which we shall set equal to~, wherehas the dimensions of an angular frequency.
Setting the left-hand side equal to~, we obtain for()a dierential equation
which can easily be integrated to give:
() =e (D-5)
In the same way,(r)must satisfy the equation:
~
2
2
(r) +(r)(r) =~(r) (D-6)
If we set= 1in equation (D-5) [which is possible if we incorporate, for example, the
constantin(r)], we achieve the following result: the function
(r) =(r) e (D-7)
is a solution of the Schrödinger equation, on the condition that(r)is a solution of (D-6).
The time and space variables are said to have been separated.
A wave function of the form (D-7) is called astationary solution of the Schrödinger
equation: it leads to a time-independent probability density(r)
2
=(r)
2
. In a sta-
tionary function, only one angular frequencyappears; according to the Planck-Einstein
relations,a stationary state is a state with a well-dened energy=~(energy eigen-
state). In classical mechanics, when the potential energy is time-independent, the total
24

D. PARTICLE IN A TIME-INDEPENDENT SCALAR POTENTIAL
energy is a constant of the motion; in quantum mechanics, there exist well-determined
energy states.
Equation (D-6) can therefore be written:
~
2
2
+(r)(r) =(r) (D-8)
or:
(r) =(r) (D-9)
whereis the dierential operator:
=
~
2
2
+(r)
(D-10)
is a linear operator since, if1and2are constants, we have:
[11(r) +22(r)] =1 1(r) +2 2(r) (D-11)
Equation (D-9) is thus theeigenvalue equationof the linear operator: the application of
to the eigenfunction(r)yields the same function, multiplied by the corresponding
eigenvalue.The allowed energies are therefore the eigenvalues of the operator. We
shall see later that equation (D-9) has square-integrable solutions(r)only for certain
values of(cf. I): this is the origin ofenergy
quantization.
Comment:
Equation (D-8) [or (D-9)] is sometimes called the time-independent Schrödinger
equation, as opposed to the time-dependent Schrödinger equation (D-1). We stress
their essential dierence: equation (D-1) is a general equation which gives the evolution
of the wave function, whatever the state of the particle; on the other hand, the eigenvalue
equation (D-9) enables us to nd, amongst all the possible states of the particle, those
which are stationary.
D-1-b. Superposition of stationary states
In order to distinguish between the various possible values of the energy(and
the corresponding eigenfunctions(r)), we label them with an index. Thus we have:
(r) = (r) (D-12)
and the stationary states of the particle have as wave functions:
(r) =(r) e
~
(D-13)
(r)is a solution of the Schrödinger equation (D-1). Since this equation is linear, it
has a whole series of other solutions of the form:
(r) = (r) e
~
(D-14)
25

CHAPTER I INTRODUCTION TO THE BASIC IDEAS OF QUANTUM MECHANICS
where the coecientsare arbitrary complex constants. In particular, we have:
(r0) = (r) (D-15)
Inversely, assume that we know(r0), the state of the particle at= 0. We shall
see later that any function(r0)can always be decomposed in terms of eigenfunctions
of, as in (D-15). The coecientsare therefore determined by(r0). The cor-
responding solution(r)of the Schrödinger equation is then given by (D-14). It is
simply obtained by multiplying each term of (D-15) by the factore
~
, whereis
the eigenvalue associated with(r). We stress the fact that these phase factors dier
from one term to another. It is only in the case of stationary states that the-dependence
involves only one exponential [formula (D-13)].
D-2. One-dimensional square potentials. Qualitative study
We said at the beginning of
going to consider potentials that varied considerably over small distances. We shall limit
ourselves here to a qualitative study, so as to concentrate on the simple physical ideas. A
more detailed study is presented in the complements of this chapter (ComplementI). To
simplify the problem, we shall consider a one-dimensional model, in which the potential
energy depends only on(the justication for such a model is given in ComplementI).
D-2-a. Physical meaning of a square potential
We consider a one-dimensional problem with a potential of the type shown in
Figure-a. Theaxis is divided into a certain number of constant-potential regions.
At the border of two adjacent regions the potential makes an abrupt jump (discontinuity).
Actually, such a function cannot really represent a physical potential, which must be
continuous. We use it to represent schematically a potential energy()which actually
has the shape shown in Figure-b: there are no discontinuities, but()varies very
rapidly in the neighborhood of certain values of. When the intervals over which these
variations occur are much smaller than all other distances involved in the problem (in
particular, the wavelength associated with the particle), we can replace the true potential
by the square potential of Figure-a. This is an approximation, which would cease to
be valid, for example, for a particle having too high an energy, whose wavelength would
be very short.
The predictions of classical mechanics concerning the behavior of a particle in a
potential such as that of Figure
()is the gravitational potential energy. Figure-b then represents the real prole
of the terrain on which the particle moves: the corresponding discontinuities are sharp
slopes, separated by horizontal plateaus. Notice that, if we x the total energyof the
particle, the domains of theaxis where are forbidden to it (its kinetic energy
= must be positive).
26

D. PARTICLE IN A TIME-INDEPENDENT SCALAR POTENTIAL‘‘Square’’
potential
Real
potential
ForceF
0
0
0
x
x
x
a
b
c
Figure 7: Square potential (g. a) which schematically represents a real potential (g. b)
for which the force has the shape shown in gure c.
Comment:
The force exerted on the particle is() =
d()
d
. In Figure-c, we have
depicted this force, obtained from the potential()of Figure-b. It can be
seen that this particle, in all the regions where the potential is constant, is not
subjected to any force. Its velocity is then constant. It is only in the frontier zones
between these plateaus that a force acts on the particle and, depending on the
case, accelerates it or slows it down.
D-2-b. Optical analogy
We are going to consider the stationary states ( ) of a particle in a one-
dimensional square potential.
In a region where the potential has a constant value, the eigenvalue equation
(D-9) is written:
~
2
2
d
2
d
2
+ () =() (D-16)
or:
d
2
d
2
+
2
~
2
( )() = 0 (D-17)
Now, in optics, there exists a completely analogous equation. Consider a trans-
parent medium whose indexdepends neither onrnor on time. In this medium, there
can be electromagnetic waves whose electric eldE(r)is independent ofandand
has the form:
E(r) =e() e

(D-18)
27

CHAPTER I INTRODUCTION TO THE BASIC IDEAS OF QUANTUM MECHANICS
whereeis a unit vector perpendicular to;()must then satisfy:
d
2
d
2
+
2

2
2
() = 0 (D-19)
We see that equations (D-17) and (D-19) become identical if we set:
2
~
2
( ) =
2

2
2
(D-20)
Moreover, at a pointwhere the potential energy[and, consequently, the in-
dexgiven by (D-20)] is discontinuous, the boundary conditions for()and()are
the same: these two functions, as well as their rst derivatives, must remain continuous
(cf.ComplementI, ). The structural analogy between the two equations (D-17)
and (D-19) thus enables us to associate with a quantum mechanical problem, correspond-
ing to the potential of Figure-a, an optical problem: the propagation of an electromag-
netic wave of angular frequencyin a medium whose indexhas discontinuities of
the same type. According to (D-20), the relation between the optical and mechanical
parameters is:
() =
1
~2
2
( ) (D-21)
For the light wave, a region where corresponds to a transparent medium
whose index is real. The wave is then of the forme.
What happens when ? Formula (D-20) gives a pure imaginary index. In
relation (D-19),
2
is negative and the solution is of the forme: it is the analogue
of an evanescent wave. Certain aspects of the situation recall the propagation of an
electromagnetic wave in a metallic medium
7
.
Thus we can transpose the well-known results of wave optics to the problems which
we are studying here. It is important, however, to realize that this is merely an analogy.
The interpretation that we give for the wave function is fundamentally dierent from
that which classical wave optics attributes to the electromagnetic wave.
D-2-c. Examples
. Potential step and barrier
Consider a particle of energywhich, coming from the region of negative, arrives
at the potential step of height0shown in Figure.
If 0(the case in which the classical particle clears the potential step and
continues towards the right with a smaller velocity), the optical analogy is the following:
a light wave propagates from left to right in a medium of index1:
1=
~2 (D-22)
at=1, there is a discontinuity, and the index, for1, is:
2=
~2( 0) (D-23)
7
This analogy should not be pushed too far, since the indexof a metallic medium has both a real
and a complex part (in a metal, an optical wave continues to oscillate as it damps out).
28

D. PARTICLE IN A TIME-INDEPENDENT SCALAR POTENTIAL0
x
1
V
0
V(x)
x
Figure 8: Potential step.
We know that the incident wave coming from the left splits into a reected wave and a
transmitted wave. Let us transpose this result to quantum mechanics:the particle has
a certain probabilityPof being reected, and only the probability1Pof pursuing
its course towards the right. This result is contrary to what is predicted by classical
mechanics.
When 0, the index2, which corresponds to the region 1, becomes
pure imaginary, and the incident light wave is totally reected. The quantum prediction
therefore coincides at this point with that of classical mechanics. Nevertheless, the
existence, for 1, of an evanescent wave, shows that the quantum particle has a
non-zero probability of being found in this region.
The role of this evanescent wave is more striking in the case of a potential barrier
(Fig.). For 0, a classical particle would always turn back. But, in the corre-
sponding optical problem, we would have a layer of nite thickness, with an imaginary
index, surrounded by a transparent medium. If this thickness is not much greater than
the range1of the evanescent wave, part of the incident wave is transmitted into the
region 2. Therefore, even for 0, we nd anonzero probability of the particle
crossing the barrier. This is called the tunnel eect.
. Potential well
The function()now has the form shown in Figure. The predictions of
classical mechanics are the following: when the particle has a negative energy(but
greater than0), it can only oscillate between1and2, with kinetic energy=
+0; when the particle has a positive energy and arrives from the left, it undergoes anV(x)
x
1
x
2
x
V
0
0
Figure 9: Potential barrier.
29

CHAPTER I INTRODUCTION TO THE BASIC IDEAS OF QUANTUM MECHANICSV(x)
– V
0
0
x
1
x
2
x
Figure 10: Potential well.
abrupt acceleration at1, then an equal deceleration at2, and then continues towards
the right.
In the optical analogue of the case0 0, the indices1and3, which
correspond to the regions 1and 2, are imaginary, while the index2, which
characterizes the interval[12], is real. Thus we have the equivalent of a layer of air,
for example, between two reecting media. The dierent waves reected successively at
1and2destroy each other through interference, except for certain well-determined
frequencies (normal modes) that allow stable stationary waves to be established. From
the quantum point of view, this implies thatthe negative energies are quantized
8
, while,
classically, all values included between0and0are possible.
For 0, the indices1,2and3are real:
1=3=

1
~2 (D-24)
2=

1
~2(+0) (D-25)
Since2is greater than1and3, the situation is analogous to that of a layer of glass in
air. In order to obtain the reected wave for1, or the transmitted wave in the region
2, it is necessary to superpose an innite number of waves that arise from successive
reections at1and2(multiple wave interferometer analogous to a Fabry-Perot). We
then nd that, for certain incident frequencies, the wave is entirely transmitted. From
the quantum point of view, the particle thus has, in general, a certain probability of
being reected. However, there exist energy values, calledresonance energies, for which
the probability of transmission is 1 and, consequently, the probability of reection is 0.
These few examples show how much the predictions of quantum mechanics can
dier from those of classical mechanics. They also clearly stress the primordial role of
potential discontinuities (which represent, schematically, rapid variations).
CONCLUSION
In this chapter we have introduced and discussed, in a qualitative and intuitive
manner, certain fundamental ideas of quantum mechanics. We shall later return to
8
The allowed energy values are not given by the well-known condition:2 1= 22, for it is
necessary to take into account the existence of evanescent waves, which introduce a phase shift upon
reection at=1and=2(cf. ComplementI, ).
30

D. PARTICLE IN A TIME-INDEPENDENT SCALAR POTENTIAL
these ideas (Chap.) so as to present them in a more precise and systematic way.
Nevertheless, it is already clear that the quantum description of physical systems diers
radically from the one given by classical mechanics (although the latter constitutes, in
numerous cases, an excellent approximation). We have limited ourselves in this chapter
to the case of physical systems composed of only one particle. The description of these
systems at a given time is, in classical mechanics, founded on the specication of six
parameters, which are the components of the positionr()and the velocityv()of the
particle. All the dynamical variables (energy, linear momentum, angular momentum) are
determined by the specication ofr()andv(). Newton's laws enable us to calculate
r()through the solution of second-order dierential equations with respect to time.
Consequently, they x the values ofr()andv()for any timewhen they are known
for the initial time.
Quantum mechanics uses a more complicated description of phenomena. The
dynamic state of a particle, at a given time, is characterized by a wave function. It
no longer depends on only six parameters, but on an innite number [the values of
(r)at all pointsrof space]. Moreover, the predictions of the measurement results
are now only probabilistic (they yield only the probability of obtaining a given result
in the measurement of a dynamical variable). The wave function is a solution of the
Schrödinger equation, which enables us to calculate(r)from(r0). This equation
implies a principle of superposition which leads to wave eects.
This upheaval in our conception of mechanics was imposed by experiment. The
structure and behavior of matter on an atomic level are incomprehensible in the frame-
work of classical mechanics. The theory has thereby lost some of its simplicity, but it
has gained a great deal of unity, since matter and radiation are described in terms of
the same general scheme (wave-particle duality). We stress the fact that this general
scheme, although it runs counter to our ideas and habits drawn from the study of the
macroscopic domain, is perfectly consistent. No one has ever succeeded in imagining
an experiment that could violate the uncertainty principle (cf.ComplementIof this
chapter). In general, no observation has, to date, contradicted the fundamental princi-
ples of quantum mechanics. Nevertheless, at present, there is no global theory including
quantum phenomena within general relativity (gravity) and, of course, nothing prevents
the possibility of a new upheaval.
References and suggestions for further reading:
Description of physical phenomena which demonstrate the necessity of introducing
quantum mechanical concepts: see the subsection Introductory work quantum
physics of section 1 of the bibliography; in particular, Wichmann (1.1) and Feynman
III (1.2), Chaps. 1 and 2.
History of the development of quantum mechanical concepts: references of sec-
tion 4 of the bibliography, in particular, Jammer (4.8); also see references (5.11) and
(5.12), which contain numerous references to the original articles.
Fundamental experiments: references to the original articles can be found in sec-
tion 3 of the bibliography.
The problem of interpretation in quantum mechanics: section 5 of the bibliogra-
phy; in particular, the Resource Letter (5.11), which contains many references.
31

CHAPTER I INTRODUCTION TO THE BASIC IDEAS OF QUANTUM MECHANICS
Analogies and dierences between matter waves and electromagnetic waves: Bohm
(5.1), Chap. 4; in particular, the table Summary on Probabilities at the end of the
chapter.
See also the articles by Schrödinger (1.25), Gamow (1.26), Born and Biem (1.28),
Scully and Sargent (1.30).
32

COMPLEMENTS OF CHAPTER I, READER'S GUIDE
AI: ORDER OF MAGNITUDE OF THE WAVE-
LENGTHS ASSOCIATED WITH MATERIAL
PARTICLES
BI: CONSTRAINTS IMPOSED BY THE UNCER-
TAINTY RELATION
CI: HEISENBERG RELATION AND ATOMIC
PARAMETERS
These three complements provide very simple but
fundamental comments on the order of magnitude
of quantum parameters.
DI: AN EXPERIMENT ILLUSTRATING THE
HEISENBERG RELATIONS
Discussion of a simple thought-experiment that
attempts to invalidate the complementarity
between the particle and the wave aspects of light
(easy, but could be reserved for further study).
EI: A SIMPLE TREATMENT OF A TWO-
DIMENSIONAL WAVE PACKET
FI: THE RELATION BETWEEN ONE- AND
THREE-DIMENSIONAL PROBLEMS
GI: ONE-DIMENSIONAL GAUSSIAN WAVE
PACKET: SPREADING OF THE WAVE PACKET
Complements on wave packets ( ):
EI: reveals in a simple, qualitative way the
relation that exists between the lateral extension
of a two-dimensional wave packet and the angular
dispersion of the wave vectors (easy).
FI: generalization to three dimensions of the
results of ; shows how the study
of a particle in three-dimensional space can, in
certain cases, be reduced to one-dimensional
problems (a little more dicult).
GI: treats in detail a special case of wave
packets for which one can calculate exactly the
properties and the evolution (some diculties in
the calculation, but conceptually simple).
HI: STATIONARY STATES OF A PARTICLE IN
ONE-DIMENSIONAL SQUARE POTENTIALS
Takes up in a more quantitative way the ideas
of . Strongly recommended,
since square potentials are often used to illustrate
simply the implications of quantum mechanics
(numerous complements and excercizes proposed
later in this book rely on the results of HI).
JI: BEHAVIOR OF A WAVE PACKET AT A
POTENTIAL STEP
More precise study, for a special case, of the
quantum behavior of a particle in a square
potential. Since the particle is suciently well
localized in space (wave packet), one can follow
its motion (average diculty; important for the
physical interpretation of the results).
KI: EXERCISES
33

ORDER OF MAGNITUDE OF THE WAVELENGTHS ASSOCIATED WITH MATERIAL PARTICLES
Complement AI
Order of magnitude of the wavelengths associated with material
particles
De Broglie's relation:
=
(1)
shows that, for a particle of massand speed, the smallerand, the longer the
corresponding wavelength.
To show that the wave properties of matter are impossible to detect in the macro-
scopic domain, take as an example a dust particle, of diameter 1and mass
10
15
kg. Even for such a small mass and a speed of1mm/s, formula (1) gives:
6610
34
10
15
10
3
meter= 6610
16
meter= 6610
6
A (2)
Such a wavelength is completely negligible on the scale of the dust particle.
Consider, on the other hand, a thermal neutron, that is, a neutron
( 16710
27
kg) with a speedcorresponding to the average thermal energy
at the (absolute) temperature.is given by the relation:
1
2
2
=
2
2
3
2
(3)
whereis the Boltzmann constant (13810
23
joule/degree). The wavelength
corresponding to such a speed is:
=
=
3
(4)
For 300K, we nd:
14

A (5)
that is, a wavelength which is of the order of the distance between atoms in a crystal
lattice. A beam of thermal neutrons falling on a crystal therefore gives rise to diraction
phenomena analogous to those observed with X-rays.
Let us now examine the order of magnitude of the de Broglie wavelengths associated
with electrons (0910
30
kg). If one accelerates an electron beam through a
potential dierence(expressed in volts), one gives the electrons a kinetic energy:
= = 1610
19
joule (6)
(= 1610
19
coulomb is the electron charge.) Since=
2
2
, the associated
wavelength is equal to:
=
=
2
(7)
35

COMPLEMENT A I
that is, numerically:
=
6610
34
20910
30
1610
19
meter
123

A (8)
With potential dierences of several hundreds of volts, one again obtains wavelengths
comparable to those of X-rays, and electron diraction phenomena can be observed with
crystals or crystalline powders.
The large accelerators currently available are able to impart considerable energy
to particles. This takes us out of the non-relativistic domain to which we have thus
far conned ourselves. For example, electron beams are easily obtained for which the
energy
1
exceeds 1 GeV=10
9
eV (1 eV=1 electron-volt= 1610
19
joule), while the
electron rest mass is equal to
2
0510
6
eV. This means that the corresponding
speed is very close to the speed of light. Consequently, the non-relativistic quantum
mechanics which we are studying here does not apply. However, the relations:
= (9-a)
=
(9-b)
remain valid in the relativistic domain. On the other hand, relation (7) must be modied
since, relativistically, the energyof a particle of rest mass0is no longer
2
20, but
instead:
=
22
+
2
0
4
(10)
In the example considered above (an electron of energy 1 GeV),
2
is negligible com-
pared to, and we obtain:
=
6610
34
310
8
1610
10
m= 1210
15
m= 12fermi (11)
(1 fermi = 10
15
m). With electrons accelerated in this way, one can explore the struc-
ture of atomic nuclei and, in particular, the structure of the proton; nuclear dimensions
are of the order of a fermi.
Comments:
(i) We want to point out a common error in the calculation of the wavelength
of a material particle of mass0= 0, whose energyis known. This error
consists of calculating the frequencyusing (9-a) and, then, by analogy with
electromagnetic waves, of takingfor the de Broglie wavelength. Obviously, the
correct reasoning consists of calculating, for example from (10) (or, in the non-
relativistic domain, from the relation=
2
2) the momentumassociated
with the energyand then using (9-b) to nd.
1
Translator's note: In the United States, the unit Gev is sometimes written BeV.
36

ORDER OF MAGNITUDE OF THE WAVELENGTHS ASSOCIATED WITH MATERIAL PARTICLES
(ii) According to (9-a), the frequencydepends on the origin chosen for the en-
ergies. The same is true for the phase velocity=
=. Note, on the other
hand, that the group velocity=
d
d
= 2
d
d
does not depend on the choice of
the energy origin. This is important in the physical interpretation of.
(iii) Strictly speaking, relation (1) would predict a wavelength that always tends
to innity when the velocity of the particle tends to zero, whatever its mass is.
Let us nevertheless consider again the dust particle introduced above. Even when
that particle has a velocity as low as10
9
mms = 10
6
, its wavelength
is already of the order of its diameter (1). It is obviously impossible to ensure
that the velocity of a dust particle does not exceed such a low limit; therefore,
even if it is at rest, its wavelength remains negligible with respect to its size. By
contrast, for a particle such as the electron or the neutron, the quantum eects
can appear much more easily.
References and suggestions for further reading:
Wichmann (1.1), Chap. 5; Eisberg and Resnick (1.3), 3.1.
37

CONSTRAINTS IMPOSED BY THE UNCERTAINTY RELATIONS
Complement BI
Constraints imposed by the uncertainty relations
1 Macroscopic system
2 Microscopic system
We saw in
simultaneously dened with arbitrary precision: the corresponding uncertaintiesand
must satisfy the uncertainty relation:
&~ (1)
Here we intend to evaluate numerically the importance of this constraint. We shall show
that it is completely negligible in the macroscopic domain and that it becomes, on the
other hand, crucial on the microscopic level.
1. Macroscopic system
Let us take up again the example of a dust particle (cf. ComplementI), whose diameter
is on the order of1and whose mass 10
15
kg, having a speed= 10
3
m/sec.
Its momentum is then equal to:
= 10
18
joule secm (2)
If its position is measured to within001, for example, the uncertaintyin the
momentum must satisfy:
~
10
34
10
8
= 10
26
joule secm (3)
Thus the uncertainty relation introduces practically no restrictions in this case
since, in practice, a momentum measurement device is incapable of attaining the required
relative accuracy of 10
8
.
In quantum terms, the dust particle is described by a wave packet whose group
velocity is= 10
3
m/sec and whose average momentum is= 10
18
joule sec/m. But
one can then choose such a small spatial extensionand momentum dispersionthat
they are both totally negligible. The maximum of the wave packet then representsthe
position of the dust particle, and its motion is identical to that of the classical particle.
39

COMPLEMENT B I
2. Microscopic system
Now let us consider an atomic electron. The Bohr model describes it as a classical
particle. The allowed orbits are dened by quantization rules which are assumeda
priori: for example, the radiusof a circular orbit and the momentum= of the
electron travelling in it must satisfy:
=~ (4)
whereis an integer.
For us to be able to speak in this way of an electron trajectory in classical terms,
the uncertainties in its position and momentum must be negligible compared toand
respectively:
(5-a)
(5-b)
which would mean that:
1 (6)
Now the uncertainty relation imposes:
&
~
(7)
If we use formula (4) to replaceby~on the right-hand side, this inequality can be
written as:
&
1
(8)
We then see that (8) is incompatible with (6), unless 1. The uncertainty
relation thus makes us reject the semi-classical picture of the Bohr orbits (see
Chapter).
References and suggestions for further reading:
Bohm (5.1), Chap. 5, 14.
40

HEISENBERG RELATION AND ATOMIC PARAMETERS
Complement CI
Heisenberg relation and atomic parameters
The Bohr orbit has no physical reality when coupled with the Heisenberg relation (cf.
ComplementI). We shall study later (Chap.) the quantum theory of the hydrogen
atom. However, we can already show how the Heisenberg uncertainty relation enables
one to understand the stability of atoms and even to derive simply the order of magnitude
of the dimensions and the energy of the hydrogen atom in its ground state.
Let us consider an electron in the Coulomb eld of a proton, which we shall assume
to be stationary at the origin of the coordinate system. When the two particles are
separated by a distance, the potential energy of the electron is:
() =
2
40
1
(1)
whereis its charge (exactly opposite to that of the proton). We shall set:
2
40
=
2
(2)
Assume that the state of the electron is described by a spherically symmetric wave
function whose spatial extent is characterized by0(this means that the probability of
presence is practically zero beyond20or30). The potential energy corresponding to
this state is then on the order of:
2
0
(3)
For it to be as low as possible, it is necessary to take0as small as possible. This means
that the wave function must be as concentrated as possible about the proton.
But it is also necessary to take the kinetic energy into account. This is where
the uncertainty principle comes in: if the electron is conned within a volume of linear
dimension0, the uncertaintyin its momentum is at least of the order of~0. In
other words, even if the average momentum is zero, the kinetic energy
associated with
the state under consideration is not zero:
&=
1
2
()
2
~
2
2
2
0
(4)
If we reduce0in order to decrease the potential energy, the minimum kinetic energy (4)
increases.
The lowest total energy compatible with the uncertainty relation is thus the min-
imum of the function:
=
+=
~
2
2
2
0
2
0
(5)
This minimum is obtained for:
0=0=
~
2
2
(6)
41

COMPLEMENT C IT + V
T
0
a
0
r
0
V
Figure 1: Variation with respect to0(extension of the wave function) of the potential
energy
, the kinetic energy, and the total energy+of a hydrogen atom. The
functions
andvary inversely, so the total energy passes through a minimum value for
some value of
and. The corresponding value0of0gives the order of magnitude
of the hydrogen atom's size.
and is equal to:
0=
4
2~
2
(7)
Expression (6) is the one found in the Bohr model for the radius of the rst
orbit, and (7) gives correctly the energy of the ground state of the hydrogen atom (see
Chap.; the wave function of the ground state is indeed
0
). Such quantitative
agreement can only be accidental, since we have been reasoning on the basis of orders
of magnitude. However, the preceding calculation reveals an important physical idea:
because of the uncertainty relations, the smaller the extension of the wave function, the
greater the kinetic energy of the electron. The ground state of the atom results from a
compromise between the kinetic energy and the potential energy.
We stress the fact that this compromise, based on the uncertainty relations, is
totally dierent from what would be expected in classical mechanics. If the electron
moved in a classical circular orbit of radius0, its potential energy would be equal to:
=
2
0
(8)
The corresponding kinetic energy is obtained by equating the electrostatic force and the
centrifugal force
1
:
2
2
0
=
2
0
(9)
1
In fact, the laws of classical electromagnetism indicate that an accelerated electron radiates, which
already forbids the existence of stable orbits.
42

HEISENBERG RELATION AND ATOMIC PARAMETERS
which gives:
=
1
2
2
=
1
2
2
0
(10)
The total energy would then be equal to:
= + =
1
2
2
0
(11)
The most favorable energetic situation would occur at0= 0, which would give an in-
nite binding energy. Thus, we can say that it is the uncertainty relations that enables
us to understand, as it were, the existence of atoms.
References and suggestions for further reading:
Feynman III (1.2), 2-4. The same type of reasoning applied to molecules:
Schi (1.18), rst section of 49.
43

AN EXPERIMENT ILLUSTRATING THE HEISENBERG RELATIONS
Complement DI
An experiment illustrating the Heisenberg relations
Young's double-slit experiment, which we analyzed in , led us to the
following conclusions: both wave and particle aspects of light are needed to explain the
observed phenomena; but they seem to be mutually exclusive, in the sense thatit is
impossible to determine through which slit each photon has passed without destroying, by
this very operation, the interference pattern. The wave and particle aspects are sometimes
said to becomplementary.
We are going to consider Young's double-slit experiment again to demonstrate
how complementarity and uncertainty relations are intimately related. To try to cast
doubt on the uncertainty relation, one can imagine more subtle devices than the one of
Chapter, which used photomultipliers placed behind the slits. We shall now analyze
one of these devices.
Assume that the plateP, in which the slits are pierced, is mounted so that it
can move vertically in the same plane. Thus, it is possible to measure the vertical
momentum transferred to it. Consider (Fig.) a photon that strikes the observation
screenEat point(for simplicity, we choose a sourceSat innity). The momentum
of this photon changes when it crossesP. Conservation of momentum implies that the
platePabsorbs the dierence. But the momentum thus transferred toPdepends on
the path of the photon; depending on whether it passed through1or2, the vertical
transferred momentum is equal to:
1=
sin1 (1)
or:
2=
sin2 (2)
(
is the photon's momentum,1and2are the angles made by1and2with
the incident direction.).
We then allow the photons to arrive one by one and gradually construct the inter-
ference pattern on the screenE. For each one, we determine through which slit it has
passed by measuring the momentum acquired by the plateP. It therefore seems that
interference phenomena can still be observed onEalthough we know through which slit
each photon has passed.
Actually, we shall see that the interference fringes are not visible with this device.
The error in the preceding argument consists of assuming that only the photons have
a quantum character. In reality, it must not be forgotten that quantum mechanics
also applies to the plateP(macroscopic object). If we want to know through which
hole a photon has passed, the uncertaintyin the vertical momentum ofPmust be
suciently small for us to be able to measure the dierence between1and2:
2 1 (3)
45

COMPLEMENT D IF
2
F
1
d
O
M
x
θ
1
θ
2
Figure 1: Diagram of a device using a movable platePwhose momentum is measured
before and after the passage of the photon to determine whether the photon passed through
1or through2before arriving at pointon the screen.
But then the uncertainty relation implies that the position ofPis only known to within
, with:
&
2 1
(4)
If we designate bythe separation of the two slits and bythe distance between the
platePand the screenE, and if we assume that1and2are small( 1), we nd
(Fig.):
sin1 1
2
sin2 2
+2
(5)
(denotes the position of the point of impactonE). Formulas (1) and (2) then give:
2 1
2 1 (6)
where=
is the wavelength of light. Substituting this value into formula (4), we
obtain:
&
(7)
46

AN EXPERIMENT ILLUSTRATING THE HEISENBERG RELATIONS
But
is precisely the fringe separation we expect to nd onE. If the vertical position
of the slits1and2is dened only to within an uncertainty greater than the fringe
separation, it is impossible to observe the interference pattern.
The preceding discussion clearly shows that it is impossible to construct a quantum
theory that is valid for light and not for material systems without running into serious
contradictions. Thus, in the above example, if we could treat the platePas a classical
material system, we could invalidate the complementarity of the two aspects of light,
and, consequently, the quantum theory of radiation. Inversely, a quantum theory of
matter alone would come up against analogous diculties. In order to obtain an overall
coherence, we must apply quantum ideas to all physical systems.
References and suggestions for further reading:
Bohm (5.1), Chaps. 5 and 6; Messiah (1.17), Chap. IV III; Schi (1.18),
4; Jammer (5.12), Chaps. 4 and 5; also see reference (5.7).
47

A SIMPLE TREATMENT OF A TWO-DIMENSIONAL WAVE PACKET
Complement EI
A simple treatment of a two-dimensional wave packet
1 Introduction
2 Angular dispersion and lateral dimensions
3 Discussion
1. Introduction
In , we studied the shape of one-dimensional wave packets, obtained
by superposing plane waves which all propagate in the same direction [formula (C-7)].
If this direction is that of theaxis, the resulting function is independent ofand.
It has a nite extension along, but is not limited in the perpendicular directions: its
value is the same at all points of a plane parallel to.
We intend to examine here another simple type of wave packets: the plane waves
which we are going to combine have coplanar wave vectors, which are (nearly) equal in
magnitude but have slightly dierent directions. The goal is to show how the angular
dispersion leads to a limitation of the wave packet in the directions perpendicular to the
average wave vector.
We saw in
waves of the one-dimensional packet, one can understand the most important aspects
of the phenomena. In particular, this led to the fundamental relation (C-18) of that
chapter. We are going to limit ourselves here to a simplied model of this type. The
generalization of the results which we are going to nd can be carried out in the same
way as in Chapter I).
2. Angular dispersion and lateral dimensions
Consider three plane waves, whose wave vectorsk1,k2andk3are shown in Figure.
All three are in theplane;k1is directed along;k2andk3are symmetric with
respect tok1, the angle between each of them andk1being, which we assume to be
small. Finally, the projections ofk1,k2andk3on are equal:
1 2=3 k1= (1)
The magnitudes of these three vectors dier only by terms which are second order in,
which we shall neglect. Their components along theaxis are:
1= 0
2= 3
(2)
We shall choose, as in , real amplitudes(k)for waves, which satisfy
49

COMPLEMENT E Ik
y
k
x
k
2
k
1
k
3
Δk
y
Δθ
Δθ0
Figure 1: Arrangement of the wave vectors
k1,k2andk3associated with three plane
waves that will be superposed to construct a
two-dimensional wave packet.
the relations:
(k2) =(k3) =
1
2
(k1) (3)
This model represents schematically a more complex situation, in which one would
have a real wave packet, as in equation (C-6) of Chapter, with the following charac-
teristics: all the wave vectors are perpendicular toand have the same projection on
(only the component alongvaries); the function(k)has, with respect to this
single variable, the shape shown in Figure; its widthis related very simply to
the angular dispersion2:
= 2 (4)k Δθ– k Δθ
0
1
1
2
g(k)
k
y
Figure 2: The three values chosen forrep-
resent very schematically a peaked function
(k)(dashed line).
The superposition of the three waves dened above yields:
() =
3
=1
(k) e
kr
=(k1)e+
1
2
e
(+ )
+
1
2
e
( )
=(k1) e[1 + cos()] (5)
(there is no-dependence, which is why this is called a two-dimensional wave packet). In
order to understand what happens, we can use Figure, where we represent, for each of
50

A SIMPLE TREATMENT OF A TWO-DIMENSIONAL WAVE PACKETx
y
0
–
Δk
y
2
Δk
y
2
Figure 3: Equal phase planes of the three waves associated with the threekvectors of
Figure: these waves are in phase at= 0, but interfere destructively at=2.
the three components, the successive wave fronts corresponding to phase dierences of2.
The function()has a maximum at= 0: the three waves interfere constructively
on theaxis. When we move away from this axis,()decreases (the phase shift
between the components increases) and goes to zero at=
2
, whereis given by:
cos
2
=1 (6)
that is, for:
= 2 (7)
The phases of the (k2) and (k3) waves are then in opposition with that of the (k1) wave
(Fig.). Using (4), we can rewrite (7) in a form analogous to that of relation (C-11) of
Chapter:
= 4 (8)
Thus an angular dispersion of the wave vectors limits the lateral dimensions of
the wave packets. Quantitatively, this limitation has the form of an uncertainty relation
[formulas (7) and (8)].
3. Discussion
Consider a plane wave with wave vectorkpropagating along. Any attempt to limit its
extension perpendicular tocauses an angular dispersion to appear, that is, transforms
it into a wave packet analogous to the ones we are studying here.
51

COMPLEMENT E IΔy
Figure 4: When the uncertaintyis de-
creased, the diraction of the wave by the
diaphragm increases the uncertainty.
Imagine, for example, that we place in the path of the plane wave a screen pierced
by a slit of width. This will give rise to a diracted wave (cf.Fig.). We know that
the angular width of the diraction pattern is given by:
2 2
(9)
where=
2
k
is the incident wavelength. This is indeed the same situation as above:
formulas (7) and (9) are identical.
52

THE RELATIONSHIP BETWEEN ONE- AND THREE-DIMENSIONAL PROBLEMS
Complement FI
The relationship between one- and three-dimensional problems
1 Three-dimensional wave packet
1-a Simple case
1-b General case
2 Justication of one-dimensional models
The space in which a classical or quantum particle moves is, of course, three-dimensional.
This is why we wrote the Schrödinger equation (D-1) in Chapter
(r)that depends on the three components,,ofr. Nevertheless, we have repeatedly
used in this chapter a one-dimensional model, in which only the-variable is considered,
without justifying this model in a very precise way. Therefore, this complement has
two purposes: rst (), to generalize to three dimensions the results given in
Chapter; then (), to show how one can, in certain cases, rigorously justify the one-
dimensional model.
1. Three-dimensional wave packet
1-a. Simple case
Let us begin by considering a very simple case, for which the following two hy-
potheses are satised:
the wave packet is free[(r)0]; it can therefore be written as in equation (C-6)
of Chapter:
(r) =
1
(2)
32
(k) e
[kr(k)]
d
3
(1)
moreover, the function(k)is of the form:
(k) =1() 2() 3() (2)
Recall the expression for(k)in terms ofk:
(k) =
~k
2
2
=
~
2
(
2
+
2
+
2
) (3)
Substitute (2) and (3) into (1). It is possible to separate the three integrations with
respect to,andto obtain:
(r) =1() 2() 3() (4)
53

COMPLEMENT F I
with:
1() =
1
2
+
1() e
[ ()]
d
() =
~
2
2
(5)
and analogous expressions for2()and3().
1()indeed has the form of a one-dimensional wave packet. In this particular
case,(r)is thus obtained simply by taking the product (4) of three one-dimensional
wave packets, each evolving in a totally independent way.
1-b. General case
In the general case, where the potential(r)is arbitrary, formula (1) is no longer
valid. It is then useful to introduce the three-dimensional Fourier transform(k)of
the function(r)by writing:
(r) =
1
(2)
32
(k) e
kr
d
3
(6)
A priori, the-dependence of(k), which brings in(r), is arbitrary. Moreover, there
is no reason in general why we should be able to express(k)in the form of a product,
as in (2). In order to generalize the results of , we make the following
hypothesis about itsk-dependence:(k)is (at a given time) a function that has a
very pronounced peak for values ofkclose tok0and takes on a negligible value when
the tip ofkleaves a domaincentered atk0and of dimensions,,. As
above, we set:
(k) =(k)e
(k)
(7)
so that the phase of the wave dened by the vectorkcan be written:
(kr) =(k) + + + (8)
We can set forth an argument similar to that of . First of all, the
wave packet attains a maximum when all the waves, for which the tip ofkis in, are
practically in phase, that is, whenvaries very little within. In general,(kr)can
be expanded aboutk0. Its variation betweenk0andkis, to rst order ink=kk0:
(kr)
(kr)
k=k0
+(kr)
k=k0
+
(kr)
k=k0
(9)
that is, more concisely
1
, using (8):
(kr)k[k(kr)]
k=k0
kr+ [k(k)]
k=k0
(10)
1
The symboldesignates a gradient: by denition,( )is the vector whose coordinates are
, , . The indexkinkmeans that, as in (9), the dierentiations must be performed
with respect to the variables,and.
54

THE RELATIONSHIP BETWEEN ONE- AND THREE-DIMENSIONAL PROBLEMS
We see from (10) that the variation of(kr)within the domainwill be minimal
for:
r=r() =[k(k)]
k=k0
(11)
We have seen that, under these conditions,(r)is maximum. Relation (11) therefore
denes the positionr()of the center of the wave packet and constitutes the general-
ization to three dimensions of equation (C-15) of Chapter.
In what domain, centered atrand of dimensions,,, does the wave
packet (6) take on non-negligible values?(r)becomes much smaller than(r)
when the variouskwaves destroy each other by interference, that is, when the variation of
(kr)within the domainis of the order of2(or roughly, of the order of 1 radian).
Setr=rr; if (11) is taken into account, relation (10) can be written:
(kr)kr (12)
The condition(kr)&1immediately gives us the relations which exist between the
dimensions ofand those of:
&1
&1
&1
(13)
The Heisenberg relations then follow directly from the relationp=~k:
&~
&~
&~ (14)
These inequalities constitute the generalization to three dimensions of (C-23) of Chap-
ter.
Finally, note that the group velocityVof the wave packet can be obtained by
dierentiating (11) with respect to:
V=
d
d
[k(k)]
k=k0
(15)
In the special case of a free wave packet which does not, however, necessarily satisfy (2),
we have:
(k) =(k0)(k) (16)
where(k)is given by (3). Formula (15) then gives:
V= [k(k)]
k=k0
=
~k0
(17)
which is the generalization of equation (C-31) of Chapter.
55

COMPLEMENT F I
2. Justication of one-dimensional models
When the potential is time-independent, we saw in
to separate the time and space variables in the Schrödinger equation. This leads to the
eigenvalue equation (D-8). We intend to show here how it is possible, in certain cases,
to extend this method further and to separate as well the,,variables in (D-8).
Assume that the potential energy(r)can be written:
(r) =( ) =1() +2() +3() (18)
and let us see if there exist solutions of the eigenvalue equation of the form:
( ) =1() 2() 3() (19)
An argument analogous to the one set forth in Chapter ) shows that this is
possible if:
~
2
2
d
2
d
2
+1()1() =11() (20)
and if we have two other similar equations whereis replaced by(or),1by2(or
3), and1by2(or3). In addition, it is also necessary that the relation:
=1+2+3 (21)
be satised.
Equation (20) is of the same type as (D-8), but in one dimension. The,and
variables are separated
2
.
What happens, for example, if the potential energy(r)of a particle depends
only on?(r)can then be written in the form (18), where1=and2=3= 0.
Equations (20) inandcorrespond to the case already studied, in ,
of the free particle in one dimension; their solutions are plane wavese ande. All
that remains is to solve equation (20), which amounts to considering a problem in only
one dimension; nevertheless, the total energy of the particle in three dimensions is now:
=1+
~
2
2
2
+
2
(22)
The one-dimensional models studied in Chapter
in three dimensions moving in a potential(r)that depends only on. The solutions
2()and3()are then very simple and correspond to particles which are free along
or along. This is why we have concentrated all our attention on the study of the
-equation.
2
It can be shown (cf.Chap., ) that, when(r)has the form (18), all the solutions of
the eigenvalue equation (D-8) are linear combinations of those we nd here.
56

ONE-DIMENSIONAL GAUSSIAN WAVE PACKET: SPREADING OF THE WAVE PACKET
Complement GI
One-dimensional Gaussian wave packet: spreading of the wave packet
1 Denition of a Gaussian wave packet
2 Calculation of and ; uncertainty relation
3 Evolution of the wave packet
3-a Calculation of(). . . . . . . . . . . . . . . . . . . . . . .
3-b Velocity of the wave packet
3-c Spreading of the wave packet
In this complement, we intend to study a particular (one-dimensional) free wave packet,
for which the function()is Gaussian. The reason why this example is interesting lies
in the fact that the calculations can be carried out exactly and to the very end. Thus,
we can rst verify, in this special case, the various properties of wave packets which we
pointed out in . We shall then use these properties to study the variation
in time of the width of this wave packet, which will reveal the phenomenon of spreading
over time.
1. Denition of a Gaussian wave packet
Consider, in a one-dimensional model, a free particle[()0]whose wave function at
time= 0is:
(0) =
(2)
34
+
e
2
4
( 0)
2
ed (1)
This wave packet is obtained by superposing plane wavesewith the coecients:
1
2
(0) =
(2)
34
e
2
4
( 0)
2
(2)
which correspond to a Gaussian function centered at=0(and multiplied by a numer-
ical coecient which normalizes the wave function). This is why the wave packet (1) is
called Gaussian.
In the calculations that follow, we shall repeatedly come upon integrals of the type:
() =
+
e
2
(+)
2
d (3)
whereandare complex numbers [note that, for the integral (3) to converge, we must
have Re
2
0]. The method of residues enables us to show that this integral does not
57

COMPLEMENT G I
depend on:
() =(0) (4)
and that, when the condition 4Arg +4is fullled (which is always
possible if Re
2
0),(0)is given by:
(0) =
1
(10) (5)
Now all that remains is to evaluate(10), which can be done classically, through a
double integration in theplane and a change into polar coordinates:
(10) =
+
e
2
d=
(6)
Thus we have:
+
e
2
(+)
2
d=
(7)
with: 4Arg +4.
Let us now calculate(0). To do this, let us group, in the exponents of (1), the
-dependent terms into a perfect square, by writing them in the form:
2
4
( 0)
2
+ =
2
4
0
2
2
2
+0
2
2
(8)
We can then use (7), which yields:
(0) =
2
2
14
e
0
e
22
(9)
We nd, as could be expected, that the Fourier transform of a Gaussian function is also
a Gaussian (cf.Appendix).
At time= 0, the probability density of the particle is therefore given by:
(0)
2
=
2
2
e
2
22
(10)
The curve representing(0)
2
is the familiar bell-shaped curve. The center of the
wave packet [the maximum of(0)
2
] is situated at the point= 0. This is indeed
what we could have found if we had applied the general formula (C-16) of Chapter
since, in this particular case, the function()is real.
2. Calculation of and; uncertainty relation
It is convenient, when one is studying a Gaussian function() = e
22
, to dene its
width by:
=
2
(11)
58

ONE-DIMENSIONAL GAUSSIAN WAVE PACKET: SPREADING OF THE WAVE PACKET
Whenvaries from 0 to ,()is reduced by a factor of1
. This denition,
which is, of course, arbitrary, has the advantage of coinciding with that of the root-
mean-square deviation of thevariable (cf.Chap., ).
With this convention, we can calculate the widthof the wave packet (10),
which is equal to:
=
2
(12)
We can proceed in the same way to calculate the width, since(0)
2
is also a
Gaussian function. This gives:
=
1
(13-a)
or:
=
~
(13-b)
Thus we obtain:
=
~
2
(14)
a result which is entirely compatible with the Heisenberg relation.
3. Evolution of the wave packet
3-a. Calculation of ()
In order to calculate the wave function()at time, all we need to do is use
the general formula (C-6) of Chapter, which gives the wave function of a free particle.
We obtain:
() =
(2)
34
+
e
2
4
( 0)
2
e
[ ()]
d (15)
with() =~
2
2(dispersion relation for a free particle). Let us show that, at time,
the wave packet still remains Gaussian. Expression (15) can be transformed by grouping,
as above, all the-dependent terms in the exponents into a perfect square. We can then
use (7), and we nd:
() =
2
2
14
e
4
+
4~
22
2
14
e
0
exp
~0
2
2
+
2~
(16-a)
whereis real and independent of:
=
~
2
0
2
withtan 2=
2~
2
(16-b)
59

COMPLEMENT G I
Let us calculate the probability density()
2
of the particle at time. We obtain:
()
2
=
2
2
1
1 +
4~
2224
exp
2
2
~0
2
4
+
4~
222
(17)
Let us show that the norm of the wave packet,
+
()
2
d, is not time-
dependent (we shall see in Chapter
Hamiltonianof the particle is Hermitian). We could, to this end, use (7) again in order
to integrate expression (17) from to+. It is quicker to observe from expression
(15) that the Fourier transform of()is given by:
() = e
()
(0) (18)
()therefore obviously has the same norm as(0). Now the Parseval-Plancherel
equality [cf.relation (45) of Appendix] tells us that()and()have the same
norm, as do(0)and(0). From this we deduce that()has the same norm as
(0).
3-b. Velocity of the wave packet
We see in (17) that the probability density()
2
is a Gaussian function, cen-
tered at=0, where the velocity0is dened by:
0=
~0
(19)
We could have expected this result, in view of the general expression (C-32) of
Chapter, which gives the group velocity.
3-c. Spreading of the wave packet
Let us take up formula (17) again. The width()of the wave packet at time,
from denition (11), is equal to:
() =
2
1 +
4~
2224
(20)
We see (cf.Fig.) that the evolution of the wave packet is not purely a simple
displacement at a velocity0. The wave packet also undergoes a deformation. When
increases from to 0, the width of the wave packet decreases, reaching a minimum
at= 0. Then, ascontinues to increase,()grows without bound (spreading of the
wave packet).
It can be seen from (17) that the height of the wave packet also varies, but in
opposition to the width, so the norm of()remains constant.
The properties of the function()are completely dierent. In fact [cf.formula
(18)]:
()=(0) (21)
60

ONE-DIMENSIONAL GAUSSIAN WAVE PACKET: SPREADING OF THE WAVE PACKETt < 0 t = 0 t > 0
0
x
ψ(x, t)
2
Figure 1: For negative, the Gaussian wave packet decreases in width as it propagates.
At time= 0, it is a minimum wave packet: the productis equal to}2. Then,
for0, the wave packet spreads again as it propagates.
Therefore, the average momentum of the wave packet (~0) and its momentum dispersion
(~) do not vary in time. We shall see later (cf.Chap.) that this arises from the
fact that the momentum is a constant of the motion for a free particle. Physically, it
is clear that since the free particle encounters no obstacle, the momentum distribution
cannot change.
The existence of a momentum dispersion=~=~means that the velocity
of the particle is only known to within=
=
~
. Imagine a group of classical
particles starting at time= 0from the point= 0, with a velocity dispersion equal
to. At time, the dispersion of their positions will be= =
~
; this
dispersion increases linearly with, as shown in Figure. Let us draw on the same graph
the curve which gives the evolution in time of(); whenbecomes innite,()
practically coincides with[the branch of the hyperbola which represents()has
for its asymptotes the straight lines which correspond to]. Thus, we can say that,
whenis very large, there exists a quasi-classical interpretation of the width. On
the other hand, whenapproaches 0,()takes on values which dier more and more
from . The quantum particle must indeed constantly satisfy the Heisenberg relation
>~2which, sinceis xed, imposes a lower limit on. This corresponds
to what can be seen in Figure.
Comments:
(i) The spreading of a packet of free waves is a general phenomenon which is not
limited to the special case studied here. It can be shown that, for an arbitrary free
wave packet, the variation in time of its width has the shape shown in Figure
(cf.exercise 4 of ComplementIII).
(ii)In Chapter, a simple argument led us in (C-17) to 1, without making any
particular hypothesis about(). We simply assumed that()has a peak of width
whose shape is that of Figure
Then how did we obtain 1(for example, for a Gaussian wave packet when
is large)?
Of course, this is only an apparent contradiction. In Chapter, in order to nd
61

COMPLEMENT G I0
t
δx
cl
x
Figure 2: Variation in time of the widthof the wave packet of Figure. For large,
approaches the dispersionof the positions of a group of classical particles which
left= 0at time= 0with a velocity dispersion .
1, we assumed in (C-13) that the argument()of()could be approximated
by a linear function in the domain. Thus we implicitly assumed a supplementary
hypothesis: that the nonlinear terms make a negligible contribution to the phase of()
in the domain. For example, for the terms which are of second order in (0), it
is necessary that:
2d
2
d
2
=0
2 (22)
If, on the contrary, the phase()cannot be approximated in the domainby a linear
function with an error much smaller than2, we nd when we return to the argument
of Chapter C-17).
In the case of the Gaussian wave packet studied in the present complement,
we have
1
and() =
~
2
2
. Consequently, condition (22) can be written
1
2
~
2. Indeed, we can verify from (20) that, as long as this condition is fullled,
the product is approximately equal to 1.
62

STATIONARY STATES OF A PARTICLE IN ONE-DIMENSIONAL SQUARE POTENTIALS
Complement HI
Stationary states of a particle in one-dimensional square potentials
1 Behavior of a stationary wave function (). . . . . . . . .
1-a Regions of constant potential energy
1-b Behavior of()at a potential energy discontinuity
1-c Outline of the calculation
2 Some simple cases
2-a Potential steps
2-b Potential barriers
2-c Bound states: square well potential
We saw in Chaptercf.) the interest in studying the motion of a particle in a
square potential whose rapid spatial variations for certain values ofintroduce purely
quantum eects. The shape of the wave functions associated with the stationary states
of the particle was predicted by considering an optical analogy which enabled us to
understand very simply how these new physical eects appear.
In this complement, we outline the quantitative calculation of the stationary states
of the particle. We shall give the results of this calculation for a certain number of simple
cases, and discuss their physical implications. We limit ourselves to one-dimensional
models (cf.ComplementI).
1. Behavior of a stationary wave function()
1-a. Regions of constant potential energy
In the case of a square potential,()is a constant function() =in certain
regions of space. In such a region, equation (D-8) of Chapter
d
2
d
2
() +
2
~
2
( )() = 0 (1)
We shall distinguish between several cases:
(i)
Let us introduce the positive constant, dened by
=
~
22
2
(2)
The solution of equation (1) can then be written:
() =e+e (3)
63

COMPLEMENT H I
whereandare complex constants.
(ii)
This condition corresponds to regions of space which would be forbidden to the
particle by the laws of classical mechanics. In this case, we introduce the positive constant
dened by:
=
~
22
2
(4)
and the solution of (1) can be written:
() =e+e (5)
whereandare complex constants.
()=. In this special case,()is a linear function of.
1-b. Behavior of ()at a potential energy discontinuity
How does the wave function behave at a point=1, where the potential()
is discontinuous? One might expect the wave function()to behave strangely at this
point, becoming itself discontinuous, for example. The aim of this section is to show that
this is not the case:()andddare continuous, and it is only the second derivative
d
2
d
2
that is discontinuous at=1.
Without giving a rigorous proof, let us try to understand this property. To do this, recall
that a square potential must be considered (cf.Chap., ) as the limit, when0, of
a potential()equal to()outside the interval[1 1+], and varying continuously
within this interval. Then consider the equation:
d
2
d
2
() +
2
~
2
[ ()]() = 0 (6)
where()is assumed to be bounded, independently of, within the interval[1 1+].
Choose a solution()which, for 1 , coincides with a given solution of (1). The
problem is to show that, when0,()tends towards a function()which is continuous
and dierentiable at=1. Let us grant that()remains bounded
1
, whatever the value of
, in the neighborhood of=1. Physically, this means that the probability density remains
nite. Integrating (6) between1 and1+, we obtain:
d
d
(1+)
d
d
(1) =
2
~
2
1+
1
[()]() d (7)
At the limit where 0, the function to be integrated on the right-hand side of this expression
remains bounded, owing to our previous assumption. Consequently, iftends towards zero, the
integral also tends towards zero, and:
d
d
(1+)
d
d
(1)
0
0 (8)
Thus, at this limit,ddis continuous at=1, and so is()(since it is the integral of a
continuous function). On the other hand,d
2
d
2
is discontinuous, and, as can be seen directly
1
This point could be proved mathematically from the properties of the dierential equation (1).
64

STATIONARY STATES OF A PARTICLE IN ONE-DIMENSIONAL SQUARE POTENTIALS
from (1), makes a jump at=1, which is equal to
2
~
2
(1)[where represents the
change in()at=1].
Comment:
It is essential, in the preceding argument, that()remain bounded. In certain
exercises of ComplementI, for example, the case is considered for which() =
(), an unbounded function whose integral remains nite. In this case,()
remains continuous, butdddoes not.
1-c. Outline of the calculation
The procedure for determining the stationary states in a square potential is
therefore the following: in all regions where()is constant, write()in whichever
of the two forms (3) or (5) is applicable; then match these functions by requiring the
continuity of()and ofddat the points where()is discontinuous.
2. Some simple cases
Let us now carry out the quantitative calculation of the stationary states, performed ac-
cording to the method described above, for all the forms of the potential()considered
in . This will ensure that the form of the solutions is indeed the one
predicted by the optical analogy.
2-a. Potential steps0
I II
V
0
x
V(x)
Figure 1: Potential step.
. Case where 0; partial reection
Set:
2~
2
=1 (9)
2( 0)~
2
=2 (10)
65

COMPLEMENT H I
The solution of (1) has the form (3) in the two regions I(0)and II(0):
I() =1e
1
+
1e
1
(11)
II() =2e
2
+
2e
2
(12)
Since equation (1) is homogeneous, the calculation method of
us to determine the ratios
11,21and
21. In fact, the two matching conditions
at= 0do not suce for the determination of these three ratios. This is why we shall
choose
2= 0, which amounts to limiting ourselves to the case of an incident particle
coming from= . The matching conditions then give:
1
1
=
1 2
1+2
(13)
2
1
=
21
1+2
(14)
I()is the superposition of two waves. The rst one (the term in1) corresponds
to an incident particle, with momentum=~1, propagating from left to right. The
second one (the term in
1) corresponds to a reected particle, with momentum~1,
propagating in the opposite direction. Since we have chosen
2= 0,II()consists of
only one wave, which is associated with a transmitted particle. We shall see in Chapter
(cf. ) how it is possible, using the concept of a probability current, to dene the
transmission coecientand the reection coecientof the potential step (see also
III). These coecients give the probability for the particle, arriving
from= , to pass the potential step at= 0or to turn back. Thus we nd:
=
1
1
2
(15)
and, for
2
:
=
2
1
2
1
2
(16)
Taking (13) and (14) into account, we then have:
= 1
412
(1+2)
2
(17)
=
412
(1+2)
2
(18)
2
The physical origin of the factor21, appearing inis discussed in I.
66

STATIONARY STATES OF A PARTICLE IN ONE-DIMENSIONAL SQUARE POTENTIALS
It is easy to verify that+= 1: it is certain that the particle will be either transmitted
or reected. Contrary to the predictions of classical mechanics, the incident particle has
a non-zero probability of turning back. This point was explained in Chapter, using the
optical analogy and considering the reection of a light wave from a plane interface (with
1 2). Furthermore, we know that in optics, no phase delay is created by such a
reection; equations (13) and (14) do indeed show that the ratios
11and21are
real. Therefore, the quantum particle is not slowed down by its reection or transmission
(cf.ComplementI, ). Finally, using (9), (10) and (18) it is easy to verify that, if
0, we have 1: when the energy of the particle is suciently large compared
to the height of the potential step, the particle clears this step as if it did not exist.
. Case where 0; total reection
We then replace (10) and (12) by:
2(0 )~
2
=2 (19)
II() =2e
2
+
2e
2
(20)
For the solution to remain bounded when+, it is necessary that:
2= 0 (21)
The matching conditions at= 0yield in this case:
1
1
=
1 2
1+2
(22)
2
1
=
21
1+2
(23)
The reection coecientis then equal to:
=
1
1
2
=
1 2
1+2
2
= 1 (24)
As in classical mechanics, the particle is always reected (total reection). Nevertheless,
there is an important dierence, which has already been pointed out in Chapter. Because
of the existence of the evanescent wavee
2
, the particle has a non-zero probability
of presence in the region of space which, classically, would be forbidden to it. This
probability decreases exponentially withand becomes negligible whenis greater
than the range12of the evanescent wave. Note also that the coecient
11is
complex. A certain phase shift appears upon reection, which, physically, is due to the
fact that the particle is delayed when it penetrates the0region (cf.ComplementI,
III, ). This phase shift is analogous to the one that appears when light
is reected from a metallic type of substance; however, there is no analogue in classical
mechanics.
67

COMPLEMENT H I
Comment:
When 0 +,2 +, so that (22) and (23) yield:
1 1
2 0
(25)
In the0region, the wave, whose range decreases without bound, tends towards
zero. Since(1+
1) 0, the wave function()goes to zero at= 0, so
that it remains continuous at this point. On the other hand, its derivative, which
changes abruptly from the value2 1to zero, is no longer continuous. This is
due to the fact that since the potential jump is innite at= 0, the integral of
(7) no longer tends towards zero whentends towards 0.
2-b. Potential barriersI II III
0 l
x
V
0
V(x)
Figure 2: Square potential barrier.
. Case where 0; resonances
3
Using notations () and (10), we nd, in the three regions I(0), II(0 )
and III( )shown in Fig.:
I() = 1e
1
+
1e
1
(26-a)
II() =2e
2
+
2e
2
(26-b)
III() =3e
1
+
3e
1
(26-c)
Let us choose, as above,
3= 0(incident particle coming from= ). The
matching conditions at=then give2and
2in terms of3, and those at= 0
give1and
1in terms of2and
2(and, consequently, in terms of3). Thus we nd:
1=cos2
2
1+
2
2
212
sin2e
1
3
1=
2
2
2
1
212
sin2e
1
3 (27)
3
V0can be either positive (the case of a potential barrier like the one shown in Figure) or negative
(a potential well).
68

STATIONARY STATES OF A PARTICLE IN ONE-DIMENSIONAL SQUARE POTENTIALST
1
0
/k
2 2/k
2
l
4E(E V
0
)
4E(E V
0
) + V
2
0
Figure 3: Variations of the transmission coecientof the barrier as a function of its
width (the height0of the barrier and the energyof the particle are xed). Resonances
appear each time thatis an integral multiple of the half-wavelength2, in region II.
11and31enable us to calculate the reection coecientand the transmission
coecientof the barrier:
=
1
1
2
=
(
2
1
2
2)
2
sin
2
2
4
2
1
2
2
+ (
2
1
2
2
)
2
sin
2
2
(28-a)
=
3
1
2
=
4
2
1
2
2
4
2
1
2
2
+ (
2
1
2
2
)
2
sin
2
2
(28-b)
It is then easy to verify that+= 1. Taking (9) and (10) into account, we have:
=
4( 0)
4( 0) +
2
0
sin
22( 0)~
(29)
The variations with respect toof the transmission coecientare shown in
Figure and0xed):oscillates periodically between its minimum value,
1 +
2
0
4( 0)
1
, and its maximum value, which is 1. This function is the analogue
of the one describing the transmission of a Fabry-Perot interferometer. As in optics, the
resonances (obtained when= 1, that is, when2=) correspond to the values of
which are integral multiples of the half-wavelength of the particle in region II. When
0, the reection of the particle at each of the potential discontinuities occurs
without a phase shift of the wave function (cf. ). This is why the resonance
condition2= corresponds to the values offor which a system of standing waves
can exist in region II. On the other hand, far from the resonances, the various waves which
are reected at= 0and=destroy each other by interference, so that the values
of the wave function are small. A study of the propagation of a wave packet (analogous
to the one in ComplementI) would show that, if the resonance condition is satised,
the wave packet spends a relatively long time in region II. In quantum mechanics this
phenomenon is calledresonance scattering.
69

COMPLEMENT H I
. Case where 0; tunnel eect
We must now replace (26-b) by (20),2still being given by (19). The matching
conditions at= 0and=enable us to calculate the transmission coecient of the
barrier. In fact, it is unnecessary to perform the calculations again: all we must do is
replace, in the equations obtained in , the wave vector2by 2. We then have:
=
3
1
2
=
4(0 )
4(0 ) +
2
0
sinh
22(0 )~
(30)
with, of course,= 1 . When21, we have:
16(0 )
2
0
e
22
(31)
We have already seen in Chapter
has a non-zero probability of crossing the potential barrier. The wave function in region II
is not zero, but has the behavior of an evanescent wave of range12. When.12,
the particle has a considerable probability of crossing the barrier by the tunnel eect.
This eect has numerous physical applications: the inversion of the ammonia molecule
(cf.ComplementIV), the tunnel diode, the Josephson eect, the-decay of certain
nuclei, etc...
For an electron, the range of the evanescent wave is:
1
2
1.96
0

A (32)
whereand0are expressed in electron-volts (this formula is easily obtained by replacing, in
formula (8) of ComplementI,= 2 by2 2). Now consider an electron of energy 1 eV
which encounters a barrier for which0= 2eV and= 1

A. The range of the evanescent wave
is then 1.96

A, that is, of the order of: the electron must then have a considerable probability
of crossing the barrier. Indeed, formula (30) gives in this case:
0.78 (33)
The quantum result is radically dierent from the classical result: the electron has approximately
8 chances out of 10 of crossing the barrier.
Let us now assume that the incident particle is a proton (whose mass is about 1 840 times
that of the electron). The range12then becomes:
1
2
1.96
1 840(0 )

A
4.6
0
10
2
A (34)
If we retain the same values:= 1eV,0= 2eV,= 1

A, we nd a range12, much smaller
than. Formula (31) then gives:
410
19
(35)
Under these conditions, the probability of the proton's crossing the potential barrier is negligible.
This is all the more true if we apply (31) to macroscopic objects, for which we nd such small
probabilities that they cannot possibly play any role in physical phenomena.
70

STATIONARY STATES OF A PARTICLE IN ONE-DIMENSIONAL SQUARE POTENTIALS
2-c. Bound states: square well potential
. Well of nite depth V
0
V(x)
x
I II III
0
a
2
+
a
2
Figure 4: Square well potential.
We shall limit ourselves to studying the case0 0(the case 0was
included in the calculations of the preceding section ).
In regions I
2
, II
2
66
2
, and III
2
shown in Fig., we
have respectively:
I() = 1e+
1e (36-a)
II() =2e+
2e (36-b)
III() =3e+
3e (36-c)
with
=
2~
2
(37)
=
2(+0)~
2
(38)
Since()must be bounded in region I, we must have:
1= 0 (39)
The matching conditions at=
2
then give:
2= e
(+)2
+
2
1
2=e
(+)2
2
1 (40)
and those at=2:
71

COMPLEMENT H I
3
1
=
e
4
(+)
2
e ( )
2
e
3
1
=
2
+
2
2
sin (41)
But()must also be bounded in region III. Therefore, it is necessary that3= 0,
that is:
+
2
= e
2
(42)
Sinceanddepend on, equation (42) can only be satised for certain values
of. Imposing a bound on()in all regions of space thus entails the quantization of
energy. More precisely, two cases are possible:
(i) if:
+
=e (43)
we have:
= tan
2
(44)
Set:
0=
2 0~
2
=
2
+
2
(45)
We then obtain:
1
cos
2
2
= 1 + tan
2
2
=
2
+
2
2
=
0
2
(46)
Equation (43) is thus equivalent to the system of equations:
cos
2
=
0
tan
2
0
(47a)
(47b)
The energy levels are determined by the intersection of a straight line, having a slope10,
with sinusoidal arcs (long dashed lines in Figure). Thus we obtain a certain number
of energy levels, whose wave functions are even. This becomes clear if we substitute
(43) into (40) and (41); it is easy to verify that
3= 1and that2=
2, so that
() =().
72

STATIONARY STATES OF A PARTICLE IN ONE-DIMENSIONAL SQUARE POTENTIALSy
P
P
P
0
/a 2/a 3/a 4/a 5/a
I
I
k
k
0
Figure 5: Graphic solution of equation (42), giving the energies of the bound states of a
particle in a square well potential. In the case shown in the gure, there exist ve bound
states, three even (associated with the pointsof the gure), and two odd (points).
(ii) if:
+
= e (48)
a calculation of the same type leads to:
sin
2
=
0
tan
2
0
(49a)
(49b)
The energy levels are then determined by the intersection of the same straight line as
before with other sinusoidal arcs (cf.short dashed lines in Figure). The levels thus
obtained fall between those found in (i). It can easily be shown that the corresponding
wave functions are odd.
Comment:
If06
, that is, if:
061=
2
~
2
2
2
(50)
Figure
an even wave function. Then, if16041, a rst odd level appears, and so on: when
0increases, there appear alternatively even and odd levels. If0 1, the slope10
of the straight line of Figure
have:
=
(51)
whereis an integer, and consequently:
=
22
~
2
2
2
0 (52)
73

COMPLEMENT H I
. Innitely deep well
Assume()to be zero for0 and innite everywhere else. Set:
=
2~
2
(53)
According to the comment made at the end of of this complement,()must be
zero outside the interval[0], and continuous at= 0, as well as at=. Now for
0 :
() =e+e (54)
Since(0) = 0, it can be deduced that= , which leads to:
() = 2sin (55)
Moreover,() = 0, so that:
=
(56)
whereis an arbitrary positive integer. If we normalize function (55), taking (56) into
account, we then obtain the stationary wave functions:
() =
2
sin (57)
with energies:
=
22
~
2
2
2
(58)
The quantization of the energy levels is thus, in this case, particularly simple.
Comments:
(i) Relation (56) simply expresses the fact that the stationary states are determined
by the condition that the widthof the well must contain an integral number of
half-wavelengths,. This is not the case when the well has a nite depth (cf.
); the dierence between the two cases arises from the phase shift of the
wave function that occurs upon reection from a potential step (cf. ).
(ii) It can easily be veried from (51) and (52) that, if the depth0of a nite well
approaches innity, we nd the energy levels of an innite well.
References and suggestions for further reading:
Eisberg and Resnick (1.3), Chap. 6; Ayant and Belorizky (1.10), Chap. 4;
Messiah (1.17), Chap. III; Merzbacher (1.16), Chap. 6; Valentin (16.1), annex V.
74

BEHAVIOR OF A WAVE PACKET AT A POTENTIAL STEP
Complement JI
Behavior of a wave packet at a potential step
1 Total reection: 0. . . . . . . . . . . . . . . . . . . . . .
2 Partial reection: 0. . . . . . . . . . . . . . . . . . . . .In ComplementI, we determined the stationary states of a particle in various
square potentials. For certain cases (a step potential, for example), the stationary
states obtained consist of unbounded plane waves (incident, reected and transmitted).
Of course, since they cannot be normalized, such wave functions cannot really represent a
physical state of the particle. However, they can be linearly superposed to form normal-
izable wave packets. Moreover, since such a wave packet is expanded directly in terms
of stationary wave functions, its time evolution is very simple to determine. All we need
to do is multiply each of the coecients of the expansion by an imaginary exponential
e
~
with a well-dened frequency
(chap., ).
We intend, in this complement, to construct such wave packets and study their time
evolution for the case where the potential presents a step of height0, as in Figure
ComplementI. In this way, we shall be able to describe precisely the quantum behavior
of the particle when it arrives at the potential step by determining the motion and the
deformation of its associated wave packet. This will also enable us to conrm various
results obtained inIthrough the study of the stationary states alone (reection and
transmission coecients, delay upon reection, etc...).
We shall set:
2~
2
=
2 0~
2
= 0 (1)
and, as in ComplementI, we shall distinguish between two cases, corresponding to
smaller or greater than0.
1. Total reection: 0
In this case, the stationary wave functions are given by formulas (11) and (20) of Com-
plementI(1will be simply calledhere), the coecients1,
1,2and
2of these
formulas being related by equations (21), (22) and (23) ofI.
We are going to construct a wave packet from these stationary wave functions
by linearly superposing them. We shall choose only values ofless than0, so that
the waves forming the packet undergo total reection. To ensure this, we shall choose
a function()(which characterizes the wave packet) which is zero for0. We
75

COMPLEMENT J I
are going to focus our attention on the negative region of the-axis, to the left of the
potential barrier. In ComplementI, relation (22) shows that the coecients1and
1
of expression (11) for a stationary wave in this region have the same modulus. Therefore,
we can set:
1()
1()
= e
2()
(2)
with [cf.formula (19) ofI]:
tan() =
2
0
2
(3)
Finally, the wave packet which we are going to consider can be written, at time= 0,
for negative:
(0) =
1
2
0
0
d()[e+ e
2()
e] (4)
As in , we assume that()has a pronounced peak of widthabout
the value=0 0.
In order to obtain the expression for the wave function()at any time, we
simply use the general relation (D-14) of Chapter:
() =
1
2
0
0
d() e
[ ()]
+
1
2
0
0
d() e
[+()+2()]
(5)
where() =~
2
2. By construction,this expression is valid only for negative. Its
rst term represents the incident wave packet; its second term, the reected packet. For
simplicity, we shall assume()to be real. The stationary phase condition (cf.Chap.,
) then enables us to calculate the positionof the center of the incident wave
packet. If, at=0, we set the derivative with respect toof the argument of the rst
exponential equal to zero, we obtain:
=
d
d
=0
=
~0
(6)
In the same way, the positionof the center of the reected packet is obtained by
dierentiating the argument of the second exponential. Dierentiating equation (3), we
nd:
[1 + tan
2
] d=1 +
2
0
2
2
d
=
d
2
2
0
2
d2
0
2
(7)
that is:
2
0
2
d=
2
0
2
1
2
0
2
d (8)
76

BEHAVIOR OF A WAVE PACKET AT A POTENTIAL STEP
Thus we have:
=
d
d
+ 2
d
d
=0
=
~0
+
2
2
0
2
0
(9)
Formulas (6) and (9) enable us to describe more precisely the motion of the particle,
localized in a region of small widthcentered ator.
First of all, let us consider what happens for negative. The centerof the
incident wave packet propagates from left to right with a constant velocity~0. On
the other hand, we see from formula (9) thatis positive, that is, situated outside the
region0where expression (5) for the wave function is valid. This means that, for all
negative values of, the various waves of the second term of (5) interfere destructively:
for negative,there is no reected wave packet, but only an incident wave packet like
those we studied in .
The center of the incident wave packet arrives at the barrier at time= 0. During
a certain interval of time around= 0, the wave packet is localized in the region0
where the barrier is, and its form is relatively complicated. But,whenissuciently
large, we see from (6) and (9) thatit is the incident wave packet which has disappeared,
and we are left with only the reected wave packet. It is nowwhich is positive,
whilehas become negative: the waves of the incident packet interfere destructively
for all negative values of, while those of the reected packet interfere constructively
for= 0. The reected wave packet propagates towards the left at a speed of
~0, opposite to that of the incident packet, whose mirror image it is; its form is
unchanged
1
. Moreover, formula (9) shows that thereection has introduced a delay,
given by:
=2
dd
dd
=0
=
2
~0
2
0
2
0
(10)
Contrary to what is predicted by classical mechanics, the particle is not instantaneously
reected. Note that the delayis related to the phase shift2()between the incident
wave and the reected wave for a given value of. Nevertheless, it should be observed that
the delay of the wave packet is not simply proportional to(0), as would be the case for
an unbounded plane wave, but to thederivativeddevaluated at=0. Physically,
this delay is due to the fact that, forclose to zero, the probability of presence of the
particle in the region0, which is classically forbidden, is not zero [evanescent wave,
see comment (i) below]. It can be said, metaphorically, that the particle spends a time
of the order ofin this region before retracing its steps. Formula (10) shows that the
closer the average energy
~
22
0
2
of the wave packet is to the height0of the barrier, the
longer the delay.
Comments:
(i)Here we have focussed on the behavior of the wave packet for0, but it is also
possible to study what happens for0. In this region, the wave packet can be written:
1
We assume to be small enough for the spreading of the wave packet to be negligible during the
time interval considered.
77

COMPLEMENT J I
() =
1
2
0
0
d()
2() e
()
e
()
(11)
where:
() =
2
0
2
(12)
2()is given by equation (23) of ComplementIwhen we replace1by 1,1by
and2by. An argument analogous to the one in
that the modulus()of expression (11) is maximum when the phase of the function
to be integrated overis stationary. Now, according to expressions (22) and (23) of
HI, the argument of
2is half that of
1, which, according to (2), is equal to2().
Consequently, if we expand()and()in the neighborhood of=0, we obtain, for
the phase of the function to be integrated overin (11
d
d =0
d
d =0
( 0) =
~0
( 0)
2
(13)
[we have used (10) and the fact that()is assumed real]. From this we can deduce
that()is maximum in the 0region for
2
=
2
. The time at which the wave
packet turns back is therefore2, which gives us the same delayupon reection that
we obtained above. We also see from expression (13) that, as soon as
2
exceeds the
timedened by:
~0
1 (14)
where is the width of(), the waves go out of phase and expression (11) for()
becomes negligible. Thus, the wave packet as a whole remains in the0region during
an interval of timeof the order of:
=
1
~0
(15)
which corresponds approximately to the time it takes, in the0region, to travel a
distance comparable to its width1.
(ii) Sinceis assumed to be much smaller than0and0, the comparison of (10) and
(15) shows that:
(16)
The delay upon reection thus involves, for the reected wave packet, a displacement
which is much smaller than its width.
2
Note that the phase (13) does not depend on, contrary to what we found in Chapter
wave packet. It follows that, in the0region,()does not have a pronounced peak that moves
with respect to time.
78

BEHAVIOR OF A WAVE PACKET AT A POTENTIAL STEP
2. Partial reection: 0
We now consider a function()of width, centered at a value=0 0, which
is zero for 0. The wave packet is formed in this case by superposing, with coe-
cients(), the stationary wave functions whose expressions are given by formulas (11)
and (12) of ComplementI. We shall choose
2= 0so as to have the particle arrive
at the barrier from the negative region of theaxis, and we shall take1= 1. The
coecients
1()and2()are obtained from formulas (13) and (14) of ComplementI
(in which1is replaced by 1,1by, and2by
2 2
0
).
In order to describe the wave packet by a single expression, valid for all values of, we
can use the Heaviside step function()dened by:
() = 0if0
() = 1if0 (17)
The wave packet we are studying can then be written:
() =()
1
2
+
0
d() e
[ ()]
+()
1
2
+
0
d()
1() e
[+()]
+()
1
2
+
0
d()2() e
[
2 2
0
()]
(18)
It is composed of three wave packets: incident, reected and transmitted. As in

,and. Since
1()and2()are real, we nd:
=
~0
(19-a)
=
~0
(19-b)
=
~
2
0
2
0
(19-c)
A discussion analogous to that of (6) and (9) leads to the following conclusions:for
negative,only the incident wave packet exists; for suciently large positive,only the
reected and transmitted wave packets exist(Fig.). Note that there is no delay, either
upon reection or upon transmission (this is due to the fact that the coecients
1()
and2()are real).
The incident and reected wave packets propagate with velocities of~0and
~0respectively. Let us assumeto be suciently small that, within the interval
0
2
0+
2
, we can neglect the variation of
1()compared to that of().
We can then, in the second term of (18), replace
1()by
1(0)and take it outside
the integral. It is then easy to see that the reected wave packet has the same form
as the incident wave packet, being its mirror image. Its amplitude is smaller, however,
79

COMPLEMENT J IV(x)
ψ(x)
2
ψ(x)
2
ψ(x)
2
0
0
0
0
x
d
c
b
a
x
x
x
Figure 1: Behavior of a wave packet at a potential step, in the case0. The potential
is shown in gurea. In gureb, the wave packet is moving towards the step. Figurec
shows the wave packet during the transitory period in which it splits in two. Interference
between the incident and reected waves are responsible for the oscillations of the wave
packet in the0region. After a certain time (g.d), we nd two wave packets. The
rst one (the reected wave packet) is returning towards the left; its amplitude is smaller
than that of the incident wave packet, and its width is the same. The second one (the
transmitted wave packet) propagates towards the right; its amplitude is slightly greater
than that of the incident wave packet, but it is narrower.
since, according to formula (13) of ComplementI,
1(0)is less than 1. The reection
coecientis, by denition, the ratio between the probabilities of nding the particle in
the reected wave packet and in the incident packet. Therefore, we have=
1(0)
2
,
which indeed corresponds to equation (15) of ComplementI[recall that we have chosen
1(0) = 1].
The situation is dierent for the transmitted wave packet. We can still use the fact
that is very small in order to simplify its expression: we replace2()by2(0),
and
2 2
0
by the approximation:2 2
0
2
0
2
0
+ ( 0)
d2 2
0d
=0
0+ ( 0)
0
0
(20)
80

BEHAVIOR OF A WAVE PACKET AT A POTENTIAL STEP
with:
0=
2
0
2
0
(21)
The transmitted wave packet can then be written:
() 2(0) e
0
1
2
+
0
d() e
( 0)
0
0
()
(22)
Let us compare this expression to the one for the incident wave packet:
() = e
0
1
2
+
0
d() e
[( 0) ()]
(23)
We see that:
() 2(0)
0
0
(24)
The transmitted wave packet thus has a slightly greater amplitude than that of the
incident packet: according to formula (14) of ComplementI,2(0)is greater than 1.
However, its width is smaller, since, if()has a width, formula (24) shows that
the width of()is:
()=
0
0
(25)
The transmission coecient (the ratio between the probabilities of nding the particle
in the transmitted packet and in the incident packet) is thus the product of two factors:
=
0
0
2(0)
2
(26)
This indeed corresponds to formula (16) of ComplementI, since1(0) = 1. Finally,
note that, taking into account the contraction of the transmitted wave packet along the
axis, we can nd its velocity:
=
~0
0
0
=
~0
(27)
References and suggestions for further reading:
Schi (1.18), Chap. 5, Figs. 16, 17, 18, 19; Eisberg and Resnick (1.3), 6-3,
Fig. 6-8; also see reference (1.32).
81

EXERCISES
Complement KI
Exercises
1. A beam of neutrons of mass( 167 10
27
kg), of constant velocity and
energy, is incident on a linear chain of atomic nuclei, arranged in a regular fashion as
shown in the gure (these nuclei could be, for example, those of a long linear molecule).
We callthe distance between two consecutive nuclei, and, their size (). A
neutron detectoris placed far away, in a direction which makes an angle ofwith the
direction of the incident neutrons.θ
l
D
a) Describe qualitatively the phenomena observed atwhen the energyof the
incident neutrons is varied.
b) The counting rate, as a function of, presents a resonance about= 1.
Knowing that there are no other resonances for1, show that one can determine.
Calculatefor= 30and1= 13 10
20
joule.
c) At about what value ofmust we begin to take the nite size of the nuclei into
account?
2. Bound state of a particle in a delta function potential
Consider a particle whose Hamiltonian[operator dened by formula (D-10) of
Chapter] is:
=
~
2
2
d
2
d
2
()
whereis a positive constant whose dimensions are to be found.
a) Integrate the eigenvalue equation ofbetweenand+. Lettingapproach
0, show that the derivative of the eigenfunction()presents a discontinuity at= 0
and determine it in terms of,and(0).
b) Assume that the energyof the particle is negative (bound state).()can
then be written:
0 () =1e+
1e
0 () =2e+
2e
Express the constantin terms ofand. Using the results of the preceding question,
calculate the matrixdened by:
2
2
=
1
1
83

COMPLEMENT K I
Then, using the condition that()must be square-integrable, nd the possible values
of the energy. Calculate the corresponding normalized wave functions.
c) Plot these wave functions on a graph. Give an order of magnitude for their
width.
d) What is the probability
dP()that a measurement of the momentum of the
particle in one of the normalized stationary states calculated above will give a result
included betweenand+ d? For what value ofis this probability maximum? In
what domain, of dimension, does it take on non-negligible values? Give an order of
magnitude for the product .
3. Transmission of a delta function potential barrier
Consider a particle placed in the same potential as in the preceding exercise. The
particle is now propagating from left to right along theaxis, with a positive energy
.
a) Show that a stationary state of the particle can be written:
if 0 () = e+e
if 0 () =e
where,andare constants which are to be calculated in terms of the energy, of
and of(watch out for the discontinuity in
d
d
at= 0).
b) Set =
2
2~
2
(bound state energy of the particle). Calculate, in
terms of the dimensionless parameter, the reection coecientand the trans-
mission coecientof the barrier. Study their variations with respect to; what
happens when ? How can this be interpreted? Show that, if the expression of
is extended for negative values of, it diverges when , and discuss this result.
4.Return to exercise 2, using this time the Fourier transform.
a) Write the eigenvalue equation ofand the Fourier transform of this equation.
Deduce directly from this the expression for
(), the Fourier transform of(), in terms
of,,and(0). Then show that only one value of, a negative one, is possible.
Only the bound state of the particle, and not the ones in which it propagates, is found
by this method; why? Then calculate()and show that one can nd in this way all
the results of exercise 2.
b) The average kinetic energy of the particle can be written (cf.Chap.):
=
1
2
+
2
()
2
d
Show that, when
()is a suciently smooth function, we also have:
=
~
2
2
+
()
d
2
d
2
d
84

EXERCISES
These formulas enable us to obtain, in two dierent ways, the energyfor a particle
in the bound state calculated ina). What result is obtained? Note that, in this case,
()is not regular at= 0, where its derivative is discontinuous. It is then necessary
to dierentiate()in the sense of distributions, which introduces a contribution of the
point= 0to the average value we are looking for. Interpret this contribution physically:
consider a square well, centered at= 0, whose widthapproaches 0 and whose depth
0approaches innity (so that0=), and study the behavior of the wave function in
this well.
5. Well consisting of two delta functions
Consider a particle of masswhose potential energy is
() = () ( ) 0
whereis a constant length.
a) Calculate the bound states of the particle, setting=
~
22
2
. Show that the
possible energies are given by the relation
e= 1
2 whereis dened by=
2
~
2
. Give a graphic solution of this equation.
()Ground state. Show that this state is even (invariant with respect to reection
about the point=2), and that its energyis less than the energyintroduced
in problem 3. Interpret this result physically. Represent graphically the corresponding
wave function.
()Excited state. Show that, whenis greater than a value to be specied, there
exists an odd excited state, of energygreater than . Find the corresponding
wave function.
() Explain how the preceding calculations enable us to construct a model which
represents an ionized diatomic molecule (
+
2
, for example) whose nuclei are separated
by a distance. How do the energies of the two levels vary with respect to? What
happens at the limit where0and at the limit where ? If the repulsion of
the two nuclei is taken into account, what is the total energy of the system? Show that
the curve that gives the variation with respect toof the energies thus obtained enables
us to predict in certain cases the existence of bound states of
+
2
, and to determine
the value ofat equilibrium. The calculation provides a very elementary model of the
chemical bond.
b) Calculate the reection and transmission coecients of the system of two delta
function barriers. Study their variations with respect to. Do the resonances thus ob-
tained occur whenis an integral multiple of the de Broglie wavelength of the particle?
Why?
85

COMPLEMENT K I
6.Consider a square well potential of widthand depth0(in this exercise, we shall
use systematically the notation of of ComplementI). We intend to study the
properties of the bound state of a particle in this well when its widthapproaches zero.
a) Show that there indeed exists only one bound state and calculate its energy
(we nd
2
0
2
2~
2
, that is, an energy which varies with the square of the area0
of the well).
b) Show that 0and that2=
2 12. Deduce from this that, in the
bound state, the probability of nding the particle outside the well approaches 1.
c) How can the preceding considerations be applied to a particle placed, as in ex-
ercise 2, in the potential() = ()?
7.Consider a particle placed in the potential
() = 0if
() = 0if0
with()innite for negative. Let()be a wave function associated with a station-
ary state of the particle. Show that()can be extended to give an odd wave function
which corresponds to a stationary state for a square well of width2and depth0(cf.
ComplementI, ). Discuss, with respect toand0, the number of bound states
of the particle. Is there always at least one such state, as for the symmetric square well?
8.Consider, in a two-dimensional problem, the oblique reection of a particle from a
potential step dened by:
() = 0if0
() =0if0
Study the motion of the center of the wave packet. In the case of total reection, interpret
physically the dierences between the trajectory of this center and the classical trajectory
(lateral shift upon reection). Show that, when0 +, the quantum trajectory
becomes asymptotic to the classical trajectory.
86

Chapter II
The mathematical tools of
quantum mechanics
A Space of the one-particle wave function
A-1 Structure of the wave function spaceF. . . . . . . . . . . .
A-2 Discrete orthonormal bases inF:(r). . . . . . . . . . .
A-3 Introduction of bases not belonging toF. . . . . . . . . .
B State space. Dirac notation
B-1 Introduction
B-2 Ket vectors and bra vectors
B-3 Linear operators
B-4 Hermitian conjugation
C Representations in state space
C-1 Introduction
C-2 Relations characteristic of an orthonormal basis
C-3 Representation of kets and bras
C-4 Representation of operators
C-5 Change of representations
D Eigenvalue equations. Observables
D-1 Eigenvalues and eigenvectors of an operator
D-2 Observables
D-3 Sets of commuting observables
E Two important examples of representations and observables
E-1 The randprepresentations
E-2 TheRandPoperators
F Tensor product of state spaces
F-1 Introduction
F-2 Denition and properties of the tensor product
F-3 Eigenvalue equations in the product space
F-4 Applications
Quantum Mechanics, Volume I, Second Edition. C. Cohen-Tannoudji, B. Diu, and F. Laloë.
© 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

CHAPTER II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
This chapter is intended to be a general survey of the basic mathematical tools
used in quantum mechanics. We shall give a simple condensed presentation aimed at
facilitating the study of subsequent chapters for readers unfamiliar with these tools. We
make no attempt to be mathematically complete or rigorous. We feel it preferable to limit
ourselves to a practical point of view, uniting in a single chapter the various concepts
useful in quantum mechanics. In particular, we wish to stress the convenience of the
Dirac notation for carrying out the various calculations to be performed.
In this spirit, we shall try to simplify the discussion as much as possible. Neither
the general denitions nor the rigorous proofs which would be required by a mathemati-
cian are to be found here. For example, we shall sometimes speak of innite-dimensional
spaces and reason as if they had a nite number of dimensions. Moreover, many terms
(square-integrable function, basis, etc...) will be employed with a meaning which, al-
though commonly used in physics, is not exactly the one used in pure mathematics.
We begin in . We
show that these wave functions belong to an abstract vector space, which we call the
wave function spaceF. This study will be carried out in detail as it introduces some
basic concepts of the mathematical formalism of quantum mechanics: scalar products,
linear operators, bases, etc... Starting in , we shall develop a more general formalism,
characterizing the state of a system by a state vector belonging to a vector space: the
state spaceE. Dirac notation, which greatly simplies calculations in this formalism,
is introduced.
recommended to the reader who is unfamiliar with the diagonalization of an operator: this
operation will be constantly useful to us in what follows. In , we treat two important
examples of representations. In particular, we show how the wave functions studied in

introduce in
concretely by a simple example in ComplementIV.
A. Space of the one-particle wave function
The probabilistic interpretation of the wave function(r)of a particle was given in
the preceding chapter:(r)
2
d
3
represents the probability of nding, at time, the
particle in a volumed
3
= dddabout the pointr. As the total probability of nding
the particle somewhere in space is equal to 1, we must have:
d
3
(r)
2
= 1 (A-1)
where the integration extends over all space.
Thus, we are led to studying the set of square-integrable functions. These are
functions for which the integral (A-1) converges
1
.
From a physical point of view, it is clear that the set
2
is too wide in scope:
given the meaning attributed to(r)
2
, the wave functions that are actually used
possess certain properties of regularity. We can only retain the functions(r)which
are everywhere dened, continuous, and innitely dierentiable (for example, to state
that a function is really discontinuous at a given point in space has no physical meaning,
1
This set is called
2
by mathematicians and it has the structure of a Hilbert space
88

A. SPACE OF THE ONE-PARTICLE WAVE FUNCTION
since no experiment enables us to have access to real phenomena on a very small scale,
say of 10
30
m). It is also possible to conne ourselves to wave functions that have a
bounded domain (which makes it certain that the particle can be found within a nite
region of space, for example inside the laboratory). We shall not try to give a precise,
general list of these supplementary conditions: we shall callFthe set of wave functions
composed of suciently regular functions of
2
(Fis a subspace of
2
).
A-1. Structure of the wave function spaceF
A-1-a. Fis a vector space
It can easily be shown thatFsatises all the criteria of a vector space. As an
example, we demonstrate that if1(r)and2(r)F, then
2
:
(r) =11(r) +22(r)F (A-2)
where1and2are two arbitrary complex numbers.
In order to show that(r)is square-integrable, expand(r)
2
:
(r)
2
= 1
2
1(r)
2
+ 2
2
2(r)
2
+
121(r)2(r) +121(r)
2(r) (A-3)
The rst two terms of the right-hand side of (A-3) are square integrable, since1and2belong
toF. The sum of the third and fourth terms is real, and its modulus has an upper limit:
212 1(r)2(r) 12 1(r)
2
+ 2(r)
2
(this inequality is obtained by writing that[1(r) +e 2(r)]
2
is necessarily positive, whatever
the real value of). The function(r)
2
is therefore smaller that a sum of functions whose
integrals converge, so that its integral also converges.
A-1-b. The scalar product
. Denition
With each pair of elements ofF,(r)and(r), taken in this order, we associate
acomplex number, denoted by(), which, by denition, is equal to:
() =d
3
(r)(r) (A-4)
()is thescalar product of(r)by(r)[this integral always converges ifand
belong toF].
. Properties
They follow from denition (A-4):
() = () (A-5)
( 11+22) =1( 1) +2( 2) (A-6)
(11+22) =
1(1) +
2(2) (A-7)
2
The symbolsignies: belongs to.
89

CHAPTER II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
The scalar product islinearwith respect to the second function of the pair,antilinear
with respect to the rst one. If() = 0,(r)and(r)are said to beorthogonal.
() =d
3
(r)
2
(A-8)
is areal, positivenumber, which iszero if and only if(r)0.
()is called the norm of(r)[it can easily be veried that this number has all
the properties of a norm]. The scalar product chosen above thus permits the denition
of a norm inF.
Let us nally mention theSchwarz inequality(cf. ComplementII):
(12)6
(11)(22) (A-9)
This becomes an equality if and only if the two functions1and2are proportional.
A-1-c. Linear operators
. Denition
A linear operatoris, by denition, a mathematical entity which associates with
every function(r)Fanother function(r), the correspondence being linear:
(r) =(r) (A-10-a)
[11(r) +22(r)] =1 1(r) +2 2(r) (A-10-b)
Let us cite some simple examples of linear operators:
the parity operator, whose denition is:
( ) =( ) (A-11)
the operator that performs a multiplication by, which we shall call, and
which is dened by:
( ) =( ) (A-12)
nally, the operator that we shall call, which dierentiates with respect to,
and whose denition is:
( ) =
( )
(A-13)
[the two operatorsand, acting on a function(r)F, can transform it into a
function which is no longer necessarily square-integrable].
. Product of operators
Letandbe two linear operators. Their productis dened by:
()(r) =[(r)] (A-14)
90

A. SPACE OF THE ONE-PARTICLE WAVE FUNCTION
is rst allowed to act on(r), which gives(r) =(r), thenoperates on the new
function(r).
In general,= . We call thecommutatorofandthe operator written
[]and dened by
[] = (A-15)
We shall calculate, as an example, the commutator[ ]. In order to do this, we shall
take an arbitrary function(r):
[ ](r) =
(r)
=
(r)[(r)]
=
(r) (r)(r) = (r) (A-16)
Since this is true for all(r), it can be deduced that:
[ ] =1 (A-17)
A-2. Discrete orthonormal bases inF:(r)
A-2-a. Denition
Consider a countable set of functions ofF, labeled by a discrete index(= 1, 2,
...,, ...):
1(r)F 2(r)F (r)F
The set(r)isorthonormalif:
( ) =d
3
(r)(r) = (A-18)
where, the Kronecker delta function, is equal to 1 for=and to 0 for=.
It constitutes a basis
3
if every function(r)Fcan be expanded in one and
only one way in terms of the(r):
(r) = (r) (A-19)
A-2-b. Components of a wave function in the (r)basis
Multiply the two sides of (A-19) by(r)and integrate over all space. From (A-6)
and (A-18)
4
:
3
When the set (r)constitutes a basis, it is sometimes said to be a complete set of functions. It
must be noted that the word complete is used with a meaning dierent from the one it usually has in
mathematics.
4
To be totally rigorous, one should make certain that one can interchangeandd
3
. We shall
systematically ignore this kind of problem.
91

CHAPTER II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
( ) = = ( )
= = (A-20)
that is:
= () =d
3
(r)(r) (A-21)
The componentof(r)on(r)is therefore equal to the scalar product of(r)
by(r). Once the(r)basis has been chosen, it is equivalent to specify(r)or the
set of its componentswith respect to the basis functions. The set of numbersis said
torepresent(r)in the(r)basis.
Comments:
(i)Note the analogy with an orthonormal basise1e2e3of the ordinary three-
dimensional space,
3
. The fact thate1,e2ande3are orthogonal and unitary can
indeed be expressed by:
ee= (= 123) (A-22)
Any vectorVof
3
can be expanded in this basis:
V=
3
=1
e (A-23)
with
=eV (A-24)
Formulas (A-18), (A-19) and (A-21) thus generalize, as it were, the well-known formulas,
(A-22), (A-23) and (A-24). However, it must be noted that theare real numbers, while
theare complex numbers.
(ii) The same function(r)obviously has dierent components in two dierent bases.
We shall study the problem of a change in basis later.
(iii) We can also, in the(r)basis, represent a linear operatorby a set of numbers
which can be arranged in the form of a matrix. We shall take up this question again in
, after we have introduced Dirac notation.
A-2-c. Expression for the scalar product in terms of the components
Let(r)and(r)be two wave functions which can be expanded as follows:
(r) = (r)
(r) = (r) (A-25)
92

A. SPACE OF THE ONE-PARTICLE WAVE FUNCTION
Their scalar product can be calculated by using (A-6), (A-7) and (A-18):
() = = ( )
=
that is:
() = (A-26)
In particular:
() =
2
(A-27)
The scalar product of two wave functions (or the square of the norm of a wave
function) can thus be very simply expressed in terms of the components of these func-
tions in the(r)basis.
Comment:
LetVandWbe two vectors of
3
, with componentsand. The analytic expression
of their scalar product is well-known:
VW=
3
=1
(A-28)
Formula (A-26) can therefore be considered to be a generalization of (A-28).
A-2-d. Closure relation
Relation (A-18), called the orthonormalization relation, expresses the fact that the
functions of the set(r)are normalized to 1 and orthogonal with respect to each
other. We are now going to establish another relation, called the closure relation, which
expresses the fact that this set constitutes a basis.
If(r)is a basis ofF, there exists an expansion such as (A-19) for every
function(r)F. Substitute into (A-19) expression (A-21) for the various components
[the name of the integration variable must be changed, sinceralready appears in
(A-19)]:
(r) = (r) =()(r)
= d
3
(r)(r)(r) (A-29)
93

CHAPTER II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
Interchangingandd
3
, we obtain:
(r) =d
3
(r) (r)(r) (A-30)
(r)(r)is therefore a function(rr)ofrand ofrsuch that, for every function
(r), we have:
(r) =d
3
(r)(rr) (A-31)
Equation (A-31) is characteristic of the function(rr)(cf. Appendix). From this
it can be deduced that:
(r)(r) =(rr) (A-32)
Reciprocally, if an orthonormal set(r)satises the closure relation (A-32), it
constitutes a basis. Any function(r)can indeed be written in the form:
(r) =d
3
(r)(rr) (A-33)
Substituting (A-32) for(rr)into this expression, we obtain formula (A-30). To return
to (A-29), all we must do is again interchange summation and integration. This equation
then expresses the fact that(r)can always be expanded in terms of the(r)and gives
the coecients of this expansion.
Comment:
We shall re-examine the closure relation using the Dirac notation in , and we shall
see that it can be given a simple geometric interpretation.
A-3. Introduction of bases not belonging toF
The (r)bases studied above are composed of square-integrable functions. It
can also be convenient to introduce bases of functions not belonging to eitherFor
2
, but in terms of which any wave function(r)can nevertheless be expanded. We are
going to give examples of such bases and we shall show how it is possible to extend to
them the important formulas established in the preceding section.
A-3-a. Plane waves
For simplicity, we treat the one-dimensional case. We shall therefore study square-
integrable functions()which depend only on thevariable. In Chapter
94

A. SPACE OF THE ONE-PARTICLE WAVE FUNCTION
advantage of using the Fourier transform
()of():
() =
1
2~
+
d
() e
~
(A-34-a)() =
1
2~
+
d() e
~
(A-34-b)
Consider the function(), dened by:
() =
1
2~
e
~
(A-35)
()is a plane wave, with the wave vector~. The integral over the wholeaxis of
()
2
=
1
2~
diverges. Therefore()F. We shall designate by()the set
of all plane waves, that is, of all functions()corresponding to the various values of
. The number, which varies continuously betweenand+, will be considered
as acontinuous indexwhich permits us to label the various functions of the set()
[recall that the indexused for the set(r)considered above was discrete].
Formulas (A-34) can be rewritten using (A-35):
() =
+
d
()() (A-36)() = () =
+
d ()() (A-37)
These two formulas can be compared to (A-19) and (A-21). Relation (A-36) expresses
the idea that every function()Fcan be expanded in one and only one way in
terms of the(), that is, the plane waves. Since the indexvaries continuously and not
discretely, the summationappearing in (A-19) must be replaced by an integration
over. Relation (A-37), like (A-21), gives the component
()of()on()in the
form of a scalar product
5
(). The set of these components, which correspond to the
various possible values of, constitutes a function of,
(), the Fourier transform of
().
Thus,
()is the analogue of. These two complex numbers, which depend either
onor on, represent thecomponents of the same function()in two dierent bases:
()and ().
This point also appears clearly if we calculate the square of the norm of().
According to Parseval's relation [Appendix, formula (45)], we have:
() =
+
d
()
2
(A-38)
5
We have only dened the scalar product for two square-integrable functions, but this denition can
easily be extended to cases like this one, provided that the corresponding integral converges.
95

CHAPTER II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
a formula which resembles (A-27), if we replaceby
()and byd.
Let us show that the()satisfy a closure relation. Using the formula [cf. Ap-
pendix, equation (34)]:
1
2
+
de=() (A-39)
we nd:
+
d()() =
1
2
d
~
e
~
( )
=( )
(A-40)
This formula is the analogue of (A-32) with, again, the substitution ofdfor.
Finally, let us calculate the scalar product( )in order to see if there exists
an equivalent of the orthonormalization relation. Again using (A-39), we obtain:
( ) =
+
d ()()
that is:
( ) =
1
2
d
~
e
~
( )
=( )
(A-41)
Compare (A-41) and (A-18). Instead of having two discrete indicesandand a Kro-
necker delta, we now have two continuous indicesandand a delta function of the
dierence between the indices,( ). Note that if we set=, the scalar product
( )diverges; again we see that()F. Although this constitutes a misuse of
the term, we shall call (A-41) an orthonormalization relation. It is also sometimes said
that the()are orthonormalized in the Dirac sense.
The generalization to three dimensions presents no diculties. We consider the
plane waves:
p(r) =
1
2~
32
e
pr~
(A-42)
The functions of the(r)basis now depend on the three continuous indices,,
, condensed into the notationp. It is then easy to show that:
(r) =d
3
(p)p(r) (A-43)(p) = (p) =d
3
p(r)(r) (A-44)
() =d
3
(p)
(p) (A-45)
d
3
p(r)
p(r) =(rr) (A-46)
96

A. SPACE OF THE ONE-PARTICLE WAVE FUNCTION
(pp) =(pp) (A-47)
They represent the generalizations of (A-36), (A-37), (A-38), (A-40) and (A-41).
Thus the(r)can be considered to constitute a continuous basis. All the formu-
las established above for the discrete basis(r)can be extended to this continuous
basis, using the correspondence rules summarized in table (II-1).
p
d
3
(pp)
Table (II-1)
A-3-b. Delta functions
In the same way, let us introduce a set of functions ofr,r0(r), labeled by the
continuous indexr0(condensed notation for0,0,0) and dened by:
r0
(r) =(rr0) (A-48)
r0
(r)represents the set of delta functions centered at the various pointsr0of space;
r0
(r)is obviously not square-integrable:r0
(r)F.
Then consider the following relations, which are valid for every function(r)F:
(r) =d
3
0(r0)(rr0) (A-49)
(r0) =d
3
(r0r)(r) (A-50)
They can be rewritten, using (A-48), in the form:
(r) =d
3
0(r0)r0(r) (A-51)
(r0) = (r0
) =d
3
r0
(r)(r) (A-52)
(A-51) expresses the fact that every function(r)Fcan be expanded in one and only
one way in terms of ther0(r). (A-52) shows that the component of(r)on the function
r0(r)(we are dealing here with real basis functions) is precisely the value(r0)of(r)
at the pointr0. (A-51) and (A-52) are analogous to (A-19) and (A-21): we simply replace
the discrete indexby the continuous indexr0, andbyd
3
0.
(r0)is therefore the equivalent of: these two complex numbers, which depend
either onr0or on, represent the components of the same function(r)in two dierent
bases:r0
(r)and (r).
97

CHAPTER II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
Formula (A-26) becomes here:
() =d
3
0(r0)(r0) (A-53)
We see that the application of (A-26) to the case of the continuous basisr0
(r)results
in the denition (A-4) of the scalar product.
Finally, note that ther0
(r)satisfy orthonormalization and closure relations of
the same type as those for thep(r). Thus we have [formula (28) of Appendix]:
d
3
0r0
(r)
r0
(r) =d
3
0(rr0)(rr0) =(rr) (A-54)
and:
(r0r
0
) =d
3
(rr0)(rr
0) =(r0r
0) (A-55)
All the formulas established for the discrete basis(r)can be generalized for
the continuous basisr0
(r), using the correspondence rules summarized in Table (II-2).
r0
d
3
0
(r0r
0)
Table (II-2)
Important comment:
The usefulness of the continuous bases that we have just introduced is revealed
more clearly in what follows. However, we must not lose sight of the following
point:a physical state must always correspond to a square-integrable wave function.
In no case canp(r)orr0
(r)represent the state of a particle. These functions are
nothing more than intermediaries, very useful in calculations involving operations
on the wave functions(r)which are used to describe a physical state.
An analogous situation is encountered in classical optics, where the plane
monochromatic wave is a mathematically very useful, but physically unrealizable,
idealization. Even the most selective lters always permit the passage of a fre-
quency band, which may be very small but is never exactly zero.
The same holds true for the functionsr0(r). We can imagine a square-
integrable wave function, localized aboutr0, for example:
()
r0
(r) =
()
(rr0) =
()
( 0)
()
( 0)
()
( 0)
where the
()
are functions which have a peak of widthand amplitude
1
,
centered at0,0or0, such that
+
()
( 0) d= 1(see
98

A. SPACE OF THE ONE-PARTICLE WAVE FUNCTION
for examples of such functions). When 0,
()
r0(r) r0
(r), which is no
longer square-integrable. However, it is impossible to have a physical state that
corresponds to this limit: as localized as the physical state of a particle may be,
is never exactly zero.
A-3-c. Generalization: continuous orthonormal bases
. Denition
Generalizing the results obtained in the two preceding paragraphs, we shall call a
continuousorthonormal basis, a set of functions ofr, (r), labeled by a continuous
index, which satisfy the two following relations, calledorthonormalizationandclosure
relations:
( ) =d
3
(r)(r) =( ) (A-56)
d (r)(r) =(rr) (A-57)
Comments:
(i)If=,( )diverges. Therefore,(r)F.
(ii)can represent several indices, as is the case forr0andpin the above examples.
(iii) It is possible to imagine a basis that includes both functions(r), labeled by a
discrete index, and functions(r), labeled by a continuous index. In this case, the set
of(r)does not form a basis; the set of(r)must be added to it.
Let us cite an example of this situation. Consider the case of the square well studied in
I). As we shall see later, the set of stationary
states of a particle in a time-independent potential constitutes a basis. For0, we
have discrete energy levels, to which correspond square-integrable wave functions labeled
by a discrete index. But these are not the only possible stationary states. Equation (D-
17) of Chapter 0, by solutions which are bounded but which
extend over all space and are thus not square-integrable.
In the case of a mixed (discrete and continuous) basis,(r)(r), the or-
thonormalization relations are:
( ) =
( ) =( ) (A-58)
( ) = 0
And the closure relation becomes:
(r)(r) +d (r)(r) =(rr) (A-59)
99

CHAPTER II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
. Components of a wave function(r)
We can always write:
(r) =d
3
(r)(rr) (A-60)
Using the expression for(rr)given by (A-57), and assuming that we can reverse the
order ofd
3
andd, we obtain:
(r) =d d
3
(r)(r)(r) (A-61)
that is:
(r) =d()(r) (A-62)
with:
() = ( ) =d
3
(r)(r) (A-63)
(A-62) expresses the fact that every wave function(r)has a unique expansion in terms
of the(r). The component()of(r)on(r)is equal, according to (A-63), to the
scalar product( ).
. Expression for the scalar product and the norm in terms of the components
Let(r)and(r)be two square-integrable functions whose components in terms
of the(r)are known:
(r) =d()(r) (A-64)
(r) =d()(r) (A-65)
Calculate their scalar product:
() =d
3
(r)(r)
=d d ()()d
3
(r)(r) (A-66)
The last integral is given by (A-56):
() =d d ()()( )
100

A. SPACE OF THE ONE-PARTICLE WAVE FUNCTION
that is:
() =d ()() (A-67)
In particular:
() =d()
2
(A-68)
All the formulas of
of table (II-3).d
( )
Table (II-3)
The most important formulas established in this section are assembled in table
(II-4). Actually, it is not necessary to remember them in this form: we shall see that the
introduction of Dirac notation enables us to rederive them very simply.
101

CHAPTER II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
Table (II-4)
Discrete basis(r) Continuous basis(r)Ortho-
normalization
relation
( ) = ( ) =( )
Closure
relation
(r)(r) =(rr) d (r)(r) =(rr)
Expansion
of a wave
function(r)
(r) = (r) (r) =d()(r)
Expression
for the
components
of(r)
= () =d
3
(r)(r)()=( )=d
3
(r)(r)
Scalar product () = () =d ()()Square of the
norm
() =
2
() =d()
2
B. State space. Dirac notation
B-1. Introduction
In Chapter, we stated the following postulate: the quantum state of a particle is
dened, at a given instant, by a wave function(r). The probabilistic interpretation of
this wave function requires that it be square-integrable. This requirement led us to study
theF-space (). We then found, in particular, that the same function(r)can be
represented by several distinct sets of components, each one corresponding to the choice
of a basis [table (II-5)]. This result can be interpreted in the following manner:, or
(p), or(), characterizes the state of a particle just as well as the wave function(r)
[if the basis being used has been specied previously]. Furthermore,(r)itself appears,
in table (II-5), on the same footing as,
(p)and(): the value(r0)which the
wave function takes on at a pointr0of space can be considered as its component with
respect to a specic functionr0
(r)of a particular basis (thefunction basis).
102

B. STATE SPACE. DIRAC NOTATION
Basis Components of (r)
(r) = 12
p(r)
(p)
r0
(r) (r0)
(r) ()
Table (II-5)
We thus nd ourselves in a situation which is analogous to the one encountered in
ordinary space,
3
: the position of a point in space can be described by a set of three
numbers, which are its coordinates with respect to a system of axes dened in advance.
If one changes axes, another set of coordinates corresponds to the same point. But the
geometrical vector concept and vector calculation enable us to avoid referring to a system
of axes; this considerably simplies both formulas and reasoning.
We are going to use a similar approach here: each quantum state of a particle will
be characterized by astate vector, belonging to an abstract space,Er, called thestate
spaceof a particle. The fact that the spaceFis a subspace of
2
means thatEris a
subspace of a Hilbert space. We are going to dene the notation and the rules of vector
calculation inEr.
Actually, the introduction of state vectors and the state space does more than
merely simplify the formalism. It also permits a generalization of the formalism. Indeed,
there exist physical systems whose quantum description cannot be given by a wave func-
tion: we shall see in Chapters
of freedom are taken into account, even for a single particle. Consequently, the rst
postulate that we shall set forth in Chapter the quantum state
of any physical system is characterized by a state vector, belonging to a spaceEwhich is
the state space of the system.
Therefore, in the rest of this chapter, we are going to develop a vector calculus inE.
The concepts which we are going to introduce and the results which we shall obtain are
valid for whatever physical system we might consider. Nevertheless, to illustrate these
concepts and results, we shall apply them to the simple case of a (spinless) particle, since
this is the case we have previously considered.
We shall begin, in this paragraph, by dening theDirac notation, which will prove
to be very useful in the formal manipulations which we shall have to perform.
B-2. Ket vectors and bra vectors
B-2-a. Elements of E: kets
. Notation
Any element, or vector, of spaceEis called aket vector, or, more simply, aket. It
is represented by the symbol, inside which is placed a distinctive sign which enables
us to distinguish the corresponding ket from all others, for example:.
In particular, since the concept of a wave function is now familiar to us, we shall
dene the spaceErof the states of a particle by associating with every square-integrable
103

CHAPTER II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
function(r)a ket vectorofEr:
(r)F Er (B-1)
Afterwards, we shall transpose intoErthe dierent operations that we introduced for
F. AlthoughFandErare isomorphic, we shall carefully distinguish between them
in order to avoid confusion and to reserve the possibilities of generalization mentioned
above in . We stress the fact that anr-dependence no longer appears in; only
the letterappears, to remind us with which function it is associated.(r)will be
interpreted () as the set of the components of the ketin a particular basis,r
playing the role of an index [cf.
which we are adopting here consists in initially characterizing a vector by its components
in a privileged coordinate system, which will later be treated on the same footing as all
other coordinate systems.
We shall designate byEthe state space of a (spinless) particle in only one dimen-
sion, that is, the abstract space constructed as in (B-1), but using wave functions that
depend only on thevariable.
. Scalar product
With each pair of ketsand , taken in this order, we associate a complex
number, which is their scalar product,( ), and which satises the various proper-
ties described by equations (A-5), (A-6) and (A-7). We shall later rewrite these formulas
in Dirac notation after we have introduced the concept of a bra.
InEr, the scalar product of two kets will coincide with the scalar product dened
above for the associated wave functions.
B-2-b. Elements of the dual space EofE: bras
. Denition of the dual spaceE
Recall, rst of all, the denition of alinear functionaldened on the ketsof
E. A linear functionalis a linear operation which associates a complex number with
every ket:
E number()
(11+22) =1(1) +2(2) (B-2)
Linear functional and linear operator must not be confused. In both cases, one is dealing
with linear operations, but the former associates each ket with a complex number, while the
latter associates another ket.
It can be shown that the set of linear functionals dened on the ketsE
constitutes a vector space, which is called thedual spaceofEand which will be symbolized
byE.
. Bra notation for the vectors ofE
Any element, or vector, of the spaceEis called abra vector, or, more simply, a
bra. It is symbolized by. For example, the bradesignates the linear functional
104

B. STATE SPACE. DIRAC NOTATION
and we shall henceforth use the notation to denote thenumberobtained by
causing the linear functionalEto act on the ket E:
() = (B-3)
The origin of this terminology is the word bracket, used to denote the symbol.
Hence the name bra for the left-hand side, and the name ket for the right-hand side
of this symbol.
B-2-c. Correspondence between kets and bras
. To every ket corresponds a bra
The existence of a scalar product inEwill now enable us to show that we can
associate, with every ketE, an element ofE, that is, a bra, which will be denoted
by .
The ket does indeed enable us to dene a linear functional: the one that
associates (in a linear way), with each ketE, a complex number equal to the
scalar product( )of by . Let be this linear functional; it is thus
dened by the relation:
= ( ) (B-4)
. This correspondance is antilinear
In the spaceE, the scalar product is antilinear with respect to the rst vector. In
the notation of (B-4), this is expressed by:
(11+22 ) =
1(1 ) +
2(2 )
=
1 1 +
2 2
= (
1 1+
2 2) (B-5)
It appears from (B-5) that the bra associated with the ket11+22is the
bra
1 1+
2 2:
11+22=
1 1+
2 2 (B-6)
The ket=bra correspondence is therefore antilinear.
Comment:
Ifis a complex number, anda ket, is a ket (Eis a vector space). We
are sometimes led to write it as:
= (B-7)
One must then be careful to remember that represents the bra associated
with the ket. Since the correspondence between a bra and a ket is antilinear,
we have:
= (B-8)
105

CHAPTER II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS0
1
ε
ε
x
0
x
0
x
(ε)
ξ
Figure 1:
()
0()is a function having a peak
at=0(of widthand amplitude1),
whose integral betweenand+is equal
to 1.
. Dirac notation for the scalar product
We now have at our disposal two distinct notations for designating the scalar
product ofby :( )or , being the bra associated with the ket
. Henceforth we shall use only the (Dirac) notation:. Table (II-6) summarizes,
in Dirac notation, the properties of the scalar product, already given in .
= (B-9)
11+22=1 1+2 2 (B-10)
11+22 =
1 1 +
2 2 (B-11)
real, positive; zero if and only if= 0 (B-12)
Table (II-6)
. Is there a ket to correspond to every bra?
Although to every ket there corresponds a bra, we shall see, in two examples chosen in
F, that it is possible to nd bras that have no corresponding kets. We shall later show why
this diculty does not hinder us in quantum mechanics.
(i) Counter-examples chosen inF
For simplicity, we shall reason in one dimension.
Let
()
0()be a suciently regular real function, such that
+
d
()
0
() = 1, and
having the form of a peak of widthand amplitude1, centered at=0[see Fig.;
()
0()
is, for example, one of the functions considered in ]. If= 0,
()
0()F
(the square of its norm is of the order of1). Denote by
()
0the corresponding ket:
()
0
()
()
0
(B-13)
106

B. STATE SPACE. DIRAC NOTATION
If= 0,
()
0 E. Let
()
0be the bra associated with this ket; for everyE, we
have:
()
0
= (
()
0
) =
+
d
()
0
()() (B-14)
Now letapproach zero. On the one hand:
Lim
0
()
0
() =
0()F (B-15)
[the square of the norm of
()
0(), which is of the order of1, diverges when 0]; therefore:
Lim
0
()
0
E (B-16)
On the other hand, when 0, integral (B-14) approaches a perfectly well-dened limit,
(0)[since, for suciently small,()can be replaced in (B-14) by(0)and removed from
the integral]. Consequently,
()
0approaches a bra which we shall denote by
0:
0is
the linear functional which associates, with every ketofE, the value(0)taken on by
the associated wave function at the point0:
Lim
0
()
0
=
0E
If E
0 =(0) (B-17)
Thus we see thatthe bra
0exists, but no ket corresponds to it.
In the same way, let us consider a plane wave which is truncated outside an interval of
width:
()
0
() =
1
2~
e
0~
if
2
66+
2
(B-18)
with the function
()
0()going rapidly to zero outside this interval (while remaining continuous
and dierentiable). We shall denote by
()
0the ket associated with
()
0():
()
0
()F
()
0
E (B-19)
The square of the norm of
()
0, which is practically equal to2~, diverges if .
Therefore:
Lim
()
0
E (B-20)
Now let us consider the bra
()
0associated with
()
0. For every E, we have:
()
0
= (
()
0
)
1
2~
+2
2
de
0~
() (B-21)
When ,
()
0 has a limit: the value
(0)of the Fourier transform()of()
for=0. Therefore, when ,
()
0tends towards a perfectly well-dened bra
0:
Lim
()
0
=
0E
If E
0 =
(0) (B-22)
107

CHAPTER II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
Here again,no ket corresponds to the bra
0.
(ii) Physical resolution of the preceding diculties
This dissymmetry of the correspondence between kets and bras is related, as the preceding
examples show, to the existence of continuous bases forF. Since the functions constituting
these bases do not belong toF, we cannot associate a ket ofEwith them. However, their
scalar product with an arbitrary function ofFis dened, and this permits us to associate
with them a linear functional inE, that is, a bra belonging toE. The reason for using such
continuous bases lies in their usefulness in certain practical calculations. The same reason
(which will become more apparent in what follows) leads us here to reestablish the symmetry
between kets and bras by introducing generalized kets, dened using functions that are not
square-integrable, but whose scalar product with every function ofFexists. In what follows,
we shall work with kets such as
0or
0, associated with
0()or
0(). It must not be
forgotten that these generalized kets cannot, strictly speaking, represent physical states. They
are merely intermediaries, useful in calculations involving certain operations to be performed on
the true kets of the spaceE, which actually characterize realizable quantum states.
This method poses a certain number of mathematical problems, which can be avoided
by adopting the following physical point of view:
0(or
0) actually denotes
()
0(or
()
0) whereis very small (oris very large) compared to all the other lengths in the problem
we are considering. In all the intermediary calculations where
()
0(or
()
0) appears, the
limit= 0(or ) is never attained, so that one is always working inE. The physical
result obtained at the end of the calculation depends very little on the value of, as long as
is suciently small with respect to all the other lengths: it is then possible to neglect, that is,
to set= 0, in the nal result (the procedure to be used foris analogous).
The objection could be raised that, unlike
0()and
0(),
()
0()and
()
0()
are not orthonormal bases, insofar as they do not rigorously satisfy the closure relation. In fact,
they fulll it approximately. For example, the expressiond0
()
0
()
()
0
()is a function of
( )which can serve as an excellent approximation for( ). Its graphical representation
is practically a triangle of base2and height
1
, centered at = 0(Appendix, 1-c-).
Ifis negligible compared to all the other lengths in the problem, the dierence between this
expression and( )is physically inappreciable.
In general, the dual spaceEand the state spaceEare not isomorphic, except,
of course, ifEis nite-dimensional
6
: although to each ketofEthere corresponds a
bra inE, the converse is not true. Nevertheless, we shall agree to use, in addition
to vectors belonging toE(whose norm is nite),generalized ketswith innite norms but
whose scalar product with every ket ofEis nite. Thus, to each braofE, there will
correspond a ket. But generalized kets do not represent physical states of the system.
B-3. Linear operators
B-3-a. Denitions
They are the same as those of .
A linear operatorassociates with every ketEanother ket E, the
6
It is true that the Hilbert space
2
and its dual space are isomorphic; however, we have taken for
the wave function spaceFa subspace of
2
, which explains whyFis larger thanF.
108

B. STATE SPACE. DIRAC NOTATION
correspondence being linear:
= (B-23)
(1 1+2 2) =1 1+2 2 (B-24)
The product of two linear operatorsand, written, is dened in the fol-
lowing way:
() =( ) (B-25)
rst acts onto give the ket ;then acts on the ket . In general,
= . The commutator[]ofandis, by denition:
[] = (B-26)
Let and be two kets. We call thematrix elementofbetween and
, the scalar product:
( ) (B-27)
Consequently, this isa numberwhich depends linearly onand antilinearly on.
B-3-b. Examples of linear operators: projectors
. Important comment about Dirac notation
We have begun to sense, in the preceding, the simplicity and convenience of the
Dirac formalism. For example,denotes a linear functional (a bra), and12, the
scalar product of two kets1and 2. The number associated by the linear functional
with an arbitrary ketis then written simply by juxtaposing the symbols
and : . This is the scalar product ofby the ketcorresponding to
(which is why it is useful to have a one-to-one correspondence between kets and bras).
Now assume that we writeand in the opposite order:
(B-28)
We shall see that if we abide by the rule of juxtaposition of symbols, this expression
represents an operator. Choose an arbitrary ketand consider:
(B-29)
We already know that is a complex number; consequently,(B-29) is a ket, obtained
by multiplyingby the scalar . , applied to an arbitrary ket, gives
another ket: it is an operator.
Thus we see thatthe order of the symbols is of critical importance.Only complex
numbers can be moved about with impunity, because of the linearity of the spaceEand
of the operators which we shall use. Indeed, ifis a number:
=
=
= (whereis a linear operator)
= =
109

CHAPTER II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
(B-30)
But, for kets, bras and operators, the order must always be carefully respected in writing
the formulas: this is the price that must be paid for the simplicity of the Dirac formalism.
. The projector onto a ket
Let be a ket which is normalized to one:
= 1 (B-31)
Consider the operator, dened by:
= (B-32)
and apply it to an arbitrary ket:
= (B-33)
, acting on an arbitrary ket, gives a ket proportional to. The coecient of
proportionality is the scalar product ofby .
The geometrical signicance ofis therefore clear: it is the orthogonal pro-
jection operator onto the ket.
This interpretation is conrmed by the fact that
2
= (projecting twice in
succession onto a given vector is equivalent to projecting a single time). To see this, we
write:
2
= = (B-34)
In this expression, is a number, which is equal to 1 [formula (B-31)]. Therefore:
2
= = (B-35)
. Projector onto a subspace
Let 1,2, ...,, benormalized vectors which are orthogonal to each
other:
= ; = 12 (B-36)
We denote byEthe subspace ofEspanned by thesevectors.
Letbe the linear operator dened by:
=
=1
(B-37)
Calculating
2
:
2
=
=1=1
(B-38)
110

B. STATE SPACE. DIRAC NOTATION
we get, using (B-36):
2
=
=1=1
=
=1
= (B-39)
is therefore a projector. It is easy to see thatprojects onto the subspaceE,
since for any E:
=
=1
(B-40)
acting ongives the linear superposition of the projections ofonto the various
, that is, the projection ofonto the subspaceE.
B-4. Hermitian conjugation
B-4-a. Action of a linear operator on a bra
Until now, we have only dened the action of a linear operatoron kets. We are
now going to see that it is also possible to dene the action ofon bras.
Let be a well-dened bra, and consider the set of all kets. With each of
these kets can be associated the complex number( ), already dened above as
the matrix element ofbetween and . Sinceis linear and the scalar product
depends linearly on the ket, the number( )depends linearly on. Thus, for
xed and, we can associate with every keta number which depends linearly
on . The specication ofandtherefore denes a new linear functional on the
kets ofE, that is, a new bra belonging toE. We shall denote this new bra by.
The relation which denes can thus be written:
( ) = ( ) (B-41)
The operatorassociates with every braa new bra . Let us show that
the correspondence is linear. In order to do this, consider a linear combination of bras
1and 2:
=1 1+2 2 (B-42)
(which means that =1 1 +2 2). From (B-41), we have:
( ) = ( )
=1 1( ) +2 2( )
=1(1)+2(2)) (B-43)
Since is arbitrary, it follows that:
= (1 1+2 2)
=1 1+2 2 (B-44)
111

CHAPTER II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICSA
ψ ψ A ψ
ψ ψ A
†
ψ
A
†
Figure 2: Denition of the adjoint operatorof an operatorusing the correspondence
between kets and bras.
Equation (B-41) therefore denes a linear operation on bras. The brais the
bra which results from the action of the linear operatoron the bra.
Commments:
(i)From denition (B-41) of , we see that the place of the parenthesis in
the symbol dening the matrix element ofbetween and is of no im-
portance. Therefore, we shall henceforth designate this matrix element by the
notation :
= ( ) = ( ) (B-45)
(ii) The relative order ofandis very important in the notation(cf.
3-b-a above). One must write and not : acting on a ket
gives a number ; is therefore indeed a bra. On the other hand,
, acting on a ket, would give , that is, an operator (the operator
multiplied by the number ). We have not dened any mathematical
object of this sort:therefore has no meaning.
B-4-b. The adjoint operator of a linear operator
We are now going to see that the correspondence between kets and bras, studied
in , enables us to associate with every linear operatoranother linear operator
, called the adjoint operator (or Hermitian conjugate) of.
Let then be an arbitrary ket ofE. The operatorassociates with it another
ket = ofE(Fig.).
To the ketcorresponds a bra; in the same way, tocorresponds.
This correspondence between kets and bras thus permits us to dene the action of the
operatoron the bras: the operatorassociates with the bracorresponding to the
ket, the bracorresponding to the ket= . We write: = .
Let us show that the relation= is linear. We know that, to the bra
1 1+2 2, corresponds the ket
11+
22(the correspondence between a
112

B. STATE SPACE. DIRAC NOTATION
bra and a ket is antilinear). The operatortransforms
11+
22into
1 1+
2 2=
11+
22. Finally, to this ket corresponds the bra:1 1+
2 2=1 1 +2 2. From this we conclude that:
(1 1+2 2)=1 1 +2 2 (B-46)
is therefore a linear operator, dened by the formula:
= = (B-47)
From (B-47), it is easy to deduce another important relationship satised by the
operator. Using the properties of the scalar product, one can always write:
= (B-48)
where is an arbitrary ket ofE. Using expressions (B-47) for and , we
obtain:
= (B-49)
a relation which is valid for alland .
Comment about notation:
We have already mentioned a notation which can lead to confusion:
and , whereis a scalar [formulas (B-7) and (B-8)]. The same problem
arises with the expressionsand , whereis a linear operator.
is another way of designating the ket:
= (B-50)
is the bra associated with the ket. Using (B-50) and (B-57), we see
that:
= (B-51)
When a linear operatoris taken outside the bra symbol, it must be replaced by
its adjoint(and placed to the right of the bra).
B-4-c. Correspondence between an operator and its adjoint
By using (B-47) or (B-49), it is easy to show that:
() = (B-52)
() = (whereis a number) (B-53)
(+)=+ (B-54)
113

CHAPTER II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
Now let us calculate(). To do this, consider the ket= . Write it
in the form= , setting= . Then:
= ()= =
since= . From this, we deduce that:
()= (B-55)
Note thatthe order changes when one takes the adjoint of a product of operators.
Comment:
Since()=, we can write, using (B-51):
= ()=
Thus the left-hand side of (B-41) can be rewritten in the form . In the
same way, the right-hand side of this same equation can be put, with the notation of
(B-50), into the form . From this results the following relation, sometimes
used to dene the adjoint operatorof:
= (B-56)
B-4-d. Hermitian conjugation in Dirac notation
In the preceding section, we introduced the concept of an adjoint operator by using
the correspondence between kets and bras. A ketand its corresponding braare
said to be Hermitian conjugates of each other. The operation of Hermitian conjugation
is represented by the wavy arrows in Figure; we see that it associateswith. This
is the reason whyis also called the Hermitian conjugate operator of.
The operation of Hermitian conjugation changes the order of the objects to which
it is applied. Thus we see in Figure becomes . The ket is
changed into, the operatorinto, and the order is reversed. In the same way,
we saw in (B-55) that the Hermitian conjugate of a product of two operators is equal
to the product of the Hermitian conjugates taken in the opposite order. Finally, let us
show that:
( )= (B-57)
(is replaced by, and by, and the order is changed). Applying relation
(B-49) to the operator , we nd:
( ) = [( )] (B-58)
Now, if we use property (B-9) of the scalar product:
[( )]= =
= ( ) (B-59)
114

B. STATE SPACE. DIRAC NOTATION
By comparing (B-58) and (B-59), we can derive (B-57).
The result of the operation of Hermitian conjugation on a constant remains to be
found. We see from (B-6) and (B-53) that this operation simply transformsinto
(complex conjugation). This is in agreement with the fact that= .
To summarize, the Hermitian conjugate of a ket is a bra, and vice versa; that of
an operator is its adjoint; that of a number, its complex conjugate. In Dirac notation,
the operation of Hermitian conjugation is very simple to perform; it suces to apply the
following rule:
RULE
To obtain the Hermitian conjugate (or the adjoint) of any expression
composed of constants, kets, bras and operators, one must:
Replace
Reverse the orderof the factors (the position of the constants, nevertheless,
is of no importance).
the constants by their complex conjugates
the kets by the bras associated with them
the bras by the kets associated with them
the operators by their adjoints
EXAMPLES
is an operator (and are numbers). The adjoint
of this operator is obtained by using the preceding rule: , which
can also be written , changing the position of the numbersand
.
In the same way, is a ket (and are constants). The conjugate
bra is , which can also be written .
B-4-e. Hermitian operators
An operatoris said to be Hermitian if it is equal to its adjoint, that is, if:
= (B-60)
Combining (B-60) and (B-49), we see that a Hermitian operator satises the rela-
tion:
= (B-61)
which is valid for alland .
Finally, for a Hermitian operator, (B-56) becomes
= (B-62)
We shall treat Hermitian operators in more detail later, when we consider the
problem of eigenvalues and eigenvectors. Moreover, we shall see in Chapter
Hermitian operators play a fundamental role in quantum mechanics.
115

CHAPTER II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
If formula (B-57) is applied to the case where= = , we see that the
projector= is Hermitian:
= = (B-63)
Comment:
The product of two Hermitian operatorsandis Hermitian only if[] = 0. Indeed,
if= and=, it can be shown using (B-55) that()= = , which
is equal toonly if[] = 0.
C. Representations in state space
C-1. Introduction
C-1-a. Denition of a representation
Choosing a representation means choosing an orthonormal basis, either discrete
or continuous, in the state spaceE. Vectors and operators are then represented in this
basis bynumbers: components for the vectors, matrix elements for the operators. The
vectorial calculus introduced in
The choice of a representation is, in theory, arbitrary. Actually, it obviously depends on
the particular problem being studied: in each case, one chooses the representation that
leads to the simplest calculations.
C-1-b. Aim of section
Using the Dirac notation, and for any arbitraryEspace, we are going to treat
again all the concepts introduced in
F.
We shall write the two characteristic relations of a basis in Dirac notation: the
orthonormalization and closure relations. Then we shall show how, using these two
relations, it is possible to solve all specic problems involving a representation and the
transformation from one representation to another.
C-2. Relations characteristic of an orthonormal basis
C-2-a. Orthonormalization relation
A set of kets, discreteor continuous , is said to beorthonormalif
the kets of this set satisfy the orthonormalization relation:
= (C-1)
or
=( ) (C-2)
116

C. REPRESENTATIONS IN STATE SPACE
It can be seen that, for a continuous set,does not exist: thehave
an innite norm and therefore do not belong toE. Nevertheless, the vectors ofEcan be
expanded on the . It is useful, consequently, to accept theas generalized kets
(see the discussions in ).
C-2-b. Closure relation
A discrete set, , or a continuous one, , constitutes abasisif every
ket belonging toEhas a unique expansion on theor the :
= (C-3)
=d() (C-4)
Let us assume, moreover, that the basis is orthonormal. Then perform the scalar
multiplication on both sides of (C-3) with , and on both sides of (C-4) with .
We obtain, using (C-1) or (C-2), expressions for the componentsor():
= (C-5)
=() (C-6)
Then replace in (C-3)by , and in (C-4)()by :
= =
= = (C-7)
=d() =d
=d = d (C-8)
[since, in (C-7), we can place the number after the ket; in the same way, in
(C-8), we can place the number after the ket].
Thus, we see two operators appear, andd . They act
on every ket belonging toEto give the same ket. Since is arbitrary, it
follows that:
= = (C-9)
=d = (C-10)
wheredenotes the identity operator inE. Relation (C-9), or (C-10), is called the
closure relation. Conversely, let us show that relations (C-9) and (C-10) express the fact
117

CHAPTER II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
that the sets and constitute bases. For everybelonging toE, we
can write:
= = =
= (C-11)
with:
= (C-12)
In the same way:
= = =d
=d() (C-13)
with:
() = (C-14)
Thus, every kethas a unique expansion on theor on the . Each of
these two sets therefore forms a basis, a discrete one or a continuous one. We also see
that relation (C-9), or (C-10), spares us the need of memorizing expressions (C-12) and
(C-14) for the componentsand().
Comments:
(i) We shall see later () that, in the case of theF-space, relations ( ) and (A-57)
can easily be deduced from (C-9) and (C-10).
(ii) Geometrical interpretation of the closure relation.
From the discussion of , we see that is a projector: the
projector onto the subspaceEspanned by 1 2 If the form a
basis, every ket ofEcan be expanded on the ; the subspaceEis then identical
with theE-space itself. Consequently, it is reasonable for to be equal to
the identity operator: projecting ontoEa ket which belongs toEdoes not modify this
ket. The same argument can be applied tod .
We can now nd an equivalent of the closure relation for the three-dimensional
space of ordinary geometry,
3
. Ife1,e2ande3are three orthonormal vectors of this
space, and1,2and3are the projectors onto these three vectors, the fact that {e1,
e2,e3} forms a basis in
3
is expressed by the relation
1+2+3= (C-15)
On the other hand, {e1,e2} constitutes an orthonormal set but not a basis of
3
.
This is expressed by the fact that the projector1+2(which projects onto the plane
spanned bye1ande2) is not equal to; for example:(1+2)e3= 0.
118

C. REPRESENTATIONS IN STATE SPACE
Table (II-7) summarizes the only fundamental formulas required for any calculation
in the or representation.
representation representation= =( )
= =
=d =
Table (II-7)
C-3. Representation of kets and bras
C-3-a. Representation of kets
In the basis, the ketis represented by the set of its components, that
is, by the set of numbers= . These numbers can be arranged vertically to
form a one-column matrix (with, in general, a countable innity of rows):
1
2
.
.
.
.
.
.
(C-16)
In the continuous basis, the ketis represented by a continuous innity
of numbers,() = , that is, by a function of. It is then possible to draw a
vertical axis, along which are placed the various possible values of. To each of these
values corresponds a number, :
.
.
.
.
.
.
.
.
.
.
.
.
(C-17)
C-3-b. Representations of bras
Let be an arbitrary bra. In the basis, we can write:
= = = (C-18)
has a unique expansion on the bras. The components of, , are the
complex conjugates of the components= of the ketassociated with.
119

CHAPTER II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
In the same way, we obtain, in the basis:
= = =d (C-19)
The components of, , are the complex conjugates of the components() =
of the ketassociated with.
We have agreed to arrange the components of a ket vertically. Before describing
how to arrange the components of a bra, let us show how the closure relation enables us to
nd simply the expression for the scalar product of two kets in terms of their components.
We know that we can always placebetween and in the expression for the scalar
product:
= =
= = (C-20)
In the same way:
= =
=d =d ()() (C-21)
Let us arrange the components of the brahorizontally, to form a row
matrix (having one row and an innite number of columns):
1 2 (C-22)
Using this convention, is the matrix product of the column matrix which repre-
sents and the row matrix which represents. The result is a matrix having one
row and one column, that is, anumber:
In the basis,has a continuous innity of components . The
various values ofare placed along a horizontal axis. To each of these values corresponds
a component of bra:
(C-23)
Comment:
In a given representation, the matrices which represent a ketand the associated bra
are Hermitian conjugates of each other (in the matrix sense): one passes from one
matrix to the other by interchanging rows and columns and taking the complex conjugate
of each element.
120

C. REPRESENTATIONS IN STATE SPACE
C-4. Representation of operators
C-4-a. Representation of by a square matrix
Given a linear operator, we can, in a or basis, associate with
it a series of numbers dened by:
= (C-24)() = (C-25)or
These numbers depend on two indices and can therefore be arranged in a square
matrix having a countable or continuous innity of rows and columns. The usual con-
vention is to have the rst index x the rows and the second, the columns. Thus, in the
basis, the operatoris represented by the matrix:
1112 1
2122 2
.
.
.
.
.
.
.
.
.
1 2
.
.
.
.
.
.
.
.
.
(C-26)
We see that theth column is made up of the components, in the basis, of the
transform of the basis vector.
For a continuous basis, we draw two perpendicular axes. To a point that has
for its abscissa andfor its ordinate, there corresponds the number():
.
.
.
.
.
.
()
(C-27)
Let us use the closure relation to calculate the matrix which represents the operator
in the basis:
=
=
= (C-28)
The convention chosen above for the arrangement of the elements[or()] is
therefore consistent with the one relating to the product of two matrices: (C-28) expresses
the fact that the matrix representing the operatoris the product of the matrices
associated withand.
121

CHAPTER II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
C-4-b. Matrix representation of the ket =
The problem is the following: knowing the components ofand the matrix
elements ofin a given representation, how can we calculate the components of=
in the same representation?
In the basis, the coordinateof are given by:
= = (C-29)
If we simply insert the closure relation betweenand , we obtain:
= =
=
= (C-30)
For the basis, we obtain, in the same way:
() = =
=d
=d ()() (C-31)
The matrix expression for= is therefore very simple. We see, for example
from (C-30), that the column matrix representingis equal to the product of the
column matrix representingand the square matrix representing:
1
2
.
.
.
.
.
.
.
.
.
=
1112 1
2122 2
.
.
.
.
.
.
.
.
.
1 2
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1
2
.
.
.
.
.
.
.
.
.
(C-32)
C-4-c. Expression for the number
By inserting the closure relation betweenandand again betweenand
, we obtain:
for the basis:
=
=
= (C-33)
122

C. REPRESENTATIONS IN STATE SPACE
for the basis:
=
= dd
= dd ()()() (C-34)
In the matrix formalism, the interpretation of these formulas is as follows:
is a number, that is, a matrix with one row and one column, obtained by multiplying
the column matrix representingrst by the square matrix representingand then
by the row matrix representing. For example, in the basis:
=
12
1112 1
2122 2
.
.
.
.
.
.
.
.
.
1 2
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1
2
.
.
.
.
.
.
.
.
.
(C-35)
Comments:
(i) It can be shown in the same way that the brais represented by a row matrix,
the product of the square matrix representingby the row matrix representing[the
rst two matrices of the right-hand side of (C-35)]. Again we see the importance of the
order of the symbols: the expression would lead to a matrix operation which is
undened (the product of a row matrix by a square matrix).
(ii) From a matrix point of view, equation (B-41) which denes merely expresses
the associativity of the product of the three matrices that appear in (C-35).
(iii) Using the preceding conventions, we express by a square matrix:
1
2
.
.
.
.
.
.
12 =
1112 1
2122 2
.
.
.
.
.
.
.
.
.
1 2
.
.
.
.
.
.
.
.
.
(C-36)
This is indeed an operator, while , the product of a column matrix by a row
matrix, is a number.
C-4-d. Matrix representation of the adjoint of
Using (B-49), we obtain easily:
()= = = (C-37)
123

CHAPTER II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
or
() = = =() (C-38)
Therefore, the matrices representingand in a given representation are Hermitian
conjugates of each other, in the matrix sense:one passes from one to the other by
interchanging rows and columns and then taking the complex conjugate.
Ifis Hermitian,=, and we can then replace()by in (C-37), and
()by()in (C-38):
= (C-39)
() =() (C-40)
A Hermitian operator is therefore represented by a Hermitian matrix, that is, one in
whichany two elements which are symmetric with respect to the principal diagonal are
complex conjugates of each other. In particular, for=or=, (C-39) and (C-40)
become:
= (C-41)
() =() (C-42)
The diagonal elements of a Hermitian matrix are therefore always real numbers.
C-5. Change of representations
C-5-a. Outline of the problem
In a given representation, a ket (or a bra, or an operator) is represented by a
matrix. If we change representations, that is, bases, the same ket (or bra, or operator)
will be represented by a dierent matrix. How are these two matrices related?
For the sake of simplicity, we shall assume here that we are going from one discrete
orthonormal basis to another discrete orthonormal basis. In , we shall
study an example of changing from one continuous basis to another continuous basis.
The change of basis is dened by specifying the componentsof each of the
kets of the new basis in terms of each of the kets of the old one. We shall set:
= (C-43)
is the matrix of the basis change (transformation matrix). Ifdenotes its Hermitian
conjugate:
()= ()= (C-44)
The following calculations can be performed very easily, and without memorization,
by using the two closure relations:
= = (C-45)
= = (C-46)
124

C. REPRESENTATIONS IN STATE SPACE
and the two orthonormalization relations:
= (C-47)
= (C-48)
Comment:
The transformation matrix,, is unitary (ComplementII). That is, it satises:
= = (C-49)
whereis the unit matrix. Indeed, we see that:
()= =
= = (C-50)
In the same way:
()= =
= = (C-51)
C-5-b. Transformation of the components of a ket
To obtain the components of a ketin the new basis from its compo-
nents in the old basis, one simply inserts (C-45) between and :
= =
=
= (C-52)
The inverse expressions can be derived in the same way, using (C-46):
= =
=
= (C-53)
125

CHAPTER II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
C-5-c. Transformation of the components of a bra
The principle of the calculation is exactly the same. For example:
= =
=
= (C-54)
C-5-d. Transformation of the matrix elements of an operator
If, in , we insert (C-45) between and, and again betweenand
, we obtain:
=
= (C-55)
that is:
= (C-56)
In the same way:
= =
=
= (C-57)
D. Eigenvalue equations. Observables
D-1. Eigenvalues and eigenvectors of an operator
D-1-a. Denitions
is said to be an eigenvector (or eigenket) of the linear operatorif:
= (D-1)
whereis a complex number. We are going to study a certain number of properties of
equation (D-1),the eigenvalue equation of the linear operator. In general, this equation
possesses solutions only whentakes on certain values, calledeigenvaluesof. The set
of the eigenvalues is called thespectrumof.
Note that, ifis an eigenvector ofwith the eigenvalue, (whereis
an arbitrary complex number) is also an eigenvector ofwith the same eigenvalue:
( ) = = =( ) (D-2)
126

D. EIGENVALUE EQUATIONS. OBSERVABLES
To rid ourselves of this ambiguity, we could agree to normalize the eigenvectors to 1:
= 1 (D-3)
But this does not completely remove the ambiguity, sincee , whereis an arbitrary real number,
has the same norm as . We shall see below that, in quantum mechanics, the physical predictions
obtained using ore are identical.
The eigenvalueis callednon-degenerate(orsimple) when its corresponding eigen-
vector is unique to within a constant factor, that is, when all its associated eigenkets are
collinear. On the other hand, if there exist at least two linearly independent kets which
are eigenvectors ofwith the same eigenvalue, this eigenvalue is said to bedegenerate.
Itsdegree (or order) of degeneracyis then the number of linearly independent eigen-
vectors associated with it (the degree of degeneracy of an eigenvalue can be nite or
innite). For example, ifis-fold degenerate, there correspond to itindependent
kets (= 12 )such that:
= (D-4)
But then every ketof the form:
=
=1
(D-5)
is an eigenvector ofwith the eigenvalue, whatever the coecients, since:
=
=1
=
=1
= (D-6)
Consequently, the set of eigenkets ofassociated withconstitutes a-dimensional
vector space(which can be innite-dimensional), called the eigensubspace of the eigen-
value. In particular, it is equivalent to say thatis non-degenerate or to say that its
degree of degeneracy is= 1.
To illustrate these denitions, let us choose the example of a projector ( ):=
(with = 1). Its eigenvalue equation is written:
=
that is:
= (D-7)
The ket on the left-hand side is always collinear with, or zero. Consequently, the eigenvectors of
are: on the one hand, itself, with an eigenvalue of= 1; on the other hand, all the kets
orthogonal to, for which the associated eigenvalue is= 0. The spectrum of therefore includes
only two values: 1 and 0. The rst one is simple, the second, innitely degenerate (if the state space
considered is innite-dimensional). The eigensubspace associated with= 0is the supplement
7
of
(see
7
In a vector spaceE, two subspacesE1andE2are said to be supplementary if all ketsofEcan
be written = 1+ 2where 1and 2belong, respectively, toE1andE2, and ifE1andE2
are disjoint (no common non-zero ket; the expansion= 1+ 2is then unique). Actually, there
127

CHAPTER II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
Comments:
(i) Taking the Hermitian conjugate of both sides of equation (D-1), we obtain:
= (D-8)
Therefore, ifis an eigenket ofwith an eigenvalue, it can also be said that
is an eigenbra ofwith an eigenvalue. However, let us stress the fact
that, except in the case whereis Hermitian ( ), nothing can be saida
prioriabout .
(ii)To be completely rigorous, one should solve the eigenvalue equation (D-1) in the
spaceE. That is, one should consider only those eigenvectorswhich have a nite
norm. In fact, we shall be obliged to use operators for which the eigenkets do not satisfy
this condition (). Therefore, we shall grant that vectors which are solutions of (D-1)
can be generalized kets.
D-1-b. Finding the eigenvalues and eigenvectors of an operator
Given a linear operator, how does one nd all its eigenvalues and the correspond-
ing eigenvectors? We are concerned with this question from a purely practical point of
view. We shall consider the case where the state space is of nite dimension, and we
shall grant that the results can be generalized to an innite-dimensional state space.
Let us choose a representation, for example, and let us project the vector
equation (D-1) onto the various orthonormal basis vectors:
= (D-9)
Inserting the closure relation betweenand , we obtain:
= (D-10)
With the usual notation:
=
= (D-11)
equations (D-10) can be written:
= (D-12)
or
[ ]= 0 (D-13)
(D-13) can be considered to be a system of equations where the unknowns are the, the
components of the eigenvector in the chosen representation. This system is linear and
homogeneous.
exists an innity of sub-subspacesE2supplementary to a given subspaceE1. One can xE2by forcing
it to be orthogonal toE1. This shall be done throughout this book, even though the word orthogonal
will not be explicitly written before supplement.
Example: In ordinary three-dimensional space, ifE1is a plane,E2can be any arbitrary straight
line, not contained in. The orthogonal supplement ofE1is the straight line passing through the origin
and orthogonal to.
128

D. EIGENVALUE EQUATIONS. OBSERVABLES
. The characteristic equation
The system (D-13) consists ofequations(= 12 )withunknowns
(= 12 ). Since it is linear and homogeneous, it has a non-trivial solution (the
trivial solution is the one for which all theare zero) if and only if the determinant of
the coecients is zero. This condition is written:
Det[A ] = 0 (D-14)
whereAis the matrix of elementsandis the unit matrix.
Equation (D-14), called the characteristic equation (or secular equation), enables
us to determine all the eigenvalues of the operator, that is, its spectrum. (D-14) can
be written explicitly in the form:
11 12 13 1
21 22 23 2
.
.
.
.
.
.
.
.
.
.
.
.
1 2 3
= 0 (D-15)
This is anth order equation in; consequently, it hasroots, real or complex,
distinct or identical. It is easy to show, by performing an arbitrary change of basis, that
the characteristic equation is independent of the representation chosen. Therefore,the
eigenvalues of an operator are the roots of its characteristic equation.
. Determination of the eigenvectors
Now let us choose an eigenvalue0, a solution of the characteristic equation (D-14),
and let us look for the corresponding eigenvectors. We are going to distinguish between
two cases:
(i) First, assume that0is a simple root of the characteristic equation. We can
then show that the system (D-13), when=0, is comprised of( 1)independent
equations, theth one following from the preceding ones and hence redundant. But
we haveunknowns; there is therefore an innite number of solutions, but all the
can be determined in a unique way in terms of one of them, say1. If we x1, we
obtain for the(1)othera system of(1)linear, inhomogeneous equations (the
right-hand side of each equation is the term in1) with a non-zero determinant [the
( 1)equations are independent]. The solution of this system is of the form:
=
0
1 (D-16)
since the initial system (D-13) is linear and homogeneous.
0
1is, of course, equal to 1
by denition, and the( 1)coecients
0
for= 1are determined from the matrix
elementsand0. The eigenvectors associated with0dier only by the value chosen
for1. They are therefore all given by:
0(1)=
0
1 =1 0 (D-17)
with:
0=
0
(D-18)
129

CHAPTER II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
Therefore, when0is a simple root of the characteristic equation, only one eigenvector
corresponds to it (to within a constant factor): it is a non-degenerate eigenvalue.
(ii) When0is a multiple root of order1of the characteristic equation, there
are two possibilities:
in general, when=0, the system (D-13) is still composed of( 1)inde-
pendent equations. Only one eigenvector then corresponds to the eigenvalue0. The
operatorcannot be diagonalized in this case: the eigenvectors ofare not suciently
numerous for one to be able to construct with them alone a basis of the state space.
nevertheless, when=0, it may happen that the system (D-13) has only
( )independent equations (whereis a number greater than 1 but not larger than
). To the eigenvalue0there then corresponds an eigensubspace of dimension, and0
is a-fold degenerate eigenvalue. Let us assume, for example, that, for=0, (D-13)
is composed of( 2)linearly independent equations. These equations enable us to
calculate the coecientsin terms of any two of them, for example1and2:
=
0
1+
0
2 (D-19)
(obviously:
0
1=
0
2= 1;
0
1=
0
2= 0). All the eigenvectors associated with0are then
of the form:
0(12)=1
1
0+2
2
0 (D-20)
with:
1
0=
0
2
0=
0
(D-21)
The vectors0(12)do indeed constitute a two-dimensional vector space, this being
characteristic of a two-fold degenerate eigenvalue.
When an operator is Hermitian, it can be shown that the degree of degeneracy
of an eigenvalueis always equal to the multiplicityof the corresponding root in
the characteristic equation. Since, in most cases, we shall be studying only Hermitian
operators, we shall only need to know the multiplicity of each root of (D-14) to obtain
immediately the dimension of the corresponding eigensubspace. Thus, in a space of nite
dimension, a Hermitian operator always haslinearly independent eigenvectors (we
shall see later that they can be chosen to be orthonormal): this operator can therefore
be diagonalized (
D-2. Observables
D-2-a. Properties of the eigenvalues and eigenvectors of a Hermitian operator
We shall now consider the very important case in which the operatoris Hermi-
tian:
= (D-22)
(i)The eigenvalues of a Hermitian operator are real.
130

D. EIGENVALUE EQUATIONS. OBSERVABLES
Taking the scalar product of the eigenvalue equation (D-1) by, we obtain:
= (D-23)
But is a real number ifis Hermitian, as we see from:
= = (D-24)
where the last equation follows from hypothesis (D-22). Since and are
real, equation (D-23) implies thatmust also be real.
Ifis Hermitian, we can, in (D-8), replacebyandby, since we have
just shown thatis real. Thus we obtain:
= (D-25)
which shows that is also an eigenbra ofwith the real eigenvalue. Therefore,
whatever the ket:
= (D-26)
The Hermitian operatoris said to act on the left in (D-26).
(ii)Two eigenvectors of a Hermitian operator corresponding to two dierent eigenvalues
are orthogonal.
Consider two eigenvectorsand of the Hermitian operator:
= (D-27-a)
= (D-27-b)
Sinceis Hermitian, (D-27-b
= (D-28)
Then multiply (D-27-a) by on the left and (D-28) by on the right:
= (D-29-a)
= (D-29-b)
Subtracting (D-29-b) from (D-29-a), we nd:
( ) = 0 (D-30)
Consequently, if( )= 0, and are orthogonal.
D-2-b. Denition of an observable
WhenEis nite-dimensional, we have seen ( ) that it is always possible
to form a basis with the eigenvectors of a Hermitian operator. WhenEis innite-
dimensional, this is no longer necessarily the case. This is why it is useful to introduce
a new concept, that of an observable.
131

CHAPTER II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
Consider a Hermitian operator. For simplicity, we shall assume that the set of
its eigenvalues forms a discrete spectrum {;=1, 2, ... }, and we shall indicate later
the modications that must be made when all or part of this spectrum is continuous.
The degree of degeneracy of the eigenvaluewill be denoted by(if= 1,is
non-degenerate). We shall denote by (= 1, 2, ...)linearly independent
vectors chosen in the eigensubspaceEof:
= ;= 12 (D-31)
We have just shown that every vector belonging toEis orthogonal to every vector
of another subspaceE, associated with=; therefore:
= 0for=and arbitraryand (D-32)
Inside each subspaceE, the can always be chosen orthonormal, that is, such
that:
= (D-33)
If such a choice is made, the result is anorthonormal system of eigenvectors of:
the satisfy the relations:
= (D-34)
obtained by regrouping (D-32) and (D-33).
By denition, the Hermitian operatoris anobservableif this orthonormal system
of vectorsforms a basisin the state space. This can be expressed by the closure relation:
=1=1
= (D-35)
Comments:
(i) Since thevectors (= 1, 2, ...,) which span the eigensubspaceE
ofare orthonormal, the projectoronto this subspaceEcan be written (cf.
):
=
=1
(D-36-a)
The observableis then given by:
= (D-36-b)
(it is easy to verify that the action of both sides of this equation on all the kets
gives the same result).
132

D. EIGENVALUE EQUATIONS. OBSERVABLES
(ii)Relation (D-35) can be generalized to include cases where the spectrum of eigenvalues
is continuous by using the rules given in table (II-3). For example, consider a Hermitian
operator whose spectrum is composed of a discrete part {(degree of degeneracy)}
and a continuous part()(assumed to be non-degenerate):
= ; = 12
= 12 (D-37-a)
=() ; 1 2 (D-37-b)
The vectors can always be chosen in such a way that they form an orthonormal system:
=
=( )
= 0 (D-38)
will be said to be an observable if this system forms a basis, that is, if:
=1
+
2
1
d = (D-39)
D-2-c. Example: the projector
Let us show that= (with = 1) is an observable. We have
already pointed out ( ) that it is Hermitian, and that its eigenvalues are 1 and 0
( ); the rst one is simple (associated eigenvector:), the second one is innitely
degenerate (associated eigenvectors: all kets orthogonal to).
Consider an arbitrary ketin the state space. It can always be written in the
form:
= + ( ) (D-40)
Now, is an eigenket ofwith the eigenvalue 1. Since
2
=, we have:
( ) =
2
= (D-41)
Moreover,( ) is also an eigenket of, but with the eigenvalue 0, as we see
from:
( ) = (
2
) = 0 (D-42)
Every ket can thus be expanded on these eigenkets of; therefore,is an ob-
servable.
We shall see in
D-3. Sets of commuting observables
D-3-a. Important theorems
. Theorem I
If two operatorsandcommute, and if is an eigenvector of, is
also an eigenvector of, with the same eigenvalue.
133

CHAPTER II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
We know that, ifis an eigenvector of, we have:
= (D-43)
Applyingto both sides of this equation, we obtain:
= (D-44)
Since we assumed thatandcommute, we also have, replacingon the left-hand
side by:
( ) =( ) (D-45)
This equation expresses the fact thatis an eigenvector of, with the eigenvalue
; the theorem is therefore proved.
Two cases may then arise:
(i) Ifis a nondegenerate eigenvalue, all the eigenvectors associated with it are
by denition colinear, andis necessarily proportional to. Thus is also an
eigenvector of.
(ii) Ifis a degenerate eigenvalue, it can only be said thatbelongs to the
eigensubspaceEof, corresponding to the eigenvalue. Therefore, for any E,
we have:
E (D-46)
Eis said to be globally invariant (or stable) under the action of. Theorem I can also
be stated in another form:
Theorem I': If two operatorsandcommute, every eigensubspace ofis
globally invariant under the action of.
. Theorem II
If two observablesandcommute, and if1and 2are two eigenvectors
ofwith dierent eigenvalues, the matrix element1 2is zero.
If1and 2are eigenvectors of, we can write:
1=1 1
2=2 2 (D-47)
According to theorem I, the fact thatandcommute means that 2is an
eigenvector of, with the eigenvalue2. 2is therefore (cf. ) orthogonal to
1(eigenvector of eigenvalue1=2), which can be written:
1 2= 0 (D-48)
The theorem is then proved. Another proof can be given, which does not involve theorem
I: since the operator[]is zero, we have:
1( )2= 0 (D-49)
134

D. EIGENVALUE EQUATIONS. OBSERVABLES
Using (D-47) and the Hermiticity of[cf. equation (D-25)], we obtain:
1 2=1 1 2
1 2=2 1 2 (D-50)
and (D-49) can be rewritten in the form:
(1 2)1 2= 0 (D-51)
Since, by hypothesis,(1 2)is not zero, we can deduce (D-48) from this equality.
. Theorem III (fundamental)
If two observablesandcommute, one can construct an orthonormal basis of
the state space with eigenvectors common toand.
Consider two commuting observables,and. In order to simplify the notation,
we shall assume that their spectra are entirely discrete. Sinceis an observable, there
exists at least one orthonormal system of eigenvectors ofwhich forms a basis in the
state space. We shall denote these vectors by:
= ; = 12
= 12 (D-52)
is the degree of degeneracy of the eigenvalue, that is, the dimension of the corre-
sponding eigensubspaceE. We have:
= (D-53)
What does the matrix look like which representsin the basis? We know (cf.
theorem II) that the matrix elements are zero when=(on the other
hand, we can say nothinga prioriabout what happens for=and=). Let us
arrange the basis vectorsin the order:
1
1
2
1
1
1
;
1
2
2
2
;
1
3
We then obtain fora block-diagonal matrix, that is, of the form:
1 2 3
1
2
3
.
.
.
.
.
.
.
.
.
.
.
.
0 0 0
0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0 0
0 0
.
.
.0
0 0 0
.
.
.
.
.
.
.
.
.
.
.
.
(D-54)
135

CHAPTER II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
(only the dotted square parts contain non-zero matrix elements). The fact that the
eigensubspacesEare globally invariant under the action of(cf. ) is evident
from this matrix.
Two cases can then arise:
(i) When is a nondegenerate eigenvalue of, there exists only one eigenvector
of, of eigenvalue(the indexin is then unnecessary): the dimension
ofEis then equal to 1. In the matrix (D-54), the corresponding block, then reduces
to a 11 matrix, that is, to a simple number. In the column associated with, all
the other matrix elements are zero. This expresses the fact (cf. -i) that
is an eigenvector common toand.
(ii) Whenis a degenerate eigenvalue of( 1), the block which represents
inEis not, in general, diagonal: theare not, in general, eigenvectors of.
It can be seen, nevertheless, that, since the action ofon each of thevectors
reduces to a simple multiplication by, the matrix representing the restriction of
to withinEis equal to(whereis the unit matrix). This expresses the
fact that an arbitrary ket ofEis an eigenvector ofwith the eigenvalue. The choice
inEof a basis such as ;= 12 is therefore arbitrary. Whatever
this basis, the matrix representing the operatorinEis always diagonal and equal to
. We shall use this property to obtain a basis ofEcomposed of vectors that are also
eigenvectors of.
The matrix representinginE, when the basis chosen is:
;= 12 ,
has for its elements:
()
= (D-55)
This matrix is Hermitian
()
=
()
, sinceis a Hermitian operator. It is therefore
diagonalizable, that is,one can nd inEa new basis ;= 12 in
whichis represented by a diagonal matrix:
=
()
(D-56)
This means that the new basis vectors inEare eigenvectors of:
=
()
(D-57)
As we saw above, these vectors are automatically eigenvectors ofwith an eigenvalue
since they belong toE. Let us stress the fact thatthe eigenvectors ofassociated with
degenerate eigenvalues are not necessarily eigenvectors of. What we have just shown
is thatit is always possible to choose, in every eigensubspace of,a basis of eigenvectors
common toand.
If we perform this operation in all the subspacesE, we obtain a basis ofE, formed
by eigenvectors common toand. The theorem is therefore proved.
Comments:
(i) From now on, we shall denote bythe eigenvectors common toand.
=
= (D-58)
136

D. EIGENVALUE EQUATIONS. OBSERVABLES
The indicesandwhich appear in enable us to specify the eigenvalues
andofand. The additional indexwill eventually be used to distinguish
between the dierent basis vectors which correspond to the same eigenvalues
and(
(ii) The converse of theorem III is very simple to prove: if there exists a basis of
eigenvectors common toand, these two observables commute. From (D-58),
it is easy to deduce:
= =
= = (D-59)
and, subtracting these equations:
[] = 0 (D-60)
This relation is valid for all,and. Since, by hypothesis, the vectors
form a basis, (D-60) entails[] = 0.
(iii) We shall occasionally solve the eigenvalue equation of an observablesuch
that:
=+ avec[] = 0 (D-61)
whereandare also observables.
When one has found a basis of eigenvectors common toand,
the problem is solved, since we see immediately thatis also an eigenvector
of, with an eigenvalue+. The fact that constitutes a basis
is obviously essential: this allows us, for example, to show simply that all the
eigenvalues ofare of the form+.
D-3-b. Complete
8
sets of commuting observables (C.S.C.O.)
9
Consider an observableand a basis ofEcomposed of eigenvectorsof.
If none of the eigenvalues ofis degenerate, the various basis vectors ofEcan be
labelled by the eigenvalue(the indexin being in this case unnecessary). All
the eigensubspacesEare then one-dimensional. Therefore, specifying the eigenvalue
determines in a unique way the corresponding eigenvector (to within a constant factor).
In other words, there exists only one basis ofEformed by the eigenvectors of(we shall
not consider here as distinct two bases whose vectors are proportional). It is then said
that the observableconstitutes, by itself, a C.S.C.O.
If, on the other hand, one or several eigenvalues ofare degenerate, the situation
is dierent. Specifyingis no longer always sucient to characterize a basis vector,
since several independent vectors correspond to any degenerate eigenvalue. In this case,
8
The word complete is used here in a sense which is totally unrelated to those referred to in the
note , p.. This use of the word complete is customary in quantum mechanics.
9
To have a good understanding of the important concepts introduced in this section, the reader
should apply them to a concrete example such as the one discussed in ComplementII(solved exercises
11 and 12).
137

CHAPTER II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
the basis of eigenvectors ofis obviously not unique. One can choose any basis inside
each of the eigensubspacesEof dimension greater than 1.
Let us then choose another observablewhich commutes with, and let us
construct an orthonormal basis of eigenvectors common toand. By denition,
andform a C.S.C.O. if this basis is unique (to within a phase factor for each of the
basis vectors), that is, if, to each of the possible pairs of eigenvalues, there
corresponds only one basis vector.
Comment:
In , we constructed a basis of eigenvectors common toandby solving the
eigenvalue equation ofinside each eigensubspaceEofE. Forandto constitute
a C.S.C.O., it is necessary and sucient that, inside each of these subspaces, all the
eigenvalues ofbe distinct. Since all the vectors ofEcorrespond to the same eigenvalue
of, thevectors can then be distinguished by the eigenvalue ofwhich is
associated with them. Note that it is not necessary that all the eigenvalues ofbe
non-degenerate: vectorsbelonging to two distinct subspacesEcan have the same
eigenvalue for. Moreover, if all the eigenvalues ofwere non-degenerate,alone
would constitute a C.S.C.O.
If, for at least one of the possible pairs, there exist several independent
vectors which are eigenvectors ofandwith these eigenvalues, the set is
not complete. Let us add to it, then, a third observable, which commutes with both
and. We can then use the same argument as in
the following way. When to a pair , there corresponds only one vector, this
vector is necessarily an eigenvector of. If there are several vectors, they form an
eigensubspaceE, in which it is possible to choose a basis formed by vectors which are
also eigenvectors of. One can thus construct, in the state space, an orthonormal basis
formed by eigenvectors common to,and.,andform a C.S.C.O. if this
basis is unique (to within multiplicative factors). Specifying a possible set of eigenvalues
of,,then characterizes only one of the vectors of this basis. If this is
not the case, one adds to,,an observablewhich commutes with each of these
three operators, and so on. In general, we are thus led to the following:
By denition, a set of observables,,... is called a complete set of commuting
observables if
(i)all the observables,,... commute by pairs,
(ii)specifying the eigenvalues of all the operators,,... determines a unique
(to within a multiplicative factor)common eigenvector.
An equivalent way of saying this is the following:
A set of observables,,... is a complete set of commuting observables if there exists
a unique orthonormal basis of common eigenvectors(to within phase factors).
C.S.C.O.'s play an important role in quantum mechanics. We shall see numerous
examples of them (see, in particular,
138

E. TWO IMPORTANT EXAMPLES OF REPRESENTATIONS AND OBSERVABLES
Comments:
(i) If is a C.S.C.O., another C.S.C.O. can be obtained by adding to it
any observable, on the condition, of course, that it commutes withand.
However, it is generally understood that one is conned to minimal sets, that
is, those which cease to be complete when any one of the observables is omitted.
(ii) Let be a C.S.C.O.. Since the specication of the eigenvalues,
,... determines a unique ket of the corresponding basis (to within a constant
factor), this ket is sometimes denoted by .
(iii) For a given physical system, there exist several distinct complete sets of
commuting observables. We shall see a particular example of this in .
E. Two important examples of representations and observables
In this paragraph, we shall return to theF-space of wave functions of a particle, or, more
exactly, to the state spaceErwhich is associated with it, and which we shall dene in
the following way. Let there correspond to every wave function(r)a ketbelonging
toEr; this correspondence is linear. Moreover, the scalar product of two kets coincides
with that of the functions which are associated with them:
=d
3
(r)(r) (E-1)
Eris thus the state space of a (spinless) particle.
We are going to dene and study, in this space, two representations and two
operators which are particularly important. In Chapter
the position and the momentum of the particle under consideration. They will enable us,
moreover, to apply and illustrate the concepts which we have introduced in the preceding
sections.
E-1. The randprepresentations
E-1-a. Denition
In , we introduced two particular bases ofF:r0
(r)and
p0
(r). They are not composed of functions belonging toF:
r0
(r) =(rr0) (E-2-a)
p0
(r) = (2~)
32
e
~
p0r
(E-2-b)
However, every suciently regular square-integrable function can be expanded in one or
the other of these bases.
This is why we shall remove the quotation marks and associate a ket with each of
the functions of these bases (cf. ). The ket associated withr0
(r)will be denoted
simply byr0, and that associated withp0
(r), byp0:
r0(r) r0 (E-3-a)
p0(r) p0 (E-3-b)
139

CHAPTER II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
Using the basesr0
(r)and p0
(r)ofF, we thus dene inErtwo represen-
tations: ther0representation and thep0representation. A basis vector of
the rst one is characterized by three continuous indices0,0and0, which are the
coordinates of a point in three-dimensional space; for the second, the three indices are
also the components of an ordinary vectorp0.
E-1-b. Orthonormalization and closure relations
Let us calculater0r0. Using the denition of the scalar product inEr:
r0r0=d
3
r0
(r)r0
(r) =(r0r0) (E-4-a)
where relation (A-55) has been used. In the same way:
p0p
0=d
3
p0
(r)p
0
(r) =(p0p
0) (E-4-b)
using (A-47). The bases which we have just dened are therefore orthonormal in the
extended sense.
The fact that the set of ther0or that of thep0constitutes a basis inErcan be
expressed by a closure relation inEr. This is written in an analogous manner to (C-10),
integrating here, however, over three indices instead of one.
We therefore have the fundamental relations:
r0r0=(r0r0)(a) p0p
0=(p0p
0)(c)
d
3
0r0r0= (b) d
3
0p0p0= (d)
(E-5)
E-1-c. Components of a ket
Consider an arbitrary ket, corresponding to the wave function(r). The
preceding closure relations enable us to write it in either of these two forms:
=d
3
0r0r0 (E-6-a)
=d
3
0p0p0 (E-6-b)
The coecientsr0 andp0 can be calculated using the formulas:
r0 =d
3
r0
(r)(r) (E-7-a)
p0 =d
3
p0
(r)(r) (E-7-b)
We then nd:
r0 =(r0) (E-8-a)
p0 =
(p0) (E-8-b)
140

E. TWO IMPORTANT EXAMPLES OF REPRESENTATIONS AND OBSERVABLES
where
(pis the Fourier transform of(r).
The value(r0)of the wave function at the pointr0is thus shown to be the com-
ponent of the keton the basis vectorr0of ther0representation. The wave
function in momentum space
(p)can be interpreted analogously. The possibility of
characterizingby(r)is thus simply a special case of the results of .
For example, for=p0, formula (E-8-a) gives:
r0p0=p0
(r0) = (2~)
32
e
~
p0r0
(E-9)
For =r0, the result is indeed in agreement with the orthonormalization relation
(E-5-a):
r0r0=r0
(r0) =(r0r0) (E-10)
Now that we have reinterpreted the wave function(r)and its Fourier transform
(p), we shall denote the basis vectors of the two representations we are studying here
byrandp, instead ofr0andp0. Formulas (E-8) can then be written:
r =(r) (E-8-a)
p =
(p) (E-8-b)
and the orthonormalization and closure relations (E-5) become:
rr=(rr)(a) pp=(pp)(c)
d
3
rr= (b) d
3
pp= (d) ( E-5)
Of course,randpare still considered to represent two sets ofcontinuous indices, {,,
} and {,,}, which x the basis kets of therandprepresentations
respectively.
Now let (r)be an orthonormal basis ofF. With each(r)is associated a ketofEr.
The set forms an orthonormal basis inEr; it therefore satises the closure relation:
= (E-11)
Evaluate the matrix element of both sides of (E-11) betweenrandr:
r r=rr=rr (E-12)
According to (E-8-a) and (E-5-a), this relation can be written:
(r)(r) =(rr) (E-13)
The closure relation for the(r)[formula (A-32)] is therefore simply the expression in ther
representation of the vectorial closure relation (E-11).
E-1-d. The scalar product of two vectors
We have dened the scalar product of two kets ofEras being equal to that of the
associated wave functions inF[equation (E-1)]. In light of the discussion in ,
141

CHAPTER II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
this denition appears simply as a special case of formula (C-21). (E-1) can, in fact, be
derived by inserting the closure relation (E-5-b) betweenand :
=d
3
rr (E-14)
and by interpreting the componentsr andr) as in (E-8-a).
If we place ourselves in theprepresentation, a well-known property of the
Fourier transform is demonstrated (Appendix, ):
=d
3
pp
=d
3
(p)
(p) (E-15)
E-1-e. Changing from the rrepresentation to theprepresentation
This is accomplished using the method indicated in , the only dierence
arising from the fact that we are dealing here with two continuous bases. Changing from
one basis to the other brings in the numbers:
rp=pr= (2~)
32
e
~
pr
(E-16)
A given ketis represented byr =(r)in therrepresentation and
byp =
(p)in theprepresentation. We already know [relations (E-2-b) and
(E-7-b)] that(r)and
(p)are related by a Fourier transform. This is indeed what the
formulas for the representation change yield:
r =d
3
rpp
that is:
(r) = (2~)
32
d
3
e
~
pr
(p) (E-17)
Inversely:
p =d
3
prr
that is:
(p) = (2~)
32
d
3
e
~
pr
(r) (E-18)
By applying the general formula (C-56), one can easily pass from the matrix ele-
mentsr r=(rr)of an operatorin therrepresentation to the matrix
elementsp p=(pp)of the same operator in theprepresentation:
(pp) = (2~)
3
d
3
d
3
e
~
(prpr)
(rr) (E-19)
An analogous formula enables one to calculate(rr)from(pp).
142

E. TWO IMPORTANT EXAMPLES OF REPRESENTATIONS AND OBSERVABLES
E-2. TheRandPoperators
E-2-a. Denition
Let be an arbitrary ket ofErand letr =(r) ( )be the
corresponding wave function. Using the denition of the operator, the ket:
= (E-20)
is represented, in therbasis, by the functionr =(r) ( )such
that:
( ) = ( ) (E-21)
In therrepresentation, theoperator therefore coincides with the operator
which multiplies by. Although we characterizeby the way in which it transforms
the wave functions, it is an operator which acts in the state spaceEr. We can introduce
two other operators,and, in an analogous manner. Thus we dene,andby
the formulas:
r =r (E-22-a)
r =r (E-22-b)
r =r (E-22-c)
where the numbers,,are precisely the three indices which label the ketr.,
andwill be considered to be the components of a vector operatorR: for the
moment, we shall treat this simply as a condensed notation, suggested by the fact that
,,are the components of the ordinary vectorr.
Manipulation of the,,operators is particularly simple in therrep-
resentation. For example, in order to calculate the matrix element , all we
need to do is insert the closure relation (E-5-b) between andand use denition
(E-22-a):
=d
3
rr
=d
3
(r)(r) (E-23)
Similarly, we dene the vector operatorPby its components,,, whose
action, in theprepresentation, is given by:
p = p (E-24-a)
p = p (E-24-b)
p = p (E-24-c)
where,,are the three indices which appear in the ketp.
Let us ascertain how thePoperator acts in therrepresentation. To do so
(cf. ), we use the closure relation (E-5-d) and the transformation matrix (E-16)
143

CHAPTER II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
to obtain:
r =d
3
rpp
= (2~)
32
d
3
e
~
pr
(p) (E-25)
We recognize in (E-25) the Fourier transform of
(p), that is
~(r)[Appendix,
relation (38a)]. Therefore:
rP =
~
rr (E-26)
In therrepresentation, thePoperator coincides with the dierential operator
(~)rapplied to the wave functions. The calculation of a matrix element such as
in therrepresentation is therefore performed in the following manner:
=d
3
rr
=d
3
(r)
~
(r) (E-27)
Placing ourselves in therrepresentation, we can also calculate the commu-
tators between the,,,,,operators. For example:
r[ ] =r( )
=r
~
r
=
~
r
~
r
=~r (E-28)
This calculation is valid for alland for any ket of therbasis. Thus one nds
10
:
[ ] =~ (E-29)
In the same way, we nd all the other commutators between the components ofRand
those ofP. The result can be written in the form:
[ ] = 0
[ ] = 0
[ ] =~
= 123 (E-30)
where1,2,3, and1,2,3designate respectively,,and,,.
Formulas (E-30) are calledcanonical commutation relations.
10
The commutator[ ]is an operator, and it should, actually, be written[ ] =~. However,
we shall often replace the identity operatorby the number 1, except when it is important to make the
distinction.
144

E. TWO IMPORTANT EXAMPLES OF REPRESENTATIONS AND OBSERVABLES
E-2-b. RandPare Hermitian
In order to show that, for example, is a Hermitian operator, we can use formula
(E-23):
=d
3
(r)(r) = d
3
(r) (r)
= (E-31)
From , we know that equation (E-31) is characteristic of a Hermitian operator.
Similar proofs show thatandare also Hermitian. For,and, the
prepresentation can be used, and the calculations are then analogous to the pre-
ceding ones.
It is interesting to show thatPis Hermitian by using equation (E-26), which gives its
action in therrepresentation. Consider, for example, formula (E-27) and integrate it by
parts:
=
~
dd
+
d (r)(r)
=
~
dd (r)(r)
=+
=
+
d(r)(r) (E-32)
Since the integral which yields the scalar productis convergent,(r)(r)approaches
zero when . The rst term on the right hand side of (E-32) is therefore equal to zero,
and:
=
~
d
3
(r)(r)
=
~
d
3
(r)(r)
= (E-33)
It can be seen that the presence of the imaginary numberis essential. The dierential oper-
ator
, acting on the functions ofF, isnotHermitian, because of the sign change which is
introduced by the integration by parts. However,
is Hermitian, as is
~
.
E-2-c. Eigenvectors ofRandP
Consider the action of theoperator on the ketr0; according to (E-22-a), we
have:
r r0=rr0=(rr0) =0(rr0) =0rr0 (E-34)
This equation expresses the fact that the components, in therrepresentation,
of the ketr0are equal to those of the ketr0multiplied by0. We therefore have:
r0=0r0 (E-35)
145

CHAPTER II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
An analogous argument shows that the ketsr0are also eigenvectors of theand
operators. Omitting the index zero which then becomes unnecessary, we can write:
r=r
r=r
r=r
(E-36)
The ketsrare therefore the eigenkets common to,and. Thus the notation
rwhich we chose above is justied: each eigenvector is labelled by a vectorr, whose
components,,represent three continuous indices which correspond to the eigenvalues
of,,.
Similar arguments can be elaborated for thePoperator, placing ourselves, this
time, in the representationp. We then obtain:
p= p
p= p
p= p
(E-37)
Comment:
This result can also be derived from equation (E-26), which gives the action ofP
in therrepresentation. Using (E-9), we nd:
r p=
~
rp=
~
(2~)
32
e
~
pr
=(2~)
32
e
~
pr
= rp (E-38)
All the components of the ketpin therrepresentation can be
obtained by multiplying those ofpby the constant:pis an eigenket of
with the eigenvalue.
E-2-d. RandPare observables
Relations (E-5-b) and (E-5-d) express the fact that thervectors and the
pvectors constitute bases inEr. Therefore,RandPare observables.
Moreover, the specication of the three eigenvalues0,0,0of,,uniquely
determines the corresponding eigenvectorr0: in therrepresentation, its coordi-
nates are( 0)( 0)( 0). The set of the three operators,,therefore
constitutes a C.S.C.O. inEr.
It can be shown in the same way that the three components,,ofPalso
constitute a C.S.C.O. inEr.
Note that, inEr,does not constitute a C.S.C.O. by itself. When the0index
is xed,0and0can take on any real values. Thus, each eigenvalue0is innitely
degenerate. On the other hand, in the state spaceEof a one-dimensional problem,
constitutes a C.S.C.O.: the eigenvalue0uniquely determines the corresponding eigenket
0, its coordinates being( 0)in the representation.
146

F. TENSOR PRODUCT OF STATE SPACES
Comment:
We have found two C.S.C.O.'s inEr, and . We shall encounter
others later. Consider, for example, the set : these three observables com-
mute (equations (E-30)); moreover, if the three eigenvalues0,0and0are xed,
there corresponds to them only one ket, whose associated wave function is written:
00 0( ) =( 0)
1
2~
e
~
(0+0)
(E-39)
F. Tensor product of state spaces
11
F-1. Introduction
We introduced the state space of a physical system using the concept of a one-
particle wave function. However, our reasoning has involved sometimes one- and some-
times three-dimensional wave functions. Now it is clear that the space of square-integrable
functions is not the same for functions of one variable()as for functions of three vari-
ables(r):ErandEare therefore dierent spaces. Nevertheless,Erappears to be
essentially a generalization ofE. Does there exist a more precise relation between these
two spaces?
In this section, we are going to dene and study the operation of taking the tensor
product of vector spaces
12
, and apply it to state spaces. This will answer, in particular,
the question we have just asked:Ercan be constructed fromEand two other spaces,
EandE, which are isomorphic to it (
In the same way, we shall be concerned later (Chapters ) with the
existence, for certain particles, of an intrinsic angular momentum or spin. In addition
to the external degrees of freedom (position, momentum), which are treated using the
observablesRandPdened inEr, it will be necessary to take into account the internal
degrees of freedom and to introduce spin observables which act in a spin state spaceE.
The state spaceEof a particle with spin will then be seen to be the tensor product of
ErandE.
Finally, the concept of a tensor product of state spaces allows us to solve the
following problem. Let(1)and(2)be two isolated physical systems (they are, for
example, suciently far apart that their interactions are perfectly negligible). The state
spaces which correspond to(1)and(2)are, respectively,E1andE2. Now let us
assume that we consider the set of these two systems to form one physical system()
(this becomes indispensable when they are close enough to interact). What is then the
state spaceEof the global system()?
It can be seen from these examples how useful the denitions and results of this
section are in quantum mechanics.
F-2. Denition and properties of the tensor product
LetE1andE2be two
13
spaces, of dimension1and2respectively (1and2
can be nite or innite). Vectors and operators of these spaces will be assigned an index,
11
This section is not necessary for the understanding of Chapter. One can study it later when it
becomes necessary to use tensor products (ComplementIV, or Chapter).
12
This operation is sometimes called the Kronecker product
13
The following denitions can casily be extended to the tensor product of a nite number of spaces.
147

CHAPTER II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
(1)or(2), depending on whether they belong toE1orE2.
F-2-a. Tensor product space E
. Denition
By denition, the vector spaceEis called thetensor product ofE1andE2:
E=E1E2 (F-1)
if there is associated with each pair of vectors,(1)belonging toE1and(2)be-
longing toE2, a vector ofE, denoted by
14
:
(1) (2) (F-2)
which is called the tensor product of(1)and(2), this correspondence satisfying
the following conditions:
(i) It islinearwith respect to multiplication by complex numbers:
(1) (2)= (1) (2)
(1) (2)= (1) (2) (F-3)
(ii) It isdistributivewith respect to vector addition:
(1) 1(2)+ 2(2)=(1) 1(2)+(1) 2(2)
1(1)+ 2(1) (2)= 1(1) (2)+ 2(1) (2) (F-4)
(iii) When a basis has been chosen in each of the spacesE1andE2, (1)for
E1and (2)forE2, the set of vectors(1) (2)constitutes a basis inE. If
1and2are nite, the dimension ofEis consequently12.
. Vectors ofE
(i) Let us rst consider atensor product vector,(1) (2). Whatever(1)
and(2)may be, they can be expressed in the(1)and (2)bases respec-
tively:
(1)= (1)
(2)= (2) (F-5)
Using the properties described in , the expansion of the vector(1) (2)
in the(1) (2)basis can be written:
(1) (2)= (1) (2) (F-6)
14
This vector can be written either(1) (2)or(2) (1); the order of the two vectors
is of no importance.
148

F. TENSOR PRODUCT OF STATE SPACES
Therefore,the components of a tensor product vector are the products of the com-
ponents of the two vectors of the product.
(ii)There exist inEvectors which are not tensor productsof a vector ofE1by a
vector ofE2. Since(1) (2)constitutes by hypothesis a basis inE, the most
general vector ofEis expressed by:
= (1) (2) (F-7)
Given12arbitrary complex numbers, it is not always possible to put them in
the form of products,, of1numbersand2numbers. Therefore, in general,
vectors(1)and(2)of which is the tensor product do not exist. However,
an arbitrary vector ofEcan always be decomposed into a linear combination of tensor
product vectors, as is shown by formula (F-7).
. The scalar product inE
The existence of scalar products inE1andE2permits us to dene one inEas well.
We rst dene the scalar product of(1)(2)=(1) (2)by(1)(2)=
(1) (2)by setting:
(1)(2)(1)(2)= (1)(1) (2)(2) (F-8)
For two arbitrary vectors ofE, we simply use the fundamental properties of the scalar
product [equations (B-9), (B-10) and (B-11)], since each of these vectors is a linear
combination of tensor product vectors.
Notice, in particular, that the basis(1)(2)= (1) (2)is or-
thonormal if each of the bases(1)and (2)is:
(1)(2)(1)(2)= (1)(1) (2)(2)
= (F-9)
F-2-b. Tensor product of operators
(i) First, consider a linear operator(1)dened inE1. We associate with it a
linear operator(1)acting inE, which we call theextension of(1)inE, and which
is characterized in the following way: when(1)is applied to a tensor product vector
(1) (2), one obtains, by denition:
(1)(1) (2)=(1)(1) (2) (F-10)
The hypothesis that(1)is linear is then sucient for determining it completely.
An arbitrary vectorofEcan be written in the form (F-7). Denition (F-10) then
gives the action of(1)on :
(1)= (1)(1) (2) (F-11)
We obtain in an analogous manner the extension(2)of an operator(2)initially
dened inE2.
149

CHAPTER II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
(ii) Now let(1)and(2)be two linear operators acting respectively inE1and
E2. Theirtensor product(1)(2)is the linear operator inE, dened by the following
relation which describes its action on the tensor product vectors:
(1)(2) (1) (2)=(1)(1) (2)(2) (F-12)
Here also, this denition is sucient for characterizing(1)(2).
Comments:
(i) The extensions of operators are special cases of tensor products: if(1)and(2)are
the identity operators inE1andE2respectively,(1)and(2)can be written:
(1) =(1)(2)
(2) =(1)(2) (F-13)
Inversely, the tensor product(1)(2)coincides with the ordinary product of
two operators(1)and(2)ofE:
(1)(2) =(1)(2) (F-14)
(ii) It is easy to show that two operators such as(1)and(2)commute inE:
(1)(2)= 0 (F-15)
We must verify that(1)(2)and(2)(1)yield the same result when they act on an
arbitrary vector of the(1) (2)basis:
(1)(2)(1) (2)=(1)(1) (2)(2)
=(1)(1) (2)(2) (F-16)
(2)(1)(1) (2)=(2)(1)(1) (2)
=(1)(1) (2)(2) (F-17)
(iii) The projector onto the tensor product vector(1)(2)=(1) (2), which
is an operator acting inE, is obtained by taking the tensor product of the projectors
onto(1)and(2):
(1)(2)(1)(2)=(1)(1) (2)(2) (F-18)
This relation follows immediately from the denition of the scalar product inE.
(iv) Just as with vectors, there exist operators inEwhich are not tensor products of an
operator ofE1and an operator ofE2.
F-2-c. Notations
In quantum mechanics, the notation generally used is a simplied version of the
one which we have dened here. This is the one we shall adopt, but it is important to
interpret it correctly in the light of the preceding discussion.
150

F. TENSOR PRODUCT OF STATE SPACES
First of all, the symbolwhich indicates the tensor product is omitted, and the
vectors or operators which are to be multiplied tensorially are simply juxtaposed:
(1)(2)means (1) (2) (F-19)
(1)(2) means(1)(2) (F-20)
Moreover, the extension inEof an operator ofE1orE2is written in the same way as
this operator itself:
(1)means (1)or(1) (F-21)
No confusion is possible in (F-19): until now we have never written two kets one
after the other as we do here. Notice in particular that the expression, where
and belong to the same spaceE, is not dened in this space: it represents a
vector of the space which is the tensor product ofEby itself.
On the other hand, the notation in (F-20) and (F-21) is slightly ambiguous, espe-
cially in the latter, where two dierent operators are represented by the same symbol.
However, it will be possible in practice to distinguish between them by the vector to
which this symbol is applied: depending on whether it is a vector ofEor ofE1, we shall
be dealing with(1), or with(1)in a strict sense. As for formula (F-20), it poses no
problem whenE1andE2are dierent, since we have, until now, dened only products
of operators which act in the same space. Moreover,(1)(2)can be considered to be
an ordinary product of operators ofE, if(1)and(2)are interpreted as designating,
in fact,(1)and(2)[equation (F-14)].
F-3. Eigenvalue equations in the product space
The vectors ofEwhich are tensor products of a vector ofE1and a vector ofE2
play an important role in the discussion above. We shall see that this is also the case for
the extensions toEof operators acting inE1andE2.
F-3-a. Eigenvalues and eigenvectors of extended operators
. Eigenvalue equation of(1)
Consider an operator(1), for which we know, inE1, all the eigenstates and
eigenvalues. We shall assume, for example, that the whole spectrum of(1)is discrete:
(1)(1)= (1);= 12 (F-22)
We want to solve the eigenvalue equation of the extension of(1)inE:
(1)= ; E (F-23)
It can immediately be seen, from (F-10), that every vector of the form(1)(2)
is an eigenvector of(1)with the eigenvalue, whatever(2)may be, since:
(1)(1)(2)=(1)(1)(2)
= (1)(2) (F-24)
151

CHAPTER II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
Let us show that, when(1)is an observable inE1, all the solutions of (F-23) can
be obtained in this way. The set of the(1)then forms a basis inE1. Consequently,
the orthonormal system of vectorssuch that:
= (1)(2) (F-25)
(where (2)is a basis ofE2) forms a basis inE. We therefore have an orthonormal
basis constituted by the eigenvectors of(1)inE, , which means that equation
(F-23) is solved.
The following conclusions can be drawn:
If(1)is an observable inE1, it is also an observable inE. This results from the
fact that the extension of(1)is Hermitian and from the fact that constitutes
a basis inE.
The spectrum of(1)is the same inEas inE1: the same eigenvaluesappear
in (F-22) and in (F-24).
Nevertheless, an eigenvaluewhich is-fold degenerate inE1has, inE, a
degree of degeneracy2 . We known that the eigensubspace associated withis
spanned inEby the kets = (1)(2)withxed and= 1, 2, ...,;=
1, 2, ...,2. Therefore, even ifis simple inE1, it is (2-fold) degenerate inE.
The projector onto the eigensubspace corresponding to an eigenvalueis written, inE
[cf. (F-18)]:
= (1) (1) (2)(2)
= (1) (1) (2) (F-26)
using inE2the closure relation relative to the(2)basis. It is therefore the extension of
the projector(1) = (1) (1)which is associated withinE1.
. Eigenvalue equation of(1)+(2)
We shall often need to solve, in a tensor product space such asE, eigenvalue
equations for operators of the form:
=(1) +(2) (F-27)
where(1)and(2)are observables whose eigenvalues and eigenvectors are known in
E1andE2respectively:
(1)(1)= (1)
(2)(2)= (2) (F-28)
[to simplify the notation, we assume the spectra of(1)and(2)to be discrete and
non-degenerate inE1andE2].
152

F. TENSOR PRODUCT OF STATE SPACES
(1)and(2)commute [formulas (F-16) and (F-17)], and the(1) (2),
which form a basis inE, are eigenvectors common to(1)and(2):
(1)(1) (2)= (1) (2)
(2)(1) (2)= (1) (2) (F-29)
They are also eigenvectors of:
(1) (2)= (+)(1) (2) (F-30)
This gives us directly the solution of the eigenvalue equation of.
Therefore:the eigenvalues of=(1) +(2)are the sums of an eigenvalue of
(1)and an eigenvalue of(2). One can nd a basis of eigenvectors ofwhich are
tensor products of an eigenvector of(1)and an eigenvector of(2).
Comment:
Equation (F-30) shows that the eigenvalues ofare all of the form= +. If
two dierent pairs of values ofandwhich give the same value fordo not exist,
is non-degenerate (recall that we have assumedandto be non-degenerate in
E1andE2respectively). The corresponding eigenvector ofis necessarily the tensor
product (1) (2). If, on the other hand, the eigenvalueis, for example, two-
fold degenerate (there existandsuch that = ), all that can be asserted is
that every eigenvector ofcorresponding to this eigenvalue is written:
(1) (2)+ (1) (2) (F-31)
whereandare arbitrary complex numbers. In this case, therefore, there exist eigen-
vectors ofwhich are not tensor products.
F-3-b. Complete sets of commuting observables inE
We are nally going to show that if a C.S.C.O. has been chosen in both spacesE1
andE2, obtaining one inEis straightforward.
As an example, let us consider the case where(1)constitutes a C.S.C.O. by itself
inE1, and the C.S.C.O. inE2is composed of two observables,(2)and(2). This means
(cf. ) that all the eigenvaluesof(1)are nondegenerate inE1:
(1)(1)= (1) (F-32)
the ket(1)being unique to within a constant factor. On the other hand, inE2, some
of the eigenvaluesof(2)are degenerate, as are some of the eigenvalues()of(2).
Nevertheless, the basis of eigenvectors common to(2)and(2)is unique inE2, since
there exists only one ket (to within a constant factor) which is an eigenvector of(2)
and of(2)with the eigenvaluesand, xed:
(2)(2)= (2)
(2)(2)= (2)
(2)unique to within a constant factor
(F-33)
153

CHAPTER II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
InE, each of the eigenvaluesis2-fold degenerate (cf. ). Therefore,
(1)no longer forms a C.S.C.O. by itself. Similarly, there exist1linearly independent
kets which are eigenvectors of(2)and(2)with the eigenvaluesandrespectively,
and the set(2)(2)is not complete either. However, we saw in
eigenvectors which are common to the three commuting observables(1),(2)and(2)
are the(1)(2)= (1) (2):
(1)(1)(2)= (1)(2)
(2)(1)(2)= (1)(2)
(2)(1)(2)= (1)(2) (F-34)
The system (1)(2)constitutes a basis inE, since this is the case for(1)
and (2)inE1andE2respectively. Moreover, if a set of three eigenvalues
is chosen, only one vector(1)(2)corresponds to it.(1),(2)
and(2)therefore constitute a C.S.C.O. inE.
The preceding argument can be generalized without diculty:by joining two sets
of commuting observables which are complete inE1andE2respectively, one obtains a
complete set of commuting observables inE.
F-4. Applications
F-4-a. One- and three-dimensional particle states
. State spaces
Consider again, in the light of the preceding discussion, the problem posed in the
introduction (): how areEandErrelated?
Eis the state space of a particle moving in one dimension, that is, the state space
associated with the wave functions(). InE, the observablewhich was studied in
); its eigenvectors are the basis kets of the
representation. A vectorofEis characterized, in this representation, by a
wave function() = ; in particular, the basis ket0corresponds to the wave
function
0
() =( 0).
In the same way, it is possible to introduce the spacesEandEassociated with
the wave functions()and(). The observableforms a C.S.C.O. inE, as does
inE. The corresponding eigenvectors are the basis kets of theand repre-
sentations ofEandErespectively. A vectorofE(or ofE) is characterized
in the (or representation by a function() = (or() = ).
The function which corresponds to the basis ket0(or0) is( 0)(or( 0)).
Let us then form the tensor product:
E=EEE (F-35)
We obtain a basis inEfrom the tensor product of the , and
bases. We shall denote it by , with:
= (F-36)
154

F. TENSOR PRODUCT OF STATE SPACES
The basis kets are simultaneous eigenvectors of the,andoperators extended into
E:
=
=
= (F-37)
Therefore,Ecoincides withEr, the state space of a three-dimensional particle, and
withr:
r= (F-38)
where,,are precisely the cartesian coordinates ofr.
There exist inErkets = that are the tensor products of three
kets, one ofE, one ofEand one ofE. The components in therrepresentation
are then [cf. formula (F-8)]:
r = (F-39)
The associated wave functions are thus factorized:()()(). This is the case for
the basis vectors themselves:
rr0=(rr0) =( 0)( 0)( 0) (F-40)
Note that the most general state ofEris not such a product. It is written:
=ddd( ) (F-41)
In( ) = , the-,- and-dependences cannot, in general, be factorized:
each of the wave functions associated with the kets ofEris a wave function with three
variables.
The results of , which constitutes a
C.S.C.O. by itself inE, no longer has this property inEr(cf. ): the eigenvalues
of its extension inErare the same as inE, but they become innitely degenerate
becauseEandEare innite-dimensional. Starting with a C.S.C.O. inE,EandE,
we construct one forEr: , for example, but also sinceforms a
C.S.C.O. inE, or , etc...
. An important application
Let us try to solve inErthe eigenvalue equation of an operatorsuch that:
= + + (F-42)
where,andare the extensions of observables acting respectively inE,Eand
E. In practice, one recognizes that, for example, is the extension of an observable of
Ebecause it is constructed using only the operatorsand. Using the reasoning of
155

CHAPTER II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
, one rst looks for the eigenvalues and eigenvectors ofinE,inEand
inE:
=
=
= (F-43)
The eigenvalues ofare then all of the form:
= ++ (F-44)
with an eigenvector that is the tensor product ; the wave function asso-
ciated with this vector is the product:
()()() =
This is the type of situation that was considered in ComplementI() for
the justication of the study of one-dimensional models. There, we were dealing with
dierential operators acting on wave functions:
=
~
2
2
+(r) (F-45)
This equation can be decomposed as in (F-42) in the particular case where the potential
can be written:
(r) =1() +2() +3() (F-46)
F-4-b. States of a two-particle system
Consider a physical system which is made up of two (spinless) particles. We shall
distinguish between them by numbering them (1) and (2). To describe the system quan-
tum mechanically, we can generalize the concept of a wave function, introduced for the
case of one particle. A state of the system can be characterized, at a given time, by
a function of six spatial variables(r1r2) (111;222). The probabilis-
tic interpretation of such a two-particle wave function is the following: the probability
dP(r1r2), at the given time, of nding particle (1) in the volumed
3
1= d1d1d1
situated at the pointr1,andparticle (2) in the volumed
3
2= d2d2d2aboutr2, is:
dP(r1r2) = (r1r2)
2
d
3
1d
3
2 (F-47)
The normalization constantis obtained by imposing the condition that the total proba-
bility must be equal to 1 (conservation of the number of particles;cf. ):
1
=d
3
1d
3
2(r1r2)
2
(F-48)
and the observables1,1,1can be dened inEr1
. Similarly, in the state spaceEr2
of particle (2), we introduce ther2representation and the observables2,2,2.
Take the tensor product:
Er1r2
=Er1
Er2
(F-49)
156

F. TENSOR PRODUCT OF STATE SPACES
The set of vectors:
r1r2=r1r2 (F-50)
forms a basis inEr1r2. Consequently, every ketof this space can be written:
=d
3
1d
3
2(r1r2)r1r2 (F-51)
with
(r1r2) =r1r2 (F-52)
Moreover, the square of the norm ofis equal to:
=d
3
1d
3
2(r1r2)
2
(F-53)
For it to be nite,(r1r2)must be square-integrable. Therefore, a wave function
(r1r2)is associated with each ket ofEr1r2
:the state space of a two-particle system
is the tensor product of the spaces which correspond to each of the particles. A C.S.C.O.
is obtained inEr1r2by joining, for example,1,1,1and2,2,2.
Assume that the state of the system is described by a tensor product ket:
= 1 2 (F-54)
The corresponding wave function can then be factorized:
(r1r2) =r1r2 =r1 1r2 2=1(r1)2(r2) (F-55)
In this case, one says thatthere is no correlationbetween the two particles. We shall analyze
later (ComplementIII) the physical consequences of such a situation.
The preceding can be generalized: when a physical system is composed of the union
of two or several simpler systems, its state space is the tensor product of the spaces which
correspond to each of the component systems.
References and suggestions for further reading:
Section 10 of the bibliography contains references to a certain number of mathematical
texts, listed by subject. Under each heading, they are ranked, as much as possible, in order of
increasing diculty. See also the quantum mechanics texts (sections 1 and 2 of the bibliogra-
phy), which treat the mathematical problems at many dierent levels. They also contain other
references.
For a very simple approach to the fundamental mathematical concepts needed to under-
stand Chapter
consult, for example: Arfken (10.4), Chap. 4; Bak and Lichtenberg (10.3), Chap. I; Bass (10.1),
vol. I, Chap. II to V. A more explicit application to quantum mechanics can be found in Jackson
(10.5) (see, in particular, Chap. 5), Butkov (10.8), Chap. 10 (nite-dimensional linear spaces)
and Chap. 11 (innite-dimensional vector spaces, spaces of functions). See also Meijer and
Bauer (2.18), Chap. 1, particularly the table at the end of this chapter.
157

COMPLEMENTS OF CHAPTER II, , READER'S GUIDE
AII: THE SCHWARZ INEQUALITY
BII: REVIEW OF SOME USEFUL PROPERTIES
OF LINEAR OPERATORS
CII: UNITARY OPERATORS
Review of some denitions and useful mathemati-
cal results (elementary level) intended for readers
unfamiliar with these concepts; will serve as a
reference later (especially BII).
DII: A MORE DETAILED STUDY OF THE r
AND pREPRESENTATIONS
EII: SOME GENERAL PROPERTIES OF TWO OB-
SERVABLES, AND , WHOSE COMMUTATOR
IS EQUAL TO}
Complete .
DII: remains at the level of Chapter
be read immediately after it.
EII: adopts a more general and a slightly more
formal point of view. Introduces, in particular,
the translation operator. May be reserved for
later study.
FII: THE PARITY OPERATOR Discussion of the parity operator, particularly
important in quantum mechanics; at the same
time, a simple illustration of the concepts of
Chapter
GII: AN APPLICATION OF THE PROPERTIES
OF THE TENSOR PRODUCT: THE TWO-
DIMENSIONAL INFINITE WELL
A simple application of the tensor product
( ); can be considered as a worked
exercise.
HII: EXERCISES Solutions are given for exercises 11 and 12; their
aim is to familiarize the reader with the properties
of commuting observables and the concept of
a C.S.C.O. in a very simple special case. It is
recommended that these exercises be done during
the reading of .
159

THE SCHWARZ INEQUALITY
Complement AII
The Schwarz inequality
For any ketbelonging to the state space, we have:
real0 (1)
being equal to zero only whenis the null vector [cf.equation (B-12) of Chap-
ter]. Using inequality (1), we shall derive the Schwarz inequality. This inequality states
that, if1and2are any arbitrary vectors of, then:
12
2
11 22 (2)
the equality being realized if and only if1and2are proportional.
Given1and2, consider the ketdened by:
= 1+ 2 (3)
whereis an arbitrary parameter. Whatevermay be:
= 11+ 12+ 21+ 220 (4)
Let us chose forthe value:
=
21
22
(5)
In (4), the second and third terms of the right-hand side are then equal, and opposite in
value to the fourth term, so that (4) reduces to:
11
12 21
22
0 (6)
Since22is positive, we can multiply this inequality by22, to obtain:
11 22 12 21 (7)
which is precisely (2). In (7), the equality can only be realized if= 0, that is,
according to (3), if1= 2. The kets1and2are then proportional.
References:
Bass I (10.1), 5-3; Arfken (10.4), 9-4.
161

REVIEW OF SOME USEFUL PROPERTIES OF LINEAR OPERATORS
Complement BII
Review of some useful properties of linear operators
1 Trace of an operator
1-a Denition
1-b The trace is invariant
1-c Important properties
2 Commutator algebra
2-a Denition
2-b Properties
3 Restriction of an operator to a subspace
4 Functions of operators
4-a Denition and simple properties
4-b An important example: the potential operator
4-c Commutators involving functions of operators
5 Derivative of an operator
5-a Denition
5-b Dierentiation rules
5-c Examples
5-d An application: a useful formula
The aim of this complement is to review a certain number of denitions and useful
properties of linear operators.
1. Trace of an operator
1-a. Denition
The trace of an operator, written Tr, is the sum of its diagonal matrix elements.
When a discrete orthonormal basis,, is chosen for the space, one has, by
denition:
Tr= (1)
For the case of a continuous orthonormal basis, one has:
Tr=d (2)
Whenis an innite-dimensional space, the trace of the operatoris dened only if
expressions (1) and (2) converge.
163

COMPLEMENT B II
1-b. The trace is invariant
The sum of the diagonal elements of the matrix which represents an operatorin
an arbitrary basis does not depend on this basis.
Let us derive this property for the case of a change from one discrete orthonormal
basis to another discrete orthonormal basis. We have:
= (3)
(where we have used the closure relation for thestates). The right-hand side of (3)
is equal to:
= (4)
(since it is possible to change the order of two numbers in a product). We can then replace
in (4) by(closure relation for thestates), and we obtain, nally:
= (5)
We have therefore demonstrated the property of invariance for this case.
Comment:
If the operatoris an observable, Trcan therefore be calculated in a basis of
eigenvectors of. The diagonal matrix elements are then the eigenvaluesof
(degree of degeneracy) and the trace can be written:
Tr= (6)
1-c. Important properties
Tr = Tr (7a)
Tr = Tr = Tr (7b)
In general, the trace of the product of any number of operators is invariant when a cyclic
permutation is performed on these operators.
Let us prove, for example, relation (7a):
Tr = =
= = = Tr (8)
(twice using the closure relation on thebasis). Relation (7a) is thus proved; its
generalization (7b) presents no diculty.
164

REVIEW OF SOME USEFUL PROPERTIES OF LINEAR OPERATORS
2. Commutator algebra
2-a. Denition
The commutator [,] of two operators is, by denition:
[] = (9)
2-b. Properties
[] =[] (10)
[(+)] = [] + [] (11)
[ ] = []+[] (12)
[[]] + [[]] + [[]] = 0 (13)
[]= [ ] (14)
The derivation of these properties is straightforward: it suces to compare both sides of
each equation after having written them out explicitly.
3. Restriction of an operator to a subspace
Letbe the projector onto the-dimensional subspacespanned by theorthonormal
vectors:
=
=1
(15)
By denition, the restriction
^
of the operatorto the subspaceis:
^
= (16)
Ifis an arbitrary ket, it follows from this denition that:
^
=
^
(17)
where:
^
= (18)
is the orthogonal projection ofonto. Consequently, to make
^
act on an arbitrary
ket, one begins by projecting this ket onto; then one lets the operatoract on
this projection, retaining only the projection inof the resulting ket. The operator
^
,
which transforms any ket ofinto a ket belonging to this same subspace, is therefore
an operator whose action has been restricted to.
What can be said about the matrix which represents
^
? Let us choose a basis
whose rstvectors belong to(they are, for example, the), the others
belonging to the supplementary subspace. We have:
^
= (19)
165

COMPLEMENT B II
that is:
^
=
if
0if one of the two indicesoris greater than
(20)
Therefore, the matrix which represents
^
is, as it were, cut out of the one which
represents. One retains only the matrix elements ofassociated with basis vectors
and , both belonging to, the other matrix elements being replaced by zeros.
4. Functions of operators
4-a. Denition and simple properties
Consider an arbitrary linear operator. It is not dicult to dene the operator
: it is the operator which corresponds tosuccessive applications of the operator
. The denition of the operator
1
, the inverse of, is also well known:
1
is the
operator (if it exists) which satises the relations:
1
=
1
= (21)
How can we dene, in a more general way, an arbitrary function of an operator?
To do this, let us consider a functionof a variable. Assume that, in a certain domain,
can be expanded in a power series in:
() =
=0
(22)
By denition, the corresponding function of the operatoris the operator()dened
by a series which has the same coecients:
() =
=0
(23)
For example, the operatoreis dened by:
e=
=0
!
=++
2
2! ++ ! + (24)
We shall not consider the problems concerning the convergence of the series (23), which
depends on the eigenvalues ofand on the radius of convergence of the series (22).
Note that if()is a real function, the coecientsare real. If, moreover,is
Hermitian, we see from (23) that()is Hermitian.
Let be an eigenvector ofwith eigenvalue:
= (25)
Applying the operatortimes in succession, we obtain:
= (26)
166

REVIEW OF SOME USEFUL PROPERTIES OF LINEAR OPERATORS
Now let us apply series (23) to; we obtain:
()=
=0
=() (27)
This leads to the following rule:when is an eigenvector ofwith the eigenvalue,
is also an eigenvector of(),with the eigenvalue().
This property leads to a second denition of a function of an operator. Let us
consider a diagonalizable operator(this is always the case ifis an observable),
and let us choose a basis where the matrix associated withis actually diagonal (its
elements are then the eigenvaluesof).()is, by denition, the operator which is
represented, in this same basis, by the diagonal matrix whose elements are().
For example, if, is the matrix
=
1 0
0 1
(28)
it follows directly that:
e=
e0
0 1e
(29)
Comment:
Attention should be given, when functions of operators are used, with respect to
the order of the operators. For example, the operatorsee,ee, ande
+
are
not, in general, equal whenandare operators and not numbers. Consider:
ee=
!!
=
!!
(30)
ee=
!!
=
!!
(31)
e
+
=
(+)
!
(32)
When andare arbitrary, the right-hand sides of (30), (31) and (32) have no
reason to be equal (see exercise 7 of ComplementII). However,whenand
commute, we have:
[] = 0 =ee= ee= e
+
(33)
(an obvious relation if the diagonal matrices that representeandeare con-
sidered in a basis of eigenvectors common toand).
167

COMPLEMENT B II
4-b. An important example: the potential operator
In one-dimensional problems, we shall often have to consider potential operators
()(so called because they correspond to the classical potential energy()of a
particle placed in a force eld), where()is a function of the position operator.
It follows from the preceding section that()has as eigenvectors the eigenvectors
of, and we have simply:
()=() (34)
The matrix elements of()in the representation are therefore:
()=()( ) (35)
Applying (34) and using the fact that()is Hermitian (the function()is
real), we obtain:
()=() =()() (36)
This equation shows that in therepresentation, the action of the operator()is
simply multiplication by().
The generalization of (34), (35) and (36) to three-dimensional problems can be
performed without diculty; in this case, we obtain:
(R)r=(r)r (37)
r(R)r=(r)(rr) (38)
r(R)=(r)(r) (39)
4-c. Commutators involving functions of operators
Denition (23) shows thatcommutes with every function of:
[()] = 0 (40)
Similarly, ifandcommute, so do()and:
[] = 0 =[()] = 0 (41)
What will be the commutator of an operator with a function of another operator
that does not commute with it? We shall restrict ourselves here to the case of theand
operators, whose commutator is equal to:
[] =~ (42)
Using relation (12), we can calculate:
[
2
] = [ ] = []+[] = 2~ (43)
More generally, let us show that:
[ ] =~
1
(44)
168

REVIEW OF SOME USEFUL PROPERTIES OF LINEAR OPERATORS
If we assume that this equation is veried, we obtain:
[
+1
] = [ ] = []+[ ]
=~+~
1
=~(+ 1) (45)
Relation (44) is therefore established by recurrence.
Now let us calculate the commutator[()]:
[()] =[ ] = ~
1
(46)
If()denotes the derivative of the function(), we recognize in (46) the denition
of the operator(). Therefore:
[()] =~() (47)
An analogous argument would have enabled us to obtain the symmetric relation:
[()] =~() (48)
Comments:
()The preceding argument is based on the fact that()(or()) depends
only on(or on). It is more dicult to calculate a commutator such as
[( )], where( )is an operator which depends on bothand
: the diculties arise from the fact thatanddo not commute.
()Equations (47) and (48) can be generalized to the case of two operators
andwhich both commute with their commutator. An argument modeled
on the preceding one shows that, if we have:
[] = [] = 0 (49)
with:
= [] (50)
then:
[()] = []() (51)
5. Derivative of an operator
5-a. Denition
Let()be an operator which depends on an arbitrary variable. By denition,
the derivative
d
d
of()with respect tois given by the limit (if it exists):
d
d
= lim
0
(+ )()

(52)
169

COMPLEMENT B II
The matrix elements of()in an arbitrary basis of-independent vectorsare
functions of:
= () (53)
Let us call
d
d
=
d
d
the matrix elements of
d
d
. It is easy to verify the
relation:
d
d
=
d
d
(54)
Thus we obtain a very simple rule: to obtain the matrix elements representing
d
d
, all we
must do is take the matrix representingand dierentiate each of its elements (without
changing their places).
5-b. Dierentiation rules
They are analogous to the ones for ordinary functions:
d
d
(+) =
d
d
+
d
d
(55)
d
d
() =
d
d
+
d
d
(56)
Nevertheless, care must be taken not to modify the order of the operators in formula
(56).
Let us prove, for example, the second of these equations. The matrix elements of
are:
= (57)
We have seen that the matrix elements of d()/dare the derivatives with respect to
of those of(). Thus we have, taking the derivative of the right-hand side of (57):
d
d
() =
d
d
+
d
d
=
d
d
+
d
d
(58)
This equation is valid for anyand. Formula (56) is thus established.
5-c. Examples
Let us calculate the derivative of the operatore. By denition, we have:
e=
=0
()
!
(59)
170

REVIEW OF SOME USEFUL PROPERTIES OF LINEAR OPERATORS
Taking the derivative of the series term by term, we obtain:
d
d
e=
=0
1
!
=
=1
()
1
(1)!
=
=1
()
1
(1)!
(60)
We recognize inside the brackets the series that denese(taking as the summation
index= 1). The result is therefore:
d
d
e=e= e (61)
In this simple case involving only one operator, it is unnecessary to pay attention to the
order of the factors:eandcommute.
This is not the case if one is interested in taking the derivative of an operator such
asee. Applying (56) and (61), we obtain:
d
d
(ee) =ee+ ee (62)
The right-hand side of this equation can be transformed intoee+ eeor
ee+ ee, for example. However, we can never obtain (unless, of course,
andcommute) an expression such as(+)ee. In this case, the order of the
operators is therefore important.
Comment:
Even when the function involves only one operator, taking the derivative can-
not always be performed according to the rules valid for ordinary functions. For
example, when()has an arbitrary time-dependence, the derivative
d
d
e
()
is
generally not equal to
d
d
e
()
. It can be seen by expandinge
()
in a power series
in()that()and
d
d
must commute for the equality to hold.
5-d. An application: a useful formula
Consider two operatorsandwhich, by hypothesis, both commute with their
commutator. In this case, we shall derive the relation:
ee= e
+
e
1
2
[]
(63)
(sometimes called Glauber's formula).
171

COMPLEMENT B II
Let us dene the operator(), a function of the real variable, by:
() = ee (64)
We have:
d
d
=ee+ ee= (+ ee)() (65)
Sinceandcommute with their commutator, formula (51) can be applied in order to
calculate:
[e] =[]e (66)
Therefore:
e =e+[]e (67)
Multiply both sides of this equation on the right bye. Substituting the relation so
obtained into (65), we obtain:
d
d
= ++[]() (68)
The operators+and[]commute by hypothesis. We can therefore integrate the
dierential equation (68) as if+and[]were numbers. This yields:
() =(0)e
(+)+
1
2
[]
2
(69)
Setting= 0in (64), we see that(0) =, and we obtain for any time:
() = e
(+)+
1
2
[]
2
(70)
Let us then set= 1; we obtain equation (63), which is thus proven.
Comment:
When the operatorsandare arbitrary, equation (63) is not in general valid:
it is necessary that bothandcommute with[]. This condition may seem
very restrictive. Actually, in quantum mechanics, one often encounters operators
whose commutator is a number: for example,and, or the operatorsand
of the harmonic oscillator (cf.Chap. V).
References:
See the subsections General texts and Linear algebra Hilbert spaces of section
10 of the bibliography.
172

UNITARY OPERATORS
Complement CII
Unitary operators
1 General properties of unitary operators
1-a Denition and simple properties
1-b Unitary operators and change of bases
1-c Unitary matrices
1-d Eigenvalues and eigenvectors of a unitary operator
2 Unitary transformations of operators
3 The innitesimal unitary operator
1. General properties of unitary operators
1-a. Denition and simple properties
By denition, an operatoris unitary if its inverse
1
is equal to its adjoint:
= = (1)
Consider two arbitrary vectors1and 2of, and their transforms
~
1and
~
2under the action of:
~
1= 1
~
2= 2 (2)
Let us calculate the scalar product
~
1
~
2; we obtain:
~
1
~
2= 1 2= 12 (3)
The unitary transformation associated with the operatortherefore conserves the scalar
product (and, consequently, the norm) in. Whenis nite-dimensional, moreover, this
property is characteristic of a unitary operator.
Comments:
()Ifis a Hermitian operator, the operator= eis unitary, since:
= e = e (4)
and therefore:
= ee=
= ee= (5)
(obviously,commutes with).
173

COMPLEMENT C II
()The product of two unitary operators is also unitary. Ifandare unitary,
we have:
= =
= = (6)
Let us now calculate:
()() = = =
()()= = = (7)
These equations indeed show that the product operatoris unitary. This
property, moreover, was foreseeable: when two transformations conserve the
scalar product, so does the successive application of these two transforma-
tions.
()In the ordinary three-dimensional space of real vectors, we are familiar with
operators which conserve the norm and the scalar product: rotations, sym-
metry operations with respect to a point, to a plane, etc. In this case, where
the space is real, these operators are said to be orthogonal. Unitary operators
constitute the generalization of orthogonal operators to complex spaces (with
an arbitrary number of dimensions).
1-b. Unitary operators and change of bases
Let be an orthonormal basis of the state space, assumed to be discrete.
Call~the transform of the vectorunder the action of a unitary operator:
~= (8)
Since the operatoris unitary, we have:
~~= = (9)
The~vectors are therefore orthonormal. Let us show that they constitute a basis of.
To do so, consider an arbitrary vectorof. Since the set constitutes a basis,
the vector can be expanded on the:
= (10)
Applying the operatorto this equation, we obtain:
= (11)
and, therefore:
= ~ (12)
174

UNITARY OPERATORS
This equation expresses the fact that any vectorcan be expanded on the vectors~,
which therefore constitute a basis. Thus we can state the following result: a necessary
condition for an operatorto be unitary is that the vectors of an orthonormal basis of
, transformed by, constitute another orthonormal basis.
. Conversely, let us show that this condition is sucient. By hypothesis, we then
have:
~=
~~=
~~= (13)
and therefore:
~ =~ (14)
Let us calculate:
= ~= ~
= ~~=
= (15)
Relation (15), which is valid for all, expresses the fact that the operatoris the
identity operator. Let us show, in the same way, that=. To do this, consider the
action ofon a vector:
=
= ~ (16)
We then have:
= ~
= ~~
= (17)
We deduce from this that=: the operatoris therefore unitary.
175

COMPLEMENT C II
1-c. Unitary matrices
Let:
= (18)
be the matrix elements of. How can one see from the matrix representingif this
operator is unitary?
Relation (1) gives us:
= (19)
that is:
= (20)
When a matrix is unitary, the sum of the products of the elements of one column and
the complex conjugates of the elements of another column is

Let us cite some examples in which this rule can be easily veried.
Examples:
()The matrix which represents a rotation through an angleabout, in ordinary
three-dimensional space:
() =
cossin0
sincos0
0 0 1
(21)
()The rotation matrix in the state space of a spin
1
2
particle (cf.Chap.):
(12)
( ) =
e
2
(+)
cos2
e
2
( )
sin2
e
2
( )
sin2
e
2
(+)
cos2
(22)
1-d. Eigenvalues and eigenvectors of a unitary operator
Let be a normalized eigenvector of the unitary operatorwith eigenvalue:
= (23)
The square of the norm of the vectoris:
= = (24)
176

UNITARY OPERATORS
Since the unitary operator conserves the norm, we have, necessarily,= 1. The
eigenvalues of a unitary operator must therefore be complex numbers of modulus 1:
= e where is real (25)
Consider two eigenvectorsand of; we then have:
= =
= e
( )
(26)
When the eigenvaluesandare dierent, we see from (26) that the scalar product
must be zero: two eigenvectors of a unitary operator corresponding to dierent
eigenvalues are orthogonal.
2. Unitary transformations of operators
We saw in 1-b that a unitary operatorpermits the construction, starting with one
orthonormal basis of, of another one,~. In this section, we are going to
dene a transformation that acts, not on the vectors, but on the operators.
By denition, the transform
~
of the operatorwill be the operator which, in the
~basis, has the same matrix elements as the operatorin the basis:
~
~
~= (27)
Substituting (8) into this equation, we obtain:
~
= (28)
Sinceandare arbitrary, we have:
~
= (29)
or, multiplying this equation on the left byand on the right by:
~
= (30)
Equation (30) can be taken to be the denition of the transform
~
of the operator
by the unitary transformation. In quantum mechanics, such transformations are often
used: a rst example is given in ComplementIIof this chapter ().
How can the eigenvectors of
~
be obtained from those of? Let us consider an
eigenvectorof, with an eigenvalue:
= (31)
Let~be the transform ofby the operator:~= . We then have:
~
~= ( ) = ()
= =
=~ (32)
177

COMPLEMENT C II
~is therefore an eigenvector of
~
, with eigenvalue. This can be generalized to the
following rule: the eigenvectors of the transform
~
ofare the transforms~of the
eigenvectorsof; the eigenvalues are unchanged.
Comments:
()The adjoint of the transform
~
ofbyis the transform ofby:
(
~
)= ( )= =
~
(33)
In particular, it follows from this relation that, ifis Hermitian,
~
is also.
()Similarly, we have:
(
~
)
2
= = =
~2
and, in general:
(
~
)=
~
(34)
Using denition (23) of ComplementII, we can easily show that:
~
() =(
~
) (35)
where()is a function of the operator.
3. The innitesimal unitary operator
Let()be a unitary operator which depends on an innitely small real quantity; by
hypothesis,() when 0. Expand()in a power series in:
() =+ + (36)
We then have:
() =+ + (37)
and:
()() =()() =+(+) + (38)
Since()is unitary, the rst-order terms inon the right-hand side of (38) are zero;
we therefore have:
+= 0 (39)
This relation expresses the fact that the operatoris anti-Hermitian. It is convenient
to set:
= (40)
178

UNITARY OPERATORS
so as to obtain the equation:
= 0 (41)
which states thatis Hermitian. An innitesimal unitary operator can therefore be
written in the form:
() = (42)
whereis a Hermitian operator.
Substituting (42) into (30), we obtain:
~
= ( )(+ ) = ( )(+ ) (43)
and, therefore:
~
= [] (44)
The variation of the operatorunder the transformationis, to rst order in, pro-
portional to the commutator[].
179

A MORE DETAILED STUDY OF THE R AND P REPRESENTATIONS
Complement DII
A more detailed study of therandprepresentations
1 The rrepresentation
1-a TheRoperator and functions ofR. . . . . . . . . . . . . .
1-b ThePoperator and functions ofP. . . . . . . . . . . . . . .
1-c The Schrödinger equation in therrepresentation
2 The prepresentation
2-a ThePoperator and functions ofP. . . . . . . . . . . . . . .
2-b TheRoperator and functions ofR. . . . . . . . . . . . . .
2-c The Schrödinger equation in theprepresentation
1. The rrepresentation
1-a. The Roperator and functions ofR
Let us calculate the matrix elements, in therrepresentation, of the,,
operators. Using formula (E-2-c) of Chapter
ketsr, we immediately obtain:
rr=(rr)
rr=(rr)
rr=(rr) (1)
These three equations can be condensed into one:
rRr=r(rr) (2)
The matrix elements, in therrepresentation, of a function(R)are also very
simple [cf.equation (27) of ComplementII]:
r(R)r=(r)(rr) (3)
181

COMPLEMENT D II
1-b. The Poperator and functions ofP
Let us calculate the matrix elementrr:
rr=d
3
rppr
=d
3
rppr
= (2~)
3
d
3
e
~
p(rr)
=
1
2~
+
d e
~
( )
1
2~
+
de
~
( )
1
2~
+
de
~
( )
(4)
From this, it follows that, using the integral form of the delta function and its derivative
[cf.Appendix, equations (34) and (53)]:
rr=
~
( )( )( ) (5)
The matrix elements of the other components ofPcould be obtained in an analogous
fashion.
Let us verify that the action ofin therrepresentation can indeed be derived
from formula (5). To do so, let us calculate:
r =d
3
rrr (6)
From (5):
r =
~
( )d ( ) d ( )( ) d (7)
Using the relation (48) of Appendix:
()() d= ()() d=(0) (8)
and taking= , we obtain:
r =
~
( ) (9)
which is indeed equation (E-26) of Chapter.
182

A MORE DETAILED STUDY OF THE R AND P REPRESENTATIONS
What is the value of the matrix elementr(P)rof a function(P)of theP
operator? An analogous calculation gives us:
r(P)r=d
3
r(P)ppr
= (2~)
3
d
3
(p) e
p(rr)~
= (2~)
32~
(rr) (10)
where
~
(r)is the inverse Fourier transform of the function(p):
~
(r) = (2~)
32
d
3
e
~
pr
(p) (11)
1-c. The Schrödinger equation in the rrepresentation
In Chapter, we shall introduce the Schrödinger equation, which is of fundamen-
tal importance in quantum mechanics:
~
d
d
()= () (12)
whereis the Hamiltonian operator, which we shall dene in that chapter. For a
(spinless) particle in a scalar potential(r)[cf.equation (B-42) of Chapter]:
=
1
2
P
2
+(R) (13)
Let us write this equation in therrepresentation, that is, using the wave
function(r), dened by:
(r) =r() (14)
Projecting (12) ontor, in the case whereis given by formula (13), we obtain:
~
r()=
1
2
rP
2
()+r(R)() (15)
The quantities involved in this equation can be expressed in terms of(r), since:
r()=(r) (16)
r(R)()=(r)(r) (17)
The matrix elementrP
2
can be calculated by using the fact thatPacts like
~
rin
therrepresentation:
rP
2
()=r(
2
+
2
+
2
)()
=~
2
2
2
+
2
2
+
2
2
( )
=~
2
(r) (18)
183

COMPLEMENT D II
The Schrödinger equation then becomes:
~
(r) =
~
2
2
+(r)(r)
(19)
This is indeed the wave equation introduced in Chapter).
2. The prepresentation
2-a. The Poperator and functions ofP
We obtain without diculty formulas analogous to (2) and (3):
pPp=p(pp) (20)
p(P)p=(p)(pp) (21)
2-b. The Roperator and functions ofR
Arguments analogous to those of 5) and
(10):
pp=~( )( )( ) (22)
and:
p(R)p= (2~)
32
(pp) (23)
with:

(p) = (2~)
32
d
3
e
~
pr
(r) (24)
2-c. The Schrödinger equation in the prepresentation
Let us introduce the wave function in theprepresentation by:
(p) =p() (25)
Using (12), we shall look for the equation giving the time evolution of
(p). Projecting
(12) onto the ketp, we obtain:
~
p()=
1
2
pP
2
()+p(R)() (26)
Now we have:
p()=
(p) (27)
pP
2
()=p
2
(p) (28)
184

A MORE DETAILED STUDY OF THE R AND P REPRESENTATIONS
The quantity which remains to be calculated is:
p(R)()=d
3
p(R)pp() (29)
Using (23), we nd:
p(R)()= (2~)
32
d
3
(pp)(p) (30)
where
(p)is the Fourier transform of(r):(p) = (2~)
32
d
3
e
~
pr
(r) (31)
The Schrödinger equation in theprepresentation is therefore written:
~
(p) =
p
2
2(p) + (2~)
32
d
3
(pp)(p)
(32)
Comment:
Since
(p)is the Fourier transform of(r)[cf.formula (E-18) of Chapter],
it would have been possible to nd equation (32) by taking the Fourier transforms
of both sides of equation (19).
185

TWO OBSERVABLES, AND , WHOSE COMMUTATOR IS EQUAL TO ~
Complement EII
Some general properties of two observables,and, whose
commutator is equal to~
1 The operator (): denition, properties
2 Eigenvalues and eigenvectors of . . . . . . . . . . . . . . .
2-a Spectrum of . . . . . . . . . . . . . . . . . . . . . . . . . .
2-b Degree of degeneracy
2-c Eigenvectors
3 The qrepresentation
3-a The action ofQin theqrepresentation
3-b The action of()in theqrepresentation; the translation
operator
3-c The action ofPin theqrepresentation
4 The prepresentation. The symmetric nature of the P
andQobservables
In quantum mechanics, one often encounters two operators whose commutator is
equal to~. This is the case, for example, when these two operators correspond to the two
classical conjugate quantitiesand(, the coordinate in a system of orthonormal
axes, and the conjugate momentum=
cf. ). In quantum
mechanics, one associates withandoperatorsandthat satisfy the relation:
[ ] =~ (1)
In , we encountered such operators:and. In this comple-
ment, we shall take a more general point of view and show that it is possible to establish
a whole series of important properties relative to two observablesandwhose com-
mutator is equal to~. All these properties are just consequences of the commutation
relation (1).
1. The operator (): denition, properties
We shall consider two observablesand, satisfying the relation:
[] =~ (2)
and we shall dene the operator(), which depends on the real parameter, by:
() = e
~
(3)
This operator is unitary; it is easy to verify the relations:
() =
1
() =() (4)
187

COMPLEMENT E II
Let us calculate the commutator[()]. We can apply formula (51) of Comple-
mentII, since[] =~commutes withand:
[()] =~
~
e
~
= () (5)
This relation can also be written:
() =()[+] (6)
Finally, note that:
()() =(+) (7)
2. Eigenvalues and eigenvectors of
2-a. Spectrum of
Assume thathas a non-zero eigenvector, with eigenvalue:
= (8)
Apply equation (6) to the vector. This yields:
()=()(+)
=()(+)= (+)() (9)
This equation expresses the fact that()is another non-zero eigenvector of,
with an eigenvalue of(+)(()is non-zero because()is unitary). Thus, starting
with an eigenvector of, one can, by applying(), construct another eigenvector of,
with any real eigenvalue (can indeed take on any real value). The spectrum ofis
therefore a continuous spectrum, composed of all possible values on the real axis
1
.
2-b. Degree of degeneracy
From now on, we shall assume, for simplicity, that the eigenvalueofis non-
degenerate (the results which we shall derive can be generalized to the case whereis
degenerate). Let us show that ifis non-degenerate, all the other eigenvalues ofare
also non-degenerate. Let us assume, for example, that the eigenvalue+is two-fold
degenerate, and we shall show that we arrive at a contradiction. There would then exist
two orthogonal eigenvectors,+ and+ , corresponding to the eigenvalue
+:
+ + = 0 (10)
1
This shows that in a spaceof nite dimension, there are no observablesand, whose
commutator is equal to~. The number of eigenvalues ofcould not be simultaneously less than or
equal toand innite.
This result can be derived directly, moreover, by taking the trace of relation (2):Tr Tr =
Tr~. When is nite, the traces on the left-hand side of this equation exist: they are nite and equal
numbers [cf.ComplementII, formula (7a)]. The equation becomes0 = Tr~= ~, which is
impossible.
188

TWO OBSERVABLES, AND , WHOSE COMMUTATOR IS EQUAL TO ~
Consider the two vectors()+ and()+ . They are, according to
(9), two eigenvectors of, with an eigenvalue of+ =. They are not collinear,
since they are orthogonal; their scalar product can be written, using the fact that()
is unitary:
+ ()()+ =+ + = 0 (11)
We reach the conclusion thatis at least two-fold degenerate, which is contrary to the
initial hypothesis. Consequently, all the eigenvalues ofmust have the same degree of
degeneracy.
2-c. Eigenvectors
We shall x the relative phases of the dierent eigenvectors ofwith respect to
the eigenvector0, of eigenvalue 0, by setting:
=()0 (12)
Applying()to both sides of (12) and using (7), we obtain:
()=()()0=(+)0=+ (13)
The adjoint expression of (13) is written:
() =+ (14)
or, using (4) and replacingby:
() = (15)
3. The qrepresentation
Sinceis an observable, the set of its eigenvectorsconstitutes a basis of. It is
possible to characterize each ket by its wave function in therepresentation:
() = (16)
3-a. The action of Qin theqrepresentation
Let us calculate, in therepresentation, the wave function associated with the
ket . It is written:
= =() (17)
[using (8) and the fact thatis Hermitian]. The action ofin the representation
is therefore simply a multiplication by.
189

COMPLEMENT E II
3-b. The action of ()in theqrepresentation; the translation operator
The wave function in the representation associated with the ket()is
written [formula (15)]:
()= =( ) (18)
The action of the operator()in the representation is therefore a translation of
the wave function over a distanceparallel to the-axis
2
. For this reason,()is called
thetranslation operator.
3-c. The action of Pin theqrepresentation
Whenis an innitely small quantity, we have:
() = e
~
=+
~
+(
2
) (19)
Consequently:
()=() +
~
+(
2
) (20)
On the other hand, equation (18) yields:
()=(+) (21)
Comparison of (20) and (21) shows that:
(+) =() +
~
+(
2
) (22)
It follows that:
=
~
Lim 0
(+)()
=
~
d
d
() (23)
The action ofin the representation is therefore that of
~
d
d
. Equation (E-26) of
Chapter
4. The prepresentation. The symmetric nature of thePandQobservables
Relation (23) enables us to obtain easily the wave function()associated, in the
representation, with the eigenvectorofwith an eigenvalue of:
() = = (2~)
12
e
~ (24)2
The function( )is the function which, at the point=0+, takes on the value(0). It
is therefore the function obtained from()by a translation of+.
190

TWO OBSERVABLES, AND , WHOSE COMMUTATOR IS EQUAL TO ~
We can therefore write:
= (2~)
12
+
de
~ (25)
A ketcan be dened by its wave function in therepresentation:
() = (26)
Using the adjoint relation of (25), we obtain:
() = (2~)
12
+
de
~() (27)()is therefore the Fourier transform of().
The action of theoperator in the representation corresponds to a multi-
plication by; that of theoperator corresponds, as can easily be shown using (27), to
the operation~
d
d
.
Thus we obtain symmetrical results in theand representations. This
is not surprising: in our hypotheses, it is possible to exchange theandoperators,
simply changing the sign of the commutator in relation (2). Instead of introducing the
operator(), we could therefore have considered()dened by:
() = e
~
(28)
and we could have developed the same arguments, replacingbyandbyevery-
where.
References:
Messiah (1.17), Vol. I, VIII-6; Dirac (1.13), 25; Merzbacher (1.16), Chap. 14,
7.
191

THE PARITY OPERATOR
Complement FII
The parity operator
1 The parity operator
1-a Denition
1-b Simple properties of. . . . . . . . . . . . . . . . . . . . . .
1-c Eigensubspaces of. . . . . . . . . . . . . . . . . . . . . . .
2 Even and odd operators
2-a Denitions
2-b Selection rules
2-c Examples
2-d Functions of operators
3 Eigenstates of an even observable B+. . . . . . . . . . . . .
4 Application to an important special case
1. The parity operator
1-a. Denition
Consider a physical system whose state space isr. The parity operatoris
dened by its action on the basis vectors
1
rofr:
r= r (1)
The matrix elements ofare therefore, in therrepresentation:
rr=rr=(r+r) (2)
Consider an arbitrary vectorofr:
=d
3
(r)r (3)
If the variable changer=ris performed,can be written:
=d
3
(r)r (4)
Now calculate; we obtain:
=d
3
(r)r (5)
1
Care must be taken not to confuse r0andr0. The former is an eigenvector ofR, with
eigenvaluer0and wavefunction r0
(r) =(r+r0). The latter is an eigenvector ofRwith eigenvalue
r0and wavefunction r0
(r) =(rr0).
193

COMPLEMENT F II
Comparison of (3) and (5) shows that the action ofin therrepresentation is to
changertor:
r=(r) (6)
Now let us consider a physical systemSwhose state vector is;describes
the physical system obtained fromSby reection through the origin of the axes.
1-b. Simple properties of
The operator
2
is the identity operator. From (1) we have:

2
r= (r) = r=r (7)
that is, since the ketsrform a basis ofr:

2
= (8a)
or:
=
1
(8b)
It is easy to show by recurrence that the operatoris:
equal towhenis even
equal towhenis odd
We can rewrite (6) in the form:
r=r (9)
Since this equation is valid for all, it can be deduced that:
r =r (10)
Moreover, the Hermitian conjugate expression of (1) is written:
r=r (11)
Since the ketsrform a basis, it follows from (10) and (11) thatis Hermitian:
= (12)
Combining this equation with (8b), we obtain:

1
= (13)
is therefore unitary as well.
194

THE PARITY OPERATOR
1-c. Eigensubspaces of
Let be an eigenvector of, with an eigenvalue of. Applying (8a), we
obtain:
=
2
=
2
(14)
We therefore have
2
= 1: the eigenvalues ofare limited to 1 and1. Since the space
ris innite-dimensional, we immediately see that these eigenvalues are degenerate. An
eigenvector ofwith the eigenvalue+1 will be said to be even; an eigenvector with the
eigenvalue1, odd.
Consider the two operators+and dened by:
+=
1
2
(+ )
=
1
2
() (15)
These operators are Hermitian; using (8a), it is easy to show that:
2
+=+
2
= (16)
+and are thus the projectors onto two subspaces ofr, which we shall call+and
. Let us calculate the products+and +; we obtain:
+=
1
4
(+
2
) = 0
+=
1
4
( +
2
) = 0 (17)
The two subspaces+and are therefore orthogonal. Let us show that they are also
supplementary. We see immediately from denition (15) that:
++ = (18)
For all , we have, therefore:
= (++)= ++ (19)
with:
+=+
= (20)
Let us calculate the products+and; we obtain:
+=
1
2
(+ ) =
1
2
( +) =+
=
1
2
() =
1
2
() = (21)
195

COMPLEMENT F II
These equations enable us to show that the vectors+and introduced in (20) are
even and odd, respectively:
+= +=+= +
= = = (22)
The spaces+and are therefore the eigensubspaces of, with the eigenvalues+1
and1. In therrepresentation, equations (22) can be written:
r+=+(r) =r+=+(r)
r =(r) =r = (r) (23)
The wave functions+(r)and(r)are even and odd, respectively.
Relation (19) expresses the fact that any ketofrcan be decomposed into a
sum of two eigenvectors of,+and , belonging respectively to the even subspace
+and the odd subspace. Therefore,is an observable.
2. Even and odd operators
2-a. Denitions
In II, we dened the concept of a unitary transformation of
operators. In the case of[which is indeed unitary; see (13)], the transformed operator
of an arbitrary operatoris written:
~
= (24)
and satises the relation [cf.equation (27) of ComplementII]:
r
~
r=r r (25)
The operator
~
is said to be the parity transform of.
In particular, if:
~
= +the operatoris said to be even
if:
~
= the operatoris said to be odd.
An even operator+therefore satises:
+= + (26)
or, multiplying this equation on the left byand using (8a):
+=+ (27)
[+] = 0 (28)
An even operator is therefore an operator that commutes with. It can be seen, similarly,
that an odd operatoris an operator that anticommutes with:
+ = 0 (29)
196

THE PARITY OPERATOR
2-b. Selection rules
Let+be an even operator. Let us calculate the matrix element+; by
hypothesis, we have:
+=+= + (30)
with:
=
= (31)
If one of the two kets,and, is even and the other odd (= , = ),
relation (30) yields:
+= += 0 (32)
Hence the rule: the matrix elements of an even operator are zero between vectors of
opposite parity.
If, now,is odd, relation (30) becomes:
= (33)
which is zero whenand are both either even or odd. Hence the rule: the matrix
elements of an odd operator are zero between vectors of the same parity. In particular, the
diagonal matrix element (the mean value ofin the state;cf.Chapter,
) is zero ifhas a denite parity.
2-c. Examples
. The ,,operators
In this case, we have:
r= =
= = r (34)
and:
r= r=
= = r (35)
Adding these two equations together, we obtain:
(+)r= 0 (36)
or, since the vectorsrform a basis:
+ = 0 (37)
is therefore odd.
The proofs are the same forand;Ris therefore an odd operator.
197

COMPLEMENT F II
. The ,,operators
Let us calculate the ketp; we obtain:
p= (2~)
32
d
3
e
pr~
r
= (2~)
32
d
3
e
pr~
r
= (2~)
32
d
3
e
pr~
r
= p (38)
We then have, using an argument analogous to the one developed in:
p= p
p= p (39)
and:
+ = 0 (40)
ThePoperator is odd.
. The parity operator
obviously commutes with itself; it is an even operator.
2-d. Functions of operators
Let+be an even operator. Using relation (8a), we obtain:

+ =
(+)(+)(+)
=
+
factors
(41)
An even operator raised to theth power is even. Hence, any operator(+)is even.
Let be an odd operator; let us calculate the operator:
=
()()()
= (1)()
factors
(42)
An odd operator raised to theth power is even ifis even, odd ifis odd. Consider
an operator(); this operator is even if the corresponding function()is even, odd
if it is odd. In general,()has no denite parity.
198

THE PARITY OPERATOR
3. Eigenstates of an even observableB+
Let us consider an arbitrary even observable+and an eigenvectorof+with an
eigenvalue. Since+is even, it commutes with. Applying the theorems of
of Chapter, we obtain the following results:
. Ifis a non-degenerate eigenvalue,is necessarily an eigenvector of; it
is therefore either an even or an odd vector. The mean value of any odd
observable, such asR,P, etc..., is then zero.
Ifis a degenerate eigenvalue corresponding to the eigensubspace, the vectors
ofdo not all necessarily have a denite parity. may be a vector which is non-
collinear with; it is nevertheless a vector which has the same eigenvalue. Moreover,
it is possible to nd a basis of eigenvectors common toand+in every subspace.
4. Application to an important special case
We shall often need to nd the eigenstates of a Hamiltonian operator, acting inr, of
the form:
=
P
2
2
+(R) (43)
Since thePoperator is odd, theP
2
operator is even. When, in addition, the
function(r)is even ((r) =(r)), the operatoris even. According to what we
have just seen, it is then possible to look for the eigenstates ofamong the even or odd
states. This often simplies the calculations considerably.
We have already encountered a certain number of cases where the Hamiltonian
is even: the square well, the innite well (cf.ComplementI). We shall study others:
the harmonic oscillator, the hydrogen atom, etc... It is easy to verify in all these special
cases the properties which we have derived.
Comment:
Ifis even, and if one of its eigenstateswhich has no denite parity (i.e.
is non-collinear to) has been found, it can be asserted that the corre-
sponding eigenvalue is degenerate: sincecommutes with, is an eigen-
vector ofwith the same eigenvalue as.
References and suggestions for further reading:
Schi (1.18), 29; Roman (2.3), 5-3 d; Feynman I (6.3), Chap. 52; Sakurai (2.7),
Chap. 3; articles by Morrison (2.28), Feinberg and Goldhaber (2.29), Wigner (2.30).
199

PROPERTIES OF THE TENSOR PRODUCT AND THE TWO-DIMENSIONAL INFINITE WELL
Complement GII
An application of the properties of the tensor product: the
two-dimensional innite well
1 Denition; eigenstates
2 Study of the energy levels
2-a Ground state
2-b First excited states
2-c Systematic and accidental degeneracies
In ComplementI() we have already studied, in a one-dimensional problem,
the stationary states of a particle placed in an innite potential well. By using the concept
of a tensor product (cf.Chap., ), we shall be able to generalize this discussion to
the case of a two-dimensional innite well (the introduction of a third dimension would
not involve any additional theoretical diculty).
1. Denition; eigenstates
We shall consider a particle of mass, restricted to a plane, inside a square box
of edge: its potential energy()becomes innite when one of its coordinatesor
leaves the interval[0]:
() =() +() (1)
with:
() = 0 if 0
= + if 0 or (2)
The Hamiltonian of the quantum particle is then (Chap., ):
=
1
2
(
2
+
2
) +() +() (3)
which can be written:
= + (4)
with:
=
1
2
2
+()
=
1
2
2
+() (5)
201

COMPLEMENT G II
We thus nd ourselves in the important special case mentioned in Chapter ),
and we can consider the eigenstates ofin the form:
= (6)
with:
= ;
= ; (7)
We then have:
=
with:
= + (8)
We have therefore reduced a two-dimensional problem to a one-dimensional prob-
lem, which, moreover, has already been solved (cf.ComplementI). Applying the
results of this complement, and formulas (7) and (8), we therefore see that:
the eigenvalues ofare of the form:
=
1
2
2
(
2
+
2
)
2
~
2
(9)
whereandare positive integers.
to these energies correspond eigenstates which can be written in the form
of tensor products:
= (10)
whose normalized wave function is:
() =()()
=
2
sinsin (11)
It is easy to verify that these wave functions vanish at the edges of the square box (
or= 0or), where the potential energy becomes innite.
2. Study of the energy levels
2-a. Ground state
andare strictly positive integers
1
. The ground state is therefore obtained when
= 1,= 1. Its energy is:
11=
2
~
2
2
(12)
This value is attained only for== 1. The ground state is therefore not degenerate.
1
The values= 0or= 0are excluded as they give null wave functions (therefore impossible to
normalize).
202

PROPERTIES OF THE TENSOR PRODUCT AND THE TWO-DIMENSIONAL INFINITE WELL
2-b. First excited states
The rst excited state is obtained either for= 1and= 2or for= 2and
= 1. Its energy is:
12=21=
5
2
2
~
2
2
(13)
This state is two-fold degenerate, since12and21are independent.
The second excited state corresponds to== 2; it is not degenerate, and its
energy is:
22= 4
2
~
2
2
(14)
The third excited state corresponds to= 1,= 3and= 3,= 1, etc.
2-c. Systematic and accidental degeneracies
The general observation can be made that all levels for which=are degenerate, since:
= (15)
This degeneracy is related to a symmetry of the problem. The square well under consid-
eration is symmetric with respect to the rst bisectrix of theplane. This is expressed by
the fact that the Hamiltonianis invariant under the exchange
2
:
(16)
If an eigenstate ofis known whose wave function is(), the state which corresponds to
() = ()is also an eigenstate ofwith the same eigenvalue. Consequently, if the
function()is not symmetric with respect toand, the eigenvalue associated with it is
necessarily degenerate. This is the origin of the degeneracy (15): for=,()is not
symmetric with respect toand[formula (11)]. This interpretation is corroborated by the
fact that if the symmetry is destroyed by choosing a well whose widths alongand along
are dierent (being equal toandrespectively), the corresponding degeneracy disappears,
and formula (9) becomes:
=
2
~
2
2
2
2
+
2
2
(17)
which implies:
= (18)
Such degeneracies, whose origin lies in a symmetry of the problem, are calledsystematic
degeneracies.
2
In the state space, an operator could be dened to correspond to a reection about the rst bisectrix.
It could then be shown that, in the present case, this operator commutes with.
203

COMPLEMENT G II
Comment:
The other symmetries of the two-dimensional square well do not create systematic degen-
eracies because the eigenstates ofare all invariant with respect to them. For example,
for arbitraryand,()is simply multiplied by a phase factor ifis replaced
by( )andby( )(symmetry with respect to the center of the well).
Degeneracies may also arise which are not directly related to the symmetry of the prob-
lem. They are calledaccidental degeneracies. For example, in the case which we have
discussed, it so happens that55= 71and74= 81
204

EXERCISES
Complement HII
Exercises
Dirac notation. Commutators. Eigenvectors and eigenvalues
1. are the eigenstates of a Hermitian operator(is, for example, the Hamiltonian
of some physical system). Assume that the statesform a discrete orthonormal basis.
The operator()is dened by:
() =
Calculate the adjoint()of().
Calculate the commutator[ ()].
Prove the relation:
()() = ()
Calculate Tr(), the trace of the operator()
Letbe an operator, with matrix elements= . Prove the relation:
= ()
Show that = Tr ().
2.In a two-dimensional vector space, consider the operator whose matrix, in an
orthonormal basis12, is written:
=
0
0
IsHermitian? Calculate its eigenvalues and eigenvectors (giving their normalized
expansion in terms of the12basis).
Calculate the matrices that represent the projectors onto these eigenvectors. Then
verify that they satisfy the orthogonality and closure relations.
Same questions for the matrices:
=
2
22 3
and, in a three-dimensional space
=
~
2
0
2 02 02
0
2 0
205

COMPLEMENT H II
3.The state space of a certain physical system is three-dimensional. Let1 2 3
be an orthonormal basis of this space. The kets0and1are dened by:
0=
1
2
1+
2
2+
1
2
3
1=
1
3
1+
3
3
Are these kets normalized?
Calculate the matrices0and1representing, in the1 2 3basis, the
projection operators onto the state0and onto the state1. Verify that these
matrices are Hermitian.
4.Letbe the operator dened by= , where and are two
vectors of the state space.
Under what condition isHermitian?
Calculate
2
. Under what condition isa projector ?
Show thatcan always be written in the form= 12whereis a constant
to be calculated and1and2are projectors.
5.Let1be the orthogonal projector onto the subspace1,2the orthogonal
projector onto the subspace2. Show that, for the product12to be an orthogonal
projector as well, it is necessary and sucient that1and2commute. In this case,
what is the subspace onto which12projects?
6.Thematrix is dened by:
=
0 1
1 0
Prove the relation:
e=cos+ sin
whereis the22unit matrix.
7.Establish, for thematrix given in exercise 2, a relation analogous to the one
proved forin the preceding exercise. Generalize for all matrices of the form:
= +
with:
2
+
2
= 1
206

EXERCISES
Calculate the matrices representinge
2
,(e)
2
ande
(+)
. Ise
2
equal to
(e)
2
?e
(+)
toee?
8.Consider the Hamiltonianof a particle in a one-dimensional problem, dened
by:
=
1
2
2
+()
whereandare the operators dened in , which satisfy the relation:
[ ] =~. The eigenvectors ofare denoted by: = , whereis a
discrete index.
Show that:
=
whereis a coecient which depends on the dierence betweenand .
Calculate(hint: consider the commutator[ ]).
From this, deduce, using the closure relation, the equation:
( )
2 2
=
~
2
2
2
9.Letbe the Hamiltonian operator of a physical system. Denote bythe
eigenvectors of, with eigenvalues:
=
For an arbitrary operator, prove the relation:
[ ]= 0
Consider a one-dimensional problem, where the physical system is a particle of
masswith potential energy(). In this case,is written:
=
1
2
2
+()
In terms of,and(), nd the commutators:[ ],[ ]and[ ].
Show that the matrix element (which we shall interpret in Chapter
as the mean value of the momentum in the state) is zero.
Establish a relation between=
2
2
(the mean value of the kinetic
energy in the state) and
d
d
. Since the mean value of the potential
energy in the stateis (), how is it related to the mean value of the
kinetic energy when:
() =0
(= 246;00)?
207

COMPLEMENT H II
10.Using the relation= (2~)
12
e
~
, nd the expressions of
and in terms of(). Can these results be found directly by using the fact
that in the representation,acts like
~
d
d
?
Complete sets of commuting observables, C.S.C.O.
11.Consider a physical system whose three-dimensional state space is spanned by the
orthonormal basis formed by the three kets1,2,3. In the basis of these three
vectors, taken in this order, the two operatorsandare dened by:
=~0
1 0 0
01 0
0 01
=
1 0 0
0 0 1
0 1 0
where0andare real constants.
AreandHermitian?
Show thatandcommute. Give a basis of eigenvectors common toand.
Of the sets of operators:,, ,
2
, which form a C.S.C.O.?
12.In the same state space as that of the preceding exercise, consider two operators
anddened by:
1= 1 2= 0 3= 3
1= 3 2= 2 3= 1
Write the matrices that represent, in the1 2 3basis, the operators,
2
,,
2
. Are these operators observables?
Give the form of the most general matrix which represents an operator which
commutes with. Same question for
2
, then for
2
.
Do
2
andform a C.S.C.O.? Give a basis of common eigenvectors.
Solution of exercise 11
andare Hermitian because the matrices which correspond to them are sym-
metric and real.
1is an eigenvector common toand. We therefore have 1= 1.
We see, then, that forandto commute, it is sucient that the restrictions of
these operators to the subspace2, spanned by2and3, commute. Now, in
this subspace, the matrix representingis equal to~0(whereis the22
unit matrix), which commutes with all22matrices.andtherefore commute
208

EXERCISES
(this result could, of course, be obtained by calculating directly the matrices
and ). The restriction ofto2is written:
2 2=
0 1
1 0
The normalized eigenvectors of this22matrix are easy to obtain; they are:
2=
1
2
[2+3](eigenvalue +)
3=
1
2
[2 3](eigenvalue )
These vectors are automatically eigenvectors ofsince2is the eigensubspace of
corresponding to the eigenvalue~0. To summarize, the eigenvectors common
toandare given by:
eigenvalue ofeigenvalue of
1= 1
2=
1
2
[2+3]
3=
1
2
[2 3]
~0
~0
~0
These vectors are the only (to within, of course, a phase factor) normalized eigen-
vectors common toand.
It can be seen from the above table thathas a two-fold degenerate eigenvalue; it
is therefore not a C.S.C.O. Similarly,also has a two-fold degenerate eigenvalue
and is therefore not a C.S.C.O.: an eigenvector ofwith the eigenvaluecan be
1, or2, or
1
3
1+
1
3
2+
1
3
3, for example. On the other hand, the
set of the two operatorsanddoes constitute a C.S.C.O. We see from the above
table that no two vectorshave the same eigenvalues for bothand. This
is why, as has already been pointed out, the system of normalized eigenvectors
common to andis unique (to within phase factors). Note that within the
eigensubspace2ofassociated with the eigenvalue~0, the eigenvalues of
are distinct (and). Similarly, in the eigensubspace ofspanned by1and
2, the eigenvalues ofare distinct (~0and~0).
2
has three eigenvectors with the eigenvalue~
22
0,1,2and3. It is easy
to see that
2
anddo not constitute a C.S.C.O., since two linearly independent
eigenvectors1and2correspond to the pair of eigenvalues~
22
0.
209

COMPLEMENT H II
Solution of exercise 12
Let us use the rule for constructing the matrix of an operator: in theth column
of the matrix, write the components of the operator transform of theth basis
vector. We obtain easily:
=
1 0 0
0 0 0
0 01
=
0 0 1
0 1 0
1 0 0
2
=
1 0 0
0 0 0
0 0 1
2
=
1 0 0
0 1 0
0 0 1
These matrices are symmetric and real, and therefore Hermitian. Since the space
is nite-dimensional, they can be diagonalized and therefore represent observables.
Letbe an operator that commutes with.cannot (cf.Chap.,
a) have any matrix elements between1and2, or between2and3, or
between1and3(eigenvectors ofwith dierent eigenvalues). The matrix
which representsis therefore necessarily diagonal, that is, of the form:
[ ] = 0 =
110 0
0 220
0 0 33
Letbe an operator that commutes with
2
. The matrix representingcan
have elements between1and3(eigenvectors of
2
with the same eigenvalue),
but none between2and1or3.is therefore written:
[
2
] = 0 =
110 13
0 220
310 33
It is therefore less restrictive to impose the condition that an operator commute
with
2
than with:is not necessarily a diagonal matrix. It can only be said
thatdoes not mix the vectors of the subspace2spanned by1and3with
those of the one-dimensional subspace spanned by2. This property, moreover,
appears very clearly if the matrixwhich represents the operatoris written
in the1 3 2basis (changing the order of the basis vectors):
=
11130
31330
0 0 22
Finally, since
2
is the identity operator, any33matrix commutes with
2
, and
its most general form is:
[
2
] = 0 =
111213
212223
313233
210

EXERCISES
2is an eigenvector common to
2
and. In the subspace2spanned by1
and3,
2
andare written:
2
2
2
=
1 0
0 1
2 2
=
0 1
1 0
The eigenvectors of the latter matrix are:
2=
1
2
[1+3]
3=
1
2
[1 3]
and the basis of eigenvectors common to
2
andis:
vector eigenvalue of
2
eigenvalue of
1= 2
2=
1
2
[1+3]
3=
1
2
[1 3]
0
1
1
1
1
1
No two lines are alike in the table of eigenvalues of
2
and: these two operators
therefore form a C.S.C.O. (this is not, however, the case for either one of them
taken alone).
211

Chapter III
The postulates
of quantum mechanics
A Introduction
B Statement of the postulates
B-1 Description of the state of a system
B-2 Description of physical quantities
B-3 The measurement of physical quantities
B-4 Time evolution of systems
B-5 Quantization rules
C The physical interpretation of the postulates concerning
observables and their measurement
C-1 The quantization rules are consistent with the probabilistic
interpretation of the wave function
C-2 Quantization of certain physical quantities
C-3 The measurement process
C-4 Mean value of an observable in a given state
C-5 The root mean square deviation
C-6 Compatibility of observables
D The physical implications of the Schrödinger equation
D-1 General properties of the Schrödinger equation
D-2 The case of conservative systems
E The superposition principle and physical predictions
E-1 Probability amplitudes and interference eects
E-2 Case in which several states can be associated with the same
measurement result
Quantum Mechanics, Volume I, Second Edition. C. Cohen-Tannoudji, B. Diu, and F. Laloë.
© 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

CHAPTER III THE POSTULATES OF QUANTUM MECHANICS
A. Introduction
In classical mechanics, the motion of any physical system is determined if the position
r( )and velocityv( ___)of each of its points are known as a function of time. In
general (Appendix), to describe such a system, one introduces generalized coordinates
() (= 12 ), whose derivatives with respect to time,_(), are the generalized
velocities. Specifying the()and_()enables us to calculate, at any given instant, the
position and velocity of any point of the system. Using the Lagrangian(_), one
denes the conjugate momentumof each of the generalized coordinates:
=
_
(A-1)
The()and() (= 12 )are called the fundamental dynamical variables.
All the physical quantities associated with the system (energy, angular momentum, etc.)
can be expressed in terms of the fundamental dynamical variables. For example, the total
energy of the system is given by the classical Hamiltonian( ). The motion of
the system can be studied by using either Lagrange's equations or the Hamilton-Jacobi
canonical equations, which are written:
d
d
= (A-2a)
d
d
= (A-2b)
In the special case of a system consisting of a single physical point of mass, the
are simply the three coordinates of this point, and the_are the components of its
velocityv. If the forces acting on this particle can be derived from a scalar potential
(r), the three conjugate momenta of its positionr(that is, the components of its
linear momentump) are equal to the components of its mechanical momentumv. The
total energy is then written:
=
p
2
2
+(r) (A-3)
and the angular momentum with respect to the origin:
=rp (A-4)
Since(rp) = (p
2
2) +(r), the Hamilton-Jacobi equations (A-2) take on the
well-known form:
dr
d
=
p
(A-5a)
dp
d
=r (A-5b)
214

B. STATEMENT OF THE POSTULATES
The classical description of a physical system can therefore be summarized as
follows:
()The state of the system at a xed time0is dened by specifyinggeneralized
coordinates(0)and theirconjugate momenta(0).
()The value, at a given time, of the various physical quantities is completely deter-
mined when the state of the system at this time is known: knowing the state of the
system, one can predict with certainty the result of any measurement performed
at time0
()The time evolution of the state of the system is given by the Hamilton-Jacobi equa-
tions. Since these are rst-order dierential equations, their solution()()
is unique if the value of these functions at a given time0is xed,(0)(0).
The state of the system is known for all time if its initial state is known.
In this chapter, we shall study the postulates on which the quantum description of
physical systems is based. We have already introduced them, in a qualitative and partial
way, in Chapter. Here we shall discuss them explicitly, within the framework of the
formalism developed in Chapter. These postulates will provide us with an answer to
the following questions (which correspond to the three points enumerated above for the
classical description):
()How is the state of a quantum system at a given time described mathematically?
()Given this state, how can we predict the results of the measurement of various
physical quantities?
()How can the state of the system at an arbitrary timebe found when the state at
time0is known?
We shall begin by stating the postulates of quantum mechanics (). Then we
shall analyze their physical content and discuss their consequences (,,).
B. Statement of the postulates
B-1. Description of the state of a system
In Chapter, we introduced the concept of the quantum state of a particle. We
rst characterized this state at a given time by a square-integrable wave function. Then,
in Chapter, we associated a ket of the state spacerwith each wave function: choosing
belonging toris equivalent to choosing the corresponding function(r) =r.
Therefore, the quantum state of a particle at a xed time is characterized by a ket of the
spacer. In this form, the concept of a state can be generalized to any physical system.
First Postulate:At a xed time0, the state of an isolated physical system is
dened by specifying a ket(0)belonging to the state space.
It is important to note that, sinceis a vector space, this rst postulate implies a
superposition principle: a linear combination of state vectors is a state vector. We shall
discuss this fundamental point and its relations to the other postulates in .
215

CHAPTER III THE POSTULATES OF QUANTUM MECHANICS
B-2. Description of physical quantities
We have already used, in , a dierential operatorrelated to
the total energy of a particle in a scalar potential. This is simply a special case of the
second postulate.
Second Postulate:Every measurable physical quantityis described by an
operatoracting in; this operator is an observable.Comments:
()The fact thatis an observable (cf.Chap., ) will be seen below
() to be essential.
()Unlike classical mechanics (cf.), quantum mechanics describes in a fun-
damentally dierent manner the state of a system and the associated physical
quantities: a state is represented by a vector, a physical quantity by an op-
erator.
B-3. The measurement of physical quantities
B-3-a. Possible results
The connection between the operatorand the total energy of the particle ap-
peared in
eigenvalues of the operator. Here as well, this relation can be extended to all physical
quantities.
Third Postulate:The only possible result of the measurement of a physical
quantityis one of the eigenvalues of the corresponding observable.Comments:
()A measurement ofalways gives a real value, sinceis by denition Her-
mitian.
()If the spectrum ofis discrete, the results that can be obtained by measuring
are quantized ().
B-3-b. Principle of spectral decomposition
We are going to generalize and discuss in more detail the conclusions of
Chapter, where we analyzed a simple experiment performed on polarized photons.
Consider a system whose state is characterized, at a given time, by the ket,
assumed to be normalized to 1:
= 1 (B-1)
216

B. STATEMENT OF THE POSTULATES
We want to predict the result of the measurement, at this time, of a physical quantity
associated with the observable. This prediction, as we already know, is of a proba-
bilistic sort. We are now going to give the rules that allow us to calculate the probability
of obtaining any given eigenvalue of.
. Case of a discrete spectrum
First, let us assume that the spectrum ofis entirely discrete. If all the eigenvalues
ofare non-degenerate, there is associated with each of them a unique (to within a
constant factor) eigenvector:
= (B-2)
Sinceis an observable, the set of the, which we shall take to be normalized,
constitutes a basis in, and the state vectorcan be written:
= (B-3)
We postulate that the probability()of ndingwhenis measured is:
() =
2
=
2
(B-4)
Fourth Postulate (case of a discrete non-degenerate spectrum):When the
physical quantityis measured on a system in thenormalizedstate,
the probability()of obtaining thenon-degenerate eigenvalueof the
corresponding observableis:
() =
2
where is the normalized eigenvector ofassociated with the eigenvalue
If, now, some of the eigenvaluesare degenerate, several orthonormalized eigen-
vectors correspond to them:
= ;= 12 (B-5)
can still be expanded on the orthonormal basis:
=
=1
(B-6)
In this case, the probability()becomes:
() =
=1
2
=
=1
2
(B-7)
217

CHAPTER III THE POSTULATES OF QUANTUM MECHANICS
(B-4) is then seen to be a special case of (B-7), which can therefore be considered to be
the general formula.
Fourth Postulate (case of a discrete spectrum):When the physical quantity
is measured on a system in thenormalizedstate, the probability()of
obtaining the eigenvalueof the corresponding observableis:
() =
=1
2
whereis the degree of degeneracy ofand (= 12 )is an
orthonormal set of vectors which forms a basis in the eigensubspace
associated with the eigenvalueof.
For this postulate to make sense, it is obviously necessary that, if the eigenvalue
is degenerate, the probability()be independent of the choice of thebasis
in. To verify this, consider the vector:
=
=1
(B-8)
where the coecientsare the same as those appearing in the expansion (B-6) of:
= (B-9)
is the part ofwhich belongs to, that is, the projection ofonto. This
is, moreover, what we nd when we substitute (B-9) into (B-8):
=
=1
= (B-10)
where:
=
=1
(B-11)
is the projector onto( ). Let us now calculate the square of the
norm of . From (B-8):
=
=1
2
(B-12)
Therefore,()is the square of the norm of= , the projection ofonto
. From this expression, it is clear that a change in the basis indoes not aect
(). This probability is written:
() = (B-13)
or, using the fact thatis Hermitian(=)and that it is a projector(
2
=):
() = (B-14)
218

B. STATEMENT OF THE POSTULATES
. Case of a continuous spectrum
Now let us assume that the spectrum ofis continuous and, for the sake of sim-
plicity, non-degenerate. The system, orthonormal in the extended sense, of eigenvectors
of:
= (B-15)
forms a continuous basis in, in terms of whichcan be expanded:
=d() (B-16)
Since the possible results of a measurement ofform a continuous set, we must dene a
probability density, just as we did for the interpretation of the wave function of a particle
( ). The probabilityd()of obtaining a value included betweenand
+ dis given by:
d() =() d
with:
() =()
2
=
2
(B-17)
Fourth Postulate (case of a continuous non-degenerate spectrum):When the
physical quantityis measured on a system in thenormalizedstate,
the probabilityd()of obtaining a result included betweenand+ d
is equal to:
d() =
2
d
where is the eigenvector corresponding to the eigenvalueof the
observableassociated with.
Comments:
()It can be veried explicitly, in each of the cases considered above, that the
total probability is equal to 1. For example, starting with formula (B-7), we
nd:
() =
=1
2
= = 1 (B-18)
sinceis normalized. This last condition is therefore indispensable if the
statements we have made are to be coherent. Nevertheless, it is not essential:
if it is not fullled, it suces to replace (B-7) and (B-17), respectively, by:
() =
1
=1
2
(B-19)
219

CHAPTER III THE POSTULATES OF QUANTUM MECHANICS
and:
() =
1
()
2
(B-20)
()For the fourth postulate to be coherent, it is necessary for the operator
associated with any physical quantity to be an observable: it must be possible
to expand any state on the eigenvectors of.
()We have not given the fourth postulate in its most general form. Starting
with the discussion of the cases we have envisaged, it is simple to extend the
principle of spectral decomposition to any situation (continuous degenerate
spectrum, partially continuous and partially discrete spectrum, etc...). In
, and later in Chapter, we shall apply this fourth postulate to a certain
number of examples, pointing out certain implications of the superposition
principle mentioned in .
. An important consequence
Consider two ketsand such that:
= e (B-21)
whereis a real number. Ifis normalized, so is:
=ee = (B-22)
The probabilities predicted for an arbitrary measurement are the same forand
since, for any:
2
=e
2
=
2
(B-23)
Similarly, we can multiplyby a constant factor:
=e (B-24)
without changing any of the physical results: since each coecient, or(), is multipled
by the same factor in both the numerator and denominator of (B-19) and (B-20), there
appear two factors of
2
that cancel each other. Therefore,two proportional state vectors
represent the same physical state.
Care must be taken to interpret this result correctly. For example, let us assume
that:
=11+22 (B-25)
where1and2are complex numbers. It is true thate
1
1represents, for all real1,
the same physical state as1, ande
2
2represents the same state as2. But, in
general:
=1e
1
1+2e
2
2 (B-26)
220

B. STATEMENT OF THE POSTULATES
does not describe the same state as(we shall see in relativephases of
the expansion coecients of the state vector play an important role). This is not true
for the special case where1=2+ 2, that is, where:
= e
1
[11+22] = e
1
(B-27)
In other words:a global phase factor does not aect the physical predictions, but the
relative phases of the coecients of an expansion are signicant.
B-3-c. Reduction of the wave packet
We have already introduced this concept in speaking of the measurement of the
polarization of photons in the experiment described in . We are now
going to generalize it, conning ourselves, nevertheless, to the case of a discrete spectrum
(we shall take up the case of a continuous spectrum in ).
Assume that we want to measure, at a given time, the physical quantity. If the
ket, which represents the state of the system immediately before the measurement, is
known, the fourth postulate allows us to predict the probabilities of obtaining the vari-
ous possible results. But when the measurement is actually performed, it is obvious that
only one of these possible results is obtained. Immediately after this measurement, we
cannot speak of the probability of having obtained this or that value: we know which
one was actually obtained. We therefore possess additional information, and it is under-
standable that the state of the system after the measurement, which must incorporate
this information, should be dierent from.
Let us rst consider the case where the measurement ofyields a simple eigenvalue
of the observable. We then postulate that the state of the system immediately after
this measurement is the eigenvectorassociated with:
()
= (B-28)
Comments:
()We have been speaking about states immediately before the measurement
() and immediately after (). The precise meaning of these expres-
sions is the following: assume that the measurement takes place at the time
00, and that we know the state(0)of the system at the time= 0.
The sixth postulate (see ) describes how the system evolves over time,
that is, enables us to calculate from(0)the state(0)immediately be-
fore the measurement. If the measurement has yielded the non-degenerate
eigenvalue, the state(1)at a time1 0must be calculated from
(0)= , the state immediately after the measurement, using the
sixth postulate to determine the evolution of the state vector between the
times0and1(Fig.).
()If we perform a second measurement ofimmediately after the rst one
(that is, before the system has had time to evolve), we shall always nd the
same result, since the state of the system immediately before the second
measurement is, and no longer.
221

CHAPTER III THE POSTULATES OF QUANTUM MECHANICS0
Measurement giving
the result a
n
t
0
t
1
t
ψ(t
0
) ψ(0)
u
n
ψ(t
1
)
Figure 1: When a measurement at time0of the observablegives the result, the
state vector of the system undergoes an abrupt modication and becomes. This new
initial state then evolves.
When the eigenvaluegiven by the measurement is degenerate, postulate (B-28)
can be generalized as follows. If the expansion of the stateimmediately before the
measurement is written, with the same notation as in section :
=
=1
(B-29)
the modication of the state vector due to the measurement is written:
()
=
1
=1
2
=1
(B-30)
=1
is the vectordened above [formula (B-8)], that is, the projection of
onto the eigensubspace associated with. In (B-30), we normalized this vector since it
is always more convenient to use state vectors of norm 1 [comment () of
With the notation of (B-10) and (B-11), we can therefore write (B-30) in the form:
()
=
(B-31)
Fifth Postulate:If the measurement of the physical quantityon the system
in the stategives the result, the state of the system immediately after the
measurement is the normalized projection,
, ofonto the
eigensubspace associated with.
The state of the system immediately after the measurement is therefore always
an eigenvector ofwith the eigenvalue. We stress the fact, however, thatit is not
222

B. STATEMENT OF THE POSTULATES
an arbitrary ket of the subspace, but the part ofthat belongs to(suitably
normalized, for convenience). In the light of above, equation (B-28) can be
seen to be a special case of (B-30). When= 1, the summation overdisappears from
(B-30), which becomes:
1
= e
Arg()
(B-32)
This ket indeed describes the same physical state as.
B-4. Time evolution of systems
We have already presented, in , the Schrödinger equation for
one particle. Here we shall write it in the general case.
Sixth Postulate:The time evolution of the state vector()is governed
by the Schrödinger equation:
~
d
d
()=()()
where()is the observable associated with the total energy of the system.
is called theHamiltonian operatorof the system, as it is obtained from the
classical Hamiltonian (Appendix
B-5. Quantization rules
We are nally going to discuss how to construct, for a physical quantityalready
dened in classical mechanics, the operatorwhich describes it in quantum mechanics.
B-5-a. Statement
Let us rst consider a system composed of a single particle, without spin, subjected
to a scalar potential. In this case:
With the positionr( )of the particle is associated the observable
R( ). With the momentump( )of the particle is associated the
observableP( ).
Recall that the components ofRandPsatisfy the canonical commutation relations
[Chap., equations (E-30)]:
[ ] = [] = 0
[ ] =~ (B-33)
Any physical quantityrelated to this particle is expressed in terms of the funda-
mental dynamical variablesrandp:(rp). To obtain the corresponding observable
223

CHAPTER III THE POSTULATES OF QUANTUM MECHANICS
, one could simply replace
1
, in the expression for(rp), the variablesrandpby
the observablesRandP:
() =(RP) (B-34)
However, this mode of action would be, in general, ambiguous. Assume, for exam-
ple, that in(rp)there appears a term of the form:
rp= + + (B-35)
In classical mechanics, the scalar productrpis commutative, and one can just as well
write:
pr= + + (B-36)
But whenrandpare replaced by the corresponding observablesRandP, the operators
obtained from (B-35) and (B-36) are not identical [see relations (B-33)]:
RP=PR (B-37)
Moreover, neitherRPnorPRis Hermitian:
(RP)= ( + + )=PR (B-38)
To the preceding postulates, therefore, must be added a symmetrization rule. For exam-
ple, the observable associated withrpwill be:
1
2
(RP+PR) (B-39)
which is indeed Hermitian. For an observable which is more complicated thanRP, an
analogous symmetrization is to be performed.
The observablewhich describes a classically dened physical quantity
is obtained by replacing, in the suitably symmetrized expression for,randp
by the observablesRandPrespectively.
We shall see, however, that there exist quantum physical quantities that have
no classical equivalent and which are therefore dened directly by the corresponding
observables (this is the case, for example, for particle spin).
Comment:
The preceding rules, and commutation rules (B-33) in particular, are valid only in
cartesian coordinates. It would be possible to generalize them to other coordinate
systems; however, they would no longer have the same simple form as they do
above.
1
See, in ComplementII, the denition of a function of an operator.
224

B. STATEMENT OF THE POSTULATES
B-5-b. Important examples
. The Hamiltonian of a particle in a scalar potential
Consider a (spinless) particle of chargeand mass, placed in an electric eld
derived from a scalar potential(r). The potential energy of the particle is therefore
(r) =(r), and the corresponding classical Hamiltonian is written [Appendix,
formula (29)]:
(rp) =
p
2
2
+(r) (B-40)
with:
p=
dr
d
=v (B-41)
wherevis the particle's velocity.
No diculties are presented by the construction of the quantum operatorcor-
responding to. No symmetrization is necessary, since neitherP
2
=
2
+
2
+
2
nor
(R)involves products of non-commuting operators. We therefore have:
=
P
2
2
+(R) (B-42)
(R)is the operator obtained by replacingrbyRin(r)(cf.ComplementII, ).
In this particular case, the Schrödinger equation, given in the sixth postulate,
becomes:
~
d
d
()=
P
2
2
+(R)() (B-43)
. The Hamiltonian of a particle in a vector potential
If the particle is now placed in an arbitrary electromagnetic eld, the classical
Hamiltonian becomes [Appendix, relation (66)]:
(rp) =
1
2
[pA(r)]
2
+(r) (B-44)
where(r)andA(r)are the scalar and vector potentials which describe the electro-
magnetic eld, and wherepis given by:
p=
dr
d
+A(r) =v+A(r) (B-45)
Once again, sinceA(r)depends only onrand the parameter(and not onp),
construction of the corresponding quantum operatorA(R)presents no problem. The
Hamiltonian operatoris then given by:
() =
1
2
[PA(R)]
2
+(R) (B-46)
with:
(R) =(R) (B-47)
225

CHAPTER III THE POSTULATES OF QUANTUM MECHANICS
and the Schrödinger equation is written:
~
d
d
()=
1
2
[PA(R)]
2
+(R)() (B-48)
Comment:
Care must be taken not to confusep(the momentum of the particle, also called the
conjugate momentum ofr) withv(the mechanical momentum of the particle):
the dierence between these two quantities appears clearly in (B-45). In quantum
mechanics, there of course exists an operator associated with the velocity of the
particle which is written here:
=
1
(PA) (B-49)
is then given by:
() =
1
2
2
+(R) (B-50)
It is the sum of two terms, one corresponding to the kinetic energy and the other
to the potential energy of the particle.
However, it is the conjugate momentumpand not the mechanical momentum
vthat becomes in quantum mechanics the operatorPsatisfying the canonical
commutation relations (B-33).
C. The physical interpretation of the postulates concerning observables and their
measurement
C-1. The quantization rules are consistent with the probabilistic interpretation of the
wave function
It is natural to associate the observablesRandP, whose action was dened in
, with the position and momentum of a particle. First of all, each of
the observables and possesses a continuous spectrum, and experiments
indeed show that all real values are possible for the six position and momentum variables.
Moreover, we shall see that applying the fourth postulate to the case of these observables
enables us to re-derive the probabilistic interpretation of the wave function as well as
that of its Fourier transform (see ).
Let us consider, for simplicity, the one-dimensional problem. If the particle is in
the normalized state, the probability that a measurement of its position will yield a
result included betweenand+ dis equal to [formula (B-17)]:
d() =
2
d (C-1)
whereis the eigenket ofwith the eigenvalue. We again nd that the square of the
modulus of the wave function() = is the particle's position probability density.
Now, to the eigenvectorof the observablecorresponds the plane wave:
=
1
2~
e
~ (C-2)
226

C. THE PHYSICAL INTERPRETATION OF THE POSTULATES CONCERNING OBSERVABLES AND THEIR
MEASUREMENT
and we have seen ( ) that the de Broglie relations associate with this
wave a well-dened momentum which is precisely. In addition, the probability of
nding, for a particle in the state, a momentum betweenand+ dis:
d() =
2
d=
()
2
d (C-3)
This is indeed what we found in .
C-2. Quantization of certain physical quantities
As we have already pointed out, the third postulate enables us to explain the
quantization observed for certain quantities, such as the energy of atoms. But it does
not imply that all quantities are quantized, since observables exist whose spectrum is
continuous. The physical predictions based on the third postulate are therefore not at
all obviousa priori. For example, when we study the hydrogen atom (Chap.), we
shall start from the total energy of the electron in the Coulomb potential of the proton,
from which we shall deduce the Hamiltonian operator. Solving its eigenvalue equation,
we shall nd that the bound states of the system can only correspond to certain discrete
energies which we shall calculate. Thus we shall not only explain the quantization of the
levels of the hydrogen atom, but also predict the possible energy values, which can be
measured experimentally. We stress the fact that these results will be obtained using the
same fundamental interaction law used in classical mechanics in the macroscopic domain.
C-3. The measurement process
The fourth and fth postulates pose a certain number of fundamental problems
which we shall not consider here (see section 5 of the bibliography of volumes I and II,
or for instanceDo we really understand quantum mechanics?by F. Laloë, Cambridge
University Press, 2019). There is, in particular, the question of the fundamental per-
turbation involved in the observation of a quantum system (cf.Chap., ).
The origin of these problems lies in the fact that the system under study is treated in-
dependently from the measurement device, although their interaction is essential to the
observation process. One should actually consider the system and the measurement de-
vice together as a whole. This raises delicate questions concerning the details of the
measurement process.
We shall content ourselves with pointing out that the nondeterministic formulation
of the fourth and fth postulates is related to the problems that we have just mentioned.
For example, the abrupt change from one state vector to another due to the measurement
corresponds to the fundamental perturbation of which we have spoken. But it is impossi-
ble to predict what this perturbation will be, since it depends on the measurement result,
which is not known with certainty in advance
2
.
We shall consider here only ideal measurements. To understand this concept, let
us return, for example, to the experiment of
is clear that when we grant that all photons polarized in a certain direction traverse the
analyzer, we assume that the analyzer is perfect. In practice, obviously, it also absorbs
some of the photons that it should let through. We shall therefore make the hypothesis,
2
Except, obviously, in the case where one is sure of the result that will be found (probability equal
to 1: the measurement does not modify the state of the system).
227

CHAPTER III THE POSTULATES OF QUANTUM MECHANICS
in the general case, that the measurement devices used are perfect: this amounts to
assuming that the perturbation they provoke is due only to the quantum mechanical
aspect of the measurement. Of course, real devices always present imperfections that
aect the measurement and the system; but one can, in principle, constantly ameliorate
them and thus approach the ideal limit dened by the postulates which we have stated.
C-4. Mean value of an observable in a given state
The predictions deduced from the fourth postulate are expressed in terms of proba-
bilities. To verify them, it would be necessary to perform a large number of measurements
under identical conditions. This means measuring the same quantity in a large number
of systems which are all in the same quantum state. If these predictions are correct,
the proportion ofidentical experiments resulting in a given event will approach, as
, the theoretically predicted probabilityof this event. Such a verication can
only be carried out in the limit where ; in practice,is of course nite, and
statistical techniques must be used to interpret the results.
The mean value of the observable
3
in the state, which we shall denote by
, or more simply by, is dened as the average of the results obtained when a
large numberof measurements of this observable are performed on systems which are
all in the state. When is given, the probabilities of nding all the possible results
are known. The mean value can therefore be predicted. We shall show that if
is normalized,is given by the formula:
= (C-4)
First consider the case where the entire spectrum ofis discrete. Out of
measurements of(the system being in the stateeach time), the eigenvaluewill
be obtained()times, with:
()
() (C-5)
and:
() = (C-6)
The mean value of the results of theseexperiments is the sum of the values found
divided by(whenexperiments have yielded the same result, this result will clearly
appeartimes in this sum). It is therefore equal to:
1
() (C-7)
Using (C-5), we see that when , this mean value approaches:
= () (C-8)
3
We shall henceforth use the word observable to designate a physical quantity as well as the
associated operator.
228

C. THE PHYSICAL INTERPRETATION OF THE POSTULATES CONCERNING OBSERVABLES AND THEIR
MEASUREMENT
Now substitute into this formula expression (B-7) for():
=
=1
(C-9)
Since:
= (C-10)
(C-9) can be written in the form:
=
=1
=
=1
(C-11)
Since the form an orthonormal basis of, the expression in brackets is equal to
the identity operator (closure relation), and we obtain formula (C-4).
The argument is completely analogous for the case where the spectrum ofis
continuous (for simplicity, we shall continue to assume it to be non-degenerate). Consider
identical experiments, and calld()the number of experiments which have yielded
a result included betweenand+ d. We have, similarly:
d()
d() (C-12)
The mean value of the results obtained is
1
d(), which, when , ap-
proaches:
= d() (C-13)
Substitute into (C-13) the expression ford()given by (B-17):
= d (C-14)
We can use the equation:
= (C-15)
to transform (C-14) into:
= d
= d (C-16)
Using the closure relation satised by the states, we again nd formula (C-4).
229

CHAPTER III THE POSTULATES OF QUANTUM MECHANICS
Comments:
(), the average over a set of identical measurements, must not be confused
with the time averages sometimes taken when dealing with time-dependent
phenomena.
()If the ketrepresenting the state of the system is not normalized, formula
(C-4) becomes [cf.comment (i) of ]:
=
(C-17)
()In practice, to calculateexplicitly, one often places oneself in a particular
representation. For example:
=
=d
3
rr
=d
3
(r)(r) (C-18)
using the denition of theoperator [cf.Chap., relations (E-22)]. Simi-
larly:
=
=d
3
(p)(p) (C-19)
or, using therrepresentation:
=d
3
rr
=d
3
(r)
~
(r) (C-20)
sincePis then represented by
~
r[formula (E-26) of Chapter].
C-5. The root mean square deviation
indicates the order of magnitude of the values of the observablewhen the
system is in the state. However, this mean value does not give any idea of the
dispersion of the results we expect when measuring. Assume, for example, that the
spectrum ofis continuous and that, for a given state, the curve representing the
variation with respect toof the probability density() =
2
has the shape
shown in Figure. For a system in the state, nearly all the values that can be
found whenis measured are included in an interval of widthcontaining, where
230

C. THE PHYSICAL INTERPRETATION OF THE POSTULATES CONCERNING OBSERVABLES AND THEIR
MEASUREMENTO
ρ()
m
α
A
Figure 2: Variation with respect toof the
probability density(). The mean value
is the abscissa of the center of gravity
of the area under the curve (it does not nec-
essarily coincide with the abscissaof the
maximum of the function).
the quantitycharacterizes the width of the curve: the smaller, the more the
measurement results are concentrated about.
How can we dene, in a general way, a quantity which characterizes the dispersion
of the measurement results about? We might envisage the following method: for each
measurement, take the dierence between the value obtained and; then calculate the
average of these deviations, dividing their sum by the numberof experiments. It is
easy to see, however, that the result obtained would be zero; we have, obviously:
= = 0 (C-21)
By the very denition of, the average of the negative deviations balances exactly the
average of the positive ones.
To avoid this compensation, it suces to denesuch that()
2
is the mean
of the squares of the deviations:
()
2
=( )
2
(C-22)
By denition, we therefore introduce theroot mean square deviationby setting:
=
( )
2
(C-23)
Using the expression for the mean value given in (C-4), we then have:
=
( )
2
(C-24)
This relation can also be written in a slightly dierent way:
( )
2
=(
2
2 +
2
)
=
2
2
2
+
2
=
2 2
(C-25)
The root-mean-square deviationis therefore also given by:
=
2 2
(C-26)
231

CHAPTER III THE POSTULATES OF QUANTUM MECHANICS
For example, in the case of the continuous spectrum of the observableconsidered
above,is given by:
()
2
=
+
[ ]
2
() d
=
+
2
() d
+
() d
2
(C-27)
If denition (C-23) is applied to the observablesRandP, it can be shown (Com-
plementIII), using their commutation relations, that for any state, one has:
~2
~2
~2
(C-28)
In other words, we nd the Heisenberg relations ( ) again, but with a
precise lower limit, which arises from the precise denition of the uncertainties.
C-6. Compatibility of observables
C-6-a. Compatibility and commutation rules
Consider two observablesandwhich commute:
[] = 0 (C-29)
We shall assume for simplicity that both of their spectra are discrete. According to the
theorem proved in , there exists a basis of the state space composed
of eigenkets common toand, which we shall denote by :
=
= (C-30)
(the indexallows us to distinguish, if necessary, between the dierent vectors corre-
sponding to one pair of eigenvalues). Therefore, for anyand(chosen, respectively,
in the spectra ofand), there exists at least one statefor which a measure-
ment ofwill always giveand a measurement ofwill always give. Two such
observablesandwhich can be simultaneously determined are said to be compatible.
On the other hand, ifanddo not commute, a state cannot in general
4
be a
simultaneous eigenvector of these two observables. They are said to be incompatible.
Let us examine more closely the measurement of two compatible observables on a system
which is initially in an arbitrary (normalized) state. This state can always be written:
= (C-31)
4
Some kets may be simultaneous eigenvectors ofand. But there would not be a sucient
number of them to form a basis, as would be the case ifandcommuted.
232

C. THE PHYSICAL INTERPRETATION OF THE POSTULATES CONCERNING OBSERVABLES AND THEIR
MEASUREMENT
First assume that we measureand then, immediately afterwards,(before the system
has had time to evolve). Let us calculate the probability( )of obtainingin the rst
measurement andin the second one. We begin by measuringin the state; the probability
of ndingis therefore:
() =
2
(C-32)
When we then measure, the system is no longer in the statebut, if we have found, in
the state:
=
1
2
(C-33)
The probability of obtainingwhen it is known that the rst measurement has yieldedis
therefore equal to:
() =
1
2
2
(C-34)
The probability( )sought corresponds to a composite event: to be in a favorable case,
we must rst ndand then, having satised this rst condition, nd. Therefore:
( ) =() () (C-35)
Substituting into this formula expressions (C-32) and (C-34), we obtain:
( ) =
2
(C-36)
Moreover, the state of the system becomes, immediately after the second measurement:
1
2
(C-37)
Therefore, if we decide to measure eitheroragain, we are sure of the result (or):
is an eigenvector common toandwith the eigenvaluesandrespectively.
Let us now return to the system in the state, and let us measure the two observables
in the opposite order (, then). What is the probability( )of obtaining the same
results as before? The reasoning is the same. We have here:
( ) =() () (C-38)
From (C-31), we see that:
() =
2
(C-39)
and that, after a measurement ofwhich yields, the state of the system becomes:
=
1
2
(C-40)
233

CHAPTER III THE POSTULATES OF QUANTUM MECHANICS
Therefore:
() =
1
2
2
(C-41)
and nally:
( ) =
2
(C-42)
If we have indeed foundand then, the system has gone into the state:
=
1
2
(C-43)
When two observables are compatible, the physical predictions are the same, what-
ever the order of performing the two measurements (provided that the time interval which
separates them is suciently small). The probabilities of obtaining eitherthenor
thenare identical:
( ) =( ) =
2
=
2
(C-44)
Moreover, the state of the system immediately after the two measurements is in both
cases (if the results areandforandrespectively):
= =
1
2
(C-45)
New measurements oforwill yield the same values again without fail.
The preceding discussion thus leads to the following result: when two observables
andare compatible, the measurement ofdoes not cause any loss of information
previously obtained from a measurement of(and vice versa) but, on the contrary, adds
to it. Moreover, the order of measuring the two observablesandis of no importance.
This last point, furthermore, enables us to envisage thesimultaneous measurementof
and. The fourth and fth postulates can be generalized to the case of such a
simultaneous measurement, as can be seen from formulas (C-44) and (C-45). To the
result correspond the orthonormal eigenvectors . From this, (C-44) and
(C-45) can be seen to be applications of postulates (B-7) and (B-30).
On the other hand, ifanddo not commute, the preceding arguments are
no longer valid. To understand this in a simple way, imagine that the state space
is replaced by the two-dimensional space of real vectors. The vectors1and2in
Figure with eigenvalues1and2respectively;1and2are
eigenvectors ofwith eigenvalues1and2respectively. Each of the two sets1 2
and 1 2forms an orthonormal basis in. We shall therefore represent them in
Figure anddo not
commute implies that these two pairs do not coincide. The physical system under study
is initially in the normalized state, which is represented in the gure by an arbitrary
234

C. THE PHYSICAL INTERPRETATION OF THE POSTULATES CONCERNING OBSERVABLES AND THEIR
MEASUREMENTK
1
H
1
H
2
K
2
O
υ
2

u
2

ψ
u
1

υ
1

Figure 3: Diagram associated with the suc-
cessive measurement of two non-compatible
observablesand. The state vector of
the system is. The eigenvectors ofare
1and2(eigenvalues1and2), which
are dierent from those of1and2
(eigenvalues1and2).
unit vector. We measureand nd, for example,1; the system goes into the state1.
We then measureand nd, for example,2; the state of the system becomes2:
(1)
= 1
(2)
= 2 (C-46)
If, on the other hand, we perform the measurements in the opposite order, obtaining the
same results:
(2)
= 2
(1)
= 1 (C-47)
The nal state of the system is not the same in both cases. We also see from Figure
that:
(12) = 1
2
2
2
(21) = 2
2
1
2
(C-48)
Although 1= 2, in general1= 2and:
(21)=(12) (C-49)
Therefore:two incompatible observables cannot be measured simultaneously. It can be
seen from (C-46) and (C-47) thatthe second measurement causes the information supplied
by the rst one to be lost. If, for example, after the sequence represented in (C-46), we
measureagain, we can no longer be sure of the result since2is not an eigenvector
of. All that was gained by the rst measurement ofis thus lost.
C-6-b. Preparation of a state
Let us consider a physical system in the stateand measure the observable
(whose spectrum we assume to be discrete).
235

CHAPTER III THE POSTULATES OF QUANTUM MECHANICS
If the measurement yields a non-degenerate eigenvalue, the state of the system
immediately after this measurement is the corresponding eigenvector. In this case,
it suces to know the result of this measurement to be able to determine unambiguously
the state of the system after this measurement, as it does not depend on the initial ket.
As we have already noted, at the end of , this is due to the fact that
represents the same physical state asitself.
The same does not hold true when the eigenvaluefound in the measurement is
degenerate. In:
=
1
2
=1
(C-50)
the absolute values of the coecientsand their relative phases are signicant (
b-). Since theare xed when the initial stateis specied, the stateafter
the measurement therefore depends on.
However, we saw in andcan be
measured simultaneously. If the result( )of this combined measurement corresponds
to only one eigenvector common to and, there is no summation overin
formula (C-37), which becomes:
=
(C-51)
This state is physically equivalent to. Again, specifying the result of the measure-
ment uniquely determines the nal state of the system, which is therefore independent
of the initial ket.
If there are associated with( )several eigenvectors ofand,
we can go back to the rst argument, measuring, at the same time asand, a third
observablewhich is compatible with both of them. We then arrive at the following con-
clusion:for the state of the system after a measurement to be completely dened uniquely
by the result obtained, this measurement must be made on a complete set of commuting
observables( ). This is the property which justies physically the
introduction of the concept of a C.S.C.O.
The methods that can be used toprepare a system in a well-dened quantum state
are analogous, in principle, to those used to obtain polarized light. When a polarizer is
placed in the path of a light beam, the outgoing light is polarized along a direction which
is characteristic of the polarizer and therefore independent of the state of polarization of
the incoming light. Similarly, we can construct devices, intended to prepare a quantum
system, in such a way that they only allow the passage of one state, corresponding to
a particular eigenvalue for each of the observables of the complete set chosen. We shall
study a concrete example of the preparation of a quantum system in Chapter).
Comment:
The measurement of a C.S.C.O. enables us to prepare only one of the basis states
associated with this C.S.C.O. However, it is obvious that changing the set of
observables allows us to obtain other states of the system. We shall see explicitly
in a concrete example, in , that we can prepare in this way
any state of the space.
236

D. THE PHYSICAL IMPLICATIONS OF THE SCHRÖDINGER EQUATION
D. The physical implications of the Schrödinger equation
The Schrödinger equation plays a fundamental role in quantum mechanics since, accord-
ing to the sixth postulate stated above, it is the equation that governs the time evolution
of the physical system. In this section, we shall study in detail the most important
properties of this equation.
D-1. General properties of the Schrödinger equation
D-1-a. Determinism in the evolution of physical systems
The Schrödinger equation:
~
d
d
()=()() (D-1)
is of rst order in. Consequently, given the initial state(0), the state()at any
subsequent timeis determined. There is no indeterminacy in the time evolution of
a quantum system. Indeterminacy appears only when a physical quantity is measured,
the state vector then undergoing an unpredictable modication (cf.fth postulate).
However, between two measurements, the state vector evolves in a perfectly deterministic
way, in accordance with equation (D-1).
D-1-b. The superposition principle
Equation (D-1) is linear and homogeneous. It follows that its solutions are linearly
superposable.
Let1()and2()be two solutions of (D-1). If the initial state of the system
is(0)=11(0)+22(0)(where1and2are two complex constants), to
it corresponds, at time, the state()=11()+22(). The correspondence
between(0)and()is therefore linear. We shall later study (ComplementIII)
the properties of the linear operator(0)which transforms(0)into():
()=(0)(0) (D-2)
D-1-c. Conservation of probability
. The norm of the state vector remains constant
Since the Hamiltonian operator()that appears in (D-1) is Hermitian, the square
of the norm of the state vector,()(), does not depend onas we shall now show:
d
d
()()=
d
d
()()+()
d
d
() (D-3)
According to (D-1), we can write:
d
d
()=
1
~
()() (D-4)
Taking the Hermitian conjugates of both sides of (D-4), we nd:
d
d
()=
1
~
()() =
1
~
()() (D-5)
237

CHAPTER III THE POSTULATES OF QUANTUM MECHANICS
since()is Hermitian (it is an observable). Substituting (D-4) and (D-5) into (D-3),
we obtain:
d
d
()()=
1
~
()()()+
1
~
()()()
= 0 (D-6)
The property of the norm conservation is very useful in quantum mechanics. For
example, it becomes indispensable when we interpret the square of the modulus(r)
2
of the wave function of a spinless particle as being the position probability density. The
fact that the state(0)of the particle is normalized at time0is expressed by the
relation:
(0)(0)=d
3
(r0)
2
= 1 (D-7)
where(r0) =r(0)is the wave function associated with(0). Equation (D-7)
means that the total probability of nding the particle somewhere in all space is equal
to 1. The property of conservation of the norm which we have just proved is expressed
by the equation:
()()=d
3
(r)
2
=(0)(0)= 1 (D-8)
where()is the solution of (D-1) corresponding to the initial state(0). In other
words, time evolution does not modify the global probability of nding the particle in all
space, which always remains equal to 1. Thus(r)
2
can be interpreted as a probability
density.
. Local conservation of probability. Probability densities and probability currents
In this paragraph, we shall conne ourselves to the case of a physical system
composed ofonly one(spinless)particle.
In this case, if(r)is normalized,
(r) =(r)
2
(D-9)
is aprobability density: the probabilityd(r)of nding, at time, the particle in an
innitesimal volumed
3
located at pointris equal to:
d(r) =(r) d
3
(D-10)
We have just shown that the integral of(r)over all space remains constant for all
time (and equal to 1 ifis normalized). This does not mean that(r)must be
independent ofat every pointr. The situation is analogous to the one encountered
in electromagnetism. If, in an isolated physical system, there is a charge distributed in
space with the volume density(r), the total charge [the integral of(r)over all
space] is conserved in time. However, within the system, the spatial distribution of this
charge may vary, giving rise to electric currents.
238

D. THE PHYSICAL IMPLICATIONS OF THE SCHRÖDINGER EQUATION
In fact, this analogy can be carried further. Global conservation of electrical charge
is based on local conservation. If the chargecontained within a xed volumevaries
over time, the closed surfacewhich limitsmust be traversed by an electric current.
More precisely, the variation dduring a time dof the charge contained withinis
equal tod, whereis the intensity of the current traversing, that is, the ux of the
vector current densityJ(r)leaving. Using classical vector analysis, we can express
local conservation of electrical charge in the form:
(r) + divJ(r) = 0 (D-11)
We are going to show that it is possible to nd a vectorJ(r), aprobability current, which
satises an equation identical to (D-11): there is thenlocalconservation of probability.
It is as if we were dealing with a probability uid whose density and motion were
described by(r)andJ(r). If the probability of nding the particle in the (xed)
volumed
3
aboutrvaries over time, it means that the probability current has a non-zero
ux accross the surface which limits this volume element.
First of all, let us assume that the particle under study is subjected to a scalar
potential(r). Its Hamiltonian is then:
=
P
2
2
+(R) (D-12)
and the Schrödinger equation is written, in therrepresentation (see ComplementII):
~
(r) =
~
2
2
(r) +(r)(r) (D-13)
(r)must be real forto be Hermitian. The complex conjugate equation of (D-13)
is therefore:
~
(r) =
~
2
2
(r) +(r)(r) (D-14)
Multiply both sides of (D-13) by(r)and both sides of (D-14) by(r). Add the
two equations thus obtained. It follows that:
~
[(r)(r)] =
~
2
2
[ ] (D-15)
that is:
(r) +
~
2
[(r)(r)(r) (r)] = 0 (D-16)
If we set:
J(r) =
~
2
[r r]
=
1
Re
~
r (D-17)
239

CHAPTER III THE POSTULATES OF QUANTUM MECHANICS
equation (D-16) can be put into the form of (D-11) since:
divJ(r) =rJ
=
~
2
(r)(r) +r
2
(r)(r) r
2
=
~
2
[ ] (D-18)
We have therefore proved the equation of local conservation of probability and have found
the expression for the probability current in terms of the normalized wave function(r).
Comment:
The form of the probability current (D-17) can be interpreted as follows.J(r)appears
as the mean value, in the state(), of an operatorK(r)given by:
K(r) =
1
2
[rrP+Prr] (D-19)
Now the mean value of the operatorrris(r)
2
, that is, the probability den-
sity(r), and
P
is the velocity operator. Therefore,Kis the quantum operator
constructed, with the help of an appropriate symmetrization, from the product of the
probability density and the velocity of the particle. This indeed corresponds to the vec-
tor current density of a classical uid (it is well known, for example, that the electrical
current density associated with a uid of charged particles is equal to the product of the
charge volume density and the velocity of the particles).
If the particle is placed in an electromagnetic eld described by the potentials
(r)andA(r), we can use the preceding argument, starting with the Hamiltonian
(B-46). We then nd, in this case:
J(r) =
1
Re
~
r A (D-20)
We see that this expression can be obtained from (D-17), using the same rule as was
used for the Hamiltonian:Pis simply replaced byPA.
Example of a plane wave.Consider a wave function of the form:
(r) =e
(kr )
(D-21)
with:~=
~
2
k
2
2
. The corresponding probability density:
(r) =(r)
2
=
2
(D-22)
is uniform throughout all space and does not depend on time. The calculation ofJ(r)
from (D-17) presents no diculties and leads to:
J(r) =
2
~k
=(r)v (D-23)
240

D. THE PHYSICAL IMPLICATIONS OF THE SCHRÖDINGER EQUATION
wherev=
~k
is the group velocity associated with the momentum~k(Chap.,
4). We see that the probability current is indeed equal to the product of the probability
density and the group velocity of the particle. In this case,andJare time-independent:
the ow of the probability uid associated with a plane wave is in asteady statecondition
(sinceandJdo not depend onreither, this state is also homogeneous and uniform).
D-1-d. Evolution of the mean value of an observable; relationship with classical mechanics
Letbe an observable. If the state()of the system is normalized (and we have
just seen that this normalization is conserved for all), the mean value of the observable
at the instantis equal to
5
:
() =()() (D-24)
We see that()depends onthrough()[and()], which evolve over time
according to the Schrödinger equation (D-4) [and (D-5)]. Moreover, the observable
may depend explicitly on time, causing an additional variation of the mean value()
with respect to.
We intend to study, in this section, the evolution of()and to show how this
enables us to relate classical mechanics to quantum mechanics.
. General formula
Dierentiating (D-24) with respect to, we obtain:
d
d
()()()=
d
d
()()()+()()
d
d
()
+()
() (D-25)
Using (D-4) and (D-5) for
d
d
()and
d
d
(), we nd:
d
d
()()()=
1
~
()[()() ()()]()
+()
() (D-26)
that is:
d
d
=
1
~
[()]+
(D-27)
Comment:
The mean value is a numberwhich depends only on. It is essential to un-
derstand how this dependence arises. For example, consider the case of a spinless
5
The notation ()means that the mean value ofis a number which depends on.
241

CHAPTER III THE POSTULATES OF QUANTUM MECHANICS
particle. Let(rp)be a classical quantity. In classical mechanics,randpde-
pend on time (they evolve according to the Hamilton equations), so that(rp)
depends onexplicitly, and implicitly throughrandp. To the classical quantity
(rp)corresponds the Hermitian operator=(RP), obtained by replac-
ing, in,randpby the operatorsRandP, and by symmetrizing the operators
if necessary (quantization rules, see ). The eigenstates and eigenvalues ofR
andPand, consequently, these observables themselves, no longer depend on.The
time dependence ofrandp, which characterizes the time evolution of the classi-
cal state,no longer appears inRandP,but in the quantum state vector(),
associated in therrepresentation with the wave function(r) =r().
In this representation, the mean value ofis written:
=d
3
(r)r
~
r (r) (D-28)
It is clear that integration overrleads to a number that only depends on. With
regard to classical mechanics, it is this number [and not the operator(r
~
r)]
that must be compared with the value taken on by the classical quantity(rp)
at time(cf. below).
. Application to the observablesRandP(Ehrenfest's theorem)
Now let us apply the general formula (D-27) to the observablesRandP. We shall
consider, for simplicity, the case of a spinless particle in a scalar stationary potential
(r). We then have:
=
P
2
2
+(R) (D-29)
so that we can write:
d
d
R=
1
~
[R]=
1
~
R
P
2
2
(D-30)
d
d
P=
1
~
[P]=
1
~
[P(R)] (D-31)
The commutator that appears in (D-30) can easily be calculated from the canonical
commutation relations; we obtain:
R
P
2
2
=
~
P (D-32)
For the one in formula (D-31), the following generalization of formula (B-33) must be
used [cf.ComplementII, formula (48)]:
[P(R)] =~r(R) (D-33)
242

D. THE PHYSICAL IMPLICATIONS OF THE SCHRÖDINGER EQUATION
wherer(R)denotes the set of three operators obtained by replacingrbyRin the
three components of the gradient of the function(r). Therefore:
d
d
R=
1
P
d
d
P=r(R)
(D-34)
(D-35)
These two equations expressEhrenfest's theorem. Their form recalls that of the classical
Hamilton-Jacobi equations for a particle (Appendix, ):
d
d
r=
1
p (D-36a)
d
d
p=r(r) (D-36b)
which reduce, in this simple case, to Newton's well-known equation:
dp
d
=
d
2
r
d
2
=r(r) (D-37)
. Discussion of Ehrenfest's theorem; classical limit
Let us analyze the physical meaning of Ehrenfest's theorem, that is, equations (D-
34) and (D-35). We shall assume that the wave function(r)describing the state of
the particle is a wave packet like those studied in Chapter.Rthen represents a set of
three time-dependent numbers . We shall call the pointR()thecenter
of the wave packet
6
at the instant. The set of points corresponding to various values
ofconstitutes thetrajectory of the center of the wave packet. Recall, however, that one
can never rigorously speak of the trajectory of the particle itself, whose state is described
by the wave packet as a whole, which inevitably has a certain spatial extension. We see,
nevertheless, that if this extension is much smaller than the other distances involved in
the problem, we can approximate the wave packet by its center. In this limiting case,
there is no appreciable dierence between the quantum and classical descriptions of the
particle.
It is therefore important to know the answer to the following question: does the
motion of the center of the wave packet obey the laws of classical mechanics? This answer
is supplied by Ehrenfest's theorem. Equation (D-34) expresses the fact that the velocity
of the center of the wave packet is equal to the average momentum of this wave packet
divided by. Consequently, the left-hand side of (D-35) can be written
d
2
d
2
R, so
that the answer to the preceding question will be armative if the right-hand side of
(D-35) is equal to the classical forceFat the point where the center of the wave packet
is situated:
F= [r(r)]
r=R (D-38)
6
The center and the maximum of a wave packet are, in general, distinct. They coincide, however, if
the wave packet has a symmetrical shape ( , Fig.).
243

CHAPTER III THE POSTULATES OF QUANTUM MECHANICS
In fact, the right-hand side of (D-35) is equal to the average of the force over the whole
wave packet, and, in general:
r(R)= [r(r)]
r=R (D-39)
(in other words, the mean value of a function is not equal to its value for the mean
value of the variable). Strictly speaking, the answer to the question we asked is therefore
negative.
Comment:
It is easy to convince ourselves of (D-39) if we consider a concrete example. Let us choose,
for simplicity, a one-dimensional model, and assume that:
() = (D-40)
whereis a real constant and, a positive integer. From this we deduce the operator
associated with():
() = (D-41)
The left-hand side of (D-39) can be written (replacingrby
d
d
)
1
. As for the
right-hand side, it is equal to:
d
d =
= [
1
]= =
1
(D-42)
Now we know that in general
1
=
1
; for example, for= 3, we have
2
=
2
(since the dierence between these two quantities enters into the calculation
of the root mean square deviation).
Note however that for= 1or 2,
1
=
1
. The two sides of (D-39) are then
equal. The same holds true, moreover, for= 0, in which case both sides are equal to
zero. For a free particle(= 0), or a particle placed in a uniform force eld(= 1)or
in a parabolic potential well (= 2, the case of a harmonic oscillator), the motion of the
center of the wave packet therefore rigorously obeys the laws of classical mechanics. We
have already established this result for the free particle(= 0)(cf.Chapter, ).
Although the two sides of (D-39) are not, in general, equal, there exist situations
(called quasi-classical) where the dierence between these two quantities is negligible:
this is the case when the wave packet is suciently localized. To see this, let us write
explicitly, in therrepresentation, the left-hand side of this equation:
r(R)=d
3
(r) [r(r)](r)
=d
3
(r)
2
r(r) (D-43)
Let us assume the wave packet to be highly localized: more precisely,(r)
2
takes on
non-negligible values only within a domain whose dimensions are much smaller than the
distances over which(r)varies appreciably. Then, within this domain, centered about
244

D. THE PHYSICAL IMPLICATIONS OF THE SCHRÖDINGER EQUATION
R,r(r)is practically constant. Therefore, in (D-43),r(r)can be replaced by its
value forr=Rand taken outside the integral, which is then equal to 1, since(r)
is normalized. Thus we nd that for suciently localized wave packets:
r(R)[r(r)]
r=R
(D-44)
In the macroscopic limit (where the de Broglie wavelengths are much smaller than the
distances over which the potential varies
7
), wave packets can be made suciently small to
satisfy (D-44) while retaining a good degree of denition for the momentum. The motion
of the wave packet is then practically that of a classical particle of massplaced in the
potential(r). The result that we have thus established is very important since it enables
us to show that the equations of classical mechanics follow from the Schrödinger equation
in certain limiting conditions satised, in particular, by most macroscopic systems.
D-2. The case of conservative systems
When the Hamiltonian of a physical system does not depend explicitly on time, the
system is said to beconservative. In classical mechanics, the most important consequence
of such a situation is theconservation of energyover time. It can also be said that the
total energy of the system is aconstant of the motion. We shall see in this section that
in quantum mechanics as well, conservative systems possess important special properties
in addition to the general properties of the preceding section.
D-2-a. Solution of the Schrödinger equation
First, let us consider the eigenvalue equation for:
= (D-45)
For simplicity, we assume the spectrum ofto be discrete;denotes the set of indices
other thanthat are necessary for characterizing a unique vector(in general,
these indices will determine the eigenvalues of operators forming a C.S.C.O. with).
Since, by hypothesis,does not depend explicitly on time, neither the eigenvalue
nor the eigenket isdependent.
First, we are going to show that, given theand the , it is very simple
to solve the Schrödinger equation, that is, to determine the time evolution of any state.
Since the form a basis (is an observable), it is always possible, for every value
of, to expand any state()of the system in terms of the:
()= () (D-46)
with:
() = () (D-47)
Since the do not depend on, all the time dependence of()is contained within
the (). To calculate the(), let us project the Schrödinger equation onto each
7
See the order of magnitude of the de Broglie wavelengths associated with a macroscopic system in
ComplementI.
245

CHAPTER III THE POSTULATES OF QUANTUM MECHANICS
of the states. This yields
8
:
~
d
d
()= () (D-48)
Sinceis Hermitian, it can be deduced from (D-45) that:
= (D-49)
so that (D-48) can be written in the form:
~
d
d
() = () (D-50)
This equation can be integrated directly to give:
() = (0) e
( 0)~
(D-51)
When does not depend explicitly on time, to nd(), given(0), proceed
as follows:
()Expand(0)in terms of a basis of eigenstates of:
(0)= (0) (D-52)
(0)is given by the usual formula:
(0) = (0) (D-53)
()Now, to obtain()for arbitrary, multiply each coecient(0)of the ex-
pansion (D-52) bye
( 0)~
, whereis the eigenvalue ofassociated with the
state :
()= (0) e
( 0)~
(D-54)
The preceding argument can easily be generalized to the case where the spectrum
ofis continuous; formula (D-54) then becomes, with obvious notation:
()= d ( 0) e
( 0)~
(D-55)
D-2-b. Stationary states
An important special case is that in which(0)is itself an eigenstate of.
Expansion (D-52) of(0)then involves only eigenstates ofwith the same eigenvalue
(for example,):
(0)= (0) (D-56)
8
In (D-48), can be placed to the right of
dd
, since does not depend on.
246

D. THE PHYSICAL IMPLICATIONS OF THE SCHRÖDINGER EQUATION
In formula (D-56), there is no summation over, and the passage from(0)to()
involves only one factore
( 0)~
, which can be taken outside the summation over:
()= (0) e
( 0)~
= e
( 0)~
(0)
= e
( 0)~
(0) (D-57)
()and(0)therefore dier only by theglobalphase factore
( 0)~
. These
two states are physically indistinguishable (cf.discussion in ). From this we
conclude that all the physical properties of a system which is in an eigenstate ofdo
not vary over time; the eigenstates ofare called, for this reason,stationary states.
It is also interesting to see how conservation of energy in a conservative system
appears in quantum mechanics. Let us assume that, at time0, we measure the energy
of such a system and we nd, for example:. Immediately after the measurement, the
system is in an eigenstate of, with an eigenvalue of(the postulate of the reduction
of the wave packet). We have just seen that the eigenstates ofare stationary states.
Therefore, the state of the system will no longer evolve after the rst measurement and
will always remain an eigenstate ofwith an eigenvalue of. It follows that a second
measurement of the energy of the system, at any subsequent time, will always yield the
same resultas the rst one.
Comment:
One passes from (D-52) to (D-54) by multiplying each coecient(0)of (D-52)
bye
( 0)~
. The fact thate
( 0)~
is a phase factor should not lead us
to believe that()and(0)are always physically indistinguishable. Actu-
ally, expansion (D-52) involves, in general,several eigenstates ofwith dierent
eigenvalues. To these dierent possible values ofcorresponddierent phase
factors. This modies therelative phasesof the expansion coecients of the state
vector and leads, consequently, to a state()which is physically distinct from
(0).
Only in the case where a single value ofenters into (D-52) [the case where
(0)is an eigenstate of] is the time evolution described by a single phase
factor, which is then a global one, of no physical importance. In other words,
there is physical evolution over time only if the energy of the initial state is not
known with certainty. We shall come back later to the relation between time
evolution and energy uncertainty (cf. ).
247

CHAPTER III THE POSTULATES OF QUANTUM MECHANICS
D-2-c. Constants of the motion
By denition, a constant of the motion is an observablewhich does not depend
explicitly on time and which commutes with:
= 0
[] = 0
(D-58)
For a conservative system,is therefore itself a constant of the motion.
Constants of the motion possess important properties which we are now going to
derive.
()If we substitute (D-58) into the general formula (D-27), we nd:
d
d
=
d
d
()()= 0 (D-59)
Whatever the state()of the physical system,the mean value ofin this statedoes
not evolve over time(hence the term constant of the motion).
()Sinceandare two observables which commute, we can always nd for
them a system of common eigenvectors, which we shall denote by :
=
= (D-60)
We shall assume for simplicity that the spectra ofandare discrete. The index
xes the eigenvalues of observables which form a C.S.C.O. withand. Since the
states are eigenstates of, they are stationary states. If the system is in the state
at the initial instant, it will therefore remain there indenitely (to within a global
phase factor). But the state is an eigenstate ofas well. Consequently, when
is a constant of the motion, there exist stationary states of the physical system (the
states ) that always remain, for all, eigenstates ofwith the same eigenvalue
. The eigenvalues ofare called, for this reason,good quantum numbers.
()Finally, let us show that for an arbitrary state(),the probability of nding
the eigenvalue, when the constant of the motionis measured,is not time-dependent.
(0)can always be expanded on the basis introduced above:
(0)= (0) (D-61)
From this we directly deduce:
()= () (D-62)
with:
() = (0) e
( 0)~
(D-63)
248

D. THE PHYSICAL IMPLICATIONS OF THE SCHRÖDINGER EQUATION
According to the postulate of spectral decomposition, the probability( 0)of
ndingwhenis measured at time0, on the system in the state(0), is equal to:
( 0) = (0)
2
(D-64)
Similarly:
() = ()
2
(D-65)
Now we see from (D-63) that ()and (0)have the same modulus. Therefore,
() =( 0), which proves the property stated above.
Comment:
If all but one of the probabilities( 0)are zero [leaving for example( 0)
non-zero and, moreover, necessarily equal to 1], the physical system at time0is
in an eigenstate ofwith an eigenvalue of. Since the()do not depend on
, the state of the system at timeremains an eigenstate ofwith an eigenvalue
of.
D-2-d. Bohr frequencies of a system. Selection rules
Letbe an arbitrary observable of the system under consideration (it does not
necessarily commute with). Formula (D-27) enables us to calculate the derivative
d
d
of the mean value of:
d
d
=
1
~
[]+ (D-66)
For a conservative system, we know the general form (D-54) of(). Therefore, in this
case, we can calculate explicitly()()(and not merely
d
d
).
The Hermitian conjugate expression of (D-54) is written (changing the summation
indices):
()= (0) e
( 0)~
(D-67)
We can then, in()(), replace()and()by expansions (D-54) and (D-67),
respectively. Thus we obtain:
()()= ()
= (0)(0) e
( )( 0)~
(D-68)
249

CHAPTER III THE POSTULATES OF QUANTUM MECHANICS
From now on, we shall assume thatdoes not depend explicitly on time: the ma-
trix elements are therefore constant. Formula (D-68) then shows that
the evolution of()is described by a series ofoscillating terms, whose frequencies
1
2~
== are characteristic of the system under consideration
but independent ofand of the initial state of the system. The frequenciesare
called theBohr frequenciesof the system. Thus, for an atom, the mean values of all the
atomic quantities (electric and magnetic dipole moments, etc...) oscillate at the various
Bohr frequencies of the atom. It is reasonable to imagine that only these frequencies can
be radiated or absorbed by the atom. This remark allows us to understand intuitively the
Bohr relation between the spectral frequencies emitted or absorbed and the dierences
in atomic energies.
It can also be seen from (D-68) that, while the frequencies involved in the motion
of()are independent of, the same does not hold true for the respectiveweightsof
these frequencies in the variation of. The importance of each frequencydepends
on the matrix elements . In particular, if these matrix elements are zero
for certain values ofand, the corresponding frequenciesare absent from the
expansion of(), whatever the initial state of the system. This is the origin of the
selection ruleswhich indicate what frequencies can be emitted or absorbed under given
conditions. To establish these rules, one must study the non-diagonal matrix elements
(=)of the various atomic operators such as the electric and magnetic dipoles, etc...
Finally, the weights of the various Bohr frequencies also depend on the initial state,
via (0)(0). In particular, if the initial state is a stationary state of energy,
the expansion of(0)contains only one value of(=) and (0)(0)can
be non-zero only for==. In this case,is not time-dependent.
Comment:
It can be directly veried, using (D-68), that the mean value of a constant of
the motion is always time-independent. We see that ifcommutes with, the
matrix elements ofare zero between two eigenstates ofthat correspond to
dierent eigenvalues (cf.Chap., ). It follows that is zero
for=. The only terms ofthat are non-zero are thus constant.
D-2-e. The time-energy uncertainty relation
We shall now see that for a conservative system, the greater the energy uncertainty,
the more rapid the time evolution. More precisely, ifis a time interval at the end of
which the system has evolved to an appreciable extent, and ifdenotes the energy
uncertainty,andsatisfy the relation:
& (D-69)
First, if the system is in an eigenstate of, its energy is perfectly well-dened:
= 0. But we have seen that such a state is stationary, meaning that it does not
evolve in time; in a sense, its evolution timeis innite [relation (D-69) indicates that
when= 0,must be innite].
250

D. THE PHYSICAL IMPLICATIONS OF THE SCHRÖDINGER EQUATION
Now let us assume that(0)is a linear superposition of two eigenstates of,
1and2, with dierent eigenvalues1and2:
(0)=11+22 (D-70)
Then:
()=1e
1( 0)~
1+2e
2( 0)~
2 (D-71)
If we measure the energy, we nd either1or2. The uncertainty ofis therefore of
the order of:
2 1 (D-72)
Now consider an arbitrary observablewhich does not commute with. The proba-
bility of nding, in a measurement ofat time, the eigenvalueassociated with the
eigenvector (we assume, for simplicity,to be non-degenerate) is given by:
() = ()
2
=1
2
1
2
+2
2
2
2
+ 2 Re
21e
(2 1)( 0)~
2 1 (D-73)
This equation shows that()oscillates between two extreme values, with the Bohr
frequency21=
2 1
. The characteristic evolution time of the system is therefore:

2 1
(D-74)
and comparison with (D-72) shows that: .
Let us now assume that the spectrum ofis continuous (and non-degenerate).
The most general state(0)can be written:
(0)=d() (D-75)
where is the eigenstate ofwith the eigenvalue. Let us assume that()
2
has non-negligible values only in a domain of widthabout0(Fig.).then
represents the energy uncertainty of the system.()is obtained by using (D-55):
()=d() e
( 0)~
(D-76)
The quantity()introduced above, which represents the probability of nding the
eigenvaluewhen the observableis measured on the system in the state(), is
here equal to:
() = ()
2
= d() e
( 0)~
2
(D-77)
In general, does not vary rapidly withwhenvaries about0. Ifis
suciently small, the variation of, in integral (D-77), can be neglected relative
251

CHAPTER III THE POSTULATES OF QUANTUM MECHANICS
to that of(). One can then replace by
0and take this quantity
outside integral (D-77):
()
0
2
d() e
( 0)~
2
(D-78)
If this approximation is valid, we thus see that()is, to within a coecient, the
square of the modulus of the Fourier transform of(). According to the properties of
the Fourier transform (cf.Appendix, ), the width inof(), that is,, is
therefore related to the widthof()
2
by relation (D-69).
Comment:
(D-69) can be established directly for a free one-dimensional wave packet. One can
associate with the momentum uncertaintyof this wave packet an energy uncertainty
=
d
d
. Since=~and=~, we have
d
d
=
d
d
=, whereis the
group velocity of the wave packet (Chap., ). Consequently:
= (D-79)
Now the characteristic evolution timeis the time taken by this wave packet, travelling
at the velocity, to pass a point in space. Ifis the spatial extension of the wave
packet, we therefore have:

(D-80)
From this we deduce, combining (D-79) and (D-80):
&~ (D-81)
Relation (D-69) is often calledthe fourth Heisenberg uncertainty relation. It is
clearly dierent, however, from the other three uncertainty relations which relate to the
three components ofRandP[formulas (14) of ComplementI]. In (D-69), only the
energy is a physical quantity likeRandP;, on the other hand,is a parameter, with
which no quantum mechanical operator is associated.ΔE
E
0
c(E)
2
0 E
Figure 4: By superposing the stationary
states with the coecients(), we ob-
tain a stateof the system where the en-
ergy is not perfectly well-dened. The cor-
responding uncertaintyis given by the
width of the curve that represents()
2
.
According to the fourth uncertainty relation,
the evolution of the state()will be signif-
icant after a timesuch that&~.
252

E. THE SUPERPOSITION PRINCIPLE AND PHYSICAL PREDICTIONS
E. The superposition principle and physical predictions
The physical meaning of the rst postulate remains to be examined. According to this
postulate, the states of a physical system belong to a vector space and are, consequently,
linearly superposable.
One of the important consequences of the rst postulate, when it is combined
with the others, is the appearance of interference eects such as those which led us to
wave-particle duality (Chap.). Our understanding of these phenomena is based on the
concept of probability amplitudes, which we shall examine here with the aid of some
simple examples.
E-1. Probability amplitudes and interference eects
E-1-a. The physical meaning of a linear superposition of states
. The dierence between a linear superposition and a statistical mixture
Let1and2be two orthogonal normalized states:
11= 22= 1
12= 0 (E-1)
(1and2could be, for example, two eigenstates of the same observableassociated
with two dierent eigenvalues1and2).
If the system is in the state1, we can calculate all the probabilities concerning
the measurement results for a given observable. For example, ifis the (normalized)
eigenvector ofwhich corresponds to the eigenvalue(assumed to be non-degenerate),
the probability1()of ndingwhenis measured on the system in the state1
is:
1() = 1
2
(E-2)
An analogous quantity2()can be dened for the state2:
2() = 2
2
(E-3)
Now consider a normalized statewhich is alinear superpositionof1and
2:
=11+22
1
2
+2
2
= 1 (E-4)
It is often said that, when the system is in the state, one has a probability1
2
of nding it in the state1and a probability2
2
of nding it in the state2.
The exact meaning of this manner of speaking is the following: if1and 2are
two eigenvectors (here assumed to be normalized) of the observablecorresponding to
dierent eigenvalues1and2, the probability of nding1whenis measured is1
2
and that of nding2is2
2
.
This could lead us to believe (wrongly, as we shall see), that a state such as (E-4)
is astatistical mixtureof the states1and 2with the weights1
2
and2
2
. In
253

CHAPTER III THE POSTULATES OF QUANTUM MECHANICS
other words, if we consider a large numberof identical systems, all in the state (E-4),
we might imagine that this set ofsystems in the statewas completely equivalent
to another set composed of1
2
systems in the state1and 2
2
systems in the
state2. Such an interpretation of the stateiserroneousand leads to inaccurate
physical predictions, as we shall see.
Assume that we are actually trying to calculate the probability()of nding
the eigenvaluewhen the observableis measured on the system in the stategiven
by (E-4). If we interpret the stateas being a statistical mixture of the states1and
2with the weights1
2
and2
2
, then we can obtain()by taking the weighted
sum of the probabilities1()and2()calculated above [formulas (E-2) and (E-3)]:
()
?
= 1
2
1() +2
2
2() (E-5)
Actually, the postulates of quantum mechanics unambiguously indicate how to calculate
(). The correct expression for this probability is:
() =
2
(E-6)
()is therefore the square of the modulus of theprobability amplitude. We see
from (E-4) that this amplitude is the sum of two terms:
=1 1+2 2 (E-7)
Thus we obtain:
() =1 1+2 2
2
= 1
2
1
2
+2
2
2
2
+ 2 Re12 1 2 (E-8)
Taking (E-2) and (E-3) into account, we nd that the correct expression for()is
therefore written:
() =1
2
1() +2
2
2() + 2 Re12 1 2 (E-9)
This result is dierent from that of formula (E-5).
It is therefore wrong to considerto be a statistical mixture of states. Such
an interpretation eliminates all theinterference eectscontained in the double product
of formula (E-9). We see that it is not only the moduli of1and2that play a role;
the relative phase
9
of1and2is just as important, since it enters explicitly into the
physical predictions, through the intermediary of12.
. A concrete example
Consider photons propagating alongwhose polarization state is represented by
the unit vector (Fig.):
e=
1
2
(e+e) (E-10)
9
Multiplyingby aglobalphase factoreis equivalent to changing1and2to1eand2e.
It can be veried from (E-9
depend only on1
2
,2
2
and1
2
.
254

E. THE SUPERPOSITION PRINCIPLE AND PHYSICAL PREDICTIONS
This state is a linear superposition of two orthogonal polarization stateseande. It
represents light which is linearly polarized at an angle of 45with respect toeande. It
would be absurd to assume thatphotons in the stateeare equivalent to
1
2
2
=
2
photons in the stateeand
1
2
2
=
2
photons in the statee. If we place in the
beam's trajectory an analyzer whose axiseis perpendicular toe, we know thatnone
of thephotons in the stateewill pass through this analyzer. But, for the statistical
mixture
2
photons in the statee,
2
photons in the statee, half the photons will
pass through the analyzer. In this concrete example, it is clear that a linear superpositione
y
O
e
x
z
e
e
Figure 5: A simple experiment which illustrates the dierence between a linear superposi-
tion and a statistical mixture of states. If all the incident photons are in the polarization
state:
e=
1
2
(e+e)
none of them will pass through an analyzer whose axiseis perpendicular toe. If we had,
on the contrary, a statistical mixture of photons polarized either alongeor alonge(in
equal proportions; i.e., natural light), half of them would pass through the analyzer.
such as (E-10), associated withlight polarizedat an angle of 45with respect toeand
e, is physically dierent from a statistical mixture of equal proportions of the statese
andeassociated withnatural light(an unpolarized beam).
We can also understand the importance of the relative phase of the expansion
255

CHAPTER III THE POSTULATES OF QUANTUM MECHANICS
coecients of the state vector, by considering the four states:
e1=
1
2
(e+e) (E-11)
e2=
1
2
(ee) (E-12)
e3=
1
2
(e+e) (E-13)
e4=
1
2
(e e) (E-14)
which dier only by the relative phase of the coecients (this phase being equal to 0,,
+
2
and
2
, respectively). These four states are physically quite dierent: the rst two
represent light which is polarized linearly along the bisectors of(ee); the second two
represent circularly polarized light (right and left respectively).
E-1-b. Summation over the intermediate states
. Prediction of measurement results in two simple experiments
()Experiment 1. Imagine that the observablehas been measured, at a given
time, on a physical system, and that the non-degenerate eigenvaluehas been found.
If is the eigenvector associated with, the physical system, immediately after the
measurement, is in the state.
Before the system has had time to evolve, we measure another observablethat
does not commute with. Using the notation introduced in , we denote by()
the probability that this second measurement will yield the result. Immediately before
the measurement of, the system is in the state. Therefore, ifis the eigenvector
ofassociated with the eigenvalue(assumed to be non-degenerate), the postulates of
quantum mechanics lead to:
() =
2
(E-15)
(ii) Experiment 2.We now imagineanotherexperiment, in which we measure
successively and very rapidly three observables,,, which do not commute with
each other (the time separating two measurements is too short for the system to evolve).
Denote by()the probability, given that the result of the rst measurement is,
that the results of the second and third will beandrespectively.()is equal to
the product of()(the probability that, the measurement ofhaving yielded, that
ofwill yield) and()(the probability that, the measurement ofhaving yielded
, that ofwill yield):
() =()() (E-16)
If all the eigenvalues ofare assumed to be non-degenerate and ifdenotes the
corresponding eigenvectors, it follows that [using for()and()formulas analogous
to (E-15)]:
() =
2 2
(E-17)
256

E. THE SUPERPOSITION PRINCIPLE AND PHYSICAL PREDICTIONSw
b
1

w
b
2

w
b
3
u
a
υ
c
Figure 6: Dierent possible paths for the state vector of the system when the system is
allowed to evolve freely (without undergoing any measurement) between the initial state
and the nal state. In this case, we must add together the probability amplitudes
associated with these dierent paths, and not their probabilities.
. The fundamental dierence between these two experiments
In both of these experiments, the state of the system after the measurement of the
observableis (the role of this measurement being to x this initial state). It then
becomes after the last measurement, that of the observable(for this reason,
will be called the nal state). It is possible in both cases to decompose the state of the
system just before the measurement ofin terms of the eigenvectorsof, and to
say that between the stateand the state, the system can pass through several
dierent intermediate states. Each of these intermediate states denes a possible
path between the initial stateand the nal state(Fig.).
The dierence between the two experiments described above is the following. In
the rst one, the path that the system has taken between the stateand the state
is not determined experimentally [we measure only the probability()that, starting
from , it ends up in]. On the other hand, in the second experiment, this path is
determined, by measuring the observable[thus enabling us to obtain the probability
()that the system, starting from, passes through a given intermediate state
and ends up nally in].
We could then be tempted, in order to relate()to(), to use the following
argument: in experiment 1, the system is free to pass through all intermediate states
; it would then seem that the global probability()should be equal to the sum
of all the probabilities()associated with each of the possible paths. Can we not
then write:
()
?
= () (E-18)
As we shall see,this formula is wrong. Let us go back to the exact formula (E-15)
for(); this formula brings in the probability amplitudewhich we can write,
257

CHAPTER III THE POSTULATES OF QUANTUM MECHANICS
using the closure relation for the states:
= (E-19)
Substitute this expression into (E-15):
() =
2
=
2 2
+
=
(E-20)
Using (E-17), we therefore obtain:
() = () +
=
(E-21)
This equation enables us to understand why formula (E-18) is wrong: all the cross
terms that appear in the square of the modulus of sum (E-19) are absent in (E-18). All
theinterference eects between the dierent possible pathsare thus missing in (E-18).
If, therefore, we want to establish a relation between these two experiments, we see
that it is necessary to reason in terms of probability amplitudes.When the intermediate
states of the system are not determined experimentally, it is the probability amplitudes,
and not the probabilities, which must be summed.
The error in the reasoning which led to the wrong relation (E-18) is obvious,
moreover, if we remember the fth postulate (reduction of the wave packet). In the second
experiment, the measurement of the observablemust, in fact, involve a perturbation
of the system under study: during the measurement its state vector undergoes an abrupt
change (projection onto one of the states). It is this unavoidable perturbation which
is responsible for the disappearance of interference eects. In the rst experiment, on
the other hand, it is incorrect to say that the physical system passes through one or
another of the states; it would be more accurate to say that it passes throughall
the states.
Comments:
()The preceding discussion resembles in every respect that of
ter
that a photon emitted by the source will arrive at a given pointof the
screen, one must rst calculate the total electric eld at. In this problem,
the electric eld plays the role of a probability amplitude. When one is not
trying to determine through which slit the photon passes, it is the electric
elds radiated by the two slits, and not their intensities, that must be added
together to obtain the total eld at(whose square yields the desired prob-
ability). In other words, the eld radiated by one of the slits at pointis
258

E. THE SUPERPOSITION PRINCIPLE AND PHYSICAL PREDICTIONS
analogous to the probability amplitude for a photon, emitted by the source,
to pass through this slit and to arrive at.
()It is not necessary to retain the assumption that the measurements ofand
in experiment 1 and of,,in experiment 2 are performed very close
together in time. If the system has had time to evolve between two of these
measurements, we can use the Schrödinger equation to determine the modi-
cation of the state of the system due to this evolution [cf.ComplementIII,
comment(ii)of ].
E-1-c. Conclusion: The importance of the concept of probability amplitudes
The two examples studied in
concept of probability amplitudes. Formulas (E-5) and (E-18), as well as the arguments
that lead to them, are incorrect since they represent an attempt to calculate a probability
directly without rst considering the corresponding probability amplitude. In both cases,
the correct expression (E-8) or (E-20) has the form of asquare of a sum(more precisely,
the square of the modulus of this sum), while the incorrect formula (E-5) or (E-18)
contains only asum of squares(all the cross terms, responsible for interference eects,
being omitted).
From the preceding discussion, we shall therefore retain the following ideas:
()The probabilistic predictions of quantum theory are always obtained by squaring
the modulus of aprobability amplitude.
()When, in a particular experiment, no measurement is made at an intermediate
stage, one must never reason in terms of the probabilities of the various results
that could have been obtained in such a measurement, but rather in terms of their
probability amplitudes.
()The fact that the states of a physical system are linearly superposable means that a
probability amplitude often presents the form of a sum of partial amplitudes. The
corresponding probability is then equal to the square of the modulus of a sum of
terms, andthe various partial amplitudes interfere with each other.
E-2. Case in which several states can be associated with the same measurement result
In the preceding section, we stressed and illustrated the fact that, in certain cases,
the probability of an event is given by the postulates of quantum mechanics in the form
of asquare of a sumof terms (more precisely, the square of the modulus of such a sum).
Now the statement of the fourth postulate [formula (B-7)] involves asum of squares(the
sum of the squares of the moduli) when the measurement result whose probability is
sought is associated with a degenerate eigenvalue. It is important to understand that
these two rules are not contradictory but, on the contrary, complementary: each term of
the sum of squares (B-7) can itself be the square of a sum. This is the rst point on which
we shall focus our attention in this section. Furthermore, this discussion will enable us to
complete the statement of the postulates: we shall consider measurement devices whose
accurary is limited (as is always, of course, the case) and see how to predict theoretically
the possible results. Finally, we shall extend to the case of continuous spectra the fth
postulate of reduction of the wave packet.
259

CHAPTER III THE POSTULATES OF QUANTUM MECHANICS
E-2-a. Degenerate eigenvalues
In the examples treated in , we always assumed that the results of the various
measurements envisaged were simple, i.e. non-degenerate, eigenvalues of the correspond-
ing observables. This hypothesis was intended to simplify these examples so that the
origin of the interference eects appeared as clearly as possible.
Now let us consider a degenerate eigenvalueof an observable. The eigenstates
associated withform a vector subspace of dimension, in which an orthonormal basis
;= 12 can be chosen.
The discussion of has yielded
is not sucient to determine the state of the physical system after this measurement. We
shall say thatseveral nal states can be associated with the same result: if the initial
state (the state before the measurement) is given, the nal state after the measurement
is perfectly well-dened; but if the initial state is changed, the nal state is, in general,
dierent (for the same measurement result). All nal states associated withare
linear combinations of theorthonormal vectors, with= 12 .
Formula (B-7) indicates unambiguously how to nd the probability()that a
measurement ofon a system in the statewill yield the result. One chooses
an orthonormal basis, for example;= 12 , in the eigensubspace which
corresponds to; one calculates the probability
2
of nding the system in each
of the states of this basis;()is then the sum of theseprobabilities. However, it
must not be forgotten that each probability
2
can be the square of the modulus of
a sum of terms. Consider, for example, the case envisaged in and assume now
that the eigenvalueof the observable, whose probability()is to be calculated,
is-fold degenerate. Formula (E-6) is then replaced by:
() =
=1
2
(E-22)
with:
=1 1+2 2 (E-23)
The discussion of remains valid for each of the terms of formula (E-22):
2
, which is obtained from (E-23), is the square of a sum;()is then the sum
of these squares.
of the observables measured are degenerate.
Before summarizing the preceding discussions, we are going to study another im-
portant situation where several nal states are associated with the same measurement
result.
E-2-b. Insuciently selective measurement devices
. Denition
Assume that, in order to measure the observablein a given physical system, we have
at our disposal a device which works in the following way:
()This device can give only two dierent answers
10
, which we shall denote, for convenience,
by yes and no.
10
The following arguments can easily be generalized to cases where the device can give several dierent
answers having characteristics similar to those described in () and ().
260

E. THE SUPERPOSITION PRINCIPLE AND PHYSICAL PREDICTIONS
()If the system is in an eigenstate ofwhose eigenvalue is included in a given interval
of the real axis, the answer is always yes; this is also the case when the state of the
system is any linear combination of eigenstates ofassociated with eigenvalues which are
all included in.
()If the state of the system is an eigenstate ofwhose eigenvalue falls outside, or any
linear combination of such eigenstates, the answer is always no.
therefore characterizes the resolving power of the measurement device under consid-
eration. If there exists only one eigenvalueofin the interval, the resolving power is
innite: when the system is in an arbitrary state, the probability(yes) of obtaining the answer
yes is equal to the probability of ndingin a measurement of; the probability(no)
of obtaining no is obviously equal to1 (yes). If, on the other hand,contains several
eigenvalues of, the device does not have a sucient resolution to distinguish between these
various eigenvalues: we shall say that it isinsuciently selective. We shall see how to calculate
(yes) and(no) in this case.
To be able to study the perturbation created by such a measurement on the state of the
system, we are going to add the following hypothesis: the device transmits without perturbation
the eigenstates ofassociated with the eigenvalues of the interval(as well as any linear
combination of these eigenstates), while it blocks the eigenstates ofassociated with the
eigenvalues outside(as well as all their linear combinations). The device thus behaves like a
perfect lter for all states associated with.
. Example
Most measurement devices used in practice are insuciently selective. For example, to
measure the abscissa of an electron propagating parallel to theaxis, one can (Fig.) place
in the plane (is perpendicular to the plane of the gure) a plate with a slit whose axis
is parallel to, the abscissas of the edges being1and2. It can then be seen that any wave
packet which is entirely included between the=1and=2planes (a superposition of
eigenstates ofhaving eigenvaluescontained within the interval [12]) will enter the region
to the right of the slit (yes answer); in this case, it will not undergo any modication. On the
other hand, any wave packet situated below the=1plane or above the=2plane will be
blocked by the plate and will not pass to the right (no answer).x
x
2
x
1
z
O
Figure 7: Schematic drawing of a device for
measuring the abscissaof a particle. Since
the interval12is necessarily non-zero,
such a device is always imperfectly selective.
. Quantum description
For such an insuciently selective device, several nal states are possible after a mea-
surement which has yielded the answer yes. They can be, for example, the various eigenstates
ofthat correspond to the eigenvalues contained in the interval.
The physical problem posed by such devices, and which we are now going to consider,
consists of predicting the answer which will be obtained when a system in an arbitrary state
261

CHAPTER III THE POSTULATES OF QUANTUM MECHANICS
enters the device. For example, for the apparatus of Figure, what happens when we are dealing
with a wave packet which is neither entirely contained between the=1and=2planes
(in which case the answer is certainly yes) nor entirely situated outside this region (in which
case the answer is certainly no)? We shall see that this is equivalent to measuring an observable
whose spectrum is degenerate.
Consider the subspacespanned by all the eigenstates ofwhose eigenvaluesare
contained within the interval. The projectoronto this subspace is written (cf.
of Chap.):
=
=1
(E-24)
(the eigenvaluesof the intervalcan be degenerate, hence the additional index; the vectors
are assumed to be orthonormal).is the subspace formed by all the possible states of
the system after a measurement which has given the result yes.
Referring to the denition of the measurement device, we see that the response will
certainly be yes for any state belonging to, that is, for any eigenstate ofwith the
eigenvalue of+1. The answer will certainly be no for any state belonging to the supplement of
, that is, for any eigenstates ofwith the eigenvalue of 0. The yes and no answers which
can be furnished by the measurement device therefore correspond to the eigenvalues+1and 0
of the observable: it could be said that the device is actually measuring the observable
rather than.
In the light of this interpretation, the case of an insuciently selective measurement device
can be treated in the framework of the postulates which we have stated. The probability(yes)
of obtaining the answer yes is equal to the probability of nding the (degenerate) eigenvalue
+1of. Now an orthonormal basis in the corresponding eigensubspace is constituted by the
set of stateswhich are eigenstates ofwith eigenvalues contained within the interval.
Applying formula (B-7) to the eigenvalue+1 of the observablewe therefore obtain (for a
system in the state):
(yes) =
=1
2
(E-25)
Since there are only two possible answers, we obviously have:
(no) = 1(yes) (E-26)
The projector onto the eigensubspace associated with the eigenvalue+1of the observable
isitself; formula (B-14) therefore gives here:
(yes) = (E-27)
[this formula is equivalent to (E-25)].
Similarly, since the device does not perturb states belonging toand blocks those of
the supplement of, we nd that the state of the system after a measurement which has given
the result yes is:
=
1
=1
2
=1
(E-28)
that is:
=
1

(E-29)
262

E. THE SUPERPOSITION PRINCIPLE AND PHYSICAL PREDICTIONS
Whencontains only one eigenvalue,reduces to: formulas (B-14) and (B-31) are
then seen to be special cases of formulas (E-27) and (E-29).
E-2-c. Recapitulation: must one sum the amplitudes or the probabilities?
There are therefore cases () where, to calculate a probability, one takes the
square of a sum, because several probability amplitudes must be added together. In other
cases (), one takes a sum of squares, because several probabilities must be added
together. It is clearly important not to confuse these dierent cases and to know, in a
given situation, if it is the probability amplitudes or the probabilities themselves which
must be summed.
Young's double-slit experiment will again furnish us with a very convenient physical
example which will enable us to illustrate and summarize the preceding discussions.
Imagine that we want to calculate the probability for a particular photon to strike the
plate anywhere between two points1and 2having abscissas of1and2(Fig.).
This probability is proportional to the total light intensity received by this portion of the
plate. It is therefore a sum of squares; more precisely, it is the integral of the intensity
()between1and2. But each term()of this sum is obtained by squaring the
electric eldE()at, which is equal to the sum of the electric eldsE()andE()
radiated atby the slitsand.()is therefore proportional toE() +E()
2
,
that is, to the square of a sum.E()andE()are the amplitudes associated with
the two possible pathsSAMandSBMwhich end at the same point; they are added
to obtain the amplitude atsince one is not trying to determine through which slit
the photon passes. Then, to calculate the total light intensity received by the interval
12, one adds the intensities which arrive at the various points of this interval.
To sum up, the fundamental idea to be retained from the discussions of this section
can be expressed schematically in the following way:
Add the amplitudes corresponding to the same nal state, then the
probabilities corresponding to orthogonal nal states.A
S
B
M
2
M
1
x
2
x
x
2
0
M
Figure 8: Young's double-slit exper-
iment. To calculate the probabil-
ity density for detecting a photon
at the point, it is necessary to
add the electric elds radiated by
the slitsand, then to square the
eld thus obtained (square of the
sum). The probability of nding
a photon in the interval[12]is
now obtained by summing this prob-
ability density between1and2
(sum of squares).
263

CHAPTER III THE POSTULATES OF QUANTUM MECHANICS
E-2-d. Application to the treatment of continuous spectra
When the observable we want to measure has a continuous spectrum, only insuciently
selective devices can be used: it is impossible to imagine a physical device that could isolate a
single eigenvalue belonging to a continuous set. We now show how the study of
us to be more precise and complete in our treatment of observables with continuous spectra.
. Example: measurement of the position of a particle
Let(r) =rbe the wave function of a (spinless) particle. What is the probability of
nding the abscissa of this particle within the interval[12]of the-axis, using, for example,
a measurement device like the one in Figure?
The subspaceassociated with this measurement result is the space spanned by the
ketsr= which are such that:1 2. Since these kets are orthonormal in the
extended sense, application of the rule stated in
(1 2) =
2
1
d
+
d
+
d
2
=
2
1
d
+
d
+
d(r)
2
(E-30)
Formula (E-27) obviously leads to the same result, since the projectoris written here:
=
2
1
d
+
d
+
d (E-31)
and we therefore have:
(1 2) =
=
2
1
d
+
d
+
d (E-32)
To know the state of the particle after such a measurement, which has yielded the
result yes, it suces to apply formula (E-29):
=
1

=
1
2
1
d
+
d
+
d (E-33)
where the normalization factor=
is known [formula (E-32)]. Let us calculate the
wave function(r) =r associated with the ket:
r=
1
2
1
d
+
d
+
drr(r) (E-34)
Nowrr=(rr) =( )( )( ). The integrations overandcan therefore
be performed immediately: they amount to replacingandbyandin the function to be
integrated. Equation (E-34) thus becomes:
( ) =
1
2
1
d( )( ) (E-35)
264

E. THE SUPERPOSITION PRINCIPLE AND PHYSICAL PREDICTIONS
If the pointis situated inside the interval of integration[12], the result is the same as if
we were integratingfrom to+:
( ) =
1
( ) for 1 2 (E-36)
On the other hand, iffalls outside the interval of integration,( )is zero for all values of
included in this interval, and:
( ) = 0 for 2and 1 (E-37)
The part of(r)that corresponds to the interval accepted by the measurement device therefore
persists, undeformed, immediately after the measurement [the factor 1/simply ensures that
(r)remains normalized]; the rest is suppressed by the measurement. The wave packet(r)
representing the initial state of the particle is, as it were, truncated by the edges of the slit.
Comments:
()This example clearly reveals the concrete meaning of the reduction of the wave
packet.
()If a large number of particles, all in the same state, enter the device successively,
the result will sometimes be yes and sometimes be no [with the probabilities(yes)
and(no)]. If the result is yes, the particle continues on its way, starting from the
truncated state; if the result is no, the particle is absorbed by the screen.
In the example we are considering here, the measurement device becomes all the more
selective as2 1becomes smaller. We see, however, that it is impossible to make it perfectly
selective because the spectrum ofis continuous: however narrow the slit may be, the interval
[12]which it denes always contains an innity of eigenvalues. Nevertheless, in the limiting
case of a slit of an innitely small width, we nd the equivalent of formula (B-17), which
was the expression of the fourth postulate in the case of a continuous spectrum.
Let us choose1=0

2
and2=0+

2
(a slit of widthcentered at0), and
assume that the wave function(r)varies very little within the interval. Then, in (E-30),
we can replace(r)
2
by(0 )
2
and perform the integration over:
0

2
0+

2

+
d
+
d(0 )
2
(E-38)
We indeed nd a probability equal to the product ofand a positive quantity which plays
the role of a probability density at the point0. The dierence with formula (B-17) lies in the
fact that the latter applies to the case of a continuous but non-degenerate spectrum, while here
the eigenvalues ofare innitely degenerate inr; this is the origin of the integrals overand
that appear in (E-38) (summation over the indices associated with the degeneracy).
. Postulate of reduction of wave packets in the case of a continuous spectrum
In , we limited ourselves, in the statement of the fth postulate, to the case of a
discrete spectrum. Formula (E-33) and its accompanying discussion enable us to understand the
form assumed by this postulate when a continuous spectrum is considered: one simply applies
the results of
Letbe an observable with a continuous spectrum (assumed, for simplicity, to be non-
degenerate). The notation is the same as in .
265

CHAPTER III THE POSTULATES OF QUANTUM MECHANICS
If a measurement ofon a system in the statehas yielded the result0to within
, the state of the system immediately after this measurement is described by:
=
1
(0)
(0) (E-39)
with:
(0) =
0+

2
0

2
d (E-40)
Figures-a and-b illustrate this statement. If the function, representingin
the basis, has the form indicated in Figure-a, the state of the system immediately after
the measurement is represented, to within a normalization factor, by the function of Figureb
[the calculation is analogous in all respects to the one which derives (E-36) and (E-37) from
(E-33)].Δ
0
0
0
0
υ

P
Δ
(
0
)

ψ υ

ψ
Figure 9: Illustration of the postulate of reduction of wave packets in the case of a con-
tinuous spectrum: one measures the observable, with eigenvectorsand eigenvalues
. The measurement device has a selectivity. If the value found is0to within,
the eect of the measurement on the wave functionis to truncate it about the
value0(to normalize the new wave function, it is obviously necessary to multiply it by
a factor larger than 1).
We see that, even ifis very small, one can never actually prepare the system in the
state
0, which would be represented, in the basis, by
0=( 0). We can
only obtain a narrow function centered at0, sinceis never exactly zero.
References and suggestions for further reading:
Development of quantum mechanical concepts: references of section 4 of the bib-
liography, particularly Jammer (4.8).
Discussion and interpretation of the postulates: references of section 5 of the
bibliography; Von Neumann (10.10), Chaps. V and VI; Feynman III (1.2), 2.6,
Chap. 3 and 8.3.
Quantization rules using Poisson brackets: Dirac (1.13), 21; Schi (1.18), 24.
Probability and statistics: see the corresponding subsection of section 10 of the
bibliography.
266

COMPLEMENTS OF CHAPTER III, READER'S GUIDE
AIII: PARTICLE IN AN INFINITE ONE-
DIMENSIONAL POTENTIAL WELL
BIII: STUDY OF THE PROBABILITY CURRENT
IN SOME SPECIAL CASES
Direct applications of Chapter
The accent is placed on the physical discussion of
the results (elementary level).
CIII: ROOT MEAN SQUARE DEVIATIONS OF
TWO CONJUGATE OBSERVABLES
A little more formal; general proof of the Heisen-
berg relations; may be skipped in a rst reading.
DIII: MEASUREMENTS BEARING ON ONLY ONE
PART OF A PHYSICAL SYSTEM
Discussion of measurements bearing on only one
part of a system; a rather simple but somewhat
formal application of Chapter; may be skipped
on a rst reading.
(more complements on the next page)
267

EIII: THE DENSITY OPERATOR
FIII: THE EVOLUTION OPERATOR
GIII: THE SCHRÖDINGER AND HEISENBERG
PICTURES
HIII: GAUGE INVARIANCE
JIII: PROPAGATOR FOR THE SCHRÖDINGER
EQUATION
Complements that serve as an introduction to a
more advanced quantum mechanics course. Aside
from FIII, which is simple, they are on a higher
level than the rest of this book, but they are
comprehensible if Chapter
be reserved for subsequent study.
EIII: denition and properties of the density op-
erator, which is used in the quantum mechanical
description of systems whose state is imperfectly
known (statistical mixture of states). Fundamen-
tal tool of quantum statistical mechanics.
FIII: introduction of the evolution operator,
which gives the quantum state of a system at an
arbitrary instantin terms of its state at the
instant0.
GIII: describes the evolution of a quantum
system in a way that is dierent from, but
equivalent to, that of Chapter . The time
dependence now appears in the observables and
not in the state of the system.
HIII: discussion of the quantum formalism in the
case where the system is subject to an electro-
magnetic eld. Although the description of the
system involves the electromagnetic potentials,
the physical properties depend only on the values
of the electric and magnetic elds; they remain
invariant when the potentials describing the same
electromagnetic eld are changed.
JIII: an introduction to a dierent way of ap-
proaching quantum mechanics, based on a prin-
ciple analogous to Huygens' principle in classical
wave optics.
KIII: UNSTABLE LEVELS, LIFETIMES Simple introduction to the important physical
concepts of instability and lifetimes; easy, but can
be skipped in a rst reading.
LIII: EXERCISES
(more complements on the next page)
268

MIII: BOUND STATES OF A PARTICLE IN A
POTENTIAL WELL OF ARBITRARY SHAPE
NIII: UNBOUND STATES OF A PARTICLE IN
THE PRESENCE OF A POTENTIAL WELL OR
BARRIER OF ARBITRARY SHAPE
OIII:QUANTUM PROPERTIES OF A PARTICLE
IN A ONE-DIMENSIONAL PERIODIC STRUCTURE
Return to one-dimensional problems, considered
from a more general point of view than in
Chapter
MIII: generalisation to an arbitray potential
well of the main results obtained in
Complement I; recommended, since easy and
physically important.
NIII: study of unbound stationary states in
an arbitrary potential; a little more formal; the
denitions and results of this complement are
necessary for complementIII.
OIII: introduction of the concept (which is funda-
mental to solid state physics) of energy bands in a
potential having a periodic structure (this concept
will be treated dierently in ComplementXI);
rather dicult, can be reserved for later reading.
269

PARTICLE IN AN INFINITE ONE-DIMENSIONAL POTENTIAL WELL
Complement AIII
Particle in an innite potential well
1 Distribution of the momentum values in a stationary state
1-a Calculation of the function
()of, and, of. . . . .
1-b Discussion
2 Evolution of the particle's wave function
2-a Wave function at the instant. . . . . . . . . . . . . . . . .
2-b Evolution of the shape of the wave packet
2-c Motion of the center of the wave packet
3 Perturbation created by a position measurement
In ComplementI( ), we studied the stationary states of a particle in a
one-dimensional innite potential well. Here we intend to re-examine this subject from a
physical point of view. This will allow us to apply some of the postulates of Chapter
to a concrete case. We shall focus in particular on the results obtained when the position
or momentum of the particle is measured.
1. Distribution of the momentum values in a stationary state
1-a. Calculation of the function
()of, and, of
We have seen that the stationary states of the particle correspond to the energies
1
:
=
22
~
2
2
2
(1)
and to the wave functions:
() =
2
sin (2)
(whereis the width of the well andis any positive integer).
Consider a particle in the state, with energy. The probability of a mea-
surement of the momentumof the particle yielding a result betweenand+ dis:
()d=
()
2
d (3)
with:
() =
1
2~02
sine
~
d (4)
1
We shall use the notation of ComplementI.
271

COMPLEMENT A III
This integral is easy to calculate; it is equal to:
() =
1
2~ 0
e
~ e
+
~d
=
1
2~
e
~ 1
~
e
+
~ 1
+
~
(5)
that is:
() =
1
2~
e
22~
~+ (1)
+1
+
~
(6)
with:
() =
sin(2~)
2~
(7)
To within a proportionality factor, the function
()is the sum (or the dierence)
of two diraction functions
~
, centered at=
~
. The width of
these functions (the distance between the rst two zeros, symmetric with respect to the
central value) does not depend onand is equal to
4~
. Their amplitude does not
depend oneither.
The function inside brackets in expression (6) is even ifis odd, and odd ifis
even. The probability density
()given in (3) is therefore an even function ofin all
cases, so that:
=
+
()d= 0 (8)
The mean value of the momentum of the particle in the energy stateis therefore zero.
Let us calculate, in the same way, the mean value
2
of the square of the mo-
mentum. Using the fact that in therepresentationacts like
~
d
d
, and performing
an integration by parts, we obtain
2
:
2
=~
2
0
d
d
2
d
=~
2
0
2
2
cos
2
d
=
~
2
(9)
2
Result (9) could also be derived from (6) by performing the integral
2
=
+()
22
d.
This calculation, which presents no theoretical diculties, is nevertheless not as direct as the one given
here.
272

PARTICLE IN AN INFINITE ONE-DIMENSIONAL POTENTIAL WELL
From (8) and (9), we get:
=
2 2
=
~ (10)
The root mean square deviation therefore increases linearly with.
1-b. Discussion
Let us plot, for dierent values of, the curves giving the probability density
(). To do this, let us begin by studying the function inside brackets in expression
(6). For the ground state (= 1), it is the sum of two functions, the centers of
these two diraction curves being separated by half their width (Fig.-a). For the rst
excited level(= 2), the distance between these centers is twice as large, and in this
case, moreover, the dierence of two functionsmust be taken (Fig.-a). Finally, for
an excited level corresponding to a large value of, the centers of the two diraction
curves are separated by a distance much greater than their width.a
πh
a
0
0
p
pp FF
p
πh

1
(p)
a
a
b
πh
a
πh
a
πh
a
πh
Figure 1: The wave function
1(), associated in therepresentation with the ground
state of a particle in an innite well, is obtained by adding two diraction functions
(dashed lines curves in gure a). Since the centers of these two functionsare
separated by half their width, their sum has the shape represented by the solid-line curve
in gure a. Squaring this sum, one obtains the probability density
1()associated with
a measurement of the momentum of the particle (g. b).
Squaring these functions, one obtains the probability density
()(cf.Fig.-
b and-b). Note that for largethe interference term between
~
and
+
~
is negligible (because of the separation of the centers of the two curves):
273

COMPLEMENT A IIIp
p
p
F
a
2πh
a
2πh
a
2πh
a
0
2πh
pF
a
2πh
a
2πh

2
(p)
a
b
Figure 2: For the rst excited level, the function
2()is obtained by taking the dierence
between two functions, which have the same width as in Figure-a but are now more
widely separated (dash-line curves in gure a). The curve obtained in this way is the
solid line in gure a. The probability density
2()then has two maxima located in the
neighborhood of=2~(g. b).
() =4~
~
+ (1)
+1
+
~
2
4~
2
~
+
2
+
~
(11)
The function
()then has the shape shown in Figure.
It can be seen that whenis large, the probability density has two symmetrical
peaks, of width
4~
, centered at=
~
. It is then possible to predict with almost
complete certainty the results of a measurement of the momentum of the particle in the
state: the value found will be nearly equal to+
~
or
~
, the relative accuracy
3
improving asincreases (the two opposite values
~
being equally probable). This is
simple to understand: for large, the function(), which varies sinusoidally, performs
numerous oscillations inside the well; it can then be considered to be practically the sum
of two progressive waves corresponding to opposite momenta=
~
.
Whendecreases, the relative accuracy with which one can predict the possible
values of the momentum diminishes. We see, for example, in Figure-b, that when= 2,
3
The absolute accuracy is independent of, since the width of the curves is always
4~
.
274

PARTICLE IN AN INFINITE ONE-DIMENSIONAL POTENTIAL WELLa
nπh nπh
a
p
p
cl
p
cl
0
!
n
(p)
Figure 3: Whenis large (a very excited level), the probability density has two pro-
nounced peaks, centered at the values= ~, which are the momenta associated
with the classical motion at the same energy.
the function
()has two peaks whose widths are comparable to their distance from
the origin. In this case, the wave function undergoes only one oscillation inside the well.
It is not surprising that, for this sinusoid truncated at= 0and=, the wavelength
(and therefore, the momentum of the particle) is poorly dened. Finally, for the ground
state, the wave function is represented by half a sinusoidal arc: the relative values of the
wavelength and momentum of the particle are then very poorly known (Fig.-b).
Comments:
()Let us calculate the momentum of a classical particle of energygiven in
(1); we have:
2
2
=
22
~
2
2
2
(12)
that is:
=
~
(13)
Whenis large, the two peaks of
()therefore correspond to the classical
values of the momentum.
()We see that, for large, although the absolute value of the momentum is well-
dened, its sign is not. This is whyis large: for probability distributions
with two maxima like that of Figure, the root mean square deviation reects
the distance between the two peaks; it is no longer related to their widths.
2. Evolution of the particle's wave function
Each of the states, with its wave function(), describes a stationary state, which
leads to time-independent physical predictions. Time evolution appears only when the
275

COMPLEMENT A III
state vector is a linear combination of several kets. We shall consider here a very
simple case, for which at time= 0the state vector(0)is:
(0)=
1
2
[1+ 2] (14)
2-a. Wave function at the instant
Apply formula (D-54) of Chapter; we immediately obtain:
()=
1
2
e
2
~
2
2
1+ e
2
2
~
2
2 (15)
or, omitting aglobalphase factor of():
()
1
2
1+ e
21
2 (16)
with:
21=
2 1
~
=
3
2
~
2
2
(17)
2-b. Evolution of the shape of the wave packet
The shape of the wave packet is given by the probability density:
()
2
=
1
2
2
1() +
1
2
2
2() +1()2() cos21 (18)
We see that the time variation of the probability density is due to the interference term
in12. Only one Bohr frequency appears,21= (2 1), since the initial state0 0
a b c φ
1
φ
2φ
1
2
φ
2
2
0
x
a
x
a
x
a
Figure 4: Graphical representation of the functions
2
1(the probability density of the
particle in the ground state),
2
2(the probability density of the particle in the rst excited
state) and12(the cross term responsible for the evolution of the shape of the wave
packet).
276

PARTICLE IN AN INFINITE ONE-DIMENSIONAL POTENTIAL WELLt = 0 0 < t < π/2ω
21
t = π/2ω
21
t = π/ω
21
t = 3π/2ω
21
t = 2π/ω
21
Figure 5: Periodic motion of a wave packet obtained by superposing the ground state and
the rst excited state of a particle in an innite well. The frequency of the motion is the
Bohr frequency212.
(14) is composed only of the two states1and2. The curves corresponding to the
variation of the functions
2
1,
2
2and12are plotted in Figures-a, b and c.
Using these gures and relation (18), it is not dicult to represent graphically the
variation in time of the shape of the wave packet (cf.Fig.): we see that the wave packet
oscillates between the two walls of the well.
2-c. Motion of the center of the wave packet
Let us calculate the mean value()of the position of the particle at time. It
is convenient to take:
= 2 (19)
since, by symmetry, the diagonal matrix elements ofare zero:
1 1
0
2
sin
2
d= 0
2 2
0
2
sin
2
2
d= 0 (20)
We then have:
() = Ree
21
1 2 (21)
277

COMPLEMENT A III
with:
1 2= 1 2
2
12
=
2
0
sinsin
2
d
=
16
9
2
(22)
Therefore:
() =
2
16
9
2
cos21 (23)0
a
X
2π/ω
21
t
Figure 6: Time variation of the mean valuecorresponding to the wave packet's
motion plotted in Figure. The dashed line represents the position of a classical particle
moving with the same period. Quantum mechanics predicts that the center of the wave
packet will turn back before reaching the wall, as explained by the action of the potential
on the edges of the wave packet.
The variation of()is represented in Figure. In dashed lines, the variation
of the position of a classical particle has been traced, for a particle moving to and fro in
the well with an angular frequency of21(since it is not subjected to any force except
at the walls, its position varies linearly withbetween 0 andduring each half-period).
We immediately notice a very clear dierence between these two types of motion,
classical and quantum mechanical. The center of the quantum wave packet, instead of
turning back at the walls of the well, executes a movement of smaller amplitude and
retraces its steps before reaching the regions where the potential is not zero. We see
again here a result of : since the potential varies innitely quickly at
= 0and=, its variation within a domain of the order of the dimension of the wave
packet is not negligible, and the motion of the center of the wave packet does not obey the
laws of classical mechanics (see also Chapter, ). The physical explanation
of this phenomenon is the following: before the center of the wave packet has touched
the wall, the action of the potential on the edges of this packet is sucient to make it
turn back.
278

PARTICLE IN AN INFINITE ONE-DIMENSIONAL POTENTIAL WELL
Comment:
The mean value of the energy of the particle in the state()calculated in (15)
is easy to obtain:
=
1
2
1+
1
2
2=
5
2
1 (24)
as is:
2
=
1
2
2
1+
1
2
2
2=
17
2
2
1 (25)
which gives:
=
3
2
1 (26)
Note in particular that,
2
andare not time-dependent; this could
have been foreseen, sinceis a constant of the motion. In addition, we see from
the preceding discussion that the wave packet evolves appreciably over a time of
the order of:

1
21
(27)
Using (26) and (27), we nd:

3
2
1
~
31
=
~
2
(28)
We again nd the time-energy uncertainty relation ( ).
3. Perturbation created by a position measurement
Consider a particle in the state1. Assume that the position of the particle is measured
at time= 0, with the result=2. What are the probabilities of the dierent results
that can be obtained in a measurement of the energy, performed immediately after this
rst measurement?
One must beware of the following false argument: after the measurement, the
particle is in the eigenstate ofcorresponding to the result found, and its wave function
is therefore proportional to( 2); if a measurement of the energy is then performed,
the various valuescan be found, with probabilities proportional to:
0
d
2
()
2
=
2
2
=
2ifis odd
0 ifis even
(29)
Using thisincorrectargument, one would nd the probabilities of all values ofcorre-
sponding to oddto be equal. This is absurd, since the sum of these probabilities would
then be innite.
279

COMPLEMENT A III
This error results from the fact that we have not taken the norm of the wave
function into account. To apply the fourth postulate of Chapter
necessary to write the wave function as normalized just after the rst measurement.
However it is not possible
4
to normalize the function( 2). The problem posed
above must be stated more precisely.
As we saw in , an experiment in which the measurement
of an observable with a continuous spectrum is performed never yields any result with
complete accuracy. For the case with which we are concerned, we can only say that:
222
+
2
(30)
wheredepends on the measurement device used but is never zero.
If we assumeto be much smaller than the extension of the wave function before
the measurement (here), the wave function after the measurement will be practically
()
2
[
()
()is the null function everywhere except in the interval dened
in (30), where it takes on the value1;cf.Appendix, ]. This wave function is
indeed normalized since:
d
()
2
2
= 1 (31)ε
2a
1 3 5 7 9 11 13 15 17
0
a
n
ε(E
n
)
2ε
Figure 7: Variation withof the probability()of nding the energyafter a
measurement of the particle's position has yielded the result2with an accuracy of
( ). The smaller, the greater the probability of nding high energy values.
What happens now if the energy is measured? Each valuecan be found with
4
We see concretely in this example that a-function cannot represent a physically realizable state.
280

PARTICLE IN AN INFINITE ONE-DIMENSIONAL POTENTIAL WELL
the probability:
() = ()
()
2
d
2
=
8
1
2
sin
2
2
ifis odd
0 if is even
(32)
The variation with respect toof(), for xedand odd, is shown in
Figure. This gure shows that the probability()becomes negligible whenis
much larger than. Therefore, however smallmay be, the distribution of probabilities
()depends strongly on. This is why, in the rst argument, where we set= 0at
the beginning, we could not obtain the correct result. We also see from the gure that the
smalleris, the more the curve extends towards large values of. The interpretation of
this result is the following: according to Heisenberg's uncertainty relations (cf.Chap.,
), if one measures the position of the particle with great accuracy, one drastically
changes its momentum. Thus kinetic energy is transferred to the particle, the amount
increasing asdecreases.
281

STUDY OF THE PROBABILITY CURRENT IN SOME SPECIAL CASES
Complement BIII
Study of the probability current in some special cases
1 Expression for the current in constant potential regions
2 Application to potential step problems
2-a Case where 0. . . . . . . . . . . . . . . . . . . . . . . .
2-b Case where 0. . . . . . . . . . . . . . . . . . . . . . . .
3 Probability current of incident and evanescent waves, in
the case of reection from a two-dimensional potential step
The probability current associated with a particle having a wave function(r)
was dened in Chapter
J(r) =
~
2
[(r)r(r)c.c.] (1)
(where c.c. is an abbreviation for complex conjugate). In this complement, we shall study
this probability current in greater detail in some special cases: one- and two-dimensional
square potentials.
1. Expression for the current in constant potential regions
Consider a one-dimensional problem, with a particle of energyplaced in a constant
potential0. In ComplementI, we distinguished between several cases.
()When 0, the wave function is written:
() =e+e (2)
with:
0=
~
22
2
(3)
Substituting (2) into (1), we obtain:
=
~
2 2
(4)
The interpretation of this result is simple: the wave function given in (2) corresponds to
two plane waves of opposite momenta=~with probability densities
2
and
2
.
()When 0, we have:
() =e+e (5)
with:
0 =
~
22
2
(6)
283

COMPLEMENT B IIIV
0
V(x)
0
x
III
Figure 1: Potential step of height0.
Substituting (5) into (1), we obtain:
=
~
[ +c.c.] (7)
In this case, we see that the two exponential waves must both have non-zero coecients
for the probability current to be non-zero.
2. Application to potential step problems
Let us apply these results to the potential step problems studied in ComplementsI,
andI. We shall therefore consider a particle of massand energypropagating in
thedirection and arriving at= 0at a potential step of height0(Fig.).
2-a. Case where 0
Apply formula (4) to wave functions (11) and (12) of ComplementI, setting, as
in that complement:
2= 0 (8)
In region I, the probability current is:
I=
~1
1
2
1
2
(9)
and in region II:
II=
~2
2
2
(10)
Iis the dierence of two terms, the rst one corresponding to the incident current
and the second, to the reected current. The ratio of these two currents gives the
reection coecientof the barrier:
=
1
1
2
(11)
which is precisely formula (15) of ComplementI.
284

STUDY OF THE PROBABILITY CURRENT IN SOME SPECIAL CASES
Similarly, the transmission coecientof the barrier is the ratio of the transmitted
currentIIto the incident current; we therefore have:
=
2
1
2
1
2
(12)
and we again nd relation (16) of ComplementI.
2-b. Case where 0
Since the expression for the wave function1()is the same as in , relation
(9) is still valid. However, in region II, the wave function is:
II() =
2e
2
(13)
[since, in equation (20) of ComplementI,2= 0]. Using (7), we thus obtain:
II= 0 (14)
The transmitted ux is zero, as is consistent with relation (24) ofI.
How should we interpret the fact that, in region II, the probability current is
zero while the probability of nding the particle in this region is not? Let us refer to
the results obtained in 1 of ComplementI. We saw that part of the incident wave
packet enters the classically forbidden region II, and then turns back, before setting out
in the negativedirection (this incursion into region II being responsible for the delay
upon reection). In the steady state, we shall therefore have two probability currents in
region II: a positive current corresponding to the entrance into this region of part of the
incident wave packet; a negative current corresponding to the return towards region I of
this part of the wave packet. These two currents are exactly equal, so the overall result
is zero.
In the case of a one-dimensional problem, the structure of the probability current
of the evanescent wave is therefore masked by the fact that the two opposite currents
balance. This is why we are going to consider a two-dimensional problem, for the case
of oblique reection, so as to obtain a non-zero current and interpret its structure.
3. Probability current of incident and evanescent waves, in the case of reection
from a two-dimensional potential step
We shall consider the following two-dimensional problem: a particle of mass, moving
in the plane, has a potential energy()which is independent ofand given by:
() = 0 if 0
() =0if 0 (15)
The present case corresponds to the one studied in 2 of ComplementI: the
potential energy()is the sum of a function1()(potential energy of a one-
dimensional step) and a function2(), which is zero here. We can therefore look for a
solution of the eigenvalue equation of the Hamiltonian in the form of a product:
() =1()2() (16)
285

COMPLEMENT B III
The functions1()and2()satisfy one-dimensional eigenvalue equations which cor-
respond respectively to1()and2()and to energies1and2such that:
1+2= (total energy of the particle) (17)
We shall assume1 0: the equation giving1()therefore corresponds to
total reection in a one-dimensional problem, and we can use formulas (11) and (20)
of ComplementI. As for the function2(), it can be obtained immediately since it
corresponds to the case of a free particle (2= 0): it is a plane wave. We therefore have,
in region I (0):
1() =e
(+ )
+e
( + )
(18)
with:
=
2 1~
2
=
2 2~
2
(19)
and, in region II (0):
II() =e e (20)
with:
=
2(0 1)~
2
(21)
Equations (22) and (23) ofIgive us the ratiosand . Introducing the
parameterdened by:
tan=
=
0 11
; 0
2
(22)
we obtain:
=
+
= e
2
(23)
and:
=
2
+
= 2 cose (24)
Let us apply relation (1), which denes the probability current. We obtain, in
region I:
JI
(I)=
~
2 2
= 0
(I)=
~
e+e
2
=
~
2
[2 + 2 cos(2+ 2)]
(25)
286

STUDY OF THE PROBABILITY CURRENT IN SOME SPECIAL CASES2k
y
k
r
k
i
Figure 2: The sum of the probability cur-
rents associated with the incident and re-
ected waves yields a probability current par-
allel to.
and in region II:
JII
(II)= 0
(II)=
~
2
e
2
=
~
4
2
cos
2
e
2
(26)
In region I, only the(I)component of the probability current is non-zero; this compo-
nent is the sum of two terms:I II
0
x
J
y
Figure 3: Because of the interference between the incident and reected waves, the prob-
ability current in region I is an oscillatory function of; in region II, it decreases expo-
nentially (evanescent wave).
the term proportional to2
2
which results from the sum of the currents of the
incident and reected waves (cf.Fig.);
the term containingcos(2+ 2), which represents an interference eect be-
tween the two waves; it is responsible for the oscillation of the probability current with
respect to(cf.Fig.).
In region II, the probability current is again parallel to. Its exponential decay
corresponds to the decay of the evanescent wave. This probability current arises from
the fact that the wave packets do enter the second region (cf.Fig.) and, before turning
back, propagate in thedirection for a time of the order of the reection delay[cf.
ComplementI, equation (10)]. This penetration is also related to the lateral shift of the
wave packet upon reection (cf.Fig.).
287

COMPLEMENT B III
Figure 4: The penetration of the particle into region II leads to a lateral shift upon
reection.
288

ROOT MEAN SQUARE DEVIATIONS OF TWO CONJUGATE OBSERVABLES
Complement CIII
Root mean square deviations of two conjugate observables
1 The Heisenberg relation for and. . . . . . . . . . . . . .
2 The minimum wave packet
Two conjugate observablesandare two observables whose commutator[]
is equal to~. We shall show in this complement that the root mean square deviations
(cf. )and, for any state vector of the system under study,
satisfy the relation:

~
2
(1)
We shall then show that if the system is in a state where the productis ex-
actly equal to~2, the wave function associated with this state in therepresentation
is a Gaussian wave packet (as is the wave function in therepresentation).
1. The Heisenberg relation forand
Consider the ket:
= (+ ) (2)
whereis an arbitrary real parameter. For all, the square of the normis positive.
This is written:
=( )(+ )
=
2
+( )+
22
=
2
+[]+
22
=
2
~+
22
0 (3)
The discriminant of this expression, of second order in, is therefore negative or zero:
~
2
4
2 2
0 (4)
and we have:
2 2
~
2
4
(5)
289

COMPLEMENT C III
Assuming to be given, let us now introduce the two observablesand
dened by:
= =
= = (6)
andare also conjugate observables, since we have:
[ ] = [] =~ (7)
Result (5), obtained above forand, is therefore also valid forand:
2 2
~
2
4
(8)
In addition, referring to denition (C-23) (Chap.) of the root mean square
deviation and using (6), we see that:
=
2
=
2
(9)
Relation (8) can therefore also be written:

~
2
(10)
Thus, if two observables are conjugate (as is the case when they correspond to a
classical positionand its conjugate momentum), there exists an exact lower bound
for the product. We thus generalize the Heisenberg uncertainty relation.
Comment:
This argument can easily be generalized to two arbitrary observablesand.
One obtains:

1
2
[] (11)
2. The minimum wave packet
When the minimum value of the productis attained:
=
~
2
(12)
the state vectoris said to correspond to a minimum wave packet for the observables
and.
290

ROOT MEAN SQUARE DEVIATIONS OF TWO CONJUGATE OBSERVABLES
According to the preceding argument, relation (12) requires that the square of the
norm of the ket:
= (+ ) (13)
be a second-order polynomial inwith a double root0. When=0, the ketis
therefore zero:
(+ 0)= [ + 0( )]= 0 (14)
On the other hand, if ~2, the polynomial which gives can never be
equal to zero (it is positive for all).
Therefore, the necessary and sucient condition for the productto take
on its minimum value~2is that the kets( )and( )be proportional.
The proportionality coecient0can easily be calculated. When=~2, the
equation:
=
2
()
2
~+ ()
2
= 0 (15)
has for its double root:
0=
~
2()
2
=
2()
2
~
(16)
Let us write relation (14) in the representation (for simplicity, we assume
that the eigenvaluesofto be non-degenerate). Using the fact (cf.ComplementII)
that in this representation,acts like
~
d
d
, we obtain:
+~0
d
d
0 () = 0 (17)
with:
() = (18)
To integrate equation (17), it is convenient to introduce the function()dened by:
() = e
~
( ) (19)
Substituting (19) into (17), we thus obtain a more simple equation:
+0~
d
d
() = 0 (20)
whose solution is:
() =e
2
20~
(21)
(whereis an arbitrary complex constant). Substituting (16) and (21) into (19), we
obtain:
() =e
~
e
2
2
(22)
291

COMPLEMENT C III
This function can be normalized by setting:
=2()
2
14
(23)
We thus arrive at the following conclusion: when the producttakes on
its minimum value~2, the wave function in the representation is a Gaussian
wave packet, obtained from the Gaussian function()by transformation (19) (which is
equivalent to two changes of the origin, one on the-axis and one on the-axis).
Comment:
This argument in the representation can be repeated in therepresen-
tation. One then nds that the wave function
()dened by:() = =
1
2~
+
de
~
() (24)
is also a Gaussian function, given by:
() =2()
2
14
e
~
e
2
2
(25)
to within a phase factor exp( }).
292

MEASUREMENTS BEARING ON ONLY ONE PART OF A PHYSICAL SYSTEM
Complement DIII
Measurements bearing on only one part of a physical system
1 Calculation of the physical predictions
2 Physical meaning of a tensor product state
3 Physical meaning of a state that is not a tensor product
The concept of a tensor product, introduced in , enabled us to
see how to construct, starting with the state spaces of two subsystems, that of the global
system obtained by considering them together. We intend to pursue this study here,
using the postulates of Chapter
of the global system is known, from measurements bearing on only one subsystem.
1. Calculation of the physical predictions
Consider a system composed of two parts (1) and (2) (for example, a system of two
electrons). If(1)and(2)are the state spaces of parts (1) and (2), the state space of the
global system(1)+(2)is the tensor product(1)(2). For example, the state of a two-
electron system is described by a wave function of six variables,(111;222),
associated with a ket ofr(1) r(2)(cf.Chap., ).
It is possible to imagine measurements that bear on only one of the two parts
[part (1), for example] of the global system. The observables
~
(1)corresponding to
these measurements are dened in(1)(2)by extending
1
the observables(1)acting
only in(1)(cf.Chap., ):
(1) =
~
(1) =(1)(2) (1)
where(2)is the identity operator in(2).
The spectrum of
~
(1)in(1)(2)is the same as that of(1)in(1). On
the other hand, we have seen that all the eigenvalues of
~
(1)are degenerate in(1)
(2), even if none of the eigenvalues of(1)is degenerate in(1)[on the condition, of
course, that the dimension of(2)be greater than 1]. When a measurement is made on
system (1) alone, the global system may therefore be in several dierent states after the
measurement, whatever the result (the state after the measurement depends not only on
the result but also on the state before the measurement). From a physical point of view,
this multiplicity of states corresponds to the degrees of freedom of system (2), about
which no information is sought in the measurement.
Let(1)be the projector, in(1), onto the eigensubspace related to the eigenvalue
of(1):
(1) =
=1
(1)(1) (2)
1
For the sake of clarity, we shall adopt throughout this complement dierent notations for(1)and
its extension
~
(1).
293

COMPLEMENT D III
where the kets(1)areorthonormal eigenvectors associated with. Let
~
(1)be
the projector, in(1)(2), onto the eigensubspace related to the same eigenvalue
of
~
(1).
~
(1)is obtained by extending(1)into(1)(2):
~
(1) =(1)(2) (3)
To write the identity operator(2)of(2), let us use the closure relation for an arbitrary
orthonormal basis(2)of(2):
(2) = (2)(2) (4)
Substituting (4) into (3) and using (2), we obtain:
~
(1) =
=1
(1)(2)(1)(2) (5)
Thus, knowing the stateof the global system (assumed to be normalized to 1),
we can calculate the probability
(1)
()of nding the resultin a measurement of
(1)on part (1) of this system. Using general formula (B-14) of Chapter, which here
gives:
(1)
() =
~
(1) (6)
we nd:
(1)
() =
=1
(1)(2)
2
(7)
Similarly, the stateof the system after the measurement can be calculated; according
to formula (B-31) of Chapter, it is given by:
=
~
(1)
~
(1)
(8)
that is, using (5):
=
=1
(1)(2)(1)(2)
=1
(1)(2)
2
(9)
Comments:
()The choice of an orthonormal basis(2)in(2)is arbitrary. We see from
(3), (6) and (8) that the predictions concerning subsystem (1) do not depend
on this choice. Physically, it is clear that if no measurement is performed on
system (2), no state or set of states of this system can play a preferential role.
294

MEASUREMENTS BEARING ON ONLY ONE PART OF A PHYSICAL SYSTEM
()If the statebefore the measurement is a tensor product:
=(1) (2) (10)
[where(1)and(2)are two normalized states of(1)and(2)respec-
tively], it is easy to see, using (3) and (8), that the stateis also a tensor
product:
=(1) (2) (11)
with:
(1)=
(1)(1)
(1)(1)(1)
(12)
The state of system (1) has therefore changed, but not that of system (2).
()If the eigenvalueof(1)is non-degenerate in(1) or, more generally, if
(1)actually represents a complete set of commuting observables of(1)
the indexis no longer necessary in formula (2) and those which follow. The
state of the system after a measurement yielding the resultcan always
be put in the form of a product of two vectors. This can be seen by writing
relation (9) in the form:
=(1) (2) (13)
where the normalized vector(2)of(2)is given by:
(2)=
(2)(1)(2)
(1)(2)
2
(14)
Therefore, whatever the stateof the global system before the measure-
ment, the state of the system after a measurement bearing on part (1) alone
is always a tensor product when this measurement is complete with respect
to part (1) [although partial as regards the global system(1) + (2)].
2. Physical meaning of a tensor product state
To see what a product state represents physically, let us apply the results of the preceding
paragraph to the particular case where the initial state of the global system is of the form
(10). We immediately obtain, using (6) and (3):
(1)
() =(1)(2)(1)(2)(1)(2) (15)
The very denition of the tensor product(1)(2)and the fact that(2)is nor-
malized then allow us to write:
(1)
() =(1)(1)(1)(2)(2)(2)
=(1)(1)(1) (16)
295

COMPLEMENT D III
(1)
()does not depend on(2), but only on(1). When the state of the global
system has the form (10), all physical predictions relating to only one of the two systems
do not, therefore, depend on the state of the other one and are expressed entirely in
terms of(1)[or of(2)], depending on whether it is system (1) alone [or system (2)
alone] that is being observed.
A product state(1) (2)can therefore be considered to represent the simple
juxtaposition of two systems, one in the state(1)and the other in the state(2).
In such a state, the two systems are also said to beuncorrelated(more precisely, the
results of the two types of measurements, bearing either on one system or on the other,
correspond toindependent random variables). Such a situation is realized when the two
systems have been separately prepared in the states(1)and(2)and then united
without interacting.
3. Physical meaning of a state that is not a tensor product
Now consider the case in which the state of the global system is not a product state,
that is, wherecannot be written in the form(1) (2). The predictions of
measurement results bearing on only one of the two systems can then no longer be
expressed in terms of a ket(1)[or(2)] in which system (1) [or (2)] would be
found. In this case, general formulas (6) and (7) must be used to nd the probabilities
of the various possible results. We assume here without proof that such a situation
generally reects the existence ofcorrelationsbetween systems (1) and (2). The results of
measurements bearing on either system (1) or system (2) correspond to random variables
which are not independent and can therefore be correlated. It can be shown, for example,
that an interaction between the two systems transforms an initial state which is a product
into one which is no longer a product: any interaction between two systems therefore
introduces, in general, correlations between them.
When the state of the global system is not a product(1) (2), how can each
partial system (1) or (2) be characterized, since the ket(1)or(2)can no longer
be associated with it? This question is very important since, in general, every physical
system has interacted with others in the past (even if it is isolated at the instant when
it is being studied). The state of the global system:system (1)+system (2) with
which it has interacted in the pastis therefore not in general a product state, and it
is not possible to associate a state vector(1)with system (1) alone. To resolve these
diculties, one must describe system (1), not by a state vector, but by an operator, called
thedensity operator. The corresponding formalism, fundamental to statistical quantum
mechanics, is introduced in ComplementIII(cf.).
However, system (1) can always be described by a state vector after a complete
set of measurements has been performed on it. We have seen that whatever the state
of the global system(1) + (2)before the measurement, a complete measurement on
system (1) places the global system in a product state [cf.formulas (13) and (14)]. The
vector associated with (1) is the unique eigenvector (to within a multiplicative factor)
associated with the results of the complete set of measurements done on it. This set of
measurements has therefore erased all correlations resulting from previous interactions
between the two systems. If, at the moment of measurement, system (2) is already far
away and no longer interacting with system (1), it can then be completely forgotten if
one is interested only in system (1).
296

MEASUREMENTS BEARING ON ONLY ONE PART OF A PHYSICAL SYSTEM
Comment:
Nervertheless, the quantum state(2)reached by system (2) depends in general
2
on the result of the measurements performed on system (1). This is easy to check
with (14), when the statebefore the measurement is not a product state. Such
a result may seem extremely surprising: how can the state of system (2) after a
series of measurements made on system (1) change as a function of the result of
measurements performed on an arbitrarily remote system (1), with which is does
not interact any longer? To this paradox, studied in detail by many physicists,
are attached the names of Einstein, Podolsky and Rosen; it is discussed in
of Chapter .
References and suggestions for further reading:
The Einstein-Podolsky-Rosen paradox/argument: see alsoDo we really under-
stand quantum mechanics?, F. Laloë, Cambridge University Press (2018), as well the
subsection Hidden variables and paradoxes of section 5 of the bibliography; Bohm
(5.1), 22.15 to 22.19; d'Espagnat (5.3), Chap. 7.
Photons produced in the decay of positronium: Feynman III (1.2), 18.3; Dicke
and Wittke (1.14), Chap. 7.
2
Recall that this is not the case whenis a product state;cf.comment()of
297

THE DENSITY OPERATOR
Complement EIII
The density operator
1 Outline of the problem
2 The concept of a statistical mixture of states
3 The pure case. Introduction of the density operator
3-a Description by a state vector
3-b Description by a density operator
3-c Properties of the density operator in a pure case
4 A statistical mixture of states (non-pure case)
4-a Denition of the density operator
4-b General properties of the density operator
4-c Populations; coherences
5 Use of the density operator: some applications
5-a System in thermodynamic equilibrium
5-b Separate description of part of a physical system. Concept of
a partial trace
1. Outline of the problem
Until now, we have considered systems whose state is perfectly well known. We have
shown how to study their time evolution and how to predict the results of various mea-
surements performed on them. To determine the state of a system at a given instant,
it suces to perform on the system a set of measurements corresponding to a C.S.C.O.
For example, in the experiment studied in , the polarization state of
the photons is perfectly well known when the light beam has traversed the polarizer.
However, in practice, the state of the system is often not perfectly determined. This
is true, for example, of the polarization state of photons coming from a source of natural
(unpolarized) light, and also for the atoms of a beam emitted by a furnace at temperature
, where the atoms' kinetic energy is known only statistically. The problem posed by
the quantum description of such systems is the following: how can we incorporate into
the formalism theincomplete informationwe possess about the state of the system, so
that our predictions make maximum use of this partial information? To do this, we shall
introduce here a very useful mathematical tool, the density operator, which facilitates
the simultaneous application of the postulates of quantum mechanics and the results of
probability calculations.
2. The concept of a statistical mixture of states
When one has incomplete information about a system, one typically appeals to the
concept of probability. For example, we know that a photon emitted by a source of
299

COMPLEMENT E III
natural light can have any polarization state with equal probability. Similarly, a system
in thermodynamic equilibrium at a temperaturehas a probability proportional to
e of being in a state of energy.
More generally, the incomplete information one has about the system usually
presents itself, in quantum mechanics, in the following way: the state of this system
may be either the state1with a probability1or the state2with a probability
2, etc... Obviously:
1+2+= = 1 (1)
We then say that we are dealing with astatistical mixtureof states1,2, ... with
probabilities1,2...
Now let us see what happens to the predictions concerning the results of measure-
ments performed on this system. If the state of the system were, we could use the
postulates stated in Chapter
measurement result. Since such a possibility (the state) has a probability of, it
is clear that the results obtained must be weighted by theand then summed over the
various values of, that is, over all the states of the statistical mixture.
Comments:
()The various states1,2, ... are not necessarily orthogonal. However,
they can always be chosen normalized; in this complement, we shall assume
that this is the case.
()Note that, in the present case, probabilities intervene at two dierent levels:

introduced probabilities at this stage: we considered the state vector to
be perfectly well known, in which case all the probabilitiesare zero,
except one, which is equal to 1);

ing to probabilistic predictions, even if the initial state of the system is
perfectly well known).
There are thus two totally dierent reasons necessitating the introduction
of probabilities: the incomplete nature of the initial information about the
state of the system (such situations are also envisaged in classical statistical
mechanics), and the (specically quantum mechanical) uncertainty related to
the measurement process.
()A system described by a statistical mixture of states (with the probability
of the state vector being) must not be confused with a system whose
stateis a linear superposition of states
1
:
= (2)
1
We assume, in this comment(), that the states are orthonormal. This hypothesis is not
essential but it simplies the discussion.
300

THE DENSITY OPERATOR
It is often said in quantum mechanics, when the state vector is the ket
given in (2), that the system has a probability
2
of being in the state
.
If we want to be precise, this must be understood to mean that if we perform
a set of measurements corresponding to a C.S.C.O. which hasas an
eigenvector, the probability of nding the set of eigenvalues associated with
is
2
. But we have stressed, in , the fact that
a system in the stategiven in (2)is not simply equivalentto a system
having the probability1
2
of being in the state1,2
2
of being in the state
2, etc... In fact, for a linear combination of, there exist, in general,
interference eects between these states (due to cross terms of the type,
obtained when the modulus of the probability amplitudes is squared) which
are very important in quantum mechanics.
We therefore see that it is impossible, in general, to describe a statistical
mixture by an average state vector which would be a superposition of the
states. As we indicated earlier, when we takea weighted sum of prob-
abilities, we can never obtain interference terms between the various states
of a statistical mixture.
3. The pure case. Introduction of the density operator
To study the behavior of a statistical mixture of states, we have envisaged one method:
calculation of the physical predictions corresponding to a possible state; weighting
the results so obtained by the probabilityassociated with this state and summation
over. Although correct in principle, this method often leads to clumsy calculations. We
have indicated [comment()] that it is impossible to associate an average state vector
with the system. Actually, it is an average operator and not an average vector which
permits a simple description of the statistical mixture of states: the density operator.
Before studying this general case, we shall return to the simple case where the
state of the system is perfectly known (all the probabilitiesare zero, except one). The
system is then said to be in apure state. We shall show that characterizing the system by
its state vector is completely equivalent to characterizing it by a certain operator acting
in the state space, the density operator. The usefulness of this operator will become
apparent in 4, where we shall show that nearly all the formulas involving this operator,
and derived for the pure case, remain valid for the description of a statistical mixture of
states.
3-a. Description by a state vector
Consider a system whose state vector at the instantis:
()= () (3)
where the form an orthonormal basis of the state space, assumed to be discrete
(extension to the case of a continuous basis presents no diculties). The coecients
301

COMPLEMENT E III
()satisfy the relation:
()
2
= 1 (4)
which expresses the fact that()is normalized.
Ifis an observable, with matrix elements:
= (5)
the mean value ofat the instantis:
() =()()= ()() (6)
Finally, the evolution of()is described by the Schrödinger equation:
~
d
d
()=()() (7)
where()is the Hamiltonian of the system.
3-b. Description by a density operator
Relation (6) shows that the coecients()enter into the mean values through
quadratic expressions of the type()(). These are simply the matrix elements of
the operator()(), the projector onto the ket()(cf.Chap., ), as
can be seen from (3):
()()=()() (8)
It is therefore natural to introduce the density operator(), dened by:
() =()() (9)
The density operator is represented in thebasis by a matrix called thedensity
matrixwhose elements are:
() = ()=()() (10)
We are going to show that the specication of()suces to characterize the
quantum state of the system; that is, it enables us to obtain all the physical predictions
that can be calculated from(). Let us write formulas (4), (6) and (7) in terms of
the operator(). According to (10), relation (4) indicates that the sum of the diagonal
elements of the density matrix is equal to 1:
()
2
= () = Tr() = 1 (11)
In addition, using (5) and (10), formula (6) becomes:
() = ()
= ()
= Tr() (12)
302

THE DENSITY OPERATOR
Finally, the time evolution of the operator()can be deduced from the Schrödinger
equation (7):
d
d
() =
d
d
() ()+()
d
d
()
=
1
~
()()()+
1
(~)
()()()
=
1
~
[()()] (13)
Therefore, in terms of the density operator, conservation of probability is expressed
by:
Tr() = 1 (14)
The mean value of an observableis calculated using the formula:
() = Tr()= Tr () (15)
and the time evolution obeys the equation:
~
d
d
() = [()()] (16)
For completeness, we must also indicate how to calculate from()the probabilities()
of the various resultswhich can be obtained in the measurement of an observable
at time. Actually, formula (15) can be used to do this. We know [see equation (B-14)
of Chapter] that()can be written as a mean value, that of the projectoronto
the eigensubspace associated with:
() =() () (17)
Using (15), we therefore obtain:
() = Tr () (18)
3-c. Properties of the density operator in a pure case
In a pure case, a system can be described just as well by a density operator as by
a state vector. However, the density operator presents a certain number of advantages.
First of all, we see from (9) that two state vectors()ande()(whereis
a real number), which describe the same physical state, correspond to the same density
operator. Using this operator therefore eliminates the drawbacks related to the existence
of an arbitrary global phase factor for the state vector. Moreover, we see from (14),
(15) and (18) that the formulas using the density operator are linear with respect to it,
while expressions (6) and (17) are quadratic with respect to(). This is an important
property which will be useful subsequently.
Finally, let us mention some properties of(), which can be deduced directly from
its denition (9):
() =() (19)
303

COMPLEMENT E III
(the density operator is Hermitian)
2
() =() (20)
Tr
2
() = 1 (21)
These last two relations, which follow from the fact that()is a projector, are true only
in a pure case. We shall see later that they are not valid for a statistical mixture of
states.
4. A statistical mixture of states (non-pure case)
4-a. Denition of the density operator
We now return to the general case described in , and consider a system for
which (at a given instant) the various probabilities12...... are arbitrary, on the
condition that they satisfy the relations:
0 12 1
= 1
(22)
Under these conditions, how does one calculate the probability()that a measurement
of the observablewill yield the result?
Let:
() = (23)
be the probability of ndingif the state vector were. To obtain the desired
probability(), one must, as we have already indicated, weight()byand
then sum over:
() = () (24)
Now, from (18), we have:
() = Tr (25)
where:
= (26)
is the density operator corresponding to the state. Substituting (25) into (24), we
have:
() = Tr
= Tr
= Tr (27)
304

THE DENSITY OPERATOR
where we have set:
= (28)
We therefore see that the linearity of the formulas which use the density operator
enables us to express all physical predictions in terms of, the average of the density
operatorswith weights;is, by denition, the density operator of the system.
Comment:
An ensemble of pure stateswith probabilitiesleads to a single density operator, but the
reverse is not true in general: the same density operator can be interpreted as several dierent
statistical mixtures of pure states. For instance, in a space of states with dimension, a
statistical mixture of thestates of a given basiswith the same probability1leads to a
density operator that is the unit operator, divided by; but the same operator may be obtained
with a statistical mixture of the kets of any other basis. These various ensembles lead to the
same probabilities(), and cannot be distinguished by measuring these probabilities. This
situation is sometimes described as the multiple preparations of the same density operator.
4-b. General properties of the density operator
Since the coecientsare real,is obviously a Hermitian operator like each of
the
Let us calculate the trace of; it is equal to:
Tr= Tr (29)
Now, as we saw in 3-b, the trace ofis always equal to 1; it follows that:
Tr= = 1 (30)
Relation (14) is therefore valid in the general case.
We have already given, in (27), the expression that enables us to calculate the
probability()in terms of. Using this expression, we can easily generalize formula
(15) to statistical mixtures:
= () = Tr
= Tr (31)
[we have used formula (D-36-b) of Chapter].
Now let us calculate the time evolution of the density operator. To do this, we shall
assume that, unlike the state of the system, its Hamiltonian()is perfectly well known.
One can then easily show that if the system at the initial time0has the probability
of being in the state, then, at a subsequent time, it has the same probabilityof
being in the state()given by:
~
d
d
()=()()
(0)=
(32)
305

COMPLEMENT E III
The density operator at the instantwill then be:
() = () (33)
with:
() =()() (34)
According to (16),()obeys the evolution equation:
~
d
d
() = [()()] (35)
The linearity of formulas (33) and (35) with respect to()implies that:
~
d
d
() = [()()] (36)
We can therefore generalize to a statistical mixture of states all the equations of
3, with the exception of (20) and (21). We see that, sinceis no longer a projector,
we have, in general
2
:
2
= (37)
and, consequently:
Tr
2
1 (38)
Moreover, it is sucient that one of the equations, (20) or (21), be satised for us to be
sure that we are dealing with a pure state.
Finally, we see from denition (28) that, for any ket, we have:
=
=
2
(39)
and consequently:
0 (40)
is therefore a positive operator.
2
Assume, for example, that the states are mutually orthogonal. In an orthonormal basis in-
cluding the ,is diagonal and its elements are the. To obtain
2
, we simply replaceby
2
. Relations (37) and (38) then follow from the fact that theare always less than 1 (except in the
particular case where only one of them is non-zero: the pure case).
306

THE DENSITY OPERATOR
4-c. Populations; coherences
What is the physical meaning of the matrix elementsofin the basis?
First, let us consider the diagonal element. According to (28), we have:
= [] (41)
that is, using (26) and introducing the components:
()
= (42)
of in the basis:
=
()
2
(43)
()
2
is a positive real number, whose physical interpretation is the following: if the
state of the system is, this number is the probability of nding, in a measurement,
this system in the state. According to (41), if we take into account the indeterminacy
of the state before the measurement,represents the average probability of nding
the system in the state. For this reason,is called thepopulationof the state
: if the same measurement is carried outtimes under the same initial conditions,
whereis a large number, systems will be found in the state. It is evident
from (43) thatis a positive real number, equal to zero only if all the
()2
are zero.
A calculation analogous to the preceding one gives the following expression for the
non-diagonal element:
=
()()
(44)
()()
is a cross term, of the same type as those studied in . It
reects the interference eects between the statesand which can appear when
the stateis a coherent linear superposition of these states. According to (44),
is the average of these cross terms, taken over all the possible states of the statistical
mixture. In contrast to the populations,can be zero even if none of the products
()()
is: while is a sum of real positive (or zero) numbers,is a sum of
complex numbers. Ifis zero, this means that the average (44) has cancelled out any
interference eects betweenand . On the other hand, ifis dierent from
zero, a certain coherence subsists between these states. This is why the non-diagonal
elements ofare often calledcoherences.
Comments:
()The distinction between populations and coherences obviously depends on
the basis chosen in the state space. Sinceis Hermitian, it is always
307

COMPLEMENT E III
possible to nd an orthonormal basiswhereis diagonal.can then
be written:
= (45)
Sinceis positive [relation (40)] and Tr= 1, we have:
0 1
= 1
(46)
can thus be considered to describe a statistical mixture of the states
with the probabilities(there are no coherences between the states).
()If the ketsare eigenvectors of the Hamiltonian, which is assumed to
be time-independent:
= (47)
we obtain directly from (36):
~
d
d
() = 0
~
d
d
() = ( )
(48)
that is:
() = constant
() = e
~
( )
(0)
(49)
The populations are constant, and the coherences oscillate at the Bohr fre-
quencies of the system.
()Using (40), one can prove the inequality:
2
(50)
It follows, for example, thatcan have coherences only between states whose
populations are not zero.
5. Use of the density operator: some applications
5-a. System in thermodynamic equilibrium
The rst example we shall consider is borrowed from quantum statistical mechan-
ics. Consider a system in thermodynamic equilibrium with a reservoir at the absolute
temperature. It can be shown that its density operator is then:
=
1
e (51)
308

THE DENSITY OPERATOR
whereis the Hamiltonian operator of the system,is the Boltzmann constant, and
is a normalization coecient chosen so as to make the trace ofequal to 1:
= Tre (52)
(is called the partition function).
In the basis of eigenvectors of, we have (cf.ComplementII, ):
=
1
e
=
1
e (53)
and:
=
1
e
=
1
e
= 0 (54)
At thermodynamic equilibrium, the populations of the stationary states are exponentially
decreasing functions of the energy (the lower the temperature, the more rapid the
decrease), and the coherences between stationary states are zero.
More details on the use of the density operator in statistical mechanics and at
thermal equilibrium are given in Appendix
5-b. Separate description of part of a physical system. Concept of a partial trace
We now return to the problem mentioned in 3 of ComplementIII. Consider two
dierent systems (1) and (2) and the global system(1) + (2), whose state space is the tensor
product:
=(1)(2) (55)
Let (1)be a basis of(1)and (2)a basis of(2); the kets(1)(2)form a
basis of.
The density operatorof the global system is an operator which acts in. We saw in
Chaptercf. ) how to extend intoan operator which acts only in(1)[or(2)].
We are going to show here how to perform the inverse operation: we shall construct froman
operator(1)[or(2)] acting only in(1)[or(2)]. This will enable us to make all the physical
predictions about measurements bearing only on system (1), or on system (2). This operation
will be called apartial tracewith respect to (2) [or (1)].
Let us introduce the operator(1)whose matrix elements are:
(1)(1)(1)= ((1)(2))((1)(2)) (56)
By denition,(1)is obtained fromby performing a partial trace on (2):
(1) = Tr2 (57)
Similarly, the operator:
(2) = Tr1 (58)
309

COMPLEMENT E III
has matrix elements:
(2)(2)(2)= (1)(2) (1)(2) (59)
It is clear why these operations are called partial traces. We know that the (total) trace ofis:
Tr= (1)(2) (1)(2) (60)
The dierence between (60) and (56) [or (59)] is the following: for the partial traces, the indices
and(orand) are not required to be equal and the summation is performed only over
(or). We have, moreover:
Tr= Tr1(Tr2) = Tr2(Tr1) (61)
(1)and(2)are therefore, like, operators whose trace is equal to 1. It can be veried from
their denitions that they are Hermitian, and that they satisfy all the general properties of a
density operator (cf.).
Now let(1)be an observable acting in(1), and
~
(1) =(1)(2)its extension in.
We obtain, using (31), the denition of the trace, and the closure relation on the(1)(2)
basis:
~
(1)= Tr
~
(1)
= (1)(2) (1)(2)
(1)(2)(1)(2)(1)(2)
= (1)(2) (1)(2)
(1)(1)(1)(2)(2) (62)
Now:
(2)(2)= (63)
We can therefore write (62) in the form:
~
(1)= (1)(2) (1)(2) (1)(1)(1) (64)
Inside the brackets on the right-hand side of (64), we recognize the matrix element of(1)
dened in (56). We therefore have:
~
(1)= (1)(1)(1) (1)(1)(1)
= (1)(1)(1)(1)
= Tr(1)(1) (65)
Let us compare this result with (31). We see that the partial trace(1)enables us to
calculate all the mean values
~
(1)as if the system (1) were isolated and had(1)for a density
operator. Making the same comment as for formula (17), we see that(1)also enables us to
obtain the probabilities of all the results of measurements bearing on system (1) alone.
310

THE DENSITY OPERATOR
Comments:
()We saw in ComplementIIIthat it is impossible to assign a state vector to system
(1) [or (2)] when the state of the global system(1) + (2)is not a product state.
We now see that the density operator is a much more simple tool than the state
vector. In all cases (whether the global system is in a product state or not, whether
it corresponds to a pure case or to a statistical mixture), one can always, thanks
to the partial trace operation, assign a density operator to subsystem (1) [or (2)].
This permits us to calculate all the physical predictions about this subsystem.
()Even ifdescribes a pure state (Tr
2
= 1), this is not in general true of the density
operators(1)and(2)obtained fromby a partial trace. It can be veried from
(56) [or (59)] that Tr
2
(1)[or Tr
2
(2)] is not generally equal to 1. This is
another way of saying that it is not in general possible to assign a state vector to
(1) [or (2)], except, of course, if the global system is in a product state.
()If the global system is in a product state:
=(1)(2) (66)
we can verify directly that the corresponding density operator is written:
=(1)(2) (67)
with:
(1) =(1)(1)
(2) =(2)(2) (68)
More generally, we can envisage states of the global system for which the density
operatorcan be factored as in () [(1)and(2)can correspond to statistical
mixtures as well as to pure cases]. The partial trace operation then yields:
Tr2(1)(2)=(1)
Tr1(1)(2)=(2) (69)
An expression such as (67) therefore represents the simple juxtaposition of a sys-
tem (1), described by the density operator(1), and a system (2), described by the
density operator(2).
()Starting with an arbitrary density operator[that cannot be factored as in ()],
let us calculate(1) = Tr2and(2) = Tr1. Then let us form the product:
=(1)(2) (70)
Unlike the case envisaged in comment(),is in general dierent from. When
the density operator cannot be factored as in (67), there is therefore a certain
correlation between systems (1) and (2), which is no longer contained in the
operatorof formula (70).
()If the evolution of the global system is described by equation (36), it is in general
impossible to nd a Hamiltonian operator relating to system (1) alone that would
enable us to write an analogous equation for(1). While the denition, at any
time, of(1)in terms ofis simple, the evolution of(1)is much more dicult to
describe.
311

COMPLEMENT E III
References and suggestions for further reading:
Articles by Fano (2.31) and Ter Haar (2.32). Using the density operator to study
relaxation phenomena: Abragam (14.1), Chap. VIII; Slichter (14.2), Chap. 5; Sargent,
Scully and Lamb (15.5), Chap. VII.
312

THE EVOLUTION OPERATOR
Complement FIII
The evolution operator
1 General properties
2 Case of conservative systems
In , we saw that the transformation of(0)(the state
vector at the initial instant0) into()(the state vector at an arbitrary instant) is
linear. There therefore exists a linear operator(0)such that:
()=(0)(0) (1)
We intend to study here the principal properties of(0), which is, by denition,
theevolution operatorof the system.
1. General properties
Since the ket(0)is arbitrary, it is clear from (1) that:
(00) = (2)
Also, substituting (1) into the Schrödinger equation, we obtain:
~
(0)(0)=()(0)(0) (3)
from which, for the same reason as above:
~
(0) =()(0) (4)
The rst-order dierential equation (4) completely denes(0), taking the initial
condition (2) into account. Note, moreover, that (2) and (4) can be condensed into a
single integral equation:
(0) =
~
0
()(0)d (5)
Now let us consider the parameter0, which appears in(0)as a variable,
just like. We then write (1) in the form:
()=()() (6)
But()can itself be obtained from a formula of the same type:
()=()() (7)
313

COMPLEMENT F III
Substitute (7) into (6):
()=()()() (8)
Since, moreover,()=()(), we deduce (()being arbitrary):
() =()() (9)
It is easy to generalize this procedure and to obtain:
( 1) =( 1)(32)(21) (10)
where12...,are arbitrary. If we assume that1 2 3 , formula (10)
is simple to interpret: to go from1to, the system progresses from1to2, then from
2to3, ..., then, nally from1to.
Set=in (9); taking (2) into consideration, we obtain:
=()() (11)
or, interchanging the roles ofand:
=()() (12)
We therefore have:
() =
1
() (13)
Now let us calculate the evolution operator between two instants separated by d.
To do this, write the Schrödinger equation in the form:
d()=(+ d) ()
=
~
()()d (14)
that is:
(+ d)=
~
()d() (15)
We then obtain, using the very denition of(+ d):
(+ d) =
~
()d (16)
(+ d)is called the innitesimal evolution operator. Since()is Hermitian,(+
d)is unitary (cf.ComplementII, ). It follows that()is also unitary since the
interval[]can be divided into a very large number of innitesimal intervals. Formula
(10) then shows that()is a product of unitary operators; it is therefore a unitary
operator. One can consequently write (13) in the form:
() =
1
() =() (17)
It is not surprising that the transformation()is unitary, that is, that it conserves
the norm of vectors on which it acts. We saw in Chaptercf. ) that the norm
of the state vector does not change over time.
314

THE EVOLUTION OPERATOR
2. Case of conservative systems
When the operatordoes not depend on time, equation (4) can easily be integrated;
taking the initial condition (2) into account, we obtain:
(0) = e
( 0)~
(18)
One can verify directly from this formula all the properties of the evolution operator
cited in 1.
It is very simple to go from formula (D-52) to (D-54) of Chapter 18). It
suces to apply the operator(0)to both sides of (D-52), noting that, since
is an eigenvector ofwith the eigenvalue:
(0) = e
( 0)~
= e
( 0)~
(19)
Comments:
()Whenis time-dependent, one might be tempted to believe, by analogy with
formula (18), that the evolution operator is equal to the operator(0)
dened by:
(0) = e
~
0
()d
(20)
Actually, this is not true, since the derivative of an operator of the form e
()
is not in general equal to()e
()
(cf.ComplementII, 5-c):
~
(0)=()(0) (21)
()Let us again consider the experiments described in .
As we have already indicated [comment()of ], it is not necessary
to assume that the measurements of the various observables,and
are made very close together in time. When the system has had the time
to evolve between two successive measurements, the variations of the state
vector can easily be taken into account by using the evolution operator. If
0,1and2designate respectively the instants at which the measurements
of,andare performed, we then replace (E-15) by:
() = (20)
2
(22)
and (E-17) by:
() = (21)
2
(10)
2
(23)
We then have, using (9):
(20)= (21)(10)
= (21) (10) (24)
Substituting (24) into (22), we see, as in (E-21), that()is not equal to
().
315

COMPLEMENT F III
References and suggestions for further reading:
The evolution operator is of fundamental importance in collision theory (see Chap-
ter ) and time-dependent perturbation theory (see Chapter ), as well as in
the study of the interactions between atoms and photons (see Chapter).
316

THE SCHRÖDINGER AND HEISENBERG PICTURES
Complement GIII
The Schrödinger and Heisenberg pictures
In the formalism developed in Chapter, it is the time-independent operators
which correspond to the observables of the system (cf.Chap., ). For example,
the position, momentum and kinetic energy operators of a particle do not depend on
time. The evolution of the system is entirely contained in that of the state vector()
[here written(), for reasons which will be evident later] and is obtained from the
Schrödinger equation. This is why this approach is called theSchrödinger picture.
Nevertheless, we know that all the predictions of quantum mechanics (probabilities,
mean values) are expressed in terms of scalar products of a bra and a ket or matrix
elements of operators. Now, as we saw in ComplementII, these quantities are invariant
when the same unitary transformation is performed on the kets and on the operators.
This transformation can be chosen so as to make the transform of the ket()a time-
independent ket. Of course, the transforms of the observables cited above then depend
on time. We thus obtain theHeisenberg picture.
To avoid confusion, in this complement, we shall systematically assign an index
to the kets and operators of the Schrödinger picture and an indexto those of
the Heisenberg picture. The indexcan be considered to be implicit in all the other
complements and chapters where only the Schrödinger picture is used.
The state vector()at the instantis expressed in terms of(0)by the
relation:
()=(0)(0) (1)
where(0)is the evolution operator (cf.ComplementIII). Since this operator is
unitary, it is sucient to perform the unitary transformation associated with the operator
(0)to obtain a constant transformed vector:
=(0)()=(0)(0)(0)
= (0) (2)
In the Heisenberg picture, the state vector, which is constant, is therefore equal to()
at time0
The transform()of an operator()is given by (ComplementII, ):
() =(0)()(0) (3)
As we have already seen,()generally depends on time, even ifdoes not.
Nevertheless, there exists an interesting special case in which, ifis time-
independent, the same is true of: the case in which the system is conservative (
does not depend on time) andcommutes with (is then a constant of the
motion;cf.Chap.,
(0) = e
( 0)~
(4)
If the operatorcommutes with , it also commutes with(0)(cf.Comple-
mentII, ), so that:
() =(0)(0)= (5)
317

COMPLEMENT G III
The operatorsand are therefore simply equal in this case (in particular,= ,
and the indicesandare, in reality, unnecessary for the Hamiltonian). Since they
are time-independent, we see that they indeed correspond to a constant of the motion.
When ()is arbitrary, let us calculate the evolution of the operator(). Using
relation (4) of ComplementIII, as well as its adjoint, we obtain:
d
d
() =
1
~
(0)()()(0) +(0)
d()
d
(0)
+
1
~
(0)()()(0) (6)
In the rst and last terms of this expression, let us insert betweenand the
product(0)(0), which is equal to the identity operator [formula (17) of Com-
plementIII]:
d
d
() =
1
~
(0)()(0)(0)()(0)
+(0)
d()
d
(0)
+
1
~
(0)()(0)(0)()(0) (7)
According to denition (3), we nally obtain:
~
d
d
() = [() ()] +~
d
d
() (8)
Comments:
()Historically, the rst picture was developed by Schrödinger, leading him to
the equation which bears his name, and the second one, by Heisenberg, who
calculated the evolution of matrices representing the various operators()
(hence the name matrix mechanics). It was not until later that the equiv-
alence of the two approaches was proved.
()Using (8), one immediately obtains equation (D-27) of Chapter
now show. In the Heisenberg picture, the evolution of the mean value
() =()()()
can be calculated, since:
() = () (9)
On the right-hand side of (9), only()depends on time, so (D-27) can be
obtained directly by dierentiation. Note, nevertheless, that equation (8) is
more general than (D-27) since, instead of expressing the equality of two mean
values (that is, two matrix elements of operators), it expresses the equality
of two operators.
318

THE SCHRÖDINGER AND HEISENBERG PICTURES
()When the system under consideration is composed of a particle of mass
under the inuence of a potential, equation (8) becomes very simple. We
then have (conning ourselves to one dimension):
() =
2
2
+( ) (10)
and therefore [cf.formula (35) of ComplementII]:
() =
2
2
+( ) (11)
Substituting (11) into (8) and using the fact that[ ] = [ ] =~,
we obtain, by an argument analogous to that of :
d
d
() =
1
()
d
d
() =( ) (12)
These equations generalize the Ehrenfest theorem [cf.Chap., relations
(D-34) and (D-35)]. They are similar to those giving the evolution of the
classical quantitiesand[cf.Chap., relations (D-36a) and (D-36b)].
An advantage of the Heisenberg picture is that it leads to equations formally
similar to those of classical mechanics.
References and suggestions for further reading:
Theinteraction picture: see exercise 15 of ComplementIIIas well as
Chapter; Messiah (1.17), Chap. VIII, 14; Schi (1.18), 24; Merzbacher (1.16),
Chap. 18, 7.
319

GAUGE INVARIANCE
Complement HIII
Gauge invariance
1 Outline of the problem: scalar and vector potentials asso-
ciated with an electromagnetic eld; concept of a gauge
2 Gauge invariance in classical mechanics
2-a Newton's equations
2-b The Hamiltonian formalism
3 Gauge invariance in quantum mechanics
3-a Quantization rules
3-b Unitary transformation of the state vector; form invariance of
the Schrödinger equation
3-c Invariance of physical predictions under a gauge transformation
1. Outline of the problem: scalar and vector potentials associated with an
electromagnetic eld; concept of a gauge
Consider an electromagnetic eld, characterized by the valuesE(r;)of the electric eld
andB(r;)of the magnetic eld at every instant and at all points in space:E(r;)and
B(r;)are not independent since they satisfy Maxwell's equations. Instead of specifying
these two vector elds, it is possible to introduce a scalar potential(r;)and a vector
potentialA(r;)such that:
E(r;) =r(r;)
A(r;)
B(r;) =rA(r;)
(1)
It can be shown from Maxwell's equations (cf.Appendix, ) that there always
exist functions(r;)andA(r;)that allowE(r;)andB(r;)to be expressed in the
form (1). All electromagnetic elds can therefore be described by scalar and vector
potentials. However, whenE(r;)andB(r;)are given,(r;)andA(r;)are not
uniquely determined. It can easily be veried that if we have a set of possible values for
(r;)andA(r;), we obtain other potentials(r;)andA(r;)describing thesame
electromagnetic eld by the transformation:
(r;) =(r;)
(r;)
A(r;) =A(r;) +r(r;) (2)
where(r;)is an arbitrary function ofrand. This can be seen by replacing(r;)
by(r;)andA(r;)byA(r;)in (1) and verifying thatE(r;)andB(r;)remain
321

COMPLEMENT H III
unchanged. Moreover, it can be shown that relations (2) give all the possible scalar and
vector potentials associated with a given electromagnetic eld.
When a particular set of potentials has been chosen to describe an electromagnetic
eld, achoice of gaugeis said to have been made. As we just mentioned, an innite num-
ber of dierent gauges can be used for the same eld, characterized byE(r;)andB(r;).
When one changes from one to another, one is said to perform agauge transformation.
It often happens in physics that the equations of motion of a system involve, not
the eldsE(r;)andB(r;), but the potentials(r;)andA(r;). We saw an example
of this in , when we wrote the Schrödinger equation for a particle
of chargein an electromagnetic eld [cf.relation (B-48) of that chapter]. The following
question can then be posed: do the physical results predicted by the theory depend
only on the values of the eldsE(r;)andB(r;)at all points in space, or do they also
depend on the gauge used to write the equations? In the latter case, it would obviously be
necessary, in order for the theory to make sense, to specify in which gauge the equations
are valid.
The aim of this complement is to answer this question. We shall see that in classical
mechanics (), as in quantum mechanics (), physical results are not modied when a
gauge transformation is performed. The scalar and vector potentials can then be seen to
be calculation tools; actually, all that counts are the values of the electric and magnetic
elds at all points in space. We shall express this result by saying that classical and
quantum mechanics possess the property ofgauge invariance.
2. Gauge invariance in classical mechanics
2-a. Newton's equations
In classical mechanics, the motion of a particle
1
of chargeand massplaced in
an electromagnetic eld can be calculated from the forcefexerted on it. This force is
given by Lorentz' law:
f=[E(r;) +vB(r;)] (3)
wherevis the velocity of the particle. To obtain the equations of motion which allow
one to calculate the positionr()of the particle at any instant, one substitutes relation
(3) into the fundamental dynamical equation (Newton's law):
d
2
d
2
r() =f (4)
In this approach, only the values of the electric and magnetic elds enter into the calcu-
lation; therefore, the problem of gauge invariance does not arise.
2-b. The Hamiltonian formalism
Instead of adopting the point of view of the preceding section, one can use other
equations of motion, the Hamilton-Jacobi equations. It is not dicult to show (cf.
1
For simplicity, we shall assume in this complement that the system under study is composed of
a single particle. Generalization to a more complex system formed by several particles placed in an
electromagnetic eld presents no diculties.
322

GAUGE INVARIANCE
Appendix) that the latter equations are completely equivalent to Newton's equations.
However, since we used the Hamiltonian formalism in Chapter
system, it is useful to study how a gauge transformation appears in this formalism.
Although the scalar and vector potentials do not enter into Newton's equations, they
are indispensable for writing those of Hamilton. The property of gauge invariance is
therefore less obvious for this second point of view.
. The dynamical variables of the system and their evolution
To determine the motion of a particle subjected to the Lorentz force written in
(3), one can use the Lagrangian
2
:
(rv;) =
1
2
v
2
[(r;)vA(r;)] (5)
This expression permits the calculation of the momentump, which is written:
p=rv(rv;) =v+A(r;) (6)
It is then possible to introduce the classical Hamiltonian:
(rp;) =
1
2
[pA(r;)]
2
+(r;) (7)
In the Hamiltonian formalism, the state of the particle at a given time is dened by its
positionrand its momentump, which we shall call the fundamental dynamical variables,
and no longer by its position and its velocity, as in
point of view). The momentump(conjugate momentum of the positionr) must not be
confused with the mechanical momentum:
=v (8)
They are indeed dierent since, according to (6):
=pA(r;) (9)
This relation allows us to calculate the mechanical momentum (and therefore the ve-
locity) whenever the values ofrandpare known. Similarly, all the other quantities
associated with the particle (kinetic energy, angular momentum, etc...) are expressed in
the Hamiltonian formalism as functions of the fundamental dynamical variablesrandp
(and, if necessary, of time).
The evolution of the system is governed by Hamilton's equations:
d
d
r() =rp[r()p();]
d
d
p() =rv[r()p();]
(10)
whereis the function ofrandpwritten in (7). These equations give the values, for all
times, of the fundamental dynamical variables if they are known at the initial instant.
2
We state without proof a certain number of results of analytical mechanics which are established
in Appendix.
323

COMPLEMENT H III
To write equations (10), it is necessary to choose a gauge, that is, a pair of po-
tentials(r;)A(r;)describing the electromagnetic eld. What happens if, instead
of this gauge, we choose another one, characterized by dierent potentials(r;)
andA(r;), but describing the same eldsE(r;)andB(r;)? We shall label with a
prime the values of the dynamical variables associated with the motion of the particle
when the gauge chosen is. As we pointed out in a, Newton's equations indicate
that the positionrand the velocityvtake on, at every instant, values independent of
the gauge. Consequently, we have:
r() =r()
() =()
(11a)
(11b)
Now, from (9):
() =p()A[r();]
() =p()A[r();] (12)
Therefore, the valuesp()andp()of the momentum in the gaugesand are
dierent; they must satisfy:
p()A[r();] =p()A[r();] (13)
If(r;)is the function appearing in formulas (2) which govern the gauge transforma-
tion fromto, the values of the fundamental dynamical variables are transformed
according to the formulas:
r() =r()
p() =p() +r[r();]
(14a)
(14b)
In the Hamiltonian formalism, the value at each instant of the dynamical variables
describing a given motion depends on the gauge chosen. Moreover, such a result is not
surprising since, in (7) and (10), the scalar and vector potentials appear explicitly in the
equations of motion for the position and momentum.
. True physical quantities and non-physical quantities
()Denitions
We have just seen, in relations (14) for example, that it is possible to distinguish
between two types of quantities associated with the particle: those which, likeror,
have identical values at all times in two dierent gauges, and those which, likep, have
values that depend on the arbitrarily chosen gauge. We are thus led to the following
general denition:
Atrue physical quantityassociated with the system under consideration is a
quantity whose value at any time does not depend (for a given motion of the system) on
the gauge used to describe the electromagnetic eld.
Anon-physical quantity, on the other hand, is a quantity whose value is modied
by a gauge transformation; thus, like the scalar and vector potentials, it is seen to be a
calculation tool, rather than an actually observable quantity.
324

GAUGE INVARIANCE
The problem then posed is the following: in the Hamiltonian formalism, all the
quantities associated with the system appear in the form of functions of the fundamental
dynamical variablesrandp; how can we know whether such a function corresponds to
a true physical quantity or not?
()Characteristic relation of true physical quantities
Let us rst assume that a quantity associated with the particle is described, in the
gauge, by a function ofrandp(which may depend on time) which we shall denote
by(rp;). If to this quantity corresponds, in another gauge, the same function
(rp;), the quantity is clearly not truly physical [except in the special case where the
functiondepends only onrand not onp; see equations (14)]. Since the values of the
momentum are dierent in the two gaugesand, the same is obviously true for the
values of the function.
To obtain the true physical quantities associated with the system,we must therefore
consider functions(rp;)whose form depends on the gauge chosen(this is why we
label these functions with an index). We have already seen an example of such a
function: the mechanical momentumis a function ofrandpvia the vector potential
A[cf.(9)]. In this case, the function is dierent in the two gaugesand; that is,
it is of the form(rp;). The denition given in()thus implies that the function
(rp;)describes a true physical quantity on the condition that:
[r()p();] =[r()p();] (15)
wherer()andp()are the values taken on by the position and momentum in the gauge
, andr()andp()are their values in the gauge. If we substitute relations (14)
into (15), we obtain:
[r()p();] =[r()p() +r(r(););] (16)
This relation must be satised at every instantand for all possible motions of the
system. Since, whenis xed, the values of the position and the momentum can be
chosen independently, both sides of (16) must in fact be the same function ofrandp,
which is written:
[rp;] =[rp+r(r;);] (17)
This relation is characteristic of the functions[rp;]associated with true phys-
ical quantities. Therefore,if one considers the function[rp;]for the gauge, and
if one replacespbyp+r(r;)[where(r;)denes, according to (2), the gauge
transformation fromto],one obtains a new function ofrandpwhich must be
identical to[rp;]. If this is not the case, the function considered corresponds to a
quantity which is not truly physical.
() Examples
Let us give some examples of functions[rp;]which describe true physical
quantities. We have already encountered two: those corresponding to the position and
to the mechanical momentum; the rst is simply equal torand the second to:
(rp;) =pA(r;) (18)
325

COMPLEMENT H III
Since relations (11) express the fact thatrandare true physical quantities, we know
a priorithat relation (17) is satised by the corresponding functions.
However, let us verify this directly in order to familiarize ourselves with the use of this
relation. As regardsr, we are dealing with a function that does not depend onpand whose
form does not depend on the gauge
3
; this immediately implies (17). As regards, relation (18)
yields:
(rp;) =pA(r;) (19)
Replace in this functionpbyp+r(r;); we obtain the function:
p+r(r;)A(r;) =pA(r;) (20)
which is none other than(rp;); relation (17) is therefore satised.
Other true physical quantities are the kinetic energy:
(rp;) =
1
2
[pA(r;)]
2
(21)
and the moment, with respect to the origin, of the mechanical momentum:
(rp;) =r[pA(r;)]
2
(22)
In general, we see that whenever a function ofrandphas the form:
(rp;) =[rpA(r;)] (23)
(whereis a function whose form is independent of the gaugechosen), we obtain a
truephysical quantity
4
. This result makes sense since (23) really expresses the fact that
the values taken on by the quantity considered are obtained from those ofrand, which
we know to be gauge-invariant.
Let us also give some examples of functions describing quantities that arenottrue
physical quantities. In addition to the momentump, we can cite the function:
(p) =
p
2
2
(24)
which must not be confused with the kinetic energy written in (21), and, in general, any
function ofpalone (and, possibly, of the time). Similarly, the angular momentum:
(rp) =rp (25)
cannot be considered to be a true physical quantity. Finally, let us cite the classical
Hamiltonian, which, according to (7), is the sum of the kinetic energy(rp;), which
is a true physical quantity, and the potential energy. Now, the latter [which should
rigorously be written in the form of a gauge-dependent function(r;)] is not a true
physical quantity since, at every point in space, its value changes when the gauge is
changed.
3
It is not dicult to verify that, in general, any function(r)that depends only onr(and, possibly,
on the time), and whose form is the same in any gaugechosen, describes a true physical quantity.
4
One could also construct functions associated with true physical quantities in which the potentials
are involved in a more complex way than in (23
and the electric eld at the position of the particle).
326

GAUGE INVARIANCE
3. Gauge invariance in quantum mechanics
In Chapter, we introduced the postulates of quantum mechanics by starting from the
Hamiltonian formulation of classical mechanics. We are thus led to ask if the problem
of gauge invariance, easily resolved in classical mechanics because of the existence of
Newton's equations, is more complex in the framework of quantum mechanics. The
following question then arises: are the postulates stated in Chapter
arbitrarily chosen gaugeor only for a particular gauge?
In answering this question, we shall be guided by the results obtained in the pre-
ceding paragraph. Following the same type of reasoning, we shall see that there exists a
close analogy between the consequences of a gauge transformation in the classical Hamil-
tonian formalism and in the quantum mechanical formalism. We shall thus establish the
gauge invariance of quantum mechanics.
To do this, we shall begin () by examining the results obtained when the
quantization rules are applied in the same way in two dierent gauges. We shall then
see () that, like in classical mechanics, where the values of the dynamical variables
generally change when the gauge is changed,a given physical system must be characterized
by a mathematical state vectorthat depends on the gauge. To pass from a state
vector corresponding to one gaugeto that of another gauge, we use aunitary
transformation. The form of the Schrödinger equation, however, always remains the
same (as do Hamilton's equations in classical mechanics). Finally, we shall examine the
behavior, under a gauge transformation, of the observables associated with the system
(). We shall see that the simultaneous modication of the state vector and the
observables is such thatthe physical content of quantum mechanics does not depend on
the gauge chosen. Moreover, we shall demonstrate this by showing that the density and
the probability current values are gauge invariant.
3-a. Quantization rules
The state space of a (spinless) particle is alwaysr. However, we are clearly led
by the results of
may be dierent in two dierent gauges. We shall therefore label these operators with
an index.
The quantization rules associate, with the positionrand the momentumpof the
particle, operatorsRandPacting inrsuch that:
[ ] = [] = [] =~ (26)
(where all the other commutators between components ofRandPare zero). In ther
representation, the operatorRacts like a multiplication byr, andPlike the dierential
operator
~
r. These rules are the same in all gauges. We can therefore write:
R=R (27a)
P=P (27b)
In fact, these equations enable us to omit the indexfor the observablesRandP, and
we shall henceforth do so.
The quantization of all other quantities associated with the particle is obtained
as follows: in a given gauge, take the function ofrandpgiving the classical quantity
327

COMPLEMENT H III
considered and (after having symmetrized, if necessary) replacerby the operatorRand
pbyP. We thus obtain the operator which, in the gauge chosen, describes this quantity.
Consider some examples:
The angular momentum operator, obtained fromrp, is the same in all gauges:
L=L (28)
The operator associated with the mechanical momentum, on the other hand,
depends on the gauge chosen. In the gauge, it is given by:
=PA(R;) (29)
If the gauge is changed, it becomes:
=PA(R;) (30)
whose action inris dierent from that of:
= r(R;) (31)
Similarly, the operator
5
:
=R=R[PA(R;)] (32)
which describes the moment of the mechanical momentum, explicitly involves the vector
potential chosen.
Finally, the Hamiltonian operator is obtained from formula (7):
=
1
2
[PA(R;)]
2
+(R;) (33)
It is obvious that in another gauge, it becomes a dierent operator, since:
=
1
2
[PA(R;)]
2
+ (R;)= (34)
3-b. Unitary transformation of the state vector; form invariance of the Schrödinger
equation
. The unitary operator()
In classical mechanics, we denoted byr()p()andr()p()the values of the
fundamental dynamical variables characterizing the state of the particle in two dierent
gaugesand. In quantum mechanics, we shall therefore denote by()and()
the state vectors relative to these two gauges, and the analogue of relations (14) is thus
given by relations between mean values:
()R ()=()R () (35a)
()P ()=()P+r(R;)() (35b)
5
It can be veried, by using the commutation relations ofRand, that it is not necessary to
symmetrize expression (32).
328

GAUGE INVARIANCE
Using (27), we immediately see that this is possible only if()and()are two
dierent kets. We shall therefore seek a unitary transformation()that enables us to
go from()to():
()=()() (36a)
()() =()() = (36b)
Taking (27) into account, we see that equations () are satised for any()on condition
that:
()R() =R (37a)
()P() =P+r(R;) (37b)
Multiplying (37a) on the left by(), we obtain:
R() =()R (38)
The desired unitary operator commutes with the three components ofR; it can therefore be
written in the form:
() = e
(R;)
(39)
where(R;)is a Hermitian operator. Relation (48) of ComplementIIthen allows us to write:
[P()] =~r(R;)() (40)
If we multiply this equation on the left by()and substitute it into (37b), we easily obtain
the relation:
~r(R;)=r(R;) (41)
which is satised when:
(R;) =0() +
~
(R;) (42)
Omitting the coecient0(), which corresponds, for the state vector(), to a global phase
factor of no physical consequence, we obtain the operator():
() = e
~
(R;)
(43)
If, in (36a),()is this operator, relations (35) are automatically satised.
Comments:
()In therrepresentation, relations (36a) and (43) imply that the wave
functions(r) =r()and(r) =r()are related by:
(r) = e
~
(r)
(r) (44)
For the wave function, the gauge transformation corresponds to a phase change
which varies from one point to another, and is not, therefore, a global phase
factor. The gauge invariance of physical predictions obtained by using the
wave functionsor, is therefore not obviousa priori.
329

COMPLEMENT H III
()If the system under study is composed of several particles having positions
r1,r2, ... and charges1,2, ..., (43) must be replaced by:
() =
(1)
()
2
()
= e
~
[1(R1)+2(R2)+]
(45)
. Time evolution of the state vector
Now let us show that if the evolution of the ket()obeys, in the gauge, to
the Schrödinger equation:
~
d
d
()= ()() (46)
the state vector()given by (36) satises an equation of the same form in the gauge
:
~
d
d
()= ()() (47)
where ()is given by (34).
To do this, let us calculate the left-hand side of (47); it is written:
~
d
d
()=~
d
d
()()
=~
d
d
()()+~()
d
d
() (48)
that is, according to (43) and (46)
6
:
~
d
d
()=(R;)()()+()()()
=
(R;) +
~
() () (49)
where
~
()designates the transform of()by the unitary operator():
~
() =()()() (50)
Equation (47) will therefore be satised if:
() =
~
()
(R;) (51)
Now
~
()is given by:
~
() =
1
2
~
P A(
~
R;)
2
+(
~
R;) (52)
6
The functiondepends onRand not onP; consequently(R)commutes with
(R;). This
is why()can be dierentiated as if(R)were an ordinary function of the time and not an operator
(cf.ComplementII, comment of ).
330

GAUGE INVARIANCE
where
~
Rand
~
Pdesignate the transforms ofRandPby the unitary operator(). According
to (37):
~
R=()R() =R (53a)
~
P=()P() =P r(R;) (53b)
These relations, substituted into (52), indicate that:
~
() =
1
2
[P A(R;)r(R;)]
2
+(R;) (54)
Using relations (2) to replace the potentials relative to the gaugeby those relative to, we
then obtain, using (34), relation (51). Therefore, the Schrödinger equation can be written in
the same way in any gauge chosen.
3-c. Invariance of physical predictions under a gauge transformation
. Behavior of the observables
Under the eect of the unitary transformation(), any observableis trans-
formed into
~
, with:
~
=() () (55)
We have already seen, in (53), that while
~
Ris simply equal toR,
~
Pis not equal toP.
Similarly,~is dierent fromsince:
~=
~
PA(
~
R;)
=Pr(R;)A(R;)
= r(R;) (56)
Taking (27a) and (31) into account, we see that relations (53a) and (56) imply that the
observablesRand, associated with true physical quantities (position and mechanical
momentum) are such that:
~
R=R
~=
(57)
On the other hand, the momentumP(which is not a true physical quantity) does not
satisfy an analogous relation, since, from (27b) and (53b):
~
P=P (58)
We shall see that this result is a general one:in quantum mechanics, for every true
physical quantity, there is an operator()that satises:
~
= () (59)
This relation is the quantum mechanical analogue of the classical relation (16). It shows
that, except for the special case ofRor a function ofRalone,the operator corresponding
331

COMPLEMENT H III
to a true physical quantity depends on the gauge. We have already seen examples of
this in (29) and (32).
To prove (59), one need only apply the quantization rules stated in Chapter
function(rp;)and use relation (17), the characteristic relation for true physical classical
quantities. We therefore replacerandpby the operatorsRandPand obtain (if necessary,
after a symmetrization with respect to these operators) the operator(). If the form of the
functiondepends on the gauge chosen, the operator()also depends on. When the
quantity associated withis a true physical quantity, we have, according to (17):
[RP;] =[RP+r(R;);] (60)
Applying the unitary transformation()to this relation, we obtain:
~
[RP;] =
~
[RP+r(R;);]
= [
~
R
~
P+r(
~
R;);] (61)
That is, taking (53) into account:
~
[RP;] =[RP;] (62)
After symmetrizing, if necessary, both sides of this relation, we indeed obtain (59).
Let us give some examples of true physical observables. In addition toRand
, we can cite the momentof the mechanical momentum [cf.(32)], or the kinetic
energy:
=

2
2
=
1
2
[PA(R;)]
2
(63)
On the other hand,PandLare not true physical quantities; neither is the Hamiltonian,
since relation (51) implies in general that:
~
()= () (64)
Comment:
In classical mechanics, it is well known that the total energy of a particle moving in
a time-independent electromagnetic eld is a constant of the motion. It is indeed
possible in this case to limit oneself to potentials which are also time-independent.
We see from (51) that one then has:
~
= (65)
In this particular case,is indeed a true physical observable which can therefore
be interpreted to be the total energy of the particle.
. Probability of the various possible results of a measurement bearing on a true
physical quantity
Assume that at timewe want to measure a true physical quantity. In the gauge
, the state of the system is described at this instant by the ket
7
, and the physical
7
We do not indicate the time dependence because all the quantities must be evaluated at the time
when we want to perform the measurement.
332

GAUGE INVARIANCE
quantity, by the observable. Let be an eigenvector of, with the eigenvalue
(assumed, for simplicity, to be non-degenerate):
= (66)
As calculated in the gaugefrom the postulates of quantum mechanics, the probability
of obtainingin the measurement envisaged is equal to:
=
2
(67)
What happens to this prediction when the gauge is changed? According to (59),
the operatorassociated with the quantity under consideration in the new gauge
can have the ket:
= (68)
as an eigenvector, with the same eigenvalueas in (66). That is:
=
= = (69)
In the gauge,therefore still appears as a possible measurement result. Moreover,
calculation of the corresponding probability yields the same value as in the gauge,
since, according to (36a) and (68):
= = (70)
We have thus veried that the postulates of quantum mechanics lead togauge-
invariant physical predictions: the possible results of any measurement and the associated
probabilities are invariant under a gauge transformation.
. Probability density and current
Let us calculate, from formulas (D-9) and (D-20) of Chapter, the probability
density(r)and probability currentJ(r)in two dierent gaugesand. For the
rst gauge, we have:
(r) =(r)
2
(71)
and:
J(r) =
1
Re (r)
~
r A(r;)(r) (72)
Relation (44) immediately shows that:
(r) =(r)
2
=(r) (73)
Moreover, it also implies that:
J(r) =
1
Ree
~
(r;)
(r)
~r A(r;)e
~
(r;)
(r)
=
1
Re (r)
~
r A(r;) +r(r;)(r) (74)
333

COMPLEMENT H III
that is, taking (2) into account:
J(r) =J(r) (75)
The probability density and current are therefore invariant under a gauge transformation.
This result could have been foreseen, moreover, from the conclusions of above,
since [cf.relation (D-19) of Chapter](r)andJ(r)can be considered to be mean
values of the operatorsrrand:
K(r) =
1
2
rr+rr (76)
It is not dicult to show that these two operators satisfy relation (59). They therefore
describe true physical quantities whose mean values are gauge-invariant.
References and suggestions for further reading:
Messiah (1.17), Chap. XXI, 20 to 22; Sakurai (2.7), 8-1.
Gauge invariance, extended to other domains, plays an important role in particle
physics; see, for example, the article by Abers and Lee (16.35).
334

PROPAGATOR FOR THE SCHRÖDINGER EQUATION
Complement JIII
Propagator for the Schrödinger equation
1 Introduction
2 Existence and properties of a propagator (21). . . . . . .
2-a Existence of a propagator
2-b Physical interpretation of(21). . . . . . . . . . . . . . . .
2-c Expression for(21)in terms of the eigenstates of. . . .
2-d Equation satised by (21). . . . . . . . . . . . . . . . . .
3 Lagrangian formulation of quantum mechanics
3-a Concept of a space-time path
3-b Decomposition of (21)into a sum of partial amplitudes
3-c Feynman's postulates
3-d The classical limit and Hamilton's principle
1. Introduction
Consider a particle described by the wave function(r). The Schrödinger equation
enables us to calculate
(r), that is, the rate of variation of(r)with respect to
. It therefore gives the time evolution of the wave function(r), using a dierential
point of view. One might wonder if it is possible to adopt a more global (but equivalent)
point of view that would allow us to determine directly the value(r0)taken on by
the wave function at a given pointr0and a given timefrom knowledge of the whole
wave function(r)at a previous time(which is not necessarily innitesimally close
to).
To consider this possibility, we can take our inspiration from another domain of
physics, electromagnetism, where both points of view are possible.Maxwell's equations
(the dierential point of view) give the rates of variation of the various components of the
electric and magnetic elds.Huygens' principle(the global point of view) permits the
direct calculation, when a monochromatic eld is known on a surface, of the eld at
any point: one sums the elds radiated at the pointbyctional secondary sources
1,2,3, ... situated on the surfaceand whose amplitude and phase are determined
by the value of the eld at1,2,3, ... (Fig.).
We intend to show in this complement that there exists an analogue of Huygens'
principle in quantum mechanics. More precisely, we can write, for2 1:
(r22) =d
3
1(r22;r11)(r11) (1)
a formula whose physical interpretation is the following: the probability amplitude of
nding the particle atr2at the instant2is obtained by summing all the amplitudes
335

COMPLEMENT J IIIM
N
3
N
2
N
1
Σ
Figure 1: In a diraction experiment, Huy-
gens' principle permits the calculation of the
electric eld at the point, as a sum of
elds radiated by secondary sources1,2,
3, ... situated on a surface.
radiated by the secondary sources(r11),(r
11)... situated in space-time on the
surface=1, each of these sources contributing to a degree proportional to(r11),
(r
11), ... (Fig.). We shall prove the preceding formula, calculate, called the
propagatorfor the Schrödinger equation, and study its properties. We shall then indicate
very qualitatively how it is possible to present all of quantum mechanics in terms of
(the Lagrangian formulation of quantum mechanics; Feynman's point of view).t = t
1
t = t
2
r
2
r
1
r
1
Figure 2: The probability amplitude(r22)
can be obtained by summing the contribu-
tions of the various amplitudes(r11),
(r
11), etc... corresponding to a given
previous instant1. With each of the ar-
rows of the gure is associated a propaga-
tor(r22;r11),(r22;r
11), etc...
2. Existence and properties of a propagator(21)
2-a. Existence of a propagator
The problem is to link directly the states of the system at two dierent times. This
is possible if we use the evolution operator introduced in ComplementIII, since we can
write:
(2)=(21)(1) (2)
Given(2), it is easy to nd the wave function(r22):
(r22) =r2(2) (3)
336

PROPAGATOR FOR THE SCHRÖDINGER EQUATION
Substituting (2) into (3) and inserting the closure relation:
d
3
1r1r1= (4)
between(21)and(1), we obtain:
(r22) =d
3
1r2(21)r1r1(1)
=d
3
1r2(21)r1(r11) (5)
The result is thus a formula identical to (1), on the condition that we set:
r2(21)r1=(r22;r11)
Moreover, since we want to use formulas of the type of (1) only for2 1, we can set
= 0for2 1. The exact denition ofthen becomes:
(r22;r11) =r2(21)r1(2 1) (6)
where(2 1)is the step function:
(2 1) = 1 if2 1
(2 1) = 0 if2 1 (7)
The introduction of(2 1)is of both physical and mathematical interest. From
the physical point of view, it is a simple way of compelling the secondary sources situated
on the surface=1of Figure
(r22;r11)as dened by (6) is called theretarded propagator. From the mathematical
point of view, we shall see later that(r22;r11), because of the factor(2 1),
obeys a partial dierential equation whose right-hand side is a delta function, which is
the equation that denes aGreen's function.
Comments:
()Note, however, that equation (5) remains valid even if2 1. It is possi-
ble, moreover, to introduce mathematically an advanced propagator which
would be dierent from zero only for2 1and which would also obey the
equation dening a Green's function. Since the physical meaning of such an
advanced propagator is not obvious at this stage, we shall not study it here.
()When no ambiguity is possible, we shall simply write (21)for
(r22;r11).
2-b. Physical interpretation of (21)
This interpretation follows very simply from denition (6):(21)represents the
probability amplitude that the particle, starting from pointr1, at time1, will arrive at
337

COMPLEMENT J III
pointr2at a later time2. If we take as the initial state at time1a state localized at
pointr1:
(1)=r1 (8)
at time2, the state vector has become:
(2)=(21)(1)=(21)r1 (9)
The probability amplitude of nding the particle at pointr2at this time is then:
r2(2)=r2(21)r1 (10)
2-c. Expression for (21)in terms of the eigenstates of
Assume that the Hamiltoniandoes not depend explicitly on time, and call
and its eigenstates and eigenvalues:
= (11)
According to formula (18) ofIII, we have:
(21) = e
(2 1)~
(12)
The closure relation:
= (13)
enables us to write (12) in the form:
(21) = e
(2 1)~
(14)
that is, taking (11) into account:
(21) =e
(2 1)~
(15)
To calculate(21), it then suces to take the matrix element of both sides of
(15) betweenr2andr1and to multiply it by(2 1). Since:
r2 =(r2) (16)
r1=(r1) (17)
this leads to:
(r22;r11) =(2 1) (r1)(r2) e
(2 1)~
(18)
338

PROPAGATOR FOR THE SCHRÖDINGER EQUATION
2-d. Equation satised by (21)
(r2) e
2~
is a solution of the Schrödinger equation. We deduce from this
that, in therrepresentation:
~
2
r2
~
r2 (r2)e
2~
= 0 (19)
wherer2is a condensed notation which designates the three operators
2
,
2
,
2
.
Let us then apply, to both sides of equation (18), the operator:
~
2
r2
~
r2
which acts only on the variablesr2and2. We know [cf.Appendix, relation (44)]
that:
2
(2 1) =(2 1) (20)
Consequently, using (19), we obtain:
~
2
r2
~
r2 (r22;r11) =
~(2 1) (r1)(r2) e
(2 1)~
(21)
Because of the presence of(2 1), we can replace2 1by zero in the sum over
appearing on the right-hand side of (21). This makes the exponential equal to 1. We are
thus left with the quantity(r2)(r1), which, according to (13), (16) and (17), is
equal to(r2r1)[taking the matrix element of (13) betweenr2andr1]. Finally,
satises the equation:
~
2
r2
~
r2 (r22;r11) =~(2 1)(r2r1) (22)
The solutions of equation (22), whose right-hand side is proportional to a four-dimensional
delta function, are calledGreen's functions. It can be shown that, to determine(21)
completely, it suces to associate with (22) the boundary condition:
(r22;r11) = 0 if 2 1 (23)
Equations (22) and (23) have interesting implications, in particular with regard to
perturbation theory, which we shall study in Chapter.
3. Lagrangian formulation of quantum mechanics
3-a. Concept of a space-time path
Let us consider, in space-time, the two points(r11)and(r22)(cf.Fig.;is
plotted as the abscissa, and the ordinate axis represents the set of the three spatial axes).
Chooseintermediate times(= 12 ), evenly spaced between1and2:
1 1 2 1 2 (24)
339

COMPLEMENT J IIIO
r
1
r
α
1
r
α
2
r
2
t
2
t
α
N
t
α
N – 1
t
α
2
t
α
1
t
1
t
r
α
N
r
α
N – 1
r
Figure 3: Diagram associated with a spacetime path: one picksintermediate times
(= 12 )evenly spaced between1and2, and chooses for each of them a value
ofr.
and, for each of them, a positionrin space. We can thus construct, whenapproaches
innity, a functionr()(which we shall assume to be continuous) such that:
r(1) =r1 (25a)
r(2) =r2 (25b)
r()is said to dene aspace-time pathbetween(r11)and(r22): such a path might be
thought of as the trajectory of a physical point leaving pointr1at time1and arriving
atr2at time2.
3-b. Decomposition of (21)into a sum of partial amplitudes
We rst return to the case where the numberof intermediate times is nite.
Formula (10) of ComplementIIIenables us to write:
(21) =(2 )(
1
)(
2 1
)(
11) (26)
We now take the matrix elements of both sides of (26) betweenr2andr1and insert
the closure relation relative to therrepresentation for each intermediate time.
According to (6) and (24), we thus obtain:
(21) =d
3
d
3
1
d
3
2
d
3
1
(2)( 1)
(21)(11)(27)
Now consider the product:
(2)( 1)(21)(11) (28)
Generalizing the argument of , we can interpret this term as being the probability
amplitude for the particle, having left point 1(r11), to arrive at point 2(r22), having
340

PROPAGATOR FOR THE SCHRÖDINGER EQUATION
passed successively through all points(r )of Figure. Note that, in formula (27),
one is summing over all possible positionsrat each time.
We now letapproach innity
1
. A series of pointsthen denes a space-time
path between 1 and 2, and the product (28) associated with it becomes the probability
amplitude for the particle to follow this path. Of course, the number of integrations in
formula (27) becomes innite. It is understandable, however, that the summation over
the set of possible positions at each time should reduce to a summation over the set of
possible paths.(21)is thus seen to be a sum (in fact, an integral) that corresponds
to the coherent superposition of the amplitudes associated with all possible space-time
paths starting from 1 and ending at 2.
3-c. Feynman's postulates
The concepts of a propagator and a space-time path permit a new formulation of
the postulate concerning the time evolution of physical systems. We shall outline here
such a formulation for the case of a spinless particle.
We dene(21)directly as being the probability amplitude for the particle,
starting fromr1at time1, to arrive atr2at time2. We then postulate that:
()(21)is the sum of an innity of partial amplitudes, one for each of the space-time
paths connecting(r11)and(r22).
()The partial amplitude(21)associated with one of these pathsis determined
in the following manner: letbe theclassical actioncalculated along, that is:
=
()
(rp) d (29)
where(rp)is the Lagrangian of the particle (cf.ppendix).(21)is then
equal to:
(21) =e
~

(30)
whereis a normalization constant (which can be determined explicitly).
It can be shown that the Schrödinger equation follows as a consequence of these
two postulates, which also lead to the canonical commutation relation between the com-
ponents of the observablesRandP. The two preceding postulates therefore permit
a formulation of quantum mechanics which is dierent from that of Chapter, but
equivalent.
3-d. The classical limit and Hamilton's principle
The formulation we have just evoked is particularly useful for discussing the rela-
tion between quantum and classical mechanics.
Consider a situation in which the actionsare much larger than~. In this
case, the variationof the action between two dierent paths, even if its relative
1
In this treatment, we make no attempt to be mathematically rigorous.
341

COMPLEMENT J III
value is small

1, is usually much larger than~. Consequently, the phase of
(21)varies rapidly, and the contributions to the global amplitude(21)of most of
the pathscancel out by interference. Let us assume, however, that there exists a path
0for which theaction is stationary(meaning that it does not vary, to rst order, when
one goes from0to another innitesimally close path). The amplitude0(21)then
interferes constructively with those of the paths next to0, since, this time, their phases
remain practically equal. Consequently, when the actionsare much larger than~,
one is in a quasi-classical situation: to obtain(21), one can ignore all the paths
except0and paths innitely close to it; it can then be said that, between points 1
and 2, the particle follows the trajectory0. Now this is indeed the classical trajectory,
dened by Hamilton's principle as being the path along which the action is minimal.
Feynman's postulates therefore include, at the classical limit, Hamilton's principle of
least action. They enable us, moreover, to associate with it the following picture: it is
the wave associated with a particle which, exploring the various possible paths, picks
the one for which the action will be the smallest.
The Lagrangian formulation of quantum mechanics presents numerous other ad-
vantages, which we shall not examine in detail. Let us point out, for example, that it
lends itself easily to relativistic generalization since one is already reasoning in space-
time. Moreover, it can be applied to any classical system (not necessarily mechanical)
governed by a variational principle (for example, a eld).
However, it has a certain number of drawbacks on the mathematical level (sum-
mation over an innite number of paths, the limit, ...).
References and suggestions for further reading:
Feynman's original article (2.38); Feynman and Hibbs (2.25); Bjorken and Drell
(2.6), Chaps. 6 and 7.
342

UNSTABLE STATES. LIFETIME
Complement KIII
Unstable states. Lifetime
1 Introduction
2 Denition of the lifetime
3 Phenomenological description of the instability of a state
1. Introduction
Consider a conservative system (a system whose Hamiltonianis time-independent).
Assume that at time= 0the state of the system is one of the eigenstatesof the
Hamiltonian, of energy:
(0)= (1)
with:
= (2)
In this case, the system remains indenitely in the same state (a stationary state,
of Chapter).
We shall study the hydrogen atom in Chapter
tion of its Hamiltonian, which is a time-independent operator. The states of the hydro-
gen atom (that is, the possible values of its energy) which we shall nd are in very good
agreement with the experimentally measured energies. However, it is known that most
of these states are actuallyunstable: if, at the instant= 0, the atom is in an excited
state (an eigenstatecorresponding to an energygreater than that of the ground
state, which is the lowest energy state), it generally falls back into this ground state
by emitting one or several photons. The stateis not really, therefore, a stationary
state in this case.
This problem arises from the fact that, in calculations of the type used in Chap-
ter, the system under study (the hydrogen atom) is treated as if it were totally
isolated, while it is actually in constant interaction with the electromagnetic eld. Al-
though the evolution of the global system atom+electromagnetic eld can be perfectly
well described by a Hamiltonian, it is not rigorously possible to dene a Hamiltonian for
the hydrogen atom alone [cf.comment()of III]. However, since
the coupling between the atom and the eld happens to be weak (it can be shown that its
force is characterized by the ne structure constant
1
137
, which we shall introduce
in Chapter), the approximation consisting of completely neglecting the existence of
the electromagnetic eld is very good, except, of course, if we are interested precisely in
the instability of the states.
343

COMPLEMENT K III
Comments:
()If, at the initial instant, a strictly conservative and isolated system is in a
state formed by a linear combination of several stationary states, it evolves
over time, and does not always remain in the same state. But its Hamiltonian
is a constant of the motion, and consequently (cf.Chapter, ), the
probability of nding one energy value or another is independent of time, as
is the mean value of the energy. On the other hand, in the case of an unstable
state, one state is transformed irreversibly into another, with a loss of energy
for the system: this energy is taken away by the photons emitted
1
.
()The instability of the excited states of an atom is caused by the spontaneous
emission of photons; the ground state is stable, since there exists no lower
energy state. Recall, nevertheless, that atoms can also absorb light energy
and so ascend to higher energy levels.
We intend to indicate here how to take the instability of a state into account
phenomenologically. The description will not be rigorous, as we shall continue to consider
the system as if it were isolated. We shall try to incorporate this instability as simply as
possible into the quantum description of the system.
ComplementXIIIpresents a more rigorous treatment of this problem, justifying
the phenomenological approach used here.
2. Denition of the lifetime
Experiments show that the instability of a state can often be characterized by just one
parameter, having the dimensions of a time, which is called thelifetimeof the state.
More precisely, if one prepares the system at time= 0in the unstable state, one
observes that the probability()of its still being excited at a later timeis equal to:
() = e (3)
This result can also be expressed in the following way. Consider a large number
of identical independent systems, all prepared at time= 0in the state. At time
, there will remain() =e in this state. Between timesand+ d, a certain
numberd()of systems leave the unstable state:
d() =() (+ d) =
d()
d
d=()
d
(4)
For each of the()systems that are still in the stateat time, a probability can
therefore be dened:
d() =
d()
()
=
d
(5)
of their leaving this state during the time intervaldfollowing the instant. We see that
d()is independent of: the system is said to have aprobability per unit time
1
of
leaving the unstable state.
1
which, moreover, may also take away linear and angular momentum.
344

UNSTABLE STATES. LIFETIME
Comments:
()Let us calculate the mean value of the time during which the system remains
in the unstable state. It is equal to:
0
e
d
= (6)
is therefore the mean time the system spends in the state; this is why
it is called the lifetime of this state.
For a stable state,()is always equal to 1, and the lifetimeis innite.
()A remarkable property of the lifetimeis that it does not depend on the
procedure used to prepare the system in the unstable state, that is, on its
previous history: the lifetime is a characteristic of the unstable state itself.
()According to the time-energy uncertainty relation ( ),
the timecharacteristic of the evolution of an unstable state is associated
with an uncertainty in the energygiven by:

~
(7)
One indeed nds that the energy of an unstable state cannot be determined
with arbitrary accuracy, but only to within an uncertainty of the order of.
is called thenatural widthof this state. For the case of the hydrogen atom,
the width of the various states is negligible compared to their separation.
This explains why we can treat them, in a rst approximation, as if they
were stable.
3. Phenomenological description of the instability of a state
First let us consider a conservative system, prepared, at the initial time, in the eigenstate
of the Hamiltonian. According to the rule (D-54) of Chapter, the state vector,
at time, becomes:
()= e
~
(8)
The probability()of nding, in a measurement at time, the system in the state
is:
() =e
~
2
(9)
Since the energyis real (being an observable), this probability is constant and
equal to 1: we again nd thatis a stationary state.
Let us examine what would happen if, in expression (9), we replaced the energy
by the complex number:
= ~
2
(10)
345

COMPLEMENT K III
The probability()then becomes:
() =e
( ~
2
)~
2
= e (11)
In this case, the probability of nding the system in the statedecreases exponentially
with time, as in formula (3). Therefore, to take into account phenomenologically the
instability of a statewhose lifetime is, it suces to add, as in (10), an imaginary
part to its energy, and set:
=
1
(12)
Comment:
When is replaced by, the norm of the state vector written in (8) becomes
e
2
and therefore varies with time. This result is not surprising. We saw
in
arose from the Hermitian nature of the Hamiltonian operator; now, an operator
whose eigenvalues are complex, as are the, cannot be Hermitian. Of course,
as we pointed out in , this is due to the fact that the system under study is
part of a larger system (it is interacting with the electromagnetic eld) and its
evolution cannot be described rigorously by means of a Hamiltonian. It is already
rather remarkable that its evolution can be simply explained by introducing a
Hamiltonian with complex eigenvalues.
346

EXERCISES
Complement LIII
Exercises
1.In a one-dimensional problem, consider a particle whose wave function is:
() =
e
0~
2
+
2
whereand0are real constants andis a normalization coecient.
Determineso that()is normalized.
The position of the particle is measured. What is the probability of nding a result
between
3
and+
3
?
Calculate the mean value of the momentum of a particle which has()for its
wave function.
2.Consider, in a one-dimensional problem, a particle of masswhose wave function at
timeis().
At time, the distanceof this particle from the origin is measured. Write, as a
function of(), the probability(0)of nding a result greater than a given
length0. What are the limits of(0)when0 0and when0 ?
Instead of performing the measurement of question, one measures the velocity
of the particle at time. Express, as a function of(), the probability of nding
a result greater than a given value0
3.The wave function of a free particle, in a one-dimensional problem, is given at time
= 0by:
(0) =
+
de
0
e
where0andare constants.
What is the probability(10)that a measurement of the momentum, performed
at time= 0, will yield a result included between1and+1? Sketch the
function(10).
What happens to this probability(1)if the measurement is performed at time
? Interpret.
What is the form of the wave packet at time= 0? Calculate for this time the
product; what is your conclusion? Describe qualitatively the subsequent
evolution of the wave packet.
347

COMPLEMENT L III
4. Spreading of a free wave packet
Consider a free particle.
Show, applying Ehrenfest's theorem, thatis a linear function of time, the mean
valueremaining constant.
Write the equations of motion for the mean values
2
and + . Integrate
these equations.
Show that, with a suitable choice of the time origin, the root mean square deviation
is given by:
()
2
=
1
2
()
2
0
2
+ ()
2
0
where()0and()0are the root mean square deviations at the initial time.
How does the width of the wave packet vary as a function of time (see
ComplementI)? Give a physical interpretation.
5. Particle subjected to a constant force
In a one-dimensional problem, consider a particle having a potential energy()
given by() = , whereis a positive constant [()arises, for example, from
a gravity eld or a uniform electric eld].
Write Ehrenfest's theorem for the mean values of the positionand the momentum
of the particle. Integrate these equations; compare with the classical motion.
Show that the root mean square deviationdoes not vary over time.
Write the Schrödinger equation in therepresentation. Deduce from it a rela-
tion between
()
2
and()
2
. Integrate the equation thus obtained;
give a physical interpretation.
6.Consider the three-dimensional wave function:
( ) =e
2
+
2
+
2
where,andare three positive lengths.
Calculate the constantwhich normalizes.
Calculate the probability that a measurement ofwill yield a result included
between 0 and.
Calculate the probability that simultaneous measurements ofandwill yield
results included respectively betweenand+, andand+.
Calculate the probability that a measurement of the momentum will yield a result
included in the elementdddcentered at the point== 0;=~.
348

EXERCISES
7.Let( ) =(r)be the normalized wave function of a particle. Express in terms
of(r)the probability for:
a measurement of the abscissa, to yield a result included between1and2;
a measurement of the componentof the momentum, to yield a result included
between1and2;
simultaneous measurements ofand, to yield:
1 2
0
simultaneous measurements of,,to yield:
1 2
3 4
5 6
Show that this probability is equal to the result ofwhen3,5 ;4,
6 +
a measurement of the component=
1
3
(++)of the position, to yield a
result included between1and2.
8.LetJ(r)be the probability current associated with a wave function(r)describing
the state of a particle of mass[Chap., relations (D-17) and (D-19)].
Show that:
d
3
J(r) =P
wherePis the mean value of the momentum.
Consider the operatorL(orbital angular momentum) dened byL=RP. Are
the three components ofLHermitian operators? Establish the relation:
d
3
[rJ(r)] =L
9.One wants to show that the physical state of a (spinless) particle is completely dened
by specifying the probability density(r) =(r)
2
and the probability currentJ(r).
Assume the function(r)known and let(r)be its argument:
(r) =
(r)e
(r)
Show that:
J(r) =
~
(r)r(r)
Deduce that two wave functions leading to the same density(r)and currentJ(r)
can dier only by a global phase factor.
349

COMPLEMENT L III
Given arbitrary functions(r)andJ(r), show that a quantum state(r)can be
associated with them only ifrv(r) = 0, wherev(r) =J(r)(r)is the velocity
associated with the probability uid.
Now assume that the particle is submitted to a magnetic eldB(r) =rA(r)
[see Chap., denition (D-20) of the probability current in this case]. Show that:
J=
(r)
[~r(r)A(r)]
and:
rv(r) =
B(r)
10. Virial theorem
In a one-dimensional problem, consider a particle with the Hamiltonian:
=
2
2
+()
where:
() =
Calculate the commutator[ ]. If there exists one or several stationary states
in the potentialshow that the mean valuesand of the kinetic and
potential energies in these states satisfy the relation:2= .
In a three-dimensional problem,is written:
=
P
2
2
+(R)
Calculate the commutator[RP]. Assume that(R)is a homogeneous function
ofth order in the variables,,. What relation necessarily exists between the
mean kinetic energy and the mean potential energy of the particle in a stationary
state?
Apply this to a particle moving in the potential() =e
2
(hydrogen atom).
Recall that a homogeneous functionofth degree in the variables,andby
denition satises the relation:
( ) = ( )
and satises Euler's identity:
++=( )
350

EXERCISES
Consider a system ofparticles of positionsRand momentaP(= 12 ).
When their potential energy is a homogeneous (th degree) function of the set of
components,,, can the results obtained above be generalized? Apply these
results to an arbitrary molecule formed of nuclei of chargesand electrons
of charge. All these particles interact by pairs through Coulomb forces. In a
stationary state of the molecule, what relation exists between the kinetic energy of
the system of particles and their energy of mutual interaction?
11. Two-particle wave function
In a one-dimensional problem, consider a system of two particles (1) and (2) with
which is associated the wave function(12).
What is the probability of nding, in a measurement of the positions1and2
of the two particles, a result such that:
1 + d
2 ?
What is the probability of nding particle (1) betweenand+ d[when no
observations are made on particle (2)]?
Give the probability of nding at least one of the particles betweenand.
Give the probability of nding one and only one particle betweenand.
What is the probability of nding the momentum of particle (1) included between
andand the position of particle (2) betweenand?
The momenta 1and2of the two particles are measured; what is the probability
of nding 1 ; 2 ?
The only quantity measured is the momentum1of the rst particle. Calculate,
rst from the results ofand then from those of, the probability of nding this
momentum included betweenand. Compare the two results obtained.
The algebraic distance1 2between the two particles is measured; what is the
probability of nding a result included betweenand+? What is the mean
value of this distance?
12. Innite one-dimensional well
Consider a particle of massplaced in the potential:
() = 0 if 0
() = + if 0 or
are the eigenstates of the Hamiltonianof the system, and their eigenvalues are
=
22
~
2
2
2
(cf.ComplementI). The state of the particle at the instant= 0is:
(0)=11+22+33+44
351

COMPLEMENT L III
What is the probability, when the energy of the particle in the state(0)is
measured, of nding a value smaller than
3
2
~
2
2
?
What is the mean value and what is the root mean square deviation of the energy
of the particle in the state(0)?
Calculate the state vector()at the instant. Do the results found inand
at the instant= 0remain valid at an arbitrary time?
When the energy is measured, the result of
8
2
~
2
2
is found. After the measurement,
what is the state of the system? What is the result if the energy is measured again?
13. Innite two-dimensional well (cf.ComplementII)
In a two-dimensional problem, consider a particle of mass; its Hamiltonian
is written:
= +
with:
=
2
2
+() =
2
2
+()
The potential energy()[or()] is zero when(or) is included in the interval [0,
] and is innite everywhere else.
Of the following sets of operators, which form a C.S.C.O.?
Consider a particle whose wave function is:
() =cos
cossin
2
sin
2
when0 and0 , and is zero everywhere else (is a constant).
What is the mean valueof the energy of the particle? If the energyis
measured, what results can be found, and with what probabilities?
The observableis measured; what results can be found, and with what
probabilities? If this measurement yields the result
2
~
2
2
2
, what will be the
results of a subsequent measurement of, and with what probabilities?
Instead of performing the preceding measurements, one now performs a simul-
taneous measurement ofand. What are the probabilities of nding:
=
9
2
~
2
2
2
and:
0 0+ d?
352

EXERCISES
14.Consider a physical system whose state space, which is three-dimensional, is spanned
by the orthonormal basis formed by the three kets1,2,3. In this basis, the
Hamiltonian operatorof the system and the two observablesandare written:
=~0
1 0 0
0 2 0
0 0 2
;=
1 0 0
0 0 1
0 1 0
; =
0 1 0
1 0 0
0 0 1
where0,andare positive real constants.
The physical system at time= 0is in the state:
(0)=
1
2
1+
1
2
2+
1
2
3
At time= 0, the energy of the system is measured. What values can be found,
and with what probabilities? Calculate, for the system in the state(0), the
mean value and the root mean square deviation.
Instead of measuringat time= 0, one measures; what results can be found,
and with what probabilities? What is the state vector immediately after the mea-
surement?
Calculate the state vector()of the system at time.
Calculate the mean values()and()ofandat time. What comments
can be made?
What results are obtained if the observableis measured at time? Same question
for the observable. Interpret.
15. Interaction picture(It is recommended that ComplementIIIand perhaps
ComplementIIIbe read before this exercise is undertaken.)
Consider an arbitrary physical system. Denote its Hamiltonian by0()and the
corresponding evolution operator by0():
~
0(0) =0()0(0)
0(00) =
Now assume that the system is perturbed in such a way that its Hamiltonian becomes:
() =0() +()
The state vector of the system in the interaction picture,(), is dened from the
state vector()in the Schrödinger picture by:
()=
0
(0)()
353

COMPLEMENT L III
Show that the evolution of()is given by:
~
d
d
()=()()
where()is the transform of operator()under the unitary transformation
associated with
0
(0):
() =
0
(0)()0(0)
Explain qualitatively why, when the perturbation()is much smaller than0(),
the motion of the vector()is much slower than that of().
Show that the preceding dierential equation is equivalent to the integral equation:
()=(0)+
1
~
0
d ()()
where:(0)= ().
Solving this integral equation by iteration, show that the ket()can be ex-
panded in a power series inof the form:
()= +
1
~
0
d () +
1
(~)
2
0
d ()
0
d () + (0)
16. Correlations between two particles
(It is recommended that the ComplementIIIbe read in order to answer question
of this exercise.)
Consider a physical system formed by two particles (1) and (2), of the same mass,
which do not interact with each other and which are both placed in an innite potential
well of width(cf.ComplementI, ). Denote by(1)and(2)the Hamiltonians
of each of the two particles and by(1)and(2)the corresponding eigenstates of
the rst and second particle, of energies
22
~
2
2
2
and
22
~
2
2
2
. In the state space of the
global system, the basis chosen is composed of the statesdened by:
= (1) (2)
What are the eigenstates and the eigenvalues of the operator=(1)+(2), the
total Hamiltonian of the system? Give the degree of degeneracy of the two lowest
energy levels.
Assume that the system, at time= 0is in the state:
(0)=
1
6
11+
1
3
12+
1
6
21+
1
3
22
What is the state of the system at time?
354

EXERCISES
The total energyis measured. What results can be found, and with what
probabilities?
Same questions if, instead of measuring, one measures(1).
Show that(0)is a tensor product state. When the system is in this state,
calculate the following mean values:(1),(2)and(1)(2). Compare
(1)(2)with(1)(2); how can this result be explained?
Show that the preceding results remain valid when the state of the system is the
state()calculated in.
Now assume that the state(0)is given by:
(0)=
1
5
11+
35
12+
1
5
21
Show that(0)cannot be put in the form of a tensor product. Answer for this
case all the questions asked in.
Write the matrix, in the basis of the vectors, that represents the density
operator(0)corresponding to the ket(0)given in. What is the density matrix
()at time? Calculate, at the instant= 0, the partial traces:
(1) = Tr2and(2) = Tr1
Do the density operators,(1)and(2)describe pure states? Comparewith
(1)(2); what is your interpretation?
Answer the same questions as in, but choosing for(0)the ket given in.
The subject of the following exercises is the density operator: they therefore assume
the concepts and results of ComplementIIIto be known.
17.Letbe the density operator of an arbitrary system, whereandare the
eigenvectors and eigenvalues of. Writeand
2
in terms of theand. What do
the matrices representing these two operators in thebasis look like rst, in the
case wheredescribes a pure state and then, in the case of a statistical mixture of states?
(Begin by showing that, in a pure case,has only one non-zero diagonal element, equal
to 1, while for a statistical mixture,has several diagonal elements included between 0
and 1.) Show thatcorresponds to a pure case if and only if the trace of
2
is equal to
1.
18.Consider a system whose density operator is(), evolving under the inuence of a
Hamiltonian(). Show that the trace of
2
does not vary over time. Conclusion: can
the system evolve so as to be successively in a pure state and a statistical mixture of
states?
19.Let (1) + (2) be a global system, composed of two subsystems (1) and (2).and
denote two operators acting in the state space(1)(2). Show that the two partial
traces Tr1 and Tr1 are equal when(or) actually acts only in the space
(1), that is, when(or) can be written:
=(1)(2) [or=(1)(2)]
355

COMPLEMENT L III
Application: if the operator, the Hamiltonian of the global system, is the sum of two
operators that act, respectively, only in(1)and only in(2):
=(1) +(2)
calculate the variation
d
d
(1)of the reduced density operator(1). Give the physical
interpretation of the result obtained.
References
Exercise 5
Flügge (1.24), 40 and 41; Landau and Lifshitz (1.19), 22.
Exercise 10
Levine (12.3), Chap. 14; Eyring et al. (12.5), 18 b
Exercise 15
See references of ComplementIII.
356

REVISITING ONE-DIMENSIONAL PROBLEMS
Now that we are more familiar with the mathematical formalism and
the physical content of quantum mechanics, we can go in more detail into
some of the results obtained in Chapter. In the three complements that
follow, we shall study in a general way the quantum properties of a particle
subjected to a scalar potential
1
of arbitrary form, conning ourselves for
simplicity to one-dimensional problems. We shall rst focus on the bound
stationary states of a particle, whose energies form a discrete spectrum
(ComplementIII), and then treat the unbound states corresponding to
an energy continuum (ComplementIII). In addition, we shall examine a
special case that is very important because of its applications, particularly
in solid state physics, that of a periodic potential (ComplémentIII).
1
The eects of a vector potentielAwill be studied later, in particular in ComplementVI.
357

BOUND STATES IN A POTENTIAL WELL OF ARBITRARY SHAPE
Complement MIII
Bound states in a potential well of arbitrary shape
1 Quantization of the bound state energies
2 Minimum value of the ground state energy
In complementI, we studied, for a special case (nite or innite square well),
the bound states of a particle in a potential well. We derived certain properties of these
bound states: a discrete energy spectrum and a ground state energy greater than the
classical minimum energy. These properties are, in fact, general, and have numerous
physical consequences, as we shall show in this complement.
When the potential energy of a particle posesses a minimum (see Figurea), the
particle is said to be placed in a potential well
1
. Before studying qualitatively the
stationary states of a quantum particle in such a well, let us recall the corresponding
motion of a classical particle. When its energytakes on the minimum possible value
= 0. (where0is the depth of the well), the particle is motionless at the point
0whose abscissa is0. In the case where0 0, the particle oscillates in
the well, with an amplitude that increases with. Finally, when 0, the particle
does not remain in the well, but moves o towards innity. The bound states of the
classical particle therefore correspond to all negative energy values between0and 0.
For a quantum particle, the situation is very dierent. Well-dened energy states
are stationary states whose wave functions()are solutions of the eigenvalue equation
of the Hamiltonian:
~
2
2
d
2
d
2
+()() =() (1)
Such a second-order dierential equation has an innite number of solutions, whatever
the value chosen for: if we pick arbitrary values of()and its derivative at any given
point, we can obtainfor any other value of. Equation (1) alone cannot, therefore,
restrict the possible energy values. However, we shall show that if, in addition, we impose
certain boundary conditions on(), only a certain number of values ofremain possible
(quantization of energy levels).
1. Quantization of the bound state energies
We shall call bound states of the particle states whose wave functions()satisfy
the eigenvalue equation (1) and aresquare-integrable[indispensable if()is actually to
describe the physical state of a particle]. These are therefore stationary states, for which
the position probability density()
2
takes on non-negligible values only in a limited
region of space [for
+
d()
2
to converge,()
2
must approach zero suciently
1
The potential energy, of course, is only dened to within a constant. Following the usual convention,
we set the potential equal to zero at innity.
359

COMPLEMENT M III
rapidly when ]. Bound states remind us of classical motion where the particle
oscillates inside the well without ever being able to emerge (energynegative, but
greater than0).
We shall see that in quantum mechanics, the fact that()is required to be
square-integrable implies that the possible energies form a discrete set of values which
are also included between0and 0. To understand this, let us return to the potential
shown in Figurea. For simplicity, we shall assume that()is identically equal to zero
outside an interval[12].
If 1(region I),() = 0, and the solution to equation (1) can immediately
be written:
0:
I() =e+e (2)
with:
=
2~
2
(3)
0:
I() =e+e (4)
with:
=
2~
2
(5)
We are looking for a square-integrable solution; we must therefore eliminate the form (2)
in whichI()is a superposition of plane waves of constant modulus which cause the
integral:
1
d I()
2
(6)
to diverge. Only possibility (4) remains, and we obtain our rst result:the bound states
of the particle all have a negative energy. In (4), we cannot retain the term ine,
which diverges when . We are therefore left with:
I() = e if 1 (7)
[We have omitted the proportionality factorsince the homogeneity of equation (1)
allows us to dene()to within a multiplicative coecient].
The value of()in the interval1 2(region II) is obtained by extending
I(): we must look for the solution of equation (1) which is equal toe
1
for=1
and whose derivative at this point is equal toe
1
. The functionII()thus obtained
depends onand, of course, on the exact expression for(). Nevertheless, since (1)
is a second-order dierential equation,II()is determined uniquely by the preceding
boundary conditions; it is, moreover, real (which enables us to trace curves such as those
in Figuresb,c andd).
360

BOUND STATES IN A POTENTIAL WELL OF ARBITRARY SHAPEx
x
x
x
x
1
x
2
V(x)
E < E
3
0
a
b
c
d
0
E = E
3
0
E > E
3
0
φ(x)
Figure 1: Potential well (g. a) situated between the points=1and=2. We
choose a solution()of the eigenvalue equation ofwhich, for 1, approaches
zero exponentially when . We then extend this solution to the entire-axis.
For an arbitrary energy value,()diverges like
~
()ewhen +: gure b
represents the case where
~
()0; gure d, that where
~
()0. However, if the
energyis chosen so as to make
~
() = 0,()approaches zero exponentially when
+(g. c), and()is square-integrable.
361

COMPLEMENT M III
All that now remains to be done is to obtain the solution when2(region III);
this solution can be written:
III() =
~
e+
~
e (8)
where
~
and
~
are real constants determined by the two continuity conditions for()
andddat the point=2.
~
and
~
depend on, as well as on the function()
We have therefore constructed a solution of equation (1), such as the one shown
in Figureb. Is this solution square-integrable? We see from (8) that, in general, it is
not, except when
~
is zero (this special case is shown in Figurec). Now, for a given
function(),
~
is a function ofthrough the intermediary of. The only values of
for which a bound state exists are therefore solutions of the equation
~
() = 0. These
solutions1,2, ... (cf.Fig.) form a discrete spectrum which, of course, depends on
the potential()chosen (we shall see in the following section that all the energies
are greater than0).E
1
E
2
E
3
– V
0
E
0
B(E)
Figure 2: Graphical representation of the function
~
(). The zeros of
~
()give the
values offor which()is square-integrable (the situation in Figurec), that is, the
energies1,2,3... of the bound states; all these energies are included between0
and 0.
We thus arrive at the following result:the bound state energy values possible for
a particle placed in a potential well of arbitrary shape form a discrete set(it is often
said that the bound state energies are quantized). This result can be compared to the
quantization of electromagnetic modes in a cavity. There is no analogue in classical
mechanics, where, as we have seen, all energy values included between0and 0 are
acceptable. In quantum mechanics, the lowest energy level1is called theground state,
the energy level2immediately above, therst excited state, the next energy level3,
the second excited state, etc. The following schematic diagram is often associated with
each of these states: inside the potential well representing(), a horizontal line is drawn
whose vertical position corresponds to the energy of the state and whose length gives an
idea of the spatial extension of the wave function (this line actually covers the points of
the axis which would be reached by a classical particle of the same energy). For the set
of energy levels, we obtain a schematic diagram of the type shown of Figure.
As we saw in Chapter, the phenomenon of energy quantization was one of the
factors which led to the introduction of quantum mechanics. Discrete energy levels appear
in a very large number of physical systems: atoms (cf.Chap., hydrogen atom), the
harmonic oscillator (cf.Chap.), atomic nuclei, etc.
362

BOUND STATES IN A POTENTIAL WELL OF ARBITRARY SHAPEV(x)
E
3
0
E
2
E
1
– V
0
x
Figure 3: Schematic representation of the bound states of a particle in a potential well.
For each of these stationary states, one draws a horizontal line whose ordinate is equal to
the energy of the corresponding level. The ends of this line are the points of intersection
with the curve representing the potential(). The line is conned to the region of
classical motion for the same energy, and gives an idea of the extension of the wave
function.
2. Minimum value of the ground state energy
We now show that the energies1,2, etc... are all greater than the minimum value
0of the potential energy(). We shall see how this result can be easily understood
using Heisenberg's uncertainty relation.
If()is a solution of (1), we obtain, multiplying this equation by()and
integrating the relation thus obtained:
~
2
2
+
d ()
d
2
d
2
() +
+
d()()
2
=
+
d()
2
(9)
For a bound state, the function()can be normalized, and equation (9) can be
written simply:
= + (10)
with:
=
~
2
2
+
d()
d
2
d
2
() =
~
2
2
+
d
d
d
()
2
(11)
[where we have performed an integration by parts and used the fact that()goes to
zero when ] and:
=
+
d()()
2
(12)
363

COMPLEMENT M III
Relation (10) shows simply thatis the sum of the mean value of the kinetic energy:
=
2
2
(13)
and that of the potential energy:
= () (14)
From relations (11) and (12), it follows immediately that:
0 (15)
+
d(0)()
2
= 0 (16)
Consequently:
= + 0 (17)
Sinceis negative, as we showed in , we see that,as in classical mechanics, the bound
state energies are always between0and0.
There exists, nevertheless, an important dierence between the classical and quan-
tum situations: while, in classical mechanics, the particle can have an energy equal to
0(case of a particle at rest at0) or slightly greater than0(case of small oscil-
lations), the same is not true in quantum mechanics, where the lowest possible energy
is the energy1of the ground state, which is necessarily greater than0(cf.Fig.).
The Heisenberg uncertainty relations enable us to understand the physical origin of this
result, as we now show.
If we try to construct a state of the particle for which the mean potential energy
is as small as possible, we see from (12) that we must choose a wave function which is
practically localized at the point0. The root mean square deviationis then very
small, sois necessarily very large. Since:
2
= ()
2
+
2
()
2
(18)
the kinetic energy=
2
2is thus also very large. Therefore, if the potential
energy of the particle approaches its minimum, the kinetic energy increases without
bound. The wave function of the ground state corresponds to a compromise, for which
the sum of these two energies is a minimum. The ground state of the quantum particle
is thus characterized by a wave function that has a certain spatial extension (cf.Fig.),
and its energy is necessarily greater than0. Unlike the situation in classical mechanics,
there exists no well-dened energy state in quantum mechanics where the particle is at
rest at the bottom of the potential well.
Comment:
Since the energy of the bound states is included between0and 0, such states
can exist only if the potential()takes on negative values in one or several
regions of the-axis. This is why we have chosen for this complement a potential
364

BOUND STATES IN A POTENTIAL WELL OF ARBITRARY SHAPEV(x)
0
V
2
V
1
– V
0
x
x
0
Figure 4: Potential well of depth0situated between two potential barriers of height1
and2(assuming, for example,1 2). Classically, there exist particle states whose
energy is between0and1that remain conned between the two barriers. In quantum
mechanics, a particle whose energy is between 0 and1can penetrate the barrier by the
tunnel eect ( of ComplementI); consequently, the bound states always have
energies between0and 0.
well like the one shown in Figurea (while in the following complement, we shall
not conne ourselves to the case of a potential well).
However, there is nothing to prevent()from being positive for certain values of
; for example, the well can be surrounded by potential barriers as is shown in
Figure
certain classical motions of positive energy remain bounded, while in quantum
mechanics, the same reasoning as above shows that the bound states always have
an energy between0and 0. Physically, this dierence arises from the fact that
a potential barrier of nite height is never able to make a quantum particle turn
back completely: the particle always has a non-zero probability of passing through
by the tunnel eect.
References and suggestions for further reading:
Feynman III (1.2), 16-6; Messiah (1.17), Chap., II; Ayant and Belorizky
(1.10), Chap. IV, 1, 2, 3; Schi (1.18), 8.
365

UNBOUND STATES OF A PARTICLE IN THE PRESENCE OF A POTENTIAL WELL OR BARRIER
Complement NIII
Unbound states of a particle in the presence of a potential well or
barrier of arbitrary shape
1 Transmission matrix (). . . . . . . . . . . . . . . . . . . .
1-a Denition of (). . . . . . . . . . . . . . . . . . . . . . . .
1-b Properties of (). . . . . . . . . . . . . . . . . . . . . . . .
2 Transmission and reection coecients
3 Example
In complementIII, we showed that bound states of a particle placed in a potential
()have negative energies
1
and that they exist only if()is an attractive potential (a
potential well which allows classical bounded motion). We had to reject positive energy
values since they led to eigenfunctions()of the Hamiltonianwhich, at innity,
behaved like superpositions of non square-integrable exponentialse. Nevertheless, we
saw as early as Chapter, that, by superposing such functions linearly, one can construct
square-integrable wave functions()(wave packets) which can therefore represent the
physical state of a particle. It is clear that, since the states thus obtained involve several
values of(that is, of the energy), they are no longer stationary states; the wave function
()therefore evolves over time, propagating and becoming deformed. However, the fact
that()is already expanded in terms of the eigenfunctions()enables us to calculate
this evolution very simply [as we did, for example, in complementI, where we used
the properties of the()to calculate the transmission and reection coecients of a
potential barrier, the delay upon reection, etc.]. This is why, despite the fact that each
of the()cannot alone represent a physical state, it is useful to study the positive
energy eigenfunctions
2
of, as we have already done, in complementI, for certain
square potentials.
In this complement, we are going to study in a general way (conning ourselves,
nevertheless, to one-dimensional problems) the eect of a potential()on the positive
energy eigenfunctions(). We shall assume nothing about the shape of(), which
may present one or several barriers, wells, etc., except that()goes to zero outside
a nite interval [1,2] of the-axis. We shall show that, in all cases, the eect of
()on the functions()can be described by a22matrix,(), which possesses
a certain number of general properties. We shall thus obtain various results that are
independent of the shape of the potential()chosen. For example, we shall see that
the transmission and reection coecients of a barrier (whether symmetrical or not) are
the same for a particle coming from the left and for a particle of the same energy coming
from the right. An additional aim of this complementIIIis to serve as the point of
1
Recall that we chose the energy origin so as to make()zero at innity.
2
One might also consider studying the non square-integrable negative energy eigenfunctions of
(those whose energies do not belong to the discrete spectrum obtained in complementIII). However,
these functions diverge very rapidly (exponentially) at innity, and one could not obtain square-integrable
wave functions by linearly superposing them.
367

COMPLEMENT N IIIl
2
0
–
l
2
x
V(x)
+
Figure 1: The potential()under consid-
eration varies in an arbitrary way within the
interval2 2and goes to zero
outside this interval.
departure for the calculations of complementIII, in which we study the properties of a
particle in a periodic potential().
1. Transmission matrix ()
1-a. Denition of ()
In a one-dimensional problem, consider a potential()which is zero outside an
interval[1 2]of length, but which varies in an arbitrary way inside this interval
(Fig.). We choose theorigin to be in the middle of the interval[1 2], so as to
have()vary only for 2. The equation satised by every wave function()
associated with a stationary state of energyis:
d
2
d
2
+
2
~
2
[ ()]() = 0 (1)
In the rest of this complement, we shall choose, to characterize the energy, the parameter
given by:
=
2~
2
(2)
In the region 2, the functionesatises equation (1); let us call()
the solution of this equation that is identical toefor
2
. When +
2
,()is
necessarily a linear combination of two independent solutionseande of (1). This
gives us:
if
2
: () = e (3a)
if +
2
: () =() e+() e (3b)
where()and()are coecients which depend on, as well as on the shape of
the potential under study. Similarly, we can introduce the solution(), which, for
2, is equal toe:
if
2
: () = e (4a)
if +
2
: () =() e+() e (4b)
368

UNBOUND STATES OF A PARTICLE IN THE PRESENCE OF A POTENTIAL WELL OR BARRIER
The most general solution()of equation (1) (of second order in), for a given
value of(that is, of), is a linear combination ofand:
() = () + () (5)
Relations (3a) and (4a) imply that:
if
2
: () =e+e (6a)
while relations (3b) and (4b) yield:
if +
2
: () =
~
e+
~
e (6b)
with:
~
=()+()
~
=()+() (7)
By denition, the matrix()is the22matrix:
() =
()()
()()
(8)
which allows us to write relations (7) in the matrix form:
~
~
=() (9)
()therefore enables us to determine, given the behavior (6a) of the wave function to
the left of the potential, its behavior (6b) to the right. We call()the transmission
matrix of the potential.
Comment:
The current associated with a wave function()is:
() =
~
2
()
d
d
()
d
d
(10)
Dierentiating, we nd:
d
d
() =
~
2
()
d
2
d
2
()
d
2
d
2
(11)
Taking (1) into account, we obtain:
d
d
() = 0 (12)
369

COMPLEMENT N III
Therefore, the current()associated with a stationary state is the same at all
points of the-axis. Note, moreover, that (12) is simply the one-dimensional
analogue of the relation:
divJ(r) = 0 (13)
which is valid, according to relation (D-11) of Chapter, for any stationary state
of a particle moving in three-dimensional space. According to (12), the current
()associated with()can therefore be calculated for any, choosing either
the form (6a) or the form (6b) of():
() =
~
2 2
=
~
~2~2
(14)
1-b. Properties of ()
It is easy to show, using the fact that the function()is real, that if()is a
solution of equation (1), so is(). Now consider the function(), which is a solution
of (1); comparison of (3a) and (4a) shows that it is identical to()when
2
. We
therefore have, for all:
() =() (15)
Substituting relations (3b) and (4b) into this relation, we obtain:
() =() (16)
() =() (17)
It follows that the matrix()can be written in the simplied form:
() =
()()
()()
(18)
We saw above [cf.(12)] that the probability current()does not depend on
for a stationary state. We must therefore have [cf.(14)]:
2 2
=
~2~2
(19)
for anyand. Now relations (9) and (18) yield:
~2~2
= [()+()] [()+()]
[()+()] [()+()]
= ()
2
()
2 2 2
(20)
Condition (19) is therefore equivalent to:
()
2
()
2
= Det() = 1 (21)
370

UNBOUND STATES OF A PARTICLE IN THE PRESENCE OF A POTENTIAL WELL OR BARRIER
Comments:
()We have made no particular assumptions about the shape of the potential. If
it is even, that is, if() =(), the matrix()possesses an additional
property: it can be shown that()is a pure imaginary.
()Relations (6) show thatand
~
are the coecients of incoming plane
waves, i.e. waves associated with particles arriving respectively from=
and= +and moving towards the zone of inuence of the potential (in-
cident particles). On the other hand,
~
and are the coecients corre-
sponding to outgoing waves, associated with particles moving away from
the potential (transmitted or reected particles). It is useful to introduce the
matrix, which allows us to calculate the amplitude of the outgoing waves
in terms of that of the incoming waves:
~
=()
~
(22)
()can easily be expressed in terms of the elements of the matrix(), as
we now show. The relations:
~
=()+() (23a)
~
=()+() (23b)
imply that:
=
1
()
~
() (24)
Substituting this relation into (23a), we obtain:
~
=
1
()
()()()()+()
~
(25)
Taking (21) into account, we can then write the matrix():
() =
1
()
1 ()
() 1
(26)
It is easy to verify, using (21) again, that:
()() =()() = 1 (27)
()is therefore unitary. This matrix plays an important role in collision
theory; we could have proved its unitary property from that of the evolution
operator (cf.ComplementIII), which simply expresses the conservation over
time of the total probability of nding the particle somewhere on theOxaxis
(norm of the wave function).
371

COMPLEMENT N III
2. Transmission and reection coecients
To calculate the reection and transmission coecients for a particle encountering the
potential(), one should (as in complementI) construct a wave packet with the
eigenfunctions ofwhich we have just studied. Consider, for example, an incident
particle of energycoming from the left. The corresponding wave packet is obtained
by superposing functions(), for which we set
~
= 0, with coecients given by a
function()which has a marked peak in the neighborhood of==
2 ~
2
.
We shall not go into these calculations in detail here; they are analogous in every way to
those of complementI. They show that the reection and transmission coecients are
equal, respectively, to()()
2
and
~
()()
2
.
Since
~
= 0, relations (22) and (26) yield:
~
() =
1
()
()
() =
()
()
() (28)
The reection and transmission coecients are therefore equal to:
1() =
()
()
2
=
()
()
2
(29a)
1() =
~
()
()
2
=
1
()
2
(29b)
[it is easy to verify that condition (21) insures that1() +1() = 1].
If we now consider a particle coming from the right, we must take= 0, which
gives:
~
() =
()
()
~
()
() =
1
()
~
() (30)
The transmission and reection coecients are now equal to:
2() =
()
~
()
2
=
1
()
2
(31a)
and:
2() =
~
()
~
()
2
=
()
()
2
(31b)
Comparison of (29) and (31) shows that1() =2()and that1() =2(): for
a given energy, the transparency of a barrier (whether symmetrical or not) is therefore
always the same for particles coming from the right and from the left.
In addition, from (21) we have:
()1 (32)
372

UNBOUND STATES OF A PARTICLE IN THE PRESENCE OF A POTENTIAL WELL OR BARRIERV(x)
l
2
–
l
2
+
0
x
Figure 2: Square potential barrier.
If (32) becomes an equality, the reection coecient is zero and the transmission coe-
cient is equal to 1 (resonance). On the other hand, the inverse situation is not possible:
since (21) imposes that() (), one can never have= 0and= 1[except in
the case whereandtend simultaneously towards innity]. Actually, such a situation
can only occur for= 0. To see this, divide the function()dened in (3) by(). If
()goes to innity, the wave function will be identically zero on the left hand side, and
hence necessarily, by extension, zero on the right hand side. However, this is impossible
unless= 0and= .
3. Example
Let us return to the square potentials studied in I: in the region
22
,()is equal
3
to a constant0(see Figure, where0has been chosen
to be positive).
First, let us assume thatis smaller than0, and set:
=
2~
2
(0 ) (33)
An elementary calculation analogous to the one in complementIyields:
() =
cosh+
2 2
2
sinhe
2
0
2
sinh
2
0
2
sinh cosh
2 2
2
sinhe
(34)
with:
0=
2 0~
2
(35)
(0is necessarily positive here, since we have assumed0).
3
In fact, we are considering here a barrier that is displaced relative to that of complementI, since
we are assuming it to be situated between= 2and= +2, instead of between= 0and=.
373

COMPLEMENT N III
If now we assume that 0, we set:
=
2~
2
( 0) (36)
and:
0=
2 0~
2
(37)
(where= +1if00and1if00). We thus obtain:
() =
cos+
2
+
2
2
sine
2
0
2
sin
2
0
2
sin cos
2 2
2
sine
(38)
It is easy to verify that the matrices()written in (34) and (38) satisfy relations (16),
(17) and (21).
References and suggestions for further reading:
Merzbacher (1.16), Chap. 6, 5, 6 and 8; see also the references of comple-
mentIII.
374

QUANTUM PROPERTIES OF A PARTICLE IN A ONE-DIMENSIONAL PERIODIC STRUCTURE
Complement OIII
Quantum properties of a particle in a one-dimensional periodic
structure
1 Passage through several successive identical potential bar-
riers
1-a Notation
1-b Matching conditions
1-c Iteration matrix(). . . . . . . . . . . . . . . . . . . . . .
1-d Eigenvalues of(). . . . . . . . . . . . . . . . . . . . . . .
2 Discussion: the concept of an allowed or forbidden energy
band
2-a Behavior of the wave function(). . . . . . . . . . . . . .
2-b Bragg reection; possible energies for a particle in a periodic
potential
3 Quantization of energy levels in a periodic potential; eect
of boundary conditions
3-a Conditions imposed on the wave function
3-b Allowed energy bands: stationary states of the particle inside
the lattice
3-c Forbidden bands: stationary states localized on the edges
In this complement, we are going to study the quantum properties of a particle
placed in a potential()having a periodic structure. The functions()which we
shall consider will not necessarily be periodic in the strict sense of the term; it suces
for them to have the shape of a periodic function in a nite region of the-axis (Fig.),
that is, to be the result of juxtaposingtimes the same motif at regular intervals [()
is truly periodic only in the limit].
Such periodic structures are encountered, for example, in the study of a linear
molecule formed byatoms (or groups of atoms) which are identical and equally spaced.
They are also encountered in solid state physics, when one chooses a one-dimensional
model in order to understand the disposition of the energy levels of an electron in a
crystal. Ifis very large (as in the case of a linear macromolecule or a macroscopic
crystal), the potential()is given in a wide region of space by a periodic function, and
the properties of the particle can be expected to be practically the same as they would be
if()were really periodic. However, from a physical point of view, the limit of innite
is never attained, and we shall be concerned here with the case whereis arbitrary.
To study the eect of the potential()on an eigenfunction()of the Hamilto-
nian, of eigenvalue, we shall introduce a22matrix, the iteration matrix, which
depends on. We shall show that the behavior of()is totally dierent depending
on whether the eigenvalues of the iteration matrix are real or imaginary. Since these
eigenvalues depend on the energychosen, we shall nd it useful to distinguish between
375

COMPLEMENT O III
domains of energy corresponding to real eigenvalues and those which lead to imaginary
eigenvalues. The concept of anallowed or forbidden energy band will thus be introduced.
Comments:
()For the sake of convenience, we shall speak of a potential barrier to des-
ignate the motif which, repeatedtimes, gives the potential()(Fig.).
However, this motif can also be a potential well or have an arbitrary shape.
()Common usage in solid state physics reserves the letterto designate a
parameter that appears in the expression of the stationary wave functions,
and which is not simply proportional to the square root of the energy. To
conform to this usage, we shall henceforth use a notation slightly dierent
from that of ComplementIII; we shall replaceby, setting:
=
2~
2
(1)
and we shall not introduce the letteruntil later (we shall see thatis
directly related to the eigenvalues of the matrixwhen they are complex).
1. Passage through several successive identical potential barriers
Consider a potential()which is obtained by juxtaposingbarriers as in Figure:
the rst barrier is centered at= 0, the second, at=, the third, at= 2, ..., the
last at= ( 1). We intend to study the behavior, during passage through this set
of barriers, of an eigenfunction()which is a solution of the eigenvalue equation of
:
d
2
d
2
+
2
~
2
[ ()]() = 0 (2)
whereandare related by (1).l
2
0
–
l
2
3l
2
5l
2
7l
2
x
V(x)
Figure 1: Potential()having a periodic structure obtained by juxtaposingtimes the
same motif (= 4in the gure).
376

QUANTUM PROPERTIES OF A PARTICLE IN A ONE-DIMENSIONAL PERIODIC STRUCTURE
1-a. Notation
To the left of thebarriers, that is, for
2
,()is zero, and the general
solution of equation (2) is:
if
2
:() =0e+
0e (3a)
Consider, as in III, the two functions()and()which
here become()and(). In the region of the rst barrier, centered at= 0, the
general solution of (2) is written:
if
22
:() =1() +
1() (3b)
Similarly, in the region of the second barrier, centered at=, we obtain:
if
2
3
2
:() =2( ) +
2( ) (3c)
and, more generally, in the region of theth barrier, centered at= (1):
if ( 1)
2
(1)+
2
:
() = [(1)] + [(1)] (3d)
Finally, to the right of thebarriers, that is, for(1)+
2
,()is again
zero, and we have:
if ( 1)+
2
:() =0e
[(1)]
+
0e
[(1)]
(3e)
We must now match these various expressions for the wave function()at
=
2
+
2
( 1)+
2
. This is what we shall do in the following section.
1-b. Matching conditions
The functionsanddepend on the form of the potential chosen. We shall
show, however, that it is simple to calculate them, and their derivatives as well, at the
two edges of each barrier, by using the results of ComplementIII.
To do so, let us imagine that all but one of the barriers are removed, leaving, for
example, theth one, centered at= (1). Solution (3d), always valid inside this
barrier, must then be extended to the left and to the right by superposing plane waves.
These waves are obtained by replacing, in formulas (6a) and (6b) ofIII,by(1)
andby, and adding an indexto,,
~
,
~
. Thus we have, if theth barrier is
isolated:
for(1)
2
:
e
[(1)]
+e
[(1)]
(4)
377

COMPLEMENT O III
for(1)+
2
:
~
e
[(1)]
+
~
e
[(1)]
(5)
with:
~
~
=() (6)
where, with the change in notation taken into account,()is the matrix()intro-
duced in ComplementIII. Consequently, at the left edge of theth barrier, the function
()dened in (3d) has the same value and the same derivative as the superposition of
plane waves (4). Similarly, at the right edge of this barrier, it has the same value and the
same derivative as (5). These results enable us to write simply the matching conditions
in the periodic structure.
At the left edge of the rst barrier (that is, at= 2), it is sucient to note
that (3a) has the same value and the same derivative as1e+
1e, which yields
directly:
0=1
0=
1
(7)
(a result which was obvious fromIII).
At the right edge of the rst barrier, which is the same as the left edge of the
second one, we must write that
~
1e+
~
1e and2e
()
+
2e
()
have
the same value and the same derivative, which yields:
2=
~
1e
2=
~
1e
(8)
Similarly, at the junction of theth and(+ 1)th barriers=
2
, we obtain,
setting equal the value and derivative of (5) and those of the expression obtained by
replacingby+ 1in (4):
+1=
~
e
+1=
~
e
(9)
Finally, at the right edge of the last barrier= ( 1)+
2
, we must write
that (3e) has the same value and the same derivative as the expression obtained by
replacingbyin (5), which yields:
0=
~
0=
~
(10)
378

QUANTUM PROPERTIES OF A PARTICLE IN A ONE-DIMENSIONAL PERIODIC STRUCTURE
1-c. Iteration matrix ()
Let us introduce the matrix()dened by:
() =
e0
0 e
(11)
It enables us to write the matching condition (9) in the form:
+1
+1
=()
~
~
(12)
that is, taking (6) into account:
+1
+1
=()() (13)
Iterating this equation and using (7), we obtain:
+1
+1
= [()()]
1
1
= [()()]
0
0
(14)
Finally, the matching condition (10) can be transformed by using (6) and (14):
0
0
=() =() [()()]
1 0
0
(15)
that is:
0
0
=()()()() ()()
matrices()
0
0
(16)
In this formula, which enables us to go from
0
0
to
0
0
, a matrix()is associated
with each barrier, and another matrix()with each interval between two successive
barriers.
Relations (13) and (14) demonstrate the importance of the role played by the
matrix:
() =()() (17)
which enters to theth power when one goes from
1
1
to
+1
+1
, that is, when
one performs a translation through a distancealong the periodic structure. For this
379

COMPLEMENT O III
reason, we shall call()the iteration matrix. Using formula (18) of ComplementIII
and expression (11) for(), we obtain:
() =
e() e ()
e () e ()
(18)
The calculation of[()]is facilitated if we change bases so as to make()
diagonal; for this reason we shall study the eigenvalues of().
1-d. Eigenvalues of ()
Letbe an eigenvalue of(). The characteristic equation of the matrix (18) is
written:
e() e () ()
2
= 0 (19)
that is, taking into account relation (21) of ComplementIII:
2
2() + 1 = 0 (20)
where()is the real part of the complex numbere():
() = Ree()=
1
2
Tr() (21)
Recall [cf.ComplementIII, relation (21)] that the modulus of()is greater than 1;
the same is therefore true ofe().
The discriminant of the second-degree equation (20) is:
= [()]
2
1 (22)
Two cases may then arise:
()If the energyis such that:
()1 (23)
(for example, if, in Figure,is between0and1), one can set:
() = cos[()] (24)
with:
0()
(25)
A simple calculation then shows that the eigenvalues of()are given by:
= e
()
(26)
There are therefore two eigenvalues, which are complex conjugates and whose modulus
is equal to 1.
380

QUANTUM PROPERTIES OF A PARTICLE IN A ONE-DIMENSIONAL PERIODIC STRUCTUREX(α)
Y(α)
α
2
α
3
α
4
+ 1– 1
α
1
α
0
O
Figure 2: Variation with respect toof the complex numbere() =() +().
Since()1, the curve obtained in the complex plane falls outside the circle of unit
radius centered at. The following discussion shows that if()is less than 1, that
is, if the value ofchosen gives a point of the curve which is between the two vertical
dashed lines of the gure, the corresponding energy falls in an allowed band; in the
opposite case, it falls in a forbidden band.
()If, on the other hand, the energygives a value ofsuch that:
()1 (27)
(for example, if, in Figure,is between1and2), one sets:
() =cosh[()] (28)
with:
()0 (29)
and= +1if()is positive,=1if()is negative. We then nd:
=e
()
(30)
In this case, both eigenvalues of()are real, and they are each other's inverse.
2. Discussion: the concept of an allowed or forbidden energy band
2-a. Behavior of the wave function ()
To apply (14), we begin by calculating the two column matrices1()and2()
associated with the eigenvectors of()and corresponding respectively to the eigenval-
381

COMPLEMENT O III
ues1and2. We then decompose the column matrix
1
1
into the form:
1
1
=1()1() +2()2() (31)
which enables us to obtain directly:
=
1
1 1()1() +
1
2 2()2() (32)
It is clear from this expression that the behavior of the wave function is very dierent
depending on whether()is smaller or greater than 1 in the energy domain of the
wave function. In the rst case, formula (26) shows that the eect of traversing successive
barriers is expressed in (32) by a phase shift in the components of the column matrix
onto1()and2(). The behavior of()here recalls that of a superposition
of imaginary exponentials. On the other hand, if the energy is such that()1,
formula (30) indicates that only one of the two eigenvalues (for example,1) has a
modulus greater than 1. Forsuciently large, we have, as a result:
1
e
(1)()
1()1() (33)
and therefore increase exponentially with[except in the special case where
1() = 0]; the wave function()then increases in modulus as it traverses the succes-
sive potential barriers, and its behavior recalls that of a superposition of real exponentials.
2-b. Bragg reection; possible energies for a particle in a periodic potential
Depending on whether()behaves like a superposition of real or imaginary
exponentials, the resulting phenomena can reasonably be expected to be dierent.
Let us evaluate, for example, the transmission coecient()of the set of
identical barriers. For thesebarriers, relation (15) shows that the matrix()[()]
1
plays a role analogous to the one played by()for a single barrier. Now, according to
relation (29b) of ComplementIII, the transmission coecient()is expressed in terms
of the element of this matrix which is placed in the rst row and the rst column [the
inverse of()is equal to the square of the modulus of this element]. What happens if
the energyof the particle is chosen so as to make the eigenvalues of()real, that is,
given by (30)? Whenbecomes suciently large, the eigenvalue1=e
()
becomes
dominant, and the matrix[()]
1
increases exponentially with[as can also be seen
from relation (33)]. Consequently, the transmission coecient decreases exponentially:
()e
2()
(34)
In this case, for large values of, the set ofpotential barriers reects the particle
practically without fail. This is explained by the fact that the waves scattered by the
dierent potential barriers interfere totally destructively for the transmitted wave, and
constructively for the reected wave. This phenomenon can therefore be likened toBragg
reection. Note, moreover, that this destructive interference for the transmitted wave
can be produced even if the energyis greater than the height of the barrier (a case
where, in classical mechanics, the particle is transmitted).
382

QUANTUM PROPERTIES OF A PARTICLE IN A ONE-DIMENSIONAL PERIODIC STRUCTURE
Nevertheless, if the transmission coecient of an isolated barrier is very close to
1, we have()1[for example, in Figure,() 1if, that is, the energy,
approaches innity]. The point representing the complex numbere()is then very
close to the circle of unit radius centered at. Figure
energy axis where()1, that is, where total reection occurs, are very narrow
and can practically be seen as isolated energy values. Physically, this is explained by
the fact that, if the energyof the incident particle is much larger than the amplitude
of variation of the potential(), its momentum is well-dened, as is the associated
wavelength. The Bragg condition=
2
(whereis an integer) then gives well-dened
energy values.
If, on the other hand, the energyof the particle falls in a domain where the
eigenvalues are of modulus 1 as in (26), the elements of the matrix[()]
1
no longer
approach innity whendoes. Under these conditions, the transmission coecient
()does not approach zero when the number of barriers is increased. We are again
dealing with a purely quantum mechanical phenomenon, related to the wave-like nature
of the wave function, which enables it to propagate in the regular periodic potential
structure without being exponentially attenuated. Note especially that the transmission
coecient()is very dierent from the product of the individual transmission coef-
cients of the barriers taken separately (this product approaches zero when
since all the factors are smaller than 1).
The quantization of energy levels for a particle placed in a series of identical and
evenly spaced potential wells (i.e. a periodic potential()) is another interesting prob-
lem, particularly in solid state physics. It will be studied in detail in , but we can
already guess the form of the spectrum of possible energies. If we assume that the energy
of the particle is such that()1, equation (33) shows that the coecientsand
become innite when . It is clear that this possibility must be rejected, since
it means that the wave function does not remain bounded. The corresponding energies
are therefore forbidden; hence, the name offorbidden bandsgiven to the energy domains
for which()1. On the other hand, if the energy of the particle is such that
()1,and remain bounded when ; the corresponding regions of
the energy axis are calledallowed bands. To sum up, the energy spectrum is composed
of nite intervals inside which all the energies are acceptable, separated by regions all of
whose energies are forbidden.
3. Quantization of energy levels in a periodic potential; eect of boundary
conditions
Consider a particle of massplaced in the potential()shown in Figure. In the region
2
+
2
,()has the form of a periodic function, composed of a series of+ 1
successive barriers of height0, centered at= 0,,2, ...,, Outside this region,()
undergoes arbitrary variations over distances comparable to, then becomes equal to a positive
constant value. In what follows, the region[0]will be called inside the lattice and the
limiting regions
2
and +
2
, ends (or edges) of the lattice.
Physically, such a function()can represent the potential seen by an electron in a linear
molecule or in a crystal (in a one-dimensional model). The potential wells at=
2
,
3
2
, ...
383

COMPLEMENT O IIIl0 Nl
x
V(x)
V
e
V
0
Figure 3: Variation with respect toof the potential seen by an electron in a one-
dimensional crystal and on its edges. Inside the crystal, the potential has a periodic
structure;()is maximum between the ions (barriers at= 0,,2, ...) and minimum
at the positions of the ions (wells at=2,32, ...). On the edges of the crystal,()
varies in a more or less complicated way over a distance comparable to, then rapidly
approaches a constant value.
then correspond to the attraction of the electron by the various ions. Far from the crystal (or
the molecule), the electron is not subjected to any attractive forces, which is why()rapidly
becomes constant outside the region
2
+
2
.
The potential()that we have chosen ts perfectly into the framework of Comple-
ment III(apart from a change in the energy origin). We already know, therefore, that the
bound states of the particle form a discrete spectrum of energies, all less thane. However, the
potential()picked here also presents the remarkable peculiarity of having a periodic struc-
ture of the type of those considered in
show that the conclusions of ComplementIIItake on a special form in this case. For example,
in ComplementIIIwe stressed the fact that it is the boundary conditions [() 0when
] that introduce the quantization of the energy levels. The boundary conditions of the
problem we are studying here, that is, the variation of the potential at the edges of the lattice,
might thus be expected to play a critical role in determining the possible energies. Actually, this
is not the case: we shall see that these energies depend practically only on the values of()
in the region where it is periodic, and not on the edge eects (on condition, of course, that the
number of potential wells is suciently large). In addition, we shall verify the result obtained
intuitively in , showing that most of the possible energies are grouped in allowed energy
bands. Only a few stationary states, localized near the edges, depend on a critical manner on
the variation of()in this region and can have an energy which falls in a forbidden band.
We shall therefore proceed essentially as in ComplementIII, rst examining precisely
the conditions imposed on the wave function()of a stationary state.
3-a. Conditions imposed on the wave function
In the region where()is periodic, relation (3d) gives the form of the wave function
(); the coecientsand are determined from (32). To write (32) more explicitly, let
384

QUANTUM PROPERTIES OF A PARTICLE IN A ONE-DIMENSIONAL PERIODIC STRUCTURE
us set:
1()1() =
1()
1
()
2()2() =
2()
2
()
(35)
We then obtain:
=1()
1
1+2()
1
2
=
1()
1
1+
2()
1
2 (36)
Now let us examine the boundary conditions on the wave function(). First of all, to
the left, far from the lattice,()is equal toand()is written in the form:
() =e
()
(37a)
with:
() =
2~
2
( ) (37b)
(we eliminate the solution ine
()
, which diverges when ). The probability current
associated with the function (37) is zero (cf.ComplementIII). Now, for a stationary state,
this current is independent of[cf.ComplementIII, relation ()]; it therefore remains zero
at all, even inside the lattice. According to relation (14) of ComplementIII, the coecients
and therefore necessarily have the same modulus. Thus, if we choose to express the
boundary conditions on the left as relations between the coecients1and
1[that is, by
writing that the expression for()for
22
is the extension of the wave function
(37)], we nd a relation of the form:
1
1
= e
()
(38a)
()is a real function of(and therefore of the energy) which depends on the precise
behavior of()at the left-hand edge of the lattice [in what follows, we shall not need the
exact expression for this function(); the essential point is that the boundary conditions on
the left have the form (38a)].
The same type of reasoning can obviously be applied on the right (+). The
boundary conditions are written:
+1
+1
= e
()
(38b)
where the real function()depends on the behavior of()on the right-hand edge of the
lattice.
To sum up, we can say that the quantization of the energy levels can be obtained in the
following manner:
we start with two coecients1and
1that satisfy (38a); this ensures that the function
()will remain bounded when . Since()is dened to within a constant factor,
we can choose, for example:
1= e
()2
1= e
()2
(39)
385

COMPLEMENT O III
we then calculate, using (36), the coecientsand so as to extend the wave
function chosen throughout all the crystal. Note that the condition (39) implies that()is
real (cf.ComplementIII, ); calculation ofand must therefore yield:
= (40)
nally, we write that the coecients+1and
+1satisfy (38b), a relation ensuring
that()will remain bounded when +. In fact, relation (40) shows that the ratio
+1 +1is automatically a complex number of unit modulus; condition (38b) therefore
amounts to an equality between the phases of two complex numbers. We thus obtain a real
equation in, which has a certain number of real solutions giving the allowed energies.
We are going to apply this method, distinguishing between two cases: real eigenvalues of
()[the case where()1] and imaginary ones [the case where()1].
3-b. Allowed energy bands: stationary states of the particle inside the lattice
First assume that the energyis in a domain where()1.
. Form of the quantization equation
Taking (26) into account, relations (36) become:
=1() e
(1)()
+2() e
(1)()
=
1() e
(1)()
+
2() e
(1)()
(41)
We also know that the choice (39) of1and
1implies that= for all. Now, it is easy
to show that relations (41) yield two complex conjugate numbers only if:
1() =
2()
2() =
1() (42)
Condition (38b) can then be written:
1() e
2 ()
+2()
2
() e
2 ()
+
1
()
= e
()
(43)
This equation ingives the quantization of the energy levels. To solve it, let us set:
() = Arg
1() e
()2
2() e
()2
1() e
()2
2
() e
()2
(44)
[()can, in principle, be calculated from(),()and the matrix()]. Equation (43)
can then be written simply:
e
2 ()
= e
()
(45)
The energy levels are therefore given by:
() =
()
2
+ (46)
with:
= 012( 1) (47)
[the other values ofmust be excluded, since condition (25) forces()to vary within an
interval of width]. We can already see that ifis very large, we can write equation (46) in
the simplied form:
()
(48)
386

QUANTUM PROPERTIES OF A PARTICLE IN A ONE-DIMENSIONAL PERIODIC STRUCTURE
. Graphical solution; locating the energy levels
If we substitute denition (24) of()into (46), we obtain an equation inthat gives
the allowed energies. To solve it graphically, let us begin by tracing the curve that represents
the function() = Re[e()]. Because of the imaginary exponentiale, we expect this
curve to have an oscillatory behavior, of the type shown in Figurea. Since()is greater
than 1 [cf.ComplementIII, relation (32)], the amplitude of the oscillation is greater than 1,
and the curve intersects the two straight lines() =1at certain values0,1,2, ... of
the variable. We then eliminate all regions of the-axis, bounded by these values, where the
condition()1is not satised. Using the set of arcs of curves thus obtained for(), we
must represent the function:
() =
1
Arc cos() (49)
Taking into account the form of the Arc cosine function (cf.Fig.), we are led to the curve
whose shape is shown in Figureb. Equation (46) indicates that the energy levels correspond to
the intersections of this curve with the curves representing the functions
()
2
+, that is,
if 1, with the horizontal lines whose equations are=
(with= 0, 1, 2, ...,1).X(α)
k(α)
0
+ 1
– 1
α
1
α
2
α
3
αα
0
α
1
α
2
α
3
α
4
αα
0
π
l
0
b
a
Figure 4: Variation with respect toof() = Re[() e](see Fig.) and of() =
1
Arc cos(). The values of(that is, of the energy) associated with stationary
states are obtained (if1) by cutting the curve which represents()with the
horizontal lines whose equations are= (= 012 1). The allowed
bands are thus revealed (intervals0 1, etc.); each includesvery close levels.
The forbidden bands are represented by the shaded areas (1 2, etc...).
The dashed-line curves correspond to the special case where() = 0(a free particle).
387

COMPLEMENT O IIIπ
Arc cos z
– 1 + 1
z
0
Figure 5: The Arc cosine function.
We thus obtain groups oflevels, associated with equidistant values of()and situated
in the allowed bands dened by0 1 2 3, etc. Between these allowed bands
are the forbidden bands (we shall examine their properties in c).
If we consider a particular allowed band, we can locate each level according to the value
of()which corresponds to it. This leads to choosingas the variable and consideringand,
consequently,as functions()and()of. The variation ofwith respect tois given
directly by the curve of Figureb, so it suces to evaluate the function
~
22
2
to obtain the
energy(). The corresponding curve has the shape shown in Figure.
Comment:
It is clear from Figure-b that, to a given value of, correspond several values of
and therefore of the energy; this is why several arcs appear in Figure. Nevertheless,0
π/l k
E(k)
Figure 6: Variation of the energy with respect
to the parameter. The solid lines correspond
to the energies for the rst two allowed bands
(the values ofwhich give the energy levels be-
ing equidistant inside the interval0 ).
The dashed lines correspond to the special case
where the potential()is zero (a free parti-
cle); the allowed bands are then contiguous, and
there are no forbidden bands.
388

QUANTUM PROPERTIES OF A PARTICLE IN A ONE-DIMENSIONAL PERIODIC STRUCTURE
if, within a given allowed band,()increases steadily from1to+1(or decreases
steadily from+1to1), only one energy level corresponds to each value offor this
band, andthis band includesenergy levels.
. Discussion
The preceding calculations show how, when we go from= 1to very high values of
, we move gradually from a set of discrete levels to allowed energy bands. Rigorously, these
bands are formed by discrete levels, but their separation is so small for a macroscopic lattice
that they practically constitute a continuum. Whenis taken as a parameter, thedensity of
states(the number of possible energies per unit interval of)is constantand equal to .
This property, which is very useful, explains whyis generally chosen as the variable.
An important point appears in going from (46) to (48): whenis large, the edge eects
of the lattice, which enter only through the intermediary of the functions(),()and, in
(46),(), no longer play any role; only the form of the periodic potential inside the lattice is
important in determining the possible energies.
It is interesting to consider the two following limiting cases:
()If() = 0(free particle), we have:
() = 1
() = cos
(50)
and we obtain:
if 0
: () =
if
2
: () =
2
etc (51)
(the corresponding broken line is shown in Figureb as a dashed line). Relation (50) shows
that the condition()1is always satised: as we know, forbidden bands do not exist for
a free particle.
Figure ()on the curve(). When
forbidden bands appear, the curves representing the energy become deformed so as to have
horizontal tangents for= 0and= (edges of the band). Unlike what happens for a free
particle, there exists a point of inection for each band where the energy varies linearly with.
()If the transmission coecient()is practically zero, we have [cf.ComplementIII,
equations (29) and (21)]:
()1
()1
(52)
In Figure, the point representing the complex numbere()is very far from the origin. We
thus see in this gure that the regions of the-axis where()1are extremely narrow. The
allowed bands therefore shrink if the transmission coecient of the elementary barrier decreases;
in the limit of zero transmission, they reduce to individual levels in an isolated well. Inversely,
as soon as the tunnel eect allows the particle to pass from one well to the next one,each of the
discrete levels of the well gives rise to an energy band, whose width increases as the transmission
coecient grows. We shall return to this property in ComplementXI.
389

COMPLEMENT O III
3-c. Forbidden bands: stationary states localized on the edges
. Form of the equations; energy levels
Let us now assume thatbelongs to a domain where()1. According to (30),
relations (36) can then be written:
=
1
1() e
(1)()
+2() e
(1)()
=
1
1() e
(1)()
+
2() e
(1)()
(53)
The fact that= for allmeans that we must have:
1() =
1()
2() =
2()
(54)
The quantization condition (38b) then takes on the form:
+1
+1
=
1() +2() e
2()
1
() +
2
() e
2()
e
()
(55)
that is:
e
2()
=() (56)
where the real function()is dened by:
() =
1() e
()2
1() e
()2
2
() e
()2
2() e
()2
(57)
Consider the case where 1; we then havee
2()
0, and equation (56) reduces
to:
() = 0 (58)
The energy levels situated in the forbidden bands are therefore given by the zeros of the function
()(cf.Fig.).enters neither into (57) nor into (58), so the number of these levels does
not depend on(unlike the number of levels situated in an allowed band). Consequently, when
1, it can be said that practically all the levels are grouped in the allowed bands.
. Discussion
The situation here is radically dierent from the one encountered in b: the number
, that is, the length of the lattice, plays no role (provided, nevertheless, that it is suciently
large); on the other hand, denition (57) of()shows that the functions()and()
play an essential role in the problem. Since we already know that these functions depend on
the behavior of()on the edges of the lattice, we expect to obtain states localized in these
regions.
This is indeed the case. Equations () and (58) oer two possibilities:
()if1()= 0, the fact that() = 0requires that:
1()
1
()
=
1()
1
()
= e
()
(59)
390

QUANTUM PROPERTIES OF A PARTICLE IN A ONE-DIMENSIONAL PERIODIC STRUCTURE0
α
1
L(α)
α
2
α
Figure 7: Variation of()with respect to
in a forbidden band. The zeros of()
give the stationary states which are localized
on the edges of the lattice.
Let us return to denition (35) of1()and
1(); we see that relation (59) shows that the
wave function constructed from the rst eigenvector of()satises the boundary conditions
on the right. This is easy to understand: if we start at= 0with an arbitrary wave function
which satises the boundary conditions on the left, the matrix
1
1
has components on the
two eigenvectors of(); the coecients+1and
+1are then (if1) essentially given
by (33), which expresses the fact that the column matrix
+1
+1
is proportional to the column
matrix of the rst eigenvector of().
Note that since the eigenvalue1()is greater than 1, the wave function grows exponen-
tially whenincreases. The stationary state given by the rst eigenvector of()is therefore
localized at the right end of the lattice.
()if1() = 0, (54) gives
1() = 0, and denitions (35) imply that1() = 0: the
corresponding stationary state is associated with the second eigenvector of(). Aside from
the fact that this state is localized at the left end of the lattice, the conclusions obtained in()
remain valid.
References and suggestions for further reading:
Merzbacher (1.16), Chap. 6, 7; Flügge (1.24), 28 and 29; Landau and Lifshitz
(1.19), 104; see also solid state physics texts (section 13 of the bibliography).
391

Chapter IV
Application of the postulates
to simple cases: spin 1/2
and two-level systems
A Spin 1/2 particle: quantization of the angular momentum
A-1 Experimental demonstration
A-2 Theoretical description
B Illustration of the postulates in the case of a spin 1/2
B-1 Actual preparation of the various spin states
B-2 Spin measurements
B-3 Evolution of a spin 1/2 particle in a uniform magnetic eld
C General study of two-level systems
C-1 Outline of the problem
C-2 Static aspect: eect of coupling on the stationary states of the
system
C-3 Dynamical aspect: oscillation of the system between the two
unperturbed states
In this chapter, we intend to illustrate the postulates of quantum mechanics, which
we stated and discussed in Chapter. We shall apply them to simple concrete cases,
in which the dimension of the state space is nite (equal to two). The interest of these
examples is not conned to their mathematical simplicity, which will allow a better
understanding of the postulates and their consequences. It is also based on their physical
importance: they exhibit typically quantum mechanical behavior which can be veried
experimentally.
In , we study the spin 1/2 case (which we shall take up again in more
detail in Chapter). First, we describe () a fundamental experiment that revealed
Quantum Mechanics, Volume I, Second Edition. C. Cohen-Tannoudji, B. Diu, and F. Laloë.
© 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

CHAPTER IV SIMPLE CASES: SPIN 1/2 AND TWO-LEVEL SYSTEMS
the quantization of a simple physical quantity, the angular momentum. We shall see that
the component alongof the angular momentum (or magnetic moment) of a neutral
paramagnetic atom can take on only certain values, which belong to adiscreteset. Thus,
for a silver atom in its ground state, there are only two possible values (+~2and~2)
for the componentof its angular momentum: a silver atom in the ground state is
said to be a spin 1/2particle. In , we indicate how quantum mechanics describes
the spin variables of such a particle. In situations where one can dispense with a
quantum treatment of the external variablesrandp, the state of the particle (spin
state space) has only two dimensions. We shall then () be able to illustrate and
discuss the quantum mechanical postulates in this particularly simple case: we shall rst
see how to prepare silver atoms in any desired arbitrary spin state, in a real experiment.
We shall then show how the measurement of the physical values of the spin on such
silver atoms enables us to verify the quantum mechanical postulates experimentally. By
integrating the corresponding Schrödinger equation, we shall study the evolution of a spin
1/2 particle in a uniform magnetic eld (Larmor precession). Finally, in , we shall
begin the study oftwo-level systems. Although these systems are not generally spin 1/2
particles, their study leads to calculations very similar to those developed in .
We shall treat in detail the eect of an external perturbation on the stationary states of a
two-level system and use this very simple model to point out important physical eects.
A. Spin 1/2 particle: quantization of the angular momentum
A-1. Experimental demonstration
First of all, we are going to describe and analyze the Stern-Gerlach experiment,
which demonstrated the quantization of the components of an angular momentum (some-
times called space quantization).
A-1-a. The Stern-Gerlach apparatus
The experiment consists of studying the deection of a beam of neutral paramag-
netic atoms (in this case, silver atoms) in a highly inhomogeneous magnetic eld. The
apparatus used is shown schematically
1
in Figure.
Silver atoms contained in a furnace, heated to a high temperature, leave through
a small opening and propagate in a straight line in the high vacuum existing inside the
whole apparatus. A collimating slitselects those atoms whose velocity is parallel
to a particular direction that we shall choose for theaxis. The atomic beam thus
constructed traverses the gap of an electromagnetbefore condensing on a plate.
Let us describe the characteristics of the magnetic eldBproduced by the elec-
tromagnet. This magnetic eld has a plane of symmetry (which we shall designate by
) that contains the initial directionof the atomic beam. In the air-gap, it is the
same at all points situated on any given line parallel to(the edges of the electromag-
net are parallel to, and we neglect edge eects).Bhas no component along. Its
largest component is along; it varies strongly with: in Figure-b, the eld lines are
much closer together close to the north pole than close to the south pole of the magnet.
1
We only indicate the most important characteristics of this equipment. A more detailed description
of the experimental technique can be found in a textbook on atomic physics.
394

A. SPIN 1/2 PARTICLE: QUANTIZATION OF THE ANGULAR MOMENTUME
South
North
F
A
a
b
O
P
N
H
y
z
z
x
Figure 1: Schematic diagram of the Stern-Gerlach experiment. Figure a shows the trajec-
tory of a silver atom emitted from the high-temperature furnace. This atom is deected
by the gradient of the magnetic eld created by the electromagnetand then condenses
aton plate.
Figure b shows a cross section in theplane of the electromagnet; the lines of
force of the magnetic eld are shown in dashed lines.has been assumed to be positive
and , negative. Consequently, the trajectory of gure a corresponds to a negative
componentMof the magnetic moment, that is, to a positive component ofS(is
negative for a silver atom).
Of course, since the magnetic eld has a conserved ux (divB= 0), it must also have a
component alongwhich varies with the distancefrom the plane of symmetry.
A-1-b. Classical calculation of the deection
2
Note, rst, that the silver atoms, being neutral, are not subjected to the Lorentz
force. On the other hand, they possess a permanent magnetic momentMMM(they are
paramagnetic atoms); the resulting forces are derived from the potential energy:
=MMMB (A-1)
The existence, for an atom, of an electronic magnetic momentMMMand an angular
momentumSSSis due to two causes: the motion of the electrons about the nucleus (the
corresponding rotation of the charges being responsible for the appearance of an orbital
magnetic moment) and the intrinsic angular momentum, or spin, (cf.Chapter) of the
electrons, with which is associated a spin magnetic moment. It can be shown (as we shall
2
We only give here an outline of the calculation.
395

CHAPTER IV SIMPLE CASES: SPIN 1/2 AND TWO-LEVEL SYSTEMSB
z
ℳ
O
θ
Figure 2: The silver atom possesses a mag-
netic momentMMMand an angular momentum
Swhich are proportional. Consequently, the
eect of a uniform maguetic eldBis to
causeMto turn aboutBwith a constant
angular velocity (Larmor precession).
assume here without proof) that, for a given atomic level,MMMandSSSare proportional
3
:
MMM=SSS (A-2)
The proportionality constantis called thegyromagnetic ratioof the level under con-
sideration.
Before the atoms traverse the electromagnet, the magnetic moments of the silver
atoms that form the atomic beam are oriented randomly (isotropically). Let us study
the action of the magnetic eld on one of these atoms, whose magnetic momentMMMhas
a given direction at the entrance of the air-gap. From expression (A-1) for the potential
energy, it is easy to deduce that the resultant of the forces exerted on the atom is:
F=r(MMMB) (A-3)
(this resultant would be equal to zero if the eldBwere uniform), and that their total
moment relative to the position of the atom is:
=MMMB (A-4)
The angular momentum theorem can be written:
dSSS
d
= (A-5)
3
In the case of silver atoms in the ground state (like those of the beam), the angular momentumS
is simply equal to the spin of the outer electron, which is therefore solely responsible for the existence
of the magnetic momentM. This is because the outer electron has a zero orbital angular momentum,
and the resultant orbital and spin angular momenta of the inner electrons are also zero. Moreover the
experimental conditions realized in practice are such that eects linked to the spin of the nucleus are
negligible. This is why the silver atom in the ground state, like the electron, has a spin 1/2.
396

A. SPIN 1/2 PARTICLE: QUANTIZATION OF THE ANGULAR MOMENTUM
that is:
dSSS
d
=SSSB (A-6)
The atom thus behaves like a gyroscope (Fig.):dSSSdis perpendicular toSSS,
and the angular momentum turns about the magnetic eld, the anglebetweenSSS
andBremaining constant. The rotational angular velocity is equal to the product of
the gyromagnetic ratioand the modulus of the magnetic eld. The components of
MMMwhich are perpendicular to the magnetic eld therefore oscillate around zero, the
component parallel toBremaining constant.
To calculate the forceF[formula (A-3)], we can, to a very good approximation,
neglect in the potential energythe terms proportional toMandMand takeM
to be constant. This is because the frequency of oscillation due to the rotation ofMMMis
so great that only the time-averaged values ofMandMcan play a role in, and
these are both zero. Consequently, it is as if the atom were submitted to the sole force:
F=r(M ) =Mr (A-7)
In addition, the components ofr alongand are zero: = 0because
the magnetic eld is independent of( = 0at all points
of the plane of symmetry.The force on the atom is therefore parallel to Oz and
proportional toM. Since it is this force that produces the deectionof the atom
(Fig.), is proportional toM(and hence, toS). Consequently, measuring
is equivalent to measuringMorS.
Since, at the entrance to the air-gap, the moments of the various atoms of the
beam are distributed isotropically (all values ofMincluded betweenMMMandMMMare
found), we expect the beam to form a single pattern, symmetrical with respect to, on
the plate. The upper bound1and the lower bound2of this pattern correspond
in principle to the maximum valueMMMand minimum valueMMMofM. In fact, the
dispersion of the velocities and the nite width of the slitcause the atoms having a
given value ofMto condense, not at the same point, but in a spot centered about the
deection corresponding to the average velocity.
A-1-c. Results and conclusions
The results of the experiment (performed for the rst time in1922by Stern and
Gerlach) are in complete contradiction with the preceding predictions.
We do not observe a single spot centered at, buttwo spots(Fig.) centered at
the points1and2, symmetrical with respect to(the width of these two spots is
due to the dispersion of the velocities, and to the width of the slit). The predictions
of classical mechanics are therefore shown to be invalidated by the experiment.
Now let us see how these experimental results can be interpreted. Of the physical
quantities associated with a silver atom, some correspond to its external degrees of free-
dom (that is, are functions of its positionrand its linear momentump), and others, to
its internal degrees of freedom (also called spin degrees of freedom)MMMorSSS.
Let us rst show that, under these experimental conditions, it is not necessary to
treat the external degrees of freedom quantum mechanically. To do this, we shall verify
that it is possible, in order to describe the motion of the silver atoms, to construct wave
397

CHAPTER IV SIMPLE CASES: SPIN 1/2 AND TWO-LEVEL SYSTEMSP
N
1
N
2
H
Figure 3: Spots observed on the platein the
Stern-Gerlach experiment. The magnetic moments
MMMof the atoms emitted from the furnaceare dis-
tributed randomly in all directions of space, so classi-
cal mechanics predicts that a measurement ofMcan
yield with equal probability all values included between
+MMMandMMM. One should therefore observe only
one large spot centered in(dashed lines in the g-
ure). In reality, the result of the experiment is com-
pletely dierent: two spots, centered at1and2,
are observed. This means that a measurement ofM
can yield only two possible results (quantization of the
measurement result).
packets whose widthand momentum dispersionare negligible.and
must satisfy the Heisenberg relation:
&~ (A-8)
Numerically, the massof a silver atom is equal to1810
25
kg.and the velocity
uncertainty= must be such that:
&
~
10
9
MKSA (A-9)
Now what are the lengths and velocities involved in the problem? The width of the slit
is about 0.1 mm and the separation12of the two spots, that is several millimeters.
The distance over which the magnetic eld varies appreciably can be deduced from
the values of the eld in the middle of the air-gap (10
4
gauss) and its gradient
(
10
5
gauss/cm), which yields1mm. In addition, the velocity of the
silver atoms leaving a furnace at an absolute temperature of1000K is of the order of
500 m/s. However well-dened the beam is, the dispersion of the velocities along
is not much less than several meters per second. It is then easy to nd uncertainties
and, which, while satisfying (A-9), are negligible on the scale of the experiment
being considered. As far as the external variablesrandpof each atom are concerned,
it is therefore not necessary to resort to quantum mechanics. It is possible to reason in
terms of quasi-pointlike wave packets moving along classical trajectories. Consequently,
it is correct to claim that measurement of the deectionconstitutes a measurement
ofMorS.
The results of the experiment thus lead us necessarily to the following conclusion:
if we measure the componentSof the intrinsic angular momentum of a silver atom in
its ground state, we can nd only one or the other of two values corresponding to the
deections 1and 2. We are therefore obliged to reject the classical image of a
vectorSSSwhose anglewith the magnetic eld can take on any value:Sis a quantized
physical quantity whose discrete spectrum includes only two eigenvalues. When we study
the quantum theory of angular momentum (Chap.), we shall see that these eigenvalues
are+~2and~2; we shall assume this here and say that the spin of the silver atom
in its ground state is 1/2.
398

A. SPIN 1/2 PARTICLE: QUANTIZATION OF THE ANGULAR MOMENTUM
A-2. Theoretical description
We are now going to show how quantum mechanics describes the degrees of freedom
of a silver atom, that is, of a spin 1/2 particle.
We do not yet possess all the necessary elements for the presentation of a deductive
and rigorous theory of the spin 1/2 particle. Such a study will be developed in Chapter,
in the framework of the general theory of angular momentum. We shall therefore be
forced here to assume without proof a small number of results which will be proved later,
in Chapter. Such a point of view is justied by the fact that the essential goal of the
present chapter is to show the reader how to handle the quantum mechanical formalism
in a simple and concrete case, and not to focus on the angular momentum aspect of
the spin 1/2. The idea is to give precise examples of kets and observables, to show how
physical predictions can be extracted from them and how to distinguish clearly between
the various stages of an experiment (preparation, evolution, measurement).
We saw in Chapter
associated, in quantum mechanics, an observable, that is, a Hermitian operator whose
eigenvectors can form a basis in the state space. We must therefore dene the state
space and the observables corresponding to the components ofSSS(S,S,Sand,
more generally,Su=Su, whereuis an arbitrary unit vector), which we know from

A-2-a. The observable and the spin state space
WithSwe must associate an observablewhich has, according to the results
of the experiment described in +~2and~2. We shall
assume (see Chap.) that these two eigenvalues are not degenerate, and we shall denote
by+and the corresponding orthonormal eigenvectors:
+= +
~
2
+
=
~
2
(A-10)
with:
++= = 1
+ = 0
(A-11)
alone therefore forms a C.S.C.O., and the spin state space is the two-dimensional
spacespanned by its eigenvectors+and. The fact that these eigenvectors form
a basis ofis expressed by the closure relation:
+++ = (A-12)
The most general (normalized) vector ofis a linear superposition of+and
:
=++ (A-13)
with:
2
+
2
= 1 (A-14)
399

CHAPTER IV SIMPLE CASES: SPIN 1/2 AND TWO-LEVEL SYSTEMSx
y
u
O
φ
θ
z
Figure 4: Denition of the polar anglesand
characterizing a unit vectoru.
In the+ basis, the matrix representingis diagonal and is written:
() =
~
2
1 0
0 1
(A-15)
A-2-b. The other spin observables
With theSandScomponents ofSwill be associated the observablesand
. The operators and must be represented in the+ basis by22
Hermitian matrices.
We shall see in Chapter
an angular momentum do not commute with each other but satisfy well-dened commu-
tation relations. This will enable us to show that, in the case of a spin 1/2, with which we
are concerned here, the matrices representingandin the basis of the eigenvectors
+and ofare the following:
() =
~
2
0 1
1 0
(A-16)
() =
~
2
0
0
(A-17)
For the moment, we shall assume this result.
As for theSucomponent ofSSSalong the unit vectoru, characterized by the polar
anglesand(Fig.), it is written:
Su=SSSu=Ssincos+Ssinsin+Scos (A-18)
400

B. ILLUSTRATION OF THE POSTULATES IN THE CASE OF A SPIN 1/2
Using (A-15), (A-16) and (A-17), we easily nd the matrix that represents the
corresponding observableu=S.uin the {+ } basis:
() = () sincos+ () sinsin+ () cos
=
~
2
cossine
i
sine
i
cos
(A-19)
In what follows, we shall need to know the eigenvalues and eigenvectors of the
observables and. The calculations that enable us to obtain them from the
matrices (A-16), (A-17) and (A-19) are not dicult. We shall only present the results
here.
The and operators have the same eigenvalues,+~2and~2, as.
This result could have been expected, since it is always possible to rotate the Stern-
Gerlach apparatus as a whole so as to make the axis dened by the magnetic eld parallel
either to, to, or tou. Since all directions of space have the same properties,
the phenomena observed on the plate of the apparatus must be unchanged under such
rotations: the measurement ofS,SorScan therefore yield only one of two results:
+~2or~2.
As for the eigenvectors of,and, we shall denote them respectively by
, and (the sign in the ket is that of the corresponding eigenvalue). Their
expansions on the basis of eigenvectorsofis written:
=
1
2
[+ ] (A-20)
=
1
2
[+i] (A-21)
+= cos
2
e
2
++ sin
2
e
2
=sin
2
e
2
++ cos
2
e
2
(A-22a)
(A-22b)
B. Illustration of the postulates in the case of a spin 1/2
Using the formalism that we have just described, we are now going to apply the postulates
of quantum mechanics to a certain number of experiments on silver atoms which can
actually be performed with the Stern-Gerlach apparatus. We shall thus be able to discuss
the consequences of these postulates in a concrete case.
B-1. Actual preparation of the various spin states
In order to make predictions about the result of a measurement, we must know the
state of the system (here, the spin of a silver atom) immediately before the measurement.
We are going to see how to prepare a beam of silver atoms so that they are all in a given
spin state.
401

CHAPTER IV SIMPLE CASES: SPIN 1/2 AND TWO-LEVEL SYSTEMSO
z
E
F
N
1
N
2
P
A
B
y
x
Figure 5: When we pierce a hole in the plateat the position of the spot1, the atoms
that pass through this hole are all in the spin state+. The Stern-Gerlach apparatus is
then acting like a polarizer.
B-1-a. Preparation of the states+and
Let us assume that we pierce a hole in the plateof the apparatus represented
in Figure-a, at the position of the spot centered at1(Fig.). The atoms which
are deected downward continue to condense about2, while some of those which are
deected upward pass through the plate(Fig.). Each of the atoms of the beam
which propagates to the right of the plate is a physical system on which we have just
performed a measurement of the observable, the result being+~2. According to the
fth postulate of Chapter, this atom is in the eigenstate corresponding to this result,
that is, in the state+(sincealone constitutes a C.S.C.O., the measurement result
suces to determine the state of the system after this measurement). The device in
Figure +. This device
acts like an atomic polarizer, since it acts the same way on atoms as an ordinary
polarizer does on photons.
Of course, if we pierced the plate around2and not around1, we would obtain
a beam all of whose atoms would be in the spin state.
B-1-b. Preparation of the states , ,
The observablealso constitutes a C.S.C.O. since none of its eigenvalues is
degenerate. To prepare one of its eigenstates, we must simply select, after a measurement
of, the atoms for which this measurement has yielded the corresponding eigenvalue.
In practice, if we rotate the apparatus of Figure +2about,
we obtain a beam of atoms whose spin state is+(Fig.).
This method can be generalized: by placing the Stern-Gerlach apparatus so that
the axis of the magnetic eld is parallel to an arbitrary unit vectoru, and piercing the
plate either at1or at2, we can prepare silver atoms in the spin state+or
4
.
B-1-c. Preparation of the most general state
We indicated above that the most general (normalized) ket of the spin state space
is of the form:
=++ (B-1)
4
The direction of the atomic beam is no longer necessarily, but this is not important in what
concerns us here.
402

B. ILLUSTRATION OF THE POSTULATES IN THE CASE OF A SPIN 1/2O
E
F
N
2
N
1
A
B
x
y
z
Figure 6: When the apparatus of Figure 90about, we obtain a
polarizer which prepares atoms in the spin state+.
with:
2
+
2
= 1 (B-2)
Is it possible to prepare atoms whose spin state is described by the corresponding ket
?
We are going to show that there exists, for all, a unit vectorusuch that
is collinear with the ket+u. We therefore choose two complex numbersandwhich
satisfy relation (B-2) but which are arbitrary in every other respect. Taking (B-2) into
account, we nd that there necessarily exists an anglesuch that:
cos
2
=
sin
2
=
(B-3)
If, in addition, we impose:
0 (B-4)
the equationtan
2
=determinesuniquely. We already know that only the dierence
of the phases ofandenters into the physical predictions. Let us therefore set:
=Arg Arg (B-5)
=Arg+Arg (B-6)
We thus have:
Arg=
1
2
+
1
2
Arg=
1
2
1
2
(B-7)
With this notation, the ketcan be written:
= e
2cos
2
e
2++ sin
2
e
2 (B-8)
403

CHAPTER IV SIMPLE CASES: SPIN 1/2 AND TWO-LEVEL SYSTEMS
If we compare this expression with formula (A-22a), we see thatdiers from the ket
+(which corresponds to the unit vectorucharacterized byand) only by the phase
factore
2
, which has no physical signicance.
Consequently, to prepare silver atoms in the state, it suces to place the Stern-
Gerlach apparatus (with its plate pierced at1) so that its axis is directed along the
vectoruwhose polar angles are determined fromandby (B-3) and (B-5).
B-2. Spin measurements
We saw in measurethe component
of the angular momentumSSSof silver atoms along a given axis. We have just pointed
out, in , that an apparatus of the same type can be used topreparean atomic beam
in a given spin state. Consequently, if we place two Stern-Gerlach magnets one after the
other, we can verify experimentally the predictions of the postulates. The rst apparatus
acts like a polarizer: the beam coming out of it is composed of a large number of silver
atoms all in the same spin state. This beam then enters the second apparatus, which
is used to measure a specied component of the angular momentumSSS: this is, as it
were, the analyzer (note the analogy with the optical experiment described in
Chapter). We shall assume in this section that the spin state of the atoms of the beam
does not evolve between the time they leave the polarizer and the time they enter the
analyzer, that is, between the preparation and the measurement. It would be easy to
forgo this hypothesis, by using the Schrödinger equation to determine the spin evolution
between the moment of preparation and the moment of measurement.
B-2-a. First experiment
Let us choose the axes of the two apparatuses parallel to(Fig.). The rst one
prepares the atoms in the state+and the second one measuresS. What is observed
on the plate of the second apparatus?E
1
F
1
B
1
B
2
A
1
P
1
A
2
P
2
Figure 7: The rst apparatus (a source composed of the furnace1and the slit1,
plus a polarizer formed by the magnet1and the pierced plate1) prepares the atoms
in the state+. The second one (an analyzer composed of the magnet2and the
plate2) measures the componentS. The result obtained is certain (+}2).
Since the state of the system under study is an eigenstate of the observablewhich
we want to measure, the postulates indicate that the measurement result iscertain: we
nd, without fail, the corresponding eigenvalue (+~2). Consequently, all the atoms of
404

B. ILLUSTRATION OF THE POSTULATES IN THE CASE OF A SPIN 1/2
the beam must condense into a single spot on the plate of the second apparatus, at the
position corresponding to+~2.
This is indeed what is observed experimentally: all the atoms strike the second
plate in the vicinity of1, none hitting near2.
B-2-b. Second experiment
Now let us place the axis of the rst apparatus along the unit vectoru, with polar
angles,=(uis therefore contained in theplane). The axis of the second
apparatus remains parallel to(Fig.). According to (A-22a), the spin state of the
atoms when they leave the polarizer is (we ignore an irrelevantfactor multiplying
whole ket):
=cos
2
++ sin
2
(B-9)
The analyzer measuresSon these atoms. What are the results?O
E
1
F
1
A
1
θθ
A
2P
1
P
2
B
1
B
2
x
y
z
Figure 8: The rst apparatus prepares the spins in the state+(uis the unit vector
of the plane that makes an anglewith). The second one measures theS
component. The possible results are+~2(probabilitycos
2
2) and~2(probability
sin
2
2).
This time, we nd that certain atoms condense at1, and others at2, although
they have all been prepared in the same way: there is an indeterminacy in the behavior
of each of the atoms taken individually. The postulate of spectral decomposition merely
enables us to predict the probability of each atom's appearance at1or2. Since
(B-9) gives the expansion of the spin state of an atom in terms of the eigenstates of
the observable being measured, we can calculate directly that these probabilities are,
respectively,cos
2
2andsin
2
2. Thus, when enough atoms have condensed on the
plate, we observe that the intensity of the spots at1and2corresponds to numbers
of atoms which are proportional, respectively, tocos
2
2andsin
2
2.
Comment:
For any value of the angle(except exactly 0 or), it is therefore always possible
to nd the two results+~2and~2in a measurement of. This prediction
405

CHAPTER IV SIMPLE CASES: SPIN 1/2 AND TWO-LEVEL SYSTEMS
may seem to a certain extent paradoxical. For example, ifis very small, the
spin at the exit of the rst apparatus points in a direction which is practically
, and yet one can nd~2as well as+~2in a measurement of(while,
in classical mechanics, the result would be(~cos)2~2). Nevertheless, the
smaller, the smaller the probability of nding~2. Moreover, we shall see later
[formula (B-11)] that the mean value of the results which would be obtained in a
large number of identical experiments is=
~
2
cos, which corresponds to the
classical result.
B-2-c. Third experiment
Let us take a polarizer positioned as in , so as to prepare atoms in the
state (B-9), and let us rotate the analyzer until its axis is directed along, so that
it measures theScomponent of the angular momentum.
To calculate the predictions of the postulates in this case, we must expand the
state (B-9) in terms of the eigenstates of the observable[formula (A-20)]. We nd:
+=
1
2
(cos
2
+ sin
2
) = cos(
42
)
=
1
2
(cos
2
sin
2
) = sin(
42
)
(B-10)
The probability of nding the eigenvalue+~2ofis thereforecos
2
42
and
that of nding(~2),sin
2
42
.
It is possible to verify these predictions by measuring the intensity of the two spots
on the plate situated at the exit of the second Stern-Gerlach apparatus.
Comment:
The fact that it is
42
that enters in here is not at all surprising: in ,
the angle between the axes of the two apparatus was; it became
2
after
rotation of the second apparatus.
B-2-d. Mean values
In the situation of , we nd experimentally that, of a great numberof
atoms,cos
2
2arrive at1andsin
2
2, at2. The measurement ofStherefore
yields+~2for each of the rst group and~2for each of the second. If we calculate
the mean value of these results, we obtain:
=
1
~
2
cos
2
2
~
2
sin
2
2
=
~
2
cos (B-11)
It is easy to verify from formulas (B-9) and (A-10) that this is indeed the value of
the matrix element .
406

B. ILLUSTRATION OF THE POSTULATES IN THE CASE OF A SPIN 1/2
Similarly, the average of the measurement results obtained in the experiment of

=
1
~
2
cos
2
42
~
2
sin
2
42
=
~
2
sin (B-12)
To calculate the matrix element , we can use the matrix (A-16) which represents
in the+ basis. In this same basis, the ketis represented by the column
vector
cos2
sin2
, and the braby the corresponding row vector. We therefore have:
=
~
2
cos2 sin2
0 1
1 0
cos2
sin2
=
~
2
sin (B-13)
The mean value ofSis indeed equal to the matrix element, in the state, of the
associated observable.
It is interesting to note that if we were dealing with aclassicalangular momentum
of modulus~2directed along the axis of the polarizer, its components alongand
would be precisely(~2) sinand(~2) cos. More generally, if we calculate [using
the same technique as in (B-13)] the mean values of,and in the state+
[formula (A-22a)], we nd:
++=
~
2
sincos
++=
~
2
sinsin
++=
~
2
cos (B-14)
These mean values are equal to the components of a classical angular momentum of
modulus~2oriented along the vectoruwhose polar angles areand. Therefore, we
can also establish here a relation between classical mechanics and quantum mechanics
through the mean values. However, we must not lose sight of the fact thata measurement
ofS,for example, on a given atom will never yield
~
2
sincos: the only results which
can be found are+~2and~2. Only in taking the average of values obtained in a
large number of identical measurements (same state of the system, here+, and same
observable measured, here) do we obtain
~
2
sincos.
Comment:
It is useful to consider again at this stage the problem of external degrees of freedom
(position, momentum).
407

CHAPTER IV SIMPLE CASES: SPIN 1/2 AND TWO-LEVEL SYSTEMSa
N
1
N
2
N
1
N
2
N
2
c
b
Figure 9: When the spin is in the state+(g. a) or(g. b), the center of the wave
packet follows a well-dened trajectory which can be calculated classically. When the spin
state is a linear superposition of+and, the wave packet splits into two parts and
it is no longer possible to say that the atom follows a classical trajectory (despite the fact
that the spread of each of the packets is much smaller than the characteristic dimensions
of the problem).
When a silver atom enters the second Stern-Gerlach apparatus in the spin stategiven
by (B-9), we have just seen that it is impossible to predict with certainty whether it will
condense at1or2. It seems dicult to reconcile this indeterminacy with the idea of
a perfectly well-determined classical trajectory, given the initial state of the system.
In fact, this is not a real paradox. To say that the external degrees of freedom can be
treated classically means only that it is possible to form wave packets which are much
smaller than all the dimensions of the problem. It does not necessarily mean, as we shall
see, that the particle itself follows a classical trajectory.
Let us rst consider a silver atom which enters the apparatus in the initial spin state+.
The wave function which describes the external degrees of freedom of this particle is a
wave packet whose spread is very small and whose center follows the classical trajectory
of Figure-a. Similarly, if the silver atom enters with the spin state, the center of
the wave packet associated with it follows the classical trajectory of Figure-b.
If we now consider an atom which enters with the spin stateof formula (B-9), the
corresponding initial state is a well-dened linear superposition of the two preceding
initial states. Since the Schrödinger equation is linear, the wave function of the particle
at a subsequent instant (Fig.-c) is a linear superposition of the two wave packets of
Figures-a and-b. The particle therefore has a certain probability amplitude of being
in one or the other of these two wave packets. We see that it does not follow a classical
trajectory at all, unlike what happens to the centers of the two wave packets. Upon
arrival on the screen, the wave function has non-zero values in two dierent regions, each
408

B. ILLUSTRATION OF THE POSTULATES IN THE CASE OF A SPIN 1/2
very localized, around the points1and 2. The particle can therefore appear either
near1or near2, and we cannot predict with certainty at which of these two points
the appearance will occur.
Note that the two wave packets of Figure-c do not represent two dierent particles;
they represent only one particle, whose wave function has two parts, each of which is
very localized about a dierent point. The two wave packets, moreover, have a well-
dened phase relation because they arise from the same initial wave packet, split into
two under the inuence of the gradient ofB. We could recombine them to form one wave
packet again by removing the screen (that is, by not performing the measurement) and
by submitting them to a new eld gradient, whose sign would be the opposite of the rst
one.
B-3. Evolution of a spin 1/2 particle in a uniform magnetic eld
B-3-a. The interaction Hamiltonian and the Schrödinger equation
Consider a silver atom in auniformmagnetic eldB0, and choose theaxis
alongB0. The classical potential energy of the magnetic momentMMM=SSSof this atom
is then:
=MMMB0=M 0= 0S (B-15)
where0is the modulus of the magnetic eld. Let us set:
0= 0 (B-16)
It is easy to see that0has the dimensions of the inverse of a time, that is, of an angular
velocity.
Since we are quantizing only the internal degrees of freedom of the particle,S
must be replaced by the operator, and the classical energy (B-15) becomes an operator:
it is the Hamiltonianwhich describes the evolution of the spin of the atom in the eld
B0:
H=0 (B-17)
Since this operator is time-independent, solving the corresponding Schrödinger
equation amounts to solving the eigenvalue equation of. We see that the eigenvectors
ofare those of:
+= +
~0
2
+
=
~0
2
(B-18)
There are therefore two energy levels,+= +~02and =~02(Fig.).
Their separation~0is proportional to the magnetic eld; they dene a single Bohr
frequency:
+=
1
(+ ) =
0
2
(B-19)
409

CHAPTER IV SIMPLE CASES: SPIN 1/2 AND TWO-LEVEL SYSTEMSE
ħω
0
E
+
E
–
| + ⟩
| – ⟩
Figure 10: Energy levels of a spin 1/2, of
gyromagnetic ratio, placed in a magnetic
eld0parallel to;0is dened by0=
0.
Comment:
()If the eldB0is parallel to the unit vectoruwhose polar angles areand
, relation (B-17
=0 (B-20)
where=Suis the component ofSalongu.
()For the silver atom,is negative;0is therefore positive, according to (B-16).
This explains the arrangement of the levels in Figure.
B-3-b. Larmor precession
Let us assume that, at time= 0, the spin is in the state:
(0)= cos
2
e
2
++ sin
2
e
2
(B-21)
(we showed in
()at an arbitrary instantt0, we apply the rule (D-54) given in Chapter. In
expression (B-21),(0)is already expanded in terms of the eigenstates of the Hamil-
tonian, and we therefore obtain:
()= cos
2
e
2e
+
~++ sin
2
e
2e
~ (B-22)
or, using the values of+and:
()= cos
2
e
(+0)2
++ sin
2
e
(+0)2
(B-23)
The presence of the magnetic eldB0therefore introduces a phase shift, proportional to
the time, between the coecients of the kets+and.
410

C. GENERAL STUDY OF TWO-LEVEL SYSTEMS
Comparing expression (B-23) for()with that for the eigenket+of the
observableSu[formula (A-22a)], we see that the directionu()along which the spin
component is+~2with certainty is dened by the polar angles:
() =
() =+0
(B-24)
The angle betweenu()and(the direction of the magnetic eldB0) therefore remains
constant, butu()revolves aboutat an angular velocity of0(proportional to the
magnetic eld). Thus, we nd in quantum mechanics the phenomenon which we de-
scribed for a classical magnetic moment in , and which bears the name ofLarmor
precession.
From expression (B-17) for the Hamiltonian, it is obvious that the observable
is a constant of the motion. It can be veried from (B-23) that the probabilities of
obtaining+~2or~2in a measurement of this observable are time-independent. Since
the modulus ofe
(+0)2
is equal to1, these probabilities are equal, respectively, to
cos
2
2andsin
2
2. The mean value ofis also time-independent:
() ()=
~
2
cos (B-25)
On the other hand,and do not commute with[it is easy to show this
by using the matrices representingand, which are given in (A-15), (A-16) and
(A-17)]. Thus, formulas (B-14) here become:
() ()=
~
2
sincos(+0)
() ()=
~
2
sinsin(+0)
(B-26)
In these expressions, we again nd the single Bohr frequency02of the system. More-
over, the mean values ofandbehave like the components of a classical angular
momentum of modulus~2undergoing Larmor precession.
C. General study of two-level systems
The simplicity of the calculations presented in
space has only two dimensions.
There exist numerous other cases in physics which, to a rst approximation, can
be treated just as simply. Consider, for example, a physical system having two states
whose energies are close together and very dierent from those of all other states of the
system. Assume that we want to evaluate the eect of an external perturbation (or of
internal interactions previously neglected) on these two states. When the intensity of the
perturbation is suciently weak, it can be shown (cf.Chap.) that its eect on the
two states can be calculated, to a rst approximation, by ignoring all the other energy
levels of the system. All the calculations can then be perfomed in a two-dimensional
subspace of the state space.
In this section, we shall study certain general properties of two-level systems (which
are not necessarily spin 1/2 particles). Such a study is interesting because it enables us,
using a mathematically simple model, to bring out some general and important physical
ideas (quantum resonance, oscillation between two levels, etc...).
411

CHAPTER IV SIMPLE CASES: SPIN 1/2 AND TWO-LEVEL SYSTEMS
C-1. Outline of the problem
C-1-a. Notation
Consider a physical system whose state space is two-dimensional (as we have al-
ready pointed out, this is usually an approximation: under certain conditions, we can
conne ourselves to a two-dimensional subspace of the state space). For a basis, we
choose the system of the two eigenstates1and 2of the HamiltonianH0whose
eigenvalues are, respectively,1and2:
01=11
02=22
(C-1)
This basis is orthonormal:
= ; = 12 (C-2)
Assume that we want to take into account an external perturbation, or interactions
internal to the system, initially neglected inH0. The Hamiltonian becomes:
=0+ (C-3)
The eigenstates and eigenvalues ofwill be denoted byand:
+=++
=
(C-4)
H0is often called the unperturbed Hamiltonian and, the perturbation or coupling.
We shall assume here thatis time-independent. In the1 2basis of eigenstates
ofH0(called unperturbed states),is represented by a Hermitian matrix:
() =
11 12
21 22
(C-5)
11and22are real. Moreover:
12= (21) (C-6)
In the absence of coupling,1and2are the possible energies of the system,
and the states1and 2are stationary states (if the system is placed in one of
these two states, it remains there indenitely). The problem consists of evaluating the
modications that appear when the couplingis introduced.
C-1-b. Consequences of the coupling
. 1and2are no longer the possible energies of the system
A measurement of the energy of the system can yield only one of the two eigenvalues
+and of, which generally dier from1and2.
The rst problem is to calculate+and in terms of12and the matrix
elements of. This amounts to studying the eect of the coupling on the energy
levels.
412

C. GENERAL STUDY OF TWO-LEVEL SYSTEMS
. 1and2are no longer stationary states
Since1and2are not generally eigenstates of the total Hamiltonian, they
are no longer stationary states. If, for example, the system at time= 0is in the
state1, there is a certain probabilityP12()of nding it in the state2at time:
therefore induces transitions between the two unperturbed states. Hence the name
coupling (between1and2) given to.
This dynamic aspect of the eect ofconstitutes the second problem with which
we shall be concerned.
Comment:
In ComplementIV, the two problems we have just cited are considered by introducing
the concept of a ctitious spin. It can indeed be shown that the Hamiltonianto be
diagonalized has the same form as that of a spin 1/2 placed in a static magnetic eld
B, whose components,and are simply expressed in terms of1,2and the
matrix elements. In other words, with every two-level system (not necessarily a spin
1/2), can be associated a spin 1/2 (called a ctitious spin) placed in a static eldB
and described by a Hamiltonian whose form is identical to that of the initial two level
system. All the results related to two-level systems which we are going to establish in
this section can be interpreted in a simple geometric way in terms of magnetic moment,
Larmor precession, and the various concepts introduced in
in connection with spin 1/2 particles. This geometrical interpretation is developed in
ComplementIV.
C-2. Static aspect: eect of coupling on the stationary states of the system
C-2-a. Expressions for the eigenstates and eigenvalues of
In the 1 2basis, the matrix representingis written:
() =
1+ 11 12
21 2+ 22
(C-7)
The diagonalization of matrix (C-7) presents no problems (it is performed in detail
in ComplementIV). We nd the eigenvalues:
+=
1
2
(1+ 11+2+ 22) +
1
2(1+ 11 2 22)
2
+ 412
2
=
1
2
(1+ 11+2+ 22)
1
2(1+ 11 2 22)
2
+ 412
2
(C-8)
(it can be veried that if= 0,+and are identical
5
to1and2). The
5
If1 2,+approaches1and approaches2when approaches zero. On the other
hand, if1 2,+approaches2and approaches1.
413

CHAPTER IV SIMPLE CASES: SPIN 1/2 AND TWO-LEVEL SYSTEMS
eigenvectors associated with+and are written:
+= cos
2
e
2
1+ sin
2
e
2
2 (C-9a)
= sin
2
e
2
1+ cos
2
e
2
2 (C-9b)
where the anglesandare dened by:
tan=
212
1+ 11 2 22
with : 0 (C-10)
21= 21e (C-11)
C-2-b. Discussion
. Graphical representation of the eect of coupling
All the interesting eects which we shall discuss later arise from the fact that the
perturbationpossesses non-diagonal matrix elements12=
21(if12= 0, the
eigenstates ofare the same as those of0, the new eigenvalues being simply1+11
and2+ 22). To simplify the discussion, we shall therefore assume from now on that
the matrix () is purely non-diagonal
6
, that is, that11= 22= 0. Formulas (C-8)
and (C-10) then become:
+=
1
2
(1+2) +
1
2(1 2)
2
+ 412
2
=
1
2
(1+2)
1
2(1 2)
2
+ 412
2
(C-12)
tan=
212
1+2
0 (C-13)
We now study the eect of the couplingon the energies+and in terms of
the values of1and2. Assume that12is xed and introduce the two parameters:
=
1
2
(1+2)
=
1
2
(1 2)
(C-14)
We see from (C-12) that the variation of+and with respect tois extremely
simple: changingreduces to shifting the origin along the energy axis. Moreover, it
can be veried from (C-9), (C-10) and (C-11) that the vectors+and do not
depend on. We are therefore concerned only with the inuence of the parameter.
Let us show on the same graph, in terms of, the four energies12+and. We
obtain for1and2two straight lines of slope+1and1(shown in dashed lines in
Figure). Substituting (C-14) into (C-12), we nd:
+= +

2
+ 12
2
(C-15)6
If11and 22are non-zero, we simply set:1= 1+ 11,2= 2+ 22. All the results
obtained in this section then remain valid if we replace1and2by1and2.
414

C. GENERAL STUDY OF TWO-LEVEL SYSTEMSEnergies
W
12
– W
12
E
+
E
–
E
1
E
m
E
2
Δ
Figure 11: Variation of the energies+and as a function of the energy dierence
= (1 2)2. In the absence of coupling, the levels cross at the origin (dashed
straight lines). Under the eect of the non-diagonal coupling, the two perturbed levels
repel each other and we obtain an anti-crossing: the curves giving+and in
terms ofare branches of a hyperbola (solid lines in the gure) whose asymptotes are
the unperturbed levels.
=

2
+ 12
2
(C-16)
Whenvaries,+and describe the two branches of a hyperbola which is sym-
metrical with respect to the coordinate axes and whose asymptotes are the two straight
lines associated with the unperturbed levels; the minimum separation between the two
branches is212(solid lines in Figure)
7
.
. Eect of the coupling on the energy levels
In the absence of coupling, the energies1and2of the two levels cross at
= 0. It is clear from Figure
each other that is, the energy values move further away from each other. The diagram
in solid lines in Figure anti-crossingdiagram.
Also, we see that, for any, we always have:
+ 1 2
This is a result that appears rather often in other domains of physics (for example, in
electrical circuit theory):the coupling separates the normal frequencies.
7
It is clear from Figure 0:
+ 1 2if 1 2
+ 2 1if 1 2
415

CHAPTER IV SIMPLE CASES: SPIN 1/2 AND TWO-LEVEL SYSTEMS
Near the asymptotes, that is, for 12, formulas (C-15) and (C-16) can be
written in the form of a limited power series expansion in
12

:
+= + 1 +
1
2
12

2
+
= 1 +
1
2
12

2
+
(C-17)
On the other hand, at the center of the hyperbola, for2=1( = 0), formulas (C-15)
and (C-16) yield:
+= + 12
= 12
(C-18)
Therefore,the eect of the coupling is much more important when the two unperturbed
levels have the same energy. The eect is then of rst order, as can be seen from (C-18),
while it is of second order when 12[formulas (C-17)].
. Eect of the coupling on the eigenstates
When (C-14) is used, formula (C-13) becomes:
tan=
12

(C-19)
It follows that, when 12(strong coupling), 2; on the other hand, when
12(weak coupling),0(assuming0).
At the center of the hyperbola, when2=1( = 0), we have:
+=
1
2
e
2
1+ e
2
2
=
1
2
e
2
1+ e
2
2
(C-20)
while near the asymptotes (that is, for 12), we have, to rst order in12:
+= e
2
1+ e
12
2
2+
= e
2
2e
12
2
1+
(C-21)
In other words, for a weak coupling (1 2 12), the perturbed states dier
very slightly from the unperturbed states. We see from (C-21) that to within a global
phase factore
2
,+is equal to the state1slightly contaminated by a small
contribution from the state2. On the other hand, for a strong coupling (1 2
12), formulas (C-20) indicate that the states+and are very dierent from
the states1and2, since they are linear superpositions of them with coecients of
the same modulus.
Thus, like the energies, the eigenstates undergo signicant modications in the
neighborhood of the point where the two unperturbed states cross.
416

C. GENERAL STUDY OF TWO-LEVEL SYSTEMS
C-2-c. Important application: the phenomenon of quantum resonance
When 1=2= , the corresponding energy of0is two-fold degenerate. As
we have just seen, the coupling12lifts this degeneracy and, in particular, gives rise to
a level whose energy is lowered by12. In other words, if the ground state of a physical
system is two-fold degenerate (and suciently far from all the other levels), any (purely
non-diagonal) coupling between the two corresponding states lowers the energy of the
ground state of the system, which thus becomes more stable.
As a rst example of this phenomenon, we shall cite the resonance stabilization of
the benzene moleculeC6H6. Experiments show that the six carbon atoms are situated at
the vertices of a regular hexagon, and we would expect the ground state to include three
double bonds between neighboring carbon atoms. Figures-a and-b represent two
possible dispositions of these bonds. The nuclei are assumed here to be xed because of
their high masses. Thus, the electronic states1and2, associated with Figures-
a and-b respectively, are dierent. If the structure of Figure-a were the only
one possible, the ground state of the electronic system would have an energy of=
1 1, whereis the Hamiltonian of the electrons in the potential created by
the nuclei. But the bonds can also be placed as shown in Figure-b. By symmetry,
we have 2 2= 1 1, and we could conclude that the ground state of the
molecule is doubly degenerate. However, the non-diagonal matrix element2 1of
the Hamiltonianis not zero. This coupling between the states1and2gives rise
to two distinct levels, one of which has an energy lower than. The benzene molecule
is therefore more stable than we would have expected. Moreover, in its true ground
state, the conguration of the molecule cannot be represented either by Figure-a or
by Figure-b: this state is a linear superposition of1and 2[the coecients of
this superposition having, as in (C-20), the same modulus]. This is what is symbolized
by the double arrow of Figure, commonly used by chemists.
Figure 12: Two possible congurations of the
double bonds in a benzene molecule.
Another example is that of the (ionized) molecule
+
2
, composed of two protons
p1andp2and one electron. The two protons, because of their large masses, can be
considered to be xed. Let us callthe distance between them and1and2, the
states where the electron is localized aroundp1or aroundp2, its wave function being
that of the hydrogen atom it would form withp1orp2(Fig.). As above, the diagonal
elements1 1and2 2of the Hamiltonian are equal because of symmetry; we
shall denote them by(). The two states1and2are not, however, stationary
states, since the matrix element1 2is not zero. Here again, we obtain an energy
level lower than()and, in the ground state, the wave function of the electron is
a linear combination of those of Figures-a and-b. The electron is thus no longer
localized about one of the two protons alone, and it is this delocalization which, by
417

CHAPTER IV SIMPLE CASES: SPIN 1/2 AND TWO-LEVEL SYSTEMSe
–
p
1 p
1
p
2
p
2
e
–
a b
Figure 13: In the
+
2
ion, the electron could logically be localized either around the
proton1(g. a) or around the proton2(g. b). In the ground state of the ion, the
wave function of the electron is a linear superposition of the wave functions associated
with gures a and b. Its probability of presence is symmetrical with respect to the plane
bisecting12.
lowering its potential energy, is responsible for the chemical bond
8
.
C-3. Dynamical aspect: oscillation of the system between the two unperturbed states
C-3-a. Evolution of the state vector
Let:
()=1()1+2()2 (C-22)
be the state vector of the system at the instant. The evolution of()in the presence
of the couplingis given by the Schrödinger equation:
~
d
d
()= (0+)() (C-23)
Let us project this equation onto the basis vectors1and2. We obtain, using
(C-5) (where we have set11= 22= 0) and (C-22):
~
d
d
1() =11() +122()
~
d
d
2() =211() +22()
(C-24)
If12= 0, these equations form a linear system of homogeneous coupled dier-
ential equations. The classical method of solving such a system reduces, in fact, to the
application of rule (D-54) of Chapter: look for the eigenvectors+(eigenvalue+)
and (eigenvalue) of the operator= 0+[whose matrix elements are the
coecients of equations (C-24)], and decompose(0)in terms of+and :
(0)= ++ (C-25)
8
A more elaborate study of the ionized molecule
+
2
will be presented in ComplementXI.
418

C. GENERAL STUDY OF TWO-LEVEL SYSTEMS
(whereandare xed by the initial conditions). We then have:
()=e
+~
++e
~
(C-26)
[which enables us to obtain1()and2()by projecting()onto1and2].
It can be shown that a system whose state vector is the vector()given in
(C-26) oscillates between the two unperturbed states1and2. To see this, we shall
assume that the system at time= 0is in the state1:
(0)= 1 (C-27)
and calculate the probabilityP12()of nding it in the state2at time.
C-3-b. Calculation ofP12(): Rabi's formula
As in (C-25), let us expand the ket(0)given in (C-27) on the +
basis. Inverting formulas (C-9), we obtain:
(0)= 1= e
2
cos
2
+sin
2
(C-28)
from which we deduce, using (C-26):
()= e
2
cos
2
e
+~
+sin
2
e
~
(C-29)
The probability amplitude of nding the system at timein the state2is then
written:
2()= e
2
cos
2
e
+~
2+sin
2
e
~
2
= e
2
sin
2
cos
2
[e
+~
e
~
]
(C-30)
which enables us to calculateP12() = 2()
2
. We thus nd:
P12() =
1
2
sin
2
1cos
+
~
= sin
2
sin
2 +
2~
(C-31)
or, using expressions (C-12) and (C-13):
P12() =
412
2
412
2
+ (1 2)
2
sin
2
412
2
+ (1 2)
2
2~
(C-32)
Formula (C-32) is sometimes called Rabi's formula.
419

CHAPTER IV SIMPLE CASES: SPIN 1/2 AND TWO-LEVEL SYSTEMSsin
2
θ
0
πħ/(E
+
– E
–
)
t
12
(t)
Figure 14: Variation with respect to time of the probabilityP12()of nding the system
in the state2when it was initially in the state1. When the states1and 2
have the same unperturbed energy, the probabilityP12()can attain the value 1.
C-3-c. Discussion
Relation (C-31) shows that the probabilityP12()oscillates over time with a fre-
quency of(+ ), which is simply the unique Bohr frequency of the system.P12()
varies between zero and a maximum value which, according to (C-31), is equal tosin
2
.
This maximum value is attained for all values ofsuch that= (2k+ 1)2(+ ),
withk=012, ... (Fig.).
The oscillation frequency(+ ), as well as the maximum valuesin
2
of
P12(), are functions of12and1 2, whose main features we are now going to
describe.
When 1=2,(+ )is equal to212, andsin
2
takes on its greatest
possible value, that is, 1: at certain times,= (2+ 1)~212, the system (which
started from the state1) is in the state2. Therefore, any coupling between two
states of equal energy causes the system to oscillate completely from one state to the
other with a frequency proportional to the coupling
9
.
When 1 2increases, so does(+ ), whilesin
2
decreases. For a weak
coupling (1 2 12),+ diers very little from1 2, andsin
2
becomes
very small. This last result is not surprising since, in the case of a weak coupling, the
state1is very close to the stationary state+[cf.formulas (C-21)]: the system,
having started in the state1, evolves very little over time.
C-3-d. Example of oscillation between two states
Let us return to the example of the
+
2
molecule. We shall assume that, at a certain
time, the electron is localized about protonp1: it is, for example, in the state shown
in Figure-a. According to the results of the preceding section, we know that it will
oscillate between the two protons with a frequency equal to the Bohr frequency associated
9
The same phenomenon is found in other domains of physics. Consider, for example, two identical
pendulums (1) and (2), suspended from the same support and having the same frequency. Let us assume
that at time= 0, pendulum (1) is set in motion. The coupling is ensured by their common support. We
then know (cf.ComplementV) that, after a certain time (which decreases if the coupling is increased),
we arrive at a situation where only pendulum (2) oscillates, with the initial amplitude of pendulum (1).
Then the motion is transferred back to pendulum (1), and so on.
420

C. GENERAL STUDY OF TWO-LEVEL SYSTEMS
with the two stationary states+and of the molecule. To this oscillation of the
electron between the two states, represented in-a and-b, corresponds an oscillation
of the mean value of the electric dipole moment of the molecule (the dipole moment is
non-zero when the electron is localized about one of the two protons, and changes sign
depending on whether the proton involved isp1orp2). Thus we see concretely how,
when the molecule is not in a stationary state, an oscillating electric dipole moment can
appear. Such an oscillating dipole can exchange energy with an electromagnetic wave
of the same frequency. Consequently, this frequency must appear in the absorption and
emission spectrum of the
+
2
ion.
Other examples of oscillations between two states are discussed in ComplementsIV,
GIVandIV.
References and suggestions for further reading:
The Stern-Gerlach experiment: original article (3.8); Cagnac and Pebay-Peyroula
(11.2), Chap. X; Eisberg and Resnick (1.3), 8-3; Bohm (5.1), 22.5 and 22.6; Frisch
(3.13).
Two-level systems: Feynman III (1.2), Chaps. 6, 10 and 11; Valentin (16.1), Annexe
XII; Allen and Eberly (15.8), particularly Chap. 3.
421

COMPLEMENTS OF CHAPTER IV, READER'S GUIDE
AIV: THE PAULI MATRICES
BIV: DIAGONALISATION OF A 2 2 HERMITIAN
MATRIX
Technical study of 2 2matrices; simple,
and important for solving numerous quantum
mechanical problems.
CIV: FICTITIOUS SPIN 1/2 ASSOCIATED WITH
A TWO-LEVEL SYSTEM
Establishes the close relation that exists between
; supplies a simple
geometrical interpretation of the properties of
two-level systems (easy, but not indispensable for
what follows).
DIV: SYSTEM OF TWO SPIN 1/2 PARTICLES Simple illustration of the tensor product
and of the postulates of quantum mechanics
(can be considered to be a set of worked exercises).
EIV: SPIN 1/2 DENSITY MATRIX Illustration, in the case of spin12particles, of
the concepts introduced in ComplementIII.
FIV: SPIN 1/2 PARTICLE IN A STATIC
MAGNETIC FIELD AND A ROTATING FIELD:
MAGNETIC RESONANCE
Study of a very important physical phenomenon
with many applications: magnetic resonance.
Can be studied later.
GIV: A SIMPLE MODEL OF THE AMMONIA
MOLECULE
Example of a physical system whose study can
be reduced, in a rst approximation, to that of a
two-level system; moderately dicult.
HIV: COUPLING BETWEEN A STABLE STATE
AND AN UNSTABLE STATE
Study of the inuence of coupling between two
levels with dierent lifetimes; easy, but requires
the concepts introduced in ComplementIII.
JIV: EXERCISES
423

THE PAULI MATRICES
Complement AIV
The Pauli matrices
1 Denition; eigenvalues and eigenvectors
2 Simple properties
3 A convenient basis of the 22matrix space
In , we introduced the matrices representing the three compo-
nents,andof a spinin the+ basis (eigenvectors of). In quantum
mechanics, it is often convenient to introduce the dimensionless operator, proportional
toS, and given by:
S=
~
2
(1)
The matrices representing the three components ofin the+ basis are called
the Pauli matrices.
1. Denition; eigenvalues and eigenvectors
Let us go back to equations (A-15), (A-16) and (A-17) of Chapter. Using (1), we see
that the denition of the Pauli matrices is:
=
0 1
1 0
=
0
0
=
1 0
0 1
(2)
These are Hermitian matrices, all three of which have the same characteristic
equation:
2
1 = 0 (3)
The eigenvalues of,andare therefore:
=1 (4)
which is consistent with the fact that those of,andare~2.
It is easy to obtain, from denition (2), the eigenvectors of,and. They
are the same, respectively, as those of,and, already introduced in
Chapter:
=
=
=
(5)
425

COMPLEMENT A IV
with:
=
1
2
[+ ]
=
1
2
[+ ] (6)
2. Simple properties
It is easy to see from denition (2) that the Pauli matrices verify the relations:
Det() =1= or (7)
Tr() = 0 (8)
(
2
) = (
2
) = (
2
) =(whereis the22unit matrix) (9)
= = (10)
as well as the equations that can be deduced from (10) by cyclic permutation of,and
.
Equations (9) and (10) are sometimes condensed into the form:
= + (11)
where is antisymmetric with respect to the interchange of any two of its indices. It
is equal to:
=
0if the indices are not all dierent
1if is an even permutation of
1if is an odd permutation of
(12)
From (10), we immediately conclude:
[ ] = 2 (13)
(and the relations obtained by cyclic permutation). This yields:
[ ] =~
[ ] =~ (14)
[ ] =~
We shall see later (cf.Chap.) that equations (14) are characteristic of an angular
momentum.
We also see from (10) that:
+ = 0 (15)
(thematrices are said to anticommute with each other) and that, taking (9) into
account:
= (16)
426

THE PAULI MATRICES
Finally, let us mention an identity which is sometimes useful in quantum mechanics.
IfAandBdenote two vectors whose components are numbers (or operators which
commute with all operators acting in the two-dimensional spin state space):
(A)(B) =AB+(AB) (17)
This identity can be demonstrated as follows. Using formula (11) and the fact thatA
andBcommute with, we can write:
(A)(B) =
= +
= + (18)
In the second term, we recognize the scalar productAB. In addition, it is easy to see
from (12) that is theth component of the vector productAB. This
proves (17). Note that ifAandBdo not commute, they must appear in the same order
on both sides of the identity.
3. A convenient basis of the22matrix space
Consider an arbitrary22matrix:
=
11 12
21 22
(19)
It can always be written as a linear combination of the four matrices:
(20)
since, using (2), we can immediately verify that:
=
11+22
2
+
11 22
2
+
12+21
2
+
12 21
2
(21)
Therefore, any22matrix can be put in the form:
=0+a (22)
where the coecients0 andare complex numbers.
Comparing (21) and (22), we see thatis Hermitian if and only if the coecients
0andaare real. These coecients can be expressed formally in terms of the matrix
in the following manner:
0=
1
2
Tr (23a)
a=
1
2
Tr (23b)
These formulas can easily be proven from (8), (9) and (10).
427

DIAGONALIZATION OF A 22HERMITIAN MATRIX
Complement BIV
Diagonalization of a22Hermitian matrix
1 Introduction
2 Changing the eigenvalue origin
3 Calculation of the eigenvalues and eigenvectors
3-a Angles and. . . . . . . . . . . . . . . . . . . . . . . . . .
3-b Eigenvalues of . . . . . . . . . . . . . . . . . . . . . . . . .
3-c Eigenvalues of . . . . . . . . . . . . . . . . . . . . . . . . .
3-d Normalized eigenvectors of . . . . . . . . . . . . . . . . . .
1. Introduction
In quantum mechanics, one must often diagonalize22matrices. When we need only the
eigenvalues, it is very easy to solve the characteristic equation since it is of second degree.
In principle, the calculation of the normalized eigenvectors is also extremely simple;
however, if it is performed clumsily, it can lead to expressions that are unnecessarily
complicated and dicult to handle.
The goal of this complement is to present a simple method of calculation which is
applicable in all cases. After having changed the origin of the eigenvalues, we introduce
the anglesand, dened in terms of the matrix elements, which enable us to write
the normalized eigenvectors in a simple easy-to-use form. The anglesandalso have
an interesting physical interpretation in the study of two-level systems, as we shall see
in ComplementIV.
2. Changing the eigenvalue origin
Consider the Hermitian matrix:
() =
1112
2122
(1)
11and22are real. Moreover:
12=
21 (2)
The matrix()therefore represents, in an orthonormal basis1 2, a certain
Hermitian operator
1
.
1
We use the letterbecause the Hermitian operator which we are trying to diagonalize is often a
Hamiltonian. Nevertheless, the calculations presented in this complement can obviously be applied to
any22Hermitian matrix.
429

COMPLEMENT B IV
Using the half-sum and half-dierence of the diagonal elements11and22, we
can write()in the following way:
() =
1
2
(11+22) 0
0
1
2
(11+22)
+
1
2
(11 22) 12
21
1
2
(11 22)
(3)
It follows that the operatoritself can be decomposed into:
=
1
2
(11+22)+
1
2
(11 22) (4)
whereis the identity operator andis the Hermitian operator represented in the
1 2basis by the matrix:
() =
1
212
11 22
221
11 22
1
(5)
It is clear from (4) thatandhave the same eigenvectors. Letbe these
eigenvectors, andand, the corresponding eigenvalues forand:
= (6)
= (7)
From (4), we immediately conclude that:
=
1
2
(11+22) +
1
2
(11 22) (8)
Actually, the rst matrix appearing on the right-hand side of (3) plays a minor
role: we could make it disappear by choosing(11+22)2for the eigenvalue origin
2
.
3. Calculation of the eigenvalues and eigenvectors
3-a. Angles and
Letandbe the angles dened in terms of the matrix elementsby:
tan=
221
11 22
with 0 (9)
2
Furthermore, this new origin is the same, whatever the basis1 2initially chosen, since
11+22= Tr()is invariant under a change of orthonormal bases.
430

DIAGONALIZATION OF A 22HERMITIAN MATRIX
21= 21e with 0 2 (10)
is the argument of the complex number21. According to (2), we have12= 21
and:
12= 12e (11)
If we use (9), (10) and (11), the matrix()becomes:
() =
1 tan e
tane 1
(12)
3-b. Eigenvalues of
The characteristic equation of the matrix (12):
Det[() ] =
2
1tan
2
= 0 (13)
directly yields the eigenvalues+and of():
+= +
1
cos
(14a)
=
1
cos
(14b)
We see that they are indeed real (property of a Hermitian matrix,cf.
Chapter). If we want to express1cosin terms of, all we need to do is use (9)
and notice thatcosandtanhave the same sign since0 :
1
cos
=
(11 22)
2
+ 412
2
11 22
(15)
3-c. Eigenvalues of
Using (8), (14) and (15), we immediately obtain:
+=
1
2
(11+22) +
1
2(11 22)
2
+ 412
2
(16a)
=
1
2
(11+22)
1
2(11 22)
2
+ 412
2
(16b)
Comments:
()As we have already said, the eigenvalues (16) can easily be obtained from the
characteristic equation of the matrix(). If we need only the eigenvalues of
(), it is therefore not necessary to introduce the anglesandas we have
done here. On the other hand, we shall see in the following section that this
method is very useful when we need to use the normalized eigenvectors of.
431

COMPLEMENT B IV
()It can be veried immediately from formulas (16) that:
++ =11+22=Tr() (17)
+ =1122 2
2
=Det() (18)
()To have+= , we must have(11 22)
2
+ 412
2
= 0; that is,
11=22and12=21= 0. A22Hermitian matrix with a degenerate
spectrum is therefore necessarily proportional to the unit matrix.
3-d. Normalized eigenvectors of
Letandbe the components of+on1and 2. According to (7), (12)
and (14a), they must satisfy:
1 tan e
tane 1
=
1
cos
(19)
which yields:
1
1
cos
+ tane = 0 (20)
that is:
sin
2
e
2
+cos
2
e
2
= 0 (21)
The normalized eigenvector+can therefore be written:
+= cos
2
e
2
1+ sin
2
e
2
2 (22)
An analogous calculation would yield:
=sin
2
e
2
1+ cos
2
e
2
2 (23)
It can be veried that+and are orthogonal.
Comment:
While the trigonometric functions of the anglecan be expressed rather simply
in terms of the matrix elements[see, for example, formulas (9) and (15)],
the same is not true of those of the angle2. Consequently, formulas (22) and
(23) for the normalized eigenvectors+and become complicated when
cos2andsin2are replaced by their expressions in terms of; they are
no longer very convenient. It is better to use expressions (22) and (23) directly,
keeping the functionscos2andsin2during the entire calculation involving the
normalized eigenvectors of. Furthermore, the nal result of the calculation often
432

DIAGONALIZATION OF A 22HERMITIAN MATRIX
involves only functions of the angle(see, for example, the calculation of
of Chapter) and, consequently, can be expressed simply in terms of the.
Expressions (22) and (23) thus enable us to carry out the intermediate calculations
elegantly, avoiding unnecessarily complicated expressions. This is the advantage
of the method presented in this complement. Another advantage concerns the
physical interpretation and will be discussed in the next complement.
433

FICTITIOUS SPIN 1/2 ASSOCIATED WITH A TWO-LEVEL SYSTEM
Complement CIV
Fictitious spin 1/2 associated with a two-level system
1 Introduction
2 Interpretation of the Hamiltonian in terms of ctitious spin
3 Geometrical interpretation
3-a Fictitious magnetic elds associated with0,and . . .
3-b Eect of coupling on the eigenvalues and eigenvectors of the
Hamiltonian
3-c Geometrical interpretation ofP12(). . . . . . . . . . . . . .
1. Introduction
Consider a two-level system whose Hamiltonianis represented, in an orthonormal
basis 1 2, by the Hermitian matrix()[formula (1) of ComplementIV]
1
. If
we choose(11+22)2as the new energy origin, the matrix()becomes:
() =
1
2
(11 22) 12
21
1
2
(11 22)
(1)
Although the two-level system under consideration is not necessarily a spin 1/2,
we can always associate with it a spin 1/2 whose Hamiltonianis represented by
the same matrix()in the+ basis of eigenstates of thecomponent of this
spin. We shall see that()can then be interpreted as describing the interaction of
this ctitious spin with a static magnetic eldB, whose direction and modulus are
very simply related to the parameters introduced in the preceding complement in the
discussion of the diagonalization of(). Thus it is possible to give a simple physical
meaning to these parameters.
Moreover, if the Hamiltonianis the sum=0+of two operators, we shall
see that we can associate with,0andthree magnetic elds,B,B0andb, such
thatB=B0+b. Introducing the couplingis equivalent, in terms of ctitious spin,
to adding the eldbtoB0. We shall show that this point of view enables us to interpret
very simply the dierent eects studied in .
2. Interpretation of the Hamiltonian in terms of ctitious spin
We saw in Chapter of the coupling between a spin 1/2 and
a magnetic eldB, of components,,can be written:
=B.S=( + + ) (2)
1
We are using the same notation as in ComplementIVand Chapter.
435

COMPLEMENT C IV
To calculate the matrix associated with this operator, we substitute into this relation the
matrices associated with [Chap., relations (A-15), (A-16), (A-17)]. This
immediately yields:
() =
~
2
+
(3)
Therefore, to make matrix (1) identical to(), we must simply choose a ctitious eld
Bdened by:
=
2
~
Re(12)
=
2
~
Im(12)
=
1
~
(22 11)
(4)
Note that the modulusof the projectionBofBonto the plane is then equal
to:
=
2
~
12
(5)
According to formulas (9) and (10) of ComplementIV, the anglesandassociated
with the matrix() = ()written in (3) are given by:
tg=
0
(+ ) = e 0 2
(6)
The gyromagnetic ratiois a simple calculation tool and can have an arbitrary value.
If we agree to choosenegative, relations (6) show that the anglesandassociated
with the matrix()are simply the polar angles of the direction of the eldB(if we had
chosenpositive, they would be those of the opposite direction).
Finally, we see that we can forget the two-level system with which we started and
consider the matrix()as representing, in the basis of the eigenstates+and of
, the Hamiltonianof a spin 1/2 placed in a eldBwhose components are given by
(4).can also be written:
=
whereis the operatorSuwhich describes the spin component along the directionu,
whose polar angles areand, andis the Larmor angular velocity:
=B (7)
The following table summarizes the various correspondences between the two-level system
and the associated ctitious spin12.
436

FICTITIOUS SPIN 1/2 ASSOCIATED WITH A TWO-LEVEL SYSTEM
Two-level system Fictitious spin 1/2
1 +
2
+ +
+ ~
Anglesandintroduced inIV Polar angles of the ctitious eldB
11 22 ~
21 ~2
3. Geometrical interpretation of the various eects discussed in
Chapter
3-a. Fictitious magnetic elds associated with 0,and
Assume, as in , thatappears as the sum of two terms:
=0+ (8)
In the 1,2basis, the unperturbed Hamiltonian0is represented by a diagonal
matrix which, with a suitable choice of the energy origin, is written:
0=
1 2
2
0
0
1 2
2
(9)
As far as the couplingis concerned, we assume, as in , that it is
purely non-diagonal:
() =
0 12
120
(10)
The discussion of the preceding section then enables us to associate with(0)and
()two eldsB0andbsuch that [cf.formulas (4) and (5)]:
0=
2 1
~
0= 0
(11)
437

COMPLEMENT C IVb
θ
B
0
B
O
z
u
Figure 1: Relative disposition of the cti-
tious elds.B0is associated with0,bwith
, andB=B0+bwith the total Hamilto-
nian=0+.
= 0
=
2
~
12
(12)
B0is therefore parallel toand proportional to(1 2)2;bis perpendicular to
and proportional to12. Since() = (0) + (), the eldBassociated with the
total Hamiltonian is the vector sum ofB0andb:
B=B0+b (13)
The three eldsB0,bandBare shown in Figure; the angleintroduced in
of Chapter B0andB, sinceB0is parallel to.
The strong coupling condition introduced in 12 1 2)
is equivalent tob B0(Fig.-a). The weak coupling condition(12 1 2)
is equivalent tob B0(Fig.-b).
3-b. Eect of coupling on the eigenvalues and eigenvectors of the Hamiltonian
1 2and+ correspond respectively to the Larmor angular velocities
0=B0and=Bin the eldsB0andB. We see in Figure B0,bandB
form a right triangle whose hypotenuse isB; we therefore always haveB B0, which
again shows that+ is always greater than1 2.
For a weak coupling (Fig.-b), the dierence betweenBandB0is very small in
relative value, being of second order inbB0. From this we deduce immediately that
+ and1 2dier in relative value by terms of second order in12(1 2).
On the other hand, for a strong coupling (Fig.-a),Bis much larger thanB0and
practically equal tob;+ is then much larger than1 2and practically
proportional to12. We thus nd again all the results of .
As far as the eect of the coupling on the eigenvectors is concerned, it can also
be understood very simply from Figures . The eigenvectors ofand0are
438

FICTITIOUS SPIN 1/2 ASSOCIATED WITH A TWO-LEVEL SYSTEMb
b
θ
θ
B
0
B
0
B
a b
B
O
O
z z
u
u
Figure 2: Relative disposition of the ctitious eldsB0,bandBin the case of strong
coupling (g. a) and weak coupling (g. b).
associated respectively with the eigenvectors of the components ofSon theand
axes. These two axes are practically parallel in the case of weak coupling (Fig.-b) and
perpendicular in the case of strong coupling (Fig.-a). The eigenvectors ofand,
and, consequently, those ofand0, are very close in the rst case and very dierent
in the second one.
3-c. Geometrical interpretation ofP12()
In terms of ctitious spin, the problem considered in
put in the following way: at time= 0, the ctitious spin associated with the two-level
system is in the eigenstate+of;bis added toB0; what is the probabilityP+()
of nding the spin in the stateat time? With the correspondences summarized in
the table,P12()must be identical toP+().
The calculation ofP+()is then very simple since the time evolution of the spin
reduces to a Larmor precession aboutB(Fig.). During this precession, the angle
between the spin and the directionofBremains constant. At time, the spin points
in the directionOn, making an anglewith; the angle formed by the (Oz, Ou) and
(Ou, On) planes is equal to. A classical formula of spherical trigonometry enables us
to write:
cos= cos
2
+ sin
2
cos (14)
Now, when the spin points in a direction that makes an angle ofwith, the probability
of nding it in the stateofis equal (cf. ) tosin
2
2 =
(1cos)2. From this we deduce, using (14), that:
P+() = sin
2
2
=
1
2
sin
2
(1cos) (15)
439

COMPLEMENT C IVb
θ
θ
α
ωt
B
B
0
O
z
u
n
Figure 3: Geometrical interpretation of
Rabi's formula in terms of ctitious spin.
Under the eect of the coupling (represented
byb), the spin, initially oriented along,
precesses aboutB; consequently, the proba-
bility of nding~2in a measurement of
itscomponent on is an oscillating
function of time.
This result is identical, when we replaceby(+ )~, to formula (C-31) of Chap-
ter
tation.
References and suggestions for further reading:
Abragam (14.1), Chap. II, F; Sargent et al. (15.5), 7-5; Allen (15.7), Chap. 2;
see also the article by Feynman et al. (1.33).
440

SYSTEM OF TWO SPIN 1/2 PARTICLES
Complement DIV
System of two spin 1/2 particles
1 Quantum mechanical description
1-a State space
1-b Complete sets of commuting observables
1-c The most general state
2 Prediction of the measurement results
2-a Measurements bearing simultaneously on the two spins
2-b Measurements bearing on one spin alone
In this complement, we intend to use the formalism introduced in
ter
cated than that of a single spin 1/2 particle. Its interest, as far as the postulates are
concerned, lies in the fact that none of the various spin observables alone constitutes a
C.S.C.O. (while this is the case for one spin alone). Thus, we shall be able to consider
measurements bearing either on one observable with a degenerate spectrum or simulta-
neously on two observables. In addition, this study provides a very simple illustration of
the concept of a tensor product, introduced in . We shall be concerned,
as in Chapter, only with the internal degrees of freedom (spin states), and we shall
moreover assume that the two particles which constitute the system are not identical
(systems of identical particles will be studied in a general way in Chapters
XV).
1. Quantum mechanical description
We saw in Chapter
1/2 particle. Thus, all we need to do is apply the results of
know how to describe systems of two spin 1/2 particles.
1-a. State space
We shall use the indices 1 and 2 to distinguish between the two particles. When
particle (1) is alone, its spin state is dened by a ket which belongs to a two-dimensional
state space(1). Similarly, the spin states of particle (2) alone form a two-dimensional
space(2). We shall designate byS1andS2the spin observables of particles (1) and (2)
respectively. In(1)[or(2)], we choose as a basis the eigenkets of1(or2), which
we shall denote by1 : +and1 :(or2 : +and2 :). The most general ket of
(1)can be written:
(1)=11 : ++11 : (1)
441

COMPLEMENT D IV
and that of(2):
(2)=22 : ++22 : (2)
(1,1,2,2are arbitrary complex numbers).
When we join the two particles to make a single system, the state spaceof such
a system is the tensor product of the two preceding spaces:
=(1) (2) (3)
In the rst place, this means that a basis ofcan be obtained by multiplying tensorially
the two bases dened above for(1)and(2). We shall use the following notation:
+ +=1 : +2 : +
+=1 : +2 :
+=1 :2 : +
=1 :2 : (4)
In the state+, for example, the component alongof the spin of particle (1) is
+~2, with absolute certainty; that of the spin of particle (2) is~2, with absolute
certainty. We shall agree here to denote by+ the conjugate bra of the ket+;
the order of the symbols is therefore the same in the ket and in the bra: the rst symbol
is always associated with particle (1) and the second, with particle (2).
The spaceis therefore four-dimensional. Since the1 :and2 :bases
are orthonormal in(1)and(2)respectively, the basis (4) is orthonormal in:
1212=
1
1
2
2
(5)
(1212are to be replaced by+ordepending on the case;is equal to 1 if
andare identical and 0 if they are dierent). The system of vectors (4) also satises
a closure relation in:
12
1212=+ ++ +++ + +
+ ++ = (6)
1-b. Complete sets of commuting observables
We extend intothe observablesS1andS2which were originally dened in(1)
and(2)(as in Chapter, we shall continue to denote these extensions byS1andS2).
Their action on the kets of the basis (4) is simple: the components ofS1, for example,
act only on the part of the ket related to particle (1). In particular, the vectors of the
basis (4) are simultaneous eigenvectors of1and2:
112=
~
2
112
212=
~
2
212
(7)
442

SYSTEM OF TWO SPIN 1/2 PARTICLES
For the other components ofS1andS2, we apply the formulas given in
ter. For example, we know from relation (A-16) of Chapter 1acts on the
kets1 ::
11 : +=
~
2
1 :
11 :=
~
2
1 : +
(8)
From this we deduce the action of1on the kets (4):
1+ +=
~
2
+
1+=
~
2
1 +=
~
2
+ +
1 =
~
2
+
(9)
It is then easy to verify that, although the three components ofS1(or ofS2) do
not commute with each other,any component ofS1commutes with any component of
S2.
In(1), the observable1alone constituted a C.S.C.O., and the same was true
of2in(2). In, the eigenvalues of1and2remain~2, but each of them
is two-fold degenerate. To the eigenvalue+~2of1, for example, correspond two
orthogonal vectors,+ +and+ [formulas (7)] and all their linear combinations.
Therefore, in, neither1nor2(taken separately) constitutes a C.S.C.O. On the
other hand, the set1 2is a C.S.C.O. in, as can be seen from formulas (7).
This is obviously not the only C.S.C.O. that can be constructed. For example,
another one is1 2. These two observables commute, as we noted above, and
each of them constitutes a C.S.C.O. in the space in which it was initially dened. The
eigenvectors which are common to1and2are obtained by taking the tensor product
of their respective eigenvectors in(1)and(2). Using relation (A-20) of Chapter,
we nd:
1 : +2 : +=
1
2
[+ +++]
1 : +2 :=
1
2
[+ + +]
1 :2 : +=
1
2
[++ ]
1 :2 :=
1
2
[+ ]
(10)
1-c. The most general state
The vectors (4) were obtained by multiplying tensorially a ket of(1)and a ket
of(2). More generally, using an arbitrary ket of(1)[such as (1)] and an arbitrary
443

COMPLEMENT D IV
ket of(2)[such as (2)], we can construct a ket of:
(1)(2)=12+ ++12++21++12 (11)
The components of such a ket in the basis (4) are the products of the components of
(1)and(2)in the bases of(1)and(2)which were used to construct (4).
Butall the kets ofare not tensor products. The most general ket ofis an
arbitrary linear combination of the basis vectors:
=+ ++++ ++ (12)
If we want to normalize, we must choose:
2
+
2
+
2
+
2
= 1 (13)
Given, it is not in general possible to nd two kets(1)and(2)of which it
is the tensor product. For (12) to be of the form (11), we must have, in particular:
= (14)
and this condition is not necessarily fullled.
2. Prediction of the measurement results
We are now going to envisage a certain number of measurements that can be performed
on a system of two spin 1/2 particles and we shall calculate the predictions provided by
the postulates for each of them. We shall assume each time that the state of the system
immediately before the measurement is described by the normalized ket (12).
2-a. Measurements bearing simultaneously on the two spins
Since any component ofS1commutes with any component ofS2, we can envisage
measuring them simultaneously (Chap., ). To calculate the predictions related
to such measurements, all we need to do is use the eigenvectors common to the two
observables.
. First example
Imagine we are simultaneously measuring1and2. What are the probabilities
of the various results that can be obtained?
Since the set1 2is a C.S.C.O., there exists only one state associated with
each measurement result. If the system is in the state (12) before the measurement, we
can therefore nd:
+
~
2
for1and +
~
2
for2 with the probability+ +
2
=
2
+
~
2
~
2
+
2
=
2
~
2
+
~
2
+
2
=
2
~
2
~
2
2
=
2
(15)
444

SYSTEM OF TWO SPIN 1/2 PARTICLES
. Second example
We now measure1and2. What is the probability of obtaining+~2for each
of the two observables?
Here again,1 2constitutes a C.S.C.O. The eigenvector common to1and
2that corresponds to the eigenvalues+~2and+~2is the tensor product of the
vector1 : +and the vector2 : +:
1 : +2 : +=
1
2
[+ ++ +] (16)
Applying the fourth postulate of Chapter, we nd that the probability we are looking
for is:
P=
1
2
[+ + +]
2
=
1
2
2
(17)
The result therefore appears in the form of a square of a sum
1
.
After the measurement, if we have actually found+~2for1and+~2for2,
the system is in the state (16).
2-b. Measurements bearing on one spin alone
It is obviously possible to measure only one component of one of the two spins. In
this case, since none of these components constitutes by itself a C.S.C.O., there exist sev-
eral eigenvectors corresponding to the same measurement result, and the corresponding
probability will be a sum of squares.
. First example
We measure only1. What results can be found, and with what probabilities?
The possible results are the eigenvalues~2of1. Each of them is doubly
degenerate. In the associated eigensubspace, we choose an orthonormal basis: we can,
for example, take+++ for+~2and + for~2. We then obtain:
P+
~
2
=+ +
2
++
2
=
2
+
2
P
~
2
= +
2
+
2
=
2
+
2
(18)
1
It must be remembered that the sign ofchanges when we go from (16) to the conjugate bra. If
this were to be forgotten, the result obtained would be incorrect (+
2
=
2
since is not
in general real).
445

COMPLEMENT D IV
Comment:
Since we are not performing any measurement on the spin (2), the choice of the
basis in(2) is arbitrary. We can, for example, choose as a basis of the eigensub-
space of1associated with the eigenvalue+~2the vectors:
1 : +2 :=
1
2
[+ + +] (19)
which again yields:
P+
~
2
=
1
2
+
2
+
1
2
2
=
2
+
2
(20)
The general proof of the fact that the probability obtained is independent (in the
case of a degenerate eigenvalue) of the choice of the basis in the corresponding
eigensubspace was given in -of Chapter.
. Second example
We now choose to measure2. What is the probability of obtaining~2? The
eigensubspace associated with the eigenvalue~2of2is two-dimensional. We can
choose as a basis in that subspace:
1 : +2 :=
1
2
[+ + +]
1 :2 :=
1
2
[+ ]
(21)
We then nd:
P=
1
2
[+ + +]
2
+
1
2
[+ ]
2
=
1
2
2
+
1
2
2
(22)
In this result, each of the terms of the sum of squares is itself the square of a sum.
If the measurement actually yields~2, the stateof the system immediately
after this measurement is the (normalized) projection ofonto the corresponding
eigensubspace. We have just calculated the components ofon the basis vectors (21)
of this subspace: they are equal, respectively, to
1
2
( )and
1
2
(). Consequently:
=
1
1
2
2
+
1
2
2
1
2
( )(+ + +) +
1
2
( )(+ )(23)
446

SYSTEM OF TWO SPIN 1/2 PARTICLES
Comment:
We have considered, in this complement, only the components ofS1andS2on
the coordinate axes. It is obviously possible to measure their componentsS1u
andS2von arbitrary unit vectorsuandv. The reasoning is the same as above.
447

SPIN12DENSITY MATRIX
Complement EIV
Spin12density matrix
1 Introduction
2 Density matrix of a perfectly polarized spin (pure case)
3 Example of a statistical mixture: unpolarized spin
4 Spin 1/2 at thermodynamic equilibrium in a static eld
5 Expansion of the density matrix in terms of the Pauli ma-
trices
1. Introduction
The aim of this complement is to illustrate the general considerations developed in Com-
plementIII, using a very simple physical system, that of a spin 1/2. We are going
to study the density matrices which describe a spin 1/2 in a certain number of cases:
perfectly polarized spin (pure case), unpolarized or partially polarized spin (statistical
mixture). This will lead us to verify and interpret the general properties stated in Com-
plementIII. In addition, we shall see that the expansion of the density matrix in terms
of the Pauli matrices can be expressed very simply as a function of the mean values of
the various spin components.
2. Density matrix of a perfectly polarized spin (pure case)
Consider a spin 1/2, coming out of an atomic polarizer of the type described in
of Chapter. We assume that it is in the eigenstate+u(eigenvalue+~2) of the
Sucomponent of the spin (recall that the polar angles of the unit vectoruareand
). The spin state is then perfectly well-known and is written [cf.formula (A-22a) of
Chapter]:
= cos
2
e
2
++ sin
2
e
2
(1)
We saw in ComplementIIIthat, by denition, such a situation corresponds to a
pure case. We shall say that the beam which leaves the polarizer is perfectly polarized.
Recall also that, for each spin, the mean valueSis equal to
~
2
u[Chap., relations (B-
14)].
It is simple to write, in the+, basis, the density matrix()correspond-
ing to the state (1). We write the matrix of the projector onto this state:
() =
cos
2
2
sin
2
cos
2
e
sin
2
cos
2
e sin
2
2
(2)
449

COMPLEMENT E IV
This matrix is generally non-diagonal. The populations++and have a very
simple physical signicance. Their dierence is equal tocos= 2 ~[cf.equations
(B-14) of Chapter], and their sum is, of course, equal to1. The populations are
therefore related to the longitudinal polarization. Similarly, the modulus of the
coherences+and +is+= +=
1
2
sin=
1
~
S(whereSis the
projection ofSonto the plane). The argument of+is, which is the angle
betweenSand: the coherences are therefore related to the transverse polarization
S.
It can also be veried that:
[()]
2
=() (3)
a relation characteristic of a pure state.
3. Example of a statistical mixture: unpolarized spin
Now let us consider the spin of a silver atom leaving a furnace, such as the one in Figure
of Chapter, and which has not passed through an atomic polarizer (the spin has
not been prepared in a particular state). The only information we then possess about
this spin is the following: it can point in any direction of space, and all directions are
equally probable. With the notation of ComplementIII, such a situation corresponds
to a statistical mixture of the states+with equal weights. Formula (28) of Comple-
mentIIIdenes the density matrixthat corresponds to this case. Nevertheless, the
discrete summust here be replaced by an integral over all possible directions:
=
1
4
d() =
1
4
2
0
d
0
sind() (4)
(the factor14insures the normalization of the probabilities associated with the various
directions). The integrals which give the matrix elements ofare simple to calculate
and lead to the following result:
=
12 0
0 12
(5)
It is easy to deduce from (5) that
2
=2, which shows that, in the case of a statistical
mixture of states,
2
is dierent from.
In addition, if we calculate from (5) the mean values of,,, we obtain:
= Tr =
1
2
Tr = 0with:= (6)
We again nd the fact that the spin is unpolarized: since all the directions are equivalent,
the mean value of the spin is zero.
450

SPIN12DENSITY MATRIX
Comments:
()
can disappear from the summation over the various states of the statistical
mixture. As we saw in 2, the coherences+and +are related to the
transverse polarizationSof the spin. Upon summing the vectorsS
corresponding to all (equiprobable) directions of theplane, we obviously
nd a null result.
()
understand the impossibility of describing a statistical mixture by an average
state vector. Assume that we are trying to chooseandso that the vector:
=++ (7)
with:
2
+
2
= 1 (8)
represents an unpolarized spin, for which, and are zero. A
simple calculation gives:
=
~
2
(+ )
=
~
2
( )
=
~
2
( )
(9)
If we want to make zero, we must chooseandso as to make
a pure imaginary; similarly,must be real forto be zero. We must
therefore have= 0, that is:
either= 0, which implies= 1and =~2
or= 0, which implies= 1and =~2
Therefore,, and cannot all be zero at the same time; conse-
quently, the state (7) cannot represent an unpolarized spin.
Furthermore, the discussion of
andthat satisfy (8), one can always associate with them two anglesand
xing a directionusuch thatis an eigenvector ofSuwith the eigenvalue
+~2. Thus we see directly that a state such as (7) always describes a spin
which is perfectly polarized in a certain direction of space.
() 5) represents a statistical mixture of the various states
+u, all the directionsubeing equiprobable (this is how we obtained it). We
could, however, imagine other statistical mixtures which would lead to the
same density matrix: for example, a statistical mixture of equal proportions
of the states+and, or a statistical mixture of equal proportions of three
states+u, such that the tips of the three corresponding vectorsuare the
vertices of an equilateral triangle centered at. Thus we see that the same
451

COMPLEMENT E IV
density matrix can be obtained in several dierent ways. In fact, since all the
physical predictions depend only on the density matrix, it is impossible to
distinguish physically between the various types of statistical mixtures that
lead to the same density matrix. They must be considered to be dierent
expressions of the same incomplete information we possess about the system.
4. Spin 1/2 at thermodynamic equilibrium in a static eld
Consider a spin 1/2 placed in a static eldB0parallel to. We saw in
Chapter +and, of energies
+~02and~02(with0= 0, whereis the gyromagnetic ratio of the spin).
If we know only that the system is in thermodynamic equilibrium at the temperature,
we can assert that it has a probability
1
e
~02
of being in the state+and
1
e
+~02
of being in the state, where= e
~02
+ e
+~02
is a normalization
factor (is called the partition function). We have here another example of a statistical
mixture, described by the density matrix:
=
1
e
~02
0
0 e
+~02
(10)
Once more, it is easy to verify that
2
=. The non-diagonal elements are zero since all
directions perpendicular toB0(that is, to) and xed by the angleare equivalent.
From (10), it is easy to calculate:
= Tr() = 0
= Tr() = 0
= Tr() =
~
2
tanh
~0
2
(11)
We see that the spin acquires a polarization parallel to the eld in which it is placed. The
larger0(that is,0) and the lower the temperature, the greater the polarization.
Sincetanh 1, this polarization is less than the value~2corresponding to a spin
that is perfectly polarized along. The density matrix (10) can therefore be said to
describe a spin which is partially polarized along.
Comment:
The magnetization is equal to. It is possible to calculate from (11) the
paramagnetic susceptibilityof the spin, dened by:
= = 0 (12)
We nd (Brillouin's formula):
=
~
20
tanh
~0
2
(13)
452

SPIN12DENSITY MATRIX
5. Expansion of the density matrix in terms of the Pauli matrices
We saw in ComplementIVthat the unit matrixand the Pauli matrices,and
form a convenient basis for expanding a22matrix. We therefore set, for the density
matrixof a spin12:
=0+a (14)
where the coecientsare given by [cf.ComplementIV, relations (23)]:
0=
1
2
Tr
=
1
2
Tr =
1
~
Tr
=
1
2
Tr =
1
~
Tr
=
1
2
Tr =
1
~
Tr (15)
Thus we have:
0=
1
2
a=
1
~
S (16)
andcan be written:
=
1
2
+
1
~
S (17)
Therefore, the density matrixof a spin 1/2 can be expressed very simply in terms
of the mean valueSof the spin.
Comment:
Let us square expression (17). We obtain, using identity (17 IV:
2
=
1
4
+
1
~
2
S
2
+
1
~
S (18)
The condition
2
=, characteristic of the pure case, is therefore equivalent, for a
spin12, to the condition:
S
2
=
~
2
4
(19)
This condition is obviously not satised for an unpolarized spin (Sis then zero)
or for a spin in thermodynamic equilibrium (we saw in 4 that in this case
S ~2). On the other hand, it can be veried, using formulas (B-14) of
Chapter, that, for a spin in the stategiven in (1),S
2
is indeed equal to
~
2
4.
References and suggestions for further reading:
Abragam (14.1), Chap. II, C.
453

SPIN 1/2 PARTICLE IN A STATIC AND A ROTATING MAGNETIC FIELDS: MAGNETIC RESONANCE
Complement FIV
Spin 1/2 particle in a static and a rotating magnetic elds:
magnetic resonance
1 Classical treatment; rotating reference frame
1-a Motion in a static eld; Larmor precession
1-b Inuence of a rotating eld; resonance
2 Quantum mechanical treatment
2-a The Schrödinger equation
2-b Changing to the rotating frame
2-c Transition probability; Rabi's formula
2-d Case where the two levels are unstable
3 Relation between the classical treatment and the quantum
mechanical treatment: evolution ofM. . . . . . . . . . . .
4 Bloch equations
4-a A concrete example
4-b Solution in the case of a rotating eld
In Chapter, we used quantum mechanics to study the evolution of a spin12
in a static magnetic eld. In this complement, we shall consider the case of a spin12
simultaneously subjected to several magnetic elds, some of which can be time dependent,
as is the case in magnetic resonance experiments. Before attacking this problem quantum
mechanically, we shall briey review several results obtained using classical mechanics.
1. Classical treatment; rotating reference frame
1-a. Motion in a static eld; Larmor precession
Consider a system with angular momentumjthat possesses a magnetic moment
m=jcollinear withj(the constantis the gyromagnetic ratio). The system is placed
in a static magnetic eldB0, which exerts a torquemB0on the system. The classical
equation of motion ofjis:
dj
d
=mB0 (1)
or:
d
d
m() =m()B0 (2)
Performing a scalar multiplication of both sides of this equation by eitherm()orB0,
we obtain:
d
d
[m()]
2
= 0 (3)
455

COMPLEMENT F IVZ
e
Z
e
x
e
X
e
y
e
Y
e
z
B
0
B
1
z
Y
O
x
X
ωt
y
Figure 1: is a xed coordinate system. The static magnetic eldB0is directed
along theaxis. The system [theaxis is in the direction of the eldB1()]
rotates aboutwith the angular velocity.
d
d
[m()B0] = 0 (4)
m()therefore evolves with a constant modulus, maintaining a constant angle withB0.
If we project equation (2) onto the plane perpendicular toB0, we see thatm()rotates
aboutB0(Larmor precession) with an angular velocity of0= 0(the rotation is
clockwise ifis positive).
1-b. Inuence of a rotating eld; resonance
Now assume that we add to the static eldB0a eldB1(), perpendicular toB0,
and which is of constant modulus and rotates aboutB0with an angular velocity(cf.
Fig.). We set:
0= 0
1= 1
(5)
We shall designate by (unit vectorse,e,e) a xed coordinate system,
whose axis is the direction of the eldB0, and by (unit vectorse,e,
e), the axes obtained fromOxyzby rotation through an angleabout[is the
direction of the rotating eldB1()]. The equation of motion ofm()in the presence of
the total eldB() =B0+B1()then becomes:
d
d
m() =m()[B0+B1()] (6)
456

SPIN 1/2 PARTICLE IN A STATIC AND A ROTATING MAGNETIC FIELDS: MAGNETIC RESONANCE
To solve this equation, it is convenient to place ourselves, not in the laboratory
reference frame, but in the rotating reference frame, with respect to which
the relative velocity of the vectorm()is:
dm
d
rel
=
dm
d
em() (7)
Let us set:
= 0 (8)
Substituting (6) into (7), we obtain:
dm
d
rel
=m()[e 1e] (9)
This equation is much simpler to solve than equation (6), since the coecients of the
right-hand side are now time-independent. Moreover, its form is analogous to that of (2):
the relative motion of the vectorm()is therefore a rotation about the eective eld
Be(which is static with respect to the rotating reference frame), given by (cf.Fig.):
Be=
1
[e 1e] (10)
To obtain the absolute motion ofm(), we must combine this precession aboutBewith
a rotation aboutof angular velocity.B
eff
B
1
m
Z
Y
X
O
ω – ω
0
γ
γ
–=
ω
1
Figure 2: In the rotating reference frame
, the eective eldBehas a xed
direction, about which the magnetic moment
m()rotates with a constant angular velocity
(precession in the rotating reference frame).
These rst results already enable us to understand the essence of the magnetic
resonance phenomenon. Let us consider a magnetic moment which, at time= 0, is
457

COMPLEMENT F IV
parallel to the eldB0(the case, for example, of a magnetic moment in thermodynamic
equilibrium at very low temperatures: it is in the lowest energy state possible in the
presence of the eldB0). What happens when we apply a weak rotating eldB1()?
If the rotation frequency2of this eld is very dierent from the natural frequency
02(more precisely, if= 0is much larger than1), the eective eld is
directed practically along. The precession ofm()aboutBethen has a very small
amplitude and hardly modies the direction of the magnetic moment. On the other hand,
if the resonance condition 0is satised ( 1), the angle between the eld
Beandis large. The precession of the magnetic moment then has a large amplitude
and, at resonance (= 0), the magnetic moment can even be completely ipped.
2. Quantum mechanical treatment
2-a. The Schrödinger equation
Let+and be two eigenvectors of the projectionof the spin onto, with
respective eigenvalues+~2and~2. The state vector of the system can be written:
()=+()++() (11)
The Hamiltonian operator()of the system is
1
:
() =MB() =S[B0+B1()] (12)
that is, expanding the scalar product:
() =0+1[cos + sin ] (13)
Using formulas (A-16) and (A-17) of Chapter, we obtain the matrix that represents
in the+ basis:
=
1
2
0 1e
1e 0
(14)
Using (11) and (14), we can write the Schrödinger equation in the form:
d
d
+() =
0
2
+() +
1
2
e ()
d
d
() =
1
2
e +()
0
2
()
(15)
1
In expression (12),MB()symbolizes the scalar product () + () + (), where
and are operators (observables of the system under study), while()()and()
are numbers (since we consider the magnetic eld to be a classical quantity whose value is imposed by
an external device, independent of the system under study).
458

SPIN 1/2 PARTICLE IN A STATIC AND A ROTATING MAGNETIC FIELDS: MAGNETIC RESONANCE
2-b. Changing to the rotating frame
Equations (15) constitute a linear homogeneous system with time-dependent coef-
cients. It is convenient to dene new functions by setting:
+() = e
2
+()
() = e
2
() (16)
Substituting (16) into (15), we obtain a system which has constant coecients:
d
d
+() =

2
+() +
1
2
()
d
d
() =
1
2
+() +

2
()
(17)
This system can also be written:
~
d
d
()= () (18)
if we introduce the ket()and the operatordened by:
()=+()++() (19)
=
~
2
1
1
(20)
Transformation (16) has led to equation (18), which is analogous to a Schrödinger
equation in which the operator, given in (20), plays the role of a time-independent
Hamiltonian.describes the interaction of the spin with axedeld, whose components
are none other than those of the eective eld introduced above in theframe
[formula (10)]. We can therefore consider that the transformation (16) is the quantum
mechanical equivalent of the change from the xedframe to the rotating
frame.
This result can be proved rigorously. According to (16), we can write:
()=()() (21)
where()is the unitary operator dened by:
() = e
~
(22)
We shall see later (cf.ComplementVI) that()describes a rotation of the coordinate system
through an angleabout. (18) is therefore indeed the transformed Schrödinger equation
in the rotating frame.
Equation (18) is very simple to solve. To determine(), given(0), all we
need to do is expand(0)on the eigenvectors of(which can be calculated exactly)
and then apply rule (D-54) of Chapter is not explicitly
time-dependent). We then go from()to()by using formulas (16).
459

COMPLEMENT F IV
2-c. Transition probability; Rabi's formula
Consider a spin which, at time= 0, is in the state+:
(0)=+ (23)
According to (16), this corresponds to:
(0)=+ (24)
What is the probabilityP+()of nding this spin in the stateat time? Since
()and()have the same modulus, we can write:
P+() = ()
2
= ()
2
=()
2
=
2
(25)
We must therefore calculate()
2
, where()is the solution of (18) that corre-
sponds to the initial condition (24).
The problem we have just posed has already been solved, in
ter. To use the calculations of that section, all we need to do is apply the following
correspondences:
1 +
2
1
~
2

2
~
2

12
~
2
1
(26)
Rabi's formula [equation (C-32) of Chapter] then becomes:
P+() =
2
1
2
1
+ ()
2
sin
2
2
1
+ ()
22
(27)
The probabilityP+()is zero at time= 0, and then varies sinusoidally with time
between the values 0 and
2
1
2
1
+()
2. Again, we have a resonance phenomenon. For
1,P+()remains almost zero (cf.Fig.-a); near resonance, the oscillation
amplitude ofP+()becomes large and, when the condition= 0is exactly satised,
we haveP+() = 1at times=
(2+1)
1
(cf.Fig.-b).
Thus we again nd the result which we have already obtained classically: at reso-
nance, a very weak rotating eld is able to reverse the direction of the spin. Note, more-
over, that the angular frequency of the oscillation ofP+()is
2
1
+ ()
2
=Be.
This oscillation corresponds, in the rotating frame, to the projection ontoof the
precession of the magnetic moment about the eective eld, sometimes called Rabi
precession [see also the calculation ofP+()in ComplementIV, ].
460

SPIN 1/2 PARTICLE IN A STATIC AND A ROTATING MAGNETIC FIELDS: MAGNETIC RESONANCEl l
l
t t
l0
0 0
a b
ω – ω
0
= 3ω
1 ω

= ω
0
2π/ω
1
+ –
(t)
+ –
(t)
Figure 3: Variation with respect to time of the transition probability between the states+
and, under the eect of a rotating magnetic eld1(). Outside resonance (g.a),
this probability remains small; at resonance (g.b), however small the eld1, there
exist times when the transition probability is equal to1.
2-d. Case where the two levels are unstable
We are now going to assume that the two statescorrespond to two sublevels of an
excited atomic level (whose angular momentum is assumed equal to12).atoms are excited
2
per unit time, all being raised to the state+. Each atom decays, by spontaneous emission of
radiation, with a probability per unit time of1, which is the same for the two sublevels.
We know that, under these conditions, an atom which was excited at timehas a probability
e of still being excited at time= 0(cf.ComplementIII).
We assume that the experiment is performed in the steady state: in the presence of the
eldsB0andB1(), the atoms are excited at a constant rateinto the state+. After a time
much longer than the lifetime, what is the numberof atoms which decay per unit time from
the state? If an atom is excited at time, the probability of nding it in the stateat
= 0iseP+(), whereP+()is given by relation (27). The total number of atoms in
the stateis obtained by taking the sum of atoms excited at all previous times, that is,
by calculating the integral:
0
eP+()d (28)
This calculation presents no diculties. Multiplying the number of atoms thus obtained by
their probability1of decay per unit time, we obtain:
=
2
2
1
()
2
+
2
1
+ (1)
2
(29)
2
In practice, this excitation can be produced, for example, by placing the atoms in a light beam.
When the incident photons are polarized, conservation of angular momentum, in certain cases, requires
that the atoms which absorb them can attain only the state+(and not the state). Similarly, by
detecting the polarization of the photons re-emitted by the atoms, one can know whether the atoms fall
back into the ground state from the state+or the state.
461

COMPLEMENT F IVN
00
B
0
– ω/γ
2L
γ
Figure 4: Resonance curve. To observe a
resonance phenomenon, we perform an ex-
periment in whichatoms are excited per
unit time into the state+. Under the ef-
fect of a eldB1(), rotating at the frequency
2, the atoms undergo transitions towards
the state. In the steady state, if we mea-
sure the numberof atoms which decay per
unit time from the state, we obtain a res-
onant variation when we scan the static eld
0about the value.
The variation ofwith respect tocorresponds to a Lorentz curve whose half-width is:
=
2
1
+ (1)
2
(30)
In the experiment described above, let us measure, for various values of the magnetic
eld0(that is, withassumed to be xed, for various values of), the number of atoms
which decay from the level. According to (29), we obtain a resonance curve which has the
shape shown in Figure.
It is very interesting to obtain such a curve experimentally, since one can use it to deter-
mine several parameters:
and measure the value
0of the eld0corresponding to the peak of
the curve, we can deduce the value of the gyromagnetic ratiothrough the relation
=
0.
, we can, by measuring the frequency2corresponding to resonance, mea-
sure the static magnetic eld0. Various magnetometers, often of very great precision,
operate on this principle. In certain cases, one can derive interesting information from
such a measurement of the eld. If, for example, the spin being considered is that of a
nucleus which belongs to a molecule or to a crystal lattice, one can nd the local eld
seen by the nucleus, its variation with the site occupied, etc.

2
of the half-width as a function of
2
1, we obtain a straight line
which, extrapolated to1= 0, gives the lifetimeof the excited level (cf.Fig.).L
2
0
1
ω
t
2
τ
2
Figure 5: The extrapolation to1= 0of the
squared half-widthof the resonance curve
of Figure
ing studied.
462

SPIN 1/2 PARTICLE IN A STATIC AND A ROTATING MAGNETIC FIELDS: MAGNETIC RESONANCE
3. Relation between the classical treatment and the quantum mechanical
treatment: evolution ofM
The results obtained in
in one case and quantum mechanics in the other. We are now going to show that this
similarity is not accidental. It arises from the fact that the quantum mechanical evolution
equations of the mean value of a magnetic moment placed in an arbitrary magnetic eld
are identical to the corresponding classical equations.
The mean value of the magnetic moment associated with a spin12is:
M() =S() (31)
To calculate the evolution ofM(), we use theorem (D-27) of Chapter:
~
d
d
M() =[M ()] (32)
where()is the operator:
() =MB() (33)
Let us calculate for example the commutator[ ()]. Using the fact that the eld
components()and()are numbers (cf.note), we nd:
[, H(t)] =
2
[ () + () + ()]
=
2
()[ ]
2
()[ ] (34)
Using relations (14) of ComplementIV, we obtain:
[ ()] =~
2
[() ()] (35)
Substituting (35) into (32):
d
d
() =[() () () ()] (36)
By cyclic permutation, we can calculate analogous expressions for the components on
and; the three equations obtained can be condensed into:
d
d
M() =M()B() (37)
Let us compare (37) with (6): the evolution of the mean valueM()obeys the
classical equations exactly, whatever the time-dependence of the magnetic eldB().
4. Bloch equations
In practice, in a magnetic resonance experiment, it is not the magnetic moment of a single
spin that is observed, but rather that of a great number of identical spins (as in the experiment
described in is detected).
Moreover, one is not concerned solely with the quantityP+(), calculated above. One can also
measure the global magnetizationMMMof the sample under study: the sum of the mean values
463

COMPLEMENT F IVE
F
A
C
Figure 6: Schematic drawing of an experimental device which supplies cellwith atoms
in the state+.
of the observableMcorresponding to each spin of the sample
3
. It is interesting, therefore, to
obtain the equations of motion ofMMM, called theBloch equations.
In order to understand the physical signicance of the various terms appearing in these
equations, we are going to derive them for a simple concrete case. The results obtained can be
generalized to other more complicated situations.
4-a. A concrete example
Consider a beam of atoms coming from an atomic polarizer of the type studied in
1-a . All the atoms of the beam
4
are in the spin state+and therefore have
their magnetic moments parallel to. They enter a cellthrough a small opening (Fig.),
rebound a certain number of times from the inside walls of the cell and, after a certain time,
escape through the same opening.
We shall denote bythe number of polarized atoms entering the cell per unit time;is
generally small and the atomic density inside the cell is low enough to allow atomic interactions
to be neglected. Moreover, if the inside walls of the cell are suitably coated, collisions with
the walls have little eect on the spin state of the atoms
5
. We shall assume that there is a
probability per unit time1 for the elementary magnetization introduced into the cell by a
polarized atom to disappear, either because of a depolarizing collision with the walls or simply
because the atom has left the cell.is called the relaxation time. The cell is placed in a
magnetic eldB()which may have a static component and a rotating component. The problem
consists of nding the equation of motion of the global magnetizationMMMMMMMMM()of the atoms which
are inside the cell at time.
First, let us write the exact expression forMMMMMMMMM():
MMMMMMMMM() =
N
=1
()
()M
()
()=
N
=1
MMM
()
() (38)
In (38), the sum is taken over theNspins which are already in the cell and which, at time,
have neither left nor undergone a depolarizing collision.
()
()is the state vector of such a
spin()at time[we are not counting, in (38), the spins which have undergone a depolarizing
3
It is possible to detect, for example, the electromotive force emf induced in a coil by the variation
ofMMMwith respect to time.
4
For example, silver or hydrogen atoms in the ground state. For the sake of simplicity, all eects
related to nuclear spin are neglected.
5
For example, for hydrogen atoms bouncing o teon walls, tens of thousands of collisions are
required for the magnetic moment of the hydrogen atom to become disoriented.
464

SPIN 1/2 PARTICLE IN A STATIC AND A ROTATING MAGNETIC FIELDS: MAGNETIC RESONANCE
collision and have not yet left the cell, since their global contribution is zero: their spins point
randomly in all directions].
Between timesand+ d,MMM()varies for three dierent reasons:
()A certain proportion,d , of theNspins undergo a depolarizing collision or leave the
compartment; these spins disappear from the sum (38) andMMM()therefore decreases by:
dMMM() =
d
MMM() (39)
()The other spins evolve freely in the eldB(). We saw in
the evolution of the mean value ofM:
MMM
()
() =
()
()M
()
()
obeys the classical law:
dMMM
()
() =MMM
()
()B()d (40)
Since the right-hand side of (40) is linear with respect toMMM
()
(), the contribution of
these spins to the variation ofMMM()is given by:
dMMM() =MMM()B() d (41)
() d, of new spins have entered the cell. Each of them adds to
the global magnetization a contribution0, equal to the mean value ofMin the state
+(0is parallel toand0=~2).MMMtherefore increases by:
dMMM() =0d (42)
The global variation ofMMMis obtained by adding (), () and (42). Dividing byd, we
obtain the equation of motion ofMMM()(Bloch equation):
d
d
MMM() =0
1
MMM() +MMM()B() (43)
We have derived (43) in a specic case, making certain hypotheses. However, the main
features of this equation remain valid for a great number of other experiments where the rate
of variation ofMMM()appears in the form of a sum of three terms:
0) which describes the preparation of the system. It would, in fact,
be impossible to observe magnetic resonance without a preliminary polarization of the
spins, which can be achieved through selection using a magnetic eld gradient (as in the
example studied here), a polarized optical excitation (as in the example studied in
above), cooling of the sample in a strong static eld, etc.
here
1
MMM()which describes the disappearance or relaxation of
the global magnetization under the eect of various processes: collisions, disappearance of
atoms, change in atomic levels through spontaneous emission (as in the example studied
in ), etc.
MMM()in the eldB()[last term of (43)].
465

COMPLEMENT F IV
4-b. Solution in the case of a rotating eld
When the eldB()is the sum of a static eldB0and a rotating eldB1(), such as
those considered above, equations (43) can be solved exactly. As in , one changes to
the rotating frame , with respect to which the relative variation ofMMM()is:
d
d
MMM
rel
=0
1
MMM+MMMBe (44)
whereBeis dened by equation (10).(
X
)
s
(
Y
)
s
(
Z
)
s
0
2
0
0
ω
1
Δω
Δω
Δω
Figure 7: Variation with respect to= 0of the stationary values of the components
ofMMMin the rotating frame. One obtains a dispersion curve for(M)and absorption
curves for(M)and(M). The three curves have the same width,2
2
1
+ (1)
2
,
which increases with1. They have been drawn assuming that 1= 1 (half-
saturation).
Projecting this equation onto, and , we obtain a system of three linear
dierential equations with constant coecients, whose stationary solution (valid after a time
466

SPIN 1/2 PARTICLE IN A STATIC AND A ROTATING MAGNETIC FIELDS: MAGNETIC RESONANCE
much greater than) is:
(M)= 0
1
()
2
+
2
1
+ (1)
2
(M)= 0
1
()
2
+
2
1
+ (1)
2
(M)= 0 1
2
1
()
2
+
2
1
+ (1)
2
(45)
The three components of the stationary magnetization(MMM), when the eld0varies, have
resonant variations about the value0= (cf.Fig.).(M)and (M)follow absorp-
tion curves (Lorentz curves of width2
2
1
+ (1)
2
).(MMM)follows a dispersion curve
(of the same width).
References and suggestions for further reading:
Feynman II (7.2), Chap. 35; Cagnac and Pebay-Peyroula (11.2), Chaps. IX 5, X
5, XI 2 to 5, XIX 3; Kuhn (11.1), VI, D.
See the references of section 14 of the bibliography, particularly Abragam (14.1)
and Slichter (14.2).
467

A SIMPLE MODEL OF THE AMMONIA MOLECULE
Complement GIV
A simple model of the ammonia molecule
1 Description of the model
2 Eigenfunctions and eigenvalues of the Hamiltonian
2-a Innite potential barrier
2-b Finite potential barrier
2-c Evolution of the molecule. Inversion frequency
3 The ammonia molecule considered as a two-level system
3-a The state space
3-b Energy levels. Removal of the degeneracy due to the trans-
parency of the potential barrier
3-c Inuence of a static electric eld
1. Description of the model
In the ammonia molecule NH3, the three hydrogen atoms form the base of a pyramid
whose apex is the nitrogen atom (cf.Fig.). We shall study this molecule by using
a simplied model with the following features: the nitrogen atom (of mass), much
heavier than its partners (of mass), is motionless; the hydrogen atoms form a rigid
equilateral triangle whose axis always passes through the nitrogen atom. The potential
energy of the system is thus a function of only one parameter, the (algebraic) distance
between the nitrogen atom and the plane dened by the three hydrogen atoms
1
. Thex
H
H
H
N0
Figure 1: Schematic drawing of the ammo-
nia molecule;is the algebraic distance be-
tween the plane of the hydrogen atoms and
the nitrogen atom, which is assumed to be
motionless.
shape of this potential energy()is given by the solid-line curve in Figure. The
1
In this one-dimensional model, eects linked to the rotation of the molecule are obviously not taken
into account.
469

COMPLEMENT G IV
symmetry of the problem with respect to the= 0plane requires()to be an even
function of. The two minima of()correspond to two symmetrical congurations of
the molecule in which, classically, it is stable; we shall choose the energy origin such that
its energy is then zero. The potential barrier at= 0, of height1, expresses the fact
that, if the nitrogen atom is in the plane of the hydrogen atoms, they repel it. Finally,
the increase in()whenis greater thancorresponds to the chemical bonding force
which insures the cohesion of the molecule.– b
V
0
V
1
V (x)
x
a a
+ b0
Figure 2: Variation with respect toof the potential energy()of the molecule.()
has two minima (classical equilibrium positions), separated by a potential barrier due to
the repulsion for smallbetween the nitrogen atom and the three hydrogen atoms. The
square potential used to approximate()is shown in dashed lines.
This model therefore reduces the problem to a one-dimensional one in which a
ctitious particle of mass(it can be shown that the reduced massof the system is
equal to
3
3 +
) is under the inuence of the potential(). Under these conditions,
what are the energy levels predicted by quantum mechanics? With respect to classical
predictions, two major dierences appear:
(i) The Heisenberg uncertainty relation forbids the molecule to have an energy
equal to the minimum of()( = 0in our case). We have already seen, in Com-
plementsIandIIIwhy this energy must be greater thanmin.
()Classically, the potential barrier at= 0cannot be cleared by a particle
whose energy is less than1: the nitrogen atom thus always remains on the same side
of the plane of the hydrogen atoms, and the molecule cannot invert itself. Quantum
mechanically, such a particle can cross this barrier by the tunnel eect (cf.Chap.,
): the inversion of the molecule is therefore always possible. We are going to
discuss the consequences of this eect.
We are concerned here only with a qualitative discussion of the physical phenomena
and not with an exact quantitative calculation which would not have much signicance
in this approximate model. For example, we shall try to demonstrate the existence of
an inversion frequency of the ammonia molecule, without giving an exact or even an
approximate value of this frequency. We shall therefore simplify the problem further by
replacing the function()by the square potential drawn in dashed lines in Figure
innite potential steps at=(+2)and a potential barrier of height0centered
at= 0and of width (2 )].
470

A SIMPLE MODEL OF THE AMMONIA MOLECULE
2. Eigenfunctions and eigenvalues of the Hamiltonian
2-a. Innite potential barrier
Before calculating the eigenfunctions and eigenvalues of the Hamiltonian corre-
sponding to the square potential of Figure, we are going to assume, in this rst
stage, that the potential barrier0is innite (in which case, no tunnel eect is possible).
This will lead us to a better understanding of the consequences of the tunnel eect across
the nite potential barrier of Figure. We shall therefore consider, to begin with, a par-
ticle in a potential()composed of two innite wells of widthcentered at=
(Fig.). If the particle is in one of these two wells, it obviously cannot go into the other
one.– b + b
V(x)
~
0
a a
Figure 3: When the height0of the poten-
tial barrier of Figure
practically innite potential wells of width
whose centers are separated by a distance of
2.
Each of the two wells of Figure I,
in . We can therefore use the results obtained in that complement. The possible
energies of the particle are:
=
~
22
2
(1)
with:
=
(2)
(whereis a positive integer). Each of the energy values is twofold degenerate, since
two wave functions correspond to it:
1() =
2
sin +
2
if
2
+
2
0 everywhere else
2() =
2
sin +
2
+ if
2
+
2
0 everywhere else
(3)
471

COMPLEMENT G IV
In the state
1, the particle is in the innite well on the right; in the state
2, it is
in the one on the left.
Figure
degenerate. The Bohr frequency(2 1)associated with these two levels corresponds,
as we saw in ComplementIII(), to the to-and-fro motion of the particle between the
two sides of the well on the right (or on the left) when its state is a linear superposition of
1
1and
2
1(or of
1
2and
2
2). Physically, such an oscillation represents a molecular
vibration of the plane of the three hydrogen atoms about its stable equilibrium position,
which corresponds to= +(or=). The frequency of this oscillation falls in the
infrared part of the spectrum.E
2
4π
2
ħ
2
2ma
2
=
E
1
E
2
–

E
1
h
E
0
ħ
2
π
2
2ma
2
=
ν =
Figure 4: First energy levels obtained in the potential wells of Figure. The oscillation
of the system in one of the two wells at the Bohr frequency= (2 1)represents
the vibration of the molecule about one of its two classical equilibrium positions.
In the rest of the calculation, it is convenient to change bases, in each of the
eigensubspaces of the Hamiltonian of the particle. Since the function()is even, this
Hamiltoniancommutes with the parity operator(cf.ComplementII, ). In this
case, we can nd a basis of eigenvectors ofthat are even or odd; the wave functions
of these vectors are the symmetrical and antisymmetrical linear combinations:
() =
1
2
[
1() +
2()]
() =
1
2
[
1()
2()]
(4)
In the statesand , the particle can be found in one or the other of the two
potential wells.
In what follows, we shall conne ourselves to the study of the ground state, for
which the wave functions
1
1(),
1
2(),
1
()and
1
() are shown in Figure.
472

A SIMPLE MODEL OF THE AMMONIA MOLECULE– b
a
b
b
x
– b
0
0
0
b
x
x
0
x
φ
1 (x)
1
φ
2 (x)
1
φ
s (x)
1
φ
a (x)
1
Figure 5: The states
1
1()and
1
2(), shown in gurea, are stationary states with
the same energy, respectively localized in the right-hand well and the left-hand well of
Figure. To use the symmetry of the problem, it is more convenient to choose as sta-
tionary states the symmetrical state
1
()and the antisymmetrical state
1
(), linear
combinations of
1
1()and
1
2()(gureb)
.
2-b. Finite potential barrier
Let us try to nd the shape of the eigenfunctions of the rst energy levels when0
has a nite value (assumed, nevertheless, to be greater than the energy of these levels).
Inside the two square potential wells (dashed lines in Figure),() = 0. The
wave function is therefore of the form:
() =sin +
2
if
2
+
2
() =sin +
2
+ if
2
+
2
(5)
473

COMPLEMENT G IV
whereis related to the energyof the level by the relation:
=
~
22
2
(6)
As in the preceding paragraph,()always goes to zero at=(+2), since()
becomes innite at these two points. On the other hand, since0is nite,()no longer
goes to zero at=( 2); consequently,no longer satises relation (2).
Once again, since()is even, we can look for eigenfunctions of the Hamiltonian,
()and(), which are respectively even and odd. Let us denote byand,
and the values of the coecientsand, introduced in (5), which correspond to
()and(). We have, obviously:
=
=
(7)
The eigenvalues associated withandwill be denoted byand, which enables
us, using (6), to dene the corresponding valuesandof the parameter.
In the interval( 2) +( 2), the wave function is no longer zero, as
it was before, since0is nite. It must be a linear combination, even or odd depending
on whether we are consideringor, of exponentials eande ;andare
dened in terms ofand0by:
=
2~
2
(0 ) =
2 2
(8)
with:
0=
~
22
2
(9)
Therefore, for( 2) ( 2), the functionsandare written:
() =cosh()
() =sinh()
(10)
Finally, we must match the eigenfunctions and their derivatives at=( 2).
The even solution()must therefore satisfy the conditions:
sin() =cosh
2
cos() = sinh
2
(11)
Sinceandcannot be zero simultaneously, we can take the ratio of equations (11)
to obtain:
tan() =
coth
2
(12)
For the odd solution(), we obtain in the same way:
tan() =
tanh
2
(13)
474

A SIMPLE MODEL OF THE AMMONIA MOLECULE
Ifandare replaced by their values in terms ofand, relations (12) and (13)
can be written:
tan() =
2 2
coth
2 2
2
(14)
and:
tan() =
2 2
tanh
2 2
2
(15)
In theory, therefore, the problem is solved. Relations (14) and (15) express the
energy quantization since they give the possible values ofandand therefore, thanks
to relation (6), the energiesand (with the condition that they be less than0).
The transcendental equations (14) and (15) can be solved graphically. A certain number
of roots are found:
1
,
2
,...,
1
,
2
... The rootis dierent from, since equations
(14) and (15) are not the same:the energiesand are therefore dierent. Of
course, when0becomes very large,andboth approach the value found in
the preceding section; this can be seen by lettingapproach innity in equations (14)
and (15), which yieldstan() = 0, an equation equivalent to (2). The energies
and therefore approach the value=~
222
2
2
calculated in the preceding
section for0approaching innity. Finally, it is easy to see that, the more0exceeds
, the closer together the two energiesand will be.
The exact values ofandare of little importance to us here. We shall content
ourselves with sketching the shape of the energy spectrum in Figure, which shows what
happens to the energies of levels1and2of Figure 0of the
potential barrier is taken into account. We see that the tunnel eect across this barrier
removes the degeneracy of1and2, giving rise to doublets, (
1
,
1
) and (
2
,
2
)
(assuming, of course, that all these energies are less than0). Since the (
1
,
1
) doublet
is the deeper one, it is clear that
1 1 2 2
. Finally, the distance between
the doublets is much greater than the spacing within each doublet (experimentally, their
ratio is of the order of a thousand). These spacings enable us, moreover, to dene new
Bohr (angular) frequencies:
1=
1 1
~
2=
2 2
~
whose physical signicance we shall study in the next paragraph (the corresponding
transitions are represented by arrows in Figure).
Figure
1
()and
1
(), which are given by
equations (5), (7) and (10) once
1
and
1
have been determined from (14) and (15). We
see that they greatly resemble the functions
1
()and
1
()of Figure, the essential
dierence being that the wave function is no longer zero in the interval( 2)
( 2). The reason for introducing the
1
and
1
basis in the preceding paragraph can
now be understood: the eigenfunctions
1
and
1
, in the presence of the tunnel eect,
resemble
1
and
1
much more than
1
1and
1
2.
475

COMPLEMENT G IV
2-c. Evolution of the molecule. Inversion frequency
Assume that at time= 0, the molecule is in the state:
(= 0)=
1
2
1
+
1
(16)
The state vector()at timecan be obtained by using the general formula (D-54) of
Chapter; we obtain:
()=
1
2
e
1
+
1
2~ e
+121
+ e
121
(17)
From this we deduce the probability density:
()
2
=
1
2
1
()
2
+
1
2
1
()
2
+ cos(1)
1
()
1
() (18)
The variation with respect to time of this probability density is simple to obtain graph-
ically from the curves of Figure. They are shown in Figure. For= 0(Fig.-a),
we see that the initial state chosen in (16) corresponds to a probability density which is
concentrated in the right-hand well (in the left-hand well, the functions
1
and
1
are
of opposite sign and very close in absolute value, so their sum is practically zero). It
can therefore be said that the particle, initially, is practically in the right-hand well. At
time=21(Fig.-b), it has moved appreciably, through the tunnel eect, into the
left-hand well, is practically there at time=1(Fig.-c), and then performs the
process in reverse (Figures-d and-e).E
0
ħΩ
2
ħΩ
1
E
a
2
E
s
2
E
a
1
E
s
1
Figure 6: When one takes the nite height
0of the barrier into account, one nds that
the energy spectrum of Figure
ed: each level splits into two distinct ones.
The Bohr frequencies12and22cor-
responding to tunnelling from one well to
the other are the inversion frequencies of the
ammonia molecule for the rst two vibration
levels. The tunnel eect is more important
in the higher vibration level, so21.
476

A SIMPLE MODEL OF THE AMMONIA MOLECULE– b + b
– b
+ b
x
x
0
0
χ
s
(x)
1
χ
a
(x)
1
Figure 7: Wave functions as-
sociated with the levels
1
and
1
in Figure. Note the anal-
ogy with the functions in Fig-
ure-b; however, these new
functions do not vanish on the
interval+2 2.
The ctitious particle therefore moves from one side of the potential barrier to
the other with the frequency12, which means that the plane of the hydrogen atoms
continually passes from one side of the nitrogen atom to the other. This is why the
frequency12is called theinversion frequencyof the molecule. Note that this inversion
frequency has no classical analogue, since its existence is related to the tunnel eect of
the ctitious particle across the potential barrier.
Since the nitrogen atom tends to attract the electrons of the three hydrogen atoms,
the ammonia molecule possesses an electric dipole moment which is proportional to the
mean valueof the position of the ctitious particle we have studied; we see in Figure
that this dipole moment is an oscillating function of time. Under these conditions,
the ammonia molecule is capable of emitting or absorbing electromagnetic radiation of
frequency12.
Experimentally, this is indeed observed; the value of1falls in the domain of
centimeter waves. In radioastronomy, ammonia molecules in interstellar space have been
shown to emit and absorb electromagnetic waves of this frequency. Let us also point out
that the principle of the ammonia maser is based on the stimulated emission of these
waves by the NH3molecule.
3. The ammonia molecule considered as a two-level system
We see in Figure
in the introduction of . The system under study possesses two levels,
1
and
1
, which are very close to each other and very far from all other levels
2
,
2
, ... If we are interested only in the two levels
1
and
1
, we can forget all the
others (the exact justication for such an approximation will be given in the framework
of perturbation theory in Chapter).
We are going to return to the preceding discussion with a slightly dierent point
of view and show that the general considerations of Chapter
477

COMPLEMENT G IVx
0
ψ(x, t)
2
x
0
x
0
x
0
x
0
a
b
c
d
e
Figure 8: Evolution of a wave packet obtained by superposing the two stationary wave
functions of Figure. The particle, initially, is in the right-hand well (g.a), tunnels
into the left-hand well (g.b) and, after a certain time, becomes localized there (g.c);
then it returns to the right-hand well (g.d) and the initial state (g.e), and so on.
systems can be applied to the ammonia molecule. This point of view will also enable us
to study very simply the eect of a static external electric eld on this molecule.
3-a. The state space
The state space we are going to consider is spanned by the two orthogonal vectors
1
1and
1
2, whose wave functions are given by (3). As we explained above, we shall
ignore the other states
1and
2for which1. In the states
1
1and
1
2, the
nitrogen atom is either above or below the plane of the hydrogen atoms. We introduced
in (4) a second orthonormal basis of the state space, composed of the even and odd
vectors:
1
=
1
2
1
1+
1
2
1
=
1
2
1
1
1
2
(19)
There is the same probability in these two states of nding the nitrogen atom above or
below the plane of the hydrogen atoms.
478

A SIMPLE MODEL OF THE AMMONIA MOLECULE
3-b. Energy levels. Removal of the degeneracy due to the transparency of the potential
barrier
When the height0of the potential barrier is innite, the states
1
1and
1
2
have the same energy (as do the states
1
and
1
), so that0, the Hamiltonian of
the system, is written:
0=1 (20)
(whereis the identity operator in the two-dimensional state space).
To take into account phenomenologically the fact that the barrier is not innite,
we add to0a perturbationwhich is non-diagonal in the
1
1
1
2basis and is
represented by the matrix:
=
0 1
1 0
(21)
whereis a real positive coecient
2
.
If we want to nd the stationary states of the molecule, we must diagonalize the
total Hamiltonian operator=0+, whose matrix is written:
=
1
1
(22)
A simple calculation gives the eigenvalues and eigenvectors of:
1+ corresponding to the eigenket
1
1
1
(23)
We see that, under the eect of the perturbation, the two levels, which were degenerate
whenwas zero, now split; an energy dierence, equal to2, appears, and the new
eigenstates are the states
1
and
1
. We again nd the results of .
If, at time= 0, the molecule is in the state
1
1:
(= 0)=
1
1=
1
2
1
+
1
(24)
the state vector at timewill be:
()=
1
2
e
1~
e
~1
+ e
~1
= e
1~
cos
~
1
1+sin
~
1
2 (25)
2
We are forced to assume 0in order to obtain the relative disposition of the
1
and
1
levels
of Figure 23)].
479

COMPLEMENT G IV
In a measurement performed at time, we therefore have a probabilitycos
2
(~)of
nding the molecule in the state
1
1(the nitrogen atom above the plane of the hydrogen
atoms) andsin
2
(~)of nding it in the state
1
2(the nitrogen atom below). Thus
we again nd that, under the eect of the coupling, the ammonia molecule inverts
periodically.
480

A SIMPLE MODEL OF THE AMMONIA MOLECULE
Comment:
The perturbation[given in (21)] describes (phenomenologically) the fact that
the potential barrier is nite. This approach is less precise than the discussion
above, since we obtain here eigenfunctions
1
()and
1
()which, unlike
1
and
1
, go to zero in the region (+2) ( 2). This much more simple
description nevertheless explains two fundamental physical eects: the removal
of the degeneracy of1and the periodic oscillation of the molecule between the
states
1
1and
1
2(inversion).
3-c. Inuence of a static electric eld
We saw above that, in the states
1
1and
1
2, the electric dipole moment of the
molecule takes on two opposite values, which we shall denote by+and. If we call
the observable associated with this physical quantity, we can therefore assume that
is represented in the {
1
1
1
2} basis by a diagonal matrix whose eigenvalues are+
and:
=
0
0
(26)
When the molecule is placed in a static electric eld
3
E, the interaction energy
with this eld is:
(E) =E (27)
This term of the Hamiltonian
4
is represented in the {
1
1
1
2} basis by the matrix:
(E) =E
1 0
0 1
(28)
Let us then write the matrix which represents, in the {
1
1
1
2} basis, the total Hamil-
tonian operator of the molecule,0++(E):
0++(E) =
1E
1+E
(29)
This matrix can easily be diagonalized; its eigenvalues+and and its eigen-
vectors+and are given by:
+=1+
2
+
2
E
2
=1
2
+
2
E
2
(30)
3
For the sake of simplicity, we assume here that this eld is parallel to theaxis of Figure
(one-dimensional model).
4
In(E),is an observable, whileEis a classical quantity which is externally imposed (cf.note,
page).
481

COMPLEMENT G IV
and:
+= cos
2
1
1sin
2
1
2
= sin
2
1
1+ cos
2
1
2
(31)
where we have set:
tan=
E
(0 ) (32)
[cf.ComplementIV, relations (9), (10), (22) and (23); sinceis real and negative, the
angleintroduced in that complement is here equal to].
WhenEis zero,=2, and we again obtain the results of , since:
+(E= 0) =1+
(E= 0) =1
(33)
with:
+(E= 0)=
1
(E= 0)=
1
(34)
When, for an arbitraryE,is zero (a perfectly opaque potential barrier), we
obtain:
+(= 0) =1+E
(= 0) =1 E
(35)
with, ifEis positive
5
:
(= 0)=
1
1
+(= 0)=
1
2
(36)
In this case, the energies therefore vary linearly withE(dashed straight lines in Figure).
Physically, results (35) and (36) are easy to understand: when the electric eld alone
acts on the molecule, it pulls the positively charged hydrogen atoms above or below
the nitrogen atom; this is why the stationary states are
1
1and
1
2.
When the electric eldEand the coupling constantare both arbitrary, the states
+and are linear superpositions of the states
1
1and
1
2(and of the states
1
and
1
as well), and result from a compromise between the action of the electric
eld, which tends to pull the hydrogen atoms to one side of the nitrogen atom, and that
of the coupling, which tends to draw the nitrogen atom accross the potential barrier.
The variation of the energies+and is shown graphically in Figure, in which we
see the phenomenon of anti-crossing (cf.Chap., ) due to the coupling.+
and correspond to the two branches of a hyperbola whose asymptotes are the dashed
lines associated with the energies in the absence of coupling. Finally, we can calculate
5
IfEis negative, the roles of
1
1
and
1
2
are inverted in (36).
482

A SIMPLE MODEL OF THE AMMONIA MOLECULEE
E
+
E
–
E
1
+ A
E
1
– A
0
Figure 9: Inuence of an electric eldEon the rst two levels of the ammonia molecule
(their spacing 2in a zero eld is due to the tunnel eect coupling). For weakE,
the molecule acquires a dipole moment proportional toE, and the corresponding energy
varies withE
2
. For largeE, the dipole moment approaches a limit (corresponding to the
nitrogen atom either above or below the plane of the hydrogen atoms), and the energy
becomes a linear function ofE.
the mean values of the electric dipole momentin each of the two stationary states
+and . Using (26) and (31), we nd:
+ += =cos (37)
which, according to (32), yields:
+ += =
2
E
2
+
2
E
2
(38)
ForE= 0, these two mean values are zero. This corresponds to the fact that, in the two
states
1
, the particle has an equal probability of being in one or the other of the two
wells. On the other hand, whenE , we again nd the dipole moment+(or)
corresponding to the state
1
1(or
1
2).
When the electric eld is weak (E ), formulas (38) can be written in the
form:
+ += =
2
E (39)
We see that the molecule in the stationary state+(or ) acquires an electric dipole
moment proportional to the external eldE. If we dene an electric susceptibility of the
molecule in the stateby the relation:
=E (40)
483

COMPLEMENT G IV
we nd, according to (39), that:
=
2
(41)
(the same calculations are valid for+and yield+= ).
Comment:
In a weak eld, formulas (30) can be expanded as a power series ofE:
=1
1
2
2
E
2
+ (42a)
+=1++
1
2
2
E
2
+ (42b)
Let us now consider ammonia molecules moving in a region whereEis weak but
whereE
2
has a strong gradient in thedirection (i.e. along the axis of the
molecules):
d
d
(E
2
) = (43)
According to (42a), the molecules in the stateare subjected to a force parallel
towhich is equal to:
=
d
d
=
1
2
2
(44)
Relation (42b) indicates that the molecules in the state+are subjected to an
opposite force:
+=
d+
d
= (45)
This result is the basis of the method used in the ammonia maser to sort the
molecules and select those in the higher energy state. The device is analogous to
the Stern-Gerlach apparatus: a beam of ammonia molecules crosses a region where
there is a strong electric eld gradient, the molecules follow dierent trajectories
depending on whether they are in one state or the other; one can, using a suitable
diaphragm, isolate either one of the two states.
References and suggestions for further reading:
Feynman III (1.2), 8-6 and Chap. 9; Alonso and Finn III (1.4), 2-8; article by
Vuylsteke (1.34); Townes and Schawlow (12.10), Chap. 12; see (15.11) for references to
original articles on masers; articles by Lyons (15.14), Gordon (15.15), and Turner (12.14).
See also Encrenaz (12.11), Chap. VI.
484

EFFECTS OF A COUPLING BETWEEN A STABLE STATE AND AN UNSTABLE STATE
Complement HIV
Eects of a coupling between a stable state and an unstable state
1 Introduction. Notation
2 Inuence of a weak coupling on states of dierent energies
3 Inuence of an arbitrary coupling on states of the same
energy
1. Introduction. Notation
The eects of a couplingbetween two states1and2of energies1and2were
discussed in detail in . What modications appear when one of the
two states (1, for example) is unstable?
The concepts of an unstable state and a lifetime were introduced in Comple-
mentIII. We shall assume, for example, that1is an excited atomic state. When the
atom is in this state, it can fall back to a lower energy level through spontaneous emission
of one or several photons, with a probability11per unit time:1is the lifetime of the
unstable state1. On the other hand, we assume that in the absence of the coupling
, the state2is stable (2is innite).
We saw in ComplementIIIthat a simple way of taking the instability of a state
into account consists of adding an imaginary term to the corresponding energy. We shall
therefore replace the energy1of the state1by:
1=1
~
2
1 (1)
with:
1=
1
1
(2)
(since2is innite,2is zero and
2= 2). In the absence of coupling, the matrix
representing the Hamiltonian0of the system is now written in the {1 2} basis
1
:
0=
10
0
2
=
1
~
2
10
0 2
(3)
1
The operator0is not Hermitian and is therefore not really a Hamiltonian (see the comment at
the end of ComplementIII).
485

COMPLEMENT H IV
2. Inuence of a weak coupling on states of dierent energies
Let us assume, as in , that we add to0a perturbation, whose
matrix in the1 2basis is:
=
0 12
210
(4)
What now happens to the energies and lifetimes of the levels?
Let us calculate the eigenvalues
1and
2of the matrix:
=0+=
1
~
2
1 12
21 2
(5)
1and
2are the solutions of the equation in:
2
1+2
~
2
1+12
~
2
12 12
2
= 0 (6)
To simplify the calculation, we shall conne ourselves to the case where the coupling is
weak, i.e.12
(1+2)
2
+
~
24
2
1
; we then nd:
1 1
~
2
1+
12
2
2 1
~2
1
2 2+
12
2
2 1+
~2
1
(7)
The energies of the eigenstates in the presence of the coupling are the real parts
of
1and
2; the lifetimes are inversely proportional to their imaginary parts. We see
from (7) that the coupling changes, to second order in12, both the energies and the
lifetimes. In particular, we observe that
1and
2are both complex when 12is not
zero: in the presence of the coupling, there is no longer any stable state. We can write
2in the form:
2= 2
~
2
2 (8)
with:
2=2+
(2 1)12
2
(2 1)
2
+
~
24
2
1
(9a)
2=1
12
2
(2 1) +
~
24
2
1
(9b)
The state2therefore acquires, under the eect of the coupling, a nite lifetime whose
inverse is given in (9b) (Bethe's formula). This result is easy to understand physically:
486

EFFECTS OF A COUPLING BETWEEN A STABLE STATE AND AN UNSTABLE STATE
if the system at= 0is in the stable state2, there is a non-zero probability at
a subsequent timeof nding it in the state1, in which the system has a nite
lifetime. It is sometimes said guratively that the coupling brings into the stable state
part of the instability of the other state. Moreover, it can be seen from expressions (7)
that, as in the case studied in , the smaller the dierence between
the unperturbed energies1and2, the more eectively the perturbation acts on the
energies and lifetimes. We shall therefore study in the next section the case where this
dierence is zero.
3. Inuence of an arbitrary coupling on states of the same energy
When the energies1and2are equal, the operatoris written, if we make its trace
appear explicitly, as in IV:
= 1
~
4
1+ (10)
whereis the identity operator andis the operator which, in the1 2basis,
has for its matrix:
() =
~
4
1 12
(12)
~
4
1
(11)
The eigenvalues1and2ofare the two solutions of the characteristic equation:
2
= 12
2
~
2
16
2
1 (12)
They therefore have opposite values:
1= 2 (13)
which yields for the eigenvalues of:
1=1
~
4
1+1
2=1
~
4
1 1 (14)
The eigenvectors ofandare the same; a simple calculation
2
enables us to
obtain these vectors
1and
2:
1= 121+ 1+
~
4
1 2
2= 121+ 1+
~
4
1 2 (15)
2
For the calculation performed here, it is not indispensable to normalize
1
and
2
. Note also
that, sinceis not Hermitian,
1
and
2
are not orthogonal.
487

COMPLEMENT H IV
Assume that the system at time= 0is in the state2(which would be stable
in the absence of the coupling):
(= 0)= 2=
1
21
[
1 2] (16)
Using (14), we see that, at time, the state vector is:
()=
1
21
e
1~
e
1
4
1
[e
1~
1e
1~
2] (17)
The probabilityP21()of nding the system at timein the state1is:
P21() = 1()
2
=
1
41
2
e
12
e
1~
11e
1~
12
2
=
1
41
2
e
12
12
2
e
1~
e
1~
2
(18)
We shall distinguish between several cases:
(i) When the condition:
12
~
4
1 (19)
is satised, we obtain directly, using (12):
1= 2=
12
2
~
4
1
2
(20)
and the eigenvalues
1and
2are given by:
1=1+
12
2
~
4
1
2
~
4
1
2=2
12
2
~
4
1
2
~
4
1 (21)
1and
2have the same imaginary part, but dierent real parts. The states
1and
2therefore have the same lifetime,21, but dierent energies.
Substituting (20) into (18), we obtain:
P21() =
12
2
12
2 ~
4
1
2
e
12
sin
2
12
2
~
4
1
2
~
(22)
The form of this result recalls Rabi's formula [cf.Chap., equation (C-32)]. The
functionP21()is represented by a damped sinusoid with time constant21(Fig.).
Condition (19) thus expresses the fact that the coupling is suciently strong to make
488

EFFECTS OF A COUPLING BETWEEN A STABLE STATE AND AN UNSTABLE STATE0 t
21
(t)
Figure 1: Eect of a strong coupling between
a stable state2and an unstable state1.
If the system is initially in the state2, the
probabilityP21()of nding it in the state
1at timepresents damped oscillations.
the system oscillate between the states1and 2before the instability of the state
1can have a real eect.
(ii) If, on the other hand, the condition:
12
~
4
1 (23)
is satised, we then have:
1= 2=
~
4
1
2
12
2
(24)
and:
1=1
~
4
1
~
4
1
2
12
2
2=1
~
4
1+
~
4
1
2
12
2
(25)
The states
1and
2then have the same energy and dierent lifetimes. Formula
(18) becomes:
P21() =
12
2
~
4
1
2
12
2
e
12
sinh
2
~
4
1
2
12
2
~
(26)
This time,P21()is a sum of damped exponentials (Fig.).
This result has a simple physical interpretation: condition (23) expresses the fact
that the lifetime1is so short that the system is completely damped before the coupling
has had the time to make it oscillate between the states1and2.
(iii) Finally, let us examine the case where we have exactly:
12=
~
4
1 (27)
We see then from (14) that the states
1and
2both have the same energy1
and the same lifetime21.
489

COMPLEMENT H IV0
t
21
(t)
Figure 2: When the coupling is weak, oscil-
lations between the states1and 2do
not have time to occur.
Equations (22) and (26), in this case, take on indeterminate forms, which can be
resolved and both yield:
P21() =
12
2
~
2
2
e
12
(28)
Comment:
The preceding discussion is very similar to that of the classical motion of a damped
harmonic oscillator. Conditions (19), (23) and (27) correspond respectively to
weak, strong and critical damping.
References and suggestions for further reading:
An important application of the phenomenon discussed in this complement is the
shortening of the lifetime of a metastable state due to an electric eld. See: Lamb and
Retherford (3.11), App. II; Sobel'man (11.12), Chap. 8, 28-5.
490

EXERCISES
Complement JIV
Exercises
1.Consider a spin 1/2 particle of magnetic momentM=S. The spin state space
is spanned by the basis of the+and vectors, eigenvectors ofwith eigenvalues
+~2and~2. At time= 0, the state of the system is:
(= 0)=+
. is measured at time= 0, what results can be found, and with
what probabilities?
.
the inuence of a magnetic eld parallel to, of modulus0. Calculate, in the
+ basis, the state of the system at time.
. , we measure the observables,,. What values can we nd,
and with what probabilities? What relation must exist between0andfor the
result of one of the measurements to be certain? Give a physical interpretation of
this condition.
2.Consider a spin 1/2 particle, as in the previous exercise (using the same nota-
tion).
. = 0, we measureand nd+~2. What is the state vector(0)
immediately after the measurement?
.
allel to. The Hamiltonian operator of the spin()is then written:
() =0()
Assume that0()is zero for0and and increases linearly from 0 to0
when0 (is a given parameter having the dimension of a time). Show
that at timethe state vector can be written:
()=
1
2
[e
()
++e
()
]
where()is a real function of(to be calculated by the student).
. = , we measure. What results can we nd, and with what
probabilities? Determine the relation that must exist between0andin order
for us to be sure of the result. Give the physical interpretation.
491

COMPLEMENT J IV
3.Consider a spin 1/2 particle placed in a magnetic eldB0with components:
=
1
2
0
= 0
=
1
2
0
The notation is the same as that of exercise (1).
. + basis, the operator, the
Hamiltonian of the system.
. .
. = 0is in the state. What values can be found if the
energy is measured, and with what probabilities?
. ()at time. At this instant,is measured; what
is the mean value of the results that can be obtained? Give a geometrical interpre-
tation.
4.Consider the experimental device described in cf.Fig.):
a beam of atoms of spin 1/2 passes through one apparatus, which serves as a polarizer
in a direction which makes an anglewith in the plane, and then through
another apparatus, the analyzer, which measures thecomponent of the spin. We
assume in this exercise that between the polarizer and the analyzer, over a lengthof
the atomic beam, a magnetic eldB0is applied which is uniform and parallel to. We
callthe speed of the atoms and= the time during which they are submitted to
the eldB0. We set0= 0.
. 1of a spin at the moment it enters the analyzer?
.
ity equal to
1
2
(1 + coscos0)of nding+~2and
1
2
(1coscos0)of nding
~2. Give a physical interpretation.
.
ned in ComplementIII. The reader is also advised to refer to ComplementIV).
Show that the density matrix1of a particle which enters the analyzer is written,
in the+ basis:
1=
1
2
1 + coscos0 sin+cossin0
sin cossin0 1coscos0
Calculate Tr{1}, Tr{1} and Tr{1}. Give an interpretation. Does the
density operator1describe a pure state?
492

EXERCISES
.
is known only to within a certain uncertainty. In addition, the eld0
is assumed to be suciently strong that0 1. The possible values of the
product0are then (modulus2) all values included between 0 and2, all of
which are equally probable.
In this case, what is the density operator2of an atom at the moment it
enters the analyzer? Does2correspond to a pure case? Calculate the quantities
Tr{2}, Tr{2} and Tr{2}. What is your interpretation? In which case
does the density operator describe a completely polarized spin? A completely
unpolarized spin?
Describe qualitatively the phenomena observed at the analyzer exit when0
varies from zero to a value where the condition0 1is satised.
5. Evolution operator of a spin 1/2(cf.ComplementIII)
Consider a spin 1/2, of magnetic momentM=S, placed in a magnetic eldB0
of components= ,= ,= .
We set:
0= B0
.
(0) = e
whereis the operator:
=
1
~
[ + + ] =
1
2
[ + + ]
expressed as a function of the three Pauli matrices,and(cf.Comple-
mentIV).
Calculate the matrix representingin the+ basis of eigenvectors of.
Show that:
2
=
1
4
[
2
+
2
+
2
] =
0
2
2
.
(0) = cos
0
2
2
0
sin
0
2
. = 0is in the state(0)=+. Show that the
probabilityP++()of nding it in the state+at timeis:
P++() =+(0)+
2
493

COMPLEMENT J IV
and derive the relation:
P++() = 1
2
+
2
2
0
sin
2 0
2
Give a geometrical interpretation.
6.Consider the system composed of two spin 1/2's,S1andS2, and the basis of
four vectors dened in ComplementIV. The system at time= 0is in the state:
(0)=
1
2
+ ++
1
2
+ +
1
2
. = 0,S1zis measured; what is the probability of nding~2? What is
the state vector after this measurement? If we then measure1, what results can
be found, and with what probabilities? Answer the same questions for the case
where the measurement of1yielded+~2.
. (0)written above,1and2are measured
simultaneously. What is the probability of nding opposite results? Identical re-
sults?
.
the inuence of the Hamiltonian:
=11+22
What is the state vector()at time? Calculate at timethe mean valuesS1
andS2. Give a physical interpretation.
. S1andS2are less than~2. What must
be the form of(0)for each of these lengths to be equal to+~2?
7.Consider the same system of two spin 1/2's as in the preceding exercise; the
state space is spanned by the basis of four states.
. 44matrix representing, in this basis, the1operator. What are the
eigenvalues and eigenvectors of this matrix?
.
=+ +++ + ++
where,,andare given complex coecients.1and2are measured
simultaneously; what results can be found, and with what probabilities? What
happens to these probabilities ifis a tensor product of a vector of the state
space of the rst spin and a vector of the state space of the second spin?
494

EXERCISES
. 1and2.
. 2. Calcu-
late, rst from the results ofand then from those of, the probability of nding
~2.
8.Consider an electron of a linear triatomic molecule formed by three equidistant
atoms.A B C
We use , , to denote three orthonormal states of this electron, correspond-
ing respectively to three wave functions localized about the nuclei of atoms,,.
We shall conne ourselves to the subspace of the state space spanned by, and
.
When we neglect the possibility of the electron jumping from one nucleus to an-
other, its energy is described by the Hamiltonian0whose eigenstates are the three
states, , with the same eigenvalue0. The coupling between the states
, , is described by an additional Hamiltoniandened by:
=
=
=
whereis a real positive constant.
. =0+
. = 0is in the state. Discuss qualitatively the localization
of the electron at subsequent times. Are there any values offor which it is
perfectly localized about atom,or?
. be the observable whose eigenstates are, , with respective
eigenvalues,0,.is measured at time; what values can be found, and with
what probabilities?
.
that can appear in the evolution of? Give a physical interpretation of. What
are the frequencies of the electromagnetic waves that can be absorbed or emitted
by the molecule?
9.A molecule is composed of six identical atoms1,2,...6which form a
regular hexagon. Consider an electron which can be localized on each of the atoms. Call
the state in which it is localized on theth atom (= 126). The electron
states will be conned to the space spanned by the, assumed to be orthonormal.
495

COMPLEMENT J IVA
6
A
1
A
2
A
3
A
4
A
5
Dene an operatorby the following relations:
1= 2; 2= 3;; 6= 1
Find the eigenvalues and eigenstates of. Show that the eigenvectors ofform a
basis of the state space.
When the possibility of the electron passing from one site to another is neglected, its
energy is described by a Hamiltonian0whose eigenstates are the six states,
with the same eigenvalue0. As in the previous exercise, we describe the possibility
of the electron jumping from one atom to another by adding a perturbationto
the Hamiltonian0;is dened by:
1= 6 2; 2= 1 3;
; 6= 5 1
Show thatcommutes with the total Hamiltonian=0+. From this deduce
the eigenstates and eigenvalues of. In these eigenstates, is the electron localized?
Apply these considerations to the benzene molecule.
Exercise 9.
Reference: Feynman III (1.2), 15-4.
496

Chapter V
The one-dimensional harmonic
oscillator
A Introduction
A-1 Importance of the harmonic oscillator in physics
A-2 The harmonic oscillator in classical mechanics
A-3 General properties of the quantum mechanical Hamiltonian
B Eigenvalues of the Hamiltonian
B-1 Notation
B-2 Determination of the spectrum
B-3 Degeneracy of the eigenvalues
C Eigenstates of the Hamiltonian
C-1 The representation
C-2 Wave functions associated with the stationary states
D Discussion
D-1 Mean values and root mean square deviations ofandin
a state . . . . . . . . . . . . . . . . . . . . . . . . . . . .
D-2 Properties of the ground state
D-3 Time evolution of the mean values
A. Introduction
A-1. Importance of the harmonic oscillator in physics
This chapter is devoted to the study of a particularly important physical system:
the one-dimensional harmonic oscillator.
Quantum Mechanics, Volume I, Second Edition. C. Cohen-Tannoudji, B. Diu, and F. Laloë.
© 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

CHAPTER V THE ONE-DIMENSIONAL HARMONIC OSCILLATOR
The simplest example of such a system is that of a particle of massmoving in
a potential which depends only onand has the form:
() =
1
2
2
(A-1)
(is a real positive constant). The particle is attracted towards the= 0plane [the
minimum of(), corresponding to positions of stable equilibrium] by a restoring force:
=
d
d
= (A-2)
which is proportional to the distancebetween the particle and the= 0plane (is an
algebraic variable:70). We know that in classical mechanics, the projection onto
of the particle's motion is a sinusoidal oscillation about= 0, of angular frequency:
=
(A-3)
Actually, a large number of systems are governed (at least approximately) by the
harmonic oscillator equations. Whenever one studies the behavior of a physical system
in the neighborhood of a stable equilibrium position, one arrives at equations which, in
the limit of small oscillations, are those of a harmonic oscillator (see ). The results
we shall derive in this chapter are applicable, therefore, to a whole series of important
physical phenomena for example, the vibrations of the atoms of a molecule about their
equilibrium position, the oscillations of atoms or ions of a crystalline lattice (phonons)
1
.
The harmonic oscillator is also involved in the study of the electromagnetic eld.
We know that in a cavity, there exist an innite number of possible stationary waves
(normal modes of the cavity). The electromagnetic eld can be expanded in terms of
these modes and it can be shown, using Maxwell's equations, that each of the coecients
of this expansion (which describe the state of the eld at each instant) obeys a dierential
equation, which is identical to that of a harmonic oscillator whose angular frequencyis
that of the associated normal mode. In other words, the electromagnetic eld is formally
equivalent to a set of independent harmonic oscillators (cf.ComplementV). The
quantization of the eld is obtained by quantizing these oscillators associated with the
various normal modes of the cavity (cf. Chapter ). Recall, moreover, that it was the
study of the behavior of these oscillators at thermal equilibrium (blackbody radiation)
which, historically, led Planck to introduce, for the rst time in physics, the constant
which bears his name. We shall see (cf.ComplementV) that the mean energy of a
harmonic oscillator in thermodynamic equilibrium at the temperatureis dierent for
classical and quantum mechanical oscillators.
The harmonic oscillator also plays an important role in the description of a set
of identical particles which are all in the same quantum mechanical state (they must
obviously be bosons,cf.Chap. ). As we shall see later, this is because the energy
levels of a harmonic oscillator are equidistant, the spacing between two adjacent levels
being equal to~. With the energy level labelled by the integer(situated at a distance
~above the ground state) can then be associated a set ofidentical particles (or
1
ComplementVis devoted to a qualitative study of some physical examples of harmonic oscillators.
498

A. INTRODUCTION
quanta), each possessing an energy~(cf. Chapter). The transition of the oscillator
from levelto level+1or1corresponds to the creation or annihilation of a quantum
of energy~. In this chapter, we shall introduce the operatorsand, which enable us
to describe this transition from levelto level+1or1. These operators, respectively
called creation and annihilation operators
2
, are used throughout quantum statistical
mechanics and quantum eld theory
3
.
The detailed study of the harmonic oscillator in quantum mechanics is therefore
extremely important from a physical point of view. Moreover, we are dealing with a
quantum mechanical system for which the Schrödinger equation can be solved rigorously.
Having studied spin 1/2 and two-level systems in Chapter, we shall therefore now
consider another simple example which illustrates the general formalism of quantum
mechanics. We shall show in particular how to solve an eigenvalue equation by dealing
only with the operators and the commutation relations (this technique will also be applied
to angular momentum). We shall also study in a detailed way the motion of wave packets,
particularly at the classical limit (cf.ComplementVon quasi-classical states).
In , we shall review some results related to the classical oscillator before
stating () certain general properties of the eigenvalues of the Hamiltonian. Then,
in , we shall determine these eigenvalues and eigenvectors by introducing
creation and annihilation operators and using only the consequences of the canonical
commutation relation[] =~, as well as the particular form of.
to a physical study of the stationary states of the oscillator and wave packets formed by
linear superpositions of these stationary states.
A-2. The harmonic oscillator in classical mechanics
The potential energy()[formula (A-1)] is shown in Figure. The motion of
the particle is governed by the dynamical equation:
d
2
d
2
=
d
d
= (A-4)
The general solution of this equation is of the form:
= cos( ) (A-5)
whereis dened by (A-3), and the constants of integrationandare determined by
the initial conditions of the motion. The particle thereforeoscillates sinusoidallyabout
the point, with an amplitudeand an angular frequency.
The kinetic energy of the particle is:
=
1
2
d
d
2
=
2
2
(A-6)
2
Annihilation operators are also often called destruction operators.
3
The aim of quantum eld theory is to describe interactions between particles in the relativistic do-
main, especially the interactions between electrons, positrons and photons. It is clear that creation and
annihilation operators should play an important role, since such processes are indeed observed experi-
mentally (absorption or emission of photons, pair creation...). The quantum theory of electromagnetism
is introduced in Chapter .
499

CHAPTER V THE ONE-DIMENSIONAL HARMONIC OSCILLATOR
where=
d
d
is the momentum of the particle. The total energy is:
=+=
2
2
+
1
2
22
(A-7)
Substituting solution (A-5) into this equation, we nd:
=
1
2
22
(A-8)
The energy of the particle is therefore time-independent (this is a general property of
conservative systems) and can take on any positive (or zero) value, sinceisa priori
arbitrary.
If we x the total energy, the limits of the classical motion= can be
determined from Figure
toof ordinate. At these points= , the potential energy is at a maximum
and equal to, and the kinetic energy is zero. On the other hand, at= 0, the potential
energy is zero and the kinetic energy is maximum.
Comment:
Consider an arbitrary potential()which has a minimum at=0(Fig.).
Expanding the function()in a Taylor's series in the neighborhood of0, we
obtain:
() =+( 0)
2
+( 0)
3
+ (A-9)0– x
M x
M
x
E
V(x)
Figure 1: The potential energy()of a one-dimensional harmonic oscillator. The
amplitude of the classical motion of energyis.
500

A. INTRODUCTIONE
0
V(x)
x
1
x
0
x
2
x
Figure 2: In the neighborhood of a minimum, any potential()can be approximated by
a parabolic potential (dashed line). In the potential(), a classical particle of energy
oscillates between1and2.
The coecients of this expansion are given by:
=(0)
=
1
2
d
2
d
2
=0
=
1
3!
d
3
d
3
=0
(A-10)
and the linear term in( 0)is zero since0corresponds to a minimum of().
The force derived from the potential()is, in the neighborhood of0:
=
d
d
=2( 0)3( 0)
2
+ (A-11)
Since=0represents a minimum, the coecientis positive.
The point=0corresponds to a stable equilibrium position for the particle:
is zero for=0; moreover, for( 0)suciently small,and( 0)
have opposite signs sinceis positive.
If the amplitude of the motion of the particle about0is suciently small for
the term in( 0)
3
of (A-9) [and therefore, the corresponding term in( 0)
2
of (A-11)] to be negligible compared to the preceding ones, we have a harmonic
oscillator since the dynamical equation can then be approximated by:
d
2
d
2
2( 0) (A-12)
The corresponding angular frequencyis related to the second derivative of()
501

CHAPTER V THE ONE-DIMENSIONAL HARMONIC OSCILLATOR
at=0by the formula:
=
2
=
1
d
2
d
2
=0
(A-13)
Since the amplitude of the motion must remain small, the energy of the harmonic
oscillator will be low.
For higher energies, the particle will be inperiodic but not sinusoidal motion
between the limits1and2(Fig.). If we expand the function()in a Fourier
series giving the position of the particle, we nd, not one, but several, sinusoidal
terms; their frequencies are integral multiples of the lowest frequency. We then
say that we are dealing with ananharmonicoscillator. Note also that, in this
case, the period of the motion is not generally2, whereis given by formula
(A-13).
A-3. General properties of the quantum mechanical Hamiltonian
In quantum mechanics, the classical quantitiesandare replaced respectively
by the observablesand, which satisfy:
[] =~ (A-14)
It is then easy to obtain the Hamiltonian operator of the system from (A-7):
=
2
2
+
1
2
22
(A-15)
Sinceis time-independent (conservative system), the quantum mechanical study
of the harmonic oscillator reduces to the solution of the eigenvalue equation:
= (A-16)
which is written, in therepresentation:
~
2
2
d
2
d
2
+
1
2
22
() = () (A-17)
Before undertaking the detailed study of equation (A-16), let us indicate some
important properties that can be deduced from the form (A-1) of the potential function:
()The eigenvalues of the Hamiltonian are positive. It can be shown that, in general
(ComplementIII), if the potential function()has a lower bound, the eigen-
valuesof the Hamiltonian=
2
2
+()are greater than the minimum of
():
() requires (A-18)
For the harmonic oscillator we are studying here, we have chosen the energy origin
such thatis zero.
502

B. EIGENVALUES OF THE HAMILTONIAN
()The eigenfunctions ofhave a denite parity. This is due to the fact that the
potential()is an even function:
() =() (A-19)
We can then (cf.ComplementsIIandV) look for eigenfunctions of, in the
representation, amongst the functions which have a denite parity (in fact,
we shall see that the eigenvalues ofare not degenerate; consequently, the wave
functions associated with the stationary states are necessarily either even or odd).
()The energy spectrum is discrete. Whatever the value of the total energy, the clas-
sical motion is limited to a bounded region of theaxis (Fig.), and it can be
shown (ComplementIII) that in this case, the eigenvalues of the Hamiltonian
form a discrete set.
We shall derive these properties (in a more precise form) in the following sections.
However, it is interesting to note that they can be obtained simply by applying to
the harmonic oscillator some general theorems concerning one-dimensional prob-
lems.
B. Eigenvalues of the Hamiltonian
We are now going to study the eigenvalue equation (A-16). First of all, using only the
canonical commutation relation (A-14), we shall nd the spectrum of the Hamiltonian
written in (A-15).
B-1. Notation
We shall begin by introducing some useful notations.
B-1-a. The
^
and
^
operators
The observablesandobviously have dimensions (those of a length and a
momentum, respectively). Sincehas the dimension of the inverse of a time and~, of
an action (product of an energy and a time), it is easy to see that the observables
^
and
^
dened by:
^
=
~
^
=
1
~
(B-1)
are dimensionless.
If we use these new operators, the canonical commutation relation will be written:
[
^^
] = (B-2)
and the Hamiltonian can be put in the form:
=~
^
(B-3)
503

CHAPTER V THE ONE-DIMENSIONAL HARMONIC OSCILLATOR
with:
^
=
1
2
^2
+
^2
(B-4)
We shall therefore seek the solutions of the eigenvalue equation:
^
= (B-5)
where the operator
^
and the eigenvaluesare dimensionless. The indexcan belong
to either a discrete or a continuous set, and the additional indexenables us to distinguish
between the various possible orthogonal eigenvectors associated with the same eigenvalue
.
B-1-b. The ,andoperators
If
^
and
^
were numbers and not operators, we could write the sum
^2
+
^2
appearing in expression (B-4) for
^
in the form of a product of linear terms, and obtain
(
^^
)(
^
+
^
). In fact, since
^
and
^
are non-commuting operators,
^2
+
^2
is not
equal to(
^^
)(
^
+
^
). We shall show, however, that the introduction of operators
proportional to
^
+
^
and
^^
enables us to simplify considerably our search for
eigenvalues and eigenvectors of
^
.
We therefore set
4
:
=
1
2
(
^
+
^
) (B-6a)
=
1
2
(
^^
) (B-6b)
These formulas can immediately be inverted to yield:
^
=
1
2
(+) (B-7a)
^
=
2
( ) (B-7b)
Since
^
and
^
are Hermitian,andare not (because of the factor), but are adjoints
of each other.
The commutator ofandis easy to calculate from (B-6) and (B-2):
[] =
1
2
^
+
^^^
=
2
^^
2
^^
(B-8)
that is:
[] = 1 (B-9)
This relation is completely equivalent to the canonical commutation relation (A-14).
4
Until now, we have designated operators by capital letters. However, to conform to standard usage,
we shall use the small lettersandfor the operators (B-6).
504

B. EIGENVALUES OF THE HAMILTONIAN
Finally, we derive some simple formulas which will be useful in the rest of this
chapter. We rst calculate:
=
1
2
(
^^
)(
^
+
^
)
=
1
2
(
^2
+
^2
+
^^^^
)
=
1
2
(
^2
+
^2
1) (B-10)
Comparing this with expression (B-4), we see that:
^
= +
1
2
=
1
2
(
^^
)(
^
+
^
) +
1
2
(B-11)
Unlike the situation in the classical case,
^
cannot be put in the form of a product of
linear terms. The non-commutativity of
^
and
^
is at the origin of the additional term
1/2 that appears on the right-hand side of (B-11). Similarly, it can be shown that:
^
=
1
2
(B-12)
Let us now introduce the operatordened by:
= (B-13)
This operator is Hermitian since:
=()= = (B-14)
Moreover, according to (B-11):
^
=+
1
2
(B-15)
so thatthe eigenvectors of
^
are eigenvectors of,and vice versa.
Finally, let us calculate the commutators ofwithand:
[] = [] =[] + []=
[ ] = [ ] =[] + []= (B-16)
that is:
[] = (B-17a)
[ ] = (B-17b)
Our study of the harmonic oscillator will be based on the use of the,and
operators. We have replaced the eigenvalue equation of, which we rst wrote in the
form (B-5), by that of:
= (B-18)
505

CHAPTER V THE ONE-DIMENSIONAL HARMONIC OSCILLATOR
When this equation is solved, we shall know that the eigenvectorofis also an
eigenvector ofwith the eigenvalue= (+ 12)~[formulas (B-3) and (B-15)]:
= (+ 12)~ (B-19)
The solution of equation (B-18) will be based on the commutation relation (B-9), which is
equivalent to the initial relation (A-14), and on formulas (B-17), which are consequences
of it.
B-2. Determination of the spectrum
B-2-a. Lemmas
. Lemma I (property of the eigenvalues of)
The eigenvaluesof the operatorare positive or zero.
Consider an arbitrary eigenvectorof. The square of the norm of the vector
is positive or zero:
2
= 0 (B-20)
Let us then use denition (B-13) of:
= = (B-21)
Since is positive, comparison of (B-20) and (B-21) shows that:
0 (B-22)
. Lemma II (properties of the vector )
Let be a (non-zero) eigenvector ofwith the eigenvalue.
We shall prove the following:
()If= 0, the ket
=0is zero.
()If0, the ket is a non-zero eigenvector ofwith the eigenvalue1.
()According to (B-21), the square of the norm ofis zero if= 0; now, the
norm of a vector is zero if and only if this vector is zero. Consequently, if= 0is an
eigenvalue of, all eigenvectors
0associated with this eigenvalue satisfy the relation:
0= 0 (B-23)
Let us now show that relation (B-23) is characteristic of these eigenvectors. Con-
sider a vectorwhich satises:
= 0 (B-24)
Multiply both sides of this equation from the left by:
= = 0 (B-25)
506

B. EIGENVALUES OF THE HAMILTONIAN
Any vector which satises (B-24) is therefore an eigenvector ofwith the eigenvalue
= 0.
()Now let us assume thatis strictly positive. According to (B-21), the vector
is then non-zero, since the square of its norm is not equal to zero.
Let us show that is an eigenvector of. To do this, let us apply the operator
relation (B-17a) to the vector:
[]=
= (B-26)
=
Therefore:
= (1)[ ] (B-27)
which shows that is an eigenvector ofwith the eigenvalue1.
. Lemma III (properties of the vector )
Let be a (non-zero) eigenvector ofof eigenvalue.
We shall prove the following:
() is always non-zero.
() is an eigenvector ofwith the eigenvalue+ 1.
()It is easy to calculate the norm of the vector, using formulas (B-9) and
(B-13):
2
=
= (+ 1)
= (+ 1) (B-28)
Since, according to lemmaI,is positive or zero, the ketalways has a non-zero
norm and, consequently, is never zero.
()The proof of the fact thatis an eigenvector ofis analogous to that
of lemmaII; starting from relation (B-17b) between operators, we obtain:
[ ] =
= + = (+ 1) (B-29)
B-2-b. The spectrum of is composed of non-negative integers
Consider an arbitrary eigenvalueofand a non-zero eigenvectorassociated
with this eigenvalue.
According to lemmaI,is necessarily positive or zero. First, let us assumeto
be non-integral. We are now going to show that such a hypothesis contradicts lemmaI
and must consequently be excluded. Ifis non-integral, we can always nd an integer
0such that:
+ 1 (B-30)
507

CHAPTER V THE ONE-DIMENSIONAL HARMONIC OSCILLATOR
Now let us consider the series of vectors:
(B-31)
According to lemmaII, each of the vectors of this series (with0 ) is
non-zero and an eigenvector ofwith the eigenvalue(cf.Fig.). The proof is by
iteration:is non-zero by hypothesis;is non-zero (since0) and corresponds
to the eigenvalue1of...; is obtained whenacts on
1
, an eigenvector
ofwith the strictly positive eigenvalue+ 1, since and [cf.(B-30)].
Now letact on the ket . Since 0according to (B-30), the action
ofon (an eigenvector ofwith the eigenvalue 0) yields a non-zero
vector (lemmaII). Moreover, again according to lemmaII,
+1
is an eigenvector
ofwith the eigenvalue 1, which is strictly negative according to (B-30). If
is non-integral, we can therefore construct a non-zero eigenvector ofwith a strictly
negative eigenvalue. Since this is impossible, according to lemmaI, the hypothesis of
non-integralmust be rejected.
What now happens if:
= (B-32)
witha positive integer or zero? In the series of vectors (B-31), is non-zero and
an eigenvector ofwith the eigenvalue 0. According to lemmaII( ()), we therefore
have:
+1
= 0 (B-33)
The series of vectors obtained by repeated action of the operatoron is therefore
limited whenis integral. It is then never possible to obtain a non-zero eigenvector of
which corresponds to a negative eigenvalue.
In conclusion,can only be a non-negative integer.
LemmaIIIcan then be used to show that the spectrum ofindeed includes
all positive or zero integers. We have already constructed an eigenvector ofwith an
eigenvalue of zero (). All we must do is let()act on such a vector in order to
obtain an eigenvector ofof eigenvalue, whereis an arbitrary positive integer.0 1
α
n
φ
i
v – n v – n + 1
n – 1 n + 1n
v – 1 v
2
v α

φ
i
v φ
i
vα
n – 1
φ
i
v
Figure 3: Lettingact several times on the ket, we can construct eigenvectors of
with eigenvalues1,2etc...
508

B. EIGENVALUES OF THE HAMILTONIAN
If we then refer to formula (B-19), we conclude that the eigenvalues ofare of
the form:
= +
1
2
~ (B-34)
with= 0, 1, 2, ... Therefore, in quantum mechanics,the energy of the harmonic
oscillator is quantizedand cannot take on any arbitrary value. Note also that the smallest
value (the ground state) is not zero, but~2(see
B-2-c. Interpretation of the andoperators
If we start with an eigenstateofcorresponding to the eigenvalue=
(+ 12)~, application of the operatoryields an eigenvector associated with the
eigenvalue 1= (+ 12)~~, and application ofyields, in the same way, the
energy +1= (+ 12)~+~.
For this reason,is said to be acreation operatorandanannihilation operator
(or destruction operator); their action on an eigenvector ofmakes an energy quantum
~appear or disappear.
B-3. Degeneracy of the eigenvalues
We now show that the energy levels of the one-dimensional harmonic oscillator,
given by equation (B-34), are not degenerate.
B-3-a. The ground state is non-degenerate
The eigenstates ofassociated with the eigenvalue0=~2, that is, the eigen-
states ofassociated with the eigenvalue= 0, according to lemmaIIof ,
must all satisfy the equation:
0= 0 (B-35)
To nd the degeneracy of the0level, all we must do is see how many linearly indepen-
dent kets satisfy (B-35).
Using denition (B-6a) ofand relations (B-1), we can write (B-35) in the form:
1
2~
+
~
0= 0 (B-36)
In the representation, this relation becomes:
~
+
d
d
0() = 0 (B-37)
where:
0() =
0 (B-38)
Therefore we must solve a rst-order dierential equation. Its general solution is:
0() =e
1
2~
2
(B-39)
whereis the constant of integration. The various solutions of (B-37) are all proportional
to each other. Consequently, to within a multiplicative factor, there exists only one ket
0that satises (B-35): the ground state0=~2is not degenerate.
509

CHAPTER V THE ONE-DIMENSIONAL HARMONIC OSCILLATOR
B-3-b. All the states are non-degenerate
We have just seen that the ground state is not degenerate. Let us show by recur-
rence that this is also the case for all the other states.
All we need prove is that, if the level= (+ 12)~is not degenerate, the
level+1= (+ 1 + 12)~is not either. Let us therefore assume that there exists, to
within a constant factor, only one vectorsuch that:
= (B-40)
Then consider an eigenvector
+1corresponding to the eigenvalue+ 1:
+1= (+ 1)
+1 (B-41)
We know that the ket
+1is not zero and that it is an eigenvector ofwith the
eigenvalue(cf.lemmaII). Since this ket is not degenerate by hypothesis, there exists
a numbersuch that:
+1= (B-42)
It is simple to invert this equation by applyingto both sides:
+1= (B-43)
that is, taking (B-13) and (B-41) into account:
+1=
+ 1
(B-44)
We already knew that was an eigenvector ofwith the eigenvalue(+ 1); we
see here that all kets
+1associated with the eigenvalue(+ 1)are proportional to
. They are therefore proportional to each other: the eigenvalue(+ 1)is not
degenerate.
Thus, since the eigenvalue= 0is not degenerate (see ), the eigenvalue
= 1is not either, nor is= 2, etc...: all the eigenvalues ofand, consequently, all
those of, are non-degenerate. This enables us to write simplyfor the eigenvector
ofassociated with the eigenvalue= (+ 12)~.
C. Eigenstates of the Hamiltonian
In this section, we are going to study the principal properties of the eigenstates of the
operatorand of the Hamiltonian.
C-1. The representation
We shall assume thatandare observables, meaning their eigenvectors con-
stitute a basis in the space, the state space of a particle in a one-dimensional problem
(this could be proved by considering the wave functions associated with the eigenstates
of, which we shall calculate in (or of
) is degenerate (see ),(or) alone constitutes a C.S.C.O. in.
510

C. EIGENSTATES OF THE HAMILTONIAN
C-1-a. The basis vectors in terms of 0
The vector0associated with= 0is the vector ofthat satises:
0= 0 (C-1)
It is dened to within a constant factor; we shall assume0to be normalized, so the
indeterminacy is reduced to a global phase factor of the forme, withreal.
According to lemmaIIIof , the vector1which corresponds to= 1is
proportional to0:
1=1 0 (C-2)
We shall determine1by requiring1to be normalized and choosing the phase of
1(relative to0) such that1is real and positive. The square of the norm of1,
according to (C-2), is equal to:
11=1
2
0 0
=1
2
0(+ 1)0 (C-3)
where (B-9) has been used. Since0is a normalized eigenstate of= with the
eigenvalue zero, we nd:
11=1
2
= 1 (C-4)
With the preceding phase convention, we have1= 1and, consequently:
1= 0 (C-5)
Similarly, we can construct2from1:
2=2 1 (C-6)
We require2to be normalized and choose its phase such that2is real and positive:
22=2
2
1 1
=2
2
1(+ 1)1
= 22
2
= 1 (C-7)
Therefore:
2=
1
2
1=
1
2
()
2
0 (C-8)
if we take (C-5) into account.
This procedure can easily be generalized. If we know1(which is normalized),
then the normalized vectoris written:
= 1 (C-9)
511

CHAPTER V THE ONE-DIMENSIONAL HARMONIC OSCILLATOR
Since:
=
2
1 1
=
2
= 1 (C-10)
we choose, with the same phase conventions as above:
=
1
(C-11)
With these successive phase choices, we can obtain all thefrom0:
=
1
1=
11
1
()
2
2=
=
1
1
1
1
2
()0 (C-12)
that is:
=
1
!
()0 (C-13)
C-1-b. Orthonormalization and closure relations
Sinceis Hermitian, the ketscorresponding to dierent values ofare
orthogonal. Since each of them is also normalized, they satisfy the orthonormalization
relation:
= (C-14)
In addition,is an observable (we shall assume this here without proof); the set
of thetherefore constitutes a basis in. This is expressed by the closure relation:
= 1 (C-15)
Comment:
It can be veried directly from expression (C-13) that the ketsare orthonormal:
=
1
!!
0 0 (C-16)
But:
0=
1
()
1
0
=
1
(+ 1)
1
0
=
1
1
0 (C-17)
512

C. EIGENSTATES OF THE HAMILTONIAN
(using the fact that
1
0is an eigenstate of= with the eigenvalue1).
Thus can we reduce the exponents ofandby iteration. We obtain, nally:
if :0 0= (1) 210 0 (C-18a)
if :0 0= (1)( + 1)0() 0 (C-18b)
if=:0 0= (1) 2100 (C-18c)
The expression (C-18a) is zero because0= 0. Similarly, (C-18b) is equal to zero
because 0() 0can be considered to be the scalar product of0and the bra
associated with 0, which is zero if. Finally, if we substitute (C-18c) into
(C-16), we see that is equal to 1.
C-1-c. Action of the various operators
The observablesandare linear combinations of the operatorsand[formu-
las (B-1) and (B-7)]. Consequently, all physical quantities can be expressed in terms of
and. Now, the action ofandon the vectors is especially simple [see equations
(C-19) below]. In most cases, it is therefore desirable to use therepresentation to
calculate the matrix elements and mean values of the various observables.
With the phase conventions introduced in and
operators on the vectors of thebasis is given by:
=
+ 1 +1
=
1
(C-19a)
(C-19b)
We have already proved (C-19a): it suces to replaceby+ 1in equations (C-9)
and (C-11). To obtain (C-19b), multiply both sides of (C-9) on the left by the operator
and use (C-11):
=
1
1=
1
(+ 1) 1=
1 (C-20)
Comment:
The adjoint equations of (C-19a) and (C-19b) are:
=
+ 1 +1 (C-21a)
=
1 (C-21b)
Note thatdecreases or increasesby one unit depending on whether it acts on
the ket or on the bra. Similarly,increases or decreasesby one unit,
depending on whether it acts on the ketor on the bra.
513

CHAPTER V THE ONE-DIMENSIONAL HARMONIC OSCILLATOR
Starting with (C-19) and using (B-1) and (B-7), we immediately nd the expres-
sions for the ketsand :
=
~
1
2
(+) =
~2+ 1 +1+1 (C-22a)
=
~2
( ) =
~2+ 1 +11 (C-22b)
The matrix elements of the,,andoperators in the representation
are therefore:
=
1 (C-23a)
=
+ 1 +1 (C-23b)
=
~2+ 1 +1+1 (C-23c)
=
~2+ 1 +11 (C-23d)
The matrices representingandare indeed Hermitian conjugates of each other,
as can be seen from their explicit expressions:
() =
0
1 0 0 0
0 0
2 0 0
0 0 0
3 0
0 0 0 0 00 0 0 0 0 0
(C-24a)
514

C. EIGENSTATES OF THE HAMILTONIAN
and:
() =
0 0 0 0 0
1 0 0 0 0
0
2 0 0 0
0 0
3 0 0
0
0 0 0 0
+ 1 0
(C-24b)
As for the matrices representingand, they are both Hermitian: the matrix associated
withis, to within a constant factor, the sum of the two preceding ones; the matrix
associated withis proportional to their dierence, but the presence of the factorin
(C-22b) re-establishes its Hermiticity.
C-2. Wave functions associated with the stationary states
We shall now use the representation and write the functions() =
which then represent the eigenstates of the Hamiltonian.
We have already determined the function0()which represents the ground state
0(cf. ):
0() = 0=
~
14
e
1
2~
2
(C-25)
The constant that appears before the exponential insures the normalization of0().
To obtain the functions()associated with the other stationary states of the
harmonic oscillator, all we need to do is use expression (C-13) for the ketand the
fact that, in therepresentation,is represented by:
1
2~~
d
d
sinceis represented by multiplication by, andby
~
d
d
[formula (B-6b)]. Thus we
obtain:
() = =
1
!
()0
=
1
!
1
2~~
d
d
0() (C-26)
515

CHAPTER V THE ONE-DIMENSIONAL HARMONIC OSCILLATOR0 1
1
1/4
mω
ħ
2 0 1
1 1
2 3 0 1 2 3
x
φ
0
x x
1/4
mω
ħ
φ
1
1/4
mω
ħ
φ
2
Figure 4: Wave functions associated with the rst three levels of a harmonic oscillator.
Figure 5: Probability densities associated with the rst three levels of a harmonic oscil-
lator.
that is:
() =
1
2!
~
12
~
14
~
d
d
e
1
2~
2
(C-27)
It is easy to see from this expression that()is the product ofe
1
2~
2
and a poly-
nomial of degreeand parity(1), called aHermite polynomial(cf.ComplementsV
andV).
A simple calculation gives the rst several functions():
1() =
4
~
3
14
e
1
2~
2
2() =
4~
14
2
~
2
1e
1
2~
2
(C-28)
These functions are shown in Figure, and the corresponding probability densities in
Figure. Figure ()and that of the probability
density()
2
for= 10.
We see from these gures that whenincreases, the region of theaxis in
which()takes on non-negligible values becomes larger. This corresponds to the fact,
516

C. EIGENSTATES OF THE HAMILTONIAN
in classical mechanics, that the amplitude of the particle's motion increases with the
energy [cf.Fig. A-8)]. It follows that the mean value of the potential
energy grows with[cf.comment()of ], since(), whenis large, takes on
non-negligible values in regions of the-axis where()is large. Moreover, we see in
these gures that the number of zeros of()is(cf.ComplementV, where this
property is derived). This implies that the mean kinetic energy of the particle increases
with[cf.comment()of ], since this energy is given by:
1
2
2
=
~
2
2
+
()
d
2
d
2
() d (C-29)
When the number of zeros of()increases, the curvature of the wave function in-
creases, and, in (C-29), the second derivative
d
2
d
2
()takes on larger and larger values.
Finally, whenis large, we observe (see, for example, Figure) that the probability
density()
2
is large for [whereis the amplitude of the classical motion
of energy;cf.(A-8)]. This result is related to a feature of the motion predicted by
classical mechanics: the classical particle has a zero velocity at= ; therefore, on
the average, it spends more time in the neighborhood of these two points than in the
center of the interval .xω
xω
x
M
ωx
M
ω
mω
mω
ħ
ħ
φ
10
φ
10
1/4
1/2
2
0.6
a
b
0.4
0.3
– 1– 0 1 2 3 +
0.2
Figure 6: Shape of the wave function (g. a) and of the probability density (g. b) for
the= 10level of a harmonic oscillator.
517

CHAPTER V THE ONE-DIMENSIONAL HARMONIC OSCILLATOR
D. Discussion
D-1. Mean values and root mean square deviations of andin a state
Neithernorcommutes with, and the eigenstatesofare not eigen-
states ofor. Consequently, if the harmonic oscillator is in a stationary state,
a measurement of the observableor the observablecan,a priori, yield any result
(since the spectra ofandinclude all real numbers). We shall now calculate the mean
values ofandin such a stationary state and then their root mean square deviations
and, which will enable us to verify the uncertainty relation.
As we indicated in , we shall perform these calculations with the help of
the operatorsand. As far as the mean values ofandare concerned, the result
follows directly from formulas (C-22), which show that neithernorhas diagonal
matrix elements:
= 0
= 0 (D-1)
To obtain the root mean square deviationsand, we must calculate the mean
values of
2
and
2
:
()
2
=
2
( )
2
=
2
()
2
=
2
( )
2
=
2
(D-2)
Now, according to (B-1) and (B-7):
2
=
~
2
(+)(+)
=
~
2
(
2
+ + +
2
)
2
=
~
2
( )( )
=
~
2
(
2
+
2
) (D-3)
The terms in
2
and
2
do not contribute to the diagonal matrix elements, since
2
is proportional to2, and
2
to +2; both are orthogonal to. On the
other hand:
(+) = (2+ 1)
= 2+ 1 (D-4)
Consequently:
()
2
=
2
= +
1
2
~
(D-5a)
()
2
=
2
= +
1
2
~ (D-5b)
518

D. DISCUSSION
The productis therefore equal to:
= +
1
2
~ (D-6)
We again nd (cf.Complement III) that it is greater than or equal to~2. In fact,
this lower bound is attained for= 0, that is, for the ground state (
Comments:
()If denotes the amplitude of the classical motion whose energy is given by
= (+ 12)~, it is easy to see, using (A-8) and (D-5a), that:
=
1
2
(D-7)
Similarly, ifdenotes the oscillation amplitude of the corresponding classical
momentum:
= (D-8)
we obtain:
=
1
2
(D-9)
It is not surprising thatis of the order of the interval[ +]over
which the classical motion occurs (cf.Fig.): we saw at the end of , that it is
approximately inside this interval that()takes on non-negligible values. Fur-
thermore, it is easy to understand why, whenincreases, so does. For large,
the probability density()
2
has two symmetric peaks situated approximately
at= . The root mean square deviation cannot be much smaller than the
distance between these peaks, even if each of them is very sharp (cf.Chap.,
III). An analogous argument
can be set forth for(cf.ComplementV).
()The mean potential energy of a particle in the stateis:
()=
1
2
22
(D-10)
that is, sinceis zero [cf.(D-1)]:
()=
1
2
2
()
2
(D-11)
Similarly, we could nd the mean kinetic energy of this particle:
2
2
=
1
2
()
2
(D-12)
519

CHAPTER V THE ONE-DIMENSIONAL HARMONIC OSCILLATOR
Substituting relations (D-5) into (D-11) and (D-12), we obtain:
()=
1
2
+
1
2
~=
2
2
2
=
1
2
+
1
2
~=
2
(D-13)
The mean potential and kinetic energies are therefore equal. This is an illustration
of the virial theorem (cf.exercise 10 of ComplementIII).
()A stationary statehas no equivalent in classical mechanics: its energy is not
zero although the mean valuesand are. Nevertheless, there is a certain
analogy between the stateand that of a classical particle whose position is
given by (A-5) [whereis related to the energyby relation (A-8)], but for
which the initial phaseof the motion is chosen at random (all values included
between 0 and2have the same probability). The mean values ofandare
then zero, since:
=
1
2
2
0
cos( ) d= 0
=
1
2
2
0
sin( ) d= 0
(D-14)
Moreover, we nd, for the root mean square deviations of the position and the
momentum, values identical to those of the state[formulas (D-7) and (D-9)]:
2
=
2
1
2
2
0
cos
2
( ) d=
2
2
2
=
2
1
2
2
0
sin
2
( ) d=
2
2
(D-15)
that is:
=
2
()
2
=
2
=
2
()
2
=
2
(D-16)
D-2. Properties of the ground state
In classical mechanics, the lowest energy of the harmonic oscillator is obtained
when the particle is at rest (zero momentum and kinetic energy) at the-origin (= 0
and therefore zero potential energy). The situation is completely dierent in quantum
mechanics: the minimum energy state is0, whoseenergy is not zero, and the associated
wave function has a certainspatial extension, characterized by the root mean square
deviation=
~2.
This essential dierence between the quantum and classical results can be seen to
have its source in the uncertainty relations, which forbid the simultaneous minimization
520

D. DISCUSSION
of the kinetic energy and the potential energy. As we pointed out in ComplementsI
andIII, the ground state corresponds to a compromise in which the sum of these two
energies is as small as possible.
In the special case of a harmonic oscillator, it is possible to state these qualitative
considerations semi-quantitatively, and thus nd the order of magnitude of the energy
and the spatial extension of the ground state. If the distancecharacterizes this spatial
extension, the mean potential energy will be of the order of:
1
2
22
(D-17)
Butis then equal to about~, so the mean kinetic energy is approximately:
=
2
2
~
2
2
2
(D-18)
The order of magnitude of the total energy is therefore:
=+
~
2
2
2
+
1
2
22
(D-19)
The variation of
,andwith respect tois shown in Figure. For small
values of,
prevails over; the opposite occurs for large values of. The ground state
therefore corresponds approximately to the minimum of the function (D-19); it is easy
to see that this minimum occurs at:
~
(D-20)
and is equal to:
~ (D-21)
We again nd the correct orders of magnitude of0andin the state0.0
T
V
ξ
m
ξ
Figure 7: Variation of the potential energy
and of the kinetic energywith respect
to a parametercharacterizing the spatial
extension of the wave function about= 0.
Since the potential energy is at a minimum
at= 0,
is a function that increases with
(
2
). On the other hand, according to
Heisenberg's uncertainty relation, the kinetic
energy
is a decreasing function of. The
lowest possible total energy, obtained for=
, results from a compromise in which the
sum
+(solid line) is at a minimum.
521

CHAPTER V THE ONE-DIMENSIONAL HARMONIC OSCILLATOR
The harmonic oscillator possesses the pecularity that, because of the form of the
potential(), the productactually attains its lower bound,~2, in the ground
state0[formula (D-6)]. This is related to the fact (cf.ComplementIII) that the
wave function of the ground state is Gaussian.
D-3. Time evolution of the mean values
Consider a harmonic oscillator whose state at= 0is:
(0)=
=0
(0) (D-22)
((0)is assumed to be normalized). Its state()atcan be obtained by using rule
(D-54) of Chapter:
()=
=0
(0) e
~
=
=0
(0) e
+
1
2
(D-23)
The mean value of any physical quantityis therefore given as a function of time
by:
()()=
=0=0
(0)(0) e
( )
(D-24)
with:
= (D-25)
Sinceandare integers, the time evolution of the mean values involves only the
frequency2and its various harmonics, which constitute the Bohr frequencies of the
harmonic oscillator.
Let us consider, in particular, the mean values of the observablesand. Ac-
cording to formulas (C-22), the only non-zero matrix elementsand are those
for which= 1. Consequently, the mean values ofandinclude only terms
ine; they are sinusoidal functions of time with angular frequency. This obviously
relates to the classical solution of the harmonic oscillator problem. Moreover, as we
pointed out in the discussion of Ehrenfest's theorem (Chap., ), the form of
the harmonic oscillator potential implies that for allthe mean values ofand
rigorously satisfy the classical equations of motion. Thus, according to general formulas
(D-34) and (D-35) of Chapter:
d
d
=
1
~
[]= (D-26a)
d
d
=
1
~
[]=
2
(D-26b)
522

D. DISCUSSION
If we integrate these equations, we obtain:
() =(0) cos+
1
(0) sin
() =(0) cos+ (0) sin (D-27)
We again nd the sinusoidal form indicated by formula (D-24).
Comment:
It is important to note that this analogy with the classical situation appears only
when(0)is a superposition of statesof the type of (D-22), where several
coecients(0)are non-zero. If all these coecients except one are equal to zero,
the oscillator is in a stationary state and the mean values of all the observables
are constant over time.
It follows that, in a stationary state, the behavior of a harmonic oscillator is
totally dierent from that predicted by classical mechanics, even ifis very large
(the limit of large quantum numbers). If we want to construct a wave packet whose
average position oscillates over time, we must superpose dierent states(see
ComplementV).
References and suggestions for further reading:
Dirac (1.13), 34; Messiah (1.17), Chap. XII.
523

COMPLEMENTS OF CHAPITER V, READER'S GUIDE
AV: SOME EXAMPLES OF HARMONIC
OSCILLATORS
Demonstrates, with some examples chosen from
various elds, the importance of the quantum
mechanical harmonic oscillator in physics. Semi-
quantitative and rather simple; recommended for
a rst reading.
BV: STUDY OF THE STATIONARY STATES
IN THE REPRESENTATION. HERMITE
POLYNOMIALS
Technical study of the stationary wave functions
of the harmonic oscillator. Intended to serve as a
reference.
CV: SOLVING THE EIGENVALUE EQUATION
OF THE HARMONIC OSCILLATOR BY THE
POLYNOMIAL METHOD
Another method that yields the results of
Chapter. Reveals the relation between energy
quantization and the behavior of the wave
functions at innity. Moderately dicult.
DV: STUDY OF THE STATIONARY STATES IN
THE REPRESENTATION
Shows that, in a stationary state of the harmonic
oscillator, the momentum probability distribution
has the same form as that of the position. Fairly
simple.
EV: THE ISOTROPIC THREE-DIMENSIONAL
HARMONIC OSCILLATOR
Generalisation of the results of Chapter
three dimensions. Recommended for a rst
reading, since it is simple and important.
FV: A CHARGED HARMONIC OSCILLATOR
PLACED IN A UNIFORM ELECTRIC FIELD
A direct and simple application of the results of
Chapter , which uses the trans-
lation operator introduced in ComplementII).
Recommended for a rst reading.
GV: COHERENT QUASI-CLASSICAL STATES
OF THE HARMONIC OSCILLATOR
Detailed study of the quasi-classical states
of the harmonic oscillator, which illustrates
the relation between quantum and classical
mechanics. Important because of its applications
to the quantum theory of radiation. Moderately
dicult; can be omitted in a rst reading.
HV: NORMAL VIBRATIONAL MODES OF TWO
COUPLED HARMONIC OSCILLATORS
Study, in the very simple case of two coupled
harmonic oscillators, of the normal vibrational
modes of a system. Recommended, since it is
simple and physically important.
525

JV: VIBRATIONAL MODES OF AN INFINITE
LINEAR CHAIN OF COUPLED HARMONIC OSCIL-
LATORS; PHONONS
KV: VIBRATIONAL MODES OF A CONTIN-
UOUS PHYSICAL SYSTEM. APPLICATION TO
RADIATION; PHOTONS
JV, V: Introduction, using simplied models,
of concepts that are particularly important in
physics. Rather dicult (graduate level); can be
reserved for later study.
JV: determination of the normal vibrational
modes of a linear chain of coupled oscillators, lead-
ing to the notion of phonon, fundamental in solid
state physics.
KV: normal vibrational modes of a continuous
system. A simple way to introduce photons in the
quantum mechanical study of the electromagnetic
eld.
LV: THE ONE-DIMENSIONAL HARMONIC OS-
CILLATOR IN THERMODYNAMIC EQUILIBRIUM
AT A TEMPERATURE
Application of the density operator (introduced
in ComplementIII) to a harmonic oscillator in
thermal equilibrium. Important from a physical
point of view, but requires knowledge ofIII.
MV: EXERCISES
526

SOME EXAMPLES OF HARMONIC OSCILLATORS
Complement AV
Some examples of harmonic oscillators
1 Vibration of the nuclei of a diatomic molecule
1-a Interaction energy of two atoms
1-b Motion of the nuclei
1-c Experimental observations of nuclear vibration
2 Vibration of the nuclei in a crystal
2-a The Einstein model
2-b The quantum mechanical nature of crystalline vibrations
3 Torsional oscillations of a molecule: ethylene
3-a Structure of the ethylene molecule C2H4. . . . . . . . . . . .
3-b Classical equations of motion
3-c Quantum mechanical treatment
4 Heavy muonic atoms
4-a Comparison with the hydrogen atom
4-b The heavy muonic atom treated as a harmonic oscillator
4-c Order of magnitude of the energies and spread of the wave
functions
We mentioned in the introduction to Chapter
study of the harmonic oscillator are applicable to numerous cases in physics, especially
those concerning small oscillations of a system about a position of stable equilibrium
(where the potential energy is at a minimum). The aim of this complement is to describe
some examples of such oscillations and to point out their physical importance: vibration
of the nuclei in a diatomic molecule or a crystalline lattice, torsional oscillations in a
molecule, motion of a muoninside a heavy nucleus. We do not intend to discuss
these phenomena in great detail here. We shall conne ourselves to a simple, qualitative
discussion.
1. Vibration of the nuclei of a diatomic molecule
1-a. Interaction energy of two atoms
The formation of a molecule from two neutral atoms occurs because the interaction
energy()of these two atoms has a minimum (is the distance between them). The
form of()is shown in Figure. Whenis very large, the two atoms do not interact
and()approaches a constant which we shall choose as the energy origin. Then, as
decreases,()varies approximately like1
6
: the corresponding attractive forces are
the Van der Waals forces (which we shall study in ComplementXI). Whenbecomes
so small that the electronic wave functions overlap,()decreases faster and passes
through a minimum at=; it then increases and becomes very large asapproaches
zero.
527

COMPLEMENT A V0
V (r)
– V
0
r
e
r
Figure 1: Form of the interaction potential
between two atoms that can form a stable
molecule. Classically,0is the dissociation
energy of the molecule and, the distance
between the two nuclei in the equilibrium po-
sition. In quantum mechanics, one obtains
vibrational states (the horizontal lines inside
the well) whose energies are all greater than
0.
The minimum of()is responsible for the phenomenon of the chemical bond that
can form between the two atoms. We have already pointed out, in
(taking the molecule
+
2
as an example), that the cause of this lowering of the energy is
a delocalization phenomenon of the electronic states (quantum resonance) which allows
the electrons to prot from the attraction of the two nuclei. The rapid rise of()at
small distances is due to the repulsion of the nuclei.
If the nuclei were classical particles, they would have stable equilibrium positions
separated by=. The depth0of the potential well at=is called, classically,
the dissociation energy of the molecule: it is, in fact, the energy that must be furnished
to the two atoms in order to separate them. The larger0, the more stable the molecule.
The theoretical and experimental determination of the curve of Figure
important problem in atomic and molecular physics. We shall see that, by studying the
vibrations of the nuclei, we get a certain amount of information about this curve.
Comment:(the Born-Oppenheimer approximation)
The quantum mechanical description of a diatomic molecule is actually a very complex
problem; it involves nding the stationary states of a system of particles, the nuclei and
the electrons, all interacting with each other. In general, it is impossible to solve the
Schrödinger equation for such a system exactly. A signicant simplication arises from
the fact that the mass of the electrons is much smaller than that of the nuclei. It follows
from this that one can, in a rst approximation, study the two motions separately. One
begins by determining the motion of the electrons for a xed value of the distance
between the two nuclei; thus one obtains a series of stationary states for the electronic
system, of energies1(),2()... Then one considers the ground state, of energy1(),
of the electronic system; whenvaries because of the motion of the nuclei, the electronic
system always remains in the ground state, for all. This means that the system's wave
function adapts itself instantaneously to any change in: the electrons, which are very
mobile, are said to follow adiabatically the motion of the nuclei. In the study of this
motion, the electronic energy1()then plays the role of a potential energy of interaction
between these two nuclei. This interaction potential depends on the distance between
the nuclei,, and adds to their electrostatic repulsion12
2
(where1and2are
the atomic numbers of the two nuclei; we have set
2
=
2
40, whereis the charge
of the electron). The total potential energy()of the system of the two nuclei, which
528

SOME EXAMPLES OF HARMONIC OSCILLATORS
enables us to determine their motion, is then:
() =1() +
12
2
(1)
It is this function that is shown in Figure
1-b. Motion of the nuclei
. Separation of the rotational and vibrational motions
We are thus faced with a problem that involves the motion of two particles of masses
1and2, whose interaction is described by the potential()of Figure
only on the distance between them. The problem is complicated by the existence of
several degrees of freedom: vibrational (variation of) and rotational (variation of the
polar anglesandwhich give the direction of the axis of the molecule). In addition,
these degrees of freedom are coupled: when the molecule vibrates, its moment of inertia
changes because of the variation of, and the rotational energy is modied.
If we conne ourselves to small amplitude vibrations, it can be shown that the
coupling between vibrational and rotational degrees of freedom is negligible since the
relative variation of the moment of inertia is very small during the vibration. The
problem is then reduced (as we shall see in detail in ComplementVII) to two independent
problems: in the rst place, the study of the rotation of a dumbbell
1
composed of two
masses1and2separated by a xed distance; plus a one-dimensional problem (in
whichis the only variable) involving a ctitious particle whose massis equal to the
reduced mass of1and2(cf.Chap., ):
=
12
1+2
(2)
moving in the potential()of Figure. We must then solve the eigenvalue equation:
~
2
2
d
2
d
2
+()() = () (3)
We shall concentrate on the latter problem here.
. Vibrational states
If we conne ourselves to small amplitude oscillations, we can make a limited
expansion of()in the neighborhood of its minimum, at=:
() = 0+
1
2
()( )
2
+
1
6
()( )
3
+ (4)
The discussion in
terms in expression (4), we are left with the equation of a one-dimensional harmonic
oscillator centered at=, of angular frequency:
=
()
(5)
1
We shall study this system (also called a rigid rotator) quantum mechanically in ComplementVI,
once we have introduced angular momentum.
529

COMPLEMENT A V
The vibrational states, shown by the horizontal lines in Figure, therefore
have energies given by:
= +
1
2
~ 0 (6)
where= 0, 1, 2, ... (is used instead ofin the notation of molecular vibrations).
According to the discussion in , the mean value of the
distance between the two nuclei oscillates aboutwith a frequency of=2which
can thus be seen to be the vibrational frequency of the molecule.
Comments:
()Even in the ground state, the wave function of a harmonic oscillator has a nite
spread, of the order of
~2(cf. ). The distance between the
two nuclei of the molecule in the vibrational ground state is therefore dened only
to within
~2. An important condition for the decoupling of the vibrational
and rotational degrees of freedom is therefore that:
~2
(7)
()When the reduced mass is known, the measurement ofyields, according to
(5), the second derivative(). When the quantum numberincreases, it is no
longer possible to neglect terms in( )
3
in expression (4) (which indicate the
deviation of the potential well from a parabolic form). The oscillator then becomes
anharmonic. Studying the eects of the term in( )
3
of (4) by perturbation
theory (as we shall do in ComplementXI), one nds that the separation+1
of two neighboring states is not the same for large and small values of. Studying
the variation of+1 with respect toenables us to obtain the coecient
()of the term in( )
3
. Thus we see how the study of the frequencies of
molecular vibration enables us to dene more precisely the form of the curve()
in the neighborhood of its minimum.
. Order of magnitude of the vibrational frequencies
Molecular vibrational frequencies are commonly expressed in cm
1
, by giving the inverse
of the wavelength(expressed in cm) of an electromagnetic wave of the same frequency. Note
that 1 cm
1
corresponds to a frequency of310
10
Hertz and to an energy of12410
4
eV.
The vibrational frequencies of diatomic molecules fall between several tens and several
thousands of cm
1
. The corresponding wavelengths therefore go from a few microns to a few
hundred microns, consequently falling in the infrared.
Formula (5) shows that asdecreases,increases. This frequency also increases with
(), that is, with a greater curvature of the potential well at=. Sinceis always
of the same order of magnitude (a few

A),()increases with the depth0of the well:
therefore increases with the chemical stability. We shall consider some concrete illustrations of
the preceding observations.
The vibrational frequencies of the hydrogen and deuterium molecules (H2and D2) are,
respectively (not taking into account anharmonicity corrections):
2= 4 401 cm
1
2= 3 112 cm
1
(8)
530

SOME EXAMPLES OF HARMONIC OSCILLATORS
The curve()is the same in these two cases: the chemical bond between the two atoms
depends only on the electronic atmosphere. However, the reduced mass of H2is half as large
as that of D2. We must therefore have, according to (),
2=
2
2. This is in agreement
with the experimental values (8).
Now let us consider an example of two molecules that have about the same reduced mass
but very dierent chemical stabilities. The molecule
79
Br
85
Rb is chemically stable (halogen-
alkaline bond); its vibrational frequency is 181 cm
1
. Molecules of
84
Kr
85
Rb have been observed
recently in optical pumping experiments. Their chemical stability is much lower, because kryp-
ton, which is a rare gas, is practically inert from a chemical point of view (in fact, the cohesion
of the molecule is due only to Van der Waals forces). These molecules have been found to have
a vibrational frequency of the order of 13 cm
1
. The considerable dierence between this gure
and the preceding one is due solely to the dierence in chemical stability of the two types of
molecules since the reduced masses are, to within a few per cent, practically the same.
1-c. Experimental observations of nuclear vibration
We shall now explain how nuclear vibration can be detected experimentally. In
particular, we shall consider the interaction of the molecule with an electromagnetic
wave.
. Infrared absorption and emission
First, let us assume the molecule to beheteropolar(composed of two dierent
atoms). Since the electrons are attracted towards the more electronegative atom, the
molecule generally has a permanent dipole moment()which depends on the distance
between the two nuclei. Expanding()in the neighborhood of the equilibrium position
=, we obtain:
() =0+1( ) + (9)
where0and1are real constants.
When the molecule is in a linear superposition()of several stationary vibra-
tional states, the mean value()()()of its electric dipole moment oscillates
about the value0with a frequency of2. The oscillatory term arises from the mean
value of the term1( )of (9) ( plays the same role in our problem as the
observableof the harmonic oscillator studied in ). Now( )has
a non-zero matrix element between two statesand only when =1. This
selection rule enables us to understand why only one Bohr frequency=2appears
in the motion of()()[the harmonic frequencies evidently appear when one takes
into account the anharmonicity of the potential and terms of higher order in expansion
(9); their intensity is however much weaker].
This vibration of the electric dipole moment results in a coupling between the
molecule and the electromagnetic eld; the molecule can consequently absorb or emit
radiation of frequency. In terms of photons, the molecule can absorb a photon of
energyand move from the stateto the state+1(Fig.-a) or emit a photon
by going from to 1(Fig.-b).
531

COMPLEMENT A V
Figure 2: Absorption (g. a) or emission (g. b) of a photon by a heteropolar molecule
going from the vibrational stateto the state+1or 1.
. The Raman eect
Now let us consider ahomopolarmolecule (consisting of two identical atoms).
Because of symmetry, the permanent electric dipole moment is then zero for all, and
the molecule is inactive in the infrared.
Imagine that an optical wave of frequency2strikes this molecule. This fre-
quency, much higher than those considered previously, is able to excite the electrons
of the molecule; under the eect of the optical wave, the electrons will undergo forced
oscillation and re-emit radiation of the same frequency in all directions. This is the well-
known phenomenon of the molecular scattering of light (Rayleigh scattering)
2
. What
new phenomena are produced by the vibration of the molecule?
What happens can be explained qualitatively in the following way. The electronic
susceptibility
3
of the molecule is generally a function of the distancebetween the two
nuclei. Whenvaries (recall that this variation is slow compared to the motion of the
electrons), the amplitude of theinducedelectric dipole moment, which vibrates at a
frequency of2, varies. The time dependence of the dipole moment is therefore that
of a sinusoid of frequency2whose amplitude is modulated at the frequency of the
molecular vibration2, which is much smaller (Fig.). The frequency distribution of
the light emitted by the molecule is given by the Fourier transform of the motion of the
electric dipole shown in Figure. It is easy to see (Fig.) that there exists a central line
of frequency2(Rayleigh scattering) and two shifted lines, of frequency( )2
(Stokes Raman scattering) and frequency( +)2(anti-Stokes Raman scattering).
It is very simple to interpret these lines in terms of photons. Consider an optical
photon of energy~which strikes the molecule when it is in the state(Fig.-a).
If the molecule does not change vibrational states during the scattering process, the
scattering is elastic. Because of conservation of energy, the scattered photon has the
2
In ComplementXIII, we shall use quantum mechanics to study the forced motion of the electrons
of an atom under the eect of incident light waves.
3
Under the eect of the eldE0e

of the incident optical wave, the electronic cloud of the molecule
acquires an induced dipole momentDgiven by:
D=()E0e

()is, by denition, the electronic susceptibility of the molecule. The important point here is that
depends on.
532

SOME EXAMPLES OF HARMONIC OSCILLATORS
same energy as the incident photon (Fig.-b: Rayleigh line). However, the molecule,
during the scattering process, can make a transition from the stateto the state
+1. The molecule acquires an energy~at the expense of the scattered photon,
whose energy therefore is~( )(Fig.-c): the scattering isinelastic(Stokes Raman
line). Finally, the molecule may move from the stateto the state 1, in which
case the scattered photon will have an energy of~( +)(Fig.-d: anti-Stokes Raman
line).
Comments:
()The Raman eect can also be observed with heteropolar molecules.
()The Raman eect has enjoyed a revival of interest because of the development of
lasers. If, in the cavity of a laser oscillating at a frequency of2, one places a
cell lled with a substance that exhibits the Raman eect, one can, in certain cases,
obtain an amplication (stimulated Raman eect) and hence a laser oscillation at
the frequency( )2, whereis the vibrational frequency of the molecules in
the cell (Raman laser). Thus, by varying this substance, one can vary the oscillation
frequency of the laser.
()The study of Raman and infrared spectra of molecules is useful in chemistry because
it permits the identication of the various bonds which exist in a complex molecule.D
0
t
Figure 3: The vibration of a molecule modu-
lates the amplitude of the oscillating electric
dipole induced by an incident light wave.Ω – ω
Ω
2π
2π
Ω + ω
2π
Figure 4: Spectrum of the oscillations shown in Figure. In addition to the central line,
whose frequency is the same as that of the incident light wave (Rayleigh line), two shifted
lines appear (the Stokes and anti-Stokes Raman lines). The frequency shift is equal to
the vibrational frequency of the molecule.
533

COMPLEMENT A V
Figure 5: Schematic representation of the scattering of a photon of energy~by a
molecule which is initially in the vibrational state(g. a): Rayleigh scattering without
a change in the vibrational state (g. b); Stokes or anti-Stokes Raman scattering with a
change in the molecule's state fromto +1(g. c) or to1(g. d).
For example, the vibration frequency of a group of two carbon atoms depends on
whether the bond between them is single, double or triple.
2. Vibration of the nuclei in a crystal
2-a. The Einstein model
A crystal consists of a system of atoms (or ions) which are regularly distributed in
space, forming a periodic lattice. For simplicity, let us choose a one-dimensional model
in which we consider a linear chain of atoms.
The average position of the nucleus of theth atom is:
0
= (10)
whereis the distance between adjacent atoms (on the order of a few

A).
Let(1 2 )be the total potential energy of the crystal nuclei, which
depends on their positions1 2 If
0
is not too large, that is, if each
nucleus is not too far from its equilibrium position,(1 2 )has, in certain
cases, the following simple form:
(1 2 ) 0+
1
2
(
0
)
2
0+ (11)
where0and
0are real constants (with
00). The absence of terms linear in
0
shows that
0
is a stable equilibrium position for the nucleus()(a minimum of). We
add to (11) the total kinetic energy:
=
2
2
(12)
534

SOME EXAMPLES OF HARMONIC OSCILLATORS
whereis the momentum of the nucleus()of mass. The total Hamiltonianof the
system is, to within the constant0, a sum of Hamiltonians of one-dimensional harmonic
oscillators centered at each nucleus():
=0+
2
2
+
1
2
(
0
)
2
0 (13)
Consequently, in this simplied model, each nucleus vibrates about its equilibrium
position independently of its neighbors, with an angular frequency:
=
0
(14)
As in the case of the diatomic molecule,increases whendecreases and when the cur-
vature of the potential attracting the nucleus towards its equilibrium position increases.
Comment:
In the simple model we have just presented, each nucleus vibrates independently of the
others. This is because the proposed potentialdoes not contain any terms which are
simultaneously dependent on more than one of the variables, as it would if it described
internuclear interactions. This model is not realistic since such interactions do, in fact,
exist. In ComplementV, we shall present a more elaborate model which takes into
account the coupling between each nucleus and its two nearest neighbors. We shall see
that it is still possible, in this model, to put the total Hamiltonian of the system in the
form of a sum of Hamiltonians of independent harmonic oscillators.
2-b. The quantum mechanical nature of crystalline vibrations
Despite its very schematic character, the Einstein model enables us to understand
a certain number of phenomena related to the quantum mechanical nature of crystalline
vibrations. The low temperature behavior of the constant volume specic heat, which
cannot be explained using classical mechanics, will be described in ComplementVin
connection with the study of the properties of a harmonic oscillator in thermodynamic
equilibrium. In the present complement, we shall discuss a spectacular eect related to
the nite spread of the wave functions associated with the position of each atom in the
ground state.
At absolute zero, under a pressure of one atmosphere, all substances except helium
are solids. To solidify helium, it is necessary to apply a pressure of at least 25 atmo-
spheres. Can this peculiarity be explained qualitatively?
First let us try to understand the phenomenon of the melting of an ordinary sub-
stance. At absolute zero, the atoms are practically localized at their equilibrium posi-
tions; the spread of their wave functions about the
0
is given by [cf.formula (D-5a) of
Chapter]:

~2
=
~
2
4
0
14
(15)
535

COMPLEMENT A VCC
H
H H
H
Figure 6: Plane structure of the ethylene
molecule.
[where we have used expression (14) for].is, in general, very small. When the
crystal is heated, the nuclei move into higher and higher vibrational states: in classi-
cal language, they vibrate with a larger and larger amplitude; in quantum mechanical
language, the spread of their wave functions increases [with the square root of the vi-
brational quantum number see formula (D-5a) of Chapter]. When this spread is no
longer negligible with respect to the interatomic distance, the crystal melts (see
of ComplementV, in which this phenomenon is studied more quantitatively).
It is impossible to solidify helium at ordinary pressures. This corresponds to the
fact that, even at absolute zero, the spread of the wave function given by (15) is not
negligible compared to. This results from the fact that the mass of helium is very small
and its chemical anity, very weak (the curvature
0of the potential in the neighborhood
of each minimum is very small, since the potential wells are very shallow). Both factors
produce the same eect in formula (15): a large spread. Now, an increase in the
pressure results in an increase in
0and therefore, in; consequently,decreases.
This is due to the fact that, at high pressures, each helium atom is wedged between
its neighbors: the smaller the average distance between these neighbors (the higher the
pressure), the sharper the potential minimum (the greater
0). Thus we see how an
increase in the pressure makes the solidication of helium possible.
3. Torsional oscillations of a molecule: ethylene
3-a. Structure of the ethylene molecule C2H4
The structure of the molecule C2H4is well-known: the six atoms of the molecule
are in the same plane (Fig.) and the angles between the variousCHandCCbonds
are close to 120.
Now imagine that, without changing the relative positions of the bonds of each
carbon atom, we rotate one of the CH2groups, about theCCaxis, through an angle
with respect to the other one. Figure CC
axis: theCHbonds of one CH2group are shown in solid lines and those of the other
one, in dashed lines. How does the potential energy()of the molecule vary with
respect to?
Since the stable structure of the molecule is planar, the angle= 0must corre-
spond to a minimum of(). It is also clear that=corresponds to another minimum
536

SOME EXAMPLES OF HARMONIC OSCILLATORSH Hα
H
H
C
C
Figure 7: Torsion of the ethylene molecule (seen
along theCCaxis): one of the CH2groups has
rotated with respect to the other one through an angle
about theCCaxis.
of(), since the two structures associated with= 0and=are undistinguishable.
()therefore has the form shown in Figurevaries from2to32;(0)is
chosen as the energy origin].
The two stable positions= 0and=are separated by a potential barrier of
height0. The potential of Figure
() =
0
2
(1cos 2) (16)
Comment:
Quantum mechanics enables us to interpret all the features of the C2H4molecule which we
have just described. In this molecule, each carbon atom has four valence electrons. Three
of these electrons (electrons) are found to have wave functions that are symmetrical
about three coplanar lines making angles of 120with each other, and dening the
directions of the chemical bonds (Fig.). These wave functions overlap those of the
electrons of the neighboring atoms to a considerable extent, and it is this overlap that
insures the stability of theCHbonds and of part of theCCbond (this phenomenon
is called sp
2
hybridation and will be studied in greater detail in ComplementVII).
The last valence electron of each carbon atom (electron) has a wave function which is
symmetrical about a line passing through C and perpendicular to the plane dened by
C and its three neighbors. The overlap of the wave functions of the twoelectrons is
maximum and, consequently, the chemical stability of the double bond is greatest when0
V
0
V (α)
π/2
α
– π/2 3π/2π
Figure 8: The potential energy of the molecule depends on the torsion angle;()is
minimal for= 0and=(planar structures).
537

COMPLEMENT A Vα
1
α
2
Figure 9: To write the classical equations of
motion, we denote by1and2the angles
formed by the planes of the two CH2groups
with a xed plane.
the two lines associated with theelectrons are parallel, that is, when the six atoms of
the molecules are in the same plane. The structure of Figure
Since()can be approximated by a parabola in the neighborhood of each of its
two minima, the molecule performs torsional oscillations about its two stable equilibrium
positions. We now examine them. First, we shall review rapidly the corresponding
classical equations.
3-b. Classical equations of motion
We denote by1and2the angles formed by the planes of the two CH2groups
with a xed plane passing through theCCaxis (Fig.). The angle in Figure
obviously:
=1 2 (17)
Letbe the moment of inertia of one of the CH2groups with respect to theCC
axis. Since the potential energy depends only on=1 2, the dynamical equations
describing the rotation of each group are written:
d
2
1
d
2
=
1
(1 2) =
d
d
()
d
2
2
d
2
=
2
(1 2) = +
d
d
()
(18)
Adding and subtracting these two equations we obtain:
d
2
d
2
(1+2) = 0 (19a)
d
2
d
2
=2
d
d
() (19b)
538

SOME EXAMPLES OF HARMONIC OSCILLATORS
Equation (19a) indicates that the entire molecule can rotate freely about theCC
axis independently of the torsional motion: the angle(1+2)2of the plane bisecting
the planes of the two CH2groups is a linear function of time. Equation (19b) describes
the torsional motion (rotation of one group with respect to the other). Let us consider
this motion in the immediate neighborhood of one of the stable equilibrium positions,
= 0. We expand expression (16) in the neighborhood of= 0:
() 0
2
(20)
Substituting (20) into (19b), we obtain:
d
2
d
2
+
40
= 0 (21)
We recognize (21) as the equation of a one-dimensional harmonic oscillator (is the only
variable) of angular frequency:
= 2
0
(22)
For the C2H4molecule,is of the order of 825 cm
1
.
3-c. Quantum mechanical treatment
In the neighborhood of its two equilibrium positions= 0and=, the molecule
possesses torsional states of quantized energy= (+12)~, with= 012In
a rst approximation, each energy level= (+12)~is therefore doubly degenerate,
since for each one there are two statesand whose wave functions()and
()dier only in that one is centered at= 0and the other, at=(Fig.-a
and-b).
In fact, we must also take into account a typically quantum mechanical eect: the
tunnel eect across the potential barrier separating the two minima (Fig.). We have
already encountered a situation of this type, in ComplementIV, in connection with
the inversion of the NH3molecule. Calculations analogous to those in that complement
could show that the degeneracy between the two statesand is removed by the
tunnel eect. Thus, for each value of, two stationary states,
+and , appear (to
a rst approximation, they are symmetrical and antisymmetrical linear combinations of
and ). The larger(that is, the closer the initial energyis to0, and hence,
the more important the tunnel eect), the greater their energy dierence~. However,
~is always much smaller than the distance~between adjacent levelsand 1
(Fig.).
For the mean value of the angle, quantum mechanics therefore predicts the
following motion: rapid oscillations of frequencyabout one of the two values= 0
and=, upon which are superposed much slower oscillations between= 0and
=, at the Bohr frequencies02,12,22...
Comment:
States of course exist for which the energy is greater than the maximum height0of
the potential barrier of Figure. These states correspond to a rotational kinetic energy
539

COMPLEMENT A V0
a
b
π/2 3 π/2π
φ
1
φ
0
φ
1
φ
0
V (α)
V (α)
α
0
π/2 3 π/2π
α
Figure 10: When one neglects the tunnel eect across the potential barriers at=2
and= 32, one can nd torsional states of the molecule localized in the wells centered
at= 0(g. a) and=(g. b).
that is large enough for one of the CH2groups to be considered as rotating almost freely
with respect to the other one (while being, nevertheless, periodically slowed down and
accelerated by the potential of Figure).
The ethane molecule C2H6behaves in this way. The absence ofelectrons in this
molecule permits a much freer rotation of one of the CH3groups with respect to the
other (the potential barrier0is much lower). In this case, the potential(), which
tends to oppose the free rotation of one of the CH3groups with respect to the other, has
a period of23because of symmetry.
540

SOME EXAMPLES OF HARMONIC OSCILLATORSE
ћδ
1 { ψ
+
1
, ψ
–
1
{ ψ
+
0
, ψ
–
0
ћδ
0
Figure 11: The tunnel eect removes the de-
generacy of the energy levels shown in Fig-
ure. As one approaches the top of the
barrier, this phenomenon becomes more im-
portant(1 0).
0
+,
0
,
1
+,
1
are the new stationary states.
4. Heavy muonic atoms
The muon (sometimes called, for historical reasons, the meson) is a particle which
has the same properties as the electron except that its mass is about 207 times greater
4
.
In particular, it is not sensitive to strong interactions, and its coupling with nuclei is
essentially electromagnetic. A muonwhich has been slowed down in matter can be
attracted by the Coulomb eld of an atomic nucleus and can form a bound state with
that nucleus. The system thus constituted is called a muonic atom.
4-a. Comparison with the hydrogen atom
In Chapter), we shall study the bound states of two particles of opposite
charge and, in particular, those of the hydrogen atom. We shall see that the results of
quantum mechanics concerning the energies of bound states are the same as those of
the Bohr model (Chap., ). Similarly, the spread of the wave functions which
describe these bound states is of the order of the Bohr orbital radius. Let us therefore
begin by using this simple model to calculate the energies and spreads of the rst bound
states of a muonin the Coulomb eld of a heavy atom such as lead (= 82,= 207).
If we consider the nucleus to be innitely heavy, theth Bohr orbital has an energy
of:
=
2 4
2~
2
1
2
(23)
whereis the atomic number of the nucleus,
2
=
2
40(whereis the electron
charge), andrepresents the mass of the electron or of the muon, depending on the case.
When one goes from hydrogen to the muonic atom under study here,is multiplied
by a factor of
2
(82)
2
2071410
6
. From this we deduce that, for the
4
The muon is unstable: it decays into an electron and two neutrinos.
541

COMPLEMENT A V
muonic atom:
1 19 MeV
2 47 MeV
(24)
The radius of theth Bohr orbital is given by:
=
2
~
2
2
(25)
For hydrogen,105

A. Here, this number must be divided by , which gives:
1310
13
cm
21210
13
cm
(26)
In the preceding calculations, we have implicitly assumed the nucleus to be point-
like (in the Bohr model and in the theory presented in Chapter, , the potential
energy is taken equal to
2
). The small values found for1and2[formulas (26)]
show us that this viewpoint is not at all valid for a heavy muonic atom. The lead nucleus
has a non-negligible radius0, on the order of8510
13
cm(recall that the radius of
a nucleus increases with
13
). The preceding qualitative calculation therefore leaves us
with the impression that the spread of the wave functions of the muon may be smaller
than the nucleus
5
. Consequently, we must reconsider the problem completely and rst
calculate the potential seen by the muon on theinsideas well as on the outside of the
nuclear charge distribution.
4-b. The heavy muonic atom treated as a harmonic oscillator
We shall use a rough model of the lead nucleus: we shall assume its charge to be
evenly distributed throughout a sphere of radius08510
13
cm.
When the distanceof the muon from the center of this sphere is greater than0,
its potential energy is given by:
() =
2
for 0 (27)
For 0, one can calculate the electrostatic force acting on the muon, using Gauss's
theorem; it is directed towards the center of the sphere and its absolute value is:
2
0
3
1
2
=
2
3
0
(28)
5
For hydrogen, the spread of the wave functions, on the order of an Angström, is about 10
5
times
larger than the dimensions of the proton, which can therefore be treated like a point. The new situation
encountered here results from several factors which reinforce each other: increasedand increased,
which results in a greater electrostatic force and a larger nuclear radius.
542

SOME EXAMPLES OF HARMONIC OSCILLATORS0
–
3Ze
2
2
V (r)
r
ρ
0
ρ
0
Figure 12: Form of the potential()seen
by a muon attracted by a nucleus of ra-
dius0situated at= 0. When 0,
the variation of the potential is parabolic
(if the charge density of the nucleus is uni-
form); when 0,()varies like1
(Coulomb's law).
This force is derived from the potential energy:
() =
1
2
2
3
0
2
+ for 0 (29)
The constantis determined by the condition that expressions (27) and (29) be identical
for=0:
=
3
2
2
0
(30)
Figure .
Inside the nucleus, the potential is parabolic. The orders of magnitude we calcu-
lated in
for the ground state of the muonic lead atom since the wave function is actually concen-
trated in the region where the potential is parabolic. It is therefore certainly preferable
to consider the muon to be elastically bound to the nucleus in this case. We then have
a three-dimensional harmonic oscillator (ComplementV) whose angular frequency is:
=
2
3
0
(31)
In fact, we shall see that the wave function of the ground state of this harmonic oscillator
is not zero outside the nucleus, so the harmonic approximation is not perfect either.
Comment:
It is interesting that the physical system studied here presents many analogies with the
rst atomic model, proposed by J. J. Thomson. This physicist assumed the positive
charge of the atom to be distributed in a sphere whose radius was of the order of a
few Angströms, with the electrons moving in the parabolic potential existing inside this
charge distribution (model of the elastically bound electron). We know from Rutherford's
experiments that the nucleus is much smaller and that such a model does not correspond
to reality for atoms.
543

COMPLEMENT A V
4-c. Order of magnitude of the energies and spread of the wave functions
If we substitute into expression (31) the numerical values:
= 82 310
8
msec
2
~
1
137
207 m18610
28
kg
~10510
34
Joule sec08510
15
m
we nd:
1310
22
radsec (32)
which corresponds to an energy~on the order of:
~ 84 MeV (33)
We can compare~to the total depth of the well
3
2
2
0
, which is equal to:
3
2
2
0
21 MeV (34)
We see that~is smaller than this depth, but not small enough for us to be able to
neglect completely the non-parabolic part of().
Similarly, the spread of the ground state, if the well were perfectly parabolic, would
be on the order of:
~2
4710
13
cm (35)
The qualitative predictions of 4-a are therefore conrmed: a large part of the
wave function of the muon is inside the nucleus. Nevertheless, what happens outside
cannot be completely neglected.
The exact calculation of the energies and the wave functions is therefore more
complicated than it would be for a simple harmonic oscillator. The Schrödinger equation
corresponding to the potential of Figure
addition, spin, relativistic corrections, etc...). Such a calculation is important: the study
of the energy of photons emitted by a heavy muonic atom contributes information about
the structure of the nucleus, for example concerning the real charge distribution inside
the nuclear volume.
Comment:
In the case of ordinary atoms (with an electron instead of a muon), it is valid to neglect the
eects of the deviation of the potential from the
2
law. However, one can take this
deviation into account by using perturbation theory (cf.Chap.). In ComplementXI,
we shall study this volume eect of the nucleus on the atomic energy levels.
544

SOME EXAMPLES OF HARMONIC OSCILLATORS
References and suggestions for further reading:
Molecular vibrations: Karplus and Porter (12.1), Chap. 7; Pauling and Wilson
(1.9), Chap. X; Herzberg (12.4), Vol. I, Chap. III, 1; Landau and Lifshitz (1.19),
Chaps. XI and XIII.
Stimulated Raman eect: Baldwin (15.19), 5.2; see also Schawlow's article
(15.17).
Torsion oscillations: Herzberg (12.4), Vol. II, Chap. II, 5d; Kondratiev (11.6),
37
The Einstein model: Kittel (13.2), Chap. 6; Seitz (13.4), Chap. III; Ziman (13.3),
Chap. 2; see also Bertman and Guyer's article (13.20).
Muonic atoms: Cagnac and Pebay-Peyroula (11.2), XIX-7; Weissenberg (16.19),
4-2; see also De Benedetti's article (11.21).
545

STUDY OF THE STATIONARY STATES IN THE X REPRESENTATION. HERMITE POLYNOMIALS
Complement BV
Study of the stationary states in therepresentation. Hermite
polynomials
1 Hermite polynomials
1-a Denition and simple properties
1-b Generating function
1-c Recurrence relations; dierential equation
1-d Examples
2 The eigenfunctions of the harmonic oscillator Hamiltonian
2-a Generating function
2-b ()in terms of the Hermite polynomials
2-c Recurrence relations
We now intend to study, in a little more detail than in , the
wave functions() = associated with the stationary statesof the harmonic
oscillator. Before undertaking this study, we shall dene the Hermite polynomials and
mention their principal properties.
1. Hermite polynomials
1-a. Denition and simple properties
Consider the Gaussian function:
() = e
2
(1)
represented by the bell-shaped curve in Figure. The successive derivatives ofare
given by:
() =2e
2
(2)
() = (4
2
2) e
2
(3)
Theth-order derivative,
()
(), can be written:
()
() = (1)() e
2
(4)
where()is anth-degree polynomial in. The proof is by recurrence. This relation
is valid for= 1, 2 [cf.equations (2) and (3)]. Assume it is true for1:
(1)
() = (1)
1
1() e
2
(5)
547

COMPLEMENT B V1
– 1 1
F
z
F
F
Figure 1: Shape of the Gaussian function
()and of its rst and second derivatives
()and().
where 1()is a polynomial of degree1. We then obtain relation (4) directly by
dierentiation, if we set:
() =2
d
d
1() (6)
Since 1()is a polynomial of degree1in, we see from this last relation that
()is indeed anth-degree polynomial. The polynomial()is called thenth-degree
Hermite polynomial. Its denition is therefore:
() = (1)e
2d
n
d
e
2
(7)
We see from (2) and (3) that1()and2()are, respectively, even and odd.
Moreover, relation (6) shows that if1()has a denite parity,()has the opposite
parity. From this, we deduce that the parity of()is(1).
The zeros of()correspond to those of theth-order derivative of the function
(). We are going to show that()hasreal zeros, between which one nds those
of 1. It can be seen from Figure 1), (2) and (3) that this is
true for= 012. Arguing by recurrence, we can generalize this result: assume that
1()has 1real zeros; if1and2are two consecutive zeros of1()and
therefore of
(1)
(), Rolle's theorem shows that the derivative
()
()of
(1)
()
goes to zero at a point3between1and2; therefore,(3) = 0. Since, in addition,
(1)
()goes to zero when and when +,
()
()and()have
at leastreal zeros [and not more, because()isth-degree] between which are
interposed those of1().
1-b. Generating function
Consider the function ofand:
(+) = e
(+)
2
(8)
548

STUDY OF THE STATIONARY STATES IN THE X REPRESENTATION. HERMITE POLYNOMIALS
Taylor's formula enables us to write:
(+) =
=0
!
()
()
=
=0
!
(1)()e
2
(9)
Multiplying this relation by
2
and replacingby we obtain:
e
2
( ) =
=0
!
() (10)
that is, if we replace( )by its value:
e
2
+2
=
=0
!
() (11)
The Hermite polynomials can therefore be obtained from the series expansion inof
the functione
2
+2
, which for this reason, is called the generating function of the
Hermite polynomials.
Relation (11) gives us another denition of the Hermite polynomials():
() =
e
2
+2
=0
(12)
1-c. Recurrence relations; dierential equation
We have already obtained, in (6), one recurrence relation. It is easy to obtain
others by dierentiating relation (11). A dierentiation with respect toyields:
2e
2
+2
=
=0
!
d
d
() (13)
that is, replacinge
2
+2
by the expansion (11) and setting equal terms of the same
power in:
d
d
() = 2 1() (14)
Similarly, if we dierentiate (11) with respect to, an analogous argument yields:
() = 2 1()2(1) 2() (15)
Finally, it is not dicult to obtain a dierential equation satised by the polyno-
mials(). Dierentiating (14) and using (6), we get:
d
2
d
2
() = 2
d
d
1()
= 2[2 1() ()] (16)
that is, replacing1()by its value as given in (14):
d
2
d
2
2
d
d
+ 2 () = 0 (17)
549

COMPLEMENT B V
1-d. Examples
Denition (7) or the recurrence relation (6) (which amounts to the same thing)
enables us to calculate the rst Hermite polynomials easily:
0() = 1
1() = 2
2() = 4
2
2
3() = 8
3
12 (18)
In general:
() =2
d
d
(1) (19)
2. The eigenfunctions of the harmonic oscillator Hamiltonian
2-a. Generating function
Consider the function:
() =
=0
!
(20)
Using the relation [cf.Chap., formula (C-13)]:
=
1
!
()0 (21)
we obtain:
() =
=0
()
!
0
=e 0 (22)
We now introduce, as in Chapter, the dimensionless operators
^
and
^
:
^
=
^
=
~
(23)
where the parameter, which has the dimensions of an inverse length, is dened by:
=
~
(24)
The operator:
e= e
2
(
^^
)
(25)
550

STUDY OF THE STATIONARY STATES IN THE X REPRESENTATION. HERMITE POLYNOMIALS
can be calculated by using formula (63) of ComplementII, where we set:
=
2
^
=
2
^
(26)
We obtain:
e= e
2
^
e
2
^
e
4
2
[
^^
]
= e
2
^
e
2
^
e
2
4
(27)
Substituting this result into (22), we nd:
() = e
2
4
e
(
2)
^
e
(2)
^
0
= e
2
4
e
2
e
(2) ~
0 (28)
Now, we have [cf.ComplementII, formula (15)]:
e
~2=2 (29)
and (28) can be written:
() = e
2
4
e
2
20
= e
2
4
e
2
0(2) (30)
Using formula (C-25) of Chapter, we nally obtain:
() =
2
14
exp
22
2
+
2
2
2
(31)
According to denition (20), all we must do to nd the wave functions() =
is expand this expression in powers of:
() =
=0
!
() (32)
()is called the generating function of the().
2-b. ()in terms of the Hermite polynomials
Replacing, in formula (11),by
2andby, we obtain:
exp
2
2
+
2=
=0
2
1
!
() (33)
551

COMPLEMENT B V
Substituting this expression into (31):
() =
2
14
=0
2
1
!
e
22
2
() (34)
Setting equal the coecients of the various powers ofin (32) and (34), we obtain:
() =
2
14
1
2!
e
22
2
() (35)
The shape of the function()is therefore analogous to that of theth-order deriva-
tive of the Gaussian function()considered in 1 above;()is of parity(1)
and possesseszeros interposed between those of+1(). We mentioned in
Chapter
the stateswhenincreases.
2-c. Recurrence relations
Let us write the equations:
=
1
=
+ 1 +1
(36)
in the representation. Using the denitions ofand[cf.Chap., relations
(B-6)], we see that in therepresentation, the action of these operators is given by:
=
2
+
1
2
d
d
=
2
1
2
d
d
(37)
Equations (36) therefore become:
2
+
1
2
d
d
() =
1()2
1
2
d
d
() =
+ 1 +1()
(38)
Let us take the sum and dierence of these equations:
2() =1() ++ 1 +1() (39)2
d
d
() =
1()+ 1 +1() (40)
552

STUDY OF THE STATIONARY STATES IN THE X REPRESENTATION. HERMITE POLYNOMIALS
Comment:
If we replace the functions()in (39) and (40) by their expressions given in
(35), we obtain, after simplication (setting^=):
2^(^) = 2 1(^) + +1(^) (41)
2^(^) +
d
d^
(^)= 2 1(^) +1(^) (42)
By taking the sum and the dierence of these equations, we obtain relations (6)
and (14) of .
References
Messiah (1.17), App. B, III; Arfken (10.4), Chap. 13, 1; Angot (10.2), 7.8.
553

THE EIGENVALUE EQUATION OF THE HARMONIC OSCILLATOR BY THE POLYNOMIAL METHOD
Complement CV
Solving the eigenvalue equation of the harmonic oscillator
by the polynomial method
1 Changing the function and the variable
2 The polynomial method
2-a The asymptotic form of^(^). . . . . . . . . . . . . . . . . .
2-b The calculation of(^)in the form of a series expansion
2-c Quantization of the energy
2-d Stationary wave functions
In , the method used to calculate the energies of the harmonic
oscillator stationary statesis based on the use of the operators,and, as
well as their their commutation relations. It is possible to obtain the same results in a
completely dierent way, by solving the eigenvalue equation of the Hamiltonianin the
representation. This is what we are going to do in this complement.
1. Changing the function and the variable
In the representation, the eigenvalue equation ofis written:
~
2
2
d
2
d
2
+
1
2
22
() =() (1)
As in Chapter, let us introduce the dimensionless operators
^
and
^
:
^
=
^
=
~
(2)
where the parameter, which has the dimensions of an inverse length, is dened by:
=
~
(3)
Let us denote by^the eigenvector of
^
with eigenvalue is^:
^
^= ^^ (4)
The orthonormalization and closure relations of the kets^are written:
^^=(^^) (5)
+
d^^^= 1 (6)
555

COMPLEMENT C V
The ket^is obviously an eigenvector of, with the eigenvalue^. Therefore,
when:
^= (7)
the two ketsand^are proportional. However, they are not equal. Writing the
closure relation for the kets:
+
d = 1 (8)
and making the change of variables given in (7), we obtain:
+
d^
=
^
=
^
= 1 (9)
Comparison with (6) shows that we can, for example, set:
=
^
=
^ (10)
to orthonormalize the kets^with respect to^, since the ketsare orthonormal with
respect to.
Letbe an arbitrary ket,() = its wave function in therepresen-
tation, and^(^) =^its wave function in the^representation. According to
(10):
^(^) =^=
1
=
^
(11)
that is:
^(^) =
1
(=
^
) (12)
Ifis normalized, relation (8) yields:
=
+
d =
+
()() d= 1 (13)
and relation (6) gives:
=
+
d^^ ^ =
+
^(^) ^(^) d^= 1 (14)
The wave function()is therefore normalized with respect to the variable, as is^(^)
with respect to the variable^. [This can be shown by using the integral in (13), in which
we make the change of variables indicated in (7)].
Now, substituting (7) and (12) into (1), we obtain:
1
2
d
2
d^
2
+ ^
2
^(^) =^(^) (15)
556

THE EIGENVALUE EQUATION OF THE HARMONIC OSCILLATOR BY THE POLYNOMIAL METHOD
setting:
=
~
(16)
Equation (15) is more convenient than equation (1), since all the quantities appearing in
it are dimensionless.
2. The polynomial method
2-a. The asymptotic form of ^(^)
Equation (15) can be written:
d
2
d^
2
(^
2
2)^(^) = 0 (17)
Let us try to predict intuitively the behavior of^(^)for very large^. To do this, consider
the functions:
(^) = e
^
2
2
(18)
They are solutions of the dierential equations:
d
2
d^
2
(^
2
1) (^) = 0 (19)
When^approaches innity:
^
2
1^
2
^
2
2 (20)
and equations (17) and (19) take on the same form asymptotically. We should therefore
expect the solutions of equation (17) to behave
1
, for large^, either likee
^
2
2
or like
e
^
2
2
. From a physical point of view, the only functions^(^)of interest to us are those
that are bounded everywhere. This restricts us to solutions of (17) that behave like
e
^
2
2
(if they exist). This is why we shall set:
^(^) = e
^
2
2
(^) (21)
Substituting (21) into (17), we obtain:
d
2
d^
2
(^)2^
d
d^
(^) + (21)(^) = 0 (22)
We are going to show how this equation can be solved by expanding(^)in a power
series. Then we shall impose the condition that its solutions be physically acceptable.
1
The solutions of equation (17) are not necessarily equivalent toe
^
2
2
ore
^
2
2
when^ : the
intuitive arguments which we have given do not exclude, for example, the possibility that^(^)may
behave like the product ofe
^
2
2
ore
^
2
2
by a power of^.
557

COMPLEMENT C V
2-b. The calculation of (^)in the form of a series expansion
As we pointed out in , the solutions of equation (1) [or, which
amounts to the same thing, of (17)] can be sought amongst either even or odd functions.
Since the functione
^
2
2
is even, we can therefore set:
(^) = ^ 0+2^
2
+4^
4
++2^
2
+ (23)
with0= 0(where0^is, by denition, the rst non-zero term of the expansion).
Writing (23) in the form:
(^) =
=0
2^
2+
(24)
we easily obtain:
d
d^
(^) =
=0
(2+)2^
(2+1)
(25)
and:
d
2
d^
2
(^) =
=0
(2+)(2+ 1)2^
(2+2)
(26)
Let us substitute (24), (25) and (26) into (22). For this equation to be satised,
each term of the series expansion of the left-hand side must be equal to zero. For the
general term in^
2+
, this condition is written:
(2++ 2)(2++ 1)2+2= (4+ 22+ 1)2 (27)
The term of lowest degree is in^
2
; its coecient will be zero if:
(1)0= 0 (28)
Since0is not zero, we therefore have either= 0[the function^(^)is then even] or
= 1[^(^)is then odd].
Relation (27) can be written:
2+2=
4+ 2+ 12
(2++ 2)(2++ 1)
2 (29)
which is a recurrence relation between the coecients2. Since0is not zero, relation
(29) enables us to calculate2in terms of0, then4in terms of2, and so on.
For arbitrary, we therefore have the series expansion of two linearly independent
solutions of equation (22), corresponding respectively to= 0and= 1.
2-c. Quantization of the energy
We must now choose, from amongst all the solutions found in the preceding section,
those which satisfy the physical conditions that^(^)be bounded everywhere.
558

THE EIGENVALUE EQUATION OF THE HARMONIC OSCILLATOR BY THE POLYNOMIAL METHOD
For most values of, the numerator of (29) does not go to zero for any positive or
zero integer. Since none of the coecients2is then zero, the series has an innite
number of terms.
It can be shown that the asymptotic behavior of such a series makes it physically
unacceptable. We see from (29) that:
2+2
2
1
(30)
Now consider the power series expansion of the functione
^
2
(whereis a real parame-
ter):
e
^
2
=
=0
2^
2
(31)
with:
2=
!
(32)
For this second series, we therefore have:
2+2
2
=
!
(+ 1)!
+1
=
+ 1
(33)
If we choose the value of the parametersuch that:
0 1 (34)
we see from (30) and (33) that there exists an integersuch that the condition
implies:
2+2
2
2+2
2
0 (35)
We can deduce from this that, when condition (34) is satised, we have:
^ (^)(^)
2
2
e
^
2
(^) (36)
where(^)and(^)are polynomials of degree2given by the rst+ 1terms of
series (23) and (31). When^approaches innity, (36) gives:
(^)
^
2
2
^e
^
2
(37)
and therefore:
^(^)
^
2
2
^e
(12)^
2
(38)
Since we can choosesuch that:
12 1 (39)
559

COMPLEMENT C V
^(^)is not bounded when^ . We must therefore reject this solution, which makes
no sense physically.
There is only one possibility left: that the numerator of (29) goes to zero for a
value0of. We then have:
2= 0 if 0
2= 0 if 0
(40)
and the power series expansion of(^)reduces to a polynomial of degree20+. The
behavior at innity of^(^)is then determined by that of the exponentiale
^
2
2
, and
^(^)is physically acceptable (since it is square-integrable).
The fact that the numerator of (29) goes to zero at= 0imposes the condition:
2= 2(20+) + 1 (41)
If we set:
20+= (42)
equation (41) can be written:
==+ 12 (43)
whereis an arbitrary positive integer or zero (is an arbitrary positive integer or zero,
andis equal to 0 or 1). Condition (43) introduces the quantization of the harmonic
oscillator energy, since it implies [cf.(16)]:
= +
1
2
~ (44)
We have thus obtained relation (B-34) of Chapter.
2-d. Stationary wave functions
The polynomial method also yields the eigenfunctions associated with the various
energies, in the form:
^(^) = e
^
2
2
(^) (45)
where(^)is anth degree polynomial. According to (23) and (42),(^)is an even
function ifis even and an odd function ifis odd.
The ground state is obtained for= 0, that is, for0== 0;0(^)is then a
constant, and:
^0(^) =0e
^
2
2
(46)
560

THE EIGENVALUE EQUATION OF THE HARMONIC OSCILLATOR BY THE POLYNOMIAL METHOD
A simple calculation shows that, to normalize^0(^)with respect to the variable^, it
suces to choose:
0=
14
(47)
Then, using (12), we nd:
0() =
2
14
e
22
2
(48)
which is indeed the expression given in Chapter C-25)].
To the rst excited state1= 3~2corresponds= 1, that is,0= 0and= 1;
1(^)then has only one term, obtained by a calculation analogous to the preceding one:
^1(^) =
4
14
^e
^
2
2
(49a)
that is:
1() =
4
6
14
e
22
2
(49b)
For= 2, we have0= 1and= 0. Relation (29) then yields:
2=20 (50)
which nally leads to:
^2(^) =
1
4
14
(2^
2
1) e
^
2
2
(51a)
that is:
2() =
2
4
14
(2
22
1) e
22
2
(51b)
For arbitrary,(^)is the polynomial solution of equation (22), which can be
written, taking the quantization condition (43) into account:
d
2
d^
2
2^
d
d^
+ 2(^) = 0 (52)
We recognize (52) to be the dierential equation satised by the Hermite polynomial
(^)[see equation (17) of ComplementV]. The polynomial(^)is therefore pro-
portional to(^), where the proportionality factor is determined by normalization of
^(^). This is in agreement with formula (35) of complementV.
References
Mathematical treatment of dierential equations: Morse and Feshbach (10.13),
Chaps. 5 and 6; Courant and Hilbert (10.11), V-11.
561

STUDY OF THE STATIONARY STATES IN THE MOMENTUM REPRESENTATION
Complement DV
Study of the stationary states in therepresentation
1 Wave functions in momentum space
1-a Changing the variable and the function
1-b Determination of^
(^). . . . . . . . . . . . . . . . . . . . .
1-c Calculation of the phase factor
2 Discussion
The distribution of the possible momenta of a particle in the stateis given by
the wave function
()in theprepresentation, the Fourier transform of the wave
function()in the representation. We shall show in this complement that in
the case of the harmonic oscillator, the functionsand
are the same (to within
multiplicative factors). Thus, in a stationary state, the probability distributions of the
momentum and the position have similar forms.
1. Wave functions in momentum space
1-a. Changing the variable and the function
In ComplementV, we introduced, for simplicity, the operator:
^
= (1)
where:
=
~
(2)
as well as the eigenkets^of
^
and the wave function^(^)in the^representation.
We shall follow a similar procedure for the operator:
^
=
~
(3)
We shall therefore call^the eigenkets of
^
:
^
^= ^^ (4)
and denote by^
(^)the wave function in the^representation:
^
(^) =^ (5)
Just as the ket^is proportional to the ket= ^, the ket^is proportional to the
ket=~^. If we changeto1~[cf.(1) and (3)], equation (10) of ComplementV
shows that:
^=
~=~^ (6)
563

COMPLEMENT D V
The wave function^
(^)in the ^representation is therefore related to the wave
function
()in the representation by:
^
(^) =
~
(=~^) (7)
Furthermore, we can use (6) and relation (10) of ComplementVto obtain:
^^=
e
^^
2
(8)
We therefore have, using denition (5) and the closure relation for the^basis:
^
(^) =
+
^^ ^d^
=
1
2
+
e
^^
^(^) d^ (9)
The function^
is therefore the Fourier transform of^.
1-b. Determination of ^
(^)
We have seen [cf.equation (15) of ComplementV] that the stationary wave
functions^(^)of the harmonic oscillator satisfy the equation:
1
2
d
2
d^
2
+ ^
2
^(^) =^(^) (10)
Now, the Fourier transform of
d
2
d^
2
^(^)is^
2^
(^)and that of^
2
^(^)is
d
2
d^
2
^
(^).
The Fourier transform of equation (10) is therefore:
1
2
^
2
d
2
d^
2
^
(^) =^(^) (11)
If we compare (10) and (11), we see that the functions^and^
satisfy the same
dierential equation. Moreover, we know that this equation, when=+ 12(where
is a positive integer or zero), has only one square-integrable solution (the eigenvalues
are non-degenerate;cf.Chapter, ). We can conclude that^and^
are
proportional. Since these two functions are normalized, the proportionality factor is a
complex number of modulus 1, so that:
^
(^) = e^(^= ^) (12)
whereeis a phase factor which we shall now determine.
1-c. Calculation of the phase factor
The wave function of the ground state is given by [cf.ComplementV, formulas
(46) and (47)]:
^0(^) =
14
e
^
2
2
(13)
564

STUDY OF THE STATIONARY STATES IN THE MOMENTUM REPRESENTATION
This is a Gaussian function; its Fourier transform is therefore [cf.Appendix, relation
(50)]:
^
0(^) =
14
e
^
2
2
(14)
This implies that0is zero.
To nd, let us write, in the^and ^representations, the relation:
=
+ 1 +1 (15)
In the^representation,
^
and
^
act like^and
1
d
d^
;therefore acts like
1
2
^
d
d^
. In the^representation,
^
acts like
d
d^
and
^
like^;therefore
acts like
2
d
d^
^.
In the^representation, relation (15) therefore becomes:
^+1(^) =
1
2(+ 1)
^
d
d^
^(^) (16)
while in the^representation, it becomes:
^
+1(^) =
2(+ 1)
d
d^
^^
(^) (17)
We therefore have:
e
+1
=e (18)
that is, knowing that0= 0:
e= () (19)
Thus we obtain:
^
(^) = ()^(^= ^) (20)
or, returning to the functionsand
:() = ()
1
~
=
2
~
(21)
2. Discussion
Consider a particle in the state. When the position of the particle is measured, one
has a probability() dof nding a result betweenand+ d, where()is given
by:
() =()
2
(22)
565

COMPLEMENT D V
Similarly, in a measurement of the momentum of the particle, one has a probability
() dof nding a result betweenand+ d, with:() =()
2
(23)
Relation (21) then yields:
() =
1
= (24)
which shows that the momentum distribution in a stationary state has the same form as
the position distribution.
We see, for example (cf.Fig. ), that ifis large,
()has a peak
at each of the two values:
= = (25)
where is the maximum momentum of the classical particle moving in the potential
well with an energy. An argument analogous to the one set forth at the end of
of Chapter
particle is equal to, its acceleration is zero (its velocity is stationary), and the values
of the momentum are, averaging over time, the most probable ones. Comment(i)
of ()can easily be transposed
to this context; for example, whenis large, the root mean square deviationcan
be interpreted as being of the order of magnitude of the distance between the peaks of
()situated at= .
It is also possible to understand directly from Figure-a of Chapter
values of the momentum are highly probable whenis large. The wave function then
performs a large number of oscillations between the two peaks, analogous to those of
a sinusoid. This happens because the dierential equation for the wave function [cf.
formula (A-17) of Chapter] when
22
2becomes:
d
2
d
2
() +
2
~
2
()0 (26)
which yields, according to the denition of:
() e
~
+e
~
(27)
The wave function (whenis large) therefore looks like a sinusoid of wavelength
over a relatively large region of theaxis. This sinusoid can be considered to be the
sum of two progressive waves [cf.(27)] associated with the two opposite momenta
(corresponding to the to-and-fro motion of the particle in the well). It is not surprising,
therefore, that the probability density
()should be large in the neighborhood of the
values= .
An analogous argument also enables us to understand the order of magnitude of
the product. This product is equal to [cf.Chap., relations (D-6), (D-7) and
(D-9)]:
= +
1
2
~=
2
(28)
566

STUDY OF THE STATIONARY STATES IN THE MOMENTUM REPRESENTATION
Whenincreases, the amplitudesandof the oscillations increase, and the product
takes on values much greater than its minimum value~2. We might wonder
why this is the case, since we have seen in several examples that when the width
of a function increases, the widthof its Fourier transform decreases. This is indeed
what would happen for the functions()if, in the interval + where
they take on non-negligible values, they varied slowly, reaching, for example, a single
maximum or minimum. This is in fact the case for small values of, for which the
value of the productis indeed near its minimum. However, whenis large, the
functions()perform numerous oscillations in the interval +, where
they havezeros. One can therefore associate with them wavelengths of the order of
, corresponding to momenta of the particle situated in a domain of
dimensiongiven by:

(29)
We thus nd again that:
(30)
This situation is somewhat analogous to the one studied in 1 of ComplementIII, in
connection with the innite one-dimensional well.
567

THE ISOTROPIC THREE-DIMENSIONAL HARMONIC OSCILLATOR
Complement EV
The isotropic three-dimensional harmonic oscillator
1 The Hamiltonian operator
2 Separation of the variables in Cartesian coordinates
3 Degeneracy of the energy levels
In Chapter, we studied the one-dimensional harmonic oscillator. We now show
how to use the results of this study to treat the three-dimensional harmonic oscillator.
1. The Hamiltonian operator
Consider a spinless particle of masswhich can move in three-dimensional space. The
particle is subjected to a central force (i.e. a force that is constantly directed towards the
coordinate origin) whose absolute value is proportional to the distance of the particle
from the point:
F= r (1)
(is a positive constant).
This force eld is derived from the potential energy:
(r) =
1
2
r
2
=
1
2
2
r
2
(2)
where the angular frequencyis dened as for the one-dimensional harmonic oscillator:
=
(3)
The classical Hamiltonian is therefore:
(rp) =
p
2
2
+
1
2
2
r
2
(4)
Using the quantization rules (Chap., ), we immediately deduce the Hamiltonian
operator:
=
P
2
2
+
1
2
2
R
2
(5)
Since the Hamiltonianis time-independent, we shall solve its eigenvalue equation:
= (6)
where belongs to the state spacerof a particle in three-dimensional space.
569

COMPLEMENT E V
Comment:
Since(r)depends only on the distance=rof the particle from the origin [(r)
is consequently invariant under an arbitrary rotation], this harmonic oscillator is
said to beisotropic. Nevertheless, the calculations which follow can easily be
generalized to the case of an anisotropic harmonic oscillator, for which:
(r) =
2
22
+
22
+
22
(7)
where the three constants,andare dierent.
2. Separation of the variables in Cartesian coordinates
Recall that the state spacercan be considered (cf.Chap., ) to be the tensor
product:
r= (8)
whereis the state space of a particle moving along, that is, the space associated
with the wave functions().andare dened analogously.
Now, expression (5) for the Hamiltoniancan be written in the form:
=
1
2
2
+
2
+
2
+
1
2
2 2
+
2
+
2
= + + (9)
with:
=
2
2
+
1
2
22
(10)
and similar denitions forand. is a function only ofand: is
therefore the extension intorof an operator that actually acts in. Similarly,and
act only inandrespectively. In,is a one-dimensional harmonic oscillator
Hamiltonian. The same is true forand inand.
,and commute. Each of them therefore commutes with their sum.
Consequently, the eigenvalue equation (6) can be solved by seeking the eigenvectors of
that are also eigenvectors of,and. Now, we already know the eigenvectors
and eigenvalues ofin, as well as those ofinand ofin:
= +
1
2
~ ; (11a)
= +
1
2
~ ; (11b)
= +
1
2
~ ; (11c)
(,andare positive integers or zero). It follows (cf.Chap., ) that the
eigenstates common to,,and are of the form:
= (12)
570

THE ISOTROPIC THREE-DIMENSIONAL HARMONIC OSCILLATOR
According to equations (9) and (11):
= +++
3
2
~ (13)
The eigenvectors ofare seen to betensor products of eigenvectorsof,and
respectively, and the eigenvalues of, to besums of eigenvaluesof these three operators.
According to equation (13), the energy levelsof the isotropic three-dimensional
harmonic oscillator are of the form:
= +
3
2
~ (14)
with:
a positive integer or zero (15)
sinceis the sum++of three numbers, each of which can take on any non-
negative integral value.
Furthermore, formula (12) enables us to deduce the properties of the vectors
, common eigenstates of,, and, from those derived in
of Chapter (which are also valid forand ).
Let us introduce three pairs of creation and annihilation operators:
=
2~
+
2~
=
2~2~
(16a)
=
2~
+
2~
=
2~2~
(16b)
=
2~
+
2~
=
2~2~
(16c)
These operators are the extensions intorof operators acting in,and. The
canonical commutation relations between the components ofRandPimply that the
only non-zero commutators of the six operators dened in (16) are:
[ ] = [] = [] = 1 (17)
Note that two operators with dierent indices always commute, as is logical because they
act in dierent spaces. The action of the operatorsandon the states is
given by the formulas:
= ( )
=
1
=
1 (18a)
= ( )
=
+ 1 +1
=
+ 1 +1 (18b)
571

COMPLEMENT E V
For,and,, we have analogous relations.
We also know [cf.equation (C-13) of Chapter] that:
=
1
!
() 0 (19)
where0is the vector ofthat satises the condition:
0= 0 (20)
Inand, there are analogous expressions forand . Consequently, accord-
ing to (12):
=
1
!!!
()()() 000 (21)
where 000is the tensor product of the ground states of the three one-dimensional
oscillators, so that:
000= 000= 000= 0 (22)
Finally, recall that, since is a tensor product, the associated wave func-
tion is of the form:
r = ()()() (23)
where , and are stationary wave functions of the one-dimensional harmonic
oscillator (Chap., ). For example:
r000=
~
34
e
2~
(
2
+
2
+
2
)
(24)
3. Degeneracy of the energy levels
We showed in constitutes a C.S.C.O. in; the same is
true forinand forin. According to , is
thus a C.S.C.O. inr. Therefore, there exists (to within a constant factor) a unique ket
ofrcorresponding to a given set of eigenvalues of,and, that is,
to xed non-negative integers,and.
However,alone does not form a C.S.C.O. sincethe energy levelsare degen-
erate. If we choose an eigenvalue of,= (+ 32)~)(which amounts to xing a
non-negative integer), all the kets of the basis that satisfy:
++= (25)
are eigenvectors ofwith the eigenvalue.
The degree of degeneracyofis therefore equal to the number of dierent
sets satisfying condition (25). To nd, we can proceed as follows. With
xed, chooserst, giving it one of the values:
= 012 (26)
572

THE ISOTROPIC THREE-DIMENSIONAL HARMONIC OSCILLATOR
With thus xed, we must have:
+= (27)
There are then( + 1)possibilities for the pair:
=0 1 1 0 (28)
The degree of degeneracyofis therefore equal to:
=
=0
( + 1) (29)
This sum is easy to calculate:
= (+ 1)
=0
1
=0
=
(+ 1)(+ 2)
2
(30)
Consequently, only the ground state0=
3
2
~is non-degenerate.
Comment:
The kets constitute an orthonormal system of eigenvectors of, which
forms a basis inr. Since the eigenvaluesofare degenerate, this system
is not unique. We shall see in particular in ComplementVIIthat, in order to
solve equation (6), it is possible to use a set of constants of the motion other than
: thus we obtain a basis ofrwhich is dierent from the preceding
one, although still consisting of eigenvectors of. The kets of this new basis
are orthonormal linear combinations of the belonging to each of the
eigensubspaces of, that is, corresponding to a xed value of the sum++.
573

A CHARGED HARMONIC OSCILLATOR IN A UNIFORM ELECTRIC FIELD
Complement FV
A charged harmonic oscillator in a uniform electric eld
1 Eigenvalue equation of (E)in the representation
2 Discussion
2-a Electrical susceptibility of an elastically bound electron
2-b Interpretation of the energy shift
3 Use of the translation operator
The one-dimensional harmonic oscillator studied in Chapter
of masshaving a potential energy:
() =
1
2
22
(1)
Assume, in addition, that this particle has a chargeand that it is placed in a uniform
electric eldEparallel to. What are its new stationary states and the corresponding
energies?
The classical potential energy of a particle placed in a uniform eldEis equal to
1
:
(E) =E (2)
To obtain the quantum mechanical Hamiltonian operator(E)in the presence of the
eldE, we must therefore add to the potential energy (1) of the harmonic oscillator the
term:
(E) =E (3)
which gives:
(E) =
2
2
+
1
2
22
E (4)
We must now nd the eigenvalues and eigenvectors of this operator. To this end, we
shall use two dierent methods. First, we shall consider directly the eigenvalue equation
of(E)in the representation, as it is very simple to interpret the results obtained.
Then we shall show how the problem can be solved in a purely operator formalism.
1. Eigenvalue equation of (E)in the representation
Let be an eigenvector of(E):
(E)= (5)
1
We use the convention of zero potential energy for the particle at= 0.
575

COMPLEMENT F V
Using (4), we can write this equation in therepresentation:
~
2
2
d
2
d
2
+
1
2
22
E () = () (6)
Completing the square with respect toon the left-hand side of (6), we get:
~
2
2
d
2
d
2
+
1
2
2
E
2
2 2
E
2
2
2
() = () (7)
Let us now replace the variableby a new variable, setting:
=
E
2
(8)
Through the intermediary of,is then a function of, and equation (7) becomes:
~
2
2
d
2
d
2
+
1
2
22
() = () (9)
with:
=+
2
E
2
2
2
(10)
Thus we see that equation (9) is the same as the one used to obtain the stationary states
of the harmonic oscillator in the absence of an electric eld in therepresentation
[cf.Chap., relation (A-17)]. Therefore, we have already solved this equation, and we
know that the acceptable values ofare given by:
= +
1
2
~ (11)
(whereis a positive integer or zero).
Relations (10) and (11) show that, in the presence of the electric eld, the energies
of the stationary states of the harmonic oscillator are modied:
(E) =+
1
2
~
2
E
2
2
2
(12)
The entire spectrum of the harmonic oscillator is therefore shifted by the quantity
2
E
2
2
2
.
Now, let us show that the eigenfunctions()associated with the energies (12)
can all be obtained from the()by a translation along. The solution of (9)
corresponding to a given value ofis()[where the functionis given, for example,
by formula (35) of ComplementV]. According to (8), we have:
() =
E
2
(13)
This translation comes from the fact that the electric eld exerts a force on the particle
2
.
2
It can be seen from (13) that the function()is obtained from()by a translation ofE
2
;
if the productEis positive, the translation is performed in the positive-direction, which is indeed the
direction of the force exerted byE.
576

A CHARGED HARMONIC OSCILLATOR IN A UNIFORM ELECTRIC FIELDV
V + W
x
O
–
q
2 2
2mω
2
W
O
q
mω
2
Figure 1: The presence of a uniform electric eld has the eect of adding a linear term
to the potential energyof the harmonic oscillator; the total potential+is then
represented by a displaced parabola.
Comment:
The change of variable given in (8) allows us to reduce the case of an arbitrary
electric eld to an already solved problem, the one in whichEwas zero. The only
eect of the electric eld is to change the-origin [cf.(13)] and the energy origin
[cf.(12)]. This result can easily be understood graphically (cf.Fig.). When
Eis zero, the potential energy()is represented by a parabola centered at.
WhenEis not zero, it is necessary to add to this potential energy the quantity
E, which corresponds to the dashed line in this gure; the curve representing
+ is again a parabola. Thus, in the presence of the eldEwe still have a
harmonic oscillator. Since the two parabolas are superposable, they correspond to
the same value ofand therefore to the same energy dierence between the levels.
However, their minimaandare dierent, as is consistent with formulas (12)
and (13).
2. Discussion
2-a. Electrical susceptibility of an elastically bound electron
In certain situations, the electrons of an atom or a molecule behave, to a good
approximation, as if they were elastically bound, that is, as if each of them were a
harmonic oscillator. We shall prove this for atoms in ComplementXIII, using time-
dependent perturbation theory.
The contribution of each electron to the electric dipole moment of the atom is
described by the operator:
= (14)
577

COMPLEMENT F V
whereis the charge of the electron(0)andthe corresponding position observable.
We are going to examine the mean value ofin the model of the elastically bound
electron.
In the absence of an electric eld, the mean value of the electric dipole moment in
a stationary state of the oscillator is zero:
= = 0 (15)
[see formulas (D-1) of Chapter].
Now, let us assume that the eldEis turned on so slowly that the state of the
electron changes gradually fromto (remaining the same). The mean dipole
moment is now dierent from zero, since:
= =
+
d ()
2
(16)
Using (8) and (13), we obtain:
=
+
()
2
d+
2
E
2
+
()
2
d=
2
E
2
(17)
since the rst integral is zero by symmetry.is therefore proportional toE. In this
model, the electrical susceptibility of the atomic electron under consideration is equal to:
=
E
=
2
2
(18)
It is positive, whatever the sign of.
It is simple to interpret result (18) physically. The eect of the electric eld is
to shift the classical equilibrium position of the electron, that is, the mean value of its
position in quantum mechanics [see formula (13)]. This results in the appearance of an
induced dipole moment.decreases whenincreases because the oscillator is less easily
deformable when the restoring force (which is proportional to
2
) is larger.
2-b. Interpretation of the energy shift
Using the model just described, we can interpret formula (12) by calculating the
variation in the mean kinetic and potential energies of the electron when it passes from
the stateto the state.
The variation in the kinetic energy is, in fact, zero (as can be understood intuitively
from Figure, for example):
2
2
2
2
=
~
2
2
+
()
d
2
d
2
() d
+
()
d
2
d
2
() d= 0(19)
according to formula (13).
The variation in the potential energy can be treated in two terms:
578

A CHARGED HARMONIC OSCILLATOR IN A UNIFORM ELECTRIC FIELD
the rst term,(E), corresponds to the electrical potential energy of the dipole
in the eldE. Since the dipole is parallel to the eld, we have, according to (17):
(E)=E =
2
E
2
2
(20)
the second term,() (), arises from the electric eld modication of
the wave function of the level labeled by the quantum number. The elastic
potential energy of the particle therefore changes by a quantity:
() ()=
1
2
2
+
2
()
2
d
+
2
()
2
d (21)
The rst integral can be calculated by using (13) and the change of variable (8):
+
2
()
2
d=
+
2
()
2
d+
2E
2
+
()
2
d
+
E
2
2+
()
2
d(22)
Since()is normalized, and since the integral of()
2
is zero by symmetry, we
obtain, nally:
() ()=
2
E
2
2
2
(23)
We see why this result should be positive, since the electric eld moves the particle away
from the pointand attracts it into a region where the elastic potential energy()
is larger.
Adding (20) to (23), we again nd that the energy of the stateis less than
that of the stateby
2
E
2
2
2
.
3. Use of the translation operator
We shall see in this section that, instead of using therepresentation, as we have
done until now, we can argue directly in terms of the operator(E)given in (4). More
precisely, we are going to show that a unitary transformation (which corresponds to a
translation of the wave function along the-axis) transforms the operator=(E= 0)
into the operator(E)(to within an additive constant which does not change the
eigenvectors). Since the eigenvectors and eigenvalues ofwere determined in Chapter,
this approach enables us to solve our problem.
Therefore, consider the operator:
() = e
( )
(24)
whereis a real constant. Its adjoint()is:
() = e
( )
(25)
579

COMPLEMENT F V
It is clear that:
()() =()() = 1 (26)
()is therefore a unitary operator. Under the corresponding unitary transformation,
becomes:
~
=() ()
=~
1
2
+() () (27)
We must now calculate the operator:
() () = ~~ (28)
with:
~=()()
~=() () (29)
To obtain~and~, we use formula (63) of ComplementII, (which can be applied here
since the commutator ofandis equal to 1), which yields:
() = e
+
= eee
2
2
() = e
+
= e ee
2
2
(30)
Also, formula (51) of ComplementIIenables us to write:
e = e
e = e
(31)
that is:
e e=
e e = (32)
Thus it follows that:
~= ee e e
= e( )e= (33)
and, similarly:
~= (34)
580

A CHARGED HARMONIC OSCILLATOR IN A UNIFORM ELECTRIC FIELD
~
is therefore given by:
~
=~
1
2
+ ( )( )
=~
1
2
+ (+) +
2
= ~(+) +
2
~ (35)
Since(+)is proportional to the operator[formulas (B-1) and (B-7) of Chapter],
it suces to set:
=
E
12~
(36)
to obtain:
~
= E+
2
E
2
2
2
=(E) +
2
E
2
2
2
(37)
The two operators
~
and(E)therefore have the same eigenvectors, and their eigen-
values dier by
2
E
2
2
2
. Now, we know (cf.ComplementII, ) that if the
eigenvectors ofare the kets, those of
~
are the kets:
~=() (38)
and the corresponding eigenvalues ofand
~
are the same. The stationary states
of the harmonic oscillator in the presence of the eldEare therefore the states~given
by (38). The associated eigenvalue of(E)is, according to (37):
(E) =+
1
2
~
2
E
2
2
2
(39)
which is the same as formula (12) of the preceding section. Expression (38) for the
eigenvectors can be put into the form:
=~= e
E
~
2
(40)
using (24) and (36), as well as formulas (B-1) and (B-7) of Chapter. We interpreted, in
ComplementII, the operatore
~
as being the translation operator over an algebraic
distancealong. is therefore the state obtained fromby a translation
E
2
, just as is indicated by formula (13).
References:
The elastically bound electron: see references of ComplementXIII.
581

COHERENT QUASI-CLASSICAL STATES OF THE HARMONIC OSCILLATOR
Complement GV
Coherent quasi-classical states of the harmonic oscillator
1 Quasi-classical states
1-a Introducing the parameter 0to characterize a classical motion
1-b Conditions dening quasi-classical states
1-c Quasi-classical states are eigenvectors of the operator. . .
2 Properties of the states
2-a Expansion of on the basis of the stationary states. .
2-b Possible values of the energy in anstate
2-c Calculation of ,,andin anstate
2-d The operator (): the wave functions (). . . . . . . .
2-e The scalar product of two states. Closure relation
3 Time evolution of a quasi-classical state
3-a A quasi-classical state always remains an eigenvector of. .
3-b Evolution of physical properties
3-c Motion of the wave packet
4 Example: quantum mechanical treatment of a macroscopic
oscillator
The properties of the stationary statesof the harmonic oscillator were studied
in Chapter; for example, in , we saw that the mean valuesand of the
position and the momentum of the oscillator are zero in such a state. Now, in classical
mechanics, it is well-known that the positionand the momentum are oscillating
functions of time, which always remain zero only if the energy of the motion is also
zero [cf.Chap., relations (A-5) and (A-8)]. Furthermore, we know that quantum
mechanics must yield the same results as classical mechanics in the limiting case where
the harmonic oscillator has an energy much greater than the quantum~(limit of large
quantum numbers).
Thus, we may ask the following question: is it possible to construct quantum me-
chanical states leading to physical predictions which are almost identical to the classical
ones, at least for a macroscopic oscillator? We shall see in this complement that such
quantum states exist: they are coherent linear superpositions of all the states. We
shall call them quasi-classical states or coherent states of the harmonic oscillator.
The problem we are considering here is of great general interest in quantum me-
chanics. As we saw in the introduction to Chapter V, many
physical systems can be likened to a harmonic oscillator, at least to a rst approxima-
tion. For all these systems, it is important to understand, in the framework of quantum
mechanics, how to move gradually from the case in which the results given by the classi-
cal approximation are sucient to the case in which quantum eects are preponderant.
Electromagnetic radiation is a very important example of such a system. Depending on
the experiment, it either reveals its quantum mechanical nature (as is the case in the
583

COMPLEMENT G V
experiment discussed in , in which the light intensity is very low)
or else can be treated classically. Coherent states of electromagnetic radiation were
introduced by Glauber and are in current use in the domain of quantum optics.
The position, the momentum, and the energy of a harmonic oscillator are described
in quantum mechanics by operators which do not commute; they are incompatible phys-
ical quantities. It is not possible, therefore, to construct a state in which they are all
perfectly well-dened. We shall thus only look for a state vector such that, for all,
the mean values,and are as close as possible to the corresponding classical
values. This will lead us to a compromise in which none of these three observables is
perfectly known. We shall see, nevertheless, that the root mean square deviations,
andare, in the macroscopic limit, completely negligible.
1. Quasi-classical states
1-a. Introducing the parameter 0to characterize a classical motion
The classical equations of motion of a one-dimensional harmonic oscillator, of mass
and angular frequency, are written:
d
d
() =
1
() (1a)
d
d
() =
2
() (1b)
The quantum mechanical calculations we shall perform later will be simplied by the
introduction of the dimensionless quantities:
^() =()
^() =
1
~
()
(2)
where:
=
~
(3)
Equations (1) can then be written:
d
d
^() =^() (4a)
d
d
^() =^() (4b)
The classical state of the harmonic oscillator is determined at timewhen we know its
position()and its momentum(), that is,^()and^(). We shall therefore combine
these two real numbers into a single dimensionless complex number()dened by:
() =
1
2
[^() +^()] (5)
584

COHERENT QUASI-CLASSICAL STATES OF THE HARMONIC OSCILLATORM
0
Im α(t)
p(t)
Re α(t)
M (t)
O
– t
x(t)
√2
ˆ
ˆ
√2
Figure 1: The point(), which
corresponds to the complex number
(), characterizes the state of the
harmonic oscillator at each instant.
moves in a circle with an angu-
lar velocity. The abscissa and
ordinate ofgive the position and
momentum of the oscillator.
The set of two equations (4) is equivalent to the single equation:
d
d
() = () (6)
whose solution is:
() =0e (7)
where we have set:
0=(0) =
1
2
[^(0) +^(0)] (8)
Now consider the points0andin the complex plane that correspond to the
complex numbers0and()[Fig.].is at0at= 0and describes, with an
angular velocity, a circle centered atof radius 0.
Since, according to (5), the coordinates ofare equal to^()
2and^()2,
we thus obtain a very simple geometrical representation of the time evolution of the state
of the system. Every possible motion corresponding to given initial conditions is entirely
characterized by the point0, that is, by the complex number0(the modulus of0
gives the amplitude of the oscillation and the argument of0, its phase). According to
(5) and (7), we have:
^() =
1
2
0e+
0e (9a)
^() =
2
0e
0e (9b)
As for the classical energyof the system, it is constant in time and equal to:
=
1
2
[(0)]
2
+
1
2
2
[(0)]
2
=
~
2
[^(0)]
2
+ [^(0)]
2
(10)
585

COMPLEMENT G V
which yields, taking (8) into account:
=~ 0
2
(11)
For a macroscopic oscillator, the energyis much greater than the quantum~, so:
01 (12)
1-b. Conditions dening quasi-classical states
We are looking for a quantum mechanical state for which at every instant the mean
values,and are practically equal to the values,andwhich correspond
to a given classical motion.
To calculate, and , we use the expressions:
^
= =
1
2
(+)
^
=
1
~
=
2
( ) (13)
and:
=~ +
1
2
(14)
For an arbitrary state(), the time evolution of the matrix element() =()()
is given by (cf. ):
~
d
d
() =[]() (15)
Now:
[] =~[ ] =~ (16)
which implies:
d
d
() = () (17)
that is:
() =(0) e (18)
The evolution of() =() ()obeys the complex conjugate equation:
() =(0) e
= (0) e (19)
(18) and (19) are analogous to the classical equation (7).
586

COHERENT QUASI-CLASSICAL STATES OF THE HARMONIC OSCILLATOR
Substituting (18) and (19) into (13), we obtain:
^
() =
1
2
(0) e+ (0) e
^
() =
2
(0) e (0) e
(20)
Comparing these results with (9), we see that, in order to have at all times:
^
() = ^()
^
() = ^()
(21)
it is necessary and sucient to set, at the instant= 0, the condition:
(0) =0 (22)
where0is the complex parameter characterizing the classical motion which we are
trying to reproduce quantum mechanically. The normalized state vector()of the
oscillator must therefore satisfy the condition:
(0)(0)=0 (23)
We must now require the mean value:
=~ (0) +
~
2
(24)
to be equal to the classical energygiven by (11). Since, for a classical oscillator,0
is much greater than 1 [cf.(12)], we shall neglect the term~2(of purely quantum
mechanical origin; see ) with respect to~ 0
2
. The second condition
on the state vector can now be written:
(0) =0
2
(25)
that is:
(0) (0)= 0
2
(26)
We shall see that conditions (23) and (26) are sucient to determine the normalized
state vector(0)(to within a constant phase factor).
1-c. Quasi-classical states are eigenvectors of the operator
We introduce the operator(0)dened by:
(0) = 0 (27)
587

COMPLEMENT G V
We then have:
(0)(0) = 0 0+
00 (28)
and the square of the norm of the ket(0)(0)is:
(0)(0)(0)(0)=
(0) (0) 0(0)(0)
0(0)(0)+
00 (29)
Substituting into this relation conditions (23) and (26), we obtain:
(0)(0)(0)(0)=
00 00 00+
00= 0 (30)
The ket(0)(0), whose norm is zero, is therefore zero:
(0)(0)= 0 (31)
that is:
(0)=0(0) (32)
Conversely, if the normalized vector(0)satises this relation, it is obvious that
conditions (23) and (26) are satised.
We therefore arrive at the following result: the quasi-classical state, associated
with a classical motion characterized by the parameter0, is such that(0)is an
eigenvector of the operatorwith the eigenvalue0.
In what follows, we shall denote the eigenvector ofwith eigenvalueby:
= (33)
[we shall show later that the solution of (33) is unique to within a constant factor].
2. Properties of the states
2-a. Expansion of on the basis of the stationary states
Let us determine the ketwhich is a solution of (33) by using an expansion on
the states:
= () (34)
We then have:
= ()
1 (35)
and, substituting this relation into (33), we obtain:
+1() =
+ 1
() (36)
588

COHERENT QUASI-CLASSICAL STATES OF THE HARMONIC OSCILLATOR
This relation enables us to determine by recurrence all the coecients()in terms of
0():
() =
!
0() (37)
It follows that, when0()is xed, all the()are also xed. The vectoris therefore
unique to within a multiplicative factor. We shall choose0()real and positive and
normalize the ket, which determines it completely. In this case, the coecients()
satisfy:
()
2
= 1 (38)
that is:
0()
2
2
!
=0()
2
e
2
= 1 (39)
With the convention we have chosen:
0() = e
2
2
(40)
and, nally:
= e
2
2
!
(41)
2-b. Possible values of the energy in an state
Let us consider an oscillator in the state. We see from (41) that a measurement
of the energy can yield the result= (+ 12)~with the probability:
() =()
2
=
2
!
e
2
(42)
The probability distribution obtained,(), is therefore a Poisson distribution. Since:
() =
2
1() (43)
it is easy to verify that()reaches its maximum value when:
=the integral part of
2
(44)
To calculate the mean valueof the energy, we can use (42) and the expression:
= ()+
1
2
~ (45)
Nevertheless, it is quicker to notice that, since the adjoint relation of (33) is:
= (46)
589

COMPLEMENT G V
we have:
= (47)
and therefore:
=~ +
1
2
=~
2
+
1
2
(48)
Comparing this result to (44), we see that, when 1, is not very dierent, in
relative value, from the energywhich corresponds to the maximum value of().
Let us calculate the mean value
2
:
2
=~
22
+
1
2
2
(49)
Using (33) and the fact that[] = 1, we easily obtain:
2
=~
22 4
+ 2
2
+
1
4
(50)
from which we get:
=~ (51)
If we compare (48) and (51), we see that, ifis very large, we have:

1
1 (52)
The energy of the stateis very well-dened, since its relative uncertainty is very
small.
Comment:
Since:
= +
1
2
~ (53)
we immediately obtain from (48) and (51):
=
2
=
(54)
Thus we see that, to obtain a quasi-classical state, we must linearly superpose a
very large number of statessince 1. However, the relative value of
the dispersion overis very small:

=
1
1 (55)
590

COHERENT QUASI-CLASSICAL STATES OF THE HARMONIC OSCILLATOR
2-c. Calculation of ,,andin anstate
The mean values and can be obtained by expressingandin terms of
and[formula (13)] and using (33) and (46). We obtain:
= =
2~
Re() (56)
= =
2~Im()
An analogous calculation yields:
2
=
~
2
(+)
2
+ 1
2
=
~
2
1( )
2
(57)
and therefore:
=
~2
=
~2
(58)
Neither nor, depends on. Note also that takes on its minimum
value:
=~2 (59)
2-d. The operator (): the wave functions ()
Consider the operator()dened by:
() = e (60)
This operator is unitary, since:
() = e (61)
immediately implies:
()() =()() = 1 (62)
Since the commutator of the operatorsand is equal to, which is a number,
we can use relation (63) of ComplementIIto write:
() = e
2
2
ee (63)
Now let us calculate the ket()0; since:
e 0=1 +
2
2!
2
+ 0
= 0 (64)
591

COMPLEMENT G V
then:
()0= e
2
2
e 0
= e
2
2
()
!
0
= e
2
2
!
(65)
Comparing (41) and (65), we see that:
=()0 (66)
()is therefore the unitary transformation which transforms the ground state0into
the quasi-classical state.
Formula (66) will enable us to obtain the wave function:
() = (67)
which characterizes the quasi-classical statein the representation. To calculate:
() = ()0 (68)
we shall write the operator in terms ofand:
=
~2~
+
2
(69)
Using formula (63) of ComplementIIagain, we obtain:
() = e = e
~2e~
+
2e
2 2
4 (70)
Substituting this result into (68), we nd:
() = e
2 2
4 e~2e~
+
2
0
= e
2 2
4e~2 e~
+
2
0 (71)
Now, the operatore
~
is the translation operator ofalong (cf.Comple-
mentII):
e
~
+
2=~
2
(+) (72)
Relation (71) therefore yields:
() = e
2 2
4e~2
0~
2
(+) (73)
592

COHERENT QUASI-CLASSICAL STATES OF THE HARMONIC OSCILLATOR
If we writeandin terms of and [formulas (56)],()becomes:
() = ee
~
0( ) (74)
where the global phase factor eis dened by:
e= e
2 2
4 (75)
Relation (74) shows that()can easily be obtained from the wave function0()of
the ground state of the oscillator: it is sucient to translate this function alongby
the quantity and to multiply it by the oscillating exponentiale
~
(since the
factoreplays no physical role, it can be omitted)
1
.
If we replace0in (74) by its explicit expression, we obtain, nally:
() = e
~
14
exp
2
2
+
~
(76)
The form of the wave packet associated with thestate is therefore given by:
()
2
=
~
exp
1
2
2
(77)
For anystate, we obtain a Gaussian wave packet. This result is consitent with the
fact that the product is always minimal (cf.ComplementIII).
2-e. The scalar product of two states. Closure relation
The states are eigenvectors of the non-Hermitian operator. There is therefore
no obvious reason for these states to satisfy orthogonality and closure relations. In this
section, we shall investigate this question.
First, we shall consider two eigenketsand of the operator. Relation (41)
gives their scalar product, since:
= ()() (78)
We therefore have:
= e
2
2
e
2
2
()
!
= e
2
2
e
2
2
e (79)
from which we conclude:
2
= e
2
(80)
1
The exponentiale
~
is obviously not a global phase factor since its value depends on. The
presence of this exponential in (74) insures that the mean value ofin the state described by()be
equal to .
593

COMPLEMENT G V
This scalar product is therefore never zero.
However, we shall show that thestates do satisfy a closure relation, which is
written:
1
dRedIm= 1 (81)
To do so, we replace, on the left-hand side of (81), by its expression (41). This yields:
1
e
2
!!
dRedIm (82)
that is, going into polar coordinates in the complexplane (setting=e):
1
0
d
2
0
de
2
e
( )
+
!!
(83)
The integral overis easily calculated:
2
0
e
( )
d= 2 (84)
which yields for (83):
1
!
(85)
with:
= 2
0
de
2
2
=
0
de (86)
Integrating by parts, we nd a recurrence relation for the:
= 1 (87)
whose solution is:
=!0=! (88)
Substituting this result into (85), we see that the left-hand side of formula (81) can nally
be written:
(89)
which proves that formula.
3. Time evolution of a quasi-classical state
Consider a harmonic oscillator which, at the instant= 0, is in a particularstate:
(0)= 0 (90)
How do its physical properties evolve over time? We already know (cf.) that the
mean values()and()always remain equal to the corresponding classical values.
We shall now study other interesting properties of the state vector().
594

COHERENT QUASI-CLASSICAL STATES OF THE HARMONIC OSCILLATOR
3-a. A quasi-classical state always remains an eigenvector of
Starting with (41), we can use the general rule to obtain()when the Hamil-
tonian is not time-dependent (cf.Chap., ):
()= e
2
2 0
!
e
~
= e
2
e
0
2
2 0e
!
(91)
If we compare this result with (41), we see that, to go from(0)= 0to(), all
we must do is change0to0e and multiply the ket obtained bye
2
(which is
a global phase factor with no physical consequences):
()= e
2
=0e (92)
In other words, we see that a quasi-classical state remains an eigenvector offor all time,
with an eigenvalue0e which is nothing more than the parameter()of Figure
(corresponding to the point), which characterizes at all times the classical oscillator
whose motion is reproduced by the state().
3-b. Evolution of physical properties
Using (92) and changingto0e in (56), we immediately obtain:
() =
2~
Re[0e]
() =
2~Im[0e]
(93)
As predicted, these equations are similar to the classical equations (9).
The average energy of the oscillator is time-independent:
=~ 0
2
+
1
2
(94)
According to (51) and (58), the root mean square deviations,andare equal
to:
=~ 0 (95)
and:
=
~2
=
~2
(96)
andare not time-dependent, the wave packet remains a minimum wave packet
for all time.
595

COMPLEMENT G V
3-c. Motion of the wave packet
Let us calculate the wave function at time:
() = () (97)
where()is given by (92). From (76), we obtain:
() = e
~
14
e
2
e
()
~e
()
2
2
(98)
At, the wave packet is still Gaussian. Its form does not vary with time, since:
()
2
= 0[ ()]
2
(99)
Thus, it remains minimum for all time [cf.(96)].
Figure
(= 2) along theaxis, without becoming distorted. We saw in ComplementI
that a Gaussian wave packet, when it is free, becomes distorted as it propagates, since its
width varies (spreading of the wave packet). We see here that nothing of the sort occurs
when a wave packet is subject to the inuence of a parabolic potential. Physically, this
result arises from the fact that the tendency of the wave packet to spread is compensated
by the potential, whose eect is to push the wave packet towards the origin from regions
where()is large.
What happens to these results whenis very large? The root mean square
deviationsanddo not change, as is shown by (96). On the other hand, the
oscillation amplitudes of()and ()become much larger thanand. By
choosing a suciently large value of, one can obtain a quantum mechanical motion for
which the position and momentum of the oscillator are, in relative value, as well-dened
as might be desired. Therefore when1, anstate describes very well the motion
of a macroscopic harmonic oscillator, for which the position, the momentum and the
energy can be considered to be classical quantities.
4. Example: quantum mechanical treatment of a macroscopic oscillator
Let us consider a concrete example: a macroscopic body of mass= 1kg suspended
from a rope of length= 01m and placed in the gravitational eld (10m/s
2
). We
know that, for small oscillations, the periodof the motion is given by:
= 2
(100)
In our case, we obtain:
063 s
= 10 rds
(101)
596

COHERENT QUASI-CLASSICAL STATES OF THE HARMONIC OSCILLATOR0
ψ (x,t)
2
0
0
0
x
x
x
x
Figure 2: Motion of the Gaussian wave packet associated with anstate: under the
eect of the parabolic potential(), the wave packet oscillates without becoming dis-
torted.
Let us now assume that this oscillator performs a periodic motion of amplitude
= 1cm. What is the quantum mechanical state that best represents its oscillation?
We have seen that this state is anstate in which, according to (93),satises
the relation:
=
2~
(102)
that is, in our case:
510
15
2210
15
1 (103)
(the argument ofis determined by the initial phase of the motion).
597

COMPLEMENT G V
The root mean square deviationsandare then:
=
~2
2210
18
m
=
~2
2210
17
kg ms
(104)
The root mean square deviationof the velocity is equal to:
2210
17
ms (105)
Since the maximum velocity of the oscillator is 0.1 m/s, we see that the uncertain-
ties of its position and its velocity are completely negligible compared to the quantities
involved in the problem. For example,is less than a Fermi (10
15
m), that is, the
approximate size of a nucleus. Measuring a macroscopic length with this accuracy is
obviously out of the question.
Finally, the energy of the oscillator is known with an excellent relative accuracy,
since, according to (52):

1
0410
15
1 (106)
The laws of classical mechanics are therefore quite adequate for the study of the evolution
of the macroscopic oscillator.
References and suggestions for further reading:
Glauber's lectures in (15.2).
598

NORMAL VIBRATIONAL MODES OF TWO COUPLED HARMONIC OSCILLATORS
Complement HV
Normal vibrational modes of two coupled harmonic oscillators
1 Vibration of the two coupled in classical mechanics
1-a Equations of motion
1-b Solving the equations of motion
1-c The physical meaning of each of the modes
1-d Motion of the system in the general case
2 Vibrational states of the system in quantum mechanics
2-a Commutation relations
2-b Transformation of the Hamiltonian operator
2-c Stationary states of the system
2-d Evolution of the mean values
This complement is devoted to the study of the motion of two coupled (one-
dimensional) harmonic oscillators. Such a study is of interest because it permits the
introduction, in a very simple case, of a physically important concept: that ofnormal vi-
brational modes. This concept, encountered in quantum mechanics as well as in classical
mechanics, appears in numerous problems: for example, in the study of atomic vibrations
in a crystal (cf.ComplementV) and of the vibrations of electromagnetic radiation (cf.
ComplementV).
1. Vibration of the two coupled in classical mechanics
1-a. Equations of motion
Let us therefore consider two particles (1) and (2), of the same mass, moving
along theaxis, with abscissas1and2. To begin, we assume their potential energy
to be:
0(12) =
1
2
2
(1)
2
+
1
2
2
(2+)
2
(1)
When 1=and2= , the potential energy0(12)is minimal, and the two
particles are in stable equilibrium. If the particles move from these equilibrium positions,
they are subjected to the forces1and2, respectively:
1=
1
0(12) =
2
(1)
2=
2
0(12) =
2
(2+)
(2)
599

COMPLEMENT H V
and their motion is given by:
d
2
d
2
1() =
2
(1)
d
2
d
2
2() =
2
(2+)
(3)
Each particle therefore follows an independent sinusoidal motion, centered at its equi-
librium position. The amplitude of the motion of each particle is arbitrary
1
and can be
xed by a suitable choice of the initial conditions.
Now let us assume the potential energy(12)of the two particles to be:
(12) =0(12) +(12) (4)
with:
(12) =
2
(1 2)
2
(5)
(whereis a dimensionless positive constant which we shall call the coupling constant).
To the forces1and2written in (2), we must add, respectively, the forces
1and
2
given by:
1=
1
(12) = 2
2
(2 1)
2=
2
(12) = 2
2
(1 2)
(6)
We see that the introduction of(12)takes into account an attractive force
between the particles, which is proportional to the distance between them. The two
particles (1) and (2) are therefore no longer independent; what is their motion now?
Before attacking this problem from a quantum mechanical point of view, we shall recall
the results given by classical mechanics.
1-b. Solving the equations of motion
In the presence of the coupling(12), we must replace (3) by the system of
coupled dierential equations:
d
2
d
2
1() =
2
(1) + 2
2
(2 1)
d
2
d
2
2() =
2
(2+) + 2
2
(1 2)
(7)
We know how to solve such a system (see for example Chapter, ). We
diagonalize the matrixof the coecients appearing on the right-hand side of (7):
=
2
1 + 2 2
2 1 + 2
(8)
1
Of course, the choice of the potential (1) implies that we are not taking into account the collisions
that could occur if suciently large amplitudes were chosen.
600

NORMAL VIBRATIONAL MODES OF TWO COUPLED HARMONIC OSCILLATORS
We then replace1()and2()by linear combinations of these two functions (given
by the eigenvectors of) whose time dependence obeys uncoupled linear dierential
equations (with coecients which are the eigenvalues of).
In this case, these linear combinations are:
() =
1
2
[1() +2()] (9)
(the position of the center of mass of the two particles) and:
() =1() 2() (10)
(the abscissa of the relative particle). Substituting (9) and (10) into (7), we obtain
(taking the sum and the dierence):
d
2
d
2
() =
2
()
d
2
d
2
() =
2
[()2]4
2
() (11)
These equations can be integrated immediately:
() =
0
cos(+)
() =
2
1 + 4
+
0
cos(+)
(12)
with:
=
=
1 + 4
(13)
0
,
0
,and are integration constants xed by the initial conditions. To obtain
the motion of particles (1) and (2), all we must do is invert formulas (9) and (10):
1() =() +
1
2
()
2() =()
1
2
()
(14)
and substitute (12) into these equations.
1-c. The physical meaning of each of the modes
Through the change of functions performed in (9) and (10), we have been able to
nd the motion of the two interacting particles by associating with them twoctitious
particles()and(), of abscissas()and(). These ctitious particles do not
601

COMPLEMENT H V
interact; their motions are independent, so their amplitudes and phases can be xed
arbitrarily by a suitable choice of the initial conditions. For example, it is possible to
require one of the two ctitious particles to be motionless without this being the case for
the other one: we then say that avibrational modeof the system is excited. It must be
understood that, in a vibrational mode, the real particles (1) and (2) are both in motion
with the same angular frequency (or, depending on the mode). No solution of
the equations of motion exists for which one of the two real particles (1) or (2) remains
motionless while the other one vibrates. If, at the instant= 0, one were to give an
initial velocity to only one of the two particles, (1) or (2), the coupling force would set
the other one in motion (cf.discussion of
The simplest case is of course the one in which neither of the two modes is excited.
In formulas (12) such a situation corresponds to
0
=
0
= 0;()and()then
always remain equal to zero and2(1+4)respectively, which, according to (14), yields:
1= 2=
1 + 4
(15)
The system does not oscillate and the two particles (1) and (2) remain motionless in
their new equilibrium positions given by (15) (it can be veried that, for these values of
1and2, the forces exerted on the particles are zero; the fact that these equilibrium
positions are closer in the presence of the coupling than when= 0is due to their
mutual attraction).
To excite only the mode corresponding to(), one places the two particles (1)
and (2) at the initial instant at the same distance2(1+4)as in the preceding case, and
one gives them equal velocities. One then nds that()remains equal to2(1 + 4)
(the initial conditions require
0
to be zero). The two particles move in unison,
performing the same motion without the distance between them varying. For this mode,
the two-particle system can be treated like a single undeformable particle of mass2
on which is exerted the force1+2=2
2
(). We then see why the angular
frequency of this mode is=[cf.formula (A-3) of Chapter].
To excite only the mode corresponding to(), one chooses an initial state in
which the positions and initial velocities of the two particles are opposite. One then nds
that, at every subsequent instant,() = 0, and the two particles move symmetrically
with respect to the origin. For this mode, the distance(2 1)varies and the
attractive force between the two particles enters into the equations of motion; this is the
reason why the angular frequency of this mode is notbut=
1 + 4.
The dynamical variables()and()associated with the independent modes,
that is, with the ctitious particles()and(), are callednormal variables.
1-d. Motion of the system in the general case
In the general case, both modes are excited and the positions1()and2()are
both given by the superposition of two oscillations of dierent frequenciesand
[cf.formulas (14)]. The motion of the system is not periodic, except in the case in which
the ratio is rational
2
.
2
If = 1
1 + 4is equal to an irreducible rational fraction12, the period of the motion
is:= 21 = 22 .
602

NORMAL VIBRATIONAL MODES OF TWO COUPLED HARMONIC OSCILLATORS
Let us investigate, for example, what happens if, at the initial time0, particle
(1) is motionless at its equilibrium position1=(1 + 4), while particle (2) has a
non-zero velocity (this is, in classical mechanics, the analogue of the problem studied in
). In the absence of coupling, particle (2) would oscillate alone and
particle (1) would remain motionless. We shall show that the coupling sets particle (1)
in motion. Two dierent angular frequenciesand appear in the time evolution
of1()and2(). The two corresponding oscillations give rise to a beat phenomenon
(Fig.), whose frequency is:
=
2
=
2
[
1 + 41] (16)
If the coupling is weak (1), this frequency is negligible with respect
to and. In this case, as long as( 0)1, particle (2) is practically the
only one to oscillate; the vibrational energy is then slowly transferred to particle (1),
whose amplitude of oscillation increases, while that of (2) decreases. After a certain
time, the original situation is inverted: particle (1) oscillates strongly while particle (2)
is practically motionless. Then the amplitude of (1) slowly decreases and that of (2)
increases until the energy is again almost entirely localized in oscillator (2). The same
process is repeated indenitely. The eect of a weak coupling is to cause the energy of the
oscillator associated with particle (1) to be constantly transferred to the one associated
with particle (2) and vice versa, with a frequency proportional to the intensity of the
coupling.t
x
1
a
0
1 + 4λ
Figure 1: Oscillations of the position of particle (1), assumed to be motionless at its
equilibrium position at= 0, particle (2) having an initial velocity. A beat phenomenon
is produced between the two modes, and the amplitude of the oscillations of particle (1)
varies over time.
Comments:
()If1and2are the respective momenta of particles (1) and (2), the classical
Hamiltonian of the system under study can be written:
(1212) =
2
1
2
+
2
2
2
+0(12) +(12) (17)
603

COMPLEMENT H V
If we set:
() =1() +2()
() =
1
2
[1() 2()]
(18)
and:
= 2
=
2
(19)
it can be veried thatbecomes:
=
2
2
+
1
2
22
+
2
2
+
1
2
2
2
1 + 4
2
+
22
4
1 + 4
(20)
By a suitable change in the energy origin, one can eliminate the last term of
this expression, which is constant.can then be seen to be the sum of two
energies, each of which corresponds to one of the modes. Unlike the situation
in (17), in which the terms in12of(12)are responsible for a coupling
between the particles, there is no coupling term in (20) between the modes:
they are indeed independent.
()We have assumed, for simplicity, the masses1and2of particles (1) and
(2) to be equal. It is easy to eliminate this hypothesis by replacing (9), (10),
(18) and (19) by:
() =
11() +22()
1+2
() =1() +2()
= 1+2
(21)
(the position, momentum and mass associated with the center of mass) and:
() =1() 2()
() =
21() 12()
1+2
=
12
1+2
(22)
604

NORMAL VIBRATIONAL MODES OF TWO COUPLED HARMONIC OSCILLATORS
(the position, momentum and mass of the relative particle). One then nds
a result analogous to (20).
()In the absence of coupling, the two modes have the same angular frequency
; in the presence of coupling, two dierent angular frequenciesand
appear. This is an example of a result which is often found in physics: the
eect of a coupling between two oscillations is, in most cases, to separate
their normal frequencies (the same phenomenon would occur here if the two
oscillators originally had dierent angular frequencies). If, instead of two, we
have an innite number of oscillators (which, if isolated, would have the same
frequency), we shall see in ComplementVthat the eect of the coupling is
to create an innite number of dierent frequencies for the modes.
2. Vibrational states of the system in quantum mechanics
Let us now reconsider the problem from a quantum mechanical point of view. We must
now replace the positions1(),2()and the momenta1(),2()of the particles by
operators, which we shall denote, respectively, by1,2,1,2. We then introduce,
by analogy with (9), (10) and (18), the observables:
=
1
2
(1+2)
=1+2
(23)
=1 2
=
1
2
(1 2)
(24)
To see if the operator, the Hamiltonian of the system, can be put into a form analogous
to (20), we shall begin by examining the commutation relations of,,et.
2-a. Commutation relations
Since all the observables concerning only particle (1) commute with those concern-
ing particle (2), the only non-zero commutators involving1,2,1and2are:
[11] =~
[22] =~ (25)
In particular,1commutes with2, and we see immediately that:
[ ] = 0 (26)
Similarly:
[ ] = 0 (27)
605

COMPLEMENT H V
Calculating the commutator [ ], we obtain:
[ ] =
1
2
[11] + [12] + [21] + [22]
=
1
2
~+~=~ (28)
Similarly, one nds:
[ ] =~ (29)
The two commutators [ ] and [ ] remain to be examined; they are equal to:
[ ] =
1
4
[11][12] + [21][22]
=
1
4
~~= 0 (30)
and, similarly:
[ ] = 0 (31)
We can thus consider, and ,to be the position and momentum
operators of two distinct particles. Formulas (28) and (29) are the canonical commutation
relations for each of these particles. Moreover, relations (26), (27), (30) and (31) express
the fact that all the observables concerning one of them commute with all those which
concern the other one.
2-b. Transformation of the Hamiltonian operator
In the presence of the coupling(12), we have:
=+ (32)
with:
=
1
2
2
1+
2
2 (33)
(the kinetic energy operator) and:
=
1
2
2
(1)
2
+ (2+)
2
+ 2(1 2)
2
(34)
(the potential energy operator). Since1and2commute, (33) can be transformed as
if these operators were numbers; we nd:
=
1
2
2
+
1
2
2
(35)
606

NORMAL VIBRATIONAL MODES OF TWO COUPLED HARMONIC OSCILLATORS
whereand are dened in (19). Similarly, since1and2commute, we have, as
above [formula (20)]:
=
1
2
22
+
1
2
2
2
1 + 4
2
+
22
4
1 + 4
(36)
where and are given by (13).
Thus we see thatcan be put into a form which is analogous to (20), in which
there is no coupling term:
= + +
22
4
1 + 4
(37)
with:
=
2
2
+
1
2
22
=
2
2
+
1
2
2
2
1 + 4
2
(38)
2-c. Stationary states of the system
The state space of the system is the tensor product(1)(2)of the state spaces
of particles (1) and (2); it is also the tensor product()()of the state spaces of
the ctitious particles, the center of mass and the relative particle associated with
each of the two modes. Sinceis the sum of two operatorsand which act only
in()and()respectively (the constant
22
4
1 + 4
merely introduces a shift in
the energy origin), we know (Chap., ) that we can nd a basis of eigenvectors of
in the form:
= (39)
where and are, respectively, eigenvectors ofand in()and().
Now, and are Hamiltonians of one-dimensional harmonic oscillators, whose eigen-
vectors and eigenvalues we know. If the operatorsandare dened by:
=
1
2~~
=
1
2~~
(40a)
with:
=
2
1 + 4
(40b)
and if
0and
0denote respectively the ground states ofand, the eigen-
vectors ofare the vectors:
=
1
!
()
0 (41)
607

COMPLEMENT H V
whose eigenvalues are:
= +
1
2
~ (42)
those ofbeing:
=
1
!
()
0 (43)
with the eigenvalues:
= +
1
2
~ (44)
Thus we have here a situation which is analogous to the one encountered in the
study of a two-dimensional anisotropic (since=) harmonic oscillator. The sta-
tionary states of the system are given by:
= =
()()
!!
00 (45)
and their energies are:
= + +
22
4
1 + 4
= +
1
2
~++
1
2
~+
22
4
1 + 4
(46)
The operatorsand or (and) can thus be seen to be annihilation or
creation operators of an energy quantum in the mode corresponding to()[or()]. We
see from (45) that, through the repeated action ofand, we can obtain stationary
states of the system in which the number of quanta in each mode is arbitrary. The action
of,,oron the stationary statesis very simple:
=
+ 1 +1
=
1
=
+ 1 +1
=
1 (47)
In general, there are no degenerate levels since there do not exist two dierent
pairs of integersand such that:
+ = + (48)
(except when the ratio =
1 + 4is rational).
608

NORMAL VIBRATIONAL MODES OF TWO COUPLED HARMONIC OSCILLATORS
2-d. Evolution of the mean values
The most general state of the system is a linear superposition of stationary states
:
()= () (49)
with:
() =(0) e
~
(50)
According to relations (40) and their adjoints,() is a linear combination of
and (ofand). We then see, by using (47), thathas non-zero matrix
elements between two states and only when =1,=(for,
we would have=, =1). From this we deduce that the only Bohr frequencies
which can appear in the time evolution of()and ()are, respectively
3
:
1
~
=
1
~
= (51)
Thus we again nd that ()and()oscillate at angular frequencies ofand
, which is consistent with the classical result obtained in .
References and suggestions for further reading:
Coupling between two classical oscillators: Berkeley 3 (7.1), 1.4 and 3.3; Alonso and
Finn (6.1), Vol. I, 12.10.
3
For these frequencies actually to appear, at least one of the products
1
, or
1
must be dierent from zero.
609

VIBRATIONAL MODES OF A LINEAR CHAIN OF COUPLED HARMONIC OSCILLATORS; PHONONS
Complement JV
Vibrational modes of an innite linear chain of coupled harmonic
oscillators; phonons
1 Classical treatment
1-a Equations of motion
1-b Simple solutions of the equations of motion
1-c Normal variables
1-d Total energy and energy of each of the modes
2 Quantum mechanical treatment
2-a Stationary states in the absence of coupling
2-b Eects of the coupling
2-c Normal operators. Commutation relations
2-d Stationary states in the presence of coupling
3 Application to the study of crystal vibrations: phonons
3-a Outline of the problem
3-b Normal modes. Speed of sound in the crystal
In ComplementV, we studied the motion of a system of two coupled harmonic
oscillators. We concluded, in essence, that, while the individual dynamical variables of
each oscillator do not evolve independently, it is possible to introduce linear combinations
of them (normal variables) which possess the important property of being uncoupled.
Such variables describe vibrational normal modes of well-dened frequencies. Expressed
in terms of these normal variables, the Hamiltonian of the system appears in the form
of a sum of Hamiltonians of independent harmonic oscillators, thus making quantization
simple.
In this complement we shall show that these ideas are also applicable to a system
formed by an innite series of identical harmonic oscillators, regularly spaced along an
axis, each one coupled to its neighbors.
To do this, we shall determine the various vibrational normal modes of the system
and show that each one corresponds to a collective vibration of the system of particles,
characterized by an angular frequencyand a wave vector. The process of nding
the eigenstates and eigenvalues of the quantum mechanical Hamiltonian is then greatly
simplied by the fact that the total energy of the system is the sum of the energies
associated with each vibrational normal mode.
The results obtained will enable us to indicate how vibrations propagate in a crystal
and to introduce the concept of a phonon, a central idea in solid state physics. Of course,
in this complement, we shall emphasize the introduction and quantization of the normal
modes, and not the detailed properties of phonons, which would be treated in a solid
state physics course.
611

COMPLEMENT J V
1. Classical treatment
1-a. Equations of motion
Let us consider an innite chain of identical one-dimensional harmonic oscillators,
each one labeled by an integer(positive, negative or zero). The particleof mass
which constitutes oscillator () has its equilibrium position at the point whose abscissa
is(Fig.), whereis the unit distance of the oscillator chain. We denote bythe
(algebraic) displacement of oscillator () with respect to its equilibrium position. The
state of the system at the instantis dened by specifying the dynamical variables()
and their time derivatives_()at this instant.(q – 1)l (q + 1)lql
M
q – 1
x
q – 1
x
q + 1
x
q
M
q + 1
M
q
Figure 1: Innite chain of oscillators; the displacement of theth particle with respect
to its equilibrium positionis denoted by.
In the absence of interactions between the various particles, the potential energy
of the system is:
( 10+1) =
+
=
1
2
22
(1)
whereis the angular frequency of each oscillator. The evolution of the system is then
given by the equations:
d
2
d
2
() =
2
() (2)
whose solutions are:
() =cos( ) (3)
where the integration constantsand are xed by the initial conditions of the
motion. The oscillators therefore vibrate independently.
Now imagine that these particles are interacting. For simplicity, we shall assume
that one need take into account only the forces exerted on a particle by its two nearest
neighbors and that these forces are attractive and proportional to the distance. Thus,
particle () is subjected to two new attractive forces exerted by particles(+ 1)and
(1). These forces are proportional to +1and 1(the coecient of
proportionality being the same in both cases). The total forceto which particle()
612

VIBRATIONAL MODES OF A LINEAR CHAIN OF COUPLED HARMONIC OSCILLATORS; PHONONS
is subjected can therefore be written:
=
2 2
1[+ (+ 1) +1]
2
1[+ (1) 1]
=
2 2
1( +1)
2
1( 1) (4)
where1is a constant [having the dimensions of an inverse time] which characterizes the
intensity of the coupling. Equations (2) must now be replaced by:
d
2
d
2
() =
2
()
2
1[2() +1() 1()] (5)
It can easily be veried that the interaction forces [terms in
2
1of (4)] are derived from
the potential energy of the couplinggiven by:
( 10+1) =
1
2
2
1
+
=
( +1)
2
(6)
According to (5), the evolution ofdepends on+1and 1. Therefore, we
must solve an innite system of coupled dierential equations. Before we introduce new
variables that allow these equations to be uncoupled, it is interesting to try to nd simple
solutions of equations (5) and investigate their physical signicance.
1-b. Simple solutions of the equations of motion
. Existence of simple solutions
The innite chain of coupled oscillators we are studying is analogous to an innite
macroscopic spring. Now, we know that progressive longitudinal waves (corresponding
to expansions and compressions) can propagate along this spring. Under the inuence
of a sinusoidal wave of this type, of wave vectorand angular frequency, the point of
the spring whose abscissa isat equilibrium is found at timeat+(), with:
() =e
( )
+e
( )
(7)
Such solutions of the equations of motion (5) do indeed exist. However, since the oscillator
chain is not a continuous medium, the eects of the wave are observed only at a series of
points, corresponding to the abscissas=;()thus represents the displacement
of oscillator()at time:
() =() =e
( )
+e
( )
(8)
It is easy to verify that this expression is a solution of equations (5) ifandsatisfy:

2
=
2 2
12e e (9)
is therefore related toby the dispersion relation:
() =
2
+ 4
2
1
sin
2
2
(10)
which we shall discuss in detail later ( 1-b-).
613

COMPLEMENT J V
. Physical interpretation
In the solution (8) of the equations of motion, all the oscillators are vibrating at
the same frequency2, with the same amplitude2, but with a phase that depends
periodically on their rest positions. It is as if the displacements of the various oscillators
were determined by a progressive sinusoidal wave of wave vectorand phase velocity:
() =
()
(11)
This is easy to show. Using (8), we see that:
1+2() =
1
2
(12)
Thus, oscillator(1+2)performs the same motion as oscillator(1), shifted by the
time taken by the wave to travel, at a velocity, the distance2separating the two
oscillators. Since all the oscillators are then in motion, solutions (8) are called collective
modes of vibration of the system.
. Possible values of the wave vector
Consider two values of the wave vector,and, which dier by an integral number
of2:
=+
2
withan integer (positive or negative); (13)
We have, obviously:
e= e
() = ()
(14)
where the second relation follows directly from (10).
We see from (8) that the two progressive wavesandlead to the same motion
for the oscillators and are, consequently,physically indistinguishable. Therefore, in the
problem we are studying here, it suces to letvary over an interval of2. For reasons
of symmetry, we choose:
(15)
The corresponding interval is often called the rst Brillouin zone.
. Dispersion relation
The dispersion relation (10) which gives the angular frequency()associated
with each value ofenables us to study the propagation of vibrations in the system. If,
for example, we form a wave packet by superposing waves with dierent wave vectors,
we know that it has a group velocity given by:
=
d()
d
(16)
614

VIBRATIONAL MODES OF A LINEAR CHAIN OF COUPLED HARMONIC OSCILLATORS; PHONONS
which is dierent from.
Figure ()with respect to, wherevaries
within the rst Brillouin zone.0
ω
ω
2
+ 4 ω
1
2
Ω
+ π/l– π/l
k
Figure 2: Dispersion relation giving the variation of the angular frequency of the vi-
brational normal modes with respect to the wave numberin the rst Brillouin zone
[ +]. The dashed line corresponds to the case= 0.
It is clear from this gure that()cannot take on arbitrary values: a vibration
of frequencycan propagate freely in the medium only iffalls within the allowed
band:
2
2
+ 4
2
12
(17)
The other values ofcorrespond to forbidden bands. The two limiting frequencies of
interval (17) are often called cut-o frequencies.
The mode of lowest angular frequency,(0) =, has a zero wave vector; it
corresponds to an in-phase vibration of all the oscillators, whose particles are moving
together without changing their relative distances (Fig.). This explains why the
angular frequency of this mode is the same as in the absence of coupling (cf.Comple-
mentV, ).
As for the mode of highest angular frequency,( ) =
2
+ 4
2
1
, in the(q – 2)l (q + 1)l (q + 2)l(q – 1)l ql
x
q – 2
x
q – 1
x
q + 1
x
q
^
Figure 3: The lowest frequency mode (= 0; =) corresponds to a displacement of
the system of oscillators as a whole. This is why its frequency does not depend on the
coupling.
615

COMPLEMENT J V(q – 2)l
x
q – 2 x
q – 1
x
q + 1 x
q + 2x
q
(q + 1)l (q + 2)l(q – 1)l ql
Figure 4: The modes= are those in which two neighboring oscillators are com-
pletely out of phase; the couplingstrongly modies their frequency.
corresponding vibration of the system, two adjacent oscillators are completely out of
phase (Fig.); the eect of the attractive forces due to the couplingis then maximal.
1-c. Normal variables
. Obtaining uncoupled equations
Returning to the equations of motion (5), let us introduce new dynamical variables
(linear combinations of the) that evolve independently.
To do so, we multiply both sides of equation (5) by the quantitye and sum
over. If we notice that:
+
=
1e = e
+
=
1e
(1)
= e
+
=
e (18)
and if we set:
+
=
() e=() (19)
we see that (5) becomes:
2
2
() =
2
+
2
12e e () (20)
that is, taking (9) into account:
2
2
() =
2
()() (21)
This relation shows that the time evolution of()is independent of that of()
fordierent from. The quantities()introduced in (19) are therefore completely
uncoupled and have a remarkably simple equation of motion.
616

VIBRATIONAL MODES OF A LINEAR CHAIN OF COUPLED HARMONIC OSCILLATORS; PHONONS
Comments:
()Equations (5) are easy to uncouple because the problem is invariant under a
translation of the system of oscillators by a quantity(replacement ofby
1). This invariance is itself due to the fact that the chain is regular and
innite.
()In reality, every chain is of course nite, even if it contains a very large
number of oscillators. To nd its vibrational normal modes, one must
thus take into account the boundary conditions at the two ends of the chain,
and the problem becomes much more complicated (edge eects). Instead of
obtaining, as we do here, a continuous innity of vibrational normal modes
corresponding to the various values ofin the rst Brillouin zone, one nds
a nite number of eigenmodes, equal to the numberof oscillators. When
one is concerned only with the behavior of the chain far from the ends, one
often introduces articial boundary conditions which are dierent from the
real boundary conditions but which have the advantage of simplifying the
calculations while conserving the essential physical properties. Thus, one
requires the two end oscillators to have the same motion (periodic boundary
conditions, also called Born-Von Karman conditions). We shall have the
opportunity to return to this question in connection with the study of other
periodic structures (cf.ComplementXI; see also XIV).
Therefore we shall not dwell any further on periodic boundary conditions,
but shall continue our discussion, conning ourselves to the simple case of an
innite chain.
The function()introduced in (19) is, by denition, the sum of a Fourier series
whose coecients are the displacements(). It is a periodic function, of period2,
which is therefore perfectly well-dened when its values in the interval
are specied [this is the rst Brillouin zone, dened in (15)].()depends on the
positions of all the oscillators at time. Conversely, these positions are unambiguously
dened when the values ofin the interval (15) are given for time. This is true because
it is possible to invert relation (19) since, using:
+
de
( )
=
2
(22)
we obtain:
() =
2
+
d() e (23)
Note also that, since the displacements()are real, the function()satises:
( ) =() (24)
Similarly, one can, using the momenta() =_(), dene the function:
() = () e (25)
617

COMPLEMENT J V
and the()can be expressed as:
() =
2
+
() ed (26)
The fact that the()are real implies that:
( ) =() (27)
Dierentiating both sides of (19) term by term, and using (25) and then (21), we nally
obtain:
() =() (28a)() =
2
()() (28b)
At time, the dynamical state of the system can be characterized just as well by
the specication of the()and()for all integers (positive, negative or zero), as by
that of the normal variables()and()(wherecan take on any value in the
rst Brillouin zone). The equations of motion (28) of the normal variables corresponding
to each value ofdescribe the evolution of the position and momentum of a harmonic
oscillator of massand angular frequency(); however,andare complex.Thus we
have reduced the study of an innite but discrete chain of coupled harmonic oscillators
to that of a continuous system of ctitious independent oscillators(labeled by the index
).
Comments:
Rigorously, these ctitious oscillators are not completely independent since, ac-
cording to conditions (24) and (27), the initial values(0)and(0)must
satisfy:
(0) =(0)
(0) =(0) (29)
. Normal variables()associated with the progressive waves
It is convenient (see also V) to condense the two normal
variables()and()into one,(), dened by:
() =
1
2
^
() +^() (30)
where
^
()and^()are dimensionless quantities proportional to()and():
^
() =()()
^() =
1
~()
()
(31)
618

VIBRATIONAL MODES OF A LINEAR CHAIN OF COUPLED HARMONIC OSCILLATORS; PHONONS
To simplify the quantum mechanical calculations presented later, we shall set:
() =
()~
(32)
It is easy to show, using (30), that the two equations (28) are equivalent to the single
equation:
() = ()() (33)
which is of rst order in[()is completely dened by the specication of(0),
while()depends on(0)and(0)]. The general solution of (33) is:
() =(0) e
()
(34)
Using (19) and (25), we easily obtain the expression for()in terms of the
()and():
() =
1
2
()e () +
()
()
(35)
Let us show, conversely, that the()and()can be simply expressed in terms of the
(). According to (24) and (27):
( ) =
1
2
^
( )^( )
=
1
2
^
()^() (36)
From this we deduce:
^
() =
1
2
[() +( )] (37a)
^() =
2
[() ( )] (37b)
which allows us to write formula (23) in the form:
() =
22
+
()
()
ed+
+
( )
()
ed (38)
Changingtoin the second integral, we nally obtain [()is an even function of
]:
() =
22
+
()
()
ed+
+
()
()
ed (39)
619

COMPLEMENT J V
An analogous calculation, starting with (26), yields:
() =
22
~
+()() ed
+
()() ed (40)
The state of the system is therefore described just as well by the()as by the set of
()and().
If we replace, in (39),()by its general expression (34),()takes on the form:
() =
22
+
d
(0)
()
e
[ ()]
+ (41)
The most general solution of the problem of the chain of coupled oscillators is therefore
a linear superposition of the progressive waves introduced in 1-b (where the coecients
of this linear combination are
22
(0)
()
). These progressive waves constitute the
vibrational normal modes of the system
1
.
Comment:
For each value of, the two terms appearing on the right-hand sides of (39) and
(40) are complex conjugates of each other. This insures the reality of the()
and()without the necessity of imposing an arbitrary condition on the().
The(), consequently, are truly independent variables.
1-d. Total energy and energy of each of the modes
The total energy of the system under consideration is the sum of the kinetic energies
of each particle()and the potential energies (1) and (6):
( 10+1 10+1) =
+
=
1
2
2
+
1
2
22
+
1
2
2
1( +1)
2
(42)
We shall see in this section that this energy can be expressed very simply in terms of the
energies that can be associated with each of the modes.
Let us therefore calculate the various sums involved in (42). Since the displace-
mentsare the coecients of the Fourier series dening the function(), Parseval's
1
We could also have introduced the modes corresponding to stationary waves in the system (sum
of two progressive waves of the same frequency and opposite velocities). We would then have obtained
equivalent results but the motion of the system would have been expanded on a dierent basis. An
expansion of this type is used in ComplementV.
620

VIBRATIONAL MODES OF A LINEAR CHAIN OF COUPLED HARMONIC OSCILLATORS; PHONONS
relation [Appendix, relation (18)] immediately yields:
+
=
()
2
=
2
+()
2
d (43)
+
=
()
2
=
2
+
()
2
d (44)
The sum which, in (42), corresponds to the coupling, now remains to be calculated. To
do so, notice as in (18) that, if the displacementsare the coecients of the Fourier
series of(), the+1are those of e(). The quantities ( +1) are therefore
the coecients of the Fourier series of[1e](), and Parseval's relation yields:
+
=
( +1)
2
=
2
+(1e()
2
d
=
2
+4 sin
2
2
()
2
d (45)
Substituting (43), (44) and (45) into (42), we nally obtain:
=
2
+
2
2
+ 4
2
1sin
2
2
()
2
+
1
2
()
2
d (46)
We shall write this result in the form:
=
2
+() d (47)
with:
() =
1
2

2
()()
2
+
1
2
()
2
(48)
where()is given by (10).is thus the sum (in fact, the integral) of the energies
associated with the ctitious uncoupled harmonic oscillators for which()gives the
position and(), the momentum.
We can also express()in terms of the variables()associated with each
normal mode. Using (37), we transform expression (48) into:
() =
1
2
~()()() +( )( ) (49)
that is, taking (34) into account:
() =
1
2
~()(0)(0) +(0)(0) (50)
()is therefore time-independent, which is not surprising since()is the energy of a
harmonic oscillator. Furthermore, we again nd in (47) that the ctitious oscillators are
621

COMPLEMENT J V
independent, since the total energyis simply the sum of the energies associated with
each of them.
Substituting expression (49) into (47), we obtain:
=
2
+d
1
2
~()()() +( )( ) (51)
We can then changetoin the integral of the second term and considerto be the
sum of the energies()associated with the normal modes characterized by the():
=
2
+d() (52)
with:
() =~()()()
=~()(0)(0) (53)
2. Quantum mechanical treatment
The quantum mechanical treatment of the problem of the innite chain of coupled oscil-
lators is based, in accordance with the general quantization rules, on the replacement of
the classical quantities()and()by the observablesandwhich satisfy the
canonical commutation relations:
[
1 2
] =~
12
(54)
2-a. Stationary states in the absence of coupling
In the absence of coupling(1= 0), the Hamiltonianof the system can be
written:
(1= 0) =
1
2
22
+
1
2
2
= (55)
whereis the Hamiltonian of a one-dimensional harmonic oscillator acting in the state
space of particle().
We introduce the operatordened by:
=
1
2~
+
~
(56)
can then be written:
=
1
2
+ ~= +
1
2
~ (57)
622

VIBRATIONAL MODES OF A LINEAR CHAIN OF COUPLED HARMONIC OSCILLATORS; PHONONS
andare the creation and annihilation operators of an energy quantum for oscillator
(). We know (Chap., ) that the eigenstates ofare given by:
=
1
()!
0
(58)
where
0
is the ground state of oscillator()andis a positive integer or zero. If we
choose as the energy origin the energy of the ground state [which amounts to omitting,
in (57), the term 1/2], we obtain for the energyof the state):
=~ (59)
In the absence of coupling, the stationary states of the global system are tensor products
of the form:
1
1
0
0
1
1
(60)
and their energies are
2
:
= = [+ 1+0+1+]~ (61)
The ground state, whose energy was chosen for the origin, is not degenerate since
= 0is obtained, in (61), only for:
= 0for all (62)
The corresponding state (60) is therefore unique. On the other hand, all the other levels
are innitely degenerate. For example, to the rst level, of energy~, correspond all
states (60) for which all the numbersare zero except for one, which is itself equal
to one. All but one of the oscillators are then in their ground states. It is because the
excitation can be localized in any one of the oscillators that the level=~is innitely
degenerate.
2-b. Eects of the coupling
When the coupling is not zero, the Hamiltonian operator becomes:
=(1= 0) + (63)
with:
=
1
2
2
1( +1)
2
(64)
2
If we had not changed the energy origin of each oscillator by omitting the term 1/2 in (57), we
would have found an innite energy for the system, whatever the quantum number. This diculty
does not arise if, instead of an innite chain, one considers a chain formed by a very large but nite
number of oscillators. However, problems related to edge eects then appear.
623

COMPLEMENT J V
The states (60) are no longer, in this case, the stationary states of the system as they are
eigenstates of(1= 0), but not of. To see this, we writein terms of the operators
and:
=
1
4
~1
1
+ +1 +1
2
(65)
Now, it is clear that the action ofon a state of type (60) changes the state: the
numbersare no longer good quantum numbers since, for example,can transfer
an excitation from site()to site(+ 1)(term in
+1
).
To nd the stationary states of the system in the presence of the coupling, it
is useful, as in classical mechanics, to introduce normal variables, that is, operators
associated with the normal modes of the system.
2-c. Normal operators. Commutation relations
To the normal variables()and()correspond the operators()and()
dened by:
() = e (66a)
() = e (66b)
The domain of variation of the continuous parameteris again limited to the rst
Brillouin zone (15). Note that, since the normal variables()and()are complex,
the associated operators()and()are not Hermitian, unlikeand. The
relations corresponding to (24) and (27) are here:
() = () (67a)
() = () (67b)
The canonical commutation relations (54) enable us to calculate the commutators
of()and(). We immediately see that()and()commute, as do()and
(). As for the commutator [(),()], it can be written:
[()()] = [ ] ee
+
=~e
( )
(68)
Using formula (31) of Appendix andboth belong to interval
(15), we obtain:
[()()] =~
2
( ) (69)
624

VIBRATIONAL MODES OF A LINEAR CHAIN OF COUPLED HARMONIC OSCILLATORS; PHONONS
We saw in that it is convenient to condense the two normal variables()
and()into one,()[formula (30)]. The operator associated with()will be:
() =
1
2
() () +
~()
() (70)
where()is dened in (32). Note that the adjoint of()can be written:
() =
1
2
() ()
~()
() (71)
Using (69) and (67), we nd without diculty that:
[()()] =()()= 0 (72a)
()()=
2
( ) (72b)
To the classical quantity()dened in (48) corresponds the operator:
() =
1
2
() () +
1
2

2
() () () (73)
since()and()commute, as do()and(). To obtain the equivalent of the
classical formula (49), one must take into consideration the fact that()and()
do not commute; the order in which these operators appear must therefore be retained
throughout the calculation. Relations (37), taking (67) into account, can be written here:
() () =
1
2
() +() (74a)
1
~()
() =
2
() () (74b)
Substituting these expressions into (73), we nd:
() =
1
2
~()()() +()() (75)
As in (52), we can put the total Hamiltonianof the system in the form:
=
2
+d() (76)
with:
() =
1
2
~()()() +()() (77)
()and()thus can be seen to be annihilation and creation operators analogous to
those of a harmonic oscillator. However, sinceis a continuous index, the commutation
relations (72) involve( )instead of a Kronecker delta, so()must remain in the
symmetrical form (77). It can easily be shown that the various operators()commute:
[()()] = 0 (78)
625

COMPLEMENT J V
2-d. Stationary states in the presence of coupling
According to formulas (76) and (77), the ground state0of the system of coupled
oscillators is dened by the condition:
()0= 0 (79)
for all values of. The other stationary states can be obtained from the state0by the
action of the operators(); their corresponding energy is the integral of the energies
associated with each of the modes. A certain number of diculties arise because of the
continuous innity of normal modes; in particular, the energy of the ground state that
can be deduced from (76) and (77) is innite. We shall not discuss these diculties here;
in any case, they do not arise for a real chain, that is, a nite one (cf.footnote).
Formula (10) gives the value of the energy quantum~()associated with each
of the modes. It therefore indicates what energy quanta the system can absorb or emit:
they must correspond to frequencies situated within the allowed band (17).
3. Application to the study of crystal vibrations: phonons
3-a. Outline of the problem
Consider a solid body, composed of a large number of atoms (or ions) whose equi-
librium positions are regularly arranged in space, forming a crystalline lattice. For sim-
plicity, we shall assume that this lattice is one-dimensional and can be treated like an
innite chain of atoms. We intend to use here the results of the preceding section to
study the motion of the nuclei of these atoms about their equilibrium positions.
With this object in view, we shall make use of the same approximation as in the
study of molecular vibrations (Born-Oppenheimer approximation;cf.ComplementV,
comment of ). We shall assume that the motion of the electrons can be calculated
as if the positions of the nuclei were xed parameters. Thus, we shall solve the
corresponding Schrödinger equation (actually, this equation is too complex to be solved
exactly; in practice, one must again settle for approximations). We shall then denote the
energy of the electronic system in its ground state by( 101)where
is the displacement of nucleus()from its equilibrium position. It can be shown that
it is then possible to calculate the motion of the nuclei, to a good approximation, by
assuming that they possess a total potential energy( 101)equal to the
sum of their electrostatic interaction energy and( 101).
In fact, we shall further simplify the problem by making some reasonable hypothe-
ses about(this is indispensable, since we do not know). We shall assume that
describes essentially the interactions of each of the nuclei with its nearest neighbors
(in an innite linear chain, each nucleus has two such neighbors), that is, that the forces
exerted between non-adjacent nuclei can be neglected. In addition, we shall grant that,
in the range of values that the displacementscan attain,is well represented by an
expression of the form:
1
2
2
1( +1)
2
(80)
whereis the mass of a nucleus and1characterizes the intensity of its interaction
626

VIBRATIONAL MODES OF A LINEAR CHAIN OF COUPLED HARMONIC OSCILLATORS; PHONONS
with its neighbors. We shall not, therefore, take into account terms of higher order in
( +1), that is, the anharmonicity of the potential.
Since expression (80) is identical to (6), we can apply the results of the preceding
sections to the simple model of a solid body that we have just dened. Note, nevertheless,
that we must choose= 0because is the total potential energy of the system of
the nuclei, which interact with their neighbors, but are not elastically bound to their
equilibrium positions
3
.
3-b. Normal modes. Speed of sound in the crystal
Each of the vibrational normal modes of the crystal is characterized by a wave
vectorand an angular frequency(). In solid state physics, the energy quantum
associated with a mode is called a phonon. The phonons can be considered to be
particles of energy~()and momentum~. Actually, a phonon is not a true particle,
since its existence involves a state of collective vibration of the real particles which
constitute the crystal. The phonons are sometimes said to be quasi-particles: they
are entirely analogous to the ctitious particles, of position()and(), introduced
in ComplementV. In addition, a phonon can be created or destroyed by giving to or
taking from the crystal the corresponding vibrational energy, while (at least in the non-
relativistic domain to which we are restricting ourselves) a particle such as an electron
cannot be created or annihilated. In this connection, note that, since the number of
phonons in a given mode is arbitrary, phonons are bosons (Chap. ).
The dispersion relation giving the function()diers, for phonons, from the one
discussed in 1.b., since the angular frequencyis zero here. In this case, choosing
= 0in (10), we obtain:
() = 21sin
2
(81)
The curve representing()is given in Figure; it is composed of two half-arcs
of a sinusoid. Unlike what happens fornot equal to zero,()now goes to zero for
= 0and varies linearly whenis very small, since, as long as:
1
(82)
we have:
() 1= (83)
where:
=1 (84)
3
The Einstein model, which we described in ComplementV, is based on a dierent hypothesis:
each nucleus is assumed to see an average potential, due to its interactions with the other nuclei,
but practically independent of the exact positions of these other nuclei. To a rst approximation, this
average potential is assumed to be parabolic, and one has a system of independent harmonic oscillators.
Here, on the other hand, we are studying a somewhat more elaborate model, in which we explicitly
(although approximately) take into account interactions between the nuclei.
627

COMPLEMENT J V0
2ω
1
Ω
+ π/l– π/l
k
Figure 5: Dispersion relation for phonons (curve of Figure = 0); the slope of the
curve at the origin gives the speed of sound in the crystal.
Condition (82) means that the wavelength2 associated with the mode being
considered must be much greater than the separation between nuclei. For such wave-
lengths, the discontinuous structure of the chain is negligible, and the medium is not
dispersive: the phase velocity() is independent of, which implies that a
wave packet involving only small values of(of the same sign) propagates without being
deformed at the velocity. Since acoustical wavelengths satisfy (82),is the speed of
sound in the crystal.
When is of the order of1, the discontinuous structure of the chain becomes
important, and the angular frequency()increases less rapidly withthan formula
(83) would indicate (in Figure, the curve deviates from the straight dashed lines which
are its tangents at the origin). The medium is then dispersive, and a wave packet moves
with a group velocity:
=
d()
d
=
()
(85)
Finally, whenapproaches the edges of the rst Brillouin zone ( ),
we see from Figure
waveguide, the propagation velocity goes to zero when the cut-o frequency (here12)
is attained.
Figure ~()
in terms of their momentum~. Knowledge of such a spectrum, for a real crystal, is
very important. It gives the precise energies and momenta that the crystal can supply
or absorb when it interacts with another system. For example, the inelastic scattering of
light by a crystal (the Brillouin eect) can be interpreted as the result of the annihilation
or creation of a phonon, with a change in the energy and momentum of the incident
photon (total energy and momentum being conserved throughout the process).
628

VIBRATIONAL MODES OF A LINEAR CHAIN OF COUPLED HARMONIC OSCILLATORS; PHONONS
Comment:
The simple one-dimensional model described here has allowed us to present some
important physical concepts, which remain valid for a real crystal: energy quanta
associated with the normal modes, dispersion of the medium, allowed and forbid-
den frequency (and, therefore, energy) bands. In reality, the crystalline lattice is
three-dimensional, and a normal mode is characterized by a true wave vectork;
then depends, in general, not only on the absolute value ofk, but also on its
direction. Also, the situation may arise (as is the case for an ionic crystal) in which
the vertices of the lattice are not all occupied by identical particles, but rather, for
example, by two dierent types of particles in regular alternation
4
. Then, for each
wave vectork, several angular frequencies(k)appear. Some of them, which go
to zero whenk 0, constitute acoustic branches like the one encountered
above; the others belong to what are called optical branches
5
, in which a phonon
of zero momentum has a non-zero energy. It would be out of the question to study
all these problems here, although they are of primordial importance in solid state
physics.
References and suggestions for further reading:
Chains of coupled classical oscillators: Berkeley 3 (7.1), 2.4 and 3.5.
See section 13 of the bibliography, particularly Kittel (13.2), Chap. 5.
Other examples of collective oscillations: Feynman III (1.2), Chap. 15.
4
A real crystal also contains impurities and imperfections distributed at random. Here we are
speaking only about perfect crystals.
5
This name arises from the fact that, in an ionic crystal, the optical phonons are coupled to
electromagnetic waves like those in the visible domain, whose wavelengths are much larger than the
atomic separation.
629

VIBRATIONAL MODES OF A CONTINUOUS PHYSICAL SYSTEM. PHOTONS
Complement KV
Vibrational modes of a continuous physical system. Application to
radiation; photons
1 Outline of the problem
2 Vibrational modes of a continuous mechanical system: ex-
ample of a vibrating string
2-a Notation. Dynamical variables of the system
2-b Classical equations of motion
2-c Introduction of the normal variables
2-d Classical Hamiltonian
2-e Quantization
3 Vibrational modes of radiation: photons
3-a Notation. Equations of motion
3-b Introduction of the normal variables
3-c Classical Hamiltonian
3-d Quantization
1. Outline of the problem
In ComplementsVandV, we introduced the idea of normal variables for a system of
two or a countable innity of coupled harmonic oscillators. The aim of this complement
is to show that the same ideas can also be applied to the electromagnetic eld, which
is acontinuousphysical system (there is no natural lower bound for the wavelength of
radiation). The quantization of the electromagnetic eld will be studied in much more
detail in Chapter .
This study raises a certain number of delicate problems. Therefore, before be-
ginning, and in order to make a smooth transition between this complement and the
preceding ones,VandV, we shall start by studying in
a continuousmechanicalsystem: a vibrating string. It is obvious that, on the atomic
scale, such a system is not continuous: the string is composed of a very large number of
atoms. However, we shall ignore this atomic structure and treat the string as if it were
really continuous, since the fundamental aim of the calculation is to show how normal
variables can be introduced for a continuous system. Since, moreover, we are dealing
with a mechanical system, we can, without diculty, dene the conjugate momenta of
the normal variables, calculate the Hamiltonian of the system, and show that it indeed
appears in the form of a sum of Hamiltonians of independent one-dimensional harmonic
oscillators. We shall also discuss in detail the quantization of such a system.
The results obtained in , the problem of the
vibrational modes of radiation. We shall show that the study of radiation conned to a
parallelepiped cavity leads to equations very similar to those of the vibrating string. The
631

COMPLEMENT K V
same transformations allow the introduction of completely uncoupled normal variables
for the radiation (associated with the standing waves which can exist inside the cavity).
Then we shall generalize the results obtained in
photon (as it is not feasible here to show rigorously how one can introduce, for a non-
mechanical system like the electromagnetic eld, conjugate momenta, a Lagrangian and
a Hamiltonian).
2. Vibrational modes of a continuous mechanical system: example of a vibrating
string
2-a. Notation. Dynamical variables of the systemP
x
L
O
u(x, t)
F
Figure 1: Vibrating string passing through two xed pointsandand subjected to a
tension;()denotes the deviation with respect to the equilibrium position of the
point of the string situated at a distancefrom
The string is xed at a point(Fig.). It passes through a very small hole
pierced in a plate, and a weight exerts a tensionon it. For simplicity, we assume that
the string is inextensible (it has a constant length) and that it always remains in the
same plane, passing throughand. Its state is dened at timewhen we know at
this time the displacement()of the various points (labeled by their abscissason
), as well as the corresponding velocities
()
. The constraints imposed atand
are expressed by the boundary conditions:
(0) =() = 0 (1)
where 0 andare the abscissas of the pointsand.
It is important to remember that, in this problem, the dynamical variables are
the displacements()at each point of abscissa: there is a continuous innity of
dynamical variables. Consequently,is not a dynamical variable, but rather a continuous
index which labels the dynamical variable with which we are concerned (plays the same
role as indices 1 and 2 of ComplementVor as the indexof ComplementV).
632

VIBRATIONAL MODES OF A CONTINUOUS PHYSICAL SYSTEM. PHOTONS
2-b. Classical equations of motion
Letbe the mass per unit length of the string. If we assume the string to be
perfectly exible and if we conne ourselves to small displacements, a classical calculation
enables us to obtain the partial dierential equation satised by. We nd:
1
2
2
2
2
2
() = 0 (2)
where:
=
(3)
is the propagation velocity of a perturbation along the string.
Such an equation expresses the fact that the evolution of the variablecorrespond-
ing to the pointdepends on the variablesat innitely near points (via
2 2
).
Thus the variables()are all coupled to each other. We can then ask the following
question: is it possible, as in ComplementsVandV, to introduce new, uncoupled
variables that are linear combinations of the variables()associated with the various
points?
2-c. Introduction of the normal variables
Consider the set of functions of:
() =
2
sin (4)
whereis a positive integer:= 123The()satisfy the same boundary conditions
as():
(0) =() = 0 (5)
In addition, it is easy to verify the relations:
0
()() d= (6)
(orthonormalization relation) and:
d
2
d
2
+
22
2
() = 0 (7)
It can be shown that any function which goes to zero at= 0and=[as is
the case for()] can be expanded in one and only one way in terms of the(). We
can therefore write:
() =
=1
()() (8)
()can be obtained easily, using (6):
() =
0
()() d (9)
633

COMPLEMENT K V
The state of the string at the instantcan be dened either by the set of values
()
()corresponding to the various pointsor by the set of numbers
()_(). The new variables()just introduced are linear combinations of the
old(), as can be seen from (9). The converse is obviously also true [cf.formula (8)].
To obtain the equation satised by the(), we substitute expansion (8) into the
equation of motion (2). Using (7), we obtain, after a simple calculation:
=1
()
1
2
d
2
d
2
() +
22
2
()= 0 (10)
that is, since the()are linearly independent:
d
2
d
2
+
2
() = 0 (11)
with:
=
(12)
Thus we see that the new variables(), also called normal variables, evolve
independently: they areuncoupled. Moreover, equation (11) is identical to that of a
one-dimensional harmonic oscillator of angular frequency, so that:
() =cos( ) (13)
Each of the terms()()appearing on the right-hand side of (8) consequently repre-
sents a standing wave of frequency2and half-wavelength. Each normal variable
is therefore associated with a vibrational normal mode of the string, the most general
motion of the string being a linear superposition of these normal modes.
Comment:
In ComplementV, we started with an innite discrete set of harmonic oscillators
and introduced a continuous innity of normal variables. Here we nd ourselves
in the opposite situation: the()form a continuous set with respect to the
index, while, because of the boundary conditions, the normal variables()are
labeled by a discrete index.
2-d. Classical Hamiltonian
. Kinetic energy
The kinetic energy of the segment of string included betweenand+ dis
1
2
d
()
2
. It follows that the total kinetic energykinof the string is equal to:
kin=
2
0
()
2
d (14)
634

VIBRATIONAL MODES OF A CONTINUOUS PHYSICAL SYSTEM. PHOTONS
kincan be expressed simply in terms of the, using (8):
kin=
2
d()
d
d()
d
0
()() d (15)
which can also be written, taking (6) into account:
kin=
2
d
d
2
(16)
. Potential energy
Consider the segment of string included between the abscissasand+ d. It
makes an anglewith theaxis such that:
tan=
()
(17)
Its length is therefore equal to:
d
cos
= d1 + tan
2
1
2
(18)
Since the displacements are small,is very small, and we can write:
d
cos
= d1 +
1
2
()
2
(19)
From this we deduce that the total increase in the length of the string with respect to
its equilibrium position (which corresponds to0for all) is equal to:
=
1
2
0
()
2
d (20)
Now,represents the distance over which the end of the inextensible rope have risen,
which means that the weight has the same motion. The potential energypotof the
string, with respect to the value corresponding to the equilibrium position, is therefore
equal to:
pot==
1
2
0
()
2
d (21)
potcan also be expressed in terms of the normal variables. A simple calcula-
tion, using (8) and (4), yields:
pot=
2
22
2
2
(22)
635

COMPLEMENT K V
. Conjugate momenta of the; classical Hamiltonian
The Lagrangianof the system (cf.Appendix) can be written:
=kin pot=
2
_
2 22
(23)
From this, we deduce the expression for the conjugate momentumof:
=
_
=_ (24)
so that we nally obtain, for the Hamiltonian( )of the system, the expression:
=kin+pot=
2
2
+
1
2
22
(25)
that is:
= (26)
with:
=
2
2
+
1
2
22
(27)
Sinceandare conjugate variables, we recognizeto be the Hamiltonian of a
one-dimensional harmonic oscillator of angular frequency.is therefore a sum of
Hamiltonians of independent one-dimensional harmonic oscillators (independent, since
the normal variables are uncoupled).
It is useful to introduce, as in ComplementsVandV, the dimensionless vari-
ables:
^= (28a)
^=
1
~
(28b)
where:
=
}
(29)
is a (dimensional) constant.can then be written:
=
1
2
~^
2
+ ^
2
(30)
636

VIBRATIONAL MODES OF A CONTINUOUS PHYSICAL SYSTEM. PHOTONS
2-e. Quantization
. Preliminary comment
The calculations performed in this section are, of course, not intended to reveal
quantum eects in the motion of a macroscopic vibrating string. The vibrational frequen-
cies2which can be excited on such a string are so low (of the order of a kilohertz
at most) and the elementary energies~so much smaller than the macroscopic energy
of the string, that a classical description is quite sucient. It might be thought that
could be as large as we like becausehas no upper bound in formula (12). In fact, for
suciently small wavelengths2, the rigidity of the string can no longer be neglected,
and equation (2) is no longer valid. Furthermore, as we pointed out in the introduction,
the string is not really a continuous system, and it would make no sense to consider
wavelengths smaller than the interatomic distance.
The calculations which will be presented here must be considered as a simple rst
approach to problems posed by the quantum mechanical description of radiation. Now,
radiation constitutes a truly continuous system (no natural lower bound exists for the
wavelength) and satises an equation which is analogous to (2), whatever frequencies
and wavelengths may be involved
1
.
. Eigenstates and eigenvalues of the quantum mechanical Hamiltonian
We quantize each oscillator by associating with^and^[see formula (28)] ob-
servables
^
and
^
such that:
^^
= (31)
Since the normal variables are uncoupled, we also assume that the operators relating to
two dierent oscillators commute. We therefore have:
^^
= (32)
Let:
=
1
2
~
^2
+
^2
(33)
be the quantum mechanical Hamiltonian of oscillator. From the results of Chapter,
we know its eigenstates and eigenvalues:
= +
1
2
~ (34)
whereis a non-negative integer (to simplify the notation, we shall writeninstead
of ).
Since thecommute, we can choose the eigenstates ofto be in the form of
tensor products of the:
12 = 12 (35)
1
If we were really interested in a microscopic vibrating string (for example, a linear macro-
molecule), it would be more realistic to consider, as in ComplementV, a chain of atoms and to study
not only their longitudinal displacements but also their transversal ones (transverse phonons).
637

COMPLEMENT K V
The ground or vacuum state corresponds to all theequal to zero:
000=0 (36)
When we choose the energy origin to be the energy of the state0, the energy of state
(35) is equal to:
12
= ~ (37)
A state such as (35) can be considered to represent a set of1energy quanta~1
energy quanta~ These vibrational quanta are analogous to the phonons studied in
ComplementV.
Finally, using
^
and
^
, we can introduce, as in , creation and
annihilation operators for an energy quantum~:
=
1
2
^
+
^
(38)
whereis the adjoint of. We then have:
= (39)
and:
12 =
12 1
12 =
+ 112 + 1
(40)
All the states (35) can be expressed in terms of the vacuum state0:
12 =
(
1
)
1
1!
(
2
)
2
2!
()
!
0 (41)
. Quantum mechanical state of the system
The most general quantum mechanical state of the system is a linear superposition
of the states12 :
()=
12
12
()12 (42)
The equation of motion of()is the Schrödinger equation:
~
d
d
()= () (43)
Using (37) and (43), we easily obtain:
12
() =
12
(0) e (44)
638

VIBRATIONAL MODES OF A CONTINUOUS PHYSICAL SYSTEM. PHOTONS
. Observables associated with the dynamical variables()
When the system is quantized,()becomes an observable()which does not
depend
2
on, and which is obtained by replacing in (8)()by the observable:
() = ()
=
1
2
()+ (45)
We see that a displacement observable()can be dened for each value of, and that
it depends linearly on the creation and annihilation operatorsand.
It is interesting to compare the mean value of(),()()(), with the
classical quantity(). Since, according to (40),andcan link only states whose
energy dierence is~, we deduce from (45) that the only Bohr frequencies that can
appear in the evolution of()()are the frequencies12,22, ...,2, ...
associated, respectively, with the spatial functions1(),2(), ...()... Thus we
nd for()()a linear superposition of the standing waves that can exist on the
string. This analogy can, moreover, be pursued further. Let us calculate the derivative
2
2
()(); using the fact that [cf.ComplementV, equation (17)]:
d
d
= (46)
and relations (7) and (12), we easily nd that the mean value of()given by (45)
satises the dierential equation:
1
2
2
2
2
2
()() = 0 (47)
which is identical to (2).
Finally, note that sincedoes not commute with,()does not commute
with. The displacement and the total energy are therefore, in quantum mechanics,
incompatible physical quantities.
3. Vibrational modes of radiation: photons
3-a. Notation. Equations of motion
The classical state of the electromagnetic eld at a givenis dened when we
know, for this time, the value of the components of the electric eldEand the magnetic
eldBat each pointrof space. As in
of dynamical variables: the six components and at each pointr.
In order to stress the importance of the idea of normal variables (or normal modes)
of a eld, we shall introduce a simplication which consists of forgetting the vector nature
2
Recall that, in quantum mechanics, the time dependence is usually contained in the state vector
and not in the observables (cf.discussion of ).
639

COMPLEMENT K V
of the eldsEandB: we shall base our arguments on a scalar eld(r)which obeys
(like each of the components ofEandB) the equation:
1
2
2
2
(r) = 0 (48)
whereis the speed of light.
We shall assume the eld to be conned to a parallelepiped cavity whose inside
walls are perfectly conducting and whose edges, parallel to,,, have, respectively,
the lengths1,2,3. As boundary conditions, we require(r)to be zero on the
walls of the cavity (in the real problem, it is, for example, the tangential components of
the electric eldEthat must go to zero on these walls). We can therefore write:
(= 0 ) =(=1 ) =(= 0) =
=( =3) = 0 (49)
3-b. Introduction of the normal variables
Consider the set of functions of,,:
( ) =
8
123
sin
1
sin
2
sin
3
(50)
where,,are positive integers (= 123). The ( )go to zero on
the walls of the cavity and therefore satisfy the same boundary conditions as( ):
(= 0) = (=1 ) == ( =3) = 0 (51)
In addition, the following relations are simple to verify:
1
0
d
2
0
d
3
0
d ( ) ( ) = (52)
and:
+
2
2
1
+
2
2
2
+
2
2
3
2
( ) = 0 (53)
Any function that goes to zero on the walls of the cavity,(r)in particular, can
be expanded in one and only one way in terms of the( ). We therefore have:
( ) = () ( ) (54)
Formula (54) can easily be inverted, with the help of (52):
() =
1
0
d
2
0
d
3
0
d ( )( ) (55)
640

VIBRATIONAL MODES OF A CONTINUOUS PHYSICAL SYSTEM. PHOTONS
Thus we see that the eld at timeis described, either by the set of variables(),
or by the set of variables( ). Formulas (54) and (55) enable us to go from one
set to the other one.
Substituting (54) into (48) and using (53), we obtain, after a simple calculation:
d
2
d
2
+
2
() = 0 (56)
where:
2
=
22
2
2
1
+
2
2
2
+
2
2
3
(57)
The normal variables()are thereforeuncoupled. According to (56),()varies
likecos( ). Each of the terms() ( )of the sum (54) therefore
represents a standing wave (a vibrational normal mode of the eld in the cavity) charac-
terized by its frequency2and its spatial dependence in the three directions,
,(half-wavelengths1,2and3respectively).
Thus we have been able to generalize the results of
nevertheless, that when the vectorial nature of the electromagnetic eld is taken into
account, the structure of the modes is more complex, However, the general idea is the
same, and one reaches similar conclusions.
3-c. Classical Hamiltonian
Basing our discussion on the very close analogy between the results of
those of, we shall assume without proof that one can associate with the eld(r)
a Lagrangian, from which can be deduced the equation of motion (48), the conjugate
momenta ()of the normal variables, and nally, the expression for the Hamiltonian
of the system. The only point which is important here is that this Hamiltonian is
analogous to (30):
=
1
2
~ (^)
2
+ (^)
2
(58)
where^and^are dimensionless variables proportional to theand :
^= ^=
1
~
(59)
is a (dimensional) constant which is analogous to the one introduced in (29).
Comments:
()The equation of motion of each normal variable[established in ()] is anal-
ogous to that of a one-dimensional harmonic oscillator of angular frequency.
Thus we see why we obtain for a sum of Hamiltonians of independent one-
dimensional harmonic oscillators. It is possible, moreover, to obtain (56) from (58).
641

COMPLEMENT K V
The Hamilton-Jacobi equations (cf.Appendix) can, in fact, be written, taking
(59) into account:
d^
d
=
1
~^
d^
d
=
1
~^
(60)
that is, with the form (58) of:
d^
d
= ^
d^
d
= ^
(61a)
(61b)
Eliminating^between these two equations, we indeed nd (56).
()For the real electromagnetic eld, composed of two eldsEandB, one can also
obtain expression (58) fordirectly without using the Lagrangian. One simply
takes the total energyof the eld as the sum of the electrical and magnetic
energies contained in the cavity:
=
0
2
1
0
d
2
0
d
3
0
dE
2
+
2
B
2
(62)
and uses forEandBexpansions analogous to (54). Thus one nds that the terms
in^
2
and^
2
of (58) correspond respectively to the electrical and magnetic
energies.
3-d. Quantization
Now, starting with equation (58), we can carry out the same operations as in .
. Eigenstates and eigenvalues of
We associate with^and^two observables
^
and
^
whose commu-
tator is equal to. Since observables relating to two dierent modes commute, we can
write:
^ ^
= (63)
Let be the Hamiltonian associated with the mode ():
=
~
2
^
2
+
^
2
(64)
We know its eigenstates and eigenvalues:
= +
1
2
~ (65)
where is a non-negative integer.
642

VIBRATIONAL MODES OF A CONTINUOUS PHYSICAL SYSTEM. PHOTONS
Since the commute with each other, we can choose the eigenstates of=
to be in the form of tensor products of the:
111 211 121 112 (66)
The ground state, called the vacuum, corresponds to all theequal to zero:
0000 0 =0 (67)
With respect to the vacuum state energy, the energy of state (66) is equal to:
111 = ~ (68)
A state such as (66) can be considered to represent a set of111energy quanta~111
energy quanta~ , ... These quanta are none other than thephotons. Thus we see
that a certain type of photon is associated with each normal mode of the cavity.
We can, as in (38), introduce annihilation and creation operators for a photon of
type ():
=
1
2
^
+
^
=
1
2
^ ^
(69)
and establish formulas identical to (39), (40) and (41):
= (70)
111 =
111 1
111 =
+ 1111 + 1
(71)
111 =
111
111
111!!
0 (72)
. Quantum state of the eld
The most general state of the eld is a linear superposition of states (66):
()=
111
111 ()111 (73)
The Schrödinger equation:
~
d
d
()= () (74)
enables us to obtain the coecients
111
()in the form:
111
() =
111
(0) e (75)
643

COMPLEMENT K V
. Field operator
Upon quantization,(r)becomes an observable(r)which no longer depends
onand is obtained by replacing in (54) ()by :
(r) =
1
(r)
^
(76)
One can also, with the help of (69), express(r)in terms of the creation and annihilation
operators:
(r) =
1
2
1
(r) + (77)
The same arguments as in enable us to show, using formulas (71) and (75),
that the only Bohr frequencies that can appear in the time evolution of the mean value
of the eld:
(r)() =()(r)()
are the frequencies1112,2112, ..., 2, ... associated, respectively, with
the spatial functions111(r),211(r), ...(r)... Thus we nd for(r)()a linear
superposition of the classical standing waves that can exist in the cavity. A calculation
identical to the one in would enable us to show that(r)satises equation (48).
Finally, we nd that(r)anddo not commute. It is therefore impossible, in
quantum theory, to know simultaneously and with certainty both the number of photons
and the value of the electromagnetic eld at a point in space.
Comment:
For the electromagnetic eld, coherent states can be constructed which are analogous to
the ones introduced in ComplementVand which represent the best possible compromise
between the incompatible quantities, eld and energy.
. Vacuum uctuations
We saw in
is zero while
2
is not, and we discussed the physical meaning of this typically quantum
mechanical eect.
In the problem we are studying here,(r)presents many analogies with theoperator
of Chapter; we see from (77) that(r)is alinearcombination of creation and annihilation
operators. Consider the mean value of(r)in the ground state0of the eld, that is, the
vacuum state of photons. Since the diagonal elements ofandare zero according to (71),
we see that:
0(r)0= 0 (78)
On the other hand, the corresponding matrix element of[(r)]
2
is not zero. According to (71):
0= 0
0 = 0
0 0=
(79)
644

VIBRATIONAL MODES OF A CONTINUOUS PHYSICAL SYSTEM. PHOTONS
Therefore, a simple calculation enables us to establish, using (77), that:
0[(r)]
2
0=
1
2
1
2
[(r)]
2
(80)
From this we see that in the vacuum, that is, in the absence of photons, the electromagnetic
eld(r)at a point of space has a zero mean value but anon-zero root mean square deviation.
This means, for example, that if we performonemeasurement of(r), we can nd a non-zero
result (varying, of course, from one measurement to another), even if there is no photon present
in space. This eect has no equivalent in classical theory, in which, when the energy is zero,
the eld is rigorously zero. The preceding result is often expressed by saying that the vacuum
state of photons is subjected to uctuations of the eld, characterized by (78) and (80) and
calledvacuum uctuations.
The existence of these uctuations has interesting physical consequences for the inter-
action of an atomic system with the electromagnetic eld. Consider, for example, an atom in
a stateof energy, interacting with aclassically representedelectromagnetic wave. We
shall see in ComplementXIII, using time-dependent perturbation theory (cf.Chap. ), that
under the eect of such an excitation, the atom can move to a higher energy state (absorption)
or to a lower energy state (induced emission). But in this semi-classical treatment, if no eld
is present in space, the atom must remain indenitely in the state. In fact, we have just
established that, even in the absence of incident photons, the atom sees the vacuum uctua-
tions related to the quantum mechanical nature of the electromagnetic eld. Under the eect
of these uctuations, it can emit a photon and fall back into a lower energy state (the energy of
the global system being conserved during this process). This is the phenomenon ofspontaneous
emission, which can thus be considered to be, as it were, an emission induced by the vacuum
uctuations. (No spontaneous absorption is possible, since this would cause the atom to move
to a higher energy state, and no electromagnetic energy can be extracted from the eld, which
is in its ground state.)
It can also be shown that another eect of vacuum uctuations is to impart to the
atomic electrons an erratic motion which slightly modies the energies of the levels. The obser-
vation of this eect in the hydrogen atom spectrum (the Lamb shift) constituted the point of
departure for the development of modern quantum electrodynamics.
Comment:
In the preceding discussions, we have always chosen the energy of the vacuum state as
the energy origin. In fact, harmonic oscillator theory gives us the absolute value of the
energy of the vacuum state:
0=
1
2
~ (81)
There is obviously a close relationship between0and the electrical and magnetic energy
associated with vacuum uctuations. One of the diculties of quantum electrodynam-
ics, of which we have just given a brief overview, is that the sum (81) is in fact innite,
as is, moreover, (80)! Nevertheless, it is possible to surmount this diculty: using the
procedure called renormalization, one manages to bypass innite quantities and cal-
culate the physical eects that are actually observable, such as the Lamb shift, with
remarkable accuracy. It is obviously out of the question to consider these vast problems
here.
645

COMPLEMENT K V
References and suggestions for further reading:
Vibration modes of a continuous string in classical mechanics: Berkeley 3 (1.1),
2.1, 2.2 and 2.3.
Quantization of the electromagnetic eld: Mandl (2.9); Schi (1.18), Chap. 14;
Messiah (1.17), Chap. XXI; Bjorken and Drell (2.10), Chap. 11; Power (2.11); Heitler
(2.13).
The Lamb shift: Lamb and Retherford (3.11); Frisch (3.13); Kuhn (11.1), Chap. III,
A 5 e; Series (11.7), Chaps. VIII, IX and X.
646

ONE-DIMENSIONAL HARMONIC OSCILLATOR IN THERMODYNAMIC EQUILIBRIUM AT A
TEMPERATURE
Complement LV
One-dimensional harmonic oscillator in thermodynamic equilibrium at
a temperature
1 Mean value of the energy
1-a Partition function
1-b Calculation of . . . . . . . . . . . . . . . . . . . . . . . .
2 Discussion
2-a Comparison with the classical oscillator
2-b Comparison with a two-level system
3 Applications
3-a Blackbody radiation
3-b Bose-Einstein distribution law
3-c Specic heats of solids at constant volume
4 Probability distribution of the observable . . . . . . . . .
4-a Denition of the probability density(). . . . . . . . . . .
4-b Calculation of(). . . . . . . . . . . . . . . . . . . . . . . .
4-c Discussion
4-d Bloch's theorem
This complement is devoted to the study of the physical properties of a one-
dimensional harmonic oscillator in thermodynamic equilibrium with a reservoir at tem-
perature. We know (cf.ComplementIII) that such an oscillator is not in a pure
state (it is impossible to describe its state by a ket). The partial information which
we possess about it and the results of statistical mechanics enable us to characterize it
by a statistical mixture of stationary stateswith weights proportional toe
(: Boltzmann constant;: energy of the state). We saw in ComplementIII
() that the corresponding density operator is then written:
=
1
e (1)
whereis the Hamiltonian operator, and:
= Tr e (2)
is a normalization factor which insures that:
Tr= 1 (3)
(is the partition function,cf.Appendix).
We shall calculate the mean valueof the oscillator's energy, interpret the
result obtained, and show how it enters into numerous problems in physics (blackbody
radiation, specic heat of solids, ...). Finally, we shall establish and discuss the expression
for the probability density of the particle's position (the observable).
647

COMPLEMENT L V
1. Mean value of the energy
1-a. Partition function
The energiesof the statesare, according to the results of ,
equal to(+ 12)~. Since the energy levels are not degenerate, we have, according to
(2):
=
=0
e
=
=0
e
(+12)~
= e
~2
1 + e
~
+ e
2~
+ (4)
Inside the brackets, we recognize a geometric progression of common ratioe
~
.
Therefore:
=
e
~2
1e
~
(5)
1-b. Calculation of
According to formula (31) of ComplementIIIand expression (1) for:
= Tr() =
1
Tr(e ) (6)
Writing the trace explicitly in thebasis, we obtain:
=
1
=0
(+ 12)~e
(+12)~
(7)
To calculate this quantity, we dierentiate both sides of (4) with respect to:
d
d
=
1
2
=0
(+ 12)~e
(+12)~
(8)
We see that:
=
2
1
d
d
(9)
Using (5), we then nd, after a simple calculation:
=
~
2
+
~
e
~
1
(10)
648

ONE-DIMENSIONAL HARMONIC OSCILLATOR IN THERMODYNAMIC EQUILIBRIUM AT A
TEMPERATURE
Comments:
()Isotropic three-dimensional oscillator
Using the results and notation of ComplementV, we can write:
= + + (11)
where is given by:
=
1
Tr(e )
=
=0 =0 =0
(+ 12)~e
[(+12)+( +12)+(+12)]~
=0 =0 =0
e
[(+12)+( +12)+(+12)]~
(12)
The sums overandcan be separated out and are identical in the numerator
and denominator, so that:
=
=0
(+ 12)~e
(+12)~
=0
e
(+12)~
(13)
Aside from the replacement ofby, the nal expression is identical to the
one calculated in the preceding section;is therefore equal to the value given
in (10). It is easy to show that the same is true forand . Therefore,
we have established the following result: at thermodynamic equilibrium, the mean
energy of an isotropic three-dimensional oscillator is equal to three times that of a
one-dimensional oscillator of the same angular frequency.
()Classical oscillator
The energy()of a classical one-dimensional oscillator is equal to:
() =
2
2
+
1
2
22
(14)
In expression (14andcan take on any values betweenand+. According
to the results of classical statistical mechanics, the mean energy of this classical
oscillator is given by:
=
+ +
() e
()
dd
+ +
e
()
dd
(15)
Substituting (14) into (15), we nd, after a simple calculation:
= (16)
An argument analogous to that of comment()shows that result (16) must be
multiplied by 3 when we go from one to three dimensions.
649

COMPLEMENT L VMean energy
0
2
ћω
T
ℋ= kT
H
Figure 1: As a function of the temperature,
variation of the mean energy of a quantum
mechanical oscillator (solid line) compared
with that of a classical oscillator (straight
dashed line).
2. Discussion
2-a. Comparison with the classical oscillator
In Figure, the solid line gives the mean energyof the one-dimensional quan-
tum mechanical oscillator as a function of. The dashed line corresponds to the mean
energy of the classical oscillator.
For= 0, =~2. This result corresponds to the fact that at absolute zero,
one is sure that the oscillator is in the ground state0, with energy~2(~2is, for
this reason, sometimes called the zero point energy). As for the classical oscillator, it
is motionless (= 0) at its stable equilibrium position (= 0), and its energy is zero:
= 0.
As long asremains small more precisely, as long as ~ only the
population of the ground state is appreciable, andremains practically equal to
~2: in this region, the solid-line curve of Figure
see this directly from expression (10), which can be written, for small:
=
~
2
+~e
~
+ (17)
On the other hand, for large, that is, for ~, the same formula yields:
=
~
2
+ 1
1
2
~
+ (18)
or:
(19)
to within an innitesimal of the order of(~ )
2
. The asymptote of the curve
givingas a function ofis therefore the straight line= .
In conclusion, the quantum mechanical and classical oscillators have the same
mean energy, , at high temperatures ( ~). Striking dierences appear at
low temperatures (.~): it is no longer possible to ignore the quantization of the
oscillator's energy once the energythat characterizes the reservoir becomes of the
order of (or smaller than) the energy dierence~separating two adjacent levels of the
oscillator.
650

ONE-DIMENSIONAL HARMONIC OSCILLATOR IN THERMODYNAMIC EQUILIBRIUM AT A
TEMPERATURE0
2
E
1
E
1
+ E
2
H
T
Figure 2: Mean energy of a quantum me-
chanical system with two energy levels1
and2, in thermodynamic equilibrium at a
temperature.
2-b. Comparison with a two-level system
It is interesting to compare the preceding results with those obtained for a two-
level system. Let1and 2be the corresponding states, with energies1and2
(with1 2). For such a system, the general equation (6) yields:
=
1e
1
+2e
2
e
1 + e
2
(20)
The mean energy of a two-level system, given by (20), is shown in Figure. For
small( 2 1), the terms ine
1
are preponderant in both the numerator
and the denominator of (20) (since1 2) and we obtain:
0
1 (21)
It can be veried that the curve starts with a horizontal tangent. For large(
2 1), the asymptote of the curve is the straight line parallel to the-axis of ordinate
(1+2)2. The preceding results are easy to understand: for= 0, the system is
in its ground state1, of energy1; at high temperatures, the populations of the two
levels are practically equal, andapproaches half the sum of the two energies1and
2.
Although the solid-line curves of Figures
temperatures, we see that this is not at all true at high temperatures. For the harmonic
oscillator,is not bounded and increases linearly with, while, for a two-level system,
cannot exceed a certain value. This dierence is due to the fact that the energy
spectrum of the harmonic oscillator extends upward indenitely: whenincreases, levels
of higher and higherare occupied, and this causesto increase. On the other hand,
for a two-level system, once the populations of the two levels are equalized, an additional
increase in the temperature does not change the mean energy.
3. Applications
3-a. Blackbody radiation
We have already pointed out, in the introduction to Chapter V,
where we justied this result more precisely), that the electromagnetic eld in a cavity is equiv-
alent to a set of independent one-dimensional harmonic oscillators. Each of these oscillators is
associated with one of the standing waves that can exist inside the cavity (normal modes) and
has the same angular frequency as this wave. Let us show that this result, combined with those
651

COMPLEMENT L V
obtained above forand , leads very simply to the Rayleigh-Jeans law and the Planck
law for blackbody radiation.
Letbe the volume of the cavity, whose walls are assumed to be perfectly reecting.
The rst modes of the cavity (those of lowest frequency) depend strongly on the form of the
cavity. On the other hand, for the high-frequency modes (those whose wavelength= is
much smaller than the dimensions of the cavity), a classical electromagnetic calculation shows
that, if()ddenotes the number of modes whose frequency is betweenand+ d,()
is practically independent of the form of the cavity and equal to:
() =
8
2
3
(22)
Let()dbe the electromagnetic energy per unit volume of the cavity contained in the
frequency band(+ d)when the cavity is in thermodynamic equilibrium at a temperature
. To obtain the energy()d, one must multiply the number of modes whose frequency
is betweenand+ dby the mean energy of the corresponding harmonic oscillators. We
calculated this energy above; it is equal
1
to or ~2, depending on whether the
problem is treated classically or quantum mechanically. We then obtain, using (10), (16) and
(22):
() =
8
2
3
(23)
in a classical treatment, and:
() =
8
2
3
1
e 1
(24)
in a quantum mechanical treatment.
We recognize (23) to be Rayleigh-Jeans' law and (24) to be Planck's law, which reduces
to the preceding one in the limit of low frequencies or high temperatures (1). The
dierences between these two laws reect those which exist between the two curves of Figure.
At high frequencies, diculties arise in Rayleigh-Jeans' law: the quantity()given in (23)
approaches innity when , which is physically absurd. In order to remedy this defect,
Planck was led to postulate that the energy of each oscillator varied discontinuously, by jumps
proportional to(energy quantization); thus he obtained formula (24), which accounts perfectly
for the experimental results.
3-b. Bose-Einstein distribution law
Instead of calculating the mean valueof the energy, as we did in , let us calculate
the mean value of the operator. Since, according to formula (B-15) of Chapter:
= +
1
2
~ (25)
we deduce from result (10) that:
=
1
e 1
(26)
1
We use ~2and not for the following reason:()represents an electromagnetic energy
which can be extracted from the cavity. At absolute zero, all the oscillators are in their ground states and
no energy can be radiated outward because the system is in its lowest energy state;()must therefore
be zero at absolute zero, as experimental observations indeed show it to be. This requires us to dene
the mean energy of the eld in the cavity with respect to the value corresponding to= 0.
652

ONE-DIMENSIONAL HARMONIC OSCILLATOR IN THERMODYNAMIC EQUILIBRIUM AT A
TEMPERATURE
The fact that the levels of a one-dimensional harmonic oscillator are equidistant enables
us to associate with the oscillator in the statea set ofidentical particles (quanta) of the
same energy. In this interpretation, the operatorsand, which take into +1
or 1, create or destroy a particle.is thus the operator associated with the number of
particles (is the eigenstate ofwith the eigenvalue).
In the special case of the electromagnetic eld, the quanta associated with each harmonic
oscillator are none other thanphotons. To each mode of the cavity considered in the preceding
paragraph correspond photons of a certain type, characterized by the frequency, polarization,
and spatial distribution of the mode. Expression (26) gives the mean number of photons as-
sociated with a mode of frequencyat thermodynamic equilibrium. We recognize (26) to be
the Bose-Einstein distribution law, which can be derived in a more general way; here, we have
established it very simply by studying the harmonic oscillator and interpreting the states.
Comment:
To be rigorous, we should write the Bose-Einstein distribution law for bosons of energy
:
=
1
e
( )
1
(27)
whereis the chemical potential. In the case of photons,= 0. This is due to the fact
that the total number of photons in the global system radiation-reservoir is not xed,
because of the possibility of absorption or emission of photons by the walls.
3-c. Specic heats of solids at constant volume
We shall conne ourselves here to the Einstein model (cf.ComplementV), in which a
solid is considered to be composed ofatoms vibrating independently about their equilibrium
positions with the same angular frequency. The internal energyof the solid at the
temperatureis therefore equal to the sum of the mean energies of theisotropic three-
dimensional oscillators in thermodynamic equilibrium at this temperature. Using comment ()
of , we see that:
= 3 (28)
where is the mean energy of a one-dimensional harmonic oscillator of angular frequency
. We know that the constant volume specic heatis the derivative of the internal energy
with respect to the temperature:
=
d
d
= 3
d
d
(29)
which, taking (10) into account, yields:
= 3
~
2
e
~
[e
~
1]
2
(30)
The variation ofwithis shown in Figure. According to (29),is proportional
to the derivative of the solid-line curve of Figure. It is therefore very simple to describe the
behavior of the specic heatas a function of the temperature. In Figure, we see that
has a horizontal tangent at the origin and increases very slowly;is therefore zero for= 0
and also increases very slowly. On the other hand, for large( ~), approaches
; we deduce thatapproaches a constant,3 , independent of. The transition
region corresponds to~ 1.
653

COMPLEMENT L Vc
V
0
3k
T
Figure 3: (Constant volume) specic heat
of a solid in Einstein's model. The high-
temperature limit corresponds to the classical
Dulong-Petit law.
The asymptote of Figure
atom of any solid,is equal to Avogadro's number and the limiting value ofis equal to3
(whereis the ideal gas constant), that is, to about 6 cal. degree
1
mole
1
.
As we pointed out above, the quantum mechanical nature of crystalline vibrations mani-
fests itself at low temperatures whenbecomes of the order of~or less. Insofar asis
concerned, this means that the specic heat approaches zero whenapproaches zero. It is as if
the degrees of freedom corresponding to crystalline vibrations were frozen beneath a certain
temperature and no longer entered into the specic heat. This can be understood physically:
at absolute zero, each oscillator is in its ground state0; as long as ~, it cannot
absorb any thermal energy, since its rst excited state has an energy far greater than.
Comments:
()Comparison with the specic heat of a two-level system
We can apply an analogous argument to a sample composed of a set of two-level
systems (for example, a paramagnetic sample composed ofspin 1/2 particles):
its specic heatis given, to within a coecient, by the derivative of the curve
of Figure. For such a system, the variation ofwithis shown in Figure.
The behavior for0is the same as in the case of Figure. However, we see that
approaches zero when 2 1, since the mean energy then becomes
independent ofand is equal to(1+2)2(cf.Fig.). For a two-level system,
therefore has a maximum (Schottky anomaly) whose physical interpretation is
the following: like the harmonic oscillator, the two-level system cannot absorb any
thermal energy at very low temperatures, as long as2 1 ;is therefore
zero at the origin. Then, asincreases,2becomes populated, andincreases.
When the temperature is high enough for the two populations to be practically
equal, the system cannot absorb any more thermal energy, since the populations
can no longer change:therefore approaches zero when .c
V
0 T
Figure 4: Specic heatfor a set of two-
level systems. At high temperatures,ap-
proaches zero because the energy spectrum
has an upper bound.
654

ONE-DIMENSIONAL HARMONIC OSCILLATOR IN THERMODYNAMIC EQUILIBRIUM AT A
TEMPERATURE
()Einstein's model enables us to understand simply why the specic heatap-
proaches zero when the temperatureapproaches zero (a classically inexplicable
result). However, it is too schematic to describe the exact dependence ofat low
temperatures.
In a real crystal, the various oscillators are coupled. This gives rise to a set of
vibrational normal modes (phonons) whose frequencies go from zero to a certain
cuto frequency (cf.ComplementV). (30) must then be summed over the dierent
possible frequencies(taking into account the fact that the number of modes whose
frequencies are included betweenand+ ddepends on). Thus one nds an
expression for the specic heat which, at low temperatures, varies like
3
(this is
conrmed experimentally).
4. Probability distribution of the observable
4-a. Denition of the probability density()
Let us return to the one-dimensional harmonic oscillator in thermodynamic equi-
librium. We seek the probability()dof nding, in a measurement of the position
of the particle, a result included betweenand+ d. It is clear that()plays
an important role in a large number of physical problems. For example, for a solid de-
scribed by Einstein's model, the width of()gives an idea of the amplitude of atomic
vibrations; the study of the variation of this width with respect tomakes it possible
to understand the phenomenon of melting [which occurs when the width of()is no
longer negligible compared to the interatomic distance].
When the oscillator is in the stationary state, the corresponding probability
density()is:
() =()
2
= (31)
At thermodynamic equilibrium, the oscillator is described by a statistical mixture of the
states with the weights:
1
e . The probability density()is then:
() =
1
()e (32)
()is the weighted sum of the probability densities()associated with the various
states. Some of the()are shown in Figures . We shall
see later that the oscillations of the functions()which are visible in these gures
disappear in the summation over: we shall show that()is simply a Gaussian function.
The probability density()dened in (32) is related very simply to the density
operatorof the harmonic oscillator in thermodynamic equilibrium. Using (31) and
(32), we obtain:
() =
1
e (33)
On the right-hand side, we can bring in the operatore which, taking into account
the closure relation for the states, can be written:
e = e = e (34)
655

COMPLEMENT L V
We then see that:
() =
1
e = (35)
where the density operatoris given by formula (1).()can then be seen to be the
diagonal element ofwhich corresponds to the ket.
4-b. Calculation of ()
We know that:
=~ +
1
2
(36)
so that()can be written in the form:
() =
1
e
2
() (37)
with:
=
~
(38)
and:
() =e (39)
In order to know(), all we need to do, therefore, is evaluate this diagonal matrix element.
To do this, let us calculate the variation in()whenis changed to+ d. Since the
ket+ dis given by [cf.ComplementII, relation (20)]:
+ d=1
d
~
(40)
we obtain, substituting this relation and the adjoint relation into (39) (neglecting innitesimals
of second order ind):
(+ d) =() +
d
~
e (41)
The matrix element appearing on the right-hand side of (41) involves the operator,
which is proportional to( ). Now, it is theoperator, proportional to(+), which
acts in a simple way on the kets. We shall therefore transform[e ]so as to make
appear. We shall begin by seeking the relation betweene ande . This can be
obtained very simply in the representation:
e =
e 1 (42a)
e =
e
(1)
1 (42b)
that is:
e = ee (43)
which can also be written:
1tanh
2
e =1 + tanh
2
e (44)
656

ONE-DIMENSIONAL HARMONIC OSCILLATOR IN THERMODYNAMIC EQUILIBRIUM AT A
TEMPERATURE
Similarly, it can be shown that:
e = e e (45)
that is:
1 + tanh
2
e =1tanh
2
e (46)
We now subtract, term by term, relations (44) and (46); we obtain:
e =tanh
2
+ e
+
(47)
where the symbol[]+denotes the anticommutator:
[]+= + (48)
If we take into account the numerical factors which result from formulas (B-1) and (B-7) in
Chapter47) nally becomes:
e = tanh
2
e
+
(49)
Substituting this result into relation (41):
(+ d) () =
~
dtanh
2
e
+
=2
~
tanh
2
() d (50)
()therefore satises the dierential equation:
d
d
() +
2
2
() = 0 (51)
where, which has the dimensions of a length, is dened by:
=
~
coth
2
=
~
coth
~
2
(52)
Equation (51) can be integrated directly:
() =(0) e
22
(53)
Therefore, we know()to within a constant factor, since, according to (37):
() =
1
e
2
(0) e
22
(54)
Since we know that the integral of()over the whole-axis must be equal to 1, we obtain
nally:
() =
1
e
22
(55)
The function()is thus a Gaussian, whose width is characterized by the lengthdened in
(52).
657

COMPLEMENT L V
4-c. Discussion
Starting from the probability density (55), it is easy to calculate:
= 0
2
= ()
2
=
2
2
(56)
Figure ()
2
with respect to. We see from (52) that()
2
is equal to~2when= 0. This result is not surprising: at= 0, the oscillator
is in its ground state, and()is equal to0()
2
;is found to be the root mean
square deviation ofin the ground state [cf.formula (D-5a) of Chapter]. Then, when
increases, so does()
2
; when ~, we have:
()
2
2
(57)
In this case,()becomes identical to the probability density of a classical oscillator in
thermodynamic equilibrium at the temperature:
() =
e
()
+
e
()
d
=
1
2
2
e
22
2
(58)
which leads to()
2
=
2
(the straight dashed line in Figure). Here again,
classical and quantum mechanical predictions meet for~.
Now, let us apply the preceding results to the problem of melting of a solid body
(for simplicity, we shall choose the one-dimensional Einstein model; see ComplementV).
Experiments show that the solid melts whenis of the order of an appreciable fraction
of the interatomic distance. Consequently, the melting point temperatureis given0
(∆X)
2
ħ
2mω
T
Figure 5: Variation with respect to the temperatureof()
2
, for a harmonic oscillator
in thermodynamic equilibrium. When ,is identical to the classical value,
shown by the dashed line; at low temperatures, quantum mechanical eects (Heisenberg
uncertainty relation) preventfrom approaching zero.
658

ONE-DIMENSIONAL HARMONIC OSCILLATOR IN THERMODYNAMIC EQUILIBRIUM AT A
TEMPERATURE
approximately by:
2
2
2
2
(59)
wherecan be replaced by its expression (52), with=. Assuming thatis large
enough that ~, we can use
2
in (59) the asymptotic form (57), and we obtain
the law for:
2
22
(60)
If we set:
~= (61)
(is called the Einstein temperature) and if we note thatdoes not vary very much
from one substance to another (anyway, much less than, that is,), we nd the
approximate law:

2
(62)
The melting point temperature of a crystal is therefore approximately proportional to
the square of a vibrational frequency which is characteristic of the crystal.
4-d. Bloch's theorem
Consider the operatore , whereis a real variable. Its mean value:
e = Tr [e ] (63)
[whereis given by (1)] is a function of, which we shall denote by():
() =e (64)
In probability theory,()is called the characteristic function of the random variable.
It is easy to calculate()if we place ourselves in therepresentation:
() =
+
d e
=
+
d e
=
+
d() e (65)
To within a factor of
2,()is therefore the Fourier transform of the function()calculated
above (). Since()is a Gaussian [formula ()],()is also a Gaussian [cf.Appendix,
relation ()]:
() = e
22
4
(66)
2
This is not always possible. Recall that helium remains liquid at atmospheric pressure, even at
= 0:is never negligible compared to, whatevermay be (cf.ComplementV).
659

COMPLEMENT L V
which, according to formula (56), can be written:
e = e
2
2
2
(67)
We could perform calculations analogous to those of
ableinstead of. We would then dene the probability density
()by:() =
1
e()
2
(68)
Formula (24) of ComplementVshows that:
() =
1
= (69)
Therefore:
() =
1e
2
222
(70)
Consequently, the study ofe would lead to the same result as in (67):
e = e
2
2
2
(71)
The generalization of formulas (67) and (71) is known as Bloch's theorem: if( )is
an arbitrary linear combination of the positionand the momentum of a one-dimensional
harmonic oscillator in thermodynamic equilibrium at the temperature, then:
e = e
2
2
2
(72)
This theorem is used in solid state physics, for example in the theory of emission without recoil
by the nuclei of a crystalline lattice (the Mössbauer eect).
References and suggestions for further reading:
Specic heats: Kittel (8.2), Chap. 6, p. 91 and 100; Kittel (13.2), Chap. 6; Seitz
(13.4), Chap. III; Ziman (13.3), Chap. 2.
Blackbody radiation: Eisberg and Resnick (1.3), Chap. 1; Kittel (8.2), Chap. 15;
Reif (8.4) 9-13 to 9-15; Bruhat (8.3), Chap. XXII.
Bloch's theorem: Messiah (1.17), Chap. XII, 11-12.
660

EXERCISES
Complement MV
Exercises
1.Consider a harmonic oscillator of massand angular frequency. At time= 0,
the state of this oscillator is given by:
(0)=
where the statesare stationary states with energies(+ 12)~.
What is the probabilitythat a measurement of the oscillator's energy performed
at an arbitrary time0, will yield a result greater than2~? When= 0, what
are the non-zero coecients?
From now on, assume that only0and1are dierent from zero. Write the
normalization condition for(0)and the mean valueof the energy in terms
of0and1. With the additional requirement=~, calculate0
2
and1
2
.
As the normalized state vector(0)is dened only to within a global phase factor,
we x this factor by choosing0real and positive. We set:1=1e
1
. We assume
that =~and that:
=
1
2~
Calculate1.
With(0)so determined, write()for0and calculate the value of1at
. Deduce the mean value()of the position at.
2. Anisotropic three-dimensional harmonic oscillator
In a three-dimensional problem, consider a particle of massand of potential
energy:
( ) =
2
2
1 +
2
3
2
+
2
+1
4
3
2
whereandare constants which satisfy:
0 0
3
4
What are the eigenstates of the Hamiltonian and the corresponding energies?
Calculate and discuss, as functions of, the variation of the energy, the parity and
the degree of degeneracy of the ground state and the rst two excited states.
661

COMPLEMENT M V
3. Harmonic oscillator: two particles
Two particles of the same mass, with positions1and2and momenta1and
2, are subjected to the same potential:
() =
1
2
22
The two particles do not interact.
Write the operator, the Hamiltonian of the two-particle system. Show that
can be written:
=1+2
where1and2act respectively only in the state space of particle (1) and in that
of particle (2). Calculate the energies of the two-particle system, their degrees of
degeneracy, and the corresponding wave functions.
Doesform a C.S.C.O.? Same question for the set12. We denote by

12
the eigenvectors common to1and2. Write the orthonormalization
and closure relations for the states
12
.
Consider a system which, at= 0, is in the state:
(0)=
1
2
(00+10+01+11)
What results can be found, and with what probabilities, if at this time one mea-
sures:

4.(This exercise is a continuation of the preceding one and uses the same notation.)
The two-particle system, at= 0, is in the state(0)given in exercise 3.
At= 0, one measures the total energyand one nds the result2~.
Calculate the mean values of the position, the momentum, and the energy of
particle (1) at an arbitrary positive. Same question for particle (2).
At 0, one measures the energy of particle (1). What results can be found,
and with what probabilities? Same question for a measurement of the position
of particle (1); trace the curve for the corresponding probability density.
Instead of measuring the total energy, at= 0, one measures the energy2of
particle (2); the result obtained is~2. What happens to the answers to questions
andof?
662

EXERCISES
5.(This exercise is a continuation of exercise 3 and uses the same notation.)
We denote by
12
the eigenstates common to1and2, of eigenvalues(1+
12)~and(2+ 12)~. The two particle exchange operatoris dened by:

12
=
21
Prove that
1
= and thatis unitary. What are the eigenvalues of?
Let= be the observable resulting from the transformation byof
an arbitrary observable. Show that the condition=(invariant under
exchange of the two particles) is equivalent to[ ] = 0.
Show that:
1=2
2=1
Doescommute with? Calculate the action ofon the observables1,1,
2,2.
Construct a basis of eigenvectors common toand. Do these two operators
form a C.S.C.O.? What happens to the spectrum ofand the degeneracy of its
eigenvalues if one retains only the eigenvectorsoffor which=?
6. Charged harmonic oscillator in a variable electric eld
A one-dimensional harmonic oscillator is composed of a particle of mass, charge
and potential energy() =
1
2
22
. We assume in this exercise that the particle is
placed in an electric eldE()parallel toand time-dependent, so that to()must
be added the potential energy:
() =E()
Write the Hamiltonian()of the particle in terms of the operatorsand.
Calculate the commutators ofandwith().
Let()be the number dened by:
() =()()
where()is the normalized state vector of the particle under study. Show from
the results of the preceding question that()satises the dierential equation:
d
d
() = () +()
where()is dened by:
() =
2~
E()
Integrate this dierential equation. At time, what are the mean values of the
position and momentum of the particle?
663

COMPLEMENT M V
The ket()is dened by:
()= [ ()]()
where()has the value calculated in. Using the results of questionsand,
show that the evolution of()is given by:
~
d
d
()= [() +~]()
How does the norm of()vary with time?
Assuming that(0)is an eigenvector ofwith the eigenvalue(0), show that
()is also an eigenvector of, and calculate its eigenvalue.
Find at timethe mean value of the unperturbed Hamiltonian0=() ()
as a function of(0). Give the root mean square deviations,and0;
how do they vary with time?
Assume that at= 0, the oscillator is in the ground state0. The electric
eld acts between times 0 andand then falls to zero. When, what is the
evolution of the mean values()and()? Application: assume that between
0 and, the eldE()is given byE() =E0cos(); discuss the phenomena
observed (resonance) in terms of= . If, at, the energy is measured,
what results can be found, and with what probabilities?
7.Consider a one-dimensional harmonic oscillator of Hamiltonianand stationary
states:
= (+ 12)~
The operator()is dened by:
() = e
whereis real.
Is()unitary? Show that, for all, its matrix elements satisfy the relation:
()
2
= 1
Express()in terms of the operatorsand. Use Glauber's formula [for-
mula (63) of complementII] to put()in the form of a product of exponential
operators.
Establish the relations:
e 0= 0
e 0=
!
whereis an arbitrary complex parameter.
664

EXERCISES
Find the expression, in terms of=~
22
2and =~, for the matrix
element:
0()
What happens when approaches zero? Could this result have been predicted
directly?
8.The evolution operator(0)of a one-dimensional harmonic oscillator is written:
(0) = e
~
with:
=~ +
1
2
Consider the operators:
~() =(0)(0)
~() =(0)(0)
By calculating their action on the eigenketsof, nd the expression for~()
and~()in terms ofand.
Calculate the operators
~
()and
~
()obtained fromandby the unitary
transformation:
~
() =(0)(0)
~
() =(0)(0)
How can the relations so obtained be interpreted?
Show that
2
0 is an eigenvector ofand specify its eigenvalue. Simi-
larly, establish that
2
0 is an eigenvector of.
At= 0, the wave function of the oscillator is(0). How can one obtain from
(0)the wave function of the oscillator at all subsequent times= 2
(whereis a positive integer)?
Choose for(0)the wave function()associated with a stationary state.
From the preceding question derive the relation which must exist between()
and its Fourier transform().
Describe qualitatively the evolution of the wave function in the following cases:
()(0) = ewhere, real, is given.
()(0) = ewhereis real and positive.
665

COMPLEMENT M V
()
(0) =
=
1
if
22
= 0everywhere else
()(0) = e
22
whereis real.
666

Chapter VI
General properties of angular
momentum in quantum
mechanics
A Introduction: the importance of angular momentum
B Commutation relations characteristic of angular momentum
B-1 Orbital angular momentum
B-2 Generalization: denition of an angular momentum
B-3 Statement of the problem
C General theory of angular momentum
C-1 Denitions and notation
C-2 Eigenvalues ofJ
2
and. . . . . . . . . . . . . . . . . . . . .
C-3 Standard } representations
D Application to orbital angular momentum
D-1 Eigenvalues and eigenfunctions ofL
2
and . . . . . . . . .
D-2 Physical considerations
A. Introduction: the importance of angular momentum
The present chapter is the rst in a series of four Chapters (VI,,) devoted
to the study of angular momenta in quantum mechanics. This is an extremely important
problem, and the results we are going to establish are used in many domains of physics:
the classication of atomic, molecular and nuclear spectra, the spin of elementary parti-
cles, magnetism, etc...
We already know the important role played by angular momentum in classical
mechanics; the total angular momentum of an isolated physical system is aconstant of
Quantum Mechanics, Volume I, Second Edition. C. Cohen-Tannoudji, B. Diu, and F. Laloë.
© 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

CHAPTER VI GENERAL PROPERTIES OF ANGULAR MOMENTUM IN QUANTUM MECHANICS
the motion. Furthermore, this is also true in certain cases in which the system is not
isolated. For example, if a point particle, of mass, is moving in a central potential
(one which depends only on the distance betweenand a xed point), the force to
whichis subjected is always directed towards. Its moment with respect tois
consequently zero, and the angular momentum theorem implies that:
d
d
=0 (A-1)
whereis the angular momentum ofwith respect to. This fact has important
consequences: the motion of the particleis limited to a xed plane (the plane passing
throughand perpendicular to the angular momentum); moreover, this motion obeys
the law of constant areas (Kepler's second law).
All these properties have their equivalents in quantum mechanics. With the an-
gular momentum of a classical system is associated an observableL, actually a set
of three observables,,and, which correspond to the three components of
in a Cartesian frame. If the physical system under study is a point moving in a central
potential, we shall see in Chapter ,andare constants of the motion in a
quantum mechanical sense, that is, they commute with the Hamiltoniandescribing the
particle in the central potential(). This important property considerably simplies
the determination and classication of eigenstates of.
Also, we described the Stern-Gerlach experiment in Chapter
thequantization of angular momentum: the component, along a xed axis, of the intrinsic
angular momentum of an atom can take on only certain discrete values. We shall see that
all angular momenta are quantized in this way. This enables us to understand atomic
magnetism, the Zeeman eect, etc... Furthermore, to analyze all these phenomena, we
must introducetypically quantum mechanical angular momenta, which have no classical
equivalents(intrinsic angular momenta of elementary particles, Chap.).
From now on, we shall denote byorbital angular momentumany angular momen-
tum that has a classical equivalent (and byL, the corresponding observables), and by
spin angular momentumany intrinsic angular momentum of an elementary particle (for
which we reserve the letterS). In a complex system, such a nucleus, an atom, or a
molecule, the orbital angular momentaLof the various elementary particles which con-
stitute the system (electrons, protons, neutrons, ...) combine with each other and with
the spin angular momentaSof these same particles to form thetotal angular momen-
tumJof the system. The way in which angular momenta are combined in quantum
mechanics (coupling of angular momenta) will be studied in Chapter. Finally, let us
add thatJwill also be used to denote an arbitrary angular momentum when it is not
necessary to specify whether we are dealing with an orbital angular momentum, a spin,
or a combination of several angular momenta.
Before beginning the study of the physical problems just mentioned (energy levels
of a particle in a central potential, spin, the Zeeman eect, addition of angular mo-
menta,...), we shall establish, in this chapter, thegeneralquantum mechanical properties
associated with all angular momenta, whatever their nature.
These properties follow from commutation relations satised by the three observ-
ables,and, the components of an arbitrary angular momentumJ. The origin of
these commutation relations is discussed in : for an orbital angular momentum, they
are simply consequences of the quantization rules ( ) and the canon-
ical commutation relations [formulas (E-30) of Chapter]; for spin angular momenta,
668

B. COMMUTATION RELATIONS CHARACTERISTIC OF ANGULAR MOMENTUM
which have no classical equivalents, they actually serve as denitions of the correspond-
ing observables
1
. In , we study the consequences of these commutation relations
which are characteristic of angular momenta. In particular, we discuss space quantiza-
tion, that is, the fact that any component of an angular momentum possesses a discrete
spectrum. Finally, the general results so obtained are applied, in , to the orbital
angular momentum of a particle.
B. Commutation relations characteristic of angular momentum
B-1. Orbital angular momentum
To obtain the observables,,associated in quantum mechanics with the
three components of the angular momentumof a spinless particle, we simply apply the
quantization rules stated in . Consider for instance the component
of the classical angular momentum:
= (B-1)
We associate with the position variablesand, the observablesand, and with
the momentum variablesand, the observablesand. Although formula (B-1)
involves products of two classical variables, no precaution needs to be taken in replacing
them by the corresponding observables, sinceandcommute, as doand[see
the canonical commutation relations (E-30) of Chapter]. We therefore do not need to
symmetrize expression (B-1) in order to obtain the operator:
= (B-2)
For the same reason (andcommute, as doand), the operator thus obtained
is Hermitian.
In the same way, we nd the operatorsandcorresponding to the components
andof the classical angular momentum. This allows us to write:
L=RP (B-3)
Since we know the canonical commutation relations of the positionRand momentumP
observables, we can easily calculate the commutators of the operators,and.
For example, let us evaluate []:
[ ] = [ ]
= [ ] + [ ] (B-4)
since commutes with , and, with. We then have:
[ ] =[ ]+[ ]
=~ +~
=~ (B-5)
1
The fundamental origin of these commutation relations is purely geometrical. We shall discuss this
point in detail in ComplementVI, in which we demonstrate the intimate relation between the angular
momentum of a system with respect to a pointand thegeometrical rotationsof this system about.
669

CHAPTER VI GENERAL PROPERTIES OF ANGULAR MOMENTUM IN QUANTUM MECHANICS
Analogous calculations yield the other two commutators, and we obtain, nally:
[ ] =~
[ ] =~
[ ] =~ (B-6)
Thus we have established the commutation relations for the components of the angular
momentum of a spinless particle.
This result can be generalized to a system ofspinless particles. The total angular
momentum of such a system is, in quantum mechanics:
L=
=1
L (B-7)
with:
L=RP (B-8)
Now, each of the individual angular momentaLsatises the commutation relations (B-
6) and commutes withLwhenis not equal to(operators acting in state spaces of
dierent particles). Thus we see that relations (B-6) remain valid for the total angular
momentumL.
B-2. Generalization: denition of an angular momentum
The three operators associated with the components of an arbitrary classical an-
gular momentum therefore satisfy the commutation relations (B-6). It can be shown,
moreover (cf.ComplementVI), that the origin of these relations lies in the geometric
properties of rotations in three-dimensional space. This is why we shall adopt a more
general point of view and dene an angular momentumJas any set of three observables
,,that satises:
[ ] =~
[ ] =~
[ ] =~
(B-9)
We then introduce the operator:
J
2
=
2
+
2
+
2
(B-10)
the (scalar) square of the angular momentumJ. This operator is Hermitian, since,
andare Hermitian. We shall assume that it is an observable. Let us show thatJ
2
commutes with the three components ofJ:
[J
2
J] =0 (B-11)
We perform the calculation for, for example:
J
2
=
2
+
2
+
2
=
2
+
2
(B-12)
670

C. GENERAL THEORY OF ANGULAR MOMENTUM
sinceobviously commutes with itself and, therefore, with its square. The other two
commutators can be obtained from (B-9):
2
=[ ] + []
=~ ~ (B-13a)
2
=[ ] + []
=~ +~ (B-13b)
The sum of these two commutators, which enters into (B-12), is indeed zero.
Angular momentum theory in quantum mechanics is founded entirely on the com-
mutation relations (B-9). Note that these relations imply that it is impossible to measure
simultaneously the three components of an angular momentum; however,J
2
and any
component ofJare compatible.
B-3. Statement of the problem
Let us return to the example of a spinless particle in a central potential, mentioned
in the introduction. We shall see in Chapter
of the angular momentumLof the particle commute with the Hamiltonian; thus,
this is also true for the operatorL
2
. We then have at our disposal four constants of
the motion:L
2
,,,. But these four operators do not all commute; to form a
complete set of commuting observables with, we must pick onlyL
2
and one of the
three other operators,for example. For a particle subjected to a central potential, we
can then look for eigenstates of the Hamiltonianwhich are also eigenvectors ofL
2
and
, without restricting the generality of the problem. However, it is impossible to obtain
a basis of the state space composed of eigenvectors common to the three components of
L, as these three observables do not commute.
The situation is the same in the general case: since the components of an arbi-
trary angular momentumJdo not commute, they are not simultaneously diagonalizable.
We shall therefore seek the system of eigenvectors common toJ
2
and, observables
corresponding to the square of the absolute value of the angular momentum and to its
component along theaxis.
C. General theory of angular momentum
In this section, we shall determine the spectrum ofJ
2
andfor the general case and
study their common eigenvectors. The reasoning will be analogous to the one used in
Chapter
C-1. Denitions and notation
C-1-a. The +and operators
Instead of using the componentsandof the angular momentumJ, it is more
convenient to introduce the following linear combinations:
+=+
=
(C-1)
671

CHAPTER VI GENERAL PROPERTIES OF ANGULAR MOMENTUM IN QUANTUM MECHANICS
Like the operatorsandof the harmonic oscillator,+and are not Hermitian:
they are adjoints of each other.
In the rest of this section, we shall use only the operators+,,andJ
2
.
It is straightforward, using (B-9) and (B-11), to show that these operators satisfy the
commutation relations:
[ +] =~+ (C-2)
[ ] =~ (C-3)
[+] = 2~ (C-4)
[J
2
+] = [J
2
] = [J
2
] = 0 (C-5)
Let us calculate the products+and +. We nd:
+= (+)( )
=
2
+
2
[ ]
=
2
+
2
+~ (C-6a)
+= ( )(+)
=
2
+
2
+[ ]
=
2
+
2
~ (C-6b)
Using denition (B-10) of the operatorJ
2
, we can write these expressions in the form:
+=J
2 2
+~ (C-7a)
+=J
2 2
~ (C-7b)
Adding relations (C-7)21, we obtain:
J
2
=
1
2
(++ +) +
2
(C-8)
C-1-b. Notation for the eigenvalues ofJ
2
and
According to (B-10),J
2
is the sum of the squares of three Hermitian operators.
Consequently, for any ket, the matrix elementJ
2
is positive or zero:
J
2
=
2
+
2
+
2
=
2
+
2
+
2
0
(C-9)
Note that this could have been expected, sinceJ
2
corresponds to the square of the
absolute value of the angular momentumJ. From this we see, in particular, thatall the
eigenvalues ofJ
2
are positive or zero, since ifis an eigenvector ofJ
2
,J
2
is
the product of the corresponding eigenvalue and the square of the norm of, which is
always positive.
We shall write the eigenvalues ofJ
2
in the form(+ 1)~
2
, with theconvention:
0 (C-10)
672

C. GENERAL THEORY OF ANGULAR MOMENTUM
This notation is intended to simplify the arguments which follow; it does not inuence
the result. SinceJhas the dimensions of~, an eigenvalue ofJ
2
is necessarily of the form
~
2
, whereis a real dimensionless number. We have just seen thatmust be positive
or zero; it can then be shown that the second-degree equation in:
(+ 1) = (C-11)
always has one and only one positive or zero root. Therefore, if we use relation (C-10),
the specication ofdeterminesuniquely; any eigenvalue ofJ
2
can thus be written
(+ 1)~
2
, withpositive or zero.
As for the eigenvalues of, which have the same dimensions as~, they are tradi-
tionally written~, whereis a dimensionless number.
C-1-c. Eigenvalue equations forJ
2
and
We shall label the eigenvectors common toJ
2
andby the indicesandwhich
characterize the associated eigenvalues. However,J
2
anddo not in general constitute
a C.S.C.O. (see, for example, ), and it is necessary to introduce a
third index in order to distinguish between the dierent eigenvectors corresponding to
the same eigenvalues(+ 1)~
2
and~ofJ
2
and(this point will be expanded in
(which does not necessarily imply that it is
always a discrete index).
We shall therefore try to solve the simultaneous eigenvalue equations:
J
2
=(+ 1)~
2
=~
(C-12)
C-2. Eigenvalues ofJ
2
and
As in , we shall begin by proving three lemmas which will then
enable us to determine the spectrum ofJ
2
and.
C-2-a. Lemmas
. Lemma I (Properties of the eigenvalues ofJ
2
and)
If(+ 1)~
2
and~are the eigenvalues ofJ
2
andassociated with the same
eigenvector , thenandsatisfy the inequality:
(C-13)
To prove this, consider the vectors+ and , and note that the square
of their norms is positive or zero:
+
2
= + 0 (C-14a)
2
= + 0 (C-14b)
673

CHAPTER VI GENERAL PROPERTIES OF ANGULAR MOMENTUM IN QUANTUM MECHANICS
To calculate the left-hand sides of these inequalities, we can use formulas (C-7). We nd
(if we assume to be normalized):
+ = (J
2 2
~)
=(+ 1)~
2 2
~
2
~
2
(C-15a)
+ = (J
2 2
+~)
=(+ 1)~
2 2
~
2
+~
2
(C-15b)
Substituting these expressions into inequalities (C-14), we obtain:
(+ 1)(+ 1) = ( )(++ 1)0 (C-16a)
(+ 1)(1) = ( + 1)(+)0 (C-16b)
that is:
(+ 1) (C-17a)
+ 1 (C-17b)
These two conditions are satised simultaneously only ifsatises inequality (C-13).
. Lemma II (Properties of the vector )
Let be an eigenvector ofJ
2
andwith the eigenvalues(+ 1)~
2
and
~.
()If=, = 0.
()If , is a non-null eigenvector ofJ
2
andwith the eigenvalues
(+ 1)~
2
and(1)~.
(i) According to (C-15b), the square of the norm of the ket is equal to
~
2
[(+ 1) ( 1)]and therefore goes to zero for=. Since the norm of a
vector goes to zero if and only if this vector is a null vector, we conclude that all vectors
are null:
= = = 0 (C-18)
It is easy to establish the converse of (C-18):
= 0 = = (C-19)
Letting+act on both sides of the equation appearing in (C-19), and using (C-7a), we
obtain:
~
2
[(+ 1)
2
+] =~
2
(+)( + 1) = 0 (C-20)
Using (C-13), (C-20) has only one solution,=.
(ii) Now assumeto be greater than. According to (C-15b), is
then a non-null vector since the square of its norm is dierent from zero.
674

C. GENERAL THEORY OF ANGULAR MOMENTUM
Let us show that it is an eigenvector ofJ
2
and. The operatorsandJ
2
commute; consequently:
[J
2
] = 0 (C-21)
which can be written:
J
2
=J
2
=(+ 1)~
2
(C-22)
This relation expresses the fact thatis an eigenvector ofJ
2
with the eigenvalue
(+ 1)~
2
.
Moreover, if we apply operator equation (C-3) to :
[ ] =~ (C-23)
that is:
= ~
=~ ~
= ( 1)~ (C-24)
is therefore an eigenvector ofwith the eigenvalue(1)~.
. Lemma III (Properties of the vector+ )
Let be an eigenvector ofJ
2
andwith the eigenvalues(+ 1)~
2
and
~.
()If=,+ = 0.
()If ,+ is a non-null eigenvector ofJ
2
andwith the eigenvalues
(+ 1)~
2
and (+ 1)~.
(i) The argument is similar to that of ( ). According to (C-14a), the
square of the norm of+ is zero if=. Therefore:
== + = 0 (C-25)
The converse can be proved in the same way:
+ = 0 = (C-26)
(ii) If , an argument analogous to that of -iiyields, using formulas
(C-5) and (C-2):
J
2
+ =(+ 1)~
2
+ (C-27)
+ = (+ 1)~+ (C-28)
675

CHAPTER VI GENERAL PROPERTIES OF ANGULAR MOMENTUM IN QUANTUM MECHANICS
C-2-b. Determination of the spectrum of J
2
and
We shall now show that the three lemmas above enable us to determine the possible
values ofand.
Let be a non-null eigenvector ofJ
2
andwith the eigenvalues(+ 1)~
2
and~. According to lemmaI, . It is therefore certain that a positive or
zero integerexists such that:
+ 1 (C-29)
Now consider the series of vectors:
() (C-30)
According to lemmaII, each of the vectors() of this series (= 01 )
is a non-null eigenvector ofJ
2
andwith the eigenvalues(+ 1)~
2
and ( )~.
The proof is by iteration. By hypothesis,is non-null and corresponds to
the eigenvalues(+ 1)~
2
and~. () is obtained by the action ofon
()
1
, which is an eigenvector ofJ
2
andwith the eigenvalues(+ 1)~
2
and ( + 1)~. The latter eigenvalue is necessarily greater thansince, according
to (C-29):
+ 1 + 1 + 1 (C-31)
It follows, according to point (ii) of lemmaII, that () is a non-null eigenvector
ofJ
2
and, the corresponding eigenvalues being(+ 1)~
2
and( )~.
Now letact on() . Let us rst assume that the eigenvalue ()~
ofassociated with () is greater than~, that is, that:
(C-32)
By point()of lemmaII,() is then non-null and corresponds to the
eigenvalues(+ 1)~
2
and ( 1)~. This is in contradiction with lemmaIsince,
according to (C-29):
1 (C-33)
We must therefore have equal to. In this case,() corresponds
to the eigenvalueof, and, according to point (i) of lemmaII,()
is zero. The vector series (C-30) obtained by the repeated action ofon is
therefore limited and the contradiction with lemmaIis removed.
We have now shown that there exists a positive or zero integersuch that:
= (C-34)
An analogous argument, based on lemmaIII, would show that there exists a pos-
itive or zero integersuch that:
+= (C-35)
since the vector series:
+ (+) (C-36)
676

C. GENERAL THEORY OF ANGULAR MOMENTUM
must be limited if there is to be no contradiction with lemmaI.
Combining (C-34) and (C-35), we obtain:
+= 2 (C-37)
is therefore equal to a positive or zero integer divided by 2. It follows thatis necessarily
integral or half-integral
2
. Furthermore, if there exists a non-null vector, all the
vectors of series (C-30) and (C-36) are also non-null and eigenvectors ofJ
2
with the
eigenvalue(+ 1)~
2
, as well as ofwith the eigenvalues:
~(+ 1)~(+ 2)~ (2)~(1)~~ (C-38)
We summarize the results obtained above as follows:
LetJbe an arbitrary angular momentum, obeying the commutation relations (B-9). If
(+ 1)~
2
and~denote the eigenvalues ofJ
2
and, then:
are positive integers or half-integers or zero, that is:
0, 1/2, 1, 3/2, 2, ... (these values are the only ones possible, but they are not all
necessarily realized for all angular momenta).
, the only values possible forare the (2+ 1) numbers:
+ 1 1;is therefore integral ifis integral and half-integral if
is half-integral. All these values ofare realized once one of them is.
C-3. Standard } representations
We shall now study the eigenvectors common toJ
2
and, which form a basis of
the state space sinceJ
2
andare, by hypothesis, observables.
C-3-a. The basis states
Consider an angular momentumJacting in a state space. We shall show how
to construct an orthonormal basis incomposed of eigenvectors common toJ
2
and.
Take a pair of eigenvalues,(+1)~
2
and~, that are actually found in the case we
are considering. The set of eigenvectors associated with this pair of eigenvalues forms a
vector subspace ofwhich we shall denote by(); the dimension()of this sub-
space may well be greater than 1, sinceJ
2
anddo not generally constitute a C.S.C.O.
We choose in()an arbitrary orthonormal basis, ;= 12 ().
Ifis not equal to, there must exist another subspace(+1)incomposed
of eigenvectors ofJ
2
andassociated with the eigenvalues(+ 1)~
2
and(+ 1)~.
Similarly, ifis not equal to, there exists a subspace( 1). In the case where
is not equal toor, we shall construct orthonormal bases in(+ 1)and in
( 1), starting with the one chosen in().
First, let us show that, if1is not equal to2,+1 and+2
are orthogonal, as are 1 and 2 . We can nd the scalar product
2
A number is said to be half-integral if it is equal to an odd number divided by 2.
677

CHAPTER VI GENERAL PROPERTIES OF ANGULAR MOMENTUM IN QUANTUM MECHANICS
of 1 and 2 by using formulas (C-7):
2 1 = 2 (J
2 2
~)1
= [(+ 1)(1)]~
2
2 1 (C-39)
These scalar products are therefore zero if1=2since the basis of()is orthonor-
mal; if1=2, the square of the norm of1 is equal to:
[(+ 1)(1)]~
2
Now let us consider the set of the()vectors dened by:
+ 1=
1
~(+ 1)(+ 1)
+ (C-40)
Because of what we have just shown, these vectors are orthonormal. We shall show
that they constitute a basis in(+ 1). Assume that there exists, in(+ 1), a
vector + 1orthogonal to all the + 1obtained from (C-40). The vector
+ 1would not be null since (+ 1) cannot be equal to; it would belong to
()and would be orthogonal to all vectors +1. Now, according to relation
(C-40), the ket +1is proportional to+ , that is, to [formula
(C-7b)]. Therefore, + 1would be a non-null vector of()which would be
orthogonal to all vectors of thebasis. But this is impossible. Consequently, the
set of vectors (C-40) constitutes a basis in(+ 1).
It can be shown, using an analogous argument, that the vectors1dened
by:
1=
1
~(+ 1)(1)
(C-41)
form an orthonormal basis in( 1).
We see, in particular, that the dimension of subspaces(+ 1)and( 1)
is equal to that of(). In other words, this dimension is independent
3
of:
(+ 1) =( 1) =() =() (C-42)
We then proceed as follows. For each value ofactually found in the problem
under consideration, we choose one of the subspaces associated with this value of, for
example()corresponding to=. In this subspace, we choose an arbitrary or-
thonormal basis, ;= 12 (). Then, using formula (C-41), we construct,
by iteration, the basis to which each of the other2subspaces()will be related:
the arrows of table (VI-1) indicate the method used. By treating all the values offound
in the problem in this way, we arrive at what is called astandard basisof the state space
. The orthonormalization and closure relations for such a basis are:
= (C-43a)
+
=
()
=1
= 1 (C-43b)
3
If this dimension is innite, the result must be interpreted as follows: there is a one-to-one corre-
spondence between the basis vectors of two subspaces corresponding to the same value of.
678

C. GENERAL THEORY OF ANGULAR MOMENTUM
Comments:
()The use of formula (C-41) implies a choice of phases: the basis vectors in
( 1)are chosen to be proportional, with a real and positive coecient,
to the vectors obtained by application ofto the basis of().
()Formulas (C-40) and (C-41) are compatible, since, if we apply+to both
sides of (C-41) and take (C-7a) into account, we nd (C-40) [withreplaced
by( 1)]. This means that one is not obliged to start, as we did, with
the maximum value =and (C-41) in order to construct bases of the
subspaces()corresponding to a given value of.
()dierent values of
= 1 = 2 ()
(2+ 1)
spaces
()
(=)
(= 1)
.
.
.
()
.
.
.
()
1 2 ()
1 1 2 1 () 1
.
.
.
.
.
.
.
.
.
1 2 ()
.
.
.
.
.
.
.
.
.
1 2 ()
(= 1)(= 2) (=())
()spaces()
Table VI-1: Schematic representation of the construction of the(2+ 1)()
vectors of a standard basis associated with a xed value of. Starting with each of the
()vectors of the rst line, one uses the action ofto construct the(2+ 1)
vectors of the corresponding column.
Each subspace()is spanned by the()vectors situated in the same row.
Each subspace()is spanned by the(2+ 1)vectors of the corresponding column.
679

CHAPTER VI GENERAL PROPERTIES OF ANGULAR MOMENTUM IN QUANTUM MECHANICS
In most cases, in order to dene a standard basis, one uses observables that
commute
4
with the three components of, and form a C.S.C.O. withJ
2
and(we shall see a
concrete example of this in ):
[J] = [J] ==0 (C-44)
For the sake of simplicity, we shall assume that only one of these observablesis required
to make a C.S.C.O. withJ
2
and. Under these conditions, each of the subspaces()
dened above is globally invariant under the action of: if is an arbitrary vector of
(), is still, according to (C-44), an eigenvector ofJ
2
and:
J
2
=J
2
=(+ 1)~
2
= =~ (C-45)
with the same eigenvalues as. Thus also belongs to(). If we then choose a
value of, we can diagonalizeinside the corresponding subspace(). We denote by
the various eigenvalues found in this way: the indexindicates in which space()they were
found, and the index(assumed to be discrete, for simplicity) distinguishes between them. A
single vector (written) of()is associated with each eigenvalue, since,J
2
and
form, by hypothesis, a C.S.C.O.:
= (C-46)
The set ;xed;= 12 ()constitutes an orthonormal basis in(), from
which we construct, using the method described above, a basis in the other subspaces()
related to the value ofchosen. By applying this procedure successively for all values of, we
arrive at a standard basis, of the state space, all of whose vectors are eigenstates,
not only ofJ
2
and, but also of:
= (C-47)
This can be shown as follows. If hypothesis (C-44) is satised,commutes with , which
means that , that is 1, is an eigenvector ofwith the same eigenvalue as
:
= = (C-48)
By repeating this process, it is easy to prove relation (C-47).
Comments:
()An observable that commutes withJ
2
anddoes not necessarily commute with
and(is itself an example). Consequently, it should not be necessary, in order
to form a C.S.C.O. withJ
2
and, to choose observables that commute with the
three components ofJas in (C-44). However, ifdid not commute with+and
(that is, withand), would not necessarily be an eigenvector of
with the same eigenvalue as .
()The spectrum ofis the same in all the subspaces()associated with the same
value of. However, the eigenvaluesgenerally depend on(this point will be
illustrated by concrete examples in ).
4
An operator that commutes with the three components of the total angular momentum of a physical
system is said to be scalar (cf.ComplementVI).
680

C. GENERAL THEORY OF ANGULAR MOMENTUM
C-3-b. The spaces ()
In the preceding section, we introduced a standard basis of the state space by
starting with a basis chosen in the subspace( =)and constructing a basis of
(= 1), then one of(= 2),...,(), etc... The state space can be
considered to be the direct sum of all the orthogonal subspaces(), wherevaries
by integral jumps fromto+andtakes on all the values actually found in the
problem. This means that any vector ofcan be written in one and only one way as a
sum of vectors, each belonging to a particular subspace().
Nevertheless, the use of the subspaces()presents certain disadvantages. First
of all, their dimension()depends on the physical system being considered and is not
necessarily known. In addition, the subspaces()are not invariant under the action
ofJsince, by the very means of construction of the vectors,+and have
non-zero matrix elements between vectors of() and those of( 1).
We shall therefore introduce other subspaces of, the spaces(). Instead of
grouping together the kets with xed indicesand[which span()], we
shall now group together those for whichandhave given values, and we shall call
()the subspace which they span. This amounts to associating, in table (VI-1), the
(2+ 1) vectors of one column [instead of the()vectors of one row].
can then be seen to be the direct sum of the orthogonal subspaces(), which
have the simpler properties:
()is (2+ 1), whatever the value ofand whatever the
physical system under consideration.
()is globally invariant under the action ofJ: any component ofJ[or a
function(J)ofJ], acting on a ket of(), yields another ket also belonging
5
to(). This result is not dicult to establish, since[or(J)] can always
be expressed in terms of,+and. Now,, acting on , yields a
ket proportional to ;+, a ket proportional to+ 1; and, a ket
proportional to 1. The existence of the property in question therefore
follows from the very means of construction of the standard basis.
C-3-c. Matrices representing the angular momentum operators
Using the subspaces() considerably simplies the search for the matrix which
represents, in a standard basis, a componentofJ[or an arbitrary function(J)].
The matrix elements of such operators between two basis kets belonging to two
dierent subspaces()are zero. The matrix therefore has the following form:
5
It can also be shown that()is irreducible with respect toJ: there exists no subspace of
()other than()itself which is globally invariant under the action of the various components of
J.
681

CHAPTER VI GENERAL PROPERTIES OF ANGULAR MOMENTUM IN QUANTUM MECHANICS
() () () ...
() matrix
(2+1)(2+1)
0 0 0() 0 matrix
(2+1)(2+1)
0 0() 0 0 matrix
(2+1)(2+1)
0.
.
. 0 0 0 ...(C-49)
All we must then do is calculate the nite-dimensional matrices that represent the oper-
ator under consideration inside each of the subspaces().
Another very important simplication arises from the fact that each of these nite
submatrices is independent ofand of the physical system under study; it depends only
onand, of course, on the operator which we want to represent. To see this, note that
the denition of the [cf.(C-12), (C-40) and (C-41)] implies that:
=~
+ =~
(+ 1)(+ 1) + 1
=~
(+ 1)(1) 1
(C-50)
that is:
=~
=~
(+ 1) ( 1) 1 (C-51)
These relations show that the matrix elements representing the components ofJdepend
only onandand not on.
In order to know, in all cases, the matrix associated with an arbitrary component
in a standard basis, all we need to do is calculate, once and for all, the universal
matrices()
()
which representinside the subspaces()for all possible values of
(= 0,12,1,32,). When we study a particular physical system and its angular
momentumJ, we shall determine the values ofactually found in the problem, as well
as the number of subspaces()associated with each of them [that is, its degree of
degeneracy(2+ 1)()]. We know that the matrix representingin this particular
case has the block-diagonal form (C-49), and we can therefore construct it from the
universal matrices which we have just dened: for each value of, we shall have()
blocks identical to()
()
.
Let us give some examples of()
()
matrices:
()= 0
The subspaces(= 0)are one-dimensional, since zero is the only possible value
for. The()
(0)
matrices therefore reduce to numbers, which, according to (C-51), are
zero.
682

C. GENERAL THEORY OF ANGULAR MOMENTUM
()= 12
The subspaces(= 12)are two-dimensional (= 12or12). If we choose
the basis vectors in this order (= 12=12), we nd, using (C-51):
()
(12)
=
~
2
1 0
0 1
(C-52)
and:
(+)
(12)
=~
0 1
0 0
()
(12)
=~
0 0
1 0
(C-53)
that is, using (C-1):
()
(12)
=
~
2
0 1
1 0
()
(12)
=
~
2
0
0
(C-54)
The matrix representingJ
2
is therefore:
(J
2
)
(12)
=
3
4
~
21 0
0 1
(C-55)
We thus nd the matrices that were introduced without justication in Chapter,
.
()= 1
We now have (order of the basis vectors:= 1= 0=1):
()
(1)
=~
1 0 0
0 0 0
0 0 1
(C-56)
(+)
(1)
=~
0
2 0
0 0
2
0 0 0
()
(1)
=~
0 0 0
2 0 0
0
2 0
(C-57)
and therefore:
()
(1)
=
~
2
0 1 0
1 0 1
0 1 0
()
(1)
=
~
2
0 0
0
0 0
(C-58)
and:
(J
2
)
(1)
= 2~
2
1 0 0
0 1 0
0 0 1
(C-59)
It can be veried that matrices (C-56) and (C-58) satisfy the commutation relations
(B-9).
683

CHAPTER VI GENERAL PROPERTIES OF ANGULAR MOMENTUM IN QUANTUM MECHANICS
(iv)arbitrary
We use relations (C-51), which, according to (C-1), can also be written:
=
~
2(+ 1) (+ 1) +1+(+ 1) ( 1) 1(C-60)
and:
=
~
2(+ 1) (+ 1) +1(+ 1) ( 1) 1(C-61)
As for the matrix(
()
), it is diagonal and its elements are the(2+1)values of~. The
only non-zero matrix elements of()
()
and()
()
are those directly above and directly
below the diagonal:()
()
is symmetrical and real, and()
()
is antisymmetrical and
pure imaginary.
Since the kets are, by construction, eigenvectors ofJ
2
, we have:
J
2
=(+ 1)~
2
(C-62)
The matrix (J
2
)
()
is therefore proportional to the(2+ 1)(2+ 1)unit matrix: its
diagonal elements are all equal to(+ 1)~
2
.
Comment:
The axis which we have chosen as the quantization axis is entirely arbitrary.
All directions in space are physically equivalent, and we should expect the eigen-
values oforto be the same as those of(their eigenvectors, however, are
dierent, sinceanddo not commute with). It can indeed be veried that
the eigenvalues of the()
(12)
and()
(12)
matrices [formulas (C-54)] are
~
2
,
and that those of the()
(1)
and()
(1)
matrices [formulas (C-58)] are+~,0~.
This result is general: inside a given subspace(), the eigenvalues ofand
(like those of the component=JuofJalong an arbitrary unit vectoru)
are~(1)~(+ 1)~~. The corresponding eigenvectors (eigenvectors
common toJ
2
and,J
2
and, orJ
2
and) are linear combinations of the
withandxed.
To conclude this section devoted to standard representations, we summarize:
684

D. APPLICATION TO ORBITAL ANGULAR MOMENTUM
An orthonormal basis of the state space, composed of eigenvectors common
toJ
2
and:
J
2
=(+ 1)~
2
=~
is called a standard basis if the action of the operators+and on the basis
vectors is given by:
+ =~
(+ 1)(+ 1) + 1
=~
(+ 1)(1) 1
D. Application to orbital angular momentum
In , we studied the general properties of angular momenta, derived uniquely from
the commutation relations (B-9). We shall now return to the orbital angular momentum
Lof a spinless particle [formula (B-3)] and see how the general theory just developed
applies to this particular case. Using therrepresentation, we shall show that the
eigenvalues of the operatorL
2
are the numbers(+ 1)~
2
corresponding toall positive
integral or zero: of the possible values forfound in , the only ones allowed
in this case are the integral values, all of which are present. Then we shall indicate the
eigenfunctions common toL
2
and and their principal properties. Finally, we shall
study these eigenstates from a physical point of view.
D-1. Eigenvalues and eigenfunctions ofL
2
and
D-1-a. Eigenvalue equation in the rrepresentation
In therrepresentation, the observablesRandPcorrespond respectively to
multiplication byrand to the dierential operator(~)r. The three components of the
angular momentumLcan then be written:
=
~
(D-1a)
=
~
(D-1b)
=
~
(D-1c)
It is more convenient to work in spherical (or polar) coordinates, since, as we shall
see, the various angular momentum operators act only on the angular variablesand
, and not on the variable. Instead of characterizing the vectorrby its Cartesian
components , we label the corresponding pointin space (OM=r) by its
685

CHAPTER VI GENERAL PROPERTIES OF ANGULAR MOMENTUM IN QUANTUM MECHANICS
spherical coordinates,,(Fig.):
=sincos
=sinsin
=cos
with : (D-2)
0
0
0 2
The volume elementd
3
= dddis written in spherical coordinates:O
M
φ
θ
z
y
x
r
Figure 1: Denition of the spherical coordi-
nates,,of an arbitrary point in space.
d
3
=
2
sinddd
=
2
dd
(D-3)
where:
d = sindd (D-4)
is the solid angle element about the direction of polar anglesand.
Applying the classical technique of changing variables, we obtain, from formulas
(D-1) and (D-2), the following expressions (the calculations are rather time-consuming
686

D. APPLICATION TO ORBITAL ANGULAR MOMENTUM
but pose no great problem):
=~sin
+
cos
tan
(D-5a)
=~cos
+
sin
tan
(D-5b)
=
~
(D-5c)
which yields:
L
2
=~
2
2
2
+
1
tan
+
1
sin
2
2
2
(D-6a)
+=~e
+cot (D-6b)
=~e
+cot (D-6c)
In therrepresentation, the eigenfunctions associated with the eigenvalues(+
1)~
2
ofL
2
and~ofare the solutions of the partial dierential equations:
2
2
+
1
tan
+
1
sin
2
2
2
( ) =(+ 1)( )( ) = ( )
(D-7a)
(D-7b)
Since the general results of
already know thatis integral or half-integral and that, for xed,can take on only
the values,+ 1, ...,1,.
In equations (D-7),does not appear in any dierential operator, so we can con-
sider it to be a parameter andtake into account only the-and-dependence of. Thus,
we denote by()a common eigenfunction ofL
2
andwhich corresponds to the
eigenvalues(+ 1)~
2
and~:
L
2
() =(+ 1)~
2
() (D-8a)
() =~() (D-8b)
To be completely rigorous, we would have to introduce an additional index in order
to distinguish between the various solutions of (D-8) corresponding to the same pair of
values ofand. In fact, as we shall see further on, these equations have only one
solution (to within a constant factor) for each pair of allowed values ofand; this is
why the indicesandare sucient.
Comments:
()Equations (D-8) give the- and-dependence of the eigenfunctions ofL
2
and. Once the solution()of these equations has been found, these
eigenfunctions will be obtained in the form:
( ) =()() (D-9)
687

CHAPTER VI GENERAL PROPERTIES OF ANGULAR MOMENTUM IN QUANTUM MECHANICS
where()is a function
6
ofthat appears as an integration constant for
the partial dierential equations (D-7). The fact that()is arbitrary shows
thatL
2
anddo not form a C.S.C.O. in the spaceof functions ofr(or
of ).
()In order to normalize( ), it is convenient to normalize()
and()separately (as we shall do here). We then have, taking (D-4) into
account:
2
0
d
0
sin ()
2
d= 1 (D-10)
and:
0
2
()
2
d= 1 (D-11)
D-1-b. Values of and
. andmust be integral
Using expression (D-5c) for, we can write (D-8b) in the form:
~
() =~() (D-12)
which shows that()is equal to:
() =() e (D-13)
We can cover all space by lettingvary between0and2. Since a wave function
must be continuous at all points in space
7
, we must have, in particular:
(= 0) =(= 2) (D-14)
which implies that:
e
2
= 1 (D-15)
According to the results of ,is integral or half-integral. Relation (D-15) shows
that,in the case of an orbital angular momentum,must be an integer(e
2
would be
equal to1ifwere half-integral). But we know thatandare either both integral
or both half-integral: it follows thatmust also be an integer.
. All integral values (positive or zero) ofcan be found
Choose an integral value of(positive or zero). We know from the general theory
of ()must satisfy:
+() = 0 (D-16)
6
()must be such that ( )is square-integrable.
7
If()were not continuous for= 0, it would not be dierentiable and could not be an
eigenfunction of the dierential operators (D-5c) and (D-6a). For example,
()would produce
a function(), which is incompatible with (D-12).
688

D. APPLICATION TO ORBITAL ANGULAR MOMENTUM
which yields, taking (D-6b) and (D-13) into account:
d
d
cot () = 0 (D-17)
This rst order equation can be integrated immediately if we notice that:
cotd=
d(sin)
sin
(D-18)
Its general solution is:
() =(sin) (D-19)
whereis a normalization constant
8
.
Consequently, for each positive or zero integral value of, there exists a function
()which is unique (to within a constant factor):
() =(sin)e (D-20)
Through the repeated action of, we construct
1
. Thus we see that
there corresponds to the pair of eigenvalues(+ 1)~
2
and~(whereis an arbitrary
positive integer or zero andis another integer such that ),one and only
one eigenfunction(), which can be unambiguously calculated from (D-20). The
eigenfunctions()are calledspherical harmonics.
D-1-c. Fundamental properties of the spherical harmonics
The spherical harmonics()will be studied in greater detail in Comple-
mentVI. Here we shall conne ourselves to summarizing this study by stating without
proof its principal results.
. Recurrence relations
According to the general results of , we have:
() =~
(+ 1)(1)
1
() (D-21)
Using expressions (D-6b) and (D-6c) for the operators+and and the fact that
()is the product of a function ofalone ande, we obtain:
e(
cot)() =
(+ 1)(+ 1)
+1
() (D-22a)
e(
cot)() =
(+ 1)(1)
1
() (D-22b)
8
Inversely, one can easily show that the function obtained in this way is actually an eigenfunction of
L
2
and with eigenvalues(+ 1)~
2
and~. According to (D-5c) and (D-13), it is immediately seen
that () =~(). Then, using this equation and (D-16), as well as (C-7b), one can show
that()is also an eigenfunction ofL
2
with the expected eigenvalue.
689

CHAPTER VI GENERAL PROPERTIES OF ANGULAR MOMENTUM IN QUANTUM MECHANICS
. Orthonormalization and closure relations
Equations (D-7) determine the spherical harmonics only to within a constant fac-
tor. We now choose this factor so as to orthonormalize the()(as functions of the
angular variablesand):
2
0
d
0
sind ()() = (D-23)
Furthermore, any function ofand,(), can be expanded in terms of the
spherical harmonics:
() =
=0
+
=
() (D-24)
with:
=
2
0
d
0
sind ()() (D-25)
The spherical harmonics therefore constitute an orthonormal basis in the spaceof
functions ofand. This fact is expressed by the closure relation:
=0=
() ( ) =(coscos)( )
=
1
sin
( )( ) (D-26)
[it is(coscos), and not( ), that enters into the right-hand side of the closure
relation, because the integrations over the variableare performed using the dierential
elementsind=d(cos)].
. Parity and complex conjugation
First of all, recall that the change fromrtor(reection through the coordinate
origin) is expressed in spherical coordinates by (Fig.):
=
=
= +
(D-27)
It is simple (see ComplementVI) to show that:
( +) = (1) () (D-28)
The spherical harmonics are therefore functions with a denite parity, which is indepen-
dent of; they are even ifis even and odd ifis odd.
Also, it can easily be seen that:
[()]= (1) () (D-29)
690

D. APPLICATION TO ORBITAL ANGULAR MOMENTUMz
M
N
y
x
N′
M′
π – θ
π + φ
φ
θ
Figure 2: Transformation in spher-
ical coordinates of an arbitrary
point by reection through the ori-
gin;is not changed,becomes
, andbecomes+.
D-1-d. Standard bases of the wave function space of a spinless particle
As we have already noted [comment () of ],L
2
anddo not constitute a
C.S.C.O. in the wave function space of a spinless particle. We shall now indicate, relying
on the reasoning and results of , the form of the standard bases of this space.
Let(=)be the subspace of eigenfunctions common toL
2
and, of eigen-
values(+)~
2
and~, whereis a xed positive integer or zero. The rst step in
the construction of a standard basis (cf.) consists of choosing an arbitrary or-
thonormal basis in each of the(=). We shall denote by (r)the functions
that constitute the basis chosen in(=), the index(assumed to be discrete for
simplicity) serving to distinguish between the various functions of this basis. By repeated
application of the operatoron the (r), we then construct the functions(r)
which complete the standard basis for=; they satisfy equations (C-12) and (C-50),
which become here:
L
2
(r) =(+ 1)~
2
(r)
(r) =~ (r)
(D-30)
and:
(r) =~
(+ 1)(1) 1(r) (D-31)
But we saw in L
2
and that corre-
spond to given eigenvalues(+ 1)~
2
and~have the same angular dependence, that
of(); only their radial dependence diers. From equations (D-30), we therefore
deduce that (r)has the form:
(r) = ()() (D-32)
691

CHAPTER VI GENERAL PROPERTIES OF ANGULAR MOMENTUM IN QUANTUM MECHANICS
Let us now show that, if the(r)constitute a standard basis, the radial functions
(r)are independent of. Since the dierential operatorsdo not act on the
-dependence, we have, according to (D-21):
(r) = (r) ()
=~
(+ 1)(1) ()
1
() (D-33)
Comparison with (D-31) shows that the radial functions must satisfy, for all:
1() = () (D-34)
and are consequently independent of. The functions (r)of a standard basis of
the wave function space of a (spinless) particle are therefore necessarily of the form:
(r) =()() (D-35)
The orthonormalization relation for such a basis is:
d
3
(r) (r) =
0
2
d () ()
2
0
d
0
sind ()() = (D-36)
Since the spherical harmonics are orthonormal [formula (D-23)], we obtain, nally:
0
2
d () () = (D-37)
The radial functions()are therefore normalized with respect to the variable;
moreover, two radial functions corresponding to the same value ofbut to dierent
indicesare orthogonal.
Comments:
()Relation (D-37) is simply a consequence of the orthonormality of the functions
(r) =()(), which have been chosen as a basis in the subspace
(=). It is therefore essential that the indexbe the same for the two
functions appearing on the left-hand side. For=, (r)and
(r)are orthogonal anyway because of their angular dependence (they
are eigenfunctions of the Hermitian operatorL
2
with dierent eigenvalues).
The integral:
0
2
d () () (D-38)
may therefore take on any valuea prioriifandare dierent.
692

D. APPLICATION TO ORBITAL ANGULAR MOMENTUM
()In general, the radial functions()depend on, for the following reason.
A function of the form()()can be continuous at the coordinate origin
(= 0,andarbitrary) only if()reduces to a constant or if()goes
to zero at= 0[since if()depends onand, the limit of()()
when 0depends on the direction along which one approaches the origin
if(0)is not zero]. Consequently, if we want the basis functions(r)
to be continuous, only the radial functions corresponding to= 0can be
non-zero at= 0[
0
0()is indeed a constant]. Similarly, if we require the
(r)to be dierentiable (once or several times) at the origin, we obtain
conditions for the(r)that depend on the value of.
D-2. Physical considerations
D-2-a. Study of a state
Consider a (spinless) particle in an eigenstateofL
2
and [whose as-
sociated wave function is(r)], that is, a state in which the square of its angular
momentum and the projection of this angular momentum along theaxis have well-
dened values [(+ 1)~
2
and~respectively].
Suppose that we want to measure the component along theoraxis of the
angular momentum of this particle. Sinceanddo not commute with,
is an eigenstate neither ofnor of; we cannot, therefore, predict with certainty the
result of such a measurement. Let us calculate the mean values and root mean square
deviations ofandin the state .
These calculations can be performed very simply if we expressandin terms
of+and. We invert formulas (C-1):
=
1
2
(++)
=
1
2
(+ ) (D-39)
Thus we see that and are linear combinations of+ 1and
1; this leads to:
= = 0 (D-40)
Furthermore:
2
=
1
4
(
2
++
2
+++ +)
2
=
1
4
(
2
++
2
+ +) (D-41)
The terms in
2
+and
2
do not contribute to the result, since
2
is proportional
to 2. In addition, formula (C-8) yields:
++ += 2(L
2 2
) (D-42)
693

CHAPTER VI GENERAL PROPERTIES OF ANGULAR MOMENTUM IN QUANTUM MECHANICS
We therefore obtain:
2
=
2
=
1
2
(L
2 2
)
=
~
2
2
[(+ 1)
2
] (D-43)
Thus, in the state :
= = 0 (D-44a)
= =~
12
[(+ 1)
2
] (D-44b)
These results suggest the following picture. Consider aclassicalangular momen-
tum, whose modulus is equal to~
(+ 1)and whose projection alongis~(Fig.):
OL=~
(+ 1)OH =~ (D-45)
We denote byandthe polar angles that characterize its direction. Since the
triangle has a right angle at, and =, we have:
=
2 2
=~(+ 1)
2
(D-46)
Consequently, the components of such a classical angular momentum would be:
=~(+ 1)
2
cos =~(+ 1)
2
sin =~(+ 1) cos =~
(D-47)
Now let us assume thatOLandare known and thatis a random variable which
can take on any value in the interval [02], all these values being equally probable (an
evenly distributed random variable). We then have, averaging over:
2
0
cos d = 0 (D-48a)
2
0
sin d = 0 (D-48b)
which corresponds to (D-44a). In addition:
2
=
1
2
~
2
[(+ 1)
2
]
2
0
cos
2
d =
~
2
2
[(+ 1)
2
] (D-49)
and, similarly:
2
=
~
2
2
[(+ 1)
2
] (D-50)
694

D. APPLICATION TO ORBITAL ANGULAR MOMENTUMy
z
K
O
H
L
Θ
Φ
J
I
x
Figure 3: A classical model for the orbital angular momentum of a particle in a state
. We assume that the distance |OL| and the angleare known, but thatis a
random variable whose probability density is constant inside the interval[02]. The
classical mean values of the components of OL, as well as those of the squares of these
components, are then equal to the corresponding quantum mechanical mean values.
These mean values are identical to the ones we found in (D-44). Consequently, the
angular momentum of a particle in the state, behaves, insofar as the mean values
of its components and their squares are concerned, like a classical angular momentum of
magnitude~
(+ 1)having a projection~along, but for whichis a random
variable evenly distributed between0and2.
Of course, this picture must be used carefully: we have shown throughout this
chapter how much the quantum mechanical properties of angular momenta dier from
their classical properties. In particular, we must stress the fact that an individual mea-
surement oforon a particle in the statecannot yield an arbitrary value
between~
(+ 1)
2
and+~(+ 1)
2
, as the preceding model might lead
us to believe. The only possible results are the eigenvalues ofor(we saw at the
end of ), that is, sinceis xed here, one of
the(2+ 1)values~,(1)~, ...,(+ 1)~,~.
695

CHAPTER VI GENERAL PROPERTIES OF ANGULAR MOMENTUM IN QUANTUM MECHANICS
D-2-b. Calculation of the physical predictions concerning measurements ofL
2
and
Consider a particle whose state is described by the (normalized) wave function:
r=(r) =( ) (D-51)
We know that a measurement ofL
2
can yield only the results0,2~
2
,6~
2
,(+1)~
2
, ...,
and a measurement of, only the results0,~,2~, ~, ... How can we calculate
the probabilities of these dierent results from the wave function( )?
. General formulas
Let us denote byPL
2()the probability of nding, in a simultaneous mea-
surement ofL
2
and, the results(+ 1)~
2
and~. This probability can be obtained
by expanding(r)on a basis composed of eigenfunctions ofL
2
and; we shall choose
a standard basis of the type introduced in :
(r) =()() (D-52)
(r)can then be written:
(r) = ()() (D-53)
where the coecients can be calculated by using the usual formula:
=d
3
(r)(r)
=
0
2
d ()
2
0
d
0
sind ()( ) (D-54)
According to the postulates of Chapter, the probabilityPL
2()is given,
under these conditions, by:
PL
2() =
2
(D-55)
If we measure onlyL
2
, the probability of nding the result(+ 1)~
2
is equal to:
PL
2() =
+
=
PL
2() =
+
=
2
(D-56)
Similarly, if it is onlythat we wish to measure, the probability of obtaining~is:
P() = PL
2() =
2
(D-57)
(the restriction is automatically satised, since there are no coecientsfor
which would be greater than).
Actually, sinceL
2
and act only onand, we see that it is the- and-
dependence of the wave function(r)that count in the preceding probability calcula-
tions. To be more precise, consider( )as a function ofanddepending on the
696

D. APPLICATION TO ORBITAL ANGULAR MOMENTUM
parameter. Like any other function ofand,can then be expanded in terms of
the spherical harmonics:
( ) = ()() (D-58)
The coecients of this expansion depend on the parameterand are given by:
() =
2
0
d
0
sind ()( ) (D-59)
If we compare expressions (D-58) and (D-53), we see that theare the coecients
of the expansion of()on the functions():
() = () (D-60)
with, taking (D-54) and (D-59) into account:
=
0
2
d ()() (D-61)
Using (D-37) and (D-60), we also obtain:
0
2
d ()
2
=
2
(D-62)
The probabilityPL
2()[formula (D-55)] can therefore also be written in the form:
PL
2() =
0
2
d ()
2
(D-63)
From this we can deduce, as in (D-56) and (D-57):
PL
2() =
+
=
0
2
d ()
2
(D-64)
and:
P() =
0
2
d ()
2
(D-65)
[here again,()exists only for ]. Consequently, to obtain the physical pre-
dictions concerning measurements ofL
2
and, we may consider the wave function as
depending only onand. We then expand it in terms of the spherical harmonics as in
(D-58) and apply formulas (D-63), (D-64) and (D-65).
Similarly, sinceacts only on, it is the-dependence of the wave function(r)
that counts in the calculation ofP(). To see this, we shall use the fact that the
697

CHAPTER VI GENERAL PROPERTIES OF ANGULAR MOMENTUM IN QUANTUM MECHANICS
spherical harmonics are products of a function ofalone and a function ofalone. We
shall write them in the form:
() =()
e
2
(D-66)
so that each of the functions of the product is normalized, since we have:
2
0
d
e
2
e
2
= (D-67)
Substituting this formula into the orthonormalization relation (D-23) for the spherical
harmonics, we nd:
0
sind ()() = (D-68)
[for reasons analogous to those indicated in comment () of , the same value of
is involved in both functionsof the left-hand side].
If we consider( )to be a function ofdened in the interval[02]and
depending on the parametersand, we can expand it in a Fourier series:
( ) = ()
e
2
(D-69)
where the coecients()can be calculated from the formula:
() =
1
2
2
0
de ( ) (D-70)
If we compare formulas (D-69) and (D-70) with (D-58) and (D-59), we see that the
()for xedare the coecients of the expansion of()on the functions
corresponding to the same value of:
() = ()() (D-71)
with:
() =
0
sind ()() (D-72)
With (D-68) taken into account, expansion (D-71) requires that:
0
sind ()
2
= ()
2
(D-73)
Substituting this formula into (D-65), we obtainP()in the form:
P() =
0
2
d
0
sind ()
2
(D-74)
698

D. APPLICATION TO ORBITAL ANGULAR MOMENTUM
Therefore, as far as measurements ofalone are concerned, all we need to do is consider
the wave function as depending solely onand expand it in a Fourier series as in (D-69)
in order to calculate the probabilities of the various possible results.
We might be tempted to think that an argument analogous to the preceding ones
would givePL
2()in terms of the expansion of( )with respect to the variable
alone. In fact, this is not the case: predictions concerning a measurement ofL
2
alone
involve both the- and the-dependence of the wave function; this is related to the fact
thatL
2
acts on bothand. We must therefore use formula (D-64).
. Special cases and examples
Suppose that the wave function(r)representing the state of the particle appears
in the form of a product of a function ofalone and a function ofand:
( ) =()() (D-75)
We can always assume()and()to be separately normalized:
0
2
d()
2
= 1 (D-76a)
2
0
d
0
sind()
2
= 1 (D-76b)
To obtain the expansion (D-58) of such a wave function, all we must do is expand()
in terms of the spherical harmonics:
() = () (D-77)
with:
=
2
0
d
0
sind ()() (D-78)
In this case, therefore, the coecients()of formula (D-58) are all proportional to
():
() = () (D-79)
With (D-76a) taken into account, expression (D-63) for the probabilityPL
2()
becomes simply:
PL
2() =
2
(D-80)
This probability is totallyindependent of the radial part()of the wave function.
Similarly, let us consider the case in which the wave function( )is the
product of three functions of a single variable:
( ) =()()() (D-81)
699

CHAPTER VI GENERAL PROPERTIES OF ANGULAR MOMENTUM IN QUANTUM MECHANICS
which we shall assume to be separately normalized:
0
2
d()
2
=
0
sind()
2
=
2
0
d()
2
= 1 (D-82)
Of course, (D-81) is a special case of (D-75), and the results we have just established
apply here. But, in addition, if we are interested only in a measurement of, all we
must do is expand()in the form:
() =
e
2
(D-83)
where:
=
1
2
2
0
de () (D-84)
in order to obtain the equivalent of formula (D-69), with:
() = ()() (D-85)
According to (D-82),P()is then given by (D-74) as:
P() =
2
(D-86)
The preceding considerations can be illustrated by some very simple examples. First, let
us assume that the wave function(r)is in fact independent ofand, so that:
() =
1
2
() =
1
2
(D-87)
We then have:
() =
1
4
=
0
0() (D-88)
Thus a measurement ofL
2
or ofmust yield zero.
Now, let us modify only the-dependence, choosing:
() =
32
cos
() =
1
2
(D-89)
In this case:
() =
34
cos=
0
1() (D-90)
700

D. APPLICATION TO ORBITAL ANGULAR MOMENTUM
We are again sure of the results of measuringL
2
or. ForL
2
, we can only obtain2~
2
;
for, 0. It can be veried that this modication of the-dependence has not changed
the physical predictions concerning the measurement of.
On the other hand, if we modify the-dependence by setting, for example:
() =
1
2
() =
e
2
(D-91)
()is no longer equal to a single spherical harmonic. According to (D-86), all the
probabilitiesP()are zero except for:
P(= 1) = 1 (D-92)
But the predictions concerning a measurement ofL
2
are also changed with respect to
the case (D-87). In order to calculate these predictions, we must expand the function:
() =
1
4
e (D-93)
on the spherical harmonics. It can be veried that all the(), with oddand
= 1, actually appear in the expansion of the function (D-93). Consequently, we are
no longer sure of the result of a measurement ofL
2
(the probabilities of the various
possible results can be calculated from the expression for the spherical harmonics). We
therefore conclude from this example that, as pointed out at the end of , the-
dependence of the wave function also enters into the calculation of predictions concerning
measurements ofL
2
.
References and suggestions for further reading:
Dirac (1.13), 35 and 36; Messiah (1.17), Chap. XIII; Rose (2.19); Edmonds
(2.21).
701

COMPLEMENTS OF CHAPTER VI, READER'S GUIDE
AVI: SPHERICAL HARMONICS Detailed study of the spherical harmonics
(); establishes certain properties used in
Chapter , as well as in certain subsequent
complements.
BVI: ANGULAR MOMENTUM AND ROTATIONS Brings out the close relation that exists between
the angular momentum Jof a quantum mechan-
ical system and the spatial rotations that can
be performed on it. Shows that the relation
commutations between the components of J
express purely geometrical properties of these
rotations; introduces the concept of a scalar
or vector observable, which will reappear in
other complements (especially X). Important
theoretically; however, sometimes dicult; can be
reserved for later study.
CVI: ROTATION OF DIATOMIC MOLECULES A simple and direct application of quantum
mechanical properties of angular momentum:
pure rotational spectra of heteropolar diatomic
molecules, and Raman rotational spectra.
Elementary level. Because of the impor-
tance of the phenomena studied in physics and
chemistry, can be recommended for a rst reading.
DVI: ANGULAR MOMENTUM OF STATIONNARY
STATES OF A TWO-DIMENSIONAL HARMONIC
OSCILLATOR
Can be considered as a worked example. Studies
the stationnary states of the two-dimensional
harmonic oscillator; in order to class these states
by angular momentum, introduces the concept
of circular quanta. Not theoretically dicult.
Some results will be used in ComplementVI.
EVI: A CHARGED PARTICLE IN A MAGNETIC
FIELD: LANDAU LEVELS
A general study of the quantum mechanical
properties of a charged particle in a magnetic
eld, followed by a study of the important special
case in which the magnetic eld is uniform
(Landau levels). Not theoretically dicult.
Recommmended for a rst reading, which can,
however, be conned to ,
2-b,.
FVI: EXERCISES .
703

SPHERICAL HARMONICS
Complement AVI
Spherical harmonics
1 Calculation of spherical harmonics
1-a Determination of (). . . . . . . . . . . . . . . . . . . .
1-b General expression for (). . . . . . . . . . . . . . . .
1-c Explicit expressions for= 0,1and2. . . . . . . . . . . . .
2 Properties of spherical harmonics
2-a Recurrence relations
2-b Orthonormalization and closure relations
2-c Parity
2-d Complex conjugation
2-e Relation between the spherical harmonics and the Legendre
polynomials and associated Legendre functions
This complement is devoted to the study of the form and principal properties of
spherical harmonics. It includes the proofs of certain results that were stated without
proof in .
1. Calculation of spherical harmonics
In order to calculate the various spherical harmonics(), we shall use the method
indicated in Chapter ); starting with the expression for(), we shall
use the operatorto obtain by iteration the spherical harmonics corresponding to the
same value ofand the (2+ 1) values ofassociated with it. Recall that the operators
+and act only on the angular dependence of a wave function and can be written:
=~e
+cot (1)
1-a. Determination of ()
We have seen ( ) that()can be calculated from the
equation:
+() = 0 (2)
and from the fact that:
() =() e (3)
Thus we obtained:
() =(sin)e (4)
705

COMPLEMENT A VI
whereis an arbitrary constant.
First, let us determine the absolute value ofby requiring()to be normal-
ized with respect to the angular variablesand:
2
0
d
0
sind ()
2
=
2
2
0
d
0
sind(sin)
2
= 1 (5)
We obtain:
2
= 1(2) (6)
whereis given by:
=
0
sind(sin)
2
=
+1
1
d(1
2
) (7)
(setting= cos).can easily be calculated by recurrence since:
=
+1
1
d(1
2
)(1
2
)
1
= 1
+1
1
d
2
(
2
)
1
(8)
An integration by parts of the last integral yields:
= 1
1
2
(9)
We therefore have:
=
2
2+ 1
1 (10)
with:
0=
+1
1
d= 2 (11)
From this, we can immediately derive the value of:
=
(2)!!
(2+ 1)!!
0=
2
2+1
(!)
2
(2+ 1)!
(12)
()is then normalized if:
=
1
2!(2+ 1)!4
(13)
In order to denecompletely, we must choose its phase. It is customary to
choose:
=
(1)
2!(2+ 1)!4
(14)
We shall see later that, with this convention,
0
()(which is independent of) has a
real positive value for= 0.
706

SPHERICAL HARMONICS
1-b. General expression for ()
We shall obtain the other spherical harmonics()by successive application
of the operatorto the()we have just determined. First, we shall prove a
convenient formula that will enable us to simplify the calculations.
. The action of ()on a function of the forme ()
The action of the operators+and on a function of the forme ()(where
is any integer) is given by:
[e()] =~e
(1)
(sin)
1
d
d(cos)
[(sin) ()] (15)
More generally:
()[e()] = (~)e
( )
(sin)
d
d(cos)
[(sin) ()] (16)
First, let us prove formula (15). We know that:
d
d(cos)
=
d
d(cos)
d
d
=
1
sin
d
d
(17)
and therefore:
(sin)
1 d
d(cos)
(sin)()
= (sin)
1 1
sin
(sin)
1
cos() + (sin)
d()
d
= cot() +
d()
d
(18)
Consequently:
e
(1)
(sin)
1 d
d(cos)
[(sin)()] = cote
(1)
()
= e
+cote () (19)
We recognize expression (1) for the operators+and; relation (19) is therefore identical to
(15).
Now, to establish formula (16), we can reason by recurrence, since, for= 1, (16) reduces
to (15), which we have just proved. Let us therefore assume that relation (16) is true for (1):
()
1
[e()] = (~)
1
e
( 1)
(sin)
1 d
1
d(cos)
1
[(sin)()] (20)
and let us show that it is then also valid for. To do so, we applyto both sides of (20); for
the right-hand side, we can use formula (15), making the substitutions:
= 1
() =(sin)
1 d
1
d(cos)
1
[(sin)()] (21)
707

COMPLEMENT A VI
We then obtain:
()[e()] = (~)e
( )
(sin)
+
d
d(cos)
(sin)
+1
(sin)
1 d
1
d(cos)
1
[(sin)()]
= (~)e
( )
(sin)
d
d(cos)
[(sin)()] (22)
Formula (16) is therefore proven by recurrence.
. Calculation of ()from()
As we have already indicated (Chapter, ), the spherical harmonics
()must satisfy:
() =~
(+ 1)(1)
1
()
=~
( )( + 1)
1
() (23)
These relations automatically insure that
1
is normalized ifis. Also, they x the
relative phases of spherical harmonics corresponding to the same value ofand dierent
values of.
In particular, we can calculate()from()by using the operator
given by (1) and formula (23). Thus, we shall obtain directly a normalized function
()whose phase will be determined by the convention used for()[formula
(14)]. To go from()to(), we must apply () times the operator;
according to (23), we thus obtain:
() ()
= ()
(2)(1)(21)(2) (++ 1)( )()(24)
that is:
() =
(+)!(2)!()!~
() (25)
Finally, using expression (4) for()[where the coecientis given by (14)]
and formula (16) (with=and= ), we can write (25) explicitly in the form:
() =
(1)
2!2+4
(+)!
( )!
e(sin)
d
d(cos)
(sin)
2
(26)
. Calculation of ()from ()
In order to obtain expression (26), we started with the result of . It is, of
course, just as easy to calculate()rst and then use the operator+. The
expression thus obtained foris dierent from (26), although the two are completely
equivalent.
708

SPHERICAL HARMONICS
Let us therefore calculate()from (26)
1
. Since:
(sin)
2
= (1cos
2
) (27)
is a polynomial of degree2incosonly its highest-order term contributes to():
d
2
d(cos)
2
(sin)
2
= (1)(2)! (28)
We therefore immediately nd that:
() =
1
2!(2+)!4
e(sin) (29)
()can then be obtained by applying (+) times the operator+to
(). Using (23) and (16), we nally arrive at:
() =
(1)
+
2!2+ 14
( )!
(+)!
e(sin)
d
+
d(cos)
+
(sin)
2
(30)
1-c. Explicit expressions for= 0,1and2
General formulas (26) and (30) yield the expressions of the spherical harmonics for
the rst values of:
0
0=
1
4
(31)
1
1
() =
38
sine
0
1() =
34
cos
(32)
2
2
() =
1532
sin
2
e
2
1
2
() =
158
sincose
0
2() =
516
(3 cos
2
1)
(33)
1
We could obviously calculate from the equation:
() = 0
However, its phase would then remain arbitrary. By using (26), we shall determine()completely,
and its phase will be a consequence of the convention chosen in .
709

COMPLEMENT A VI
2. Properties of spherical harmonics
2-a. Recurrence relations
By their very construction, the spherical harmonics satisfy relations (23); that is,
using (1):
e
cot () =
(+ 1)(1)
1
() (34)
Also note the following formula, which is often useful:
cos () =
(++ 1)( + 1)(2+ 1)(2+ 3)
+1()
+
(+)( )(2+ 1)(21)
1()(35)
Here is an outline of its proof. According to (25):
cos =
(+)!(2)!( )!
cos
~
() (36)
Now, using expression (1) for, it is easy to verify that:
[ cos] =~esin (37)
and:
[ esin] = 0 (38)
Using a recurrence argument, we can then calculate the commutator of(~)andcos, since,
if we assume that:
~
1
cos= (1)esin
~
2
(39)
we obtain:
~
cos=
~
1
~
cos+
~
1
cos
~
=
~
1
esin+ (1)esin
~
1
(40)
that is:
~
cos=esin
~
1
=
~
1
esin (41)
This relation has therefore been established by recurrence. We can use it to write () in the
form:
cos =
(+)!(2)!( )!~
cos ( )
~
1
esin (42)
710

SPHERICAL HARMONICS
Using (4) and (14), we can easily show that:
esin =
2+ 12
(1cos
2
)
1
1
(43)
If we then calculate the explicit expressions for
+1and
1
+1
from the general expression (26),
we nd that:
cos =
1
2+ 3
+1 (44a)
cos
2 1
1
=
2
2+ 12+ 3
1
+1
+
1
2+ 1
1
1
(44b)
Substituting relations (43) and (44) into (42 23), we obtain (35).
2-b. Orthonormalization and closure relations
Because of the way we constructed them, the spherical harmonics constitute a set
of normalized functions; they are also orthogonal, since they are eigenfunctions of the
Hermitian operatorsL
2
andwith dierent eigenvalues. The corresponding orthonor-
malization relation is:
2
0
d
0
sind () () = (45)
It can be shown (here, we shall assume) that any square-integrable function of
andcan be expanded in one and only one way on the spherical harmonics:
() = () (46)
with:
=
2
0
d
0
sin ()() (47)
The set of spherical harmonics therefore constitutes an orthonormal basis of the space of
square-integrable functions ofand. This can be expressed by the closure relation:
() ( ) =(coscos)( ) (48)
2-c. Parity
The parity operation on a function dened in ordinary space (cf.ComplementII)
consists of replacing in this function the coordinates of any point in space by those of
the point symmetrical to it with respect to the origin of the reference frame:
r= r (49)
711

COMPLEMENT A VI
In spherical coordinates, this operation is expressed by the substitutions (Fig.
ter):
=
=
= +
(50)
Consequently, if we are using a standard basis for the wave function space of a spin-
less particle ( ), the radial part of the basis functions()
is unchanged by the parity operation. The only transformation is that of the spherical
harmonics, which we shall now describe.
First, note that in the substitution of (50):
sin=sin
cos= cos
e=(1)e
(51)
Under these conditions, the function() which we calculated in
into:
( +) = (1)() (52)
Moreover:
==
(53)
Relations (51) and (53) show that the operators+and [formulas (1)] remain un-
changed [which means that+and are even operators, in the sense dened in Com-
plementII()]. Consequently, according to result (52) and formula (25), which
enables us to calculate():
( +) = (1) () (54)
The spherical harmonics are therefore functions whose parity is well-dened and
independent of: they are even foreven and odd forodd.
2-d. Complex conjugation
Because of their-dependence, the spherical harmonics are complex-valued func-
tions. It can be seen directly, by comparing (26) and (30), that:
[()]= (1) () (55)
2-e. Relation between the spherical harmonics and the Legendre polynomials and
associated Legendre functions
The-dependence of the spherical harmonics resides in functions known as Leg-
endre polynomials and associated Legendre functions. We shall neither prove nor even
enumerate all the properties of these functions, but shall simply indicate their relation
with the spherical harmonics.
712

SPHERICAL HARMONICS
.
0
()is proportional to a Legendre polynomial
For= 0, formulas (26) and (30) yield:
0
() =
(1)
2!2+ 14
d
d(cos)
(sin)
2
(56)
which can be written in the form:
0
() =
2+ 14
(cos) (57)
setting:
() =
(1)
2!
d
d
(1
2
) (58)
According to its denition (),() is anth order polynomial inof parity
2
(1):
() = (1)() (59)
()is theth order Legendre polynomial. It is easy to show that it haszeros in the
interval[1+1], and that the numerical coecient in (58) insures that:
(1) = 1 (60)
It can also be proven that the Legendre polynomials form a set of orthogonal functions:
+1
1
d()() =
0
sind(cos)(cos) =
2
2+ 1
(61)
on which can be expanded functions ofalone:
() = (cos) (62)
with:
=
2+ 1
2
0
sind(cos)() (63)
Comment:
According to (57) and (60):
0
(0) =
2+ 14
(64)
As we pointed out in , the phase convention chosen for() gives a real
positive value to
0
(0).
2
Parity with respect to the variable. Note, however, that the parity operation in space [formulas
(50)] amounts to changingcostocos; property (59) can be expressed by:
0
( ) = (1)
0
()
which is a special case of (54).
713

COMPLEMENT A VI
. ()is proportional to an associated Legendre function
Forpositive,()can be obtained by applying+to
0
(); using (23):
() =
(1)!(1 +)!
+
~
0
() ( 0) (65)
Using formulas (1) and (16), we then nd:
() = (1)
2+ 14
( )!
(1 +)!
(cos) e ( 0) (66)
where is an associated Legendre function, dened by:
() =
(1
2
)
d
d
() ( 1 +1) (67)
()is the product of
(1
2
)and a polynomial of degree () and
parity(1);
0
()is the Legendre polynomial(). The set of()for xed
constitutes an orthogonal system of functions:
+1
1
d ()() =
0
sind (cos)(cos) =
2
2+ 1
(+)!
( )!
(68)
on which can be expanded functions ofalone.
Formula (66) is valid forpositive (or zero); for negative, it suces to use
relation (55) to obtain:
() =
2+ 14
(+)!
(1)!
(cos) e ( 0) (69)
. Spherical harmonic addition theorem
Consider two arbitrary directions in space,and , dened respectively by
the polar angles () and ( ), and call the angle between them. The following
relation can be proven:
2+ 1
4
(cos) =
+
=
(1) ( ) ( ) (70)
(whereis theth-order Legendre polynomial). It is known as the spherical harmonics
addition theorem.
We shall indicate the main steps of an elementary proof of relation (70). First of all, note
that, ifcosis expressed in terms of the polar angles () and ( ), the left-hand side of
(70) can be considered to be a function ofand; it can therefore be expanded on the spherical
harmonics ( ). The coecients of this expansion, which are, of course, functions of the
other two variables,and, can also be expanded on the spherical harmonics( ).
We must therefore have:
2+ 1
4
(cos) = ; ( ) ( ) (71)
714

SPHERICAL HARMONICS
where the problem is to calculate the coecients ; . They can be obtained by the
following process:
(i) in the rst place, these coecients are dierent from zero only for:
== (72)
To show this, rst x the direction;(cos)then depends only onand'. If theaxis
is chosen along,cos= cosand(cos) is proportional to
0
()[cf.relation (57)]. To
generalize to the case in which the direction ofis arbitrary, we perform a rotation which
takesonto this direction:cosremains unchanged, as does(cos). Since the rotation
operators (ComplementVI, ) commute withL
2
, the transform of
0
()remains an
eigenfunction ofL
2
with the eigenvalue(+ 1)~
2
, that is, a linear combination of the spherical
harmonics ( ); we therefore have=. Similarly, it can be established that=.
()under a rotation of both directionsand through an angleabout, the
angleis not changed, and neither areand, whileand become+and+.
The left-hand side of (71) therefore does not change in value, and each term of the right-hand
side is multiplied bye
(+ )
. Consequently, the only non-zero coecients in the sum of the
right-hand side are those which satisfy:
+ = 0 (73)
()combining results (72) and (73), we see that formula (71) can be written in the form:
2+ 1
4
(cos) =
+
=
(1) ( ) ( ) (74)
If we set=and=, we obtain, according to (60):
2+ 1
4
=
+
=
(1) ( ) ( ) (75)
Since(1) is simply , the integration of (75) with respect tod= sindd
yields, with the orthonormalization equation (45):
2+ 1 =
+
=
(76)
We now take the square of the modulus of both sides of (74 dand
d. Using relation (45), it is easy to see that the right-hand side yields
+
=
2
. As far
as the left-hand side is concerned, we can again take advantage of the invariance of the angle
with respect to a rotation in order to show thatd (cos)
2
is actually independent of
( ). If we then choosealongto evaluate this integral, we nd, according to relation
(61):
d (cos)
2
=d (cos)
2
= 2
2
2+ 1
(77)
Integrating overd, we nd a second relation between the coecients:
2+ 1 =
+
=
2
(78)
715

COMPLEMENT A VI
()equations (76) and (78) suce for the determination of the (2+ 1) coecients:
they are all equal to 1. To show this, consider, in a normed (2+1)-dimensional vector space, the
vectorXof components =
2+ 1and the vectorYof componentsm= 12+ 1.
The Schwartz inequality indicates that:
(XX)(YY)YX
2
(79)
where there is equality if and only ifXandYare proportional. Now, (76) and (78) show that
this is the case:and are therefore independent of, as is, and we have, necessarily,
= 1. This concludes the proof of formula (70).
References
Messiah (1.17), App. B, IV; Arfken (10.4), Chap. 12; Edmonds (2.21), Table 1;
Butkov (10.8), Chap. 9, 5 and 8; Whittaker and Watson (10.12), Chap. XV; Bateman
(10.39), Chap. III; Bass (10.1), vol. I, 17-7.
716

ANGULAR MOMENTUM AND ROTATIONS
Complement BVI
Angular momentum and rotations
1 Introduction
2 Brief study of geometrical rotations R. . . . . . . . . . . . .
2-a Denition. Parametrization
2-b Innitesimal rotations
3 Rotation operators in state space. Example: a spinless
particle
3-a Existence and denition of rotation operators
3-b Properties of rotation operators. . . . . . . . . . . . . . .
3-c Expressing rotation operators in terms
of angular momentum observables
4 Rotation operators in the state space of an arbitrary system 727
4-a System of several spinless particles
4-b An arbitrary system
5 Rotation of observables
5-a General transformation law
5-b Scalar observables
5-c Vector observables
6 Rotation invariance
6-a Invariance of physical laws
6-b Consequence: conservation of angular momentum
6-c Applications
1. Introduction
We indicated in Chapter) that the commutation relations between the compo-
nents of an angular momentum are actually the expression of the geometrical properties
of rotations in ordinary three-dimensional space. This is what we intend to show in this
complement, where we investigate the relation between rotations and angular momentum
operators.
Consider a physical system () whose quantum mechanical state, at a given time,
is characterized by the ketof the state space. We perform a rotationRon this
system; in this new position, the state of the system is described by a ketwhich is
dierent from. Given the geometrical transformationR, the problem is to determine
from. We shall see that it has the following solution: with every geometrical
rotationRcan be associated a linear operatoracting in the state spacesuch that:
= (1)
717

COMPLEMENT B VI
Let us immediately stress the necessity of distinguishing between the geometrical rotation
R, which operates in ordinary space, and its image, which acts in the state space:
R= (2)
We shall begin () by reviewing the principal properties of geometrical rotations
R. We shall not embark upon a detailed study of them; rather, we shall simply note
some results which will be useful to us later. Then, in , we shall use the example of a
spinless particle to dene the rotation operatorsprecisely, to study their most impor-
tant properties, and to determine their relation to the angular momentum operatorsL.
We shall then be able to interpret the commutation relations amongst the components of
the angular momentumLas the image, in the spacer, of purely geometrical character-
istics of rotationsR. We shall then generalize () these concepts to arbitrary quantum
mechanical systems. In , we shall examine the behavior of the observables describing
the physical quantities measurable in this system, upon rotation of the system. This
will lead us to classify observables according to how they transform under a rotation
(scalar, vector, tensor observables). Finally, in , we shall briey consider the problem
of rotation invariance and indicate some important consequences of this invariance.
2. Brief study of geometrical rotationsR
2-a. Denition. Parametrization
A rotationRis a one-to-one transformation of three-dimensional space that con-
serves a point of this space, the angles and the distances, as well as the handedness of
the reference frames
1
. We shall be concerned here with the set of rotations that conserve
a given point, which we shall choose as the origin of the reference frame. A rotation
can then be characterized by theaxis of rotation(given by its unit vectoruor its polar
anglesand) and the angle of rotation(0 2). To determine a rotation, three
parameters are required; they can be chosen to be the components of the vector:
=u (3)
whose absolute value is equal to the angle of rotation and whose direction denes the
axis of rotation. Note that a rotation can also be characterized by three angles, called
Euler angles. We shall denote byRu()the geometrical rotation through an angle
about the axis dened by the unit vectoru.
The set of rotationsRconstitutes a group: the product of two rotations (that is,
the transformation resulting from the successive application of these two rotations) is
also a rotation; there exists an identity rotation (rotation through a zero angle about
an arbitrary axis); for every rotationRu()there is an inverse rotation,Ru(). The
group of rotations is not commutative: in general, the product of two rotations depends
on the order in which they are performed
2
:
Ru()Ru()=Ru()Ru() (4)
1
This last property is imposed in order to exclude reections with respect to a point or a plane.
2
When one writesR2R1, this means that rotationR1must be performed rst,R2being applied
subsequently to the result obtained.
718

ANGULAR MOMENTUM AND ROTATIONS
Recall, however, that two rotations performed about the same axis always commute:
Ru()Ru() =Ru()Ru() =Ru(+) (5)
(if necessary,2is subtracted from+, to keep it within the interval[02]).
2-b. Innitesimal rotations
An innitesimal rotation is dened as a rotation that is innitesimally close to the
identity rotation, that is, a rotationRu(d)through an innitesimal angledabout
an arbitrary axisu. It is easy to see that the transform of a vectorOMunder the
innitesimal rotationRu(d)can be written, to rst order ind:
Ru(d)OM=OM+ duOM (6)
Every nite rotation can be decomposed into an innite number of innitesimal
rotations, since the angle of rotation can vary continuously and since, according to (5):
Ru(+ d) =Ru()Ru(d) =Ru(d)Ru() (7)
whereRu(d)is an innitesimal rotation. Thus, the study of the rotation group can be
reduced to an examination of innitesimal rotations
3
.
Before ending this rapid survey of the properties of geometrical rotations, we note
the following relation which will be useful to us later:
Re(d)Re(d)Re(d)Re(d) =Re(dd) (8)
wheree,eandedenote the unit vectors of the three coordinate axes,and
respectively. Ifdanddare rst-order innitesimal angles, this relation is correct to
the second order. It describes, in a special case, the non-commutative structure of the
rotation group.
To prove relation (8), let us apply its left-hand side to an arbitrary vectorOM. We use
formula (6) to nd the vectorOM, the transform ofOMunder the succesive action of the four
innitesimal rotations. It can be seen immediately that ifdis zero, the left-hand side of (8)
reduces to the productRe(d)Re(d), which is equal to the identity rotation [see (5)]; the
vectorOM OMmust therefore be proportional tod. For an analogous reason, it must also
be proportional tod. Consequently, the dierenceOM OMis proportional todd.
Therefore, to calculateOMto second order, we may restrict ourselves to rst order in
each of the two innitesimal anglesdandd. First of all, according to (6):
Re(d)OM=OM deOM (9)
We must then applyRe(d)to this vector; this can be done by again using (6):
Re(d)Re(d)OM
=(OM deOM) + de(OM deOM)
=OM deOM+ deOM dde(eOM) (10)
3
However, limiting ourselves to innitesimal rotations, we lose sight of a global property of the
nite rotation group: the fact that a rotation through an angle of2is the identity transformation.
The rotation operators (see 3) constructed from innitesimal operators do not always have this global
property. In certain cases (see ComplementIX), the operator associated with a2rotation is not the
unit operator but its opposite.
719

COMPLEMENT B VI
The action ofRe(d)on the vector appearing on the right-hand side of (10) results in the
addition of the following innitesimal terms to this vector:
deOM+ dde(eOM) (11)
obtained by the vector multiplication of the right-hand side of (10) byde, with only the rst
order terms indbeing retained. Therefore:
Re(d)Re(d)Re(d)OM
=OM+ deOM+ dd[e(eOM)e(eOM)](12)
Finally,OMis equal to the sum of the vector just obtained and its vector product byde.
To rst order ind, this vector product can be written simply:
deOM (13)
which means that:
Re(d)Re(d)Re(d)Re(d)OM
=OM+ dd[e(eOM)e(eOM)](14)
It is then easy to transform the double vector products; we nd:
Re(d)Re(d)Re(d)Re(d)OM=OM+ ddeOM
=Re(dd)OM (15)
Since this relation is true for any vectorOM, expression (8) is veried.
3. Rotation operators in state space. Example: a spinless particle
In this section, we consider a physical system composed of a single (spinless) particle in
three-dimensional space.
3-a. Existence and denition of rotation operators
At a given time, the quantum mechanical state of the particle is characterized, in
the state spacer, by the ketwith that is associated the wave function(r) =r.
Let us perform a rotationRon this system which associates with the pointr0(000)
of space the pointr
0(
000)such that:
r
0=Rr0 (16)
Let be the state vector of the system after rotation, and(r) =r, the cor-
responding wave function. It is natural to assume that the value of the initial wave
function(r)at the pointr0will be found, after rotation, to be the value of the nal
wave function(r)at the pointr
0given by (16):
(r
0) =(r0) (17)
that is:
(r
0) =(R
1
r
0) (18)
720

ANGULAR MOMENTUM AND ROTATIONS
Since this equation is valid for any point (r
0) in space, it can be written in the form:
(r) =(R
1
r) (19)
By denition, the operatorin the state spacerassociated with the geometrical
rotationRbeing treated is the one that acts on the statebefore rotation to yield the
stateafter the rotationR:
= (20)
is called a rotation operator. Relation (19) characterizes its action in ther
representation:
r =R
1
r (21)
whereR
1
ris the basis ket of this representation determined by the components of
the vectorR
1
r.
Comment:
If the state of the particle after rotation weree (whereis an arbitrary real number)
instead of, its physical properties would not be modied. In other words, relation
(17) could be replaced by:
(
0) = e(0) (22)
would obviously be independent of0, but could depend on the rotationR. We shall
not treat this diculty here.
3-b. Properties of rotation operators
. is a linear operator
This essential property of rotation operators follows from their very denition. If
the statebefore the rotation is a linear superposition of states, for example:
=11+22 (23)
formula (21) indicates that:
r =1R
1
r1+2R
1
r2
=1r 1+2r 2 (24)
Since this relation is true for any ket of therbasis, we deduce thatis a linear
operator:
=[11+22] =1 1+2 2 (25)
721

COMPLEMENT B VI
. is unitary
In formula (21), the ketcan be arbitrary. The action of the operatoron the
braris therefore given by:
r=R
1
r (26)
Taking the Hermitian conjugate of both sides of equation (26), we obtain:
r=R
1
r (27)
Moreover, if we recall that the ketrrepresents a state in which the particle is
perfectly localized at the pointr, we see that:
r=Rr (28)
This equation simply expresses the fact that if the particle was localized at the pointr
before the rotation, it will be localized at the pointr=Rrafter the rotation. To get
(28) from (21), we choose a basis stater0for:
rr0=R
1
rr0=[(R
1
r)r0] (29)
where we have used the orthonormalization relation of therbasis. Furthermore
4
:
[(R
1
r)r0] =[r(Rr0)] (30)
Substituting (30) into (29), we indeed nd that:
r 0=[r(Rr0)] =rRr0 (31)
that is, sinceris a basis ofr:
r0=Rr0 (32)
Starting with formulas (27) and (28), it is easy to show that:
= = 1 (33)
since the action ofor on any vector of therbasis yields the same vector;
for example:
r=R
1
r=RR
1
r=r (34)
The operatoris therefore unitary.
Comment:
The operatortherefore conserves the scalar product and the norm of vectors
that it transforms:
=
=
= = (35)
This property is very important from the physical point of view since the prob-
ability amplitudes which yield physical predictions appear in the form of scalar
products of two kets.
4
Relation (30) can easily be established by using the denition of delta functions and the fact that
a rotation conserves the innitesimal volume element.
722

ANGULAR MOMENTUM AND ROTATIONS
. The set of operatorsconstitutes a representation of the rotation group
We have pointed out () that the geometrical rotations form a group; in partic-
ular, the product of two rotationsR1andR2is always a rotation:
R2R1=R3 (36)
With the three geometrical rotationsR1,R2andR3are associated, in the state
spacer, three rotation operators1,2and3, respectively. If the three geometrical
rotations satisfy relation (36), we shall show that the corresponding rotation operators
are such that:
21=3 (37)
(21is a product of operators ofras dened in Chapter, ).
Consider a particle whose state is described by an arbitrary ketrof the basis
characterizing therrepresentation. If we perform the rotationR1on this particle,
its state becomes:
1r=R1r (38)
by denition of1. Now, we perform the rotationR2on the new state we have just
obtained; the state of the particle after this second rotation is, according to (38) and the
denition of2:
21r=2R1r=R2R1r (39)
If we take (36) into account, we see that relation (39) is equivalent to:
21r=R3r (40)
Now, the operator3, associated with the rotationR3, is such that:
3r=R3r (41)
Relation (37) is therefore proven, since the ketrunder consideration can be chosen in
an arbitrary way from the kets of therbasis.
To express the important result we have just established, one says that the corre-
spondenceR= between geometrical rotations and rotation operators conserves the
group law, or that the set of operatorsconstitutes a representation of the rotation
group. Of course, with the identity rotation, we associate the identity operator inr,
and with the rotationR
1
(the inverse of a rotationR), the operator
1
, which is the
inverse of the one corresponding toR(we showed in that
1
=).
3-c. Expression for rotation operators
in terms of angular momentum observables
. Innitesimal rotation operators
First of all, let us consider an innitesimal rotation about theaxis,Re(d).
If we apply it to a particle whose state is described by the wave function(r), we know
723

COMPLEMENT B VI
from (19) that the wave function(r)associated with the state of the particle after
rotation satises:
(r) =[R
1
e(d)r] (42)
But if () are the components ofr, those ofR
1
(d)rcan easily be calculated from
(6):
R
1
e(d)r=Re(d)r= (rder)
+d
d (43)
Equation (42) can then be written in the form:
( ) =(+d d) (44)
which yields, to rst order ind:
( ) =( ) + d
=( )d
( ) (45)
Within the brackets, we recognize, to within a factor of~, the expression in ther
representation for the operator= . We therefore obtain the result:
(r) =r=r1
~
d (46)
Now, by denition of the operator(d)associated with the rotationR(d):
=e(d) (47)
Therefore, since the original stateis arbitrary, we nally nd that:
e(d) = 1
~
d
(48)
The preceding argument can easily be generalized to an innitesimal rotation about an
arbitrary axis. We therefore have, in general:
u(d) = 1
~
dLu
(49)
Comment:
(46) can also be quickly established by using the spherical coordinates (),
sincethen corresponds to the dierential operator
~
.
724

ANGULAR MOMENTUM AND ROTATIONS
. Interpretation of the commutation relations for the components of the angular
momentumL
What then is the image in the state spacerof relation (8)? According to the
results of -and the expressions we have just obtained, this relation implies that,
to rst order with respect to each of the anglesdandd:
1 +
~
d 1
~
d 1
~
d 1 +
~
d = 1
~
dd (50)
Expanding the left-hand side and setting the coecients ofddequal, we easily nd
that relation (50) reduces to:
[ ] =~ (51)
Of course, the two other commutation relations for the components ofLcan be found,
by an analogous argument, from the formulas obtained from (8) by cyclic permutation
of the vectorse,eande.
Thus,the commutation relations of the orbital angular momentum of a particle can
be seen to be consequences of the non-commutative structure of the geometrical rotation
group.
. Finite rotation operators
Now, consider a rotationRe() through an arbitrary angleabout theaxis.
According to formula (7), the operatore()associated with such a rotation must
satisfy (again using the results of ):
e(+ d) =e()e(d) (52)
where the two operators of the right-hand side commute. But we know the expression
fore(d), so we have:
e(+ d) =e()1
~
d (53)
that is:
e(+ d) e() =
~
d e() (54)
Here again,e()andmust commute. Although we are dealing with operators, the
solution of equation (54) is formally the same as it would be if we were considering an
ordinary function of the variable:
e() = e
~
(55)
Indeed, if we recall (cf.ComplementII, ) that the exponential of an operator is dened by
the corresponding power series expansion, it is easy to verify that expression (55) is the solution
of equation (54). Moreover, the integration constant is equal to 1, since we know that:
e(0) = 1 (56)
725

COMPLEMENT B VI
As in -above, it is easy to generalize this result to a nite rotation about an arbitrary
axis:
u() = e
~
Lu
(57)
Comments:
(i) Formula (57) can be written explicitly in the form:
u() = e
~
( + + )
(58)
where,andare the components of the unit vectoru. Recall, however,
that, since,anddo not commute:
u()= e
~ e
~ e
~ (59)
()It can be seen from expression (57) that the operatoru() is unitary. Since
the components ofLare Hermitian:
[u()]= e
~
Lu
(60)
we have (asLuobviously commutes with itself):
[u()]u() =u() [u()]= 1 (61)
()In the special case envisaged in this section, we nd that:
u(2) = 1 (62)
We shall conne ourselves to proving this result for the rotation through2about
theaxis (the generalization of this proof involves no diculties). To this end,
consider an arbitrary ket, and expand it on a basis composed of eigenvectors
of the observable:
= (63)
with:
=~ (64)
(symbolizes the indices other thanthat are necessary to specify the vectors
of the basis used, such as a standard basiscf. ).
The action ofe()on is then easy to obtain:
e()= e
~
= e (65)
726

ANGULAR MOMENTUM AND ROTATIONS
But we know that, for the orbital angular momentum of a particle,is always
integral. Consequently, whenattains the value2, all the factorse become
equal to1, and:
e(2)= = (66)
Since this relation is satised for all, we deduce thate(2)is the identity
operator.
The preceding argument clearly indicates thatformula(62)would not be valid
if half-integral values ofwere not excluded. Indeed, we shall see in Comple-
mentIXthat, for a spin 1/2, the operator associated with a rotation of2is
equal to1and not 1; this result is related to the fact that we constructed the
nite rotations from innitesimal rotations (cf.footnote).
4. Rotation operators in the state space of an arbitrary system
We shall now generalize the concepts we introduced and the results we obtained for a
special case (in ).
4-a. System of several spinless particles
First of all, the arguments of
composed of several spinless particles. We shall quickly demonstrate this, choosing as an
example a system of two spinless particles, (1) and (2).
The state spaceof such a system is the tensor product of the state spacesr1
andr2
of the two particles:
=r1 r2
(67)
We shall use the same notation as in . Starting from the position
and momentum observables (R1andP1on the one hand,R2andP2on the other), we
can dene an orbital angular momentum for each of the particles:
L1=R1P1
L2=R2P2 (68)
The components ofL1, as well as those ofL2, satisfy the commutation relations charac-
teristic of angular momenta.
Consider a vector which is a tensor product of a vector ofr1and a vector ofr2:
=(1) (2) (69)
represents the state of the system formed by particle (1) in the state(1)and
particle (2) in the state(2). If we perform a rotation through an angleaboutuon
the two-particle system, the state of the system after the rotation corresponds to the two
particles in the rotated states(1)and(2)respectively:
= (1) (2)= [
1
u()(1)][
2
u()(2)] (70)
727

COMPLEMENT B VI
where
1
u()and
2
u()are the rotation operators inr1
andr2
:
1
u() = e
~
L1u
(71a)
2
u() = e
~
L2u
(71b)
Relation (70) can also be written, by denition of the tensor product of two operators
(Chap., ):
= [
1
u()
2
u()](1) (2) (72)
Since every vector ofis a linear combination of vectors analogous to (69), the rotation
transform of an arbitrary vectorofis:
= [
1
u()
2
u()] (73)
Using formula (F-14) of Chapter L1andL2commute (they are
operators relating to dierent particles), we obtain for the rotation operators in:
1
u()
2
u() = e
~
L1u
e
~
L2u
= e
~
Lu
(74)
where:
L=L1+L2 (75)
is the total angular momentum of the two-particle system. All the formulas of the preced-
ing section therefore remain valid as long asLrepresents thetotal angular momentum.
Comments:
(i)Lis an operator that acts in. In (75),L1is, rigorously, the extension of
the operatorL1acting inr1into(an analogous comment could be made for
L2). To simplify the notation, we shall not use dierent symbols forL1and its
extension into(cf.Chap., ).
()We might consider performing a rotation on only one of the two particles, for
example, particle (1). In the course of such a partial rotation, a vector such as
(69) transforms into:
[
1
u()(1)](2) (76)
where only the state of particle (1) is modied. As above, it can be shown that
the eect of a rotation performed only on particle (1) on an arbitrary stateof
is described by the operator:
1
u()(2) = e
~
L1u
(77)
whereis the unit operator inr2[in (77),L1acts in].
728

ANGULAR MOMENTUM AND ROTATIONS
4-b. An arbitrary system
The starting point of the arguments elaborated thus far is equation (19), which
gives the transformation law of the state vector of the system in terms of that of its
wave function. In the case of an arbitrary quantum mechanical system (which does not
necessarily have a classical analogue), one cannot use the same method. For example, for
a particle with spin, the operators,andno longer form a C.S.C.O., and the state
of the particle can no longer be dened by a wave function( ) (cf.Chap.). One
must reason directly in the state spaceof the system. Without going into detail, we
shall assume here that an operatoracting incan be associated with any geometrical
rotationR; if the system is initially in the state, the rotationRtakes it into the
state:
= (78)
where the operatoris linear and unitary (cf.comment of -).
As far as the group law of the rotationsRis concerned, it is conserved by the oper-
ators, but only locally: the product of two geometrical rotations, at least one of which
is innitesimal, is represented in the state spaceby the product of the correspond-
ing operators(which implies, in particular, that the image of a rotation through
an angle equal to zero is the identity operator). However, the operator associated with
a geometrical rotation through an angle2is not necessarily the identity operator [cf.
comment () of and ComplementIX].
Now, let us consider an innitesimal rotationRe(d)about theaxis. Since
the group law is conserved for innitesimal rotations, the operatore(d)is necessarily
of the form:
e(d) = 1
~
d (79)
whereis a Hermitian operator sincee(d)is unitary (cf.ComplementII, ).
This relation is thedenitionof. Similarly, the Hermitian operatorsandcan be
introduced by starting with innitesimal rotations about theand axes. The total
angular momentumJof the system is then dened in terms of its three components,
and.
Now we can use the reasoning of -: the geometrical relation (8) implies that
the components ofJsatisfy commutation relations which are identical to those of orbital
angular momenta. Thus,the total angular momentum of any quantum mechanical system
is related to the corresponding rotation operators; the commutation relations amongst its
components follow directly, which enables us to use them, as in Chapter), to
characterize any angular momentum.
Finally, let us show that, with,anddened as we have just indicated,
the operatoru(d)associated with an arbitrary innitesimal rotation is written (,
andbeing the components of the unit vectoru):
u(d) = 1
~
d( + + ) (80)
which can then be condensed into the form:
u(d) = 1
~
dJu (81)
729

COMPLEMENT B VI
Formula (80) is simply a consequence of the geometrical relation:
Ru(d) =Re(d)Re(d)Re(d) (82)
valid to rst order ind, and which can be obtained directly from formula (6).
We have thus generalized expressions (48) and (49) for innitesimal rotation opera-
tors. Since the group law is conserved locally (see above), relation (52) and the argument
following it remain valid. Consequently, the nite rotation operators have expressions
analogous to (55) and (57):
u() = e
~
Ju
(83)
5. Rotation of observables
We just showed how the vector representing the state of a quantum mechanical system
transforms under rotation. But in quantum mechanics, the state of a system and the
physical quantities are described independently. Therefore, we shall now indicate what
happens to observables upon rotation.
5-a. General transformation law
Consider an observable, relating to a given physical system; we shall assume, to
simplify the notation, the spectrum ofto be discrete and non-degenerate:
= (84)
In order to understand how this observable is aected by a rotation, we shall imagine
that we have a device which can measurein the physical system under consideration.
Now, the observable, the transform ofwith respect to the geometrical rotationR, is
by denition what is measured by the device when it has been subjected to the rotation
R.
Let us assume the system to be in the eigenstateof: the device for measuring
in this system will give the resultwithout fail. But just before performing the
measurement, we apply a rotationRto the physical system and, simultaneously, to
the measurement device; their relative positions are unchanged. Consequently, if the
observablewhich we are considering describes a physical quantity attached only to the
system which we have rotated (that is, independent of other systems or devices which we
have not rotated), then, in its new position, the measurement device will still give the
sameresultwithout fail. Now, after rotation, the device, by denition, measures,
and the system is in the state:
= (85)
We must therefore have:
= = = (86)
Combining (85) and (86), we nd:
= (87)
730

ANGULAR MOMENTUM AND ROTATIONS
that is:
= (88)
since the inverse ofis. The set of vectorsconstitutes a basis in the state space
(is an observable), so we have:
= (89)
that is:
= (90)
In the special case of an innitesimal rotationRu(d), the general expression (81),
substituted into (90), gives, to rst order ind:
=1
~
dJu 1 +
~
dJu
=
~
d[Ju ] (91)
Comments:
(i) In the case of a spinless particle, relation (90) implies that:
rr=r r (92)
Using (26) and (27), we therefore obtain:
rr=R
1
rR
1
r (93)
The transformation which enables us to obtainfromis therefore completely
analogous to the one which givesin terms of[cf.(19)].
()Consider the case in which the observableis associated with a classical
quantityA.Ais then a function of the positionsrand momentapof the
particles which constitute the system; the operatoris obtained from this function
by applying the quantization rules given in Chapter. We know how to nd the
quantityAassociated withAby a rotationRin classical mechanics: for example,
ifAis a scalar,Ais the same asA; ifAis the component along an axisof a
vectorial quantity,Ais the component of this same vectorial quantity along the
axis which is the result of the transformation ofby the rotationR. We can
also construct the quantum mechanical operator corresponding toA, by applying
the same quantization rules as above. It can be shown that this operator is the
same as the operatorgiven in (90); this is what is shown in Figure.
731

COMPLEMENT B VIR

ℛ
A A ′
′
Figure 1: Behavior, under a rotationR, of
a classical physical quantityAand of the
associated observable.
5-b. Scalar observables
An observableis said to be scalar if:
= (94)
for all. According to (91), this implies that:
[J] = 0 (95)
A scalar observable commutes with the three components of the total angular mo-
mentum.
There are numerous examples of scalar observables.J
2
is always a scalar (this re-
sults, as we saw in , from the commutation relations that characterize
an angular momentum). For a spinless particle,R
2
,P
2
andRP, which correspond to
classical scalar quantities, are scalars. It is easy to show, moreover (cf.
R
2
,P
2
andRPsatisfy (95). We shall also see later () that the Hamiltonian of an
isolated physical system is a scalar.
5-c. Vector observables
A vector observableVis a set of three observables (its Cartesian com-
ponents) that is transformed by rotation according to the characteristic law of vectors.
The transform, under a rotationR, of the component=VuofValong a given
axis(of unit vectoru) must be the componentu=VuofValong the axis
derived fromby the rotationR.
Consider, for example, the componentof such an observable. We shall examine
its behavior under innitesimal rotations about each of the coordinate axes.is obvi-
ously unchanged by a rotation about; according to (91), this can be expressed in the
form:
[ ] = 0 (96)
If we perform a rotationRe(d) about theaxis, the transform ofis the observable
()given by (91) as:
()=
~
d[ ] (97)
732

ANGULAR MOMENTUM AND ROTATIONS
Butis the component ofValong theaxis, of unit vector. The rotationRe(d)
takeseontoesuch that [formula (6)]:
e=e+ dee
=ede (98)
Consequently, ifVis a vector observable,()must be the same asVe:
()=VedVe
= d (99)
Comparing (97) and (99), we see that:
[ ] =~ (100)
For an innitesimal rotationRe(d) about theaxis, an argument analogous to the
one above leads to the relation:
[ ] =~ (101)
By studying the behavior ofandunder innitesimal rotations, one can prove the
formulas which can be derived from (96), (100) and (101) by cyclic permutation of the
indices . The set of relations obtained in this way is characteristic of a vector
observable: they imply that an arbitrary innitesimal rotation transformsVuinto
Vu, whereuis the transform ofuwith respect to the rotation under consideration.
It is clear that the angular momentumJitself is a vector observable; (96), (100)
and (101) then follow from the commutation relations characterizing angular momenta.
For a system composed of a single spinless particle,RandPare vector observables, as
can easily be veried from the canonical commutation relations. Thus the vector notation
we use forRPLandJis justied.
Comments:
(i) The scalar productVWof two vector observables, dened by the customary
formula:
VW= + + (102)
is a scalar operator. To verify this, we can calculate, for example, the commutator
ofVWwith:
[VW ] = [ ] + [ ]
=[ ] + []+[ ] + []
=~ ~ +~ +~
= 0 (103)
We have already pointed out thatJ
2
,R
2
,P
2
andRPare scalar observables.
()It is thetotal angular momentum of the systemunder study that appears in
relations (96), (100) and (101). The following example illustrates the importance
of this fact: if, for a two-particle system, we were to useL1instead ofL=L1+L2,
R2would appear to be a set of three scalar observables and not a vector observable.
733

COMPLEMENT B VI
6. Rotation invariance
The discussion presented in the preceding sections does not have as its sole purpose the
justication of the denition of angular momenta in terms of the commutation relations.
The importance of rotations in physics is essentially related to the fact that physical laws
are rotation invariant. We are going to explain in this section exactly what this means,
and we shall indicate some of the consequences of this fundamental property.
6-a. Invariance of physical laws
Consider a physical system (), classical or quantum mechanical, which we subject
to a rotationRat some given time. If we take the precaution of rotating, at the same time
as the system () under consideration, all other systems or devices that can inuence
it,the physical properties and behavior of () are not modied. This means that the
physical laws governing the system have remained the same: the physical laws are said
to be rotation invariant. Note that this property is not at all obvious: there exist
transformations those of similarity
5
, for example with respect to which the physical
laws are not invariant
6
. It is therefore advisable to consider rotation invariance to be a
postulate which is justied by the experimental verication of its consequences.
When we say that the physical properties and behavior of a system are unchanged
by a rotation performed at the time0, this statement covers two observations:
(i) the properties of the system at this time are not modied (although the descrip-
tion of the state of the system and the physical quantities are; see preceding sections).
In quantum mechanics, this implies that the transformof an arbitrary observable
has the same spectrum, and that the probability of nding one of the eigenvalues of this
spectrum in a measurement ofon the system after rotation is the same as it was for
the measurement ofon the system before rotation. From this, it can be deduced that
the operatorsdescribing rotations in state space are linear and unitary, or antilinear
and unitary (that is, anti-unitary
7
).
()the time evolution of the system is not aected. To state this point more
precisely, let us denote the state of the system by(0); under a rotation performed at
0, this state becomes:
(0)= (0) (104)
We now let the system evolve freely and compare its state()at a subsequent time
to the state()which it would have attained if it had been allowed to evolve freely
from(0). If the behavior of the system is not modied, we must have:
()= () (105)
that is, for all, the state()must be obtained from()by the same rotation as
in (104). Therefore, if()is a solution of the Schrödinger equation,()is also
5
Consider, for example, a hydrogen atom. If we multiply the distance between the proton and the
electron by a constant= 1(without modifying the charges and masses of the particles), we obtain a
system whose evolution no longer obeys either classical or quantum mechanical physical laws.
6
Let us also point out that experiments have shown that the laws governing the-decay of nuclei
are not invariant under reection with respect to a plane (non-conservation of parity).
7
All transformations which leave the physical laws invariant are described by unitary operators,
except for time reversal, with which is associated an anti-unitary operator.
734

ANGULAR MOMENTUM AND ROTATIONS
a solution of this equation:the transform of a possible motion of the system is also a
possible motion. We shall see in of the
system is a scalar observable.
The invariance of physical laws under rotation is related to thesymmetry proper-
tiesof the equations which state these laws mathematically. To understand the origin of
these symmetries, consider, for example, a system composed of a single (spinless) parti-
cle. The expression of the physical laws governing such a system explicitly involves the
parametersr( )andp( )which characterize the position of the particle and
its momentum: in classical mechanics,randpdene at each instant the state of the
particle; in quantum mechanics, although the meaning of these parameters is a little less
simple, they appear in the wave function(r)and its Fourier transform
(p). When
the particle is subjected to an instantaneous rotationR,randpare transformed intor
andpsuch that:
r=Rr
p=Rp (106)
If we replacerbyR
1
randpbyR
1
pin the equations which express the physical
laws, we obtain relations that now involverandp. The invariance of physical laws
under the rotationRthus implies thatthe form of the equations forrandpis the
same as that of the equations forrandp: simply omitting the primes labeling the new
parameters must give us back the original equations. It is clear that this considerably
restrains the number of possible forms of these equations.
Comments:
(i) What happens when we perform a rotation on a system which is not isolated? Con-
sider, for example, a particle in an external potential. If we rotate this system,without
simultaneously rotating the sources of the external potential, the subsequent evolution of
the system is, in general, modied
8
. In classical mechanics, the forces exerted on the par-
ticle are not the same in its new position. In quantum mechanics,(r) =(R
1
r)is
a solution of a Schrödinger equation in which the potential(r)is replaced by(R
1
r),
which is, in general, dierent from(r). Therefore, the transform of a possible motion
is no longer a possible motion. The presence of the external potential destroys, so to
speak, the homogeneity of the space in which the system under study evolves.
However, the external potential may present certain symmetries that allow certain ro-
tations to be performed on the physical system without its behavior being modied. If
there exist rotationsR0such that(R
1
0
r)is the same as(r), the properties of the
system are unchanged by one of these rotationsR0. This is the case, for example, for
8
If the particle is placed in a vector potential, its properties immediately after the rotation may be
profoundly modied. Consider, for example, a spinless particle in an external magnetic eld. According
to the transformation law (19), the probability current given by formula (D-20) of Chapter
in general be derived by rotation of the initial current, as it depends on the vector potential describing
the magnetic eld.The physical interpretation of this phenomenon is as follows. We can imagine that,
instead of rotating the particle, we rotate the magnetic eld rapidly and in the opposite direction. The
wave function has not had the time to change: this corresponds to formula (19). If the physical properties
are modied, it is because an induced electromotive eld has appeared and acted on the particle. This
action does not depend on the exact way in which we rotate the magnetic eld, provided that we do so
quickly enough.
735

COMPLEMENT B VI
central potentials, that is, those depending only on the distance to a xed point: the
rotationsR0are then all rotations that conserve the point(cf.Chap.).
()Let us return to the case of isolated physical systems. Thus far, we have adopted an
active viewpoint: the observer remains xed, and the physical system is rotated.
We can also dene a passive viewpoint: the observer rotates, and, without touching
the system being studied, uses a new coordinate frame, derived from the initial frame
by the given rotation. Rotation invariance is then expressed in the following way: in
his new position (that is, using his new coordinate axes), the observer describes physical
phenomena by laws that have the same form as in the old frame. Nothing allows him
to assert that one of his positions is more fundamental than the other: it is impossible
to dene an absolute orientation in space by the study of any physical phenomenon. It
is clear that, for an isolated system, a passive rotation is equivalent to the active
rotation through an equal angle about the opposite axis.
6-b. Consequence: conservation of angular momentum
We indicated in 6-a that rotation invariance is related to symmetry properties
of the equations expressing the physical laws. Here, we shall study the case of the
Schrödinger equation, and we shall show thatthe Hamiltonian of an isolated physical
system is a scalar observable.
Consider an isolated system in the state(0). We perform an arbitrary rotation
Rat time0; the state of the system becomes:
(0)= (0) (107)
whereis the image of the rotationR. If we now let the system evolve freely from
(0), its state at the instant0+ d, according to the Schrödinger equation, will be:
(0+ d)=(0)+
d
~
(0) (108)
Now, if we had not performed the rotation, the state of the system at time0+ dwould
have been:
(0+ d)=(0)+
d
~
(0) (109)
Rotation invariance implies (cf.) that:
(0+ d)= (0+ d) (110)
whereis the same as in (107). According to the two preceding equations, this implies
that:
(0)= (0) (111)
that is:
(0)= (0) (112)
Since(0)is arbitrary, it follows thatcommutes with all rotation operators. For
this to be so, it is necessary and sucient thatcommute with the innitesimal rotation
736

ANGULAR MOMENTUM AND ROTATIONS
operators, that is, with the three components of the total angular momentumJof the
system:
[J] = 0 (113)
is therefore a scalar observable.
Rotation invariance is therefore related to the fact thatthe total angular momentum
of an isolated system is a constant of the motion: conservation of angular momentum
can be seen to be a consequence of rotation invariance.
Comments:
(i) The Hamiltonian of a non-isolated system is not, in general, a scalar. However,
if certain rotations exist that leave the system invariant [comment (i) of ], the
Hamiltonian commutes with the corresponding operators. Thus, the Hamiltonian of a
particle in a central potential commutes with the operatorLassociated with the angular
momentum of the particle with respect to the center of forces.
()For an isolated system composed of several interacting particles, the Hamiltonian
commutes with thetotalangular momentum. However, it does not generally commute
with the individual angular momentum of each particle. For the transform of a possible
motion to remain a possible motion, the rotation must be performed on the whole system,
not on only some of the particles.
6-c. Applications
We have just shown that rotation invariance means that the total angular momen-
tumJof an isolated system is a constant of the motion in the quantum mechanical sense.
It is therefore useful to determine the stationary states of such a system (eigenstates of
the Hamiltonian) which are also eigenstates ofJ
2
and. We can then choose for the
state space a standard basis , composed of eigenstates common to,J
2
and
:
=
J
2
=(+ 1)~
2
=~ (114)
. Essential rotational degeneracy
Since the Hamiltonianis a scalar observable, it commutes with+and. From
this fact, we deduce that+1and 1, which are respectively proportional
to+ and , are eigenstates ofwith the same eigenvalue as
[the argument is the same as for formula (C-48) of Chapter]. Thus it can be shown
by iteration that the(2+ 1)vectors of the standard basis characterized by the given
values ofandhave the same energy. The corresponding degeneracy of the eigenvalues
ofis called essential because it arises from rotation invariance and occurs for any
form of the Hamiltonian. Of course, in certain cases, the energy levels can present
additional degeneracies, which are called accidental. We shall see an example of this
in Chapter, .
737

COMPLEMENT B VI
. Matrix elements of observables in a standard basis
When we study a physical quantity in an isolated system, the knowledge of the
behavior of the associated observable under a rotation enables us to establish some of its
properties, without having to consider its precise form. We can predict that only some of
its matrix elements in a standard basis such aswill be dierent from zero, and
we can give the relations between them. Thus, a scalar observable has matrix elements
only between two basis vectors whose values ofare equal, as are their values of[this
results from the fact that this observable commutes withJ
2
and;cf.theorems on
commuting observables, ]. Moreover, these non-zero elements are
independent of(since the scalar observable also commutes with+and). For vector
or tensor observables, these properties are contained in theWigner-Eckart theorem, which
we shall prove later in a special case (cf.ComplementX), and which is frequently used
in the areas of physics in which phenomena are treated by quantum mechanics (atomic,
molecular, and nuclear physics, elementary particle physics, etc.).
References and suggestions for further reading:
Symmetry and conservation laws: Feynman III (1.2), Chap. 17; Schi (1.18),
Chap. 7; Messiah (1.17), Chap. XV; see also the articles by Morrisson (2.28), Feinberg
and Goldhaber (2.29), Wigner (2.30).
Relation with group theory: Messiah (1.17), Appendix D; Meijer and Bauer (2.18),
Chaps. 5 and 6; Bacry (10.31), Chap. 6; Wigner (2.23), Chaps. 14 and 15. See also Omnès
(16.13), in particular Chap. III.
738

ROTATION OF DIATOMIC MOLECULES
Complement CVI
Rotation of diatomic molecules
1 Introduction
2 Rigid rotator. Classical study
2-a Notation
2-b Motion of the rotator. Angular momentum and energy
2-c The ctitious particle associated with the rotator
3 Quantization of the rigid rotator
3-a The quantum mechanical state and observables of the rotator
3-b Eigenstates and eigenvalues of the Hamiltonian
3-c Study of the observableZ. . . . . . . . . . . . . . . . . . . .
4 Experimental evidence for the rotation of molecules
4-a Heteropolar molecules. Pure rotational spectrum
4-b Homopolar molecules. Raman rotational spectra
1. Introduction
In V, we studied the vibrations of the two nuclei of a diatomic
molecule about their equilibrium position, neglecting the rotation of these two nuclei
about their center of mass. We obtained stationary vibrational states of energies
whose wave functions()depended only on the distancebetween the nuclei.
Here, we shall adopt a complementary point of view: we shall study the rotation
of the two nuclei about their center of mass, neglecting their vibrations. That is, we
shall assume that the distancebetween them remains xed and equal to(where
represents the distance between the two nuclei in the stable equilibrium position of
the molecule; see Figure V). The wave functions of the stationary
rotational states then can depend only on the polar anglesandwhich dene the
direction of the molecular axis. We shall see that these wave functions are the spherical
harmonics()[studied in Chapter VI], and that
they correspond to a rotational energythat depends only on.
Actually, in the center of mass frame, the molecule both rotates and vibrates, and
the wave functions of its stationary states must be functions of the three variables
and. We shall show in ComplementVIIthat, to a rst approximation, these wave
functions are of the form()()and correspond to the energy+. This
result justies the approach adopted here, which consists of considering only one degree
of freedom rotational or vibrational at a time
1
. We shall begin in
the classical study of a system of two masses separated by a xed distance (rigid rotator).
The quantum mechanical treatment of this problem will then be taken up in , where
1
In ComplementVII, we shall also study the corrections introduced by the coupling between the
vibrational and rotational degrees of freedom.
739

COMPLEMENT C VIM
2
M
1
r
1
r
2
θ
φ
O
y
z
x
Figure 1: Parameters dening the position
of the rigid rotator12whose center of
mass is at the originof the reference
frame; the distances1and2are xed; only
the polar anglesandcan vary.
we shall use the results of Chapter
in , we shall describe some experimental manifestations of the rotation of diatomic
molecules (pure and Raman rotational spectra).
2. Rigid rotator. Classical study
2-a. Notation
Two particles, of mass1and 2, are separated by a xed distance. Their
center of massis chosen as the origin of a coordinate framewith respect to
which the direction of the axis connecting them is dened by means of the polar angles
and(Fig.). The distances1and 2are denoted respectively by1and2;
by denition of the center of mass:
11= 22 (1)
which allows us to write:
1
2
=
2
1
=
1+2
(2)
The moment of inertiaof the system with respect tois equal to:
= 1
2
1+2
2
2 (3)
Introducing the reduced mass:
=
12
1+2
(4)
and using (2), we can putin the form:
=
2
(5)
740

ROTATION OF DIATOMIC MOLECULES
2-b. Motion of the rotator. Angular momentum and energy
If no external force acts on the rotator, the total angular momentumof the
system with respect to the pointis a constant of the motion. The rotator therefore
rotates aboutin a plane perpendicular to the xed vector, with a constant angular
velocity. The modulus ofis related toby:
= 111+222= (6)
that is, using (5):
=
2
(7)
The rotational frequency of the system= 2is proportional to the angular
momentum and inversely proportional to the moment of inertia.
In the center of mass frame, the total energyof the system reduces to the
rotational kinetic energy:
=
1
2
2
(8)
which can also be written, using (6) and (5):
=
2
2
=
2
2
2
(9)
2-c. The ctitious particle associated with the rotator
Formulas (5), (7) and (9) show that the problem we are studying here is formally
equivalent to that of a ctitious particle of massforced to remain at a xed distance
from the point, about which it rotates with the angular velocity.is the angular
momentum of this ctitious particle with respect to.
3. Quantization of the rigid rotator
3-a. The quantum mechanical state and observables of the rotator
Sinceis xed, the parameters dening the position of the rotator (or that of
the associated ctitious particle) are the polar anglesandof Figure. The quantum
mechanical state of the rotator will then be described by a wave function() which
depends only on these two parameters.()is square-integrable; we shall assume it
to be normalized:
2
0
d
0
sind()
2
= 1 (10)
The physical interpretation of() is the following:()
2
sinddrepresents
the probability of nding the axis of the rotator pointing in the solid angle element
d = sinddabout the direction of polar anglesand.
Using Dirac notation, we associate with every square-integrable function(),
a ketof the state space:
() (11)
741

COMPLEMENT C VI
The scalar product ofand is, by denition:
=d ()() (12)
where()and()are the wave functions associated withand.
The quantum mechanical Hamiltonianof the rotator (or of the associated cti-
tious particle) can be obtained by replacing
2
in expression (9) for the classical energy
by the operatorL
2
studied in D of Chapter:
=
L
2
2
2
(13)
is an operator acting in. According to formula (D-6a) of Chapter, ifis
represented by the wave function(), is represented by:
~
2
2
2
2
2
+
1
tg
+
1
sin
2
2
2
() (14)
Other observables of interest, which we shall study later, are those which corre-
spond to the three algebraic projectionsof the segment12( are also the
coordinates of the ctitious particle):
=sincos
=sinsin (15)
=cos
The importance of these variables will be seen in 4-a. The observablescor-
responding to act in. With the kets , are associated the
functions:
sincos()
sinsin() (16)
cos()
Comment:
As we have already pointed out in the introduction, the true wave functions of the
molecule depend on . Similarly, the observables of this molecule, obtained from
the corresponding classical quantities by the quantization rules of Chapter, act on
these functions of three variables and not solely on the functions ofand. In Comple-
mentVII, we shall justify the point of view we are adopting here, namely, ignoring the
radial part of the wave functions and consideringto be a xed parameter that is equal
to[cf.formulas (14) and (16)].
3-b. Eigenstates and eigenvalues of the Hamiltonian
We determined the eigenvalues of the operatorL
2
in D of Chapter: they are
of the form(+ 1)~
2
, whereis any non-negative integer. Furthermore, we know an
742

ROTATION OF DIATOMIC MOLECULES
orthonormal system of eigenfunctions ofL
2
: the spherical harmonics(), which
constitute a basis in the space of functions that are square-integrable inand( D-
l-c-of Chapter). We shall denote bythe ket ofassociated with():
() (17)
We see from (13) that:
=
(+ 1)~
2
2
2
(18)
It is customary to set:
=
~
4
=
~
4
2
(19)
is called the rotational constant and has the dimensions of a frequency
2
. The
eigenvalues ofare thus of the form:
= (+ 1) (20)
Since, for a given value of, there exist (2+ 1) spherical harmonics(), where
= + 1 , we see that each eigenvalueis (2+ 1)-fold degenerate. Figure
represents the rst energy levels of the rotator. The separation of two adjacent levels,
and1, is equal to:
1= (+ 1)(1)= 2 (21)
and increases linearly with.
The eigenstates ofsatisfy the following orthogonality and closure relations (de-
duced from those satised by the spherical harmonics, of Chapter):
=
=0
+
=
= 1 (22)
The most general quantum state of the rotator can be expanded on the states:
()=
=0
+
=
() (23)
The component:
() = ()=d ()(;) (24)
evolves in time in accordance with the equation:
() =(0) e
~
(25)
2
The speed of lightis sometimes placed in the denominator of the right-hand side of (19).then
has the dimensions of an inverse length and is expressed in cm
1
(in the system).
743

COMPLEMENT C VIl = 5
l = 4
10 Bh
8 Bh
6 Bh
4 Bh
2 Bh
l = 3
l = 2
l = 1
l = 0
Figure 2: First levels of the rigid rotator, of
energies:
= (+ 1)
(with= 012...). Each levelfor which
1is separated from the next lower level
by an energy2.
3-c. Study of the observableZ
Earlier, we introduced the observableswhich correspond to the projections
onto the three axes of the segment12. In this section, we shall study the evolution
of the mean values of these observables and compare the results obtained with those
predicted by classical mechanics. We shall conne ourselves to the calculation of()
since()and ()have analogous properties.
A Bohr frequency( )can appear in the function(), ifhas a non-
zero matrix element between a state of energyand a state of energy
. The rst problem is therefore to nd the non-zero matrix elements of. To solve
this problem, we shall use the following relation, which can be established by using the
mathematical properties of spherical harmonics [ComplementVI, formula (35)]:
cos () =
2 2
4
2
1
1() +
(+ 1)
2 24(+ 1)
2
1
+1() (26)
From this, we deduce, using (16), (17) and (22):
= 1
2 2
4
2
1
+ +1
(+ 1)
2 24(+ 1)
2
1
(27)
Comment:
According to (27), the selection rules satised byare:=1= 0. It can be
shown that forandwe have:=1,=1. Since the energies depend only
on, the Bohr frequencies are the same for and .
The operatorcan therefore connect only states belonging to two adjacent levels
of Figure ).
The only Bohr frequencies which appear in the evolution of()are thus of the form:
1=
1
= 2 (28)
They form a series of equidistant frequencies, separated by the interval2(Fig.).
744

ROTATION OF DIATOMIC MOLECULES2B0 4 B 6B 8B 10B
44220 5 3311
v
Figure 3: Frequencies appearing in the evolution of the mean value of the observable
. Because of the selection rule=1, only the Bohr frequencies2(with1),
associated with two adjacent levelsand 1in Figure, are observed.
The mean value()can evolve only at a well-dened series of frequencies, This
is unlike the classical case, in which the frequency of rotationof the rotator can take
on any value.
According to (27), if the system is in a stationary state()is always zero,
even for large. To obtain a quantum mechanical state in whichbehaves like the
corresponding classical variable, one must superpose a large number of states. If
we assume that the state of the system is given by formula (23), and that the numbers
(0)
2
have values which vary withas is shown in Figure, the most probable value
of,M, is very large; the spreadof the values ofis also very large in absolute value,
but very small in relative value:
1 (29a)

1 (29b)0
c
l, m
(0)
2
Δl
l
M
l
Figure 4: Square of the moduli
of the expansion coecients of a
quasi-classical state on the sta-
tionary statesof the rigid ro-
tator. The spreadis large; how-
ever, since the most probable value
of,M, is very large, we have
M 1, and the relative accu-
racy with respect tois very good.
It can then be shown that, in such a state:
L
2
L
2
(+ 1)~
2 2
~
2
(30)
In addition, the Bohr frequencies that appear in()are then all very close (in relative
value) to:
= 2 (31)
745

COMPLEMENT C VI
Eliminatingbetween (30) and (31), we obtain, according to (19):
2L
~
=
L
2
(32)
which is the equivalent of the classical relation (6).
Comment:
It is interesting to study in greater detail the motion of the wave packet corresponding to
the state of Figure. It is represented by a function ofandand can be considered to
evolve on the sphere of unit radius. The preceding discussion shows that this wave packet
rotates on the sphere with the average frequency. Because of the spreadofand
the corresponding spread2of the Bohr frequencies which enter into, and
, the wave packet becomes distorted over time. This distortion becomes appreciable
after a time of the order of:
1
2
(33)
Since the spread ofis small in relative value, we have:

1 (34)
The distortion of the wave packet is therefore slow, relative to its rotation.
In fact, the Bohr frequencies of the system form adiscreteseries ofequidistantfre-
quencies, separated by the interval2. The motion which results from the superposition
of these frequencies is therefore periodic, of period:
=
1
2
(35)
with, according to (29a):
1
(36)
The distortion of the wave packet is therefore not irreversible; it follows a cycle which
is repeated periodically. This is related to the fact that the wave packet evolves on the
unit sphere, which is a bounded surface. This behavior should be compared with that of
free wave packets (irreversible spreading; ComplementI) and that of the quasi-classical
states of the harmonic oscillator (oscillation without distortion; ComplementV).
4. Experimental evidence for the rotation of molecules
4-a. Heteropolar molecules. Pure rotational spectrum
. Description of the spectrum
If the molecule is composed of two dierent atoms, the electrons are attracted
by the more electronegative atom, and the molecule generally has apermanentelectric
dipole moment0, directed along the molecular axis. The projection of the electric dipole
moment onto becomes, in quantum mechanics, an observable proportional to. We
have seen that()evolves at all the Bohr frequencies2(= 123)shown in
746

ROTATION OF DIATOMIC MOLECULES
Figure. Thus we see how the molecule is coupled to the electromagnetic eld and can
absorb or emit radiation polarized parallel
3
to Oz, on the condition that the frequency
of this radiation is equal to one of the Bohr frequencies2.
The corresponding absorption or emission spectrum of the molecule is called the
pure rotational spectrum. It is composed of a series of equidistant lines, the frequency
separation between two successive lines being equal to2, as in Figure. The absorption
(or emission) of the line of frequency2corresponds to the passage of the molecule from
the level1to the level(or from the levelto the level1), at the same time that
a photon of frequency2is absorbed (or emitted). Figure
schematically [(5-a) represents the absorption and (5-b), the emission of a photon of
frequency2].
The pure rotational spectra of diatomic molecules therefore provide direct experi-
mental proof of the quantization of the observableL
2
.E
l
2Bl
a b
E
l – 1
Figure 5: Schematic representation
of the passage of the molecule from
a given rotational level to the neigh-
boring level with absorption (g. a)
or emission (g. b) of a photon.
. Comparison with the pure vibrational spectrum
In of ComplementV, we studied the pure vibrational spectrum of a
heteropolar diatomic molecule. It is interesting to compare this spectrum with the pure
rotational spectrum we are studying here.
(i) The rotational frequencies of a diatomic molecule are generally much lower than
the vibrational frequencies. The separation2 of two rotational lines varies between
a few tenths of a cm
1
and a few dozen cm
1
. For small values of, the rotational
frequencies2therefore correspond to wavelengths of the order of a centimeter or a
millimeter. Takingas an example, the separation2 is equal to 20.8 cm
1
, while
the vibration frequency, which corresponds to 2 886 cm
1
, is more than a hundred times
greater.
Pure rotational spectra therefore fall in the very far infrared or the microwave
domain.
Comment:
As we shall show in ComplementVII, the rotation of molecules is also responsible
for a ne structure of vibrational spectra (vibration-rotation spectra).2can then
be measured in a domain of wavelengths which is no longer that of microwaves.
The same comment applies to the Raman rotational eect ( 4-b below), which
appears as a rotational structure of an optical line.
3
If we study the motion of ()and (), we see that the molecule can also absorb or emit
radiation polarized parallel toor.
747

COMPLEMENT C VI
()The pure vibrational spectrum studied in ComplementVhas onlyone
vibrational line. This is due to the fact that the various vibrational levels are equidistant
(if the anharmonicity of the potential is neglected) and, consequently, only one Bohr
frequency appears in the dipole motion (selection rule=1). On the other hand,
the pure rotational spectrum consists of aseries of equidistant lines.
()We indicated in ComplementVthat the permanent electric dipole moment
of the heteropolar molecule can be expanded in powers ofin the neighborhood of
the stable equilibrium position of the molecule:
() =0+1( ) + (37)
For the pure vibrational spectrum to appear,()must vary with:1must therefore
be dierent from zero. On the other hand, even ifremains xed and equal to, the
rotation of the molecule modulates the projection of the electric dipole onto one of the
axes, provided that0is dierent from zero. Thus we see that the study of the intensity
of vibrational and rotational lines permits the separate measurement of the coecients
1and0of (37).
. Applications
The study of pure rotational spectra has some interesting applications; we shall
mention three examples.
(i) Measurement of the separation2of two neighboring lines yields the moment
of inertiaof the molecule, according to (19). If we know1and2, we can deduce,
the separation of the two nuclei in the stable equilibrium position of the molecule [is
the abscissa of the minimum of the curve()of Figure V]. Recall
that measurement of the vibrational frequency yields the curvature of()at=.
()Consider two diatomic molecules and , in which two isotopes
and of the same element are bound to the same atom. Since the distances
between the nuclei are equal in the two molecules, measurement of the ratio of the
corresponding coecients, which can be performed with great accuracy, yields the
ratio of the masses of the two isotopesand.
One could also compare the vibrational frequencies of the two molecules, but it is
preferable to use the rotational spectrum, since the rotational frequencies vary with1
[formula (19)], while the vibrational frequencies vary with1
[formula (5) ofV].
()In the study of a sample containing a great number of identical molecules,
the relative intensities of the lines (in absorption or emission) of the pure rotational
spectrum yields information about the distribution of the molecules among the various
levels. Unlike what happens in the case of the vibrational spectrum, transitions
between two given adjacent levels (arrows of Figure) occur at a particular frequency,
which is characteristic of these two levels. Thus, the intensity of the corresponding line
depends on the number of molecules that are found in the two levels.
This information can be used to determine the temperature of a medium
4
. If
thermodynamic equilibrium has been attained, we know that the probability that a
given molecule is in a particular state of energyis proportional toe ; since the
degeneracy of the rotational levelis (2+ 1), the total probabilityPof nding the
4
Actually, the vibration-rotation or Raman rotational spectra are more often used, since they fall
into more convenient frequency ranges than does the pure rotational spectrum.
748

ROTATION OF DIATOMIC MOLECULES0
0.1
0.2
0.3
1 2 3 4 5 6 7
10
hB
kT
1
=
l

l
Figure 6: PopulationPof the various rotational levelsat thermodynamic equilibrium.
The fact thatPbegins by increasing witharises from the (2+ 1)-fold degeneracy of
the levels. Whenbecomes suciently large, the Boltzmann factore prevails
and is responsible for the decrease inP.
molecule being considered in one of the states of the level(the population of the
level) is:
P=
1
(2+ 1)e
=
1
(2+ 1)e
(+1)
(38)
where:
=
=0
(2+ 1)e
(+1)
(39)
is the associated partition function. If we are studying a system containing a large
number of molecules whose interactions can be neglected,Pgives the fraction of them
whose energy is.
At ordinary temperatures,is much smaller than, so that several rotational
levels are populated. Note that the presence of the factor (2+ 1) means that it is not
the lowest levels that are the most populated: Figure Pas a
function offor a temperaturesuch that is of the order of110. Recall that
the vibrational levels, on the other hand, are non-degenerate, and that their separation
is much greater than; consequently, when the distribution of the molecules between
the two rotational levels is that of Figure, they are practically all in the vibrational
ground state (= 0).
4-b. Homopolar molecules. Raman rotational spectra
As we pointed out in of ComplementV, a homopolar molecule (that is,
a molecule composed of two identical atoms) has no permanent electric dipole moment:
in formula (37), we have0=1== 0. The vibration and rotation of the molecule
749

COMPLEMENT C VI
induce no coupling with the electromagnetic eld, and the molecule is consequently
inactive in the near infrared (vibration) and the microwave (rotation) domains. Like
the vibration (cf. ofV), the rotation of the molecule can, however, be observed
via the inelastic scattering of light (the Raman eect).
. The Raman rotational eect. Classical treatment
We have already introduced, in ComplementV, the susceptibilityof a molecule
in the optical domain: an incident light wave, whose electric eld isEe

, sets the elec-
trons of the molecule in forced motion and causes an electric dipoleDe

, oscillating at
the same frequency as the incident wave, to appear.is the coecient of proportionality
betweenDandE. IfEis parallel to the axis of the molecule,depends on the distance
between the two nuclei: when the molecule vibrates,vibrates at the same frequency.
This is the origin of the Raman vibrational eect described in ofV.
Actually, a diatomic molecule is an anisotropic system. When the angle between
the molecular axis andEis arbitrary,Dis not generally parallel toE: the relation
betweenDandEis tensorial (is the susceptibility tensor).Dis parallel toEin the
two following simple cases:Eparallel to the molecular axis (we then have=
), and
Eperpendicular to this axis (=). In the general case, we choose theaxis along
the electric eldEof the light wave (assumed to be polarized); we consider a molecule
whose axis points in the direction of polar anglesandand calculate the component
along of the dipole induced byEon this molecule.Ecan be decomposed into a
componentE
, parallel to the molecular axis, and a componentE, perpendicular to
12and contained in the plane formed byand12(Fig.). The dipole induced
on the molecule by the eldEcos is then equal to:
D= (
E
+E) cos (40)E
E
⊥
E
//
D
//θ
D
⊥
D
O
M
1
M
2
z
Figure 7: Decomposition of the electric eld
Einto a componentE
parallel to the molec-
ular axis and a componentEperpendic-
ular to this axis. These elds induce elec-
tric dipoles
E
andEcollinear with
the corresponding elds. However, since

and have dierent values (the molecule
is anisotropic), the induced electrical dipole
D=
E
+Eis not collinear withE.
Its projection ontocan be calculated immediately:
= (cos
E
+ sin E) cos
= (cos
2
+ sin
2
)cos (41)
= [+ (
) cos
2
]cos
750

ROTATION OF DIATOMIC MOLECULES
We see thatdepends on, since
andare not equal (the molecule is anisotropic).
To see what happens when the molecule rotates, we shall begin by reasoning clas-
sically. The fact that the molecule rotates at the frequency2means thatcos
oscillates at the same frequency:
cos=cos( ) (42)
whereanddepend on the initial conditions and the orientation of the angular mo-
mentum (which is xed). Thus we see that the term incos
2
of (41) gives rise to
components ofthat oscillate at frequencies of(2)2, in addition to the com-
ponent that varies at the frequency2. The fact that the rotation of the molecule
at the frequency2modulates its polarizability at twice its frequency is easy to un-
derstand: after half a rotation, performed in half a period, the molecule returns to the
same geometrical position with respect to the incident light wave. The light re-emitted
with a polarization parallel tois that which is radiated by. It has an unshifted
line of frequency2(Rayleigh line), as well as two shifted lines, one on each side of
the Rayleigh line, of frequencies(2)2(Raman-Stokes line) and( + 2)2
(Raman-anti-Stokes line).
. Quantum mechanical selection rules. Form of the Raman spectrum
Quantum mechanically, Raman scattering corresponds to an inelastic scattering
process in which the molecule goes from levelto level, while the energy~of the
photon becomes~ + (the total energy of the system is conserved during this
process).
The quantum theory of the Raman eect (which we shall not discuss here) indicates
that the probability of such a process involves the matrix elements of (
) cos
2
+
between the initial state() and the nal state() of the molecule:
d ()(
) cos
2
+ () (43)
It can be shown, using the properties of the spherical harmonics, that such a matrix
element is dierent from zero only if
5
:
= 0+22 (44)
There is only one Rayleigh line (which corresponds to=). Since the rotational
levels are not equidistant, there are several Raman-anti-Stokes lines (which correspond
to= 2), of frequencies:

2
+
+2
=

2
+ 4 +
3
2
with= 012
(45)
and several Raman-Stokes lines (which correspond to=+ 2), of frequencies:

2
+
+2
=

2
4 +
3
2
with= 012
(46)
5
The integral (43) is also zero if=. If we consider light re-emitted with a dierent polarization
state from that of the incident light wave, we obtain the following selection rules for:= 012.
751

COMPLEMENT C VI4B
3→5 2→4 1→3 0→2 l→l′2→0 3→1 4→2 5→3
6B
Rayleigh line
Raman-Stokes
lines
Raman-anti-Stokes
lines
6B 4B
v
Figure 8: Raman rotational spectrum of a molecule. This molecule, initially in the
rotational level, inelastically scatters an incident photon of energy~. After the
scattering, the molecule has moved to the rotational level, and the energy of the
photon is~ + (conservation of energy).
If=, the scattered photon has the same frequency= 2as the incident photon;
this yields the Rayleigh line. But it is also possible to have=2; if=+ 2, the
frequency of the scattered photon is lower (Stokes scattering); if= 2, it is higher
(anti-Stokes scattering). Since the rotational levelsare not equidistant (cf. Fig.),
there are as many Stokes or anti-Stokes lines as there are values of. These lines are
labeled in the gure by (with= 2).
The form of the Raman rotational spectrum is shown in Figure. The Stokes and
anti-Stokes lines occur symmetrically with respect to the Rayleigh line. The separation
between two adjacent Stokes (or anti-Stokes) lines is equal to4, that is, totwice
the separation which would be found between two adjacent lines of the pure rotational
spectrum if it existed. Moreover, since the vibrational frequency is much larger than
, the Raman-Stokes and anti-Stokes vibrational lines are situated much further to the
left and to the right of the Rayleigh line than the rotational Raman lines and hence do
not appear in the gure (these lines also have rotational structures similar to that of
Figure).
Comments:
(i) Consider a wave packet like those studied in 3-c, that is, one for which the
values ofare grouped about a very large integer(Fig.). According to (45)
and (46), the frequencies of the various Stokes and anti-Stokes lines will be very
close (in relative value) to:

2
4 (47)
that is, according to (31):

2
2 (48)
752

ROTATION OF DIATOMIC MOLECULES
where is the average rotational frequency of the molecule. Thus, the quantum
mechanical treatment, at the classical limit, yields the results of 4-b-.
()In Raman rotational spectra, the Stokes and anti-Stokes lines appear with
comparable intensities since levels of largehave large populations, asis much
smaller than. This is necessary for the observation of anti-Stokes lines, for
which the initial state of the molecule must be at least= 2. However, the
anti-Stokes vibrational line has a much smaller intensity than the Stokes line. The
vibrational energy is much larger than; the population of the vibrational ground
state= 0is much larger than the others, and Stokes processes= 0 = 1
are much more frequent than anti-Stokes processes= 1 = 0.
()The Raman rotational eect also exists for heteropolar molecules.
References and suggestions for further reading:
Karplus and Porter (12.1), 7.4; Herzberg (12.4), Vol. I, Chap. III, 1 and 2; Landau
and Lifshitz (1.19), Chaps. XI and XIII; Townes and Schawlow (12.10), Chaps. 1 to
4.
753

ANGULAR MOMENTUM OF STATIONARY STATES OF A TWO-DIMENSIONAL HARMONIC OSCILLATOR
Complement DVI
Angular momentum of stationary states of a two-dimensional
harmonic oscillator
1 Introduction
1-a Review of the classical problem
1-b The problem in quantum mechanics
2 Classication of the stationary states by the quantum num-
bers and . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-a Energies; stationary states
2-b does not constitue a C.S.C.O. in. . . . . . . . . . .
3 Classication of the stationary states in terms of their an-
gular momenta
3-a Signicance and properties of the operator. . . . . . . . .
3-b Right and left circular quanta
3-c Stationary states of well-dened angular momentum
3-d Wave functions associated with the eigenstates common to
and . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Quasi-classical states
4-a Denition of the states and . . . . . . . . . .
4-b Mean values and root mean square deviations of the various
observables
In this complement, we shall be concerned with the quantum mechanical properties
of a two-dimensional harmonic oscillator. The quantum mechanical problem is exactly
soluble and does not involve complicated calculations. Furthermore, this subject provides
an opportunity to study a simple application of the properties of the orbital angular
momentumL, since, as we shall see, the stationary states of such an oscillator can be
classied with respect to the possible values of the observable. In addition, the results
obtained will be useful in the subsequent ComplementVI.
1. Introduction
1-a. Review of the classical problem
A physical particle always moves in three-dimensional space. However, if its po-
tential energy depends only onand, the problem can be treated in two dimensions.
We shall assume here that this potential energy can be written:
() =
2
2
(
2
+
2
) (1)
whereis the mass of the particle andis a constant. The classical Hamiltonian of the
system is then:
= + (2)
755

COMPLEMENT D VI
with:
=
1
2
(
2
+
2
) +
1
2
2
(
2
+
2
)
=
1
2
2
(3)
where,,are the three components of the momentumpof the particle.is a
two-dimensional harmonic oscillator Hamiltonian.
The equations of motion can easily be integrated to yield:
() =0
() =
0
+0
(4)
() = cos( )
() = sin( )
(5)
() =cos( )
() = sin( )
(6)
where0,0,,,,are constants which depend on the initial conditions (we
assume and to be positive).D C
O
– y
M
A B y
M
y
– x
M
+ x
M
x
Figure 1: Projection of the classical trajectory of a particle in a two-dimensional har-
monic potential onto theplane; we obtain an ellipse inscribed in the rectangleABCD.
We see that the projection of the particle ontodescribes a uniform motion with
a velocity of0. The projection onto theplane describes an ellipse inscribed in
the rectangleABCDof Figure. The direction the particle takes on this ellipse depends
on the phase dierence . When =, the ellipse reduces to the lineAC.
When is betweenand 0, the particle moves clockwise on the ellipse (left-
handed motion), with the axes of the ellipse parallel toand for = 2.
When = 0, the ellipse reduces to the lineBD. Finally, when is between 0
756

ANGULAR MOMENTUM OF STATIONARY STATES OF A TWO-DIMENSIONAL HARMONIC OSCILLATOR
and, the particle moves counterclockwise on the ellipse (right-handed motion), with
the axes parallel toand for = 2. Note that the ellipse reduces to a
circle if = 2and =.
It is easy to determine several constants of the motion related to the projection of
the motion onto theplane:
the total energy, which, according to (3), (5), (6), is equal to:
=
1
2
2
(
2
+
2
) (7)
the energies:
=
1
2
22
(8a)
=
1
2
22
(8b)
of the projections of the motion ontoand;
the component of the orbital angular momentumof the particle along:
= (9)
which, according to (5) and (6), is equal to:
= sin( ) (10)
We see thatis positive or negative depending on whether the motion is counterclock-
wise(0 )or clockwise( 0).is zero for the two rectilinear
motions ( = and = 0). Finally, for a motion at a given energy, that
is, according to (7), for a xed value of
2
+
2
, is maximal when = 2
and the product is maximal, which implies=. Of all motions at a given
energy, it is the counterclockwise (clockwise) motion which corresponds to the maximal
(minimal) algebraic value of.
1-b. The problem in quantum mechanics
The quantization rules of Chapter ,,from,
,. The stationary statesof the particle are given by:
= (+)= (11)
with:
=
2
+
2
2
+
1
2
2
(
2
+
2
) (12a)
=
2
2
(12b)
According to the results of ComplementI, we know that we can choose a basis of
eigenstates ofcomposed of vectors of the form:
= (13)
757

COMPLEMENT D VI
where is an eigenvector ofin the state spaceassociated with the variables
and:
= (14)
and is an eigenvector ofin the spaceassociated with the variable:
= (15)
The total energy associated with the state (13) is then:
= + (16)
Now, equation (15), which in fact describes the stationary states of a free particle
in a one-dimensional problem, can be solved immediately; it yields:
=
1
2~
e
~
(17)
(whereis an arbitrary real constant), with:
=
2
2
(18)
The problem therefore reduces to the determination of the solutions of equation (14),
that is, the energies and stationary states of a two-dimensional harmonic oscillator. This
is the problem we shall now try to solve.
We shall see that the eigenvaluesof are degenerate:alone does not
constitute a C.S.C.O. in. We must therefore add one or several other observables
to in order to construct a C.S.C.O. In fact, we nd in quantum mechanics the
same constants of the motion as in classical mechanics:and, the energies of the
projection of the motion ontoand; and, the component alongof the orbital
angular momentumL. Sincecommutes with neithernor, we shall see that a
C.S.C.O. can be formed of,and () or ofand().
Comments:
(i) Formula (18) indicates that the eigenvaluesofare all two-fold degenerate in
the space. Furthermore, the degeneracy in= of the eigenvalues (16) of the
total Hamiltonianis not due solely to the degeneracy ofin and ofin:
two eigenvectors ofof the form (13) can have the same total energywithout their
corresponding values of(and of) being equal.
()commutes with the component ofL, but not withand. This results
from the fact that the potential energy written in (1) is rotation-invariant only about
. Moreover, of the three operators,and, only one,, acts only in.
In the study of the two-dimensional harmonic oscillator, therefore, we shall use only
the observable. In ComplementVII, we shall study the isotropic three-dimensional
harmonic oscillator, whose potential energy is invariant with respect to any rotation
about an axis which passes through the origin; we shall see that all the components ofL
then commute with the Hamiltonian.
758

ANGULAR MOMENTUM OF STATIONARY STATES OF A TWO-DIMENSIONAL HARMONIC OSCILLATOR
2. Classication of the stationary states by the quantum numbersand
2-a. Energies; stationary states
To obtain the solutions of the eigenvalue equation (14), note that, can be
written:
= + (19)
where and are both Hamiltonians of one-dimensional harmonic oscillators:
=
2
2
+
1
2
22
=
2
2
+
1
2
22
(20)
We know the eigenstates ofinand the eigenstatesofin. Their
energies are, respectively,= (+ 12)~and= (+ 12)~(whereand
are positive integers or zero). The eigenstates ofcan thus be chosen in the form:
= (21)
where the corresponding energyis given by:
= +
1
2
~+ +
1
2
~
= (++ 1)~ (22)
According to the properties of the one-dimensional harmonic oscillator,is non-
degenerate in, andin. Consequently, a vector of, which is unique
to within a constant factor, corresponds to a pair:and form a C.S.C.O.
in.
It will prove convenient to use the operatorsand(destruction operators of a
quantum, relative toand respectively), dened by:
=
1
2
+
~
=
1
2
+
~
(23)
with:
=
~
(24)
Sinceandact in dierent spaces,and, the only non-zero commutators between
the four operators,,,, are:
= = 1 (25)
759

COMPLEMENT D VI
The operators(the number of quanta relative to theaxis) and(the number
of quanta relative to theaxis) are given by:
=
= (26)
which enables us to writein the form:
= + = (+ + 1)~ (27)
We have, obviously:
=
= (28)
The ground state00is given by:
00= =0 =0 (29)
The state dened by (21) can be obtained from00by the successive appli-
cation of the operatorsand:
=
1
!!
()() 00 (30)
The corresponding wave function is the product of()and ()[cf.Comple-
mentV, formula (35)]:
() =
(2)
+
()!()!
e
2
(
2
+
2
)2
()() (31)
where theare the Hermite polynomials (ComplementV, ).
2-b. does not constitue a C.S.C.O. in
We see from (22) that the eigenvalues ofare of the form:
= = (+ 1)~ (32)
where:
=+ (33)
is a positive integer or zero. To each value of the energy correspond the various orthogonal
eigenvectors:
= =0 =1=1 =0 = (34)
Since there are(+1)of these vectors, the eigenvalueis(+1)-fold degenerate in.
alone does not, therefore, constitute a C.S.C.O. On the other hand, we have seen
that is a C.S.C.O.; this is also, obviously, true ofet .
760

ANGULAR MOMENTUM OF STATIONARY STATES OF A TWO-DIMENSIONAL HARMONIC OSCILLATOR
3. Classication of the stationary states in terms of their angular momenta
3-a. Signicance and properties of the operator
In the preceding section, we identied the stationary states by the quantum num-
bersand. But the and axes do not enjoy a privileged position in this
problem. Since the potential energy is invariant under rotation about, we could just
as well have chosen another system of orthogonal axesand in the plane;
we would have then obtained stationary states dierent from the preceding ones.
Therefore, in order to take better advantage of the symmetry of the problem, we
shall now consider the componentof the angular momentum, dened by:
= (35)
Expressingand in terms ofand, andand in terms ofand, we
get:
=~ (36)
Now, the expression forin terms of the same operators is:
= + + 1~ (37)
Since:
+ = = 0
+ = + = 0 (38)
we nd that:
[ ] = 0 (39)
We shall therefore look for a basis of eigenvectors common toand.
3-b. Right and left circular quanta
We introduce the operatorsanddened by:
=
1
2
( )
=
1
2
(+) (40)
We see from this denition that the action of(or) on yields a state which is
a linear combination of 1 and 1, that is, a stationary state which has
one less energy quantum~. Similarly, the action of(or) on yields another
stationary state which has one more energy quantum. In fact, we shall see that(or
) is similar to(or), and thatandcan be interpreted as being destruction
operators of a right and left circular quantum respectively.
761

COMPLEMENT D VI
First of all, using (40) and (25), it is simple to verify that the only non-zero
commutators between the four operators,,,are:
[ ] = [] = 1 (41)
These relations are indeed analogous to (25). Moreover,can be written, in terms of
these operators, in a way that is similar to (37); since:
=
1
2
+ +
=
1
2
+ + (42)
we have:
= + + 1~ (43)
In addition, using (36), we see that:
=~ (44)
If we introduce the operatorsand(the number of right and left circular quanta):
=
= (45)
formulas (43) and (44) become:
= (++ 1)~
=~( ) (46)
Thus, while maintainingin a form as simple as (27), we have simplied that of.
3-c. Stationary states of well-dened angular momentum
Using the operatorsand, we can now go through the same arguments we
used forand. It follows that the spectra ofandare composed of all positive
integers and zero. In addition, specifying a pairof such integers determines
uniquely (to within a constant factor) the eigenvector common toand, associated
with these eigenvalues, which is written:
=
1
()!()!
()() 00 (47)
andtherefore form a C.S.C.O. in. Thus we see, by using (46), that is
also an eigenvector ofand of, with the eigenvalues(+ 1)~and~, where
andare given by:
=+
= (48)
762

ANGULAR MOMENTUM OF STATIONARY STATES OF A TWO-DIMENSIONAL HARMONIC OSCILLATOR
Equations (48) enable us to understand the origin of the name of right or left circular
quanta. The action of the operatoron yields a state with one more quantum,
to which, sincehas increased by one, an additional angular momentum+~must be
attributed (this corresponds to a counterclockwise rotation about). Similarly,
yields a state with one more quantum, of angular momentum~(clockwise rotation).
Sinceandare positive integers (or zero), our results are in agreement with
those of the preceding section: the eigenvalues ofare of the form(+ 1)~, where
is a positive integer or zero; their degree of degeneracy is(+ 1)since, for xed, we
can have:
= ;= 0
= 1 ;= 1
.
.
.
= 0 ; =
(49)
Furthermore, we see that the eigenvalues ofare of the form~, whereis a
positive or negative integer or zero, which is the result that was established for the general
case in Chapter. In addition, table (49) tells us which values ofare associated with
a given value of. For example, for the ground state, we have= 0, and therefore,
necessarily,= 0; for the rst excited state, we can have= 1and= 0, or= 0
and= 1, which yields either= +1or=1. In general, formulas (48) and (49)
show that, for a given energy level(+ 1)~, the possible values ofare:
= 2 4 + 2 (50)
It follows that, to a pair of values ofand, there corresponds a single vector (to within
a constant factor):
=
+
2
=
2
andtherefore form a C.S.C.O. in.
Comment:
For a given value of the total energy (labeled by), the states= =0and
=0 =correspond to the maximal(~)and minimal(~)values of.
These states therefore recall the classical right and left circular motions associated
with a given value of the total energy, for whichtakes on its maximal and
minimal values (see ).
3-d. Wave functions associated with the eigenstates common to and
To conserve the symmetry of the problem with respect to rotation about, we shall
use polar coordinates, setting:
=cos 0
=sin 0 2 (51)
763

COMPLEMENT D VI
Now, what is the action of the operatorsandon a function ofand? We shall begin
by determining their action on a function ofand. Knowing that ofandand therefore
that of(and, by analogy, that of), we can use (40), which yields:
=
1
2
( ) +
1
(52)
According to the rules for dierentiating functions of several variables, we then obtain:
=
e
2
+
1
(53)
Similarly:
=
e
2
1
(54)
and:
=
e
2
+
1
+
=
e
2
1
+ (55)
To calculate the wave functions(), simply apply the dierential operators rep-
resentingandto the function00(), which is, according to (31):
00() =
e
22
2
(56)
Now it can be seen from (54) and (55) that the action of(or of) on a function of the form
e ()is given by:
e ()=
e
(+1)
2
+()
1d
d
e ()=
e
( 1)
2
()
1d
d
(57)
Through the repeated application of these relations to (56), we see that the-dependence of
()is simply given by:e
( )
. This is a general result, established in Chapter:
the-dependence of an eigenfunction ofof eigenvalue~is e.
If, in (57), we choose() =e
22
2
, then:
e e
22
2
=e
(+1) +1
e
22
2
(58)
Applying the operatorto the function00()times, we obtain:
0() =
()!
e()e
22
2
(59)
An analogous calculation yields:
0() =
()!
e ()e
22
2
(60)
764

ANGULAR MOMENTUM OF STATIONARY STATES OF A TWO-DIMENSIONAL HARMONIC OSCILLATOR
These wave functions are normalized. For a given energy level(+ 1)~, the wave functions
(59) and (60) correspond to the limiting values+and of the quantum number. Their-
dependence is particularly simple: their modulus reaches a maximum for=
. Therefore
(as in the case of a one-dimensional harmonic oscillator), the spatial spread of these wave
functions increases with the energy(+ 1)~with which they are associated.
In the same way, application of the operators(or) to (59) and (60) permits the
construction of the functions()for anyand. The results obtained for the rst
excited levels are given in Table I.
= 0 = 0 00() =
e
22
2
= 1
= 1 10() =
e
22
2
e
=101() =
e
22
2
e
= 2
= 2 20() =
2
()
2
e
22
2
e
2
= 0 11() =
()
2
1e
22
2
=202() =
2
()
2
e
22
2
e
2
Table I:Eigenfunctions common to the Hamiltonian and the observable, for the rst
levels of the two-dimensional harmonic oscillator.
Comment:
The functions 0()given in (59) are proportional toe
22
2
(e). More
generally, all their linear combinations are of the form:
() = e
22
2
(e) (61)
(whereis an arbitrary function of one variable) and are eigenfunctions ofwith the
eigenvalue zero. As expected, it can easily be shown from (55) that:
() = 0 (62)
Similarly, the subspace of eigenfunctions ofof eigenvalue zero is composed of functions
of the form:
() = e
22
2
(e) (63)
4. Quasi-classical states
Using the properties of the one-dimensional harmonic oscillator, we can easily calculate
the time evolution of the state vector and the mean values of the various observables
765

COMPLEMENT D VI
of the two-dimensional oscillator. For example, it is not dicult to show that in the
mean values()and (), as well as()and (), only the Bohr frequency
appears. Moreover, it can be shown that these mean values exactly obey the classical
equations of motion. In this section, we shall be concerned with the properties and
evolution of the quasi-classical states of the two-dimensional harmonic oscillator.
4-a. Denition of the states and
To construct a quasi-classical state of the two-dimensional harmonic oscillator, we
can base our reasoning on the one-dimensional oscillator (cf.ComplementV). Recall
that, in a quasi-classical state associated with a given classical motion, the mean values
()and()coincide at each instant with()and(). Similarly, the mean value
of the Hamiltonianis equal (to within a half-quantum~2) to the classical energy.
We showed in ComplementVthat, at any time, the quasi-classical states are eigenstates
of the destruction operatorand can be written:
= () (64)
whereis the eigenvalue of, and:
() =
!
e
2
2
(65)
In the case which concerns us here, we can use the rules of the tensor product to
obtain the quasi-classical states in the form:
= =
=0 =0
()() (66)
with:
=
= (67)
We are then sure that,, ,,, are the same as the corresponding
classical quantities. Now, returning to denition (40) and using (67), we see that:
=
= (68)
with:
=
1
2
( )
=
1
2
(+) (69)
Therefore, the state is also an eigenvector ofandwith the eigenvalues given
in (69). We shall denote by the eigenvector common toandassociated
766

ANGULAR MOMENTUM OF STATIONARY STATES OF A TWO-DIMENSIONAL HARMONIC OSCILLATOR
with the eigenvaluesand. It is easy to show that the expansion ofon the
basis has the same form as that of on the basis:
=
=0=0
()() (70)
where the coecientsare given by (65). It follows from (68) and (69) that:
= =
2
=
+
2
(71)
Because of the properties of the states(cf.ComplementV, ), we see
that if:
(0)= = (72)
the state vector at the instantwill be:
()= e e e
= e e e (73)
4-b. Mean values and root mean square deviations of the various observables
We set:
= e
= e (74)
Using formulas (93) of ComplementV, we obtain:
() =
2
cos( )
() =
2
cos( )
(75)
() =
2
sin( )
() =
2
sin( )
(76)
Comparing (75) and (76) with (5) and (6), we see that:
=
2
e
=
2
e (77)
where,,,are the parameters dening the classical motion which the state
best reproduces.
767

COMPLEMENT D VI
Also:
=
2
=
2
(78)
and:
=
2
=
1
2
2
+
2
+
=
2
=
1
2
2
+
2
(79)
that is, according to (46):
=~
2
+
2
+ 1=~
2
+
2
+ 1 (80)
and:
= 2~ sin( ) =~
2 2
(81)
According to (77), is the same as the classical value of[formula (10)].
Now let us consider the root mean square deviations of the position and momentum
and then of the energy and angular momentum in a state. Directly applying the
results of ComplementV, we obtain:
= =
1
2
= =
2
(82)
The root mean square deviations of the position and momentum are independent of
and; ifand are much greater than 1, the position and momentum of the
oscillator have a very small spread about, and , .
Finally, let us calculate the root mean square deviations for the energy and
for the angular momentum. As in ComplementV:
=
=
=
= (83)
But the Hamiltonianinvolves= +, andis proportional to . We
must now calculate, for example:
()
2
=(+)
2
(+)
2
= ()
2
+ ()
2
+ 2[ ] (84)
According to (66), the state of the system is a tensor product, which means that the
observablesand are not correlated:
= (85)
768

ANGULAR MOMENTUM OF STATIONARY STATES OF A TWO-DIMENSIONAL HARMONIC OSCILLATOR
It follows that:
()
2
= ()
2
+ ()
2
(86)
that is:
=~
2
+
2
=~
2
+
2
(87)
Similarly:
=~
2
+
2
=~2
+
2
(88)
769

A CHARGED PARTICLE IN A MAGNETIC FIELD: LANDAU LEVELS
Complement EVI
A charged particle in a magnetic eld: Landau levels
1 Review of the classical problem
1-a Motion of the particle
1-b The vector potential. The classical Lagrangian and Hamiltonian
1-c Constants of the motion in a uniform eld
2 General quantum mechanical properties of a particle in a
magnetic eld
2-a Quantization. Hamiltonian
2-b Commutation relations
2-c Physical consequences
3 Case of a uniform magnetic eld
3-a Eigenvalues of the Hamiltonian
3-b The observables in a particular gauge
3-c The stationary states
3-d Time evolution
Thus far, we have been considering, for various special cases, the properties of a
particle subjected to a scalar potential(r)(representing, for example, the eect of an
electric eld on a charged particle). Chapter
(particle subjected to a central potential) treat other examples of scalar potentials. Here
we shall be concerned with a complementary problem, that of the properties of a particle
subjected to a vector potentialA(r)(a charged particle placed in a magnetic eld). We
shall encounter a number of purely quantum mechanical eects, such as equally spaced
energy levels in a uniform magnetic eld (Landau levels)
1
. Before studying the problem
from a quantum mechanical point of view, we shall rapidly review some classical results.
1. Review of the classical problem
1-a. Motion of the particle
When a particle of positionrand chargeis subjected to a magnetic eldB(r),
the forcefexerted on it is the Lorentz force:
f=vB(r) (1)
where:
v=
dr
d
(2)
1
This equal spacing is, as we shall show, a consequence of the properties of the harmonic oscillator,
and it could have been treated in Chapter. However, we shall also see that the properties of angular
momentum are useful in the study and classication of the stationary states of the particle. This is why
this complement follows Chapter.
771

COMPLEMENT E VI
is the velocity of the particle. Its motion obeys the fundamental law of dynamics:
dv
d
=f (3)
(whereis the mass of the particle).
In the rest of this complement, we shall often be considering the case in which the
magnetic eld is uniform; we shall choose anaxis parallel to this eld. By solving
the equation of motion (3), one can show that in this case the three coordinates(),
()and()of the particle are given by:
() =0+cos( 0)
() =0+sin( 0) (4)
() =0+0
where0,0,0,,0and0are six constant parameters which depend on the initial
conditions; thecyclotron frequencyis given by:
=
(5)
Equations (4) show that the projection of the positionof the particle onto the
plane performs a uniform circular motion, of angular velocityand initial phase0,
on a circle of radiuswhose center is the point0, with coordinates0and0. The
motion of the projection ofontois simply rectilinear and uniform. It follows that
the particle moves in space along a circular helix (cf.Fig.), whose axis is parallel to
and passes through0.
If we are concerned only with the motion of the point, the projection ofonto
the plane, we study the behavior of the vector:
=e+e (6)
(whereeandeare the unit vectors of theand axes). The velocity ofis:
v=
d
d
(7)
It is therefore convenient to introduce the componentsandof the vectorC0Q:
= 0
= 0 (8)
Sinceperforms a uniform circular motion about0, we have:
v=eC0Q (9)
(whereeis the unit vector of). This implies that the coordinates0and0of0
are related to the coordinates ofand to the components ofvby:
0=
1
0=+
1
(10)
772

A CHARGED PARTICLE IN A MAGNETIC FIELD: LANDAU LEVELSz
e
z
e
y
e
x
B
O
Q
y
x
MC
0
+
Figure 1: Classical trajectory of a charged particle in a uniform magnetic eld parallel to
. The particle moves at constant velocity along a circular helix whose axis, parallel to
, passes through the point0. The gure is drawn for0(the case of the electron),
that is,0.
1-b. The vector potential. The classical Lagrangian and Hamiltonian
To describe the magnetic eldB(r), one can use a vector potentialA(r)which is,
by denition, related toB(r)by:
B(r) = A(r) (11)
For example, if the eldBis uniform, one can choose:
A(r) =
1
2
rB (12)
We know, furthermore, that whenB(r)is given, condition (11) does not determineA(r)
uniquely: a gradient of an arbitrary function ofrcan be added toA(r)without changing
2
B(r).
It can be shown (cf.Appendix, ) that the Lagrange function(rv)of
the particle is given by:
(rv) =
1
2
v
2
+vA(r) (13)
2
For example, for a uniform eld parallel to, one could choose, instead of the vectorA(r)given
by (12), the vector whose components are= 0,= ,= 0.
773

COMPLEMENT E VI
It follows thatp, the conjugate momentum of the positionr, is related tovandA(r)
by:
p=v(rv) =v+A(r) (14)
The classical Hamiltonian(rp)is then:
(rp) =
1
2
[pA(r)]
2
(15)
It will prove convenient to set:
(rp) =(rp) +
(rp) (16)
with:
(rp) =
1
2
[ (r)]
2
+ [ (r)]
2
(rp) =
1
2
[ (r)]
2
(17)
Comment:
In this case, unlike that of a scalar potential(r), relation (14) shows that the
momentumpis not equal to the mechanical momentumv. Also, comparing
(14) with (15), we see thatis equal to the kinetic energyv
2
2of the particle;
this results from the fact that since the Lorentz force written in (1) is always
perpendicular tov, it does no work during the motion. Similarly, it must be noted
that the angular momentum:
=rp (18)
is dierent from the moment of the mechanical momentumv:
=rv (19)
1-c. Constants of the motion in a uniform eld
Consider the special case in which the eldBis uniform. The motion of the particle
( 1-a) is such thatand , dened in (17), are constants of the motion
3
.
If we substitute (14) into (10), we obtain:
0=
1
[ (r)]
0=+
1
[ (r)] (20)
It follows that the radiusof the helical trajectory satises:
2
= ( 0)
2
+ ( 0)
2
=
1
2
[ (r)]
2
+ [ (r)]
2
=
2
2
(21)
3
This follows from the fact that, according to (14) and (17), and
are equal, respectively, to
the kinetic energiesv
2
2andv
2
2associated with the motions perpendicular and parallel to.
774

A CHARGED PARTICLE IN A MAGNETIC FIELD: LANDAU LEVELS
2
is therefore proportional to the Hamiltonian.
Similarly, letbe the moment of the mechanical momentum vwith respect to the
center0of the circle:
=C0M v (22)
The component of this moment can then be written, with (20) taken into account:
=[( 0) ( 0)]
=
1
[ (r)]
2
+ [ (r)]
2
=
2
(23)
is therefore a constant of the motion, as might have been expected. On the other hand, the
component of the moment of the mechanical momentumvwith respect tois not, in
general, constant, since:
=+[0() 0()] (24)
Therefore, according to (4),varies sinusoidally in time.
Finally, consider the projectionontoof the angular momentum:
= (25)
According to (14), it can be written:
=[+ (r)][+ (r)] (26)
It therefore depends explicitly on the gauge chosen, that is, on the vector potentialApicked to
describe the magnetic eld. In most cases,is not a constant of the motion. Nevertheless, if
one chooses the gauge given in (12), one obtains from (4):
=
2
2
0+
2
0
2
(27)
is then a constant of the motion.
Relation (27) does not have a simple physical interpretation, since it is valid only in a
particular gauge. However, it will prove useful to us in the following sections for the quantum
mechanical study of the problem.
2. General quantum mechanical properties of a particle
in a magnetic eld
2-a. Quantization. Hamiltonian
Consider a particle placed in an arbitrary magnetic eld described by the vector
potentialA( ). In quantum mechanics, the vector potential becomes an operator,
a function of three observables,,and. The operator, the Hamiltonian of the
particle, can be obtained from (15):
=
1
2
[PA( )]
2
(28)
According to (14), the operatorVassociated with the velocity of the particle is given
by:
V=
1
[PA( )] (29)
775

COMPLEMENT E VI
which enables us to writein the form:
=
2
V
2
(30)
2-b. Commutation relations
The observablesRandPsatisfy the canonical commutation relations:
[ ] = [] = [] =~ (31)
The other commutators between components ofRandPare zero. Two components of
Ptherefore commute. However, we see from (29) that the same is not true forV; for
example:
[ ] =
2
[ (R)] + [(R)] (32)
This expression is easy to calculate, using the rule given in ComplementII[cf.for-
mula (48)]:
[ ] =
~
2
=
~
2
(R) (33a)
Similarly, it can be shown that:
[ ] =
~
2
(R) (33b)
[ ] =
~
2
(R) (33c)
The magnetic eld therefore enters explicitly into the commutation relations for the
velocity.
However, sinceA(R)commutes with,and, relation (29) implies that:
[ ] =
1
[ ] =
~
(34a)
and, similarly:
[ ] = [] =
~
(34b)
(the other commutators between a component ofRand a component ofVare zero).
From these relations, it can be deduced (cf.ComplementIII) that:

~
2
(35)
(with analogous inequalities for the components alongand). The physical conse-
quences of the Heisenberg uncertainty relations are therefore not modied by the presence
of a magnetic eld.
776

A CHARGED PARTICLE IN A MAGNETIC FIELD: LANDAU LEVELS
Finally, let us calculate the commutation relations between the components of the
operator:
=RV (36)
associated with the moment with respect toof the mechanical momentum
4
. We obtain:
[] =
2
[ ]
=
2
[ ]+[ ]
22
[ ]
+
2
[ ] + [] (37)
that is, with (33) and (34) taken into account:
[] =~ + +
2
+ + (38)
It follows that:
[] =~+RB(R) (39)
(the other commutators can be obtained by cyclic permutation of the indices,and
). When the eldBis not zero, the commutation relations ofare completely dierent
from those ofL. The operatortherefore does not,a priori, possess the properties of
angular momenta proved in Chapter.
2-c. Physical consequences
. Evolution ofR
The time variation of the mean position of the particle is given by Ehrenfest's
theorem:
~
d
d
R=[R]=R
2
V
2
(40)
[according to formula (30)]. Equations (34) are not dicult to interpret, since, substi-
tuted into (40), they yield:
d
d
R=V (41)
As in the case in which the magnetic eld is zero, the mean velocity is therefore equal to
the derivative ofR. Equation (41) is the quantum mechanical analogue of (2).
. Evolution ofV. The Lorentz force
Let us calculate the time derivative of the mean valueVof the velocity:
~
d
d
V=V
2
V
2
(42)
4
Of course, the components of the angular momentumL=RPalways satisfy the usual commu-
tation relations.
777

COMPLEMENT E VI
Since, according to relations (33):
V
2
=
2
+
2
+
2
=[ ] + [ ]+[ ] + [ ]
=
~
2
(R) (R)+ (R) +(R) (43)
it is easy to see that:
d
d
V=F(RV) (44)
where the operatorF(RV)is dened by:
F(RV) =
2
VB(R)B(R)V (45)
The last two relations are simply the analogues of the classical relations (1) and (3).
Here, we obtain a symmetrized expression forF(RV)(cf.Chap., ), sinceR
andVdo not commute; a minus sign appears since the vector product is antisymmetric.
. Evolution of
Now let us evaluate:
~
d
d
=[] (46)
To do so, let us calculate, for example, the commutator[ ]:
[ ] =[ ] + [] [ ][]
=
~
( ) +~( ) (47)
Butandcommute, as doand. The commutator we are calculating is therefore
equal to:
[ ] =[ ] + [] [][ ]
=
~
( ) +~( ) (48)
Taking half the sum of these two expressions, we nd
d
d
in the form:
d
d
=
1
2
+ (49)
Analogous arguments give the derivative ofand; nally:
d
d
=
1
2
RF(RV)F(RV)R (50)
The classical analogue of this relation is:
d
d
=rf(rv) (51)
which expresses a well-known theorem: the time derivative of the moment of the me-
chanical momentum with respect to a xed pointis equal to the moment with respect
toof the force exerted on the particle.
778

A CHARGED PARTICLE IN A MAGNETIC FIELD: LANDAU LEVELS
3. Case of a uniform magnetic eld
When the magnetic eld is uniform, the preceding general study can easily be pursued
further. We choose the direction of the eldBas theaxis. The commutation relations
(33) then become, using denition (5):
[ ] =
~
(52a)
[ ] = [ ] = 0 (52b)
Comment:
Applying the results of ComplementIIItoand, we can see from (52a) that
their root mean square deviations satisfy:

~
2
(53)
The components of the velocityVare therefore incompatible physical quantities.
3-a. Eigenvalues of the Hamiltonian
By analogy with (16),can be written in the form:
= +
(54)
with:
=
2
2
+
2
(55a)
=
2
2
(55b)
According to (52b):
[
] = 0 (56)
We can now look for a basis of eigenvectors common to(eigenvalues) and

(eigenvalues
); they will automatically be eigenvectors of, with the eigenvalues:
= +
(57)
. Eigenvalues of

The eigenvectors of the operatorare also eigenvectors of
. Now,and
are two Hermitian operators which satisfy the relation:
[ ] =
~
(58)
779

COMPLEMENT E VI
We can therefore apply to them the results of ComplementII; in particular, the spec-
trum ofincludes all the real numbers.
Consequently, the eigenvalues of
are of the form:
=
2
2
(59)
whereis a real arbitrary constant. The spectrum of
is therefore continuous: the
energy
can take on any positive value or zero.
The interpretation of this result is obvious:
describes the kinetic energy of a
free particle moving along(as in classical mechanics; 1-a).
. Eigenvalues of
We shall assume, as an example, that the particle under consideration has a neg-
ative charge; the cyclotron frequencyis then positive [formula (5)]
5
.
We set:
^
=
~
^
=
~
(60)
Relation (52a) can then be written:
[
^^
] = (61)
and becomes:
=
~
2
^2
+
^2
(62)
then takes on the form of the Hamiltonian of a one-dimensional harmonic oscillator
[cf.Chap., relation (B-4)].
^
and
^
, which satisfy (61), play the roles of the position
^
and momentum
^
of this oscillator.
The arguments set forth in
^
and
^
can
be repeated here for
^
and
^
. For example, it can easily be shown that ifis an
eigenvector of:
= (63)
the kets:
=
1
2
^
+
^
(64a)
=
1
2
^^
(64b)
5
For a positive charge, one can keep the convention of positiveby choosing the direction of the
axis opposite to the magnetic eld.
780

A CHARGED PARTICLE IN A MAGNETIC FIELD: LANDAU LEVELS
are also eigenvectors of:
= ( ~) (65a)
= (+~) (65b)
From this we deduce that the possible values ofare given by:
= +
1
2
~ (66)
whereis a positive integer or zero.
. Eigenvalues of
According to the preceding results, the eigenvalues of the total Hamiltonianare
of the form:
() =+
1
2
~+
1
2
2
(67)
The corresponding levels are calledLandau levels.
For a given value of, all possible values of(positive integers or zero) are actually
found. From (64) and
1
2
^^
on an eigenvector ofof eigenvalue( )provides an energy state( ), where
is any integer but wherehas not changed (since
^
and
^
commute with
).
Therefore, although the energy of the motion alongis not quantized, that of the
motion projected ontois.
Comment:
We showed in Chapter) that the energy levels of the one-dimensional har-
monic oscillator are non-degenerate in. The situation is dierent here, since the
particle under study is moving in three-dimensional space. Since the destruction
operator of a quantum~is
1
2
^
+
^
=
2~
(+), the eigenvectors
ofcorresponding to= 0are solutions of the equation:
(+)= 0 (68)
On the one hand, vectors that are solutions of (68) can be eigenvectors of

with an arbitrary (positive) eigenvalue. On the other, even for a xed value of,
equation (68) is a partial dierential equation with respect toand, and has
an innite number of solutions. The energies(= 0)are therefore innitely
degenerate. By using the creation operator for a quantum, it can easily be shown
that this is true for all the levels(), for any(non-negative integer).
781

COMPLEMENT E VI
3-b. The observables in a particular gauge
In order to state the above results more precisely, we shall calculate the stationary
states of the system. This will enable us to study their physical properties. It is now
necessary to choose a gauge; we shall choose the one given by (12). The components of
the velocity are then:
=
2
=
+
2
=
(69)
. The Hamiltonians and
. Relation with the two-dimensional harmonic
oscillator
Substituting (69) into (55), we obtain:
=
2
+
2
2
+
2
+
2
8
2
+
2
(70a)
=
2
2
(70b)
whereis the component alongof the angular momentumL=RP.
In therrepresentation,
is an operator that acts only on the variable,
whileacts only on the variablesand. We can therefore nd a basis of eigenvectors
ofby solving inthe eigenvalue equation of
, and then, in, that of. All
we must then do is take the tensor products of the vectors obtained.
Actually, the eigenvalue equation of
simply leads to the wave functions:
() =
1
2~
e
~
(71)
with:
=
2
2
(72)
[we again nd (59)]. Therefore, we shall concentrate on solving the eigenvalue equation
of in; the wave functions we shall be considering now depend onand, and
not on.
Comparing (70a) with expression (12a) of ComplementVI, we see thatcan
be expressed simply in terms of the Hamiltonianof a two-dimensional harmonic
oscillator:
= +
2
(73)
if we choose for the value of the constant that enters into:
=
2
(74)
782

A CHARGED PARTICLE IN A MAGNETIC FIELD: LANDAU LEVELS
Now, in ComplementVI, we saw that and form a C.S.C.O. in, and we
constructed a basis of eigenvectors common to these two observables [cf.for-
mula (47) ofV]. The are also eigenvectors of; ComplementVItherefore
gives the solutions to the eigenvalue equation of.
Comments:
(i) In , we saw thatcan be written in a form that is analogous to that
of a Hamiltonian of a one-dimensional harmonic oscillator. Here, we nd that,
in a particular gauge, this same operatoris also simply related to the Hamil-
tonian of a two-dimensional harmonic oscillator. These two results are not
contradictory; they simply correspond to two dierent decompositions of the same
Hamiltonian, which must obviously lead to the same physical conclusions.
()One must not lose sight of the fact that the Hamiltonianinvolves a physical
problem which is completely dierent from that of the two-dimensional harmonic
oscillator: the charged particle is subjected to a vector potential (describing a
uniform magnetic eld) and not a harmonic scalar potential (which would describe,
for example, a non-uniform electric eld). It so happens that, in the gauge chosen,
the eects of the magnetic eld can be likened to those of a ctitious harmonic
scalar potential.
. Expression for the observables in terms of the creation and destruction operators
of circular quanta
First of all, we shall express the observables describing the quantities associated
with the particle in terms of the operatorsand[dened by equations (40) of Com-
plementVI] and their adjointsand(we shall also use the operators=
and= ).
Substituting relations (46) ofVIinto (73), we obtain
6
:
= +
1
2
~ (75)
The energy associated with the state is therefore:
= +
1
2
~ (76)
as we found in (66). Moreover, sinceis independent of, we see that all the
eigenvalues ofare innitely degenerate.
Using relations (23) and (40) ofVI, we can see that:
=
1
2
+++
=
2
+
(77)
6
Recall that we have assumed to be positive. Ifwere negative, the indicesandwould
have to be inverted in a certain number of the following formulas; for example, (75
= (+ 12)~.
783

COMPLEMENT E VI
where, using (74),is dened by:
=
2~
(78)
Similarly:
=
~
2
+ +
=
~
2
+ (79)
These expressions, substituted into (69), yield:
=
2
=
2
+ (80)
Sinceand do not commute with , it can be seen by using (75) that, as in
classical mechanics,andare not constants of the motion; in addition, using the
commutation relations ofand, we indeed obtain (52a).
It is also interesting to study the quantum mechanical operators associated with
the various variables introduced in the description of the classical motion (): the coor-
dinates (0,0) of the center0of the classical trajectory, the components( )of the
vectorC0Q, etc. As above, we shall denote each of these operators by the capital letter
corresponding to the small letter which designates the corresponding classical variable.
By analogy with (10), we therefore set:
0=
1
=
1
2
+ (81a)
0=+
1
=
2
(81b)
The operatorsandcommute with; it follows that0and0are constants of
the motion. Formulas (81) also imply that:
[00] =
2
2
=
~
(82)
Consequently,0and0are incompatible physical quantities, their root mean square
deviations being related by:
00
~
2
(83)
We also dene:
= 0=
1
2
+
= 0=
2
(84)
784

A CHARGED PARTICLE IN A MAGNETIC FIELD: LANDAU LEVELS
We immediately see thatand, as in classical mechanics, are not constants of the
motion; moreover,andare simply proportional toandrespectively:
=
= (85)
like the corresponding classical variables [formula (9)]. According to (53), equations (85)
imply:

~
2
(86)
Let
2
be the operator corresponding to
2
(square of the radius of the classical trajec-
tory):

2
= ( 0)
2
+ ( 0)
2
(87)
According to (), we have:

2
=
1
2
2
+
2
=
2
2
(88)

2
is therefore a constant of the motion, as is
2
in classical mechanics.
Finally, the operator associated with the moment of the mechanical momentumvwith
respect tois:
=[( 0) ( 0)] (89)
and formulas (81) indicate that:
=
2
(90)
as in (23).is therefore a constant of the motion. On the other hand, the operator, the
component along ofRV, is:
=
2
+~ + (91)
and therefore does not commute with.
3-c. The stationary states
We indicated above that the eigenvalues of the Hamiltonianare all innitely degener-
ate in. For each positive or zero integer, there exists an innite-dimensional subspace
()
of, all of whose kets are eigenvectors ofwith the same eigenvalues(+ 12)~. In this
section, we shall study dierent bases which can be chosen in each of these subspaces. First, we
shall indicate the general properties of the stationary states, valid for any basis of eigenstates
of.
785

COMPLEMENT E VI
. General properties
Relations (88) and (90) show that an arbitrary stationary state is necessarily an eigen-
vector of
2
and; the corresponding physical quantities are therefore always well-dened in
such a state and are equal to:
(2+ 1)
~
for
2
(2+ 1)~ for (92)
The values of
2
andare proportional to the energy; this corresponds to the classical
description of the motion (cf. 1).
It follows from (80) and (84) that,,and have no matrix elements inside a
given subspace
()
; it follows, for a stationary state, that:
= = 0
= = 0 (93)
Nevertheless, sinceand (and thereforeand) are not constants of the motion, the
corresponding physical quantities do not have perfectly well-dened values in a stationary state.
In fact, by using (80), (84) and the properties of the one-dimensional harmonic oscillator [cf.
Chap., relation (D-5)], it can be shown that:
==
+
12
~
==
+
12
~
(94)
in agreement with (53). Moreover, we see that the only stationary states in which the product
(or) takes on its minimal value are the ground states(= 0).
Comment:
The various ground states are solutions of the equation:
= 0 (95a)
that is, using (80):
(+)= 0 (95b)
as we found in (68).
. The states
As we saw in ComplementVI, the fact thatand form a C.S.C.O. in can
be used to construct a basis of eigenvectors common to these two observables. This basis is
composed of the vectors , since, according to (75) and formula (46) of ComplementVI:
= +
1
2
~ (96a)
= ( )~ (96b)
786

A CHARGED PARTICLE IN A MAGNETIC FIELD: LANDAU LEVELS
The subspace
()
dened by specifying the (non-negative) integeris therefore spanned
by the set of vectors such that=. The eigenvalues ofassociated with these
dierent vectors are of the form~, and, for xed,is an integer which can vary between
and(for example, all the ground states correspond to negative values of; this is related
to the hypothesis 0posed above).
The wave functions associated with the stateswere calculated in ComplementVI
().
Note that the states are eigenstates of the operator, but not of the operator
associated with the moment of the mechanical momentum. This can be seen directly from
formula (91).
In a state , the mean values0and 0are zero, according to (81). However,
neither0nor0corresponds to perfectly well-dened physical quantities, since, by using the
properties of the one-dimensional harmonic oscillator, it can easily be shown that, in a state
:
0=
+
12
~
0=
+
12
~
(97)
The minimal value of the product00is therefore attained for the states=0, that
is, the states of each energy level= (+ 12)~for whichtakes on its maximal value
~[cf.(96)].
However, let us dene the operator:

2
=
2
0+
2
0 (98)
It corresponds to the square of the distance from the center0of the trajectory to the origin.
Using (81), we easily nd:

2
=
~
+
=
~
(2+ 1) (99)
The state is therefore an eigenstate of
2
with the eigenvalue
~
(2+ 1); the fact
that this value can never go to zero is related to the non-commutativity of the operators0and
0.
Comment:
The operator, according to (75) and (99), is given by:
=~( ) =~
~
1
22~

2
+
1
2
(100)
that is, according to (88):
=
2

2

2
=
2

2

2
(101)
which is the equivalent of the classical relation (27).
787

COMPLEMENT E VI
. Other types of stationary states
Any linear combination of vectors associated with the same value ofis an
eigenstate ofand therefore possesses the properties stated in 3-c-. By a suitable choice
of the coecients of the linear combination, one can obtain stationary states that possess other
interesting properties as well.
We know, for example ( ), that0and0are constants of the motion. However,
since0and0do not commute, there are no eigenstates common to these two operators.
This means that, in quantum mechanics, it is not possible to obtain a state in which the two
coordinates of the point0are known.
To construct the eigenstates common toand 0, we can use the properties of the
one-dimensional harmonic oscillator; formula (81a) shows that0has the same expression,
to within a constant factor, as the position operatorof a one-dimensional oscillator whose
destruction operator is:
0=
1
2
^
(102)
Since we know the wave functions^(^)associated with the stationary states^of a one-
dimensional harmonic oscillator (cf.ComplementV, ), we know how to write the eigen-
vectors^of the position operator as linear combinations of the states^:
^=
=0
^^^
=
=0
^(^)^ (103)
In order to obtain the eigenstates common toand 0it suces to apply this result to the
states =; the vector:
0=
=0
^(
20) = = (104)
is a common eigenvector ofand 0with the eigenvalues(+ 12)~and0.
The eigenstates
0common to and0can be found in an analogous fashion.
Relation (81b) indicates that0is proportional to the momentum operatorof the cticious
one-dimensional oscillator just used:
0=
1
2
^
(105)
Consequently [see formula (20) of ComplementV]:
0=
=0
^(
20) = = (106)
We have just constructed the states in which either0or0is perfectly well-dened.
We can also determine the stationary states in which the product00reaches its minimal
value, given by (83). For a one-dimensional harmonic oscillator, we studied in ComplementV
the states in which the product
^

^
is minimal; these are the quasi-classical states, given
by:
=
=0
() (107)
788

A CHARGED PARTICLE IN A MAGNETIC FIELD: LANDAU LEVELS
with:
() =
!
e
2
2
(108)
In these states:

^
=
^
=
1
2
(109)
It follows that, in the case which interests us here, the state:
0=
=0
(0) = = (110)
yields, for0and0, root mean square deviations:
0= 0=
1
2
(111)
The product00is therefore minimal.
Comment:
Since the magnetic eld is uniform, the physical problem we are considering is invariant
with respect to translation. Thus far, this symmetry has been masked by the choice of
the particular gauge (12), which gives the origina privileged position with respect to
all other points in space. Consequently, neither the Hamiltoniannor its eigenstates
are invariant with respect to translation. We know, however (cf.ComplementIII) that
the physical predictions of quantum mechanics are gauge-invariant. These predictions
must remain the same if, by a change of gauge, we give a point other thana privileged
position. Consequently, the translation symmetry must reappear when we study the
physical properties of a given state.
To show this more precisely, let us assume that, at a given instant, the state of the
particle is characterized in the gauge (12) by the ketwith which the wave function
r=(r)is associated. We then perform a translationdened by the vectora,
and consider the ketdened by:
= e
~
Pa
(112)
with which, according to the results of ComplementII, is associated the wave function:
(r) =r =(ra) (113)
The same translation can be applied to the vector potential, which becomes:
A(r) =A(ra) =
1
2
(ra)B (114)
A(r)clearly describes the same magnetic eld asA(r). Since the physical properties
attached to a given state vector depend only on this state vector and the potential
Achosen, they must undergo the translationwhen(r)andA(r)are replaced by
expressions (113) and (114). It is simple to use these relations to obtain the expression
for the probability density associated with:
(r) =(r)
2
=(ra)
2
=(ra) (115)
789

COMPLEMENT E VI
and that for the currentJ(r), calculated with the vector potentialA(r):
J(r) =
1
2
(r)
~
+
2
(ra)B (r) +
=
1
2
(ra)
~
+
2
(ra)B(ra) +
=J(ra) (116)
[whereJ(r)is the probability current associated with(r)in the gauge (12)]. The ket
therefore describes,in the new gaugeA(r), a state whose physical properties are
related by the translationto those corresponding to the ketin the gaugeA(r).
Let us show, moreover, that the translation of a possible motion yields another possible
motion; this will conclude the proof of the translation invariance of the problem. To do
so, consider the Schrödinger equation in therrepresentation, in the gaugeA(r):
~
(r) =
1
2
~
A(r)
2
(r) (117)
Changingrtorain this equation, we obtain, using (113) and (114):
~
(r) =
1
2
~
A(r)
2
(r) (118)
The operator appearing on the right-hand side of (118) is none other than the Hamiltonian
in the gaugeA(r). Consequently, if(r)describes, in the gaugeA(r), a possible
motion of the system,(r)describes, in the equivalent gaugeA(r), another possible
motion, which, according to what we have just shown, is nothing more than the result
of a translation of the rst motion. In particular, if:
(r) =(r) e
~
is a stationary state [in the gaugeA(r)],
(r) =(r) e
~
is another stationnary state of the same energy [in the gaugeA(r)].
If we want to continue to use the gauge (12) after having performed the translation
on the physical state of the particle, we must describe the translated state by a mathe-
matical ket which is dierent from. According to of ComplementIII,
the ket can be obtained from by a unitary transformation:
= (119)
The operatoris given by:
= e
~
(R)
(120)
where(r)is the function characterizing the gauge transformation performed. Here, the
potential after the gauge change is:
A(r) =
1
2
rB=A(r)
1
2
aB (121)
so that:
(r) =
1
2
r(aB) (122)
790

A CHARGED PARTICLE IN A MAGNETIC FIELD: LANDAU LEVELS
Substituting (112), (120) and (122) into (119), we nally obtain:
=(a) (123)
with:
(a) = e
2~
R(aB)
e~
Pa
(124)
Therefore, if we remain in the gaugeA(r), the translation operator is(a), given by
(124).
The components ofRandPalong two perpendicular axes enter into formula (124); they
therefore commute, so we can write:
(a) = e
2~
R(aB)
~
Pa
(125)
Whenais a vector of theplane, a simple calculation, using formulas (10) and (69),
yields:
(a) = e
~
(aR0)B
(126)
with:
R0= 0e+0e (127)
The operators0and0(coordinates of the center of the circle) are therefore associated
with translations alongand respectively.
3-d. Time evolution
. Mean values of the observables
We have already encountered a certain number of physical quantities that are
constants of the motion:0,0,,
2
. Whatever the state of the system, their mean
values are time-independent.
Let us examine the time evolution of the mean values,,, and ,
. We immediately see from the expressions given in 3-b-that the corresponding
operators have matrix elements only between stateswhose values ofdier by
1(or 0). The evolution of these mean values therefore involves only one Bohr frequency,
which is none other than the cyclotron frequency2dened in (5).
This result is completely analogous to the one given by classical mechanics.
. Quasi-classical states
Assume that at= 0the state of the particle is:
(0)= (128)
where the ket is dened by expression (70) of ComplementVI. Since expression
(75) forinvolvesbut not, the state vector()at the instantis obtained
by changingtoe :
()= e
2
e (129)
[cf.expression (92) of ComplementV].
791

COMPLEMENT E VI
We set:
= e
= e (130)
Relations (80), (81) and (84) then show that:
0=
1
2
(+) =cos
0=
2
( ) =sin
(131)
() =
1
2
(e +e) =cos( )
() =
2
(e e) =sin( )
(132)
and:
() =
sin( )
() =
cos( )
(133)
Moreover, the properties of the statesimply that:
=~
2
+
1
2
= 2~
2
+
1
2

2
=
1
2
2
+
1
2
(134)
All these results are extremely close to those given by classical mechanics [cf.(4)]. We
see thatis related to the radiusof the classical trajectory, andto the initial
phase0, whileis related to the distance0, andcorresponds to the polar angle
of the vectorOC0.
Furthermore, the properties of the statescan be used to show that:
0= 0= = =
1
2
(135a)
= =
2
(135b)
(the products00,andtherefore take on their minimal values),
and:
=~ = 2~
2
=
1
2
(136)
792

A CHARGED PARTICLE IN A MAGNETIC FIELD: LANDAU LEVELS
As for the deviationsand, they can be calculated by using the fact that:
()= e
2
=
e +
2
=
e
2
(137)
[where is dened by relation (66) ofVI], which yields:
= =
~
=
1
2
(138)
(andcan easily be obtained in the same way).
If the conditions:
1 1 (139)
are satised, we see, therefore, that the various physical quantities (position, velocity,
energy, ...) are, in relative value, very well dened. The states (129) therefore represent
quasi-classical states of the charged particle placed in a uniform magnetic eld.
Comment:
If= 0, we obtain:
=
1
2
~
= 0
(140)
The states:
= (141)
therefore correspond to the ground state.
References and suggestions for further reading:
Landau and Lifshitz (1.19), Chap. XVI, 124 and 125; Ter Haar (1.23), Chap. 6.
Application to solid state physics: Mott and Jones (13.7), Chap. VI, 6; Kittel
(13.2), Chap. 8, p. 239 and Chap. 9, p. 290.
793

EXERCISES
Complement FVI
Exercises
1.Consider a system of angular momentum= 1, whose state space is spanned by the
basis+ 10 1of three eigenvectors common toJ
2
(eigenvalue2~
2
) and
(respective eigenvalues+~, 0 and~). The state of the system is:
=+ 1+0+ 1
where,,are three given complex parameters.
Calculate the mean valueJof the angular momentum in terms of,and.
Give the expression for the three mean values
2
,
2
and
2
in terms of the
same quantities.
2.Consider an arbitrary physical system whose four-dimensional state space is
spanned by a basis of four eigenvectors common toJ
2
and (= 0or 1;
+), of eigenvalues(+ 1)~
2
and~, such that:
=~
(+ 1) ( 1) 1
+ = = 0
Express in terms of the kets, the eigenstates common toJ
2
and, to be
denoted by .
Consider a system in the normalized state:
= = 1 = 1+ = 1 = 0
+ = 1 =1+ = 0 = 0
()What is the probability of nding2~
2
and~ifJ
2
andare measured simul-
taneously?
()Calculate the mean value ofwhen the system is in the state, and the
probabilities of the various possible results of a measurement bearing only on
this observable.
()Same questions for the observableJ
2
and for.
()
2
is now measured; what are the possible results, their probabilities, and
their mean value?
3.LetL=RPbe the angular momentum of a system whose state space isr.
Prove the commutation relations:
[ ] =~
[ ] =~
[P
2
] = [R
2
] = [RP] = 0
795

COMPLEMENT F VI
where,,denote arbitrary components ofL,R,Pin an orthonormal system,
and is dened by:
= 0if two (or three) of the indices,,are equal
= 1if these indices are an even permutation of,,
=1if the permutation is odd.
4. Rotation of a polyatomic molecule
Consider a system composed ofdierent particles, of positionsR1 R R,
and momentaP1 P P. We set:
J= L
with:
L=R P
Show that the operatorJsatises the commutation relations that dene an angular
momentum. Deduce from this that, ifVandVdenote two ordinary vectors of
three-dimensional space, then:
[JVJV] =~(VV)J
Calculate the commutators ofJwith the three components ofRand with those
ofP. Show that:
[JRR] = 0
Prove that:
[JJR] = 0
and deduce from this the relation:
[JR JR] =~(R R)J=~J(R R)
We set:
W= R
W= R
where the coecientsand are given. Show that:
[JWJW] =~(WW)J
Conclusion: what is the dierence between the commutation relations of the com-
ponents ofJalong xed axes and those of the components ofJalong the moving
axes of the system being studied?
796

EXERCISES
Consider a molecule which is formed byunaligned atoms whose relative distances
are assumed to be invariant (a rigid rotator).Jis the sum of the angular momenta
of the atoms with respect to the center of mass of the molecule, situated at a xed
point; the axes constitute a xed orthonormal frame. The three principal
inertial axes of the system are denoted by,and, with the ellipsoid of
inertia assumed to be an ellipsoid of revolution about(a symmetrical rotator).
The rotational energy of the molecule is then:
=
1
2
2

+
2
+
2
where,andare the components ofJalong the unit vectorsw,wand
wof the moving axes,,attached to the molecule, and
andare
the corresponding moments of inertia. We grant that:
2
+
2
+
2
=
2
+
2
+
2
=J
2
()Derive the commutation relations of,,from the results of.
()We introduce the operators= . Using the general arguments of
Chapter, show that one can nd eigenvectors common toJ
2
and, of
eigenvalues(+ 1)~
2
and~, with= + 1 1
()Express the Hamiltonianof the rotator in terms ofJ
2
and
2
. Find its
eigenvalues.
()Show that one can nd eigenstates common toJ
2
,and, to be denoted
by [the respective eigenvalues are(+ 1)~
2
,~,~]. Show that
these states are also eigenstates of.
()Calculate the commutators ofand withJ
2
,,. Derive from them
the action ofand on . Show that the eigenvalues ofare
at least2(2+ 1)-fold degenerate if= 0, and(2+ 1)-fold degenerate if
= 0.
()Draw the energy diagram of the rigid rotator (is an integer sinceJis a sum
of orbital angular momenta;cf.Chapter). What happens to this diagram
when
= (spherical rotator)?
5.A system whose state space isrhas for its wave function:
( ) =(++)e
r
2 2
where, which is real, is given andis a normalization constant.
The observablesandL
2
are measured; what are the probabilities of nding 0
and2~
2
? Recall that:
0
1() =
34
cos
797

COMPLEMENT F VI
If one also uses the fact that:
1
1
() =
38
sine
is it possible to predict directly the probabilities of all possible results of measure-
ments ofL
2
andin the system of wave function( )?
6.Consider a system of angular momentum= 1. A basis of its state space
is formed by the three eigenvectors of:+ 1,0,1, whose eigenvalues are,
respectively,+~, 0, and~, and which satisfy:
=~
2 1
+1= 1= 0
This system, which possesses an electric quadrupole moment, is placed in an electric eld
gradient, so that its Hamiltonian can be written:
=
0
~
(
2 2
)
where and are the components ofLalong the two directionsand of the
plane that form angles of 45withand;0is a real constant.
Write the matrix representingin the+ 10 1basis. What are the
stationary states of the system, and what are their energies? (These states are to
be written1,2,3, in order of decreasing energies.)
At time= 0, the system is in the state:
(0)=
1
2
[+ 1 1]
What is the state vector()at time? At,is measured; what are the
probabilities of the various possible results?
Calculate the mean values(), ()and ()at. What is the motion
performed by the vectorL?
At, a measurement of
2
is performed.
()Do times exist when only one result is possible?
()Assume that this measurement has yielded the result~
2
. What is the state of
the system immediately after the measurement? Indicate, without calculation,
its subsequent evolution.
7.Consider rotations in ordinary three-dimensional space, to be denoted byRu(),
whereuis the unit vector which denes the axis of rotation andis the angle of rotation.
Show that, ifis the transform ofunder an innitesimal rotation of angle,
then:
OM=OM+uOM
798

EXERCISES
IfOMis represented by the column vector, what is the matrix associated
withRu()? Derive from it the matrices representing the components of the oper-
atordened by:
Ru() = 1 +u
Calculate the commutators:
[ ] ; [ ] ; [ ]
What are the quantum mechanical analogues of the purely geometrical relations
obtained?
Starting with the matrix representing, calculate the one that representse;
show thatR() = e; what is the analogue of this relation in quantum me-
chanics?
8.Consider a particle in three-dimensional space, whose state vector is, and
whose wave function is(r) =r. Letbe an observable that commutes with
L=RP, the orbital angular momentum of the particle. Assuming that,L
2
and
form a C.S.C.O. inr, call their common eigenkets, whose eigenvalues are,
respectively,(the indexis assumed to be discrete),(+ 1)~
2
and~.
Let()be the unitary operator dened by:
() = e
~
whereis a real dimensionless parameter. For an arbitrary operator, we call
~
the
transform ofby the unitary operator():
~
=() ()
We set+= + ,= . Calculate
~
+ and show that+
and
~
+are proportional; calculate the proportionality constant. Same question for
and
~
.
Express
~
,
~
and
~
in terms of,and. What geometrical transformation
can be associated with the transformation ofLinto
~
L?
Calculate the commutators[ ]and[ ]. Show that the kets(
) and are eigenvectors ofand calculate their eigenvalues.
What relation must exist betweenandfor the matrix element
to be non-zero? Same question for .
By comparing the matrix elements of^ and
~
with those of and,
calculate
~
,
~
,
~
in terms of,,. Give a geometrical interpretation.
799

COMPLEMENT F VI
9.Consider a physical system of xed angular momentum, whose state space is
, and whose state vector is; its orbital angular momentum operator is denoted byL.
We assume that a basis ofis composed of2+1eigenvectors of( +),
associated with the wave functions()(). We callL=Lthe mean value
ofL.
We begin by assuming that:
= = 0
Out of all the possible states of the system, what are those for which the sum
()
2
+ ()
2
+ ()
2
is minimal? Show that, for these states, the root mean
square deviationof the component ofLalong an axis making an anglewith
is given by:
=~
2
sin
We now assume thatLhas an arbitrary direction with respect to theaxes.
We denote by a frame whose axis is directed alongL, with the
axis in theplane.
()Show that the state0of the system for which()
2
+ ()
2
+ ()
2
is minimal is such that:
(+ )0= 0
0=~0
()Let0be the angle betweenand , and0, the angle betweenand
; prove the relations:
+ = cos
20
2
e
0
+sin
20
2
e
0
sin0
= sin
0
2
cos
0
2
e
0
++ sin
0
2
cos
0
2
e
0
+ cos0
If we set:
0=
show that:
= tan
0
2
e
0
++ 1
+1
Expressin terms of,0,0and.
()To calculate, show that the wave function associated with0is0( ) =
(+)
()[whereis dened by equation (D-20) of Chapter], the
800

EXERCISES
one associated withbeing
(+)
(). By replacing,andin
this expression for0( )by their values in terms of,,, nd the
value ofand the relation:
=sin
0
2
cos
0
2
+
e
0
(2)!(+)!()!
()With the system in the state0,is measured. What are the probabilities
of the various possible results? What is the most probable result? Show that,
ifis much greater than 1, the results correspond to the classical limit.
10.LetJbe the angular momentum operator of an arbitrary physical system
whose state vector is.
Can states of the system be found for which the root mean square deviations,
andare simultaneously zero?
Prove the relation:

~
2
and those obtained by cyclic permutation of,,.
LetJbe the mean value of the angular momentum of the system. The
axes are assumed to be chosen in such a way that= = 0. Show that:
()
2
+ ()
2
~
Show that the two inequalities proven in question. both become equalities if and
only if+= 0or = 0.
The system under consideration is a spinless particle for whichJ=L=RP.
Show that it is not possible to have both=
~
2
and()
2
+
()
2
=~ unless the wave function of the system is of the form:
( ) =(sine)
11.Consider a three-dimensional harmonic oscillator, whose state vectoris:
=
where , and are quasi-classical states (cf.ComplementV) for one-
dimensional harmonic oscillators moving along,and, respectively. LetL=
RPbe the orbital angular momentum of the three-dimensional oscillator.
801

COMPLEMENT F VI
Prove:
=~
=~
2
+
2
and the analogous expressions for the components ofLalongand.
We now assume that:
= = 0 =~0
Show thatmust be zero. We then x the value of. Show that, in order to
minimize+ , we must choose:
= =
2
e
0
(where0is an arbitrary real number). Do the expressionsand()
2
+
()
2
in this case have minimum values compatible with the inequalities obtained
in question. of the preceding exercise?
Show that the state of a system for which the preceding conditions are satised is
necessarily of the form:
= () = =0 =0
with:
= =0 =0=
+
2!
=0 =0 =0
() =
!
e
2
2
; = e
0
(the results of ComplementVand of VIcan be used). Show
that the angular dependence of= =0 =0is(sine).
L
2
is measured on a system in the state. Show that the probabilities of the
various possible results are given by a Poisson distribution. What results can be
obtained in a measurement ofthat follows a measurement ofL
2
whose result
was(+ 1)~
2
?
Exercise 4 :
Reference: Landau and Lifshitz (1.19), 101; Ter Haar (1.23), 8.13 and 8.14.
802

Chapter VII
Particle in a central potential.
The hydrogen atom
A Stationary states of a particle in a central potential
A-1 Outline of the problem
A-2 Separation of variables
A-3 Stationary states of a particle in a central potential
B Motion of the center of mass and relative motion for a
system of two interacting particles
B-1 Motion of the center of mass and relative motion in classical
mechanics
B-2 Separation of variables in quantum mechanics
C The hydrogen atom
C-1 Introduction
C-2 The Bohr model
C-3 Quantum mechanical theory of the hydrogen atom
C-4 Discussion of the results
In this chapter, we shall consider the quantum mechanical properties of a parti-
cle placed in a central potential [that is, a potential()which depends only on the
distancefrom the origin]. This problem is closely related to the study of angular mo-
mentum presented in the preceding chapter. As we shall see in , the fact that()
is invariant under any rotation about the origin means that the Hamiltonianof the
particle commutes with the three components of the orbital angular momentum operator
L. This considerably simplies the determination of the eigenfunctions and eigenvalues
of, since these functions can be required to be eigenfunctions ofL
2
and as well.
This immediately denes their angular dependence, and the eigenvalue equation of
can be replaced by a dierential equation involving only the variable.
The importance of this problem derives from a property that will be established
in : a two-particle system in which the interaction is described by a potential energy
Quantum Mechanics, Volume I, Second Edition. C. Cohen-Tannoudji, B. Diu, and F. Laloë.
© 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

CHAPTER VII PARTICLE IN A CENTRAL POTENTIAL. THE HYDROGEN ATOM
that depends only on the relative positions of the particles can be reduced to a simpler
problem involving only one ctitious particle. In addition, when the interaction potential
of the two particles depends only on the distance between them, the ctitious particle's
motion is governed by a central potential. This explains why the problem considered in
this chapter is of such general interest: it is encountered in quantum mechanics whenever
we investigate the behavior of an isolated system composed of two interacting particles.
In , we shall apply the general methods already described to a special case:
that in which()is a Coulomb potential. The hydrogen atom, composed of an electron
and a proton which electrostatically attract each other, supplies the simplest example
of a system of this type. It is not the only one: in addition to hydrogen isotopes (deu-
terium, tritium), there are the hydrogenoid ions, which are systems composed of a single
electron and a nucleus, such as the ions He
+
, Li
++
, etc... (other examples will be given
in ComplementVII). For these systems, we shall explicitly calculate the energies of
the bound states and the corresponding wave functions. We also recall the fact that,
historically, quantum mechanics was introduced in order to explain atomic properties (in
particular, those of the simplest atom, hydrogen), which could not be accounted for by
classical mechanics. The remarkable agreement between the theoretical predictions and
the experimental observations constitutes one of the most spectacular successes of this
branch of physics. Finally, it should be noted that the exact results concerning the hy-
drogen atom serve as the basis of all approximate calculations relating to more complex
atoms (having several electrons).
A. Stationary states of a particle in a central potential
In this section, we consider a (spinless) particle of mass, subjected to a central force
derived from the potential()(the center of force is chosen as the origin).
A-1. Outline of the problem
A-1-a. Review of some classical results
The force acting on the classical particle situated at the point(withOM=r)
is equal to:
F=r() =
d
d
r
(A-1)
Fis always directed towards, and its moment with respect to this point is therefore
always zero. If:
=rp (A-2)
is the angular momentum of the particle with respect to, the angular momentum
theorem implies that:
d
d
=0 (A-3)
is therefore aconstant of the motion, so that the particle's trajectory is necessarily
situated in the plane passing throughand perpendicular to.
804

A. STATIONARY STATES OF A PARTICLE IN A CENTRAL POTENTIALM
O
v
v
⊥
v
r
Figure 1: Radial componentvand tangen-
tial componentvof a particle's velocity.
Now let us consider (Fig.) the position (denoted byOM=r) and velocityvof
the particle at the instant. The two vectorsrandvlie in the plane of the trajectory and
the velocityvcan be decomposed into the radial componentv(along the axis dened
byr) and the tangential componentv(along the axis perpendicular tor). The radial
velocity, the algebraic value ofv, is the time derivative of the distance of the particle
from the point:
r=
d
d
(A-4)
The tangential velocity can be expressed in terms ofand the angular momentum,
since:
rv=v (A-5)
so that the modulus of the angular momentumis equal to:
=rv=v (A-6)
The total energy of the particle:
=
1
2
v
2
+() =
1
2
v
2
+
1
2
v
2
+() (A-7)
can be written:
=
1
2
v
2
+
2
2
2
+() (A-8)
The classical Hamiltonian of the system is then:
=
2
2
+
2
2
2
+() (A-9)
where:
=
d
d
(A-10)
805

CHAPTER VII PARTICLE IN A CENTRAL POTENTIAL. THE HYDROGEN ATOM
is the conjugate momentum ofand
2
must be expressed in terms of the variables,
,and their conjugate momenta,,. One nds (cf.Appendix, ):
2
=
2
+
1
sin
2
2
(A-11)
In expression (A-9), the kinetic energy is broken into two terms: the radial kinetic
energy and the kinetic energy of rotation about. The reason is that, since()is
independent ofandin this case, the angular variables and their conjugate momenta
appear only in the
2
term. In fact, if we are interested in the evolution of, we can use
the fact thatis a constant of the motion, and replace
2
by a constant in expression
(A-9). The Hamiltonianthen appears as a function only of the radial variablesand
r(
2
plays the role of a parameter), and the result is a dierential equation involving
only one variable,:
dr
d
=
d
2
d
2
= (A-12a)
that is:
d
2
d
2
=
2
3
d
d
(A-12b)
It is just as if we had a one-dimensional problem (withvarying only between 0 and
+), with a particle of masssubjected to the eective potential:
e() =() +
2
2
2
(A-13)
We shall see that the situation is analogous in quantum mechanics.
A-1-b. The quantum mechanical Hamiltonian
In quantum mechanics, we want to solve the eigenvalue equation of the Hamiltonian
, the observable associated with the total energy. This equation is written, in ther
representation:
~
2
2
+()(r) =(r) (A-14)
Since the potentialdepends only on the distanceof the particle from the origin,
spherical coordinates (cf. ) are best adapted to the problem. We
therefore express the Laplacianin spherical coordinates
1
:
=
1
2
2
+
1
2
2
2
+
1
tan
+
1
sin
2
2
2
(A-15)
and look for eigenfunctions(r)that are functions of the variables,,.
1
Expression (A-15) gives the Laplacian only for non-zero. This is because of the privileged position
of the origin in spherical coordinates; it can be seen, moreover, that expression (A-15) is not dened for
= 0.
806

A. STATIONARY STATES OF A PARTICLE IN A CENTRAL POTENTIAL
If we compare expression (A-15) with the one for the operatorL
2
[formula (D-6a)
of Chapter], we see that the quantum mechanical Hamiltoniancan be put in a
form completely analogous to (A-9):
=
~
2
2
1
2
2
+
1
2
2
L
2
+()
(A-16)
The angular dependence of the Hamiltonian is contained entirely in theL
2
term, which
is an operator here. We could, in fact, perfect the analogy by dening an operatorr,
which would allow us to write the rst term of (A-16) like the one in (A-9).
We shall now show how one can solve the eigenvalue equation:
~
2
2
1
2
2
+
1
2
2
L
2
+()( ) =( ) (A-17)
A-2. Separation of variables
A-2-a. Angular dependence of the eigenfunctions
We know [cf.formulas (D-5) of Chapter] that the three components of the
angular momentum operatorLact only on the angular variablesand; consequently,
they commute with all operators acting only on the-dependence. In addition, they
commute withL
2
. Therefore, according to expression (A-16) for the Hamiltonian,the
three components ofLare constants of the motion
2
in the quantum mechanical sense:
[L] =0 (A-18)
Obviously,also commutes withL
2
.
Although we have at our disposition four constants of the motion (,,
andL
2
), we cannot use all four of them to solve equation (A-17) because they do not
commute with each other; we shall use onlyL
2
and. Since the three observables
L
2
andcommute, we can nd a basis of the state spacerof the particle composed of
eigenfunctions common to these three observables. We can, therefore, without restricting
the generality of the problem outlined in ( ),
solutions of equation (A-17), to be eigenfunctions ofL
2
andas well. We must then
solve the system of dierential equations:
(r) =(r) (A-19a)
L
2
(r) =(+ 1)~
2
(r) (A-19b)
(r) =~(r) (A-19c)
But we already know the general form of the common eigenfunctions ofL
2
and
(Chap., ): the solutions(r)of equations (A-19), corresponding to xed values
ofand, are necessarily products of a function ofalone and the spherical harmonic
():
(r) =()() (A-20)
2
Equation (A-18 is a scalar operator with respect to rotations about the
point(see ComplementVI). This is true because the potential energy is invariant under rotations
about.
807

CHAPTER VII PARTICLE IN A CENTRAL POTENTIAL. THE HYDROGEN ATOM
Whatever the radial function(),(r)is a solution of equations (A-19b) and (A-19c).
The only problem which remains to be solved is therefore how to determine()such
that(r)is also an eigenfunction of[equation (A-19a)].
A-2-b. The radial equation
We shall now substitute expressions (A-16) and (A-20) into equation (A-19a).
Since(r)is an eigenfunction ofL
2
with the eigenvalue(+ 1)~
2
, we see that()
is a common factor on both sides. After simplifying, we obtain the radial equation:
~
2
2
1d
2
d
2
+
(+ 1)~
2
2
2
+()() =() (A-21)
Actually, a solution of (A-21), substituted into (A-20), does not necessarily yield
a solution of the eigenvalue equation (A-14) of the Hamiltonian. As we have already
pointed out (cf.footnote), expression (A-15) for the Laplacian is not necessarily valid
at= 0. We must therefore make sure that the behavior of the solutions()of (A-21)
at the origin is suciently regular for (A-20) to be in fact a solution of (A-14).
Instead of solving the partial dierential equation (A-17) involving the three vari-
ables,,, we must now solve a dierential equation involving only the variable,
but dependent on a parameter: we are looking for eigenvalues and eigenfunctions of an
operatorwhich is dierent for each value of.
In other words, we consider separately, in the state spacer, the subspaces()
corresponding to xed values ofand(cf.Chap., ), studying the eigenvalue
equation ofin each of these subspaces (which is possible becausecommutes with
L
2
and). The equation to be solved depends on, but not on; it is therefore the
same in the(2+ 1)subspaces()associated with a given value of. We shall denote
by the eigenvalues of, that is, the eigenvalues of the Hamiltonianinside a
given subspace(). The index, which can be discrete or continuous, represents
the various eigenvalues associated with the same value of. As for the eigenfunctions of
, we shall label them with the same two indices as the eigenvalues:(). It is not
obvious that this is sucient: several radial functions might exist and be eigenfunctions
of the same operatorwith the same eigenvalue; we shall see in
not the case and that, consequently, the two indicesandare sucient to characterize
the dierent radial functions. We shall therefore rewrite equation (A-21) in the form:
~
2
2
1d
2
d
2
+
(+ 1)~
2
2
2
+() () = () (A-22)
We can simplify the dierential operator to be studied by a change in functions.
We set:
() =
1
() (A-23)
Multiplying both sides of (A-22) by, we obtain for()the following dierential
equation:
~
2
2
d
2
d
2
+
(+ 1)~
2
2
2
+() () = ()
(A-24)
808

A. STATIONARY STATES OF A PARTICLE IN A CENTRAL POTENTIAL
This equation is analogous to the one we would have to solve if, in aone-dimensional
problem, a particle of masswere moving in aneective potentiale():
e() =() +
(+ 1)~
2
2
2
(A-25)
Nevertheless, we must not lose sight of the fact that the variablecan take on only
non-negativereal values. The term(+ 1)~
2
2
2
which is added to the potential()
is always positive or zero; the corresponding force (equal to minus the gradient of this
term) always tends to repel the particle from the force center; this is why this term is
called thecentrifugal potential(or centrifugal barrier). Figure
the eective potentiale()for various values ofin the case where()is an attractive
Coulomb potential [() =
2
]: for1, the presence of the centrifugal term, which
predominates for smallvalues, causeseto be repulsive for short distances.l = 0
l = 1
l = 2
5
1
0
1
V
eff
(r)
r/a
0
a
o
e
2
×
Figure 2: Shape of the eective potential
e()for the rst values ofin the case
where() =
2
. When= 0,e()
is simply equal to(). Whentakes on
the values 1, 2, etc.,e()is obtained
by adding to()the centrifugal potential
(+1)~
2
2
2
, which approaches+when
approaches zero.
A-2-c. Behavior of the solutions of the radial equation at the origin
We have already pointed out that it is necessary to examine the behavior of the
solutions()of the radial equation (A-21) at the origin in order to know if they are
really solutions of (A-14).
We shall assume that whenapproaches zero, the potential()remains nite,
or at least approaches innity less rapidly than1(this hypothesis is true in most cases
encountered in physics and, in particular, in the case of the Coulomb potential, to be
studied in ). We shall consider a solution of (A-22) and assume that it behaves at the
origin like:
()
0
(A-26)
809

CHAPTER VII PARTICLE IN A CENTRAL POTENTIAL. THE HYDROGEN ATOM
Substituting (A-26) into (A-22), and setting the coecient of the dominant term equal
to zero, we obtain the equation:
(+ 1) +(+ 1) = 0 (A-27)
and, consequently:
either= (A-28a)
or=(+ 1) (A-28b)
For a given value of, there are therefore two linearly independent solutions
of the second-order equation (A-22), behaving at the origin likeand1
+1
, respec-
tively. But those which behave like1
+1
must be rejected, since it can be shown
3
that
1
+1
()is not a solution of the eigenvalue equation (A-14) for= 0. From this,
we see that acceptable solutions of (A-24) go to zero at the origin for all, since:
()
0
+1
(A-29)
Consequently, to equation (A-24) must be added the condition:
(0) = 0 (A-30)
Comment:
In equation (A-24),, the distance of the particle from the origin, varies only between
0 and+. However, thanks to condition (A-30), we can assume that we are actually
dealing with a one-dimensional problem, in which the particle can theoretically move
along the entire axis, but in which the eective potential is innite for all negative values
of the variable. We know that, in such a case, the wave function must be identically zero
on the negative half-axis; condition (A-30) insures the continuity of the wave function at
= 0.
A-3. Stationary states of a particle in a central potential
A-3-a. Quantum numbers
We can summarize the results of 2 as follows: the fact that the potential()is
independent ofandmakes it possible:
() to require the eigenfunctions ofto besimultaneous eigenfunctions ofL
2
and
,which determines their angular dependence:
(r) =()() =
1
()() (A-31)
()to replace the eigenvalue equation of, an equation involving partial deriva-
tives with respect to,,,by a dierential equation involving only the variableand
depending on a parameter[equation (A-24)], with condition (A-30) imposed.
3
This is because the Laplacian of
1
+1
()involves theth derivatives of(r)(cf.Appendix,
end of
810

A. STATIONARY STATES OF A PARTICLE IN A CENTRAL POTENTIAL
These results can be compared with those recalled in 1-a, of which they are the
quantum mechanical analogues.
In principle, the functions( )must be square-integrable, that is, nor-
malizable:
( )
22
dd = 1 (A-32)
Their form (A-31) allows us to separate radial and angular integrations:
( )
22
dd =
0
2
d ()
2
d ()
2
(A-33)
But the spherical harmonics()are normalized with respect toand; condition
(A-32) therefore reduces to:
0
2
d ()
2
=
0
d ()
2
= 1 (A-34)
Actually, we know that it is often convenient to accept eigenfunctions of the Hamil-
tonian that are not square-integrable. If the spectrum ofhas a continuous part, we
shall require only that the corresponding eigenfunctions be orthonormalized in the ex-
tended sense, that is, that they satisfy a condition of the form:
0
2
d ()() =
0
d ()() =( ) (A-35)
whereis a continuous index.
In (A-34) and (A-35), the integrals converge at their lower limit,= 0[condition
(A-30)]. This is physically satisfying since the probability of nding the particle in any
volume of nite dimensions is then always nite. It is therefore only because of the
behavior of the wave functions for that, in the case of a continuous spectrum,
the normalization integrals (A-35) diverge if=.
Finally, the eigenfunctions of the Hamiltonianof a particle placed in a cen-
tral potential()depend on at least three indices [formula (A-31)]: ( ) =
()()is a simultaneous eigenfunction ofL
2
and with the respective
eigenvalues,(+ 1)~
2
and~.is called theradialquantum number;, the
azimuthalquantum number; and, themagneticquantum number. The radial part
() =
1
()of the eigenfunction and the eigenvalueofare independent of
the magnetic quantum number and are given by the radial equation (A-24). The angular
part of the eigenfunction depends only onandand not on; it does not depend on
the form of the potential().
A-3-b. Degeneracy of the energy levels
Finally, we shall consider the degeneracy of the energy levels, that is, of the eigen-
values of the Hamiltonian. The(2+ 1)functions ( )withandxed and
varying fromto+are eigenfunctions ofwith the same eigenvalue[these
(2+ 1)functions are clearly orthogonal, since they correspond to dierent eigenvalues
of]. The levelis therefore at least(2+ 1)-fold degenerate. This degeneracy,
811

CHAPTER VII PARTICLE IN A CENTRAL POTENTIAL. THE HYDROGEN ATOM
which exists for all potentials(), is called anessential degeneracy: it is due to the
fact that the HamiltoniancontainsL
2
, but not, which means thatdoes not
appear in the radial equation
4
. It is also possible for one of the eigenvaluesof the
radial equation corresponding to a given value ofto be the same as an eigenvalue,
associated with another radial equation, characterized by=. This occurs only for
certain potentials(). The resulting degeneracies are calledaccidental(we shall see in

We must now show that, for a xed value of, the radial equation has at most
one physically acceptable solution for each eigenvalue. This actually results from
condition (A-30). The radial equation, since it is a second-order dierential equation,
hasa prioritwo linearly independent solutions for each value of. Condition (A-30)
eliminates one of them, so there is at most one acceptable solution for each value of
. We must also consider the behavior of the solutions forapproaching innity; if
() 0when , the negative values offor which the solution we have just
chosen is also acceptable at innity (that is, bounded) form a discrete set (see example
of VII).
It follows from the preceding considerations thatL
2
andconstitute a C.S.C.O.
5
.
If we x three eigenvalues,(+ 1)~
2
and~, there corresponds to them a single
function (r). The eigenvalue ofL
2
indicates which equation yields the radial func-
tion; the eigenvalue ofdetermines this radial function()uniquely, as we have
just seen; nally, there exists only one spherical harmonic()for a givenand.
B. Motion of the center of mass and relative motion for a system of two
interacting particles
Consider a system of two spinless particles, of masses1and2and positionsr1andr2.
We assume that the forces exerted on these particles are derived from apotential energy
(r1r2)which depends only onr1r2. This is true if there are no forces originating
outside the system (that is, the system is isolated), and if the interactions between the
two particles are derived from a potential. This potential must depend only onr1r2,
since only the relative positions of the two particles are involved. We shall show that the
study of such a system can be reduced to that of a single particle placed in the potential
(r).
B-1. Motion of the center of mass and relative motion in classical mechanics
In classical mechanics, the two-particle system is described by the Lagrangian (cf.
Appendix):
(r1_r1;r2_r2) = =
1
2
1_r
2
1+
1
2
2_r
2
2 (r1r2) (B-1)
4
This essential degeneracy appears whenever the Hamiltonian is rotation-invariant (cf.Comple-
mentVI). This is why it is encountered in numerous physical problems.
5
Actually, we have not proven that these operators are observables, that is, that the set of(r)
form a basis in the state spacer.
812

B. MOTION OF THE CENTER OF MASS AND RELATIVE MOTION FOR A SYSTEM OF TWO INTERACTING
PARTICLES
and the conjugate momenta of the six coordinates of the two particles are the components
of the mechanical momenta:
p1= 1_r1
p2= 2_r2 (B-2)
The study of the motion of the two particles is simplied by replacing the positions
rby the threecoordinates of the center of mass(or center of gravity):
r=
1r1+2r2
1+2
(B-3)
and the threerelative coordinates
6
:
r=r1r2 (B-4)
Formulas (B-3) and (B-4) can be inverted to yield:
r1=r+
2
1+2
r
r2=r
1
1+2
r (B-5)
The Lagrangian can then be written, in terms of the new variablesrandr:
(r_r;r_r) =
1
2
1_r+
2
1+2
_r
2
+
1
2
2_r
1
1+2
_r
2
(r)
=
1
2
_r
2
+
1
2
_r
2
(r) (B-6)
where:
= 1+2 (B-7)
is thetotal massof the system, and:
=
12
1+2
(B-8a)
is itsreduced mass(the geometrical mean of the two masses1and2), which is also
given by:
1
=
1
1
+
1
2
(B-8b)
The conjugate momenta of the variablesrandrare obtained by dierentiating
expression (B-6) with respect to the components of_rand_r. Using (B-3), (B-4) and
(B-2), we nd:
p=_r= 1_r1+2_r2=p1+p2 (B-9a)
p=_r=
2p1 1p2
1+2
(B-9b)
6
Denition (B-4) introduces a slight asymmetry between the two particles.
813

CHAPTER VII PARTICLE IN A CENTRAL POTENTIAL. THE HYDROGEN ATOM
or:
p
=
p1
1
p2
2
(B-9c)
pis thetotal momentumof the system, andpis called therelative momentumof the
two particles.
We can express the classical Hamiltonian of the system in terms of the new dy-
namical variables we have just introduced:
(rp;rp) =
p
2
2
+
p
2
2
+(r) (B-10)
This leads to the following equations of motion [formulas (27) of Appendix]:
_p=0 (B-11)
_p=r(r) (B-12)
The rst term of expression (B-10) represents the kinetic energy of a ctitious
particle whose masswould be the sum1+2of the masses of the two real particles,
whose position would be that of the center of mass of the system [formula (B-3)], and
whose momentumpwould be the total momentump1+p2of the system. Equation (B-
11) indicates that this ctitious particle is in uniform rectilinear motion (free particle).
This result is well known in classical mechanics: the center of mass of a system of particles
moves like a single particle whose mass is the total mass of this system, subjected to the
resultant of all the forces exerted on the various particles. Here, this resultant is zero
since the only forces present are internal ones obeying the principle of action and reaction.
Since the center of mass is in uniform rectilinear motion with respect to the initially
chosen frame, the frame in which it is at rest (p=0) is also an inertial frame. In this
center of mass frame, the rst term of (B-10) is zero. The classical Hamiltonian, that is,
the total energy of the system, then reduces to:
=
p
2
2
+(r) (B-13)
is the energy associated with therelative motionof the two particles. It is obviously
this relative motion that is the most interesting in the study of the two interacting par-
ticles. It can be described by introducing a ctitious particle, called therelative particle:
its mass is the reduced massof the two real particles, its position is characterized by the
relative coordinatesr, and its momentum is the relative momentump. Since its motion
obeys equation (B-12), it behaves as if it were subjected to a potential(r)equal to the
potential energy of interaction between the two real particles.
The study of the relative motion of two interacting particles therefore reduces to
that of the motion of a single ctitious particle, characterized by formulas (B-4), (B-8)
and (B-9c). This last equation expresses the fact that the velocitypof the relative
particle is indeed the dierence between the velocities of the two particles, that is, their
relative velocity.
B-2. Separation of variables in quantum mechanics
The considerations of the preceding section can easily be transposed to quantum
mechanics, as we shall now show.
814

B. MOTION OF THE CENTER OF MASS AND RELATIVE MOTION FOR A SYSTEM OF TWO INTERACTING
PARTICLES
B-2-a. Observables associated with the center of mass and the relative particle
The operatorsR1,P1andR2,P2, which describe the positions and momenta of
the two particles of the system, satisfy the canonical commutation relations:
[11] =~
[22] =~ (B-14)
with analogous expressions for the components alongand. All the observables
labeled by the index 1 commute with all those of index 2, and all the observables relating
to one of the axes,orcommute with those corresponding to another one of
these axes.
Now let us dene the observablesRandRby formulas similar to (B-3) and
(B-4):
R=
1R1+2R2
1+2
(B-15a)
R=R1R2 (B-15b)
and the observablesPandPby formulas similar to (B-9):
P=P1+P2 (B-16a)
P=
2P1 1P2
1+2
(B-16b)
It is easy to calculate the various commutators of these new observables. The results
are as follows:
[ ] =~ (B-17a)
[ ] =~ (B-17b)
with analogous expressions for the components alongand; all the other commu-
tators are zero. Consequently,RandP, likeRandP, satisfy canonical commutation
relations. Moreover, every observable of the setRPcommutes with every observable
of the setRP.
We can also interpretRandP,on the one hand, andRandP,on the other,
as being the position and momentum observables of two distinct ctitious particles.
B-2-b. Eigenvalues and eigenfunctions of the Hamiltonian
The Hamiltonian operator of the system is obtained from formulas (B-1) and (B-2)
and the quantization rules of Chapter:
=
P
2
1
21
+
P
2
2
22
+(R1R2) (B-18)
Since denitions (B-15) and (B-16) are formally identical to (B-3), (B-4) and (B-9), and
since all the momentum operators commute, a simple algebraic calculation yields the
equivalent of expression (B-10).
=
P
2
2
+
P
2
2
+(R) (B-19)
815

CHAPTER VII PARTICLE IN A CENTRAL POTENTIAL. THE HYDROGEN ATOM
The Hamiltonianthen appears as the sum of two terms:
= + (B-20)
with:
=
P
2
2
(B-21a)
=
P
2
2
+(R) (B-21b)
which commute, according to the results of :
[ ] = 0 (B-22)
andtherefore commute with. It follows that there exists a basis of eigenvectors
ofthat are also eigenvectors ofand; we shall therefore look for solutions of the
system:
=
= (B-23)
which immediately implies, according to (B-20):
= (B-24)
with:
= + (B-25)
Consider therrrepresentation, whose basis vectors are the eigenvectors
common to the observablesRandR. In this representation, a state is characterized by
a wave function(rr)which is a function of six variables. The action of the operators
RandRis expressed by the multiplication of the wave functions by the variablesr
andrrespectively.PandPbecome the dierential operators
~
rand
~
r(where
rdenotes the set of three operators, and ). The state spaceof
the system can then be considered to be the tensor productr rof the state space
rassociated with the observableRand the spacerassociated withR. and
then appear as the extensions intoof operators actually acting only inrandr,
respectively. We can therefore, as we saw in , nd a basis of eigenvectors
satisfying (B-23), in the form:
= r (B-26)
with:
=
r
(B-27a)
rr=rr
r r
(B-27b)
816

B. MOTION OF THE CENTER OF MASS AND RELATIVE MOTION FOR A SYSTEM OF TWO INTERACTING
PARTICLES
Writing these equations in therandrrepresentations respectively, we obtain:
~
2
2
(r) = (r) (B-28a)
~
2
2
+(r)r(r) =rr(r) (B-28b)
The rst of these equations, (B-28a), shows that the particle associated with the
center of mass of the system is free, as in classical mechanics. We know its solutions:
they are, for example, the plane waves:
(r) =
1
(2~)
32
e
~
pr
(B-29)
whose energy is equal to:
=
p
2
2
(B-30)
can take on any positive value or zero; it is the kinetic energy corresponding to a
translation of the system as a whole.
The more interesting equation from a physical point of view is the second one,
(B-28b), which concerns the relative particle. It describes the behavior of the system of
the two interacting particles in the center of mass frame. If the interaction potential of
the two real particles depends only on the distance between them,r1r2, and not on
the direction of the vectorr1r2, the relative particle is subjected to a central potential
(); the problem is then reduced to the one treated in .
Comment:
The total angular momentum of the system of the two real particles is:
J=L1+L2 (B-31)
with:
L1=R1P1
L2=R2P2 (B-32)
It can easily be shown that it can also be written:
J=L+L (B-33)
where:
L=R P
L=RP (B-34)
are the angular momenta of the ctitious particles (according to the results of
,LandLsatisfy the commutation relations that characterize angular
momenta, and the components ofLcommute with those ofL).
817

CHAPTER VII PARTICLE IN A CENTRAL POTENTIAL. THE HYDROGEN ATOM
C. The hydrogen atom
C-1. Introduction
The hydrogen atom consists of a proton, of mass:
1710
27
kg (C-1)
and charge:
1610
19
Coulomb (C-2)
and of an electron, of mass:
09110
30
kg (C-3)
and charge. The interaction between these two particles is essentially electrostatic.
The corresponding potential energy is:
()
2
40
1
=
2
(C-4)
wheredenotes the distance between the two particles, and:
2
40
=
2
(C-5)
Using the results of , we conne ourselves to the study of this system in the
center of mass frame. The classical Hamiltonian that describes the relative motion of the
two particles is then
7
:
(rp) =
p
2
2
2
(C-6)
Since [formulas (C-1) and (C-3)], the reduced massof the system is very
close to:
=
+
1 (C-7)
(the correction term is on the order of 1/1 800). This means that the center
of mass of the system is practically at the position of the proton, and that the relative
particle can be identied, to a very good approximation, with the electron. This is
why we shall adopt the slightly inaccurate convention of calling the relative particle the
electron and the center of mass the proton.
7
Henceforth, we shall omit the indexwhich was used to label in
to the relative motion.
818

C. THE HYDROGEN ATOM
C-2. The Bohr model
We shall briey review the results of the Bohr model, which relate to the hydrogen
atom. This model, which is based on the concept of a trajectory, is incompatible with
the ideas of quantum mechanics. However, it allows us to introduce, in a very simple
way, fundamental quantities such as the ionization energyof the hydrogen atom and
a parameter which characterizes atomic dimensions (the Bohr radius0). In addition, it
so happens that the energiesgiven by the Bohr theory are the same as the eigenvalues
of the Hamiltonian we shall calculate in . Finally, quantum mechanical theory is in
agreement with some of the intuitive images of the Bohr model ( ).
This semi-classical model is based on the hypothesis that the electron describes a
circular orbitof radiusabout the proton, obeying the following equations:
=
1
2
2
2
(C-8)
2
=
2
2
(C-9)
=~;a positive integer (C-10)
The rst two equations are classical ones. (C-8) expresses the fact that the total energy
of the electron is the sum of its kinetic energy
2
2and its potential energy
2
.
(C-9) is none other than the fundamental equation of Newtonian dynamics (
22
is the
Coulomb force exerted on the electron, and
2
is the acceleration of its uniform circular
motion). The third equation expresses the quantization condition, introduced empirically
by Bohr in order to explain the existence of discrete energy levels: he postulated that
only circular orbits satisfying this condition are possible trajectories for the electron. The
dierent orbits, as well as the corresponding values of the various physical quantities, are
labeled by the integerassociated with them.
A very simple algebraic calculation then yields the expressions for,and:
=
1
2
(C-11a)
=
2
0 (C-11b)
=
1
0 (C-11c)
with:
=
4
2~
2
(C-12a)
0=
~
2
2
(C-12b)
0=
2
~
(C-12c)
When this model was proposed by Bohr, it marked an important step towards the
understanding of atomic phenomena, since it yielded the correct values for the energy
levels of the hydrogen atom. These values indeed follow the1
2
(the Balmer formula)
819

CHAPTER VII PARTICLE IN A CENTRAL POTENTIAL. THE HYDROGEN ATOM
law indicated by expression (C-11a). Moreover, the experimentally measuredionization
energy(the energy which must be supplied to the hydrogen atom in its ground state in
order to remove the electron) is equal to the numerical value of:
136 eV (C-13)
Finally, theBohr radius0indeed characterizes atomic dimensions:
0052

A (C-14)
Comment:
ComplementIshows how the uncertainty principle, applied to the hydrogen
atom, explains the existence of a stable ground state and permits the evaluation
of the order of magnitude of its energy and its spatial extension.
C-3. Quantum mechanical theory of the hydrogen atom
We shall now take up the question of the determination of the eigenvalues and
eigenfunctions of the Hamiltoniandescribing the relative motion of the proton and
the electron in the center of mass frame [formula (C-6)]. In therrepresentation, the
eigenvalue equation of the Hamiltonianis written:
~
2
2

2
(r) =(r) (C-15)
Since the potential
2
is central, we can apply the results of : the eigenfunctions
(r)are of the form:
(r) =
1
()() (C-16)
()is given by the radial equation, (A-24), that is:
~
2
2
d
2
d
2
+
(+ 1)~
2
2
2
2
() = () (C-17)
We add to this equation condition (A-30):
(0) = 0 (C-18)
It can be shown that the spectrum ofincludes a discrete part (negative eigenvalues) and
a continuous part (positive eigenvalues). Consider Figure, which shows the eective potential
for a given value of(the gure is drawn for= 0, but the reasoning remains valid for= 0).
For a positive value of, the classical motion is not bounded in space: for the value
0chosen in Figure, it is limited on the left by the abscissa of point, but it is not limited
on the right
8
. As a result (cf.Complement III) equation (C-17) has acceptable solutions
for any 0. The spectrum ofis therefore continuous for0, and the corresponding
eigenfunctions are not square-integrable.
8
For a1potential, the classical trajectories are conic sections; unbounded motion follows a
hyperbola or a parabola.
820

C. THE HYDROGEN ATOM
On the other hand, for 0, the classical motion is bounded: it is conned to the
region between the abscissas of the two pointsand
9
. We shall see later that equation
(C-17) has acceptable solutions only for certain discrete values of. The spectrum ofis
therefore discrete for0, and the corresponding eigenfunctions are square-integrable.V
eff
(r)
r/a
0
B
C
A
E > 0
E < 0
1 5
0
Figure 3: For a positive value of the energy
, the classical motion is not bounded. The
spectrum of the quantum mechanical Hamil-
tonianis therefore continuous for0,
and the corresponding eigenfunctions are not
normalizable. On the other hand, for nega-
tive, the classical motion is limited to the
interval. The spectrum ofis therefore
discrete for 0, and the corresponding
eigenfunctions are normalizable.
C-3-a. Change of variables
To simplify the reasoning, we shall choose0and[formulas (C-12)] as the units
of length and energy. That is, we shall introduce the dimensionless quantities:
= 0 (C-19)
=
(C-20)
(the quantity under the radical sign is positive, since we are looking for the bound states).
With expressions (C-12a) and (C-12b) forand0taken into account, the radial
equation (C-17) becomes simply:
d
2
d
2
(+ 1)
2
+
2
2
() = 0 (C-21)
C-3-b. Solving the radial equation
In order to solve equation (C-21), we shall use the method illustrated in Comple-
mentV, expanding()in a power series.
. Asymptotic behavior
Let us determine the asymptotic behavior of()qualitatively. Whenap-
proaches innity, the terms in1and1
2
become negligible compared to the constant
9
The classical trajectory is then an ellipse or a circle.
821

CHAPTER VII PARTICLE IN A CENTRAL POTENTIAL. THE HYDROGEN ATOM
term
2
, so that equation (C-21) practically reduces to:
d
2
d
2
2
() = 0 (C-22)
whose solutions are e. This argument is not rigorous, since we have completely
neglected the terms in1and1
2
; actually, it can be shown that()is equal to
e multiplied by a power of.
We shall later be led by physical considerations to require the function()to
be bounded at innity, and hence to reject the solutions of (C-21) whose asymptotic
behavior is governed by e
+
. This is why we perform the change of function:
() = e () (C-23)
Although this change of function singles oute , it clearly does not eliminate solutions
ine
+
, which must be identied and then rejected at the end of the calculation. The
dierential equation that()must satisfy can easily be derived from (C-21):
d
2
d
2
2
d
d
+
2(+ 1)
2
() = 0 (C-24)
Condition (C-18) must be associated with this equation, that is:
(0) = 0 (C-25)
. Solutions in the form of power series
Consider the expansion of()in powers of:
() =
=0
(C-26)
By denition,0is the rst non-zero coecient of this expansion:
0= 0 (C-27)
Condition (C-25) implies thatis strictly positive.
We calculate
d
d
()and
d
2
d
2
()from (C-26):
d
d
() =
=0
(+)
+1
(C-28a)
d
2
d
2
() =
=0
(+)(+ 1)
+2
(C-28b)
To obtain the left-hand side of (C-24), we multiply expressions (C-26), (C-28a) and
(C-28b) respectively by the factors
2
(+ 1)
2
,2and 1. According to (C-24),
the series so determined must be identically zero, that is, all its coecients must be zero.
822

C. THE HYDROGEN ATOM
The lowest order term is in
2
. Taking its coecient as zero, we obtain:
[(+ 1) +(1)]0= 0 (C-29)
If we take (C-27) into account, we see thatcan take on one of two values:
=+ 1 (C-30a)
= (C-30b)
(in agreement with the general result of ). We have seen that only (C-30a) gives a
behavior at the origin that can lead to an acceptable solution [condition (C-25)]. Setting
the coecient of the general term in
+2
equal to zero, we obtain (with=+ 1) the
following recurrence relation:
(+ 2+ 1)= 2 [(+) 1]1 (C-31)
If we x0, this relation enables us to calculate1, then2, and thus by recurrence
all the coecients. Since 1approaches zero when , the corresponding
series is convergent for all. Thus we have determined, for any value of, the solution
of (C-24) that satises condition (C-25).
C-3-c. Energy quantization. Radial functions
We are now going to require the preceding solution to have a physically acceptable
asymptotic behavior (cf. ). This will involve quantization of the possible values
of.
If the term in brackets on the right-hand side of (C-31) does not go to zero for any
integer, expansion (C-26) is a true innite series, for which:
1
2
(C-32)
Now, the power series expansion of the function e
2
is written:
e
2
=
=0
=
(2)
!
(C-33)
which implies:
1
=
2
(C-34)
If we compare (C-32) and (C-34), we see
10
that, for large values of, the series being
considered behaves like e
2
. The corresponding function[formula (C-23)] is then
proportional to e
+
, which is not physically acceptable.
Consequently, we must reject all cases in which expansion (C-26) is an innite
series. The only possible values ofare those for which (C-26) has only a nite
10
The reader can nd a fuller discussion relating to an analogous problem in ComplementV.
823

CHAPTER VII PARTICLE IN A CENTRAL POTENTIAL. THE HYDROGEN ATOM
number of terms, that is, those for whichreduces to a polynomial. The corresponding
functionis then physically acceptable, since its asymptotic behavior is dominated
bye . Therefore, all we need is an integersuch that the term in brackets of the
right-hand side of (C-31) goes to zero for=: the corresponding coecientis then
zero, as are all those of higher order, since the fact thatis zero means that+1is as
well, and so on. For xed, we label the corresponding values ofby this integer
(note thatis greater than or equal to 1, since0never goes to zero). We then have,
according to (C-31):
=
1
+
(C-35)
For a given, the only negative energies possible are therefore [formula (C-20)]:
=
(+)
2
; = 123 (C-36)
We shall discuss this result in .
is therefore a polynomial, whose term of lowest order is in
+1
and whose
term of highest order is in
+
. Its various coecients can be calculated in terms of0
by solving recurrence relation (C-31), which can be written, using (C-35):
=
2( )
(+ 2+ 1)(+)
1 (C-37)
It is easy to show that:
= (1)
2
+
(1)!
( 1)!
(2+ 1)!
!(+ 2+ 1)!
0 (C-38)
()is then given by formula (C-23), and0is determined (to within a phase
factor) by normalization condition (A-34) [we must rst, of course, return to the variable
by using (C-19)]. Finally, we obtain the true function()by dividing()by.
The following three examples give an idea of the form of these radial functions:
=1=0() = 2(0)
32
e
0
(C-39a)
=2=0() = 2(20)
32
1
20
e
20
(C-39b)
=1=1() = (20)
32
1
3
0
e
20
(C-39c)
C-4. Discussion of the results
C-4-a. Order of magnitude of atomic parameters
Formulas (C-36) and (C-39) show that, for the hydrogen atom, the ionization
energy, dened by (C-12a), and the Bohr radius, given by (C-12b), play an important
role. These quantities give an order of magnitude of the energies and spatial extensions
of the wave functions associated with the bound states of the hydrogen atom.
824

C. THE HYDROGEN ATOM
Relations (C-12a) and (C-12b) can be written in the form:
=
1
2
22
(C-40a)
0=
1
- (C-40b)
whereis thene structure constant, a dimensionless constant which plays a very
important role in physics:
=
2
~
=
2
40~
1
137
(C-41)
and where-is dened by:
-=
~
(C-42)
Sinceis almost the same as, the rest mass of the electron,-is practically equal to
theCompton wavelengthof the electron, which is given by:
~
3810
3
A (C-43)
Relation (C-40b) therefore indicates that0is on the order of one hundred times
the Compton wavelength of the electron. Relation (C-40a) shows that the order of
magnitude of the binding energy of the electron is between 10
42
and10
52
, where
2
is practically equal to the rest energy of the electron:
2
05110
6
eV (C-44)
It follows that:
2
(C-45)
This justies our choice of the non-relativistic Schrödinger equation to describe the hy-
drogen atom. Of course, relativistic eects, although small, do exist; nevertheless, their
smallness allows them to be studied by perturbation theory (cf.Chap.).
C-4-b. Energy levels
. Possible values of the quantum numbers; degeneracies
For xed, there exists an innite number of possible energy values [formula (C-
36)], corresponding to= 123, ... Each of them is at least(2+ 1)-fold degenerate:
this is anessential degeneracyrelated to the fact that the radial equation depends only
on the quantum numberand not on(). But, in addition, there existaccidental
degeneracies: equation (C-36) indicates that two eigenvaluesand correspond-
ing to dierent radial equations (=) are equal if+=+. Figure, in which the
rst eigenvalues associated with= 012and 3 are shown on a common energy scale,
clearly reveals several accidental degeneracies.
825

CHAPTER VII PARTICLE IN A CENTRAL POTENTIAL. THE HYDROGEN ATOMl = 0
(s)
l = 1
(p)
l = 2
(d)
l = 3
(f)
4s 4p 4d 4f
3s 3p 3d
2s
1s
(n = 2)
(n = 1)
– E
I
E
(n = 3)
(n = 4)
0
2p
Figure 4: Energy levels of the hydrogen atom. The energyof each level depends only
on. Ifis xed, several values ofare possible:= 012 1. To each of these
values ofcorrespond(2+ 1)possible values for:
= + 1
Consequently, the levelis
2
-fold degenerate.
In the special case of the hydrogen atom,does not depend onandseparately,
but only on their sum. We set:
=+ (C-46)
The various energy states are labeled by the integer(greater than or equal to 1), and
(C-36) becomes:
=
1
2
(C-47)
According to (C-46), it is equivalent to specifyandorandto determine the
eigenfunctions. Following convention, from now on we shall use the quantum numbers
826

C. THE HYDROGEN ATOM
and. The energy is xed by, which is called theprincipal quantum number; a given
value ofcharacterizes what is called anelectron shell.
Sinceis necessarily an integer which is greater than or equal to 1 ( 3-c above),
there is only a nite number of values ofassociated with the same value of. According
to (C-46), ifis xed, one can have:
= 012 1 (C-48)
The shell characterized byis said to containsub-shells
11
, each one corresponding to
one of the values ofgiven in (C-48). Finally, each sub-shell contains(2+ 1)distinct
states, associated with the(2+ 1)possible values offor xed.
The total degeneracy of the energy levelis therefore:
=
1
=0
(2+ 1) = 2
(1)
2
+=
2
(C-49)
We shall see in Chapter
2 (if we also take into account the proton spin, which is equal to that of the electron, we
obtain another factor of 2).
. Spectroscopic notation
For historical reasons (dating from the period, before the development of quantum
mechanics, in which the study of spectra resulted in an empirical classication of the
numerous lines observed), letters of the alphabet are associated with the various values
of. The correspondence is as follows:
= 0
= 1
= 2
= 3
= 4
.
.
.
.
.
.
alphabetical order (C-50)
Therefore,spectroscopic notationlabels a sub-shell by the corresponding numberfol-
lowed by the letter that characterizes the value of. Thus, the ground level [which is
non-degenerate, according to (C-49)], sometimes called the shell, includes only the
1sub-shell; the rst excited level, or shell, includes the 2and 2sub-shells; the
second excited level (shell) includes the 3, 3and 3sub-shells, etc. (The capi-
tal letters sometimes associated with the successive shells follow an alphabetical order,
starting with the letter.)
11
The concept of a sub-shell can even be found in the semi-classical model of Sommerfeld. This model
assigns, to each valueof Bohr's quantum number,elliptical orbits of the same energy and dierent
angular momenta. One of these orbits is circular; it is the one that corresponds to the maximum value
of the angular momentum.
827

CHAPTER VII PARTICLE IN A CENTRAL POTENTIAL. THE HYDROGEN ATOM
C-4-c. Wave functions
The wave functions associated with the eigenstates common toL
2
,and the
Hamiltonianof the hydrogen atom are generally labeled, not by the three quantum
numbers,,, as we have done until now, but by,and[passage from one set
to the other simply involves use of relation (C-46)]. Since the operatorsL
2
and
constitute a C.S.C.O. (cf.), specication of the three integersand, which
is equivalent to that of the eigenvalues of,L
2
and, unambiguously determines the
corresponding eigenfunction(r).
. Angular dependence
As is the case for any central potential, the functions(r)are products of
a radial function and a spherical harmonic(). To visualize their angular de-
pendence on the axis characterized by the polar anglesand, we can measure o a
distance that is proportional to( )
2
for anyxed, that is, proportional to
()
2
. Thus, we obtain a surface of revolution about theaxis, since we know
that()depends ononly through the factore( ); con-
sequently,()
2
is independent of. We can therefore represent its cross-section
by a plane containing. This is what is done in Figure, for= 0and= 0, 1
and 2 [the corresponding spherical harmonics are given in ComplementVI, formulas
(31), (32) and (33)]:
0
0is a constant, and is therefore spherically symmetric;
0
1
2
is
proportional tocos
2
, and
0
2
2
to(3 cos
2
1)
2
.O
O O
z
zz
l = 0
m = 0
l = 1
m = 0
l = 2
m = 0
Figure 5: Angular dependence,(), of some stationary wave functions of the hy-
drogen atom, corresponding to well-dened values ofand. For each direction of polar
angles,, the value of()
2
is recorded; a surface of revolution about theaxis
is thus obtained. When= 0, this surface is a sphere centered at; it becomes more
complicated for higher values of.
828

C. THE HYDROGEN ATOM
Figure 6: Radial dependence()of wave
functions associated with the rst few levels
of the hydrogen atom. When 0,()
behaves like; only thestates (for which
= 0) have a non-zero position probability
at the origin.
. Radial dependence
The radial functions(), each of which characterizes a sub-shell, can be cal-
culated from the results of , paying attention, however, to the change in notation
introduced by formula (C-46). Figure of the
three radial functions given in (C-39):
=1=0 =1=0; =2=0 =2=0; =1=1 =2=1 (C-51)
The behavior of()in the neighborhood of= 0is that of(see discussion
of ). Consequently,only states belonging tosub-shells(= 0)give a non-zero
probability of presence at the origin. The greater, the larger the region around the
proton in which the position probability of the electron is negligible. This has a certain
number of physical consequences, particularly in the phenomenon of electron capture by
certain nuclei, and in the hyperne structure of lines (cf.Chap., ).
Finally, we can derive formula (C-11b) for the successive Bohr radii. To do so,
consider the various states for which= 1
12
. We calculate the variation withof
the probability density for each of the preceding levels in an innitesimal solid angled
about a xed direction of polar anglesand. In general, the position probability for
the electron in the volume element d
3
=
2
ddsituated at the point( )is given
by:
d
3
( ) = ( )
22
dd
= ()
22
d ()
2
d (C-52)
Here, we have xed,andd. The probability of nding the electron betweenand
+ d, inside the solid angle under consideration, is then proportional to
2
()
2
d.
The corresponding density is therefore, to within a constant factor,
2
()
2
(the
12
These states correspond to the circular orbits of the Sommerfeld theory (cf.note).
829

CHAPTER VII PARTICLE IN A CENTRAL POTENTIAL. THE HYDROGEN ATOM
factor
2
arises from the expression for the volume element in spherical coordinates). We
are interested in the case where= 1, that is,= = 1;
that the polynomial which enters into()contains only one term in(0)
1
. The
desired probability density is therefore proportional to:
() =
2
2
00
1
e
0
2
=
0
2
e
2 0
(C-53)
This function has a maximum for:
==
2
0 (C-54)
which is the radius of the Bohr orbit corresponding to the energy.
Finally, the table below gives the expressions for the wave functions of the rst
energy levels:
1level =1=0=0=
1
3
0
e
0
2level =2=0=0=
1
8
3
0
1
20
e
20
=2=1=1=
1
8
3
00
e
20
sine
2level =2=1=0=
1
42
3
00
e
20
cos
=2=1=1=
1
8
3
00
e
20
sine
References and suggestions for further reading:
Particle in a central potential: Messiah (1.17), Chap. IX; Schi (1.18), 16.
The Bohr-Sommerfeld atom and the old quantum therory: Cagnac and Pebay-
Peyroula (11.2), Chaps. V, VI and XIII; Born (11.4), Chap. V, 1 and 2; Pauling
and Wilson (1.9), Chap. II; Tomonaga (1.8), Vol. I; Eisberg and Resnick (1.3), Chap. 4.
Hydrogen-like wave functions: Levine (12.3), 6.5; Karplus and Porter (12.1),
3.8 and 3.10; Eisberg and Resnick (1.3), 7.6 and 7.7.
Degeneracy related to a1potential (dynamical group): Borowitz (1.7), 13.7;
Schi (1.18), 30; Bacry (10.31), 6.11.
Mathematical treatment of dierential equations: Morse and Feshbach (10.13),
Chaps. 5 and 6; Courant and Hilbert (10.11), Vol. I, V-11.
830

COMPLEMENTS OF CHAPTER VII, READER'S GUIDE
AVII: HYDROGEN-LIKE SYSTEMS Presentation of various hydrogen-like systems to
which the calculations of Chapter
applied directly. The accent is placed on physical
discussions and on the inuence of the masses
of the particles involved in the system. Simple,
advised for a rst reading.
BVII: A SOLUBLE EXAMPLE OF A CEN-
TRAL POTENTIAL: THE ISOTROPIC THREE-
DIMENSIONAL HARMONIC OSCILLATOR
Study of another case (a three-dimensional
harmonic oscillator) in which it is possible to
calculate exactly the energy levels of a particle in
a central potential by the method of Chapter
(solution of the radial equation). Not theoret-
ically dicult; may be considered as a worked
example.
CVII: PROBABILITY CURRENTS ASSOCIATED
WITH THE STATIONARY STATES OF THE
HYDROGEN ATOM
Completes the results of
concerning the properties of the stationary
states of the hydrogen atom by calculating their
probability currents. Short and simple, useful for
ComplementVII.
DVII: THE HYDROGEN ATOM PLACED IN A
UNIFORM MAGNETIC FIELD. PARAMAGNETISM
AND DIAMAGNETISM. THE ZEEMAN EFFECT
EVII: STUDY OF SOME ATOMIC ORBITALS.
HYBRID ORBITALS
FVII: VIBRATIONAL-ROTATIONAL LEVELS
OF DIATOMIC MOLECULES
Discussion of a certain number of physical phe-
nomena, using the results of Chapter .
DVII: the properties of an atom in a magnetic
eld (diamagnetism, paramagnetism, Zeeman
eect). Moderately dicult, important because
of the numerous applications.
EVII: complement intended to introduce the
concept of an atomic hybrid orbital, essential for
understanding certain properties of the chemical
bond. No theoretical diculty. Stresses the
geometrical aspect of wave functions.
FVII: direct application of the theory of Chap-
ter
spectrum of heteropolar diatomic molecules.
Sequel to Complements V() and VI;
moderately dicult.
GVII: EXERCISES Exercise 2 studies the inuence of a uniform
magnetic eld on the levels of a simple physical
system, in an exactly soluble case. It then
provides a concrete illustration of the general
considerations of Complements VIIand VII
concerning the inuence of the paramagnetic and
diamagnetic terms of the Hamiltonian.
831

HYDROGEN-LIKE SYSTEMS
Complement AVII
Hydrogen-like systems
1 Hydrogen-like systems with one electron
1-a Electrically neutral systems
1-b Hydrogen-like ions
2 Hydrogen-like systems without an electron
2-a Muonic atoms
2-b Hadronic atoms
The calculations of Chapter, which enabled us to determine numerous physical
properties of the hydrogen atom (energy levels, spatial distribution of the wave functions,
etc.) are based on the fact that the system under study is composed of two particles (an
electron and a proton) whose mutual attraction energy is inversely proportional to the
distance between them. There exist numerous other systems in physics that full this
condition: deuterium or tritium, muonium, positronium, muonic atoms, etc. The results
obtained in Chapter
to do is change the constants introduced in the calculations (masses and charges of the
two particles). This is what we shall do in this complement, in which we shall study,
in particular, how the Bohr radius and ionization energyare modied in each of the
systems to be considered. The wave functions associated with their stationary states and
the corresponding energies will then be obtained by replacing, in formulas (C-39) and
(C-47) of Chapter,0andby their new values, which give the order of magnitude
of the spatial extension of the wave functions and the binding energies of these systems.
We recall the expressions for0and:
0=-
1
=
~
2
2
(1)
=
1
2
22
=
4
2~
2
(2)
whereis the reduced mass of the electron-proton system:
=(H) =
+
1 (3)
and
2
characterizes the intensity of the attractive potential():
() =
2
r1r2
(4)
In the case of hydrogen, we have seen that:
0(H)052

A (5a)
136 eV2210
18
J (5b)
833

COMPLEMENT A VII
How can we obtain the corresponding values for a system of two arbitrary particles,
of masses1and2, whose attraction energy is:
() =
2
r1r2
(6)
(whereis a dimensionless parameter)? All we must do is calculate the reduced mass
of the system by replacingand by1and2in (3):
=
12
1+2
(7a)
and substitute the result obtained into (1) and (2), being careful to perform the substi-
tution:
2
=
2
(7b)
This is what we shall do in a certain number of physical examples.
1. Hydrogen-like systems with one electron
1-a. Electrically neutral systems
. Heavy isotopes of hydrogen
The physical systems closest to the hydrogen atom are its two isotopes, deuterium
and tritium. In these atoms, the proton is replaced by a nucleus having the same charge
but possessing either one or two neutrons in addition to the proton. The mass of the
deuterium nucleus is approximately2, and that of the tritium nucleus,3. The
reduced masses therefore become, in these two cases:
(deuterium) 1
2
(8a)
(tritium) 1
3
(8b)
Since:
1
1 836
1 (9)
it is clear that the reduced masses of deuterium and tritium are very close to that of
hydrogen, and that they can be replaced bywithout great inaccuracy.
If we substitute either (3), or (8a), or (8b) into formulas (1) and (2), we see that the
Bohr radii and energies of the hydrogen, deuterium and tritium atoms are practically the
same. Nevertheless, there are slight dierences, of the order of a thousandth in relative
value. These dierences can be detected experimentally. For example, with an optical
spectrograph of sucient resolution, it can be observed that the wavelengths of the
lines emitted by hydrogen atoms are slightly greater than those emitted by deuterium,
which are in turn greater than those emitted by tritium. This slight shift in the emitted
wavelengths is related to the fact that the nucleus is not innitely heavy, and does not
remain xed while the electron moves; this is called the nuclear nite mass eect.
Experiments have veried that formulas (7a), (1) and (2) account very precisely for this
eect.
834

HYDROGEN-LIKE SYSTEMS
. Muonium
The muon is a particle whose fundamental properties are the same as those of the
electron, except for a dierence in mass (the massof the muon is approximately equal
to 207). In particular, the muon is not sensitive to nuclear forces (strong interactions).
There are two types of muons, theand the
+
, whose charges are respectively equal
to those of the electronand the positron
1+
. Like all other charged particles, the
muon is sensitive to electromagnetic interactions.
We can therefore consider a physical system formed by a
+
muon and an electron
in which the electrostatic attraction is the same as for a proton and an electron.
Bound states therefore exist. This is, so to speak, a light isotope of hydrogen, in which
the
+
muon replaces the proton. This isotope is called muonium (its atomic mass is
on the order of 01).
It is not dicult to use the results in Chapter
and Bohr radius associated with muonium; (1), (2) and (7) yield:
0(muonium) =0(H)
1 +
1 +
0(H)1 +
1
200
(10a)
(muonium) =(H)
1 +
1 +
(H)1
1
200
(10b)
Since the muon is approximately ten times lighter than the proton, the nuclear nite
mass eect is about ten times greater for muonium than for hydrogen; however, since
the electron is distinctly lighter than the muon, this eect remains small (on the order
of 0.5%). For example, the wavelengths of the optical lines emitted by muonium are be
close to those of the corresponding lines for hydrogen. The frequency of the resonance
line has been measured with a very high accuracy in 1999 [cf.ref. (11.26)], and has
shown that the charges of the electron and of the muon are the same within10
9
.
Experimentally, the existence of muonium was revealed by its instability: the
+
muon decays, emitting a positron and two neutrinos, and the lifetime of muonium is
2210
6
s. The positron resulting from this decay can be detected. It is emitted
preferentially in the direction of the
+
muon spin
2
(non-conservation of parity in weak
interactions). Detection of the positrons then leads to the experimental determination
of this direction. Since, in addition, the spin of the
+
muon of a muonium atom
is coupled to that of the electron (hyperne structure coupling;cf.Chap.
complements), its precession frequency in a magnetic eld is dierent from that of a free
muon. Measurement of this frequency thus reveals the existence of muonium atoms.
The study of muonium is of very great interest, theoretically as well as experi-
mentally. The two particles which constitute this system are not subjected to strong
interactions, so that its energy levels (in particular, the hyperne structure of the ground
state 1) can be calculated with great precision, without bringing in any nuclear cor-
rection (for the hydrogen atom, on the other hand, one must take into account the
internal structure and polarizability of the proton, which are due to strong interactions).
Comparison between theoretical predictions and experimental results provides a very
severe test of quantum electrodynamics. For instance, a measurement of the hyperne
1
In addition, likeand
+
,and
+
are antiparticles of each other.
2
Like the electron, the muon has a spin of 1/2 with which is associated a magnetic momentM=
S.
835

COMPLEMENT A VII
structure of muonium can give an excellent determinations of the ne structure constant
=
2
~.
. Positronium
Positronium is a bound system composed of an electronand a positron
+
.
Like muonium, it can be said by extension to be an isotope of hydrogen, with the proton
being replaced by a positron. However, it must be noted that the situation is not quite
the same: in the hydrogen atom, the proton (which is much heavier than the electron)
remains almost motionless, while in positronium, the positron, the antiparticle of the
electron, has the same mass and consequently the same velocity as the electron when the
center of mass of positronium is xed (cf.Fig.-b).
According to (7a), the reduced mass associated with positronium is:
(positronium) =
2
(11)
Therefore:
0(positronium)20(H) (12a)
(positronium)
1
2
(H) (12b)
Thus, for a given state of positronium, the average electron-positron distance is twice the
electron-proton distance for the corresponding state of the hydrogen atom (cf.Fig.).
The dierences between the energies of the stationary states, however, are twice as small,
and the optical line spectrum emitted by positronium is obtained by doubling all the
wavelengths of that of hydrogen.
Comment:
One should not conclude from formula (12a) that the radius of positronium is
twice that of the hydrogen atom. The Bohr radius gives an idea of the extensione
+
e
–
e
–
p
a b
Figure 1: Schematic representation of the
hydrogen (electron+proton system) and
positronium (electron+positron system)
atoms. Since the proton is much heavier
than the electron, it is located practically at
the center of mass of the hydrogen atom;
the electron revolves about the proton at
a distance0(). On the other hand, the
positron is equal in mass to the electron;
these particles therefore both revolve about
their center of mass, the distance between
them being0(positronium)= 20().
836

HYDROGEN-LIKE SYSTEMS
of the wave functions associated with the relative particle (cf.Chap., ),
whose positionr1r2is related to the distance between the two particles and not
to the distance between them and the center of mass. Figure
moreover, that the hydrogen and positronium atoms are of equal size. In general,
all hydrogen-like systems for which the attractive potential is given by (6) with
= 1have exactly the same radius, since formula (B-5) of Chapter
that:
r1r=
2
1+2
r=
1
r (13)
Using (1), which gives the order of magnitude of the spatial extension of the wave
function100(r)of the ground state, we see that the radiusof the atom can
be dened by:
=
~
2
1
2
(14)
where1is the mass of the lighter particle (the heavier particle is found closer to
the center of mass). In all the systems considered until now,= 1and1=;
their radii are therefore the same. We shall see cases later in which the radius
is smaller, either because1= or because= 1.
The optical spectrum of positronium has been observed in 1975 [cf.ref. (11-23)]. As
for the (hyperne) structure of the ground state (due to the interaction between the magnetic
moments of the electron and the positron), it has been accurately determined (cf.Comple-
mentXII).
The fact that positronium, like muonium, is a purely electrodynamic system (neither the
electron nor the positron is sensitive to strong interactions) explains the importance attached
to its theoretical and experimental study.
Let us also point out that positronium is an unstable system. Since the ground state is a1
state, the electron and positron come into contact and annihilate, yielding two or three photons,
depending on the hyperne structure level they started from. The study of the corresponding
decay rate is also of great interest in quantum electrodynamics.
. Hydrogen-like systems in solid state physics
Atomic physics is not the only domain of application of the theory presented in
Chapter. For example, the donor atoms localized in semiconductors constitute
approximately hydrogen-like systems in solid state physics.
Consider a silicon crystal; in the silicon lattice, each atom uses its four valence
electrons to form four tetrahedral bonds with its neighbors. If a pentavalent atom like
phosphorus (a donor atom) is introduced into the lattice in place of a silicon atom, it
must lose a valence electron, and its overall charge becomes positive. It then behaves like
a center which can retain the electron and form a hydrogen-like system with it. Actually,
the force acting on the electron cannot be calculated directly from Coulomb's law in a
vacuum since silicon has a large dielectric constant12, so that (4) must be replaced
by:
() =
2
r1r2
(15)
837

COMPLEMENT A VII
To be completely rigorous, we should have to replace the electron mass by the eective
massof the electron in silicon, which is dierent from the free electron mass because
of interactions with the charges of the nuclei in the crystal. Nevertheless, we shall conne
ourselves to a qualitative discussion, noting that the eect of the high value ofis to
decrease
2
in (15), that is, according to (1), to increase the Bohr radius by a factor on
the order of 10. The donor atom impurity is therefore similar to a very large hydrogen
atom, whose wave functions are spread over distances much greater than the unit cell
length of the silicon lattice.
Let us briey describe another hydrogen-like system in solid state physics: the ex-
citon. Consider a semi-conductor crystal. In the absence of external perturbations, the
outside electrons of the atoms forming the crystal are all in states belonging to the va-
lence band (the temperature is assumed to be suciently low;cf.ComplementXIV).
By suitably illuminating the crystal, we can, through the absorption of a photon, cause
an electron to go into the conduction band (which contains a set of energy levels which
are higher than those of the valence band). There is then one electron missing from the
valence band. We can treat this band as if it contained a particle of charge opposite
to that of an electron, called a hole. The hole can attract an electron of the valence
band and form a bound system with it: the exciton. The exciton, like hydrogen, has
a series of energy levels between which it can undergo transitions. Its existence can be
demonstrated by a measurement of light absorption by the crystal.
1-b. Hydrogen-like ions
The neutral helium atom is composed of two electrons and a positively charged
nucleus of charge2. Such a system, which consists of three particles, cannot be
studied with the theory of Chapter. However, if an electron is somehow removed
from the helium atom, the He
+
ion is similar to a hydrogen atom; the only dierences
are in the nuclear charge, which is twice that of the proton (the total charge of the ion is
positive and equal to) and its mass (which, for
4
He, is approximately four times that
of the proton). Of course, there are other hydrogen-like ions: the Li
++
ion (the lithium
atom, when it is not ionized, has= 3electrons), the Be
+++
ion(= 4), etc...
Let us then consider a system formed by a nucleus, of massand positive charge
, and an electron. If we substitute (7b) in (1) and (2), we obtain:
0()
0()
(16)
()
2
() (17)
(since , we have neglected the dierence between the reduced mass of hydrogen
and that of the hydrogen-like ion under study; the consequences of the nuclear nite
mass eect on0and are, in eect, negligible compared to those due to the charge
variation). Hydrogen-like ions are therefore all smaller than the hydrogen atom, as one
would expect since the nucleus and the electron are more strongly bound. Moreover,
their energy increases rapidly with(quadratically): for example, the energy which
must be supplied to a Li
++
ion to remove its last electron is greater than 100 eV. This is
why the electromagnetic frequencies that can be emitted or absorbed by a hydrogen-like
ion fall into the ultraviolet, and even, whenis large enough, into the domain of X-rays.
838

HYDROGEN-LIKE SYSTEMS
2. Hydrogen-like systems without an electron
Thus far, the systems we have considered all include an electron. Nevertheless, there
exist numerous other particles having the same charge, able to form a hydrogen-like
system with a nucleus of charge. We shall give a few examples. The atoms
we are going to describe here are of course less common than the usual atoms which
appear in Mendeleev's classication. They are unstable, and, in order to study them, it
is necessary to use high-energy particle accelerators to produce the particles needed for
their formation. This is why they are sometimes called exotic atoms.
2-a. Muonic atoms
We have already mentioned some essential features of the muon and pointed out
the existence of themuon. When this particle is attracted by a positively charged
atomic nucleus, it can form a bound system called a muonic atom
3
.
Consider, for example, the simplest muonic atom, which is composed of amuon
and a proton; this is a neutral system whose Bohr radius is:
0(
+
)
~
2
2
0()
200
(18)
and whose ionization energy is:
(
+
)
4
2~
2
200() (19)
The size of this muonic atom is therefore on the order of several thousandths of an
Angström. Its spectrum is obtained from that of hydrogen by dividing the wavelengths
by 200; it therefore falls into the domain of soft X-rays.
What happens if, instead of revolving about a proton, theis captured by a
nucleuswhose charge istimes greater, like lead
4
for example, for which= 82?
Formulas (1) and (2) then yield:
0( )
0()
200
(20)
( )200
2
() (21)
Setting= 82in these formulas, we nd for the transitions of the lead muonic atom
energies equal to several MeV (1 MeV = 10
6
electron volts). However, it must be noted
that formulas (1) and (2) are no longer valid in this case, since equation (20) would yield:
0(Pb)310
5
A = 3 Fermi (22)
3
We could also imagine a bound system composed of a
+
muon and a muon. However, given
the low intensity of muon beams that can be produced, such an atom is very dicult to create.
4
Such a system can be formed by directing abeam onto a target of lead atoms. When a is
captured by a lead nucleus, it revolves about it at a distance about 200 times smaller than the distance
to the electrons of the innermost shell of the atom. The nuclear charge is therefore practically the only
one to aect the muon. Thus, in studying the states of the muonic atom, we can simply ignore the
electrons.
839

COMPLEMENT A VII
that is, a radius slightly smaller than the radius of the lead nucleus. The calculations of
Chapter 6) of
the potential(), which is correct
5
only when the particles under study, separated by
distances much greater than their dimensions, can be considered to be point particles.
This hypothesis, very well satised for hydrogen, is not valid in the case studied here.
However, (20) and (21) give the correct order of magnitude of the energies and
radius of the lead muonic atom. The physical consequences of the existence of a non-
zero spatial extension of the nucleus (volume eect) will be studied in greater detail
in ComplementXI. We take this occasion to note, however, that one of the reasons
for interest in muonic atoms is precisely related to this type of eect: themuon
explores, as it were, the internal structure of the nucleus
6
, and the energy levels of
muonic atoms depend on the electrical charge distribution and magnetism inside the
nucleus (recall that the muon is not sensitive to nuclear forces). Thus, the study of these
states can furnish information very useful in nuclear physics.
2-b. Hadronic atoms
Hadrons are those particles sensitive to strong interactions, as opposed to lep-
tons, which are not. Electrons and muons, whose bound states in a Coulomb potential
we have studied thus far, are leptons. Protons, neutrons and mesons such as themeson,
etc... are hadrons. Of the latter, those that are negatively charged can form a hydrogen-
like bound system with an atomic nucleus, called a hadronic atom. For example, the
nucleus-meson system yields a pionic atom; the nucleus-particle system, a sig-
maonic atom
7
; the nucleus-meson system, a kaonic atom; the nucleus-antiproton
system, an antiprotonic atom, etc... All of the systems just cited have actually been
observed and studied. They are all unstable, but their lifetimes are long enough for us
to observe some of their spectral lines. Hydrogen atom theory, which takes into account
only the electrostatic interaction between the two particles under consideration, does not,
of course, apply to such systems, in which strong interactions play an important role.
However, since they are of very short range, strong interactions can be neglected in the
study of excited states of the hadronic atom (other than thestates), in which the two
particles are far apart. The theory of Chapter
(1) and (2), which, in all these cases, lead to much smaller Bohr radii and much greater
energies than for hydrogen. Thus, measurement of the spectral frequencies emitted by
pionic atoms gives a very precise determination of the mass of themeson.
References and suggestions for further reading:
Exotic atoms: see the subsection Exotic atoms of section 11 of the bibliography;
see also Cagnac and Pebay-Peyroula (11.2), Chap. XIX, 7; Weissenberg (16.19),
Chap. 4, 2 and Chap. 6.
Excitons: Kittel (13.2), Chap. 17, p. 538; Ziman (13.3), 6.7.
5
Inside the nucleus, the potential is approximately parabolic (cf.ComplementsV, XI).
6
The concept of the impenetrability of two solid bodies is macroscopic. In quantum mechanics,
nothing prevents the wave functions of two particles of dierent nature from overlapping.
7
The term mesic atom is sometimes used to denote a system involving a meson. Similarly, since
theis a hyperon (a particle heavier than the proton), the sigmaonic atom is sometimes called a
hyperonic atom.
840

THE ISOTROPIC THREE-DIMENSIONAL HARMONIC OSCILLATOR
Complement BVII
A soluble example of a central potential: the isotropic
three-dimensional harmonic oscillator
1 Solving the radial equation
2 Energy levels and stationary wave functions
In this complement we shall examine a special case of a central potential for which
the radial equation is exactly soluble: the isotropic three-dimensional harmonic oscillator.
We have already treated this problem (ComplementV) by considering the state space
ras the tensor product ; this amounts, in the representation, to
separating the variables in Cartesian coordinates. We thus obtained three dierential
equations, one in the-variable, one in, and the third in. Here we intend to seek the
stationary states that are also eigenstates ofL
2
and, by separating the variables in
polar coordinates. We shall then indicate how the two bases ofrobtained by these two
dierent methods are related to each other.
We shall also study, in ComplementVIII, the stationary states of well-dened
angular momentum of a free particle. This can be considered to be another special case
of a central potential[()0]which leads to an exactly soluble radial equation.
A three-dimensional harmonic oscillator is composed of a (spinless) particle of mass
subjected to the potential:
( ) =
1
2
22
+
22
+
22
(1)
where,andare real positive constants. The oscillator is said to be isotropic if:
=== (2)
Since the potential (1) is the sum of a function ofalone, a function ofalone
and a function ofalone, we can solve the eigenvalue equation of the Hamiltonian:
=
P
2
2
+(R) (3)
by separating the variables,andin the representation. This is what was done
in ComplementV. The energy levels, for an isotropic oscillator, are then found to be
of the form:
= +
3
2
~ (4)
whereis any positive integer or zero. The degree of degeneracyof the levelis
equal to:
=
1
2
(+ 1)(+ 2) (5)
841

COMPLEMENT B VII
and the associated eigenfunctions are:
( ) =
2
34
1
2
++
!!!
e
2
2
(
2
+
2
+
2
)
()()()(6)
with:
=
~
(7)
[()denotes the Hermite polynomial of degree;cf.ComplementV]. is
an eigenfunction of the Hamiltonianwith the eigenvaluesuch that:
=++ (8)
If the oscillator under consideration is isotropic
1
, the potential (1) is a function
only of the distancebetween the particle and the origin:
() =
1
2
22
(9)
Consequently, the three components of the orbital angular momentumLare constants
of the motion. We want to nd the common eigenstates ofL
2
and. To do so, we
could proceed, as in ComplementVI, by introducing operators related to right and left
circular quanta and to longitudinal quanta corresponding to the third degree of freedom
along(an outline of this method is given at the end of this complement). However,
we prefer to use this example to illustrate the method elaborated in Chapter)
and solve the radial equation by the polynomial method.
1. Solving the radial equation
For a xed value of the quantum number, the radial functions()and energies
are given by the equation: [cf. relation (A-21) of Chapter
~
2
2
1d
2
d
2
+
1
2
22
+
(+ 1)~
2
2
2
() = () (10)
We set:
() =
1
() (11a)
=
2
~
2
(11b)
Equation (10) then becomes:
d
2
d
2
42
(+ 1)
2
+ () = 0 (12)
1
Separation of the polar variables,,is possible only for an isotropic oscillator.
842

THE ISOTROPIC THREE-DIMENSIONAL HARMONIC OSCILLATOR
[whereis the constant dened in (7)]. We must add the condition at the origin:
(0) = 0 (13)
For large, (12) virtually reduces to:
d
2
d
2
42
() 0 (14)
The asymptotic behavior of the solutions of equation (12) is therefore dominated by
e
22
2
ore
22
2
. Only the second possibility is physically acceptable. This leads us
to the change of functions:
() = e
22
2
() (15)
It is easy to nd that()must satisfy:
d
2
d
2
2
2
d
d
+
2
(+ 1)
2
= 0 (16a)
(0) = 0 (16b)
Now we shall seek()in the form of a power series in:
() =
=0
(17)
where, by denition,0is the coecient of the rst non-zero term:
0= 0 (18)
When we substitute expansion (17) into equation (16a), the term of lowest order is in
2
; its coecient is zero if:
[(1)(+ 1)]0= 0 (19)
With conditions (18) and (16b) taken into account, the only way to satisfy relation (19)
is to choose:
=+ 1 (20)
(this result could have been predicted;cf. ). The next term in the
expansion of equation (16a) is in
1
, and its coecient is equal to:
[(+ 1)(+ 1)]1 (21)
Sinceis already xed by (20), this coecient can go to zero only if:
1= 0 (22)
843

COMPLEMENT B VII
Finally, let us set the coecient of the general term in
+
equal to zero:
[(++ 2)(++ 1)(+ 1)]+2+ [
2
2
2
(+)]= 0 (23)
that is, using (20):
(+ 2)(+ 2+ 3)+2= [(2+ 2+ 3)
2
] (24)
We therefore obtain a recurrence relation for the coecientsof expansion (17).
Note, rst of all, that this recurrence relation, combined with result (22), implies
thatall coecientsof odd rankare zero. As for the coecients of even rank, they
must all be proportional to0. If the value ofis such that no integermakes the
term in brackets on the right-hand side of (24) go to zero, we nd the solutionof
(16) in the form of an innite power series, for which:
+2
2
2
(25)
This behavior is the same as that of the coecients appearing in the expansion of the
functione
22
, since:
e
22
=
=0
2
2
(26)
with:
2=
2
!
(27)
and, consequently:
2+2
2
2
(28)
Since it is 2that corresponds to the even integerof the expansion of, (28) is indeed
identical to (25). From this, we can see that if (17) really contains an innite number of
terms, the asymptotic behavior ofis dominated bye
22
, which renders this function
physically unacceptable [cf.relation (15)].
The only cases which are interesting from a physical point of view are therefore
those in which there exists an even integer, positive or zero, such that:
= (2+ 2+ 3)
2
(29)
Recurrence relation (24) indicates that the coecients of even rank greater thanare
then zero. Since all the coecients of odd rank are also zero, expansion (17) reduces to
a polynomial, and the radial function()given by (15) decreases exponentially as
goes to innity.
844

THE ISOTROPIC THREE-DIMENSIONAL HARMONIC OSCILLATOR
2. Energy levels and stationary wave functions
Using denitions (7) and (11b), relation (29) gives the energiesassociated with a
given value of:
=~ ++
3
2
(30)
whereis any even positive integer or zero. Sinceactually depends only on the
sum:
=+ (31)
accidental degeneracies appear: the energy levels of the isotropic three-dimensional har-
monic oscillator are of the form:
= +
3
2
~ (32)
is any positive integer or zero, andis any even positive integer or zero;can therefore
take on all integral values, positive or zero. This is in agreement with result (4).
We shall x an energy, that is, an integer, positive or zero. The values of
andwhich can be associated with it according to (31) are the following:
() = (0)(2 2)(22)(0) ifis even (33a)
() = (0)(2 2)(33)(11) ifis odd (33b)
From this, we can immediately get the values ofassociated with the rst values of:
= 0 := 0
= 1 := 1
= 2 := 02 (34)
= 3 := 13
= 4 := 024
Figure cf.Figure
of Chapter), the lowest energy levels of an isotropic three-dimensional harmonic
oscillator.
For each pair(), there exists one and only one radial function(), which
corresponds to(2+ 1)common eigenfunctions ofL
2
and:
(r) =
1
()() (35)
Consequently, the degree of degeneracy of the energyunder consideration is equal to:
=
=02
(2+ 1) if is even (36a)
=
=13
(2+ 1) if is odd (36b)
845

COMPLEMENT B VII0
n = 4
n = 3
4s
E
3p
1p
3f
4d
2s
0s
2d
4g
l
n = 2
n = 1
n = 0
11hω/2
9hω/2
7hω/2
5hω/2
3hω/2
1 2 3 4
Figure 1: Lowest energy levels of the three-dimensional harmonic oscillator. Whenis
even,can take on
2
+ 1values:=, 2, ...,0. Whenis odd,can take on
+ 1
2
values:= 2 1. With the possible values of( )taken into
account, the degree of degeneracy of the levelis
(+ 1)(+ 2)
2
.
These sums are simple to calculate, and we again obtain result (5):
for even: =
2
=0
(4+ 1) =
1
2
(+ 1)(+ 2) (37a)
for odd: =
(1)2
=0
(4+ 3) =
1
2
(+ 1)(+ 2) (37b)
For each of the pairs()given in (33), the results of 1 enable us to determine
the corresponding radial function()(to within the factor0) and, therefore, the
(2+ 1)common eigenfunctions ofandL
2
, of eigenvaluesand(+ 1)~
2
. We shall
calculate, for example, the wave functions associated in this way with the three lowest
energy levels.
For the ground state0=
3
2
~, we must have:
== 0 (38)
00()then reduces to0. If we choose0to be real and positive, the normalized
function===0can be written:
000(r) =
2
34
e
22
2
(39)
Since the ground state is not degenerate(0= 1),000is the same as the function
===0which is found by separating the Cartesian variables,, and[cf.formula
(6)].
846

THE ISOTROPIC THREE-DIMENSIONAL HARMONIC OSCILLATOR
With the rst excited state1=
5
2
~, which is three-fold degenerate, is again
associated a single pair():
= 0
= 1
(40)
and01=0
2
. The three functions of the basis dened byL
2
andare therefore:
01(r) =
83
32
14
e
22
2
1() = 101 (41)
We know [cf.ComplementVI, formulas (32)] that the spherical harmonics
1are such
that:
0
1() =
342
1
1
1
1=
342
1
1
+
1
1=
34
(42)
and that the Hermite polynomial of order 1 is [cf.ComplementVI, formulas (18)]:
1() = 2 (43)
Consequently, it is clear that the three functions01are related to the functions
of basis (6) by the equations:
=0 =0 =1= =0=1=0
=1 =0 =0=
1
2
[=0=1=1 =0=1=1]
=0 =1 =0=
2
[=0=1=1+ =0=1=1] (44)
Finally, consider the second excited state, of energy2=
7
2
~. It is six-fold
degenerate, and the quantum numbersandcan take on the values:
= 0 = 2 (45a)
= 2 = 0 (45b)
The function02()which corresponds to the values (45a) is simply0
3
. For the values
(45b),20contains two terms; using (24) and (29), we easily nd:
20() =01
2
3
22
(46)
The six basis functions in the eigensubspace associated with2are thus of the form:
02(r) =
1615
32
14
22
e
22
2
2() = 21012 (47a)
200(r) =
32
32
34
1
2
3
22
e
22
2
(47b)
847

COMPLEMENT B VII
Knowing the explicit expressions for the spherical harmonics [formulas (33) of Com-
plementVI] and the Hermite polynomials [formulas (18) of ComplementV], we can
easily prove the following relations:
=2=0=0=
13
=2 =0 =0+ =0 =2 =0+ =0 =0 =2
1
2
[=0=2=2+ =0=2=2] =
1
2
=2 =0 =0 =0 =2 =0
1
2
[=0=2=2 =0=2=2] = =1 =1 =0
1
2
[=0=2=1 =0=2=1] = =1 =0 =1
1
2
[=0=2=1+ =0=2=1] = =0 =1 =1
=0=2=0=
23
=0 =0 =2
1
2
=2 =0 =0
1
2
=0 =2 =0(48)
Comment:
As we pointed out in the beginning of this complement, we can apply a method
analogous to the one presented in ComplementVIto the isotropic three-dimensional
harmonic oscillator. If,andare the annihilation operators which act in
the state spaces,andrespectively, we dene:
=
1
2
( ) (49a)
=
1
2
(+) (49b)
It can be shown thatandbehave like independent annihilation operators
(ComplementVI, 3-b). The Hamiltonianand the angular momentum oper-
ators can then be expressed in terms of,,and their adjoints:
=~ +++
3
2
(50a)
=~( ) (50b)
+=~
2 (50c)
=~
2 (50d)
The common eigenvectors of the observables,and can be
obtained through the action of the creation operators,andon the ground
848

THE ISOTROPIC THREE-DIMENSIONAL HARMONIC OSCILLATOR
state000of the Hamiltonian[this state is unique to within a constant factor;
cf.formulas (6) and (39)]:
=
1
!!!
()()()000 (51)
According to (50a) and (50b), is an eigenvector ofandwith the
eigenvalues(+++32)~and( )~. The eigensubspace, associated
with a given energycan therefore be spanned by the set of vectors
such that:
++= (52)
Of these, the eigenvector ofwith the largest eigenvalue compatible withis
00, whose eigenvalue is~. This ket, according to (50c), satises:
+ 00= 0 (53)
Consequently
2
, it is an eigenvector ofL
2
with the eigenvalue(+ 1)~
2
, and it
can be identied with the ket of the basis such that:
+=
== (54)
Therefore:
=0= == = =0 =0 (55)
Application of the operator[formula (50d)] to both sides of relation (55)
yields:
0 1= 101 (56)
The eigenvalue(2)~of, unlike the two preceding ones, is two-fold
degenerate in: two orthogonal vectors,202and 110correspond to
it. Using (50d) again in order to applyto (56), we nd that:
0 2=
2(1)21
202
1
21
110 (57)
It can be shown that the action of+on the linear combination orthogonal to (57)
yields the null vector. This linear combination must therefore be an eigenvector
ofL
2
with the eigenvalue(2)(1)~
2
. This gives, to within a phase factor:
2 2 2=
1
21
202+
2(1)21
110 (58)
2
This result follows directly from relation (C-7b) of Chapter, which, applied to00, yields:
L
2
00=~
2
(
2
+) 00
849

COMPLEMENT B VII
We can thus relate, by iteration
3
, the two bases, and .
Of course, replacingandby functions ofandin (51), we can express
as a linear combination of the vectors whose wave functions
are given by (6).
References and suggestions for further reading:
Other soluble examples (spherical square well, etc.): Messiah (1.17), Chap. IX,
10; Schi (1.18), 15; see also Flügge (1.24), 58 to 79.
3
An argument analogous to the one just outlined will be used in Chapter
momenta.
850

PROBABILITY CURRENTS ASSOCIATED WITH THE STATIONARY STATES OF THE HYDROGEN ATOM
Complement CVII
Probability currents associated with the stationary states of the
hydrogen atom
1 General expression for the probability current
2 Application to the stationary states of the hydrogen atom
2-a Structure of the probability current
2-b Eect of a magnetic eld
The normalized wave functions (r)associated with the stationary states of
the hydrogen atom were determined in Chapter. (r)is the product of the
spherical harmonic()and the function()calculated in
(r) = ()() (1)
The spatial variation of the probability density:
(r) = (r)
2
(2)
was then studied, at least for the lowest energy states.
It is important, however, to understand that a stationary state is not characterized
only by the value of the probability density(r)at all points of space. We must also
associate with it a probability current, which can be expressed as:
J (r) =
~
2
(r)r (r) + (3)
[we assume here that the vector potentialA(r)is zero;denotes the mass of the
particle].
Thus, we can associate with the quantum state of a particle, a uid (called
the probability uid) whose density at each point of space is(r). This uid is not
motionless; it is in a state of ow characterized by the current densityJ. In a stationary
state,andJare not time-dependent: the uid is in a steady state of ow.
To complete the results of Chapter
stationary states of the hydrogen atom, we shall now study the probability currents
J (r).
1. General expression for the probability current
Consider an arbitrary normalized wave function(r). We introduce the real quantities
(r)[the modulus of(r)] and(r)[the argument of(r)] by setting:
(r) =(r) e
(r)
(4)
851

COMPLEMENT C VII
with:
(r)0 ; 0 (r)2 (5)
If we substitute (4) into the expressions for the probability density(r)and the
currentJ(r), we obtain [still assuming the vector potentialA(r)to be zero]:
(r) =
2
(r) (6)
J(r) =
~
2
(r)r(r) (7)
Therefore,(r)depends only on the modulus of the wave function, whileJ(r)brings in
its phase [for example,J(r)is zero if this phase is constant throughout all space].
Comment:
If the wave function(r)is given, it is obvious that(r)andJ(r)are perfectly well-
dened. Conversely, is there always one and only one function(r)corresponding to
given values of(r)andJ(r)?
According to (6), the modulus(r)of the wave function can be obtained directly
1
from
(r); the argument(r)must satisfy the equation:
r(r) =
~
J(r)
(r)
(8)
We know that such an equation has a solution only if:
r
J(r)
(r)
=0 (9)
It then has an innite number of solutions, which dier from each other by a constant.
Since this constant corresponds to a global phase factor, it follows, if condition (9) is sat-
ised, that the wave function of the particle is perfectly well-dened by the specication
of(r)andJ(r). If condition (9) is not fullled, no wave function exists that corresponds
to the values of(r)andJ(r)under consideration.
2. Application to the stationary states of the hydrogen atom
2-a. Structure of the probability current
When the wave function has the form of (1), where()is a real function and
()is the product ofeand a real function, we have:
(r) = () ()
(r) = (10)
Applying (7) and using the expression for the gradient in polar coordinates, we get:
J (r) =
~
(r)
sin
e(r) (11)
1
Of course, in order to be a probability density,(r)must be positive everywhere.
852

PROBABILITY CURRENTS ASSOCIATED WITH THE STATIONARY STATES OF THE HYDROGEN ATOM
Figure 1: Structure of the probability current
associated with a stationary stateof
the hydrogen atom (shown in a plane perpen-
dicular to). The indexrefers to the
eigenvalue~of. If0, the probabil-
ity uid rotates counterclockwise about;
if 0, clockwise. If= 0, the probabil-
ity current is zero at all points of space.
wheree(r)is the unit vector that forms withandra right-handed Cartesian coor-
dinate frame.
Probability current variations in a plane perpendicular toare shown in Figure.
According to (11), the current at each pointis perpendicular to the plane dened by
and theaxis: the probability uid rotates about. SinceJis not proportional
tosin(r), the system does not rotate as a whole. The eigenvalue~of the observable
can be interpreted as the classical angular momentum associated with this rotational
motion of the probability uid. The contribution of the volume element d
3
, situated at
the pointr, to the angular momentum with respect to the origin can be written:
d=rJ (r)d
3
(12)
By symmetry, the resultant of all these elementary momenta is directed along; it is
equal to:
= d
3
e[rJ (r)] (13)
Using expression (11) forJ (r), we easily obtain:
= d
3
J (r)sin
=~d
3
(r)
=~ (14)
2-b. Eect of a magnetic eld
The results obtained thus far are valid only if the vector potentialA(r)is zero.
Let us examine what happens when this is not the case. Imagine, for example, that we
place the hydrogen atom in a uniform magnetic eldB. This eld can be described by
the vector potential:
A(r) =
1
2
rB (15)
853

COMPLEMENT C VII
What is the probability current associated with the ground state?
For simplicity, we shall assume that the magnetic eldBdoes not modify the wave
function of the ground state
2
. The probability current can then be calculated from the
general expression forJ[cf.relation (D-20) of Chapter]. This yields:
J (r) =
1
2
(r)
~
r A(r) (r) +
=
1
(r) [~r (r)A(r)] (16)
For the ground state and a eldBdirected along, we obtain, using (15):
J100(r) =
2
100(r)er (17)
where the cyclotron frequencyis dened by:
=
(18)
Therefore, the probability current of the ground state is not zero in the presence of
a magnetic eld as it is whenB=0. Expression (17) indicates that the probability uid
rotates as a whole aboutBwith the angular velocity2. Physically, this result arises
from the fact that when the magnetic eldBis turned on, a transient electric eldE()
must exist. Under its inuence, the electron, while remaining in the ground state, goes
into rotation about the proton, with an angular velocity depending only on the value of
B(and not on the precise way in which the eld was turned on during the transition
period).
Comment:
The particular choice of gauge made in (15) allowed us, with only a negligible error, to
retain the same wave functions we used in the absence of a eld (cf.footnote). With
another gauge, the wave functions would have been dierent (cf.ComplementIII)
and, in (16), the term containingA(r)explicitly would not have been the only one to
contribute to the value ofJ(r), to rst order inB. Nevertheless, we would have found
(17) at the end of the calculation, since the physical result must not be gauge-dependent.
2
Since the Hamiltoniandepends onB, this is obviously not rigorously correct. Nevertheless, by
considering the expression for[cf.formulas (6) and (7) of ComplementVII], we can show that, for
the gauge chosen in (15) andBdirected along, the functions (r)are eigenfunctions ofto
within a term of second order inB. By using the perturbation theory of Chapter, one can show that,
for magnetic elds normally produced in the laboratory, this second-order term is negligible.
854

THE HYDROGEN ATOM PLACED IN A UNIFORM MAGNETIC FIELD
Complement DVII
The hydrogen atom placed in a uniform magnetic eld.
Paramagnetism and diamagnetism. The Zeeman eect
1 The Hamiltonian of the problem. The paramagnetic term
and the diamagnetic term
1-a Expression for the Hamiltonian
1-b Order of magnitude of the various terms
1-c Interpretation of the paramagnetic term
1-d Interpretation of the diamagnetic term
2 The Zeeman eect
2-a Energy levels of the atom in the presence of a magnetic eld
2-b Electric dipole oscillations
2-c Frequency and polarization of emitted radiation
In Chapter, we studied the quantum mechanical properties of a free hydrogen
atom, that is, of the system formed by an electron and a proton exerting an electrostatic
attraction on each other but not interacting with any external eld. This complement
is devoted to the study of the new eects that appear when this atom is placed in a
static magnetic eld. We shall conne ourselves to the case in which this eld is uniform,
as it always is, moreover, in practice, since the magnetic elds that can be produced
in the laboratory vary very little in relative value over distances comparable to atomic
dimensions.
We have already studied the behavior of an electron subjected either to an electric
eld alone (cf.for example, Chapter) or to a magnetic eld alone (cf.Comple-
mentVI). Here, we shall generalize these discussions by calculating the energy levels of
an electron subjected both to the inuence of the internal electric eld of the atom and to
an external magnetic eld. Under these conditions, the exact solution of the Schrödinger
equation may seem to be a very complicated problem. However, we shall see that this
problem can be simplied considerably by means of certain approximations. In the rst
place, we shall totally neglect the nuclear nite mass eect
1
. Then we shall use the fact
that, in practice, the eect of the external magnetic eld is much smaller than that of
the internal electric eld of the atom: the atomic level shifts due to the magnetic eld
are much smaller than the energy separations in a zero eld.
The discussion presented in this complement will enable us to introduce and explain
certain eects which are important in atomic physics. We shall see, in particular, how
atomic paramagnetism and diamagnetism appear in the quantum mechanical formalism.
1
For the hydrogen atom, such an approximation is justied by the fact that the proton is considerably
heavier than the electron. For muonium (cf.ComplementVII), the approximation is not as good and
it becomes totally inapplicable for the case of positronium. Moreover, in the presence of a magnetic
eld, it is not possible to separate the motion of the center of mass rigorously. If one wished to take the
nuclear nite mass eect into account in this complement, it would not suce to replace the mass
of the electron by the reduced massof the electron-proton system.
855

COMPLEMENT D VII
In addition, we shall be able to predict the modications occurring in the optical spectrum
emitted by hydrogen atoms when they are placed in a static magnetic eld (the Zeeman
eect).
1. The Hamiltonian of the problem. The paramagnetic term and the
diamagnetic term
1-a. Expression for the Hamiltonian
Consider a spinless particle, of massand charge, subjected simultaneously to
a scalar central potential()and a vector potentialA(r). Its Hamiltonian is:
=
1
2
[PA(R)]
2
+(R) (1)
When the magnetic eldB=rA(r)is uniform, the vector potentialAcan be
put into the form:
A(r) =
1
2
rB (2)
To substitute this expression into (1), we shall calculate the quantity:
[PA(R)]
2
=P
2
+
2
[P(RB) + (RB)P] +
2
4
(RB)
2
(3)
Now,Bis actually a constant, and not an operator. All observables therefore commute
withB, so we can write, using the rules of vector calculus:
[PA(R)]
2
=P
2
+
2
[B(PR)(RP)B] +
2
4
[R
2
B
2
(RB)
2
] (4)
On the right-hand side of this expression, the angular momentumLof the particle
appears:
L=RP=PR (5)
We can therefore writein the form:
=0+1+2 (6)
where0,1and2are dened by:
0=
P
2
2
+(R) (7a)
1=
~
LB (7b)
2=
2
B
2
8
R
2
(7c)
In these relations,denotes the Bohr magneton (whose dimensions are those of a
magnetic moment):
=
~
2
(8)
856

THE HYDROGEN ATOM PLACED IN A UNIFORM MAGNETIC FIELD
and the operatorRis the projection ofRonto a plane perpendicular toB:
R
2
=R
2
(RB)
2
B
2
(9)
If we choose a system of orthonormal axessuch thatBis parallel to, we have:
R
2
=
2
+
2
(10)
Comment:
When the eldBis zero,becomes equal to0, which is the sum of the kinetic energy
P
2
2and the potential energy(R). Nevertheless, we must not conclude from this
that whenBis not zero,P
2
2still represents the kinetic energy of the electron. We
have seen (cf.ComplementIII) that the physical meaning of operators acting in the
state space changes when the vector potential is not zero. For example, the momentum
Pno longer represents the mechanical momentum=V, and the kinetic energy is
then equal to:

2
2
=
1
2
[P A(R)]
2
(11)
The meaning of the termP
2
2, taken alone, depends on the gauge chosen. With the
one dened by (2), it can easily be shown to correspond to the relative kinetic energy

2
R2, whereRis the mechanical momentum of the particle with respect to the
Larmor frame rotating aboutBwith angular velocity= 2. The term2
then describes the kinetic energy
2
E2related to the velocity of the frame. As for
1, it corresponds to the cross termER .
1-b. Order of magnitude of the various terms
In the presence of the magnetic eldB, two new terms,1and2, therefore
appear in. Before examining their physical meaning in greater detail, we shall calculate
the order of magnitude of the energy dierences(or the frequency dierences )
associated with them.
As far as0is concerned, we already know the corresponding energy dierences
0(cf.Chap.). The associated frequencies are of the order of:
0
10
14
to10
15
Hz (12)
Also, by using (7b), we see that1is approximately given by:
1
1
~
~=
2
(13)
whereis the Larmor angular velocity
2
:
=
2
(14)
2
Note that the Larmor frequency
2
is half the cyclotron frequency.
857

COMPLEMENT D VII
A simple numerical calculation shows that, for an electron, the Larmor frequency is such
that:
=
2
14010
10
Hztesla = 140 MHzgauss (15)
Now, with the elds usually produced in the laboratory (which rarely exceed 100,000 gauss),
we have:
2
.10
11
Hz (16)
Comparing (12) and (16), we see that:
10 (17)
Let us show, similarly, that:
21 (18)
To do so, we shall evaluate the order of magnitude2of the energies associated with
2. The matrix elements of the operatorR
2
=
2
+
2
are of the same order of
magnitude as
2
0, where0=~
2 2
characterizes atomic dimensions. Thus we obtain:
2
22
2
0 (19)
We then nd the ratio:
2
1
22
2
0
1
~
= 2~
2
0
~
2
(20)
Now, according to formulas (C-12a) and (C-12b) of Chapter:
0
~
2
2
0
(21)
Relation (20) therefore yields, with (13) taken into account:
2
1
1
0
(22)
which, according to (17), proves (18).
Therefore, the eects of the magnetic eld always remain, in practice, much smaller
than those due to the internal eld of the atom. Moreover, it is generally sucient, when
we study them, to retain only the term1, compared to which2is negligible (2will
be taken into account only in the special cases in which the contribution of1is zero)
3
.
3
The Zeeman eect of a three-dimensional harmonic oscillator can be calculated without approxi-
mations (cf.problem 2 of ComplementVII). This is true because(R)and2then have analogous
forms. This example is interesting since it enables us to analyze the contributions of1and2in a
soluble case.
858

THE HYDROGEN ATOM PLACED IN A UNIFORM MAGNETIC FIELD
1-c. Interpretation of the paramagnetic term
Consider, rst of all, the term1given by (7b). We shall see that it can be
interpreted to be the coupling energyM1Bof the eldBand the magnetic moment
M1related to the revolution of the electron in its orbit.
For this purpose, we shall begin by calculating the magnetic momentclassically
associated with a chargein a circular orbit of radius(Fig.). If the speed of the
particle is, its motion is equivalent to a current:
=
2
(23)
Since the surfacedened by this current is:
=
2
(24)
the magnetic moment is given by:
= =
2
(25)q
ℳ
v
Figure 1: Classically, the motion of an elec-
tron in its orbit can be regarded as a current
loop of magnetic moment.
Introducing the angular momentum, which, since the velocity is tangential, has a
modulus of:
= (26)
we can write (25) in the form:
=
2
(27)
(this is a vector relation sinceand are parallel, as both are perpendicular to the
plane of the classical orbit).
The quantum mechanical analogue of (27) is the operator relation:
M1=
2
L (28)
859

COMPLEMENT D VII
We can therefore write1in the form:
1=M1B (29)
This conrms the interpretation given above:1corresponds to the coupling between
the magnetic eldBand the permanent atomic magnetic moment (M1is independent
ofB).1is called theparamagnetic coupling term.
Comments:
()According to (28), the eigenvalues of any component of the magnetic moment
M1are of the form:
2
(~) = (30)
whereis an integer.therefore gives the order of magnitude of the
magnetic moment associated with the orbital moment of the electron. This
is why denition (8) is useful. In the MKSA system:
92710
24
Jouletesla (31)
()As we shall see in Chapter, the electron possesses, in addition to the orbital
angular momentumL, a spin angular momentumS. With this observable is
associated a magnetic moment,M, proportional toS:
M= 2
~
S (32)
Although the magnetic eects due to the spin are important, we shall ignore
them for now (we shall return to them in ComplementXII).
()The classical argument presented above is not completely correct. We have
confused the angular momentum:
=rp (33)
with the moment of the mechanical momentum:
=r v= rA(r) (34)
In fact, the error is small. As we shall see in the next section, it simply
amounts to neglecting2relative to1.
1-d. Interpretation of the diamagnetic term
Consider a zero angular momentum state of the hydrogen atom (for example, the
ground state). The correction supplied by1to the energy of this state is also zero.
Thus, to determine the eect of the eldB, we must now take into account the presence
of2. How should the corresponding energy be interpreted?
860

THE HYDROGEN ATOM PLACED IN A UNIFORM MAGNETIC FIELD
We have seen (cf.ComplementVII, ) that, in the presence of a uniform
magnetic eld, the probability current associated with the electron is modied. This
current is cylindrically symmetric with respect toB. It corresponds to a uniform rotation
of the probability uid, clockwise ifis positive and counterclockwise ifis negative.
With the corresponding electric current is then associated a magnetic momentM2
antiparallel toB, and, therefore, a positive coupling energy, which explains the physical
origin of the term2.
To see this more precisely, we shall return to the classical argument of the preceding
section, taking into account the fact [cf.comment () of ] that the magnetic moment
is in fact proportional to=r v(and not to=rp):
=
2
=
2
[ rA(r)] (35)
When is zero,reduces, in gauge (2), to:
2=
2
4
r(rB) =
2
4
(rB)rr
2
B (36)
2is proportional to the value of the magnetic eld
4
. It therefore represents themoment
inducedbyBin the atom. Its coupling energy withBis:
2=
B
0
2(B)dB=
1
2
2(B)B
=
2
8
r
2
B
2
(rB)
2
=
2
8
r
2
B
2
(37)
as we found in (7c). Therefore, the interpretation given above has been conrmed:2
describes the coupling between the eldBand the magnetic momentM2induced in the
atom. Since the induced moment, according to Lenz' law, opposes the applied eld, the
coupling energy is positive.2is called thediamagnetic termof the Hamiltonian.
Comment:
As we have already pointed out [cf.(18)], atomic diamagnetism is a weak phe-
nomenon which is concealed by paramagnetism when both are present. As is
shown by (37) (and the calculations of ), this result is related to the small
size of the atomic radius: for magnetic elds of the type usually produced, the
magnetic ux intercepted by an atom is very small. It must not be concluded
that we can always neglect2relative to1, whatever the physical problem. For
example, in the case of a free electron (for which the radius of the classical orbit
would be innite in a zero magnetic eld), we saw in ComplementVIthat the
contribution of the diamagnetic term is as important as that of the paramagnetic
term.
4
2is not collinear withB. However, it can be shown that, in the ground state of the hydrogen
atom, the mean valueM2of the operator associated with2is antiparallel toB. This is in agreement
with the result obtained above from the structure of the probability current.
861

COMPLEMENT D VII
2. The Zeeman eect
Now that we have explained the physical signicance of the various terms appearing
in the Hamiltonian, we shall look more closely at their eects on the spectrum of the
hydrogen atom. More precisely, we shall examine the way in which the emission of the
optical line called the resonance line( 1 200

A)is modied when the hydrogen
atom is placed in a static magnetic eld. We shall see that this changes not only the
frequency, but also the polarization, of the atomic lines: this is what is usually called the
Zeeman eect.
Important comment:In reality, because of the existence of electron and proton
spins, the resonance line of hydrogen includes several neighboring components (ne and
hyperne structure;cf.Chap.). Moreover, the spin degrees of freedom profoundly
modify the eect of a magnetic eld on the various components of the resonance line
(the Zeeman eect of the hydrogen atom is sometimes called anomalous). Since we are
ignoring the eects of spin here, the following calculations do not truly correspond to the
real physical situation. However, they can easily be generalized to take spins into account
(cf.ComplementXII). Moreover, the results we shall obtain (the appearance of several
Zeeman components of dierent frequencies and polarizations) remain qualitatively valid.
2-a. Energy levels of the atom in the presence of a magnetic eld
The resonance line of hydrogen corresponds to an atomic transition between the
ground state1(= 1;== 0)and the excited state2(= 2;= 1;= +101).
While the angular momentum is zero in the ground state, it is not so in the excited state;
in calculating optical line modications in the presence of the magnetic eldB, we
therefore make a small error by neglecting the eects of the diamagnetic term2, which
amounts to taking0+1for the Hamiltonian.
We denote by the common eigenstates of0(eigenvalue=
2
),
L
2
[eigenvalue(+ 1)~
2
] and(eigenvalue~). The wave functions of these states
have been calculated in Chapter:
( ) = ()() (38)
We choose theaxis parallel toB; it is not dicult to see that the statesare
then also eigenvectors of0+1:
(0+1) = 0
~
= ( ) (39)
If we neglect the diamagnetic term, the stationary states of the atom placed in the eld
Bare therefore still the; only the corresponding energies are modied.
In particular, for the states involved in the resonance line, we see that:
(0+1)100= 100 (40a)
(0+1)21= [ +~( + )]21 (40b)
where:
=
2 1
~
=
3
4~
(41)
is the angular frequency of the resonance line in a zero eld.
862

THE HYDROGEN ATOM PLACED IN A UNIFORM MAGNETIC FIELD
2-b. Electric dipole oscillations
. Matrix elements of the operator associated with the dipole
Let:
D=R (42)
be the electric dipole operator of the atom. To calculate the mean valueDof this
dipole, we begin by evaluating the matrix elements ofD.
Under reection through the origin,Dis changed intoD: the electric dipole is
therefore an odd operator (cf.ComplementII). Now, the states also have a
well-dened parity: since their angular dependence is given by(), their parity is
+1ifis even and1ifis odd (cf.ComplementVI). It follows, in particular, that:
100D100=0
21D21=0
(43)
for alland.
The non-zero matrix elements ofDare therefore necessarily non-diagonal elements.
To calculate the matrix elements21D100, it is convenient to note that,and
can easily be expressed in terms of the spherical harmonics:
=
23
1
1
()
1
1()
=
23
1
1
() +
1
1()
=
43
0
1()
(44)
In the expressions for the desired matrix elements, we therefore have:
:
=
0
21()10()
3
d (45)
44), reduces
to a scalar product of spherical harmonics, which can be calculated directly from
their orthogonality relations. We obtain, nally:
211 100= 211 100=
6
210 100= 0
(46a)
211 100= 211 100=
6
210 100= 0
(46b)
211 100= 211 100= 0
210 100=
3
(46c)
863

COMPLEMENT D VII
. Calculation of the mean value of the dipole
The results of indicate that, if the system is in a stationary state, the mean
value of the operatorDis zero. Let us assume, rather, that the state vector of the system
is initially a linear superposition of the ground state 1and one of the 2states:
(0)= cos 100+ sin 21 (47)
with= +1, 0 or1(is a real parameter). We then immediately obtain the state
vector at time:
()= cos 100+ sine
(+ )
21 (48)
(we have omitted the global phase factore
~
, which is of no physical consequence).
To calculate the mean value of the electric dipole:
D() = ()D () (49)
we shall use results (46) and (48), and cite three cases:
()if= 1, we obtain:
1=
6
sin 2cos [( +)]
1=
6
sin 2sin [( +)]
1= 0
(50)
The vectorD1()therefore rotates in theplane about theaxis, in the coun-
terclockwise direction and with the angular velocity + .
()if= 0:
0= 0= 0
0=
3
sin 2cos
(51)
The motion ofD0()is now a linear oscillation along theaxis, of angular frequency
.
()if=1:
1=
6
sin 2cos [( )]
1=
6
sin 2sin [( )]
1= 0
(52)
The vectorD1()again rotates in theplane about, but this time in the
clockwise direction and with the angular velocity .
864

THE HYDROGEN ATOM PLACED IN A UNIFORM MAGNETIC FIELD
2-c. Frequency and polarization of emitted radiation
In the three cases (= +1, 0 and1), the mean value of the electric dipole is an
oscillating function of time. It is clear that such a dipole radiates electromagnetic energy.
Since the atomic dimensions are negligible compared to the optical wavelength,
the atom's radiation at great distances can be treated like that of a dipole. We shall
assume that the characteristics of the light emitted (or absorbed) by the atom during
transition between a state21and the ground state are correctly given by the clas-
sical calculation of radiation of a dipole
5
equal to the mean quantum mechanical value
D().
To state the problem precisely, imagine we want to study the radiation emitted
by a sample containing a great number of hydrogen atoms, which have somehow been
excited into a 2state. In most experiments actually performed, the excitation of the
atoms is isotropic, and the three states211,210and 211occur with equal
probability. Therefore, we shall begin by calculating the radiation diagram for each of
the cases cited in the preceding sections. Then we shall obtain the radiation actually
emitted by the atomic system by taking, for each spatial direction, the sum of the light
intensities emitted in each case.
()If= 1, the angular frequency of the emitted radiation is(+ ). The optical
line frequency is therefore slightly shifted by the magnetic eld. In accordance with the
laws of classical electromagnetism applied to a rotating dipole such asD1(), the radi-
ation emitted in thedirection is circularly polarized (the corresponding polarization
is called+). However, the radiation emitted in a direction of theplane is linearly
polarized (parallel to this plane). In other directions, the polarization is elliptical.
()If= 0, we must consider a dipole oscillating linearly along, with angular
frequency, that is, the same as in a zero eld. The wavelength of the radiation is
therefore not changed by the eldB. Its polarization is always linear, whatever the
propagation direction being considered. For example, for a propagation direction situated
in the plane, this polarization is parallel to(polarization). No radiation is
emitted in thedirection (an oscillating linear dipole does not radiate along its axis).
()If=1, the results are analogous to those for= 1. The only dierence
is that the angular frequency of the radiation is( )instead of( + ), and the
dipole rotates in the opposite direction; this changes, for example, the direction of the
circular polarization (polarization).
.
If we now assume that there are equal numbers of excited atoms in the three states
= +1, 0 and1, we see that:
2,(
)2. The polarization associated with the rst one is linear, and that associated
with the others is, in general, elliptical;
B, the three polarizations are linear (cf.
Fig.). The rst one is parallel toB, and the other two are perpendicular. The
intensity of the central line is twice that of each of the shifted lines [cf.formulas
5
If we wanted to treat the problem entirely quantum mechanically, we should have to use the quantum
mechanical theory of radiation. In particular, the return of the atom to the ground state by spontaneous
emission of a photon could only be understood in the framework of this theory. However, the results we
shall obtain here semi-classically would remain essentially valid as far as radiation is concerned.
865

COMPLEMENT D VII2ω
L
/2π
v
Ω/2π
Figure 2: The Zeeman components of the
resonance line of hydrogen observed in a di-
rection perpendicular to the magnetic eld
B(ignoring electron spin). We obtain a
component of unshifted frequency2, po-
larized parallel toB, and two components
shifted by2, polarized perpendicularly
toB.
(50), (51) and (52)]. In a direction parallel toB, only the two shifted frequencies
( )2are emitted, and the associated light polarizations are both circular
but opposite in direction (cf.Fig.).
Comment:
The atom therefore emits+polarized radiation in going from the state211to
the state100,in going from211to100, andin going from210
to100. Formulas (46) furnish a simple rule for nding these polarizations.
Consider the operators+ , and; their only non-zero matrix
elements between the2and1states taken in this order are:
211+ 100 211 100
and 210 100
To the+,andpolarizations, therefore, correspond the operators+ ,
and, respectively. This is a general rule: there is emission of electric
dipole radiation when the operatorDhas a non-zero matrix element between the
atom's initial state and its nal state. The polarization of this radiation is+,
ordepending on whether the non-zero matrix element
6
is that of+ ,
or.v
2ω
L
/2π
Ω/2π
Figure 3: When the observation is carried
out along the direction of the eldB, only
two Zeeman components are obtained, cir-
cularly polarized in opposite directions and
shifted by2.
6
The order of the states in the matrix element must be respected in order not to confuse+with
.
866

THE HYDROGEN ATOM PLACED IN A UNIFORM MAGNETIC FIELD
References and suggestions for further reading:
Paramagnetism and diamagnetism: Feynman II (7.2), Chaps. 34 and 35; Cagnac
and Pebay-Peyroula (11.2), Chaps. VIII and IX; Kittel (13.2), Chap. 14; Slater (1.6),
Chap. 14; Flügge (1.24), 128 and 160.
Dipole radiation: Cagnac and Pebay-Peyroula (11.2), Annex III; Panofsky and
Phillips (7.6), 14-7; Jackson (7.5), 9-2.
Angular momentum of radiation and selection rules: Cagnac and Pebay-Peyroula
(11.2), Chap. XI.
867

SOME ATOMIC ORBITALS. HYBRID ORBITALS
Complement EVII
Some atomic orbitals. Hybrid orbitals
1 Introduction
2 Atomic orbitals associated with real wave functions
2-a orbitals(= 0). . . . . . . . . . . . . . . . . . . . . . . . .
2-b orbitals(= 1). . . . . . . . . . . . . . . . . . . . . . . . .
2-c Other values of . . . . . . . . . . . . . . . . . . . . . . . . .
3 hybridization
3-a Introduction of hybrid orbitals
3-b Properties of hybrid orbitals
3-c Example: the structure of acetylene
4
2
hybridization
4-a Introduction of
2
hybrid orbitals
4-b Properties of
2
hybrid orbitals
4-c Example: the structure of ethylene
5
3
hybridization
5-a Introduction of
3
hybrid orbitals
5-b Properties of
3
hybrid orbitals
5-c Example: The structure of methane
1. Introduction
In , we determined an orthonormal basis of stationary states for the
electron of the hydrogen atom. The corresponding wave functions are:
(r) = ()() (1)
and the quantum numbers refer, respectively, to the energy=
2
, the
square of the angular momentum(+ 1)~
2
, and the-component of the angular mo-
mentum~.
By linearly superposing stationary states of the same energy, that is, of the same
quantum number, we can construct new stationary states which no longer necessarily
correspond to well-dened values ofand. In this complement, we intend to study the
properties of some of these new stationary states in particular, the angular dependence
of the associated wave functions.
The wave functions (1) are often calledatomic orbitals. A linear superposition of
orbitals of the samebut dierentandis called ahybrid orbital. We shall see that
a hybrid orbital can extend further in certain spatial directions than the (pure) orbitals
from which it is constructed. It is this property, important in the formation of chemical
bonds, which justies the introduction of hybrid orbitals.
Although the calculations presented in this complement are rigorously valid only for
the hydrogen atom, we shall also indicate qualitatively how they explain the geometrical
structure of the various bonds formed by an atom with several valence electrons.
869

COMPLEMENT E VII
2. Atomic orbitals associated with real wave functions
In expression (1), the radial function()is real. However,(), except for
= 0, is a complex function of, since:
() =() e (2)
where()is a real function of.
Atomic orbitals are therefore generally complex functions. By superposing the
(r)and (r)orbitals, one can, however, construct real orbitals whose advan-
tage is their simple angular dependence, which can be represented graphically without
having to take the square of the modulus of the wave function (as we did in of
Chapter).
2-a. orbitals(= 0)
When== 0, the wave function00(r)is real, and we say we are dealing with
an orbital. We shall denote the corresponding stationary state by. To represent
the angular dependence of theorbital, we xand measure o in each direction
of polar anglesanda line segment of length( ). The surface obtained by
varyingandis a sphere centered at(Fig.).
2-b. orbitals(= 1)
. ,,orbitals
If we use the expression for the three spherical harmonics()given in Com-
plementVI[formulas (32)], we obtain, for the three atomic orbitals1(r)corre-
sponding to= 1:
11(r) =
38
1() sine (3a)
10(r) =
34
1() cos (3b)
11(r) =
38
1() sine (3c)y
x
O
z
Figure 1: Anorbital is spherically symmet-
ric: the wave function depends on neither
nor.
870

SOME ATOMIC ORBITALS. HYBRID ORBITALSy
x
O
z
+
–
φ
φ
max
1
1
1
0.9
ba
+
–
0.9
0.6
0.6
0.2
A
B
0.2
x/a
0
z/a
0
Figure 2: Two possible representations of aorbital(= 1= 0)
g. a: angular dependence of this orbital. Withxed, we display =1=0( )
for each direction. Thus we obtain two spheres, tangential atto the plane.
The sign indicated on each of them is that of the wave function (which is real).
g. b: cross sections in theplane of a family of surfaces, each one corresponding to
a given value for=1=0( )[we have chosen values equal to 0.2, 0.6, 0.9 times
the maximum value ofat pointsand]. These are surfaces of revolution about
. The sign indicated is that of the wave function (which is real). Unlike the one in
gure a, the representation in gure b depends on the radial part of the wave function
(the one chosen here corresponds to the= 2state of the hydrogen atom).
We now form the three linear superpositions:
10(r) (4a)
1
2
[11(r) 11(r)] (4b)
2
[11(r) + 11(r)] (4c)
It is easy to see that the three preceding wave functions can also be written:
34
1() (5a)
34
1() (5b)
34
1() (5c)
871

COMPLEMENT E VIIy
x
O
z
Figure 3: Angular dependence of aor-
bital (the representation adopted is that of
Figure-a).
These are real functions ofwhich, like the1(r), are orthonormal and form
a basis in the subspace=1. The wave functions (5) will be denoted(r), (r)
and (r)respectively, and will be called ,,orbitals.
Two distinct geometrical representations enable us to visualize the form of an
orbital( ). First of all, if we are interested in the angular dependence of the
orbital, we xand measure o a line segment of length( )along each direction
of polar anglesand. Thus, the angular dependence of the2orbital is that of
= cos. Asvaries between 0 and2, andvaries between 0 and, the end of the
line segment of lengthcosdrawn in the direction of polar anglesanddescribes
two spheres centered on theaxis, tangential atto theplane and mirror images
with respect to theplane (Fig.-a). The sign indicated in the gure is that of the
wave function, which is real. Another possible representation of the orbital( )
is obtained by tracing a family of surfaces, each one corresponding to a given value of
( )(surfaces of equal probability density). This is what is done for the2orbital
in Figure-b (here again, the sign indicated is that of the wave function, which is real).
In the rest of this complement, we shall use one or the other of these two representations.
Theandorbitals can be obtained respectively from theorbitals by rota-
tions through angles of+2and 2about and (cf.Figures, which
use a geometric representation identical to the one in Figure-a).
Unlike anorbital, which is spherically symmetric, the,,orbitals therefore
point along the,,axes, respectively.
. orbitals
The choice of the,,axes is obviously arbitrary. By linearly superposing
the,andorbitals, we should therefore be able to construct aorbital having
the same form but directed along an arbitraryOuaxis.
LetOube such an axis, forming angles,,with,,. We have,
872

SOME ATOMIC ORBITALS. HYBRID ORBITALSy
x
O
z
Figure 4: Angular dependence of aor-
bital.
obviously:
cos
2
+ cos
2
+ cos
2
= 1 (6)
Consider the state:
cos + cos + cos (7)
which, according to (6), is normalized. We can, using formulas (5), put the corresponding
wave function in the form:
34
1()
cos+cos+cos
=
34
1() (8)
where:
=cos+cos+cos (9)
is the projection ofronto theOuaxis. Comparison with (5) indicates that the orbital
so constructed is indeed aorbital.
Therefore, any real and normalized linear superposition of,andorbitals:
(r) + (r) + (r) (10)
can be considered to be aorbital directed along theOudirection, dened by:
cos=
cos=
cos=
(11)
. Example: structure of the H2O and H3N molecules
To a rst approximation (cf.ComplementXIV), in a many-electron atom, each electron
can be considered to move independently of the others in a central potential()which is the
sum of the electrostatic attractive potential of the nucleus and a mean potential due to the
repulsion of the other electrons. Each electron can therefore be found in a state characterized by
873

COMPLEMENT E VII
three quantum numbers, . However, since the potential()no longer varies exactly like
1, the energy no longer depends only on, but also on. We shall see in ComplementXIV
that the energy of the2state is slightly lower than that of the2state; the3state is also
lower than the3state, which is, in turn, lower than the3state, etc.
The existence of spin and Pauli's principle (which we shall study in Chapters )
imply that the1,2, ... sub-shells can contain only two electrons; the 2, 3, ... sub-shells, six
electrons, ...; thesub-shells,2(2+ 1)electrons (the factor2+ 1arises from the degeneracy
related to, and the factor 2, from the electron spin).
Thus, for the oxygen atom, which has eight electrons, the1and2sub-shells are lled
and contain four electrons in all. The four remaining electrons are in the2sub-shell: two of
them (with opposite spins) can ll one of the three2orbitals, for example, 2; the other two
are then distributed in the remaining 2and 2orbitals. These last two electrons are the
valence electrons: they are unpaired, which means that the orbitals in which they can be found
can accept another electron. The 2and2wave functions of the valence electrons of oxygen
are therefore directed along two perpendicular axes. Now, it can be shown that, the greater
the overlapping of the wave functions of the two electrons participating in a chemical bond, the
greater the stability of this bond. The two hydrogen atoms that will bind with the oxygen atom
to form a water molecule must therefore be centered respectively on theand axes. Then
the spherical1orbital of the valence electron of each hydrogen will maximally overlap one of
the2and 2orbitals of the valence electrons of oxygen. Figure
probability clouds associated with the valence electrons of the oxygen and hydrogen atoms in
the water molecule. The graphical representation used is analogous to the one in Figure-b. We
have drawn, for each electron, a surface dened as follows: the probability density has the same
value at all points of this surface; this value is chosen in such a way that the total probability
contained inside the surface has a xed value close to 1 (0.9, for example).
The preceding argument enables us to understand the form of the H2O molecule. The
angle between the two OH bonds should be close to 90. Actually, the angle found experimentally
is 104. The deviation from the value 90arises, in part, from the electrostatic repulsion between
the two protons of the hydrogen atoms, which tends to open up the angle between the two
OH bonds
1
.
An analogous argument explains the pyramidal form of the NH3molecule. The three
valence electrons of nitrogen occupy the2,2,2orbitals, directed at right angles to each
other. Here again, the electrostatic repulsion between the protons of the three hydrogen atoms
causes the bond angle to go from 90to 108(through a slight hybridization of the2and2
orbitals).
2-c. Other values of
We have conned ourselves thus far to theandorbitals. Actually, an orthonor-
mal basis of real orbitals can be constructed for each value of. If we note that [cf.
relation (D-29) of Chapter]:
[()]= (1) () (12)
1
The opening up of the angle between the two OH bonds can be described as the result of a slight
3
hybridization of the2and2orbitals (cf. 5).
874

SOME ATOMIC ORBITALS. HYBRID ORBITALS
Figure 5: Schematic structure of the water molecule H2O. The2and2orbitals
yield bonds making an angle of approximately 90(the real angle is 104because of the
electrostatic repulsion between the two protons).
we immediately see that (for= 0) the two complex functions(r)and (r)
can be replaced by the two functions:
1
2
[ (r) + (1) (r)] (13a)
2
[ (r)(1) (r)] (13b)
which are real and orthonormal.
Thus, for= 2(orbitals), we can construct ve real orbitals, for which the
angular dependence is given by:
12
3 cos
2
1
6 sincoscos6 sincossin32
sin
2
cos 2
32
sin
2
sin 2
(3
2 2,,, 2 2,orbitals).
The form of these orbitals is a little more complicated than that of theand
orbitals to which we shall conne ourselves here. However, it is possible to apply to them
arguments of the same type as those below.
875

COMPLEMENT E VII
3. hybridization
3-a. Introduction of hybrid orbitals
Returning to the hydrogen atom, we shall consider the subspace , sub-
tended by the four real orbitals(r), (r), (r)and (r)(which correspond
to the same energy). We shall show that, by linearly superposingand orbitals,
we can construct other real orbitals, which form an orthonormal basis inand
possess some interesting properties.
We begin by linearly superposing the(r)and (r)orbitals alone, without
using (r)and (r). We therefore replace the two functions(r)and (r)
by the two real orthonormal linear combination:
cos (r) + sin (r) (14a)
sin (r)cos (r) (14b)
In addition, we require the two orbitals (14a) and (14b) to have the same geomet-
rical form. Since this form depends only on the relative amounts of theandorbitals
in the linear superposition, we see immediately that we must havesin= cos, that is,
=4. The two new orbitals we are introducing are therefore of the form:
(r) =
1
2
[(r) + (r)] (15a)
(r) =
1
2
[(r) (r)] (15b)
and correspond to what is called hybridization. Thus we have constructed a new
orthonormal basis of , composed of (r), (r), (r)and (r).
3-b. Properties of hybrid orbitals
To study the angular dependence of the(r)and (r)hybrid orbitals,
we now choose a given value0ofand set:
=
14
0(0)
=
34
1(0) (16)
Thus we obtain, using (5) and (15), the angular functions:
1
2
(+cos)
1
2
( cos)
(17)
which we shall represent, using the same method as in 2 (cf.Fig.-a), by measuring
o, along each direction of polar anglesand, a line segment of length
1
2
[+cos]
876

SOME ATOMIC ORBITALS. HYBRID ORBITALS
or
1
2
[ cos]and indicating by a plus or minus sign whether the wave function
is positive or negative. Figure plane of the
surfaces so obtained, which have cylindrical symmetry with respect to(we have
assumed 0). The (r)orbital can be transformed into the(r)by a
reection through the point. It can be seen that the(r)orbital has no simple
symmetry with respect to the point. This asymmetry is due to the fact that the
(r)and (r)orbitals of which it is formed (and which are shown in Figure-c)
have opposite parities. In the region where0,(r)and (r)have the same sign
and add, while in the region where0,(r)and (r)have opposite signs and
subtract. The conclusions are reversed for(r).+
+
+
–
–
–
x
z
O
x
z
O x
s
z
p
z
O
a b c
Figure 6: Angular dependence of the (r)(g. a) and (r)(g. b), hybrid
orbitals obtained from the(r)and (r)orbitals, which have opposite parities
(g. c). A hybrid orbital can extend further in certain directions than the pure orbitals
from which it is obtained.
The (r)orbital therefore extends further in the positive direction of the
axis than in the negative direction since, for xed, the values it takes on are greater
(in absolute value) for= 0than for=. In general, for large values of,and
are such that the values of the(r)orbital in the positive direction of theaxis
are larger than those taken on separately by the(r)and (r)orbitals [the same
conclusions are valid for the(r)orbital and the negative direction of theaxis].
This property plays an important role in the study of the chemical bond. To
understand this qualitatively, assume that, in a particular atom, one of the valence
electrons can be either in theorbital or in one of theorbitals. Then suppose that
another atomis in the neighborhood of the rst one, and callthe axis joining
and. The (r)orbital ofwill overlap more with the orbitals of the valence
electrons ofthan the(r)or (r)orbitals. Thus we see that hybridization of
the orbitals ofcan lead to a greater stability of the chemical bond, since this stability
increases, as we have already pointed out, with the overlapping of the electronic orbitals
ofandinvolved in the bond.
877

COMPLEMENT E VII
3-c. Example: the structure of acetylene
The carbon atom has six electrons. When this atom is free, two of these electrons are in
the1sub-shell, two in the2sub-shell, and two in the2sub-shell. Only these last two are
unpaired, and we should therefore expect carbon to be bivalent. This is indeed what is observed
in some of its compounds. However, carbon is usually present in a quadrivalent form. This
arises from the fact that when a carbon atom is bound to other atoms, one of its2electrons
can leave this sub-shell and place itself in the third2orbital, which is unoccupied in the free
carbon atom. There are then four unpaired electrons, whose wave functions are the result of a
hybridization of the four orbitals,2,2,2and2
Thus, in the acetylene molecule C2H2, the four valence electrons of each carbon atom
are distributed as follows: two electrons are in the2(r)and
2(r)hybrid orbitals we
have just introduced, and the other two are in the2(r)and2(r)orbitals studied in 2-
b. According to Figures-a and-b, the two electrons of each carbon atom that occupy the
2(r)and
2(r)hybrid orbitals participate in bonds separated by an angle of 180: the
rst one with the other carbon atom, and the second one with one of the two hydrogen atoms
(whose valence electrons occupy1orbitals). Thus we understand why the C2H2molecule is
linear (cf.Fig., where we have used the same type of graphical representation as in Figure).
As for the2orbitals centered on each of the carbon atoms, they present a partial
lateral overlapping, as do the two2orbitals, as is shown by the solid lines in Figure. They
contribute to the reinforcement of the chemical stability of the molecule. The two carbon atoms
thus form atriple bondbetween them. One bond is produced by the2(r)and
2(r)
hybrid orbitals, each centered on one of the two atoms and cylindrically symmetric with respect
to theaxis (bond). Two bonds are produced by the2(r)and2(r)orbitals, which
are symmetric with respect to theand planes (bonds).
Comment:
As we have already pointed out, the2sub-shell, in a many-electron atom, has an energy
greater than that of the2sub-shell. The passage of an electron from the2sub-shell to
the2sub-shell is therefore not energetically favorable. However, the energy needed for
this excitation is fully compensated by the increase in stability due to the hybrid orbitals
involved in the C H and C C bonds.
4.
2
hybridization
4-a. Introduction of
2
hybrid orbitals
We shall now return to the four orbitals(r), (r), (r)and (r), and
replace the rst three by the three following real combinations:
(r) = (r) + (r) + (r) (18a)
(r) = (r) + (r) + (r) (18b)
(r) = (r) + (r) + (r) (18c)
We require the three wave functions (18) to be equivalent, that is, to be transformable
into each other under rotation about. Consequently, the proportion of the(r)
orbital must be the same in each of them:
== (19)
878

SOME ATOMIC ORBITALS. HYBRID ORBITALSz
y
y′
x
x′
C
C
H
H
Figure 7: Schematic structure of the acetylene molecule C2H2. For each carbon atom,
two electrons are in thehybrid orbitals (cf. Fig.) and contribute to the C H and
C C bonds (bonds). In addition, two electrons are in theandorbitals and form
additional bonds between the two carbon atoms (bonds, weaker thanbonds), shown
by the vertical lines in the gure. The C C bond is therefore a triple bond.
It is always possible to choose the axes so as to make the rst orbital (18a) symmetric
about the plane. We can therefore choose:
= 0 (20)
By taking the three orbitals (18) to be normalized and orthogonal, we obtain six relations
which enable us to determine
2
the six coecients,,,,,. A simple calculation
2
Actually, the signs of,andcan be chosen arbitrarily.
879

COMPLEMENT E VII
yields:
(r) =
1
3
(r) +
23
(r) (21a)
(r) =
1
3
(r)
16
(r) +
1
2
(r) (21b)
(r) =
1
3
(r)
16
(r)
1
2
(r) (21c)
We have thus produced what is called
2
hybridization. The three hybrid orbitals
(21) and the(r)orbital form a new orthonormal basis in the space.
4-b. Properties of
2
hybrid orbitals
We shall use the same graphical representation as in Figure. The (r)
orbital has cylindrical symmetry with respect to. Figure-a represents the cross
section in theplane of the surface which describes its angular dependence for xed
. The form of the curve obtained is completely analogous to that of Figure-a: the
orbital points along the positive direction of theaxis.+
–
+
–
+
–
x
a b c
y
x
y
O O x
y
O
Figure 8: Angular dependence of the three orthogonal
2
orbitals. The ,
and orbitals can be transformed into each other by rotations through
120about.
By using expression (4b) for(r), we can easily obtain the action onof
the operator which performs a rotation through an angleabout,e
~
:
e
~
= cos + sin (22)
Also, we obviously have:
e
~
= (23)
880

SOME ATOMIC ORBITALS. HYBRID ORBITALS
Formulas (21) then indicate that:
= e
2
3
~
(24a)
= e
2
3
~
(24b)
The two orbitals (21b) and (21c) can therefore be obtained from the orbital (21a) by
rotations through angles23and23about. Figures (8-b) and (8-c) give the
cross sections in theplane of the surfaces describing their angular dependence.
4-c. Example: the structure of ethylene
As in the acetylene molecule, each of the two carbon atoms of the ethylene molecule
C2H4has four valence electrons (one electron in the2sub-shell and three electrons in the2
sub-shell).
Three of these four electrons occupy
2
hybrid orbitals of the type we just considered.
They are the ones that, for each carbon atom, form the bonds with the neighboring carbon atom
and the two hydrogen atoms of the CH2group. Thus we see why the three bonds, C C, C H,
C H, originating from one carbon atom are coplanar and form angles of 120with each other
(cf.Fig., in which we have used the same graphical representation as in Figures ).
The remaining electron of each carbon atom occupies the2orbital. The2orbitals of the
two carbons present a partial lateral overlapping, shown by the solid lines in Figure.x
z′ z
H H
C
H
C
H
Figure 9: Schematic structure of the ethylene molecule C2H4. The two carbon atoms
form a double bond with each other: onebond due to
2
orbitals of the type of those
shown in Figure
2
hybrid orbitals at 120with this one form the C H
bonds), and onebond, due to the overlapping of theorbitals.
The two carbon atoms of the ethylene molecule are therefore connected by adouble bond:
one bond formed by two hybrid orbitals of the
2
type, cylindrically symmetric with respect to
the axis joining the two carbon atoms (bond), and one bond formed by two 2orbitals,
symmetric about the plane (bond). It is the latter bond which blocks the rotation of
one CH2group with respect to the other one. If one of the CH2groups were to rotate with
respect to the other one about the axis joining the two carbons, the axes of the two orbitals
2and 2(Fig.) would no longer be parallel. This would diminish their lateral overlapping
881

COMPLEMENT E VII
and, consequently, the stability of the system. Thus we see why the six atoms of the ethylene
molecule are in the same plane.
5.
3
hybridization
5-a. Introduction of
3
hybrid orbitals
We shall now superpose the four orbitals,(r), (r), (r), (r), to
form the four hybrid orbitals:
(r) = (r) + (r) + (r) + (r) (25a)
(r) = (r) + (r) + (r) + (r) (25b)
(r) = (r) + (r) + (r) + (r) (25c)
(r) = (r) + (r) + (r) + (r) (25d)
We again require the four orbitals to have the same geometrical form, which means that:
=== (26)
We can arbitrarily choose the symmetry axis of one of the orbitals, then the plane con-
taining this axis and that of a second orbital. This reduces the number of free parameters
to 10; we can nd them by taking the four orbitals (25) to be orthonormal.
We shall content ourselves here with giving a possible set of such hybrid orbitals,
dened by:
====
1
2
= = ==
1
2
= == =
1
2
= = = =
1
2
(27)
and which can easily be shown to be orthonormal and of the same geometrical form. All
the other possible sets can be obtained from this one by rotation.
We have thus produced what is called a
3
hybridization. The four orbitals
(25) corresponding to the coecients (27) form a new orthonormal basis in the space
.
5-b. Properties of
3
hybrid orbitals
The four orbitals constructed in 5-a are analogous in form to those studied in
3 and 4. They point respectively in the directions of the vectors whose components
are:
(111)
(111)
(111)
(111)
(28)
882

SOME ATOMIC ORBITALS. HYBRID ORBITALSH
H
H
H
C
Figure 10: Schematic structure of the
methane molecule. The
3
orbitals produce
bonds arranged like the straight lines joining
the center of a tetrahedron to its four corners
(angles of 10928).
The axes of the four
3
orbitals are therefore arranged like the straight lines joining
the center of a regular tetrahedron to the four corners of this tetrahedron. The angle
between any two of these straight lines is equal to 10928.
5-c. Example: The structure of methane
In the methane molecule CH4, the four valence electrons of the carbon atom each occupy
one of the four
3
hybrid orbitals studied above. This immediately explains why the four
hydrogen atoms form the corners of a regular tetrahedron centered on the carbon atom (Fig.).
In the ethane molecule C2H6, one of the hydrogen atoms of methane is replaced by a
CH3group. The two carbon atoms are then connected by a single bond, formed by two
3
hybrid orbitals, cylindrically symmetric with respect to the straight line joining the two carbon
atoms. The absence of a double bond permits the practically free rotation of one CH3group
with respect to the other one.
References and suggestions for further reading:
Various geometrical representations of orbitals: Levine (12.3), 6.6; Karplus and
Porter (12.1), 3.10.
Hybrid orbitals: Karplus and Porter (12.1), 6.3; Alonso and Finn III (1.4), 5-5;
Eyring et al. (12.5), Chap. XII, 12 b; Coulson (12.6), Chap. VIII; Pauling (12.2),
Chap. III, 13 and 14.
883

VIBRATIONAL-ROTATIONAL LEVELS OF DIATOMIC MOLECULES
Complement FVII
Vibrational-rotational levels of diatomic molecules
1 Introduction
2 Approximate solution of the radial equation
2-a The zero angular momentum states(= 0). . . . . . . . . .
2-b General case (any positive integer)
2-c The vibrational-rotational spectrum
3 Evaluation of some corrections
3-a More precise study of the form of the eective potentiale()893
3-b Energy levels and wave functions of the stationary states
3-c Interpretation of the various corrections
1. Introduction
In this complement, we shall use the results of Chapter
cally the stationary states of the system formed by the two nuclei of a diatomic molecule.
We shall simultaneously take into account all the degrees of freedom of the system: vi-
bration of the two nuclei about their equilibrium position and rotation of the system
about the center of mass. We shall show that the results obtained in ComplementsV
andVI, in which one degree of freedom at a time was considered, are valid to a rst
approximation. In addition, a certain number of corrections due to the centrifugal dis-
tortion of the molecule and to the vibration-rotation coupling will be calculated and
interpreted.
We saw in 1-a of ComplementV(the Born-Oppenheimer approximation) that
the potential energy()of interaction between the two nuclei depends only on the
distancebetween them and has the form shown in Figure:()is attractive at large
distances, repulsive at short distances, and has a minimum at=of depth0. Let1
and2be the masses of the two nuclei. Since()depends only on, we can, according
to , study separately the motion of the center of mass (a free particle
of mass= 1+ 2) and the relative motion in the center of mass reference frame,
which is equivalent to that of a ctitious particle of mass:
=
12
1+2
(1)
placed in the potential()of Figure.
If we are interested only in the relative motion, the stationary states of the system,
according to the results of , are described by the wave functions:
( ) =
1
()() (2)
885

COMPLEMENT F VIIr
r
e
V(r)
– V
0
Figure 1: Variation of the interaction poten-
tial energy()between the nuclei of a di-
atomic molecule as a function of the distance
between them.()takes on its minimum
value 0at=. The rst vibrational
states are represented by the horizontal lines
in the potential well.
where the corresponding energiesand the radial functions()are given by the
equation:
~
2
2
d
2
d
2
+() +
(+ 1)~
2
2
2
() = () (3)
Comment:
Rigorously speaking, we implicitly assume in all of this complement (as inVand
CVI) that the projection of the total orbital angular momentum of the electrons
onto the internuclear axis is zero, as is their total spin. The total angular momen-
tum of the molecule then arises only from the rotation of the two nuclei. Such a
situation is found in virtually all diatomic molecules in their ground states. In the
general case, terms will also appear in the nuclear interaction energy which do not
depend exclusively on the distance.
2. Approximate solution of the radial equation
The radial equation has the same form as the eigenvalue equation of the Hamiltonian of a
one-dimensional problem in which a particle of massis placed in the eective potential:
e() =() +
(+ 1)~
2
2
2
(4)
2-a. The zero angular momentum states (= 0)
For= 0, the centrifugal potential(+)~
2
2
2
is zero, ande()is then the
same as(). In the neighborhood of the minimum at=,()can be expanded in
powers of :
() = 0+( )
2
( )
3
+ (5)
The coecientsandare positive, since=is a minimum and since the potential
increases faster forthan for
886

VIBRATIONAL-ROTATIONAL LEVELS OF DIATOMIC MOLECULES
We begin by neglecting the term in( )
3
and terms of higher order. The
potential is then purely parabolic, and we know the eigenstates and eigenvalues of the
Hamiltonian. If we set:
=
2
(6)
we obtain levels whose energy is:
0= 0++
1
2
~ (7)
(= 012)
with the associated wave functions (cf.Chap. V):
() =
2
14
1
2!
e
2
( )
2
2
[( )] (8)
with:
=
~
(9)
(is a Hermite polynomial). In Figure, we have represented the rst two energy
levels by horizontal lines. The length of the lines gives an idea of the extension()of
the wave functions corresponding to these states. Recall [Chap., formula (D-5a)] that:
()
+
12
~
(10)
For the preceding calculation to be valid, it is necessary, in a region of width()
about=, for the term in( )
3
of (5) to be always negligible compared to the
term in( )
2
. We must therefore have:
()=()0
+
12
(11)
where()0is the extension of the ground state:
()0=
~
(12)
This implies, in particular that:
()0 (13)
Condition (13) is always satised in practice. We shall conne ourselves in what follows
to quantum numberswhich are small enough for (11) to be satised as well.
887

COMPLEMENT F VII
Comment:
Expansion (5) is obviously not valid at= 0, where()is innite. The preceding
argument implicitly assumes that:
() (14)
In this case, the wave functions (8) are practically zero at the origin, and almost
identical to the exact solutions of the radial equation (3), which must be rigorously
zero at= 0(cf. ).
2-b. General case ( any positive integer)
. Evaluation of the eect of the centrifugal potential
At=, the centrifugal potential is equal to:
(+ 1)~
2
2
2
= (+ 1) (15)
where:
=
~
4
2
(16)
is the rotational constant introduced in ComplementVI. We have already pointed out in
that complement ( 4-a-) that the energy 2(the distance between two adjacent lines
of the pure rotational spectrum) is always very much smaller than~(the vibrational
quantum):
2 ~ (17)
We shall conne ourselves here to rotational quantum numberssuciently small that:
(+ 1)~ (18)
In a domain of small widthabout=, the variation of the centrifugal
potential is of the order of:
(+ 1)~
2
3
= 2(+ 1)

(19)
That of the potential()is approximately:
()
2
=
1
2
2
()
2
=
1
2
~
()
2
()
2
0
(20)
where we have used (12). We know from 2-a that the extensionof the wave functions
we shall be considering is negligible with respect to, but certainly at least of the order of
()0. Consequently, in the region of space in which the wave functions have signicant
amplitudes, the variation (19) of the centrifugal potential is, according to (18), much
smaller than that of()found in (20). We can then, to a rst approximation, replace
the centrifugal potential, in equation (4), by its value (15) at=. This gives for the
eective potential:
e() () +(+ 1) (21)
888

VIBRATIONAL-ROTATIONAL LEVELS OF DIATOMIC MOLECULES
. Energy levels and stationary wave functions
By using (21) and neglecting terms of order greater than two in expansion (5), we
can put the radial equation (3) in the form:
~
2
2
d
2
d
2
+
1
2
2
( )
2
() = [+0 (+ 1)]() (22)
which is completely analogous to the eigenvalue equation of a one-dimensional harmonic
oscillator.
It thus follows that the term in brackets on the right-hand side must be equal to
(+ 12)~, where= 012; this yields the possible energiesof the molecule:
= 0++
1
2
~+ (+ 1) (23)
with:
= 012
= 012
As for the radial functions, they do not depend on, since the dierential appearing on
the left-hand side of (22) does not depend on. Consequently, we have:
() =() (24)
where()was given in (8). Expression (2) for the wave functions of the stationary
states can then be written, in this approximation:
( ) =
1
()() (25)
Thus we see that the energies of the stationary states are the sums of the energies
calculated in ComplementsVandVI, in which only one degree of freedom at a time
(vibration or rotation) was taken into account. In addition, the wave functions are the
products of the wave functions found in these two complements, to within a factor of
1.
Figure = 0and= 1, with their rotational
structure due to the term(+ 1).
2-c. The vibrational-rotational spectrum
We shall conne ourselves to the study of the infrared absorption or emission
spectrum, thereby assuming the molecule to be heteropolar (calculations analogous to
those presented in l-c-ofVand 4-b ofVIcould be performed when dealing with
homopolar molecules and the Raman eect).
. Selection rules
Recall that the dipole moment()of the molecule is directed along the straight
line joining the two nuclei and can be expanded in powers ofabout:
() =0+1( ) + (26)
889

COMPLEMENT F VIIυ = 1
υ = 0
l = 3
l = 2
l = 1
l = 0
l = 3
l = 2
l = 1
l = 0
Figure 2: Diagram showing the rst two vibrational levels (= 0and= 1) of a
diatomic molecule and their rotational structure (= 012). Within the limits of the
approximations, this rotational structure is the same for the various vibrational levels.
For a heteropolar molecule, the transitions represented by the vertical arrows in the gure
yield the lines of the vibrational-rotational spectrum of the molecule. These lines fall in
the infrared. These transitions obey the selection rule= =1.
The projection of this dipole moment ontois equal to() cos(whereis the angle
between the axis of the molecule and).
We want to determine the frequency spectrum of electromagnetic waves polarized
along that the molecule can absorb or emit as a consequence of the variation of
its electric dipole. As we have done several times before, we shall look for the Bohr
frequencies that can appear in the time evolution of the mean value of() cos. All we
must do, then, is nd for what values of,,and,,the matrix element:
() cos
=
2
dd ( )() cos ( )(27)
is dierent from zero. Using expression (25) for the wave functions, we put this matrix
890

VIBRATIONAL-ROTATIONAL LEVELS OF DIATOMIC MOLECULES
element in the form:
0
d ()()() d () cos() (28)
We thus obtain a product of two integrals which have already been treated in Comple-
mentsVandVI. The second integral is dierent from zero only if:
= +11 (29)
As for the rst one, if we conne ourselves to the terms in0and1of (26), it is dierent
from zero only if:
= 0+11 (30)
The set of lines corresponding to= 0constitutes the pure rotational spectrum
studied in ComplementVI(its intensity is proportional to
2
0). As for the lines=
1, =1, of intensity proportional to
2
1, they constitute the vibrational-rotational
spectrum which we shall now briey describe.
Comment:
The selection rule= +1arises from the angular dependence of the wave functions.
It is therefore independent of the approximation used to solve the radial equation (),
while (30) is valid only in the harmonic approximation.
. Form of the spectrum
Letbe the larger of the two vibrational quantum numbers under consideration
(=+ 1). The vibrational-rotational lines can be separated into two groups:
the lines=+ 1,=+ 1 ,, of frequencies:
2
+(+ 1)(+ 2) (+ 1) =
2
+ 2(+ 1) (31)
with= 012
(these lines correspond to the transitions indicated by the arrows on the right-hand side
of Figure).
the lines=+ 1,= 1 ,, of frequencies:
2
+(+ 1)(+ 1)(+ 2) =
2
2(+ 1) (32)
with= 012
(transitions indicated by the arrows on the left-hand side of Figure).
The vibrational-rotational spectrum therefore has the form shown in Figure. It
contains two groups of equidistant lines, symmetric with respect to the vibrational fre-
quency2. All these lines together constitute a band. The group of lines correspond-
ing to frequencies (31) is called the branch of the band, and the one corresponding
to frequencies (32), the branch of the band. In each branch, the distance between
two adjacent lines is 2. The central interval separating the two branches is of width
891

COMPLEMENT F VIIP branch
3↔4 2↔3 1↔2 0↔1 1 ↔0 2↔1 3↔2 4↔3
ω/2π
ν
R branch
Figure 3: The vibrational-rotational spectrum for a heteropolar molecule. Since transi-
tions between levels of Figure are forbidden by the selection
rules, no line has the pure vibrational frequency
2
. Transitions in which the molecule
passes from the level()to the level(= 1= 1)correspond to frequencies
2
+ 2(+ 1)(lines of the branch). Transitions in which the molecule passes from
the level()to the level (= 1,=+1) correspond to frequencies
2
2(+1)
(lines of the branch). The dierent lines are labeledin the gure.
4: there is no line at the pure vibrational frequency2(there is often said to be a
missing line in the spectrum).
Comment:
The pure vibrational spectrum, studied inVand composed of a single line at
2, therefore does not exist in practice. It is only when one uses a spectroscopic
device with low resolution that one can ignore the rotational structure of the
vibrational-rotational line and treat the band of Figure
at2(recall that2 2).
3. Evaluation of some corrections
The calculations of the preceding section are based on the approximation that replaces the
centrifugal potential by its value at=in the radial equation. The eective potential
e()can then be obtained from()by a simple vertical translation [formula (21)].
In this section, we shall study the corrections that must be performed on the results
of 2 in order to take into account the slow variation of the centrifugal potential about
=. To do so, we shall use its expansion in powers of( ):
(+ 1)~
2
2
2
=
(+ 1)~
2
2
2
(+ 1)~
2
3
( ) +
3(+ 1)~
2
2
4
( )
2
+ (33)
892

VIBRATIONAL-ROTATIONAL LEVELS OF DIATOMIC MOLECULES
3-a. More precise study of the form of the eective potentiale()
If we use (5) and (33), the expansion of the eective potential (4) in the neighbor-
hood of=can be written:
e() = 0+( )
2
( )
3
+
+
(+ 1)~
2
2
2
(+ 1)~
2
3
( ) +
3(+ 1)~
2
2
4
( )
2
+ (34)
We shall see that the variation of the centrifugal potential in the neighborhood of
=produces, fordierent from zero, the following eects:
()The position~of the minimum ofe()does not coincide exactly with.
()The valuee(~)of this minimum is slightly dierent from0+ (+ 1).
()The curvature ofe()at= ~[which xes, as in formula (6), the an-
gular frequency of the equivalent harmonic oscillator] is no longer strictly given by the
coecient.
We shall evaluate these various eects by using expansion (34). As far as the rst
two are concerned, we can neglect terms of order higher than 2 in(), and those of
order higher than 1 in the centrifugal potential , since the distance~ which we shall
nd is very small [it will even be small relative to()0]. In fact, we shall be able to
verifya posteriorithat:
(~ ) (35a)
3(+ 1)~
2
2
4
(~ )
(+ 1)~
2
3
(35b)
. Position and value of the minimum ofe()
If, in expansion (34), we keep only the rst two terms of()and the rst two
terms of the centrifugal potential,~is given by:
2(~ )
(+ 1)~
2
3
(36)
that is:
~
(+ 1)~
2
2
3
=
(+ 1)
(37)
According to (6) and (12), we have:
~
()0
2(+ 1)
~
()0
1 (38)
which, with (13) and (14) taken into account, proves (35a) and (35b).
Substituting this value of~into the expansion ofe(), we nd:
e(~) 0+ (+ 1) [(+ 1)]
2
(39)
with:
=
~
3
8
26
(40)
893

COMPLEMENT F VII
. Curvature of e()at its minimum
In the neighborhood of= ~, we can therefore writee()in the form:
e() =e(~) +(~)
2
(~)
3
+ (41)
The coecientis related to the curvature ofe()at= ~:
=
1
2
d
2
d
2
e()
=~
(42)
To evaluate the dierence betweenand, we must take into account the term
in( )
3
of()in expansion (34) and, consequently, also the term in( )
2
of the
centrifugal potential. A simple calculation then yields, using (37):
2 2+
3(+ 1)~
2
4
3(+ 1)~
2
3
(43)
The angular frequencydened by (6) must therefore be replaced by:
=
2
(44)
Expanding the square root, we easily nd:
= 2 (+ 1) (45)
with:
=
3~
2
8
3
1
(46)
We could carry out an analogous calculation to determine. Actually, since the
term in(~)
3
of (41) adds only a small correction to the results obtained using the
rst two terms, we shall neglect the variation of
d
3
d
3
e()when we go fromto~,
and take .
In conclusion, in the neighborhood of its minimum, we can writee()in the
form:
e() e(~) +
1
2
2
(~)
2
(~)
3
(47)
where~,e(~),are given by (37), (39) and (45).
3-b. Energy levels and wave functions of the stationary states
With expression (47) fore(), the radial equation becomes:
2
2
d
2
d
2
+
1
2
2
(~)
2
(~)
3
() = [ e(~)]() (48)
894

VIBRATIONAL-ROTATIONAL LEVELS OF DIATOMIC MOLECULES
If, as in 2, we neglect the term in(~)
3
, we recognize the eigenvalue equa-
tion of a one-dimensional harmonic oscillator of angular frequencywhose equilibrium
position is= ~. From this, we deduce that the only possible values of the term in
brackets on the right-hand side are(+ 12)~, with= 012According to (39),
we therefore have:
= 0++
1
2
~+ (+ 1) [(+ 1)]
2
(49)
As for the wave functions of the stationary states, they have the same form as in
(25). All we need to do in expression (8) for the radial function is replaceby~and
by:
=
~
(50)
We have taken into account the term in( )
3
in the calculation of the new
angular frequency. For the calculation to be consistent, it is then necessary to eval-
uate the corrections in the eigenvalues and eigenfunctions of the radial equation due to
the presence of this term on the left-hand side of (48). We shall do this in Comple-
mentXI, using perturbation theory. Here, we shall content ourselves with stating the
result concerning the eigenvalues: we must add to expression (49) for the energy the
term:
~ +
1
2
2
+
7
60
~ (51)
where:
=
15
4
2
~
35
(52)
is a dimensionless quantity much smaller than 1 (hencemay be replaced byin this
corrective term).
3-c. Interpretation of the various corrections
. Centrifugal distortion of the molecule
The discussion of 3-a-shows that the distance between the two nuclei increases
when the molecule rotates. According to (37), this increase in distance becomes larger
when(+ 1)becomes larger, that is, when the molecule rotates faster. This is quite
comprehensible: in classical terms, one would say that the centrifugal force tends to
separate the two nuclei until it is balanced by the restoring force 2(~ )due to the
potential().
The molecule is therefore not really a rigid rotator. The variation~ of
the average distance between the nuclei produces an increase in the moment of inertia
of the molecule and, consequently, a decrease (at constant angular momentum) in the
rotational energy. This decrease is only partially compensated by the increase in the
potential energy(~) (). This is the physical origin of the energy correction
2
(+ 1)
2
which appears in (49). This correction, whose sign is negative, increases
895

COMPLEMENT F VII
much faster withthan the rotational energy(+1). This can be seen experimentally:
the lines of the pure rotational spectrum are not rigorously equidistant; the separation
of the lines decreases whenincreases.
. Vibrational-rotational coupling
We shall group the second and third terms of (49) and replaceby its expression
(45). We obtain:
+
1
2
~+ (+ 1) =+
1
2
~+ (+ 1) (+ 1)+
1
2
(53)
The rst two terms on the right-hand side of (53) are the vibrational and rotational
energies calculated in ComplementsVandVI. The third term, which depends on the
two quantum numbersand, represents the eects of the coupling of the vibrational
and rotational degrees of freedom.
We can rewrite (53) in the form:
+
1
2
~+ (+ 1) (54)
with:
= +
1
2
(55)
It looks as if each vibrational level had an eective rotational constantdepending on
theassociated.
To explain this coupling of the vibration and rotation of the molecule, we shall
argue in classical terms. The rotational constantis proportional to1
2
[formula (16)].
When the molecule vibrates,varies, and, consequently, so does. Since the vibrational
frequencies are much higher than the rotational ones, we can dene an eective rotational
constant of the molecule in a given vibrational state: this will be the average oftaken
over a time interval which is much longer than the vibrational period. We must therefore
take the time average of1
2
in the vibrational state under consideration.
In this way, we can interpret the two terms of opposite sign that appear in ex-
pression (46) for. The rst of these terms, which is proportional to, is due to the
anharmonicity of the potential(), which increases with the amplitude of vibration
(that is, in fact, with). Given the asymmetric form of()(Fig.), the molecule
spends more time in the region than in the region . It follows that
the average value of1
2
is less than1
2
: the anharmonicity decreases the eective
rotational constant. This can be seen in formulas (55) and (46). Actually, even if the
vibrational motion were perfectly symmetric with respect to(that is, ifwere zero),
the average value of1
2
would not be equal to1
2
, since:
1
2
=
1
2
(56)
This is the origin of the second term of expression (46): when the average of1
2
is
taken, small values ofare favorized, so that1
2
is greater than1
2
; hence the
sign of this second correction.
896

VIBRATIONAL-ROTATIONAL LEVELS OF DIATOMIC MOLECULES
The overall sign ofresults from the competition between the two preceding
eects. In general, the anharmonicity term dominates, so thatis positive andis
less than.
Comments:
()Vibrational-rotational coupling exists even in the vibrational ground state= 0:
0=
1
2
(57)
This is another manifestation of the nite extension()0of the wave function of
the= 0state.
()Experimentally, vibrational-rotational coupling appears in the following way: if
is positive, the rotational structure is slightly more compact in the higher vibrational
statethan in the lower vibrational state= 1. It is easy to show that the
andbranches of Figure
completely equidistant and are, on the average, closer together in thebranch
than in thebranch.
To sum up, the energy of a vibrational-rotational level of a diatomic molecule,
labeled by the quantum numbersand, is given by:
= 0++
1
2
~+ +
1
2
(+ 1)
2
(+ 1)
2
+ +
1
2
2
~+
7
60
~ (58)
0: dissociation energy of the molecule;
2: vibrational frequency;
: rotational constant given by (16);
,,: dimensionless constants given by (40), (46)) and (52).
References and suggestions for further reading:
Molecular spectra: Eisberg and Resnick (1.3), Chap. 12; Pauling and Wilson (1.9),
Chap. X; Karplus and Porter (12.1), Chap. 7; Herzberg (12.4), Vol. I, Chap. III, 2
b and 2 c; Landau and Lifshitz (1.19), Chaps. XI and XIII.
Nuclear vibration and rotation: Valentin (16.1), VII-2.
897

EXERCISES
Complement GVII
Exercises
1. Particle in a cylindrically symmetric potential
Let,,be the cylindrical coordinates of a spinless particle (=cos,=sin;
0,0 2). Assume that the potential energy of this particle depends only on
, and not onand. Recall that:
2
2
+
2
2
=
2
2
+
1
+
1
2
2
2
Write, in cylindrical coordinates, the dierential operator associated with the
Hamiltonian. Show thatcommutes withand. Show that this allows writing the
wave functions associated with the stationary states of the particle as:
( ) = () ee
where the values that can be taken on by the indicesandare to be specied.
Write, in cylindrical coordinates, the eigenvalue equation of the Hamiltonian
of the particle. Derive from it the dierential equation that()obeys.
Letbe the operator whose action, in therrepresentation, is to change
to(reection with respect to theplane). Doescommute with? Show
thatanticommutes with, and show that, as a result, is an eigenvector
of. What is the corresponding eigenvalue? What can be concluded concerning the
degeneracy of the energy levels of the particle? Could this result be predicted directly
from the dierential equation established in()?
2. Three-dimensional harmonic oscillator in a uniform magnetic eld
N.B. The object of this exercise is to study a simple physical system for which the
eect of a uniform magnetic eld can be calculated exactly. This will allow comparing
precisely the relative importance of the paramagnetic and diamagnetic terms. The
modication of the wave function of the ground state due to the eect of the diamagnetic
term is also detailed (the reader may wish to refer to ComplementsVIandVII).
Consider a particle of mass, whose Hamiltonian is:
0=
P
2
2
+
1
2
2
0R
2
(an isotropic three-dimensional harmonic oscillator), where0is a given positive con-
stant.
Find the energy levels of the particle and their degrees of degeneracy. Is it
possible to construct a basis of eigenstates common to0,L
2
,?
Now, assume that the particle, which has a charge, is placed in a uniform
magnetic eldBparallel to. We set= 2. If we choose to use the gauge
A=
1
2
rB, the Hamiltonianof the particle is:
=0+1()
899

COMPLEMENT G VII
where1is the sum of an operator which is linearly dependent on(the paramagnetic
term) and an operator which is quadratically dependent on(the diamagnetic term).
Show that the new stationary states of the system and their degrees of degeneracy can
be determined exactly.
Show that ifis much smaller than0, the eect of the diamagnetic term is
negligible compared to that of the paramagnetic term.
We now consider the rst excited state of the oscillator, that is, the states whose
energies approach5~02when 0. To rst order in 0, what are the energy
levels in the presence of the eldBand their degrees of degeneracy (the Zeeman eect for
a three-dimensional harmonic oscillator)? Same questions for the second excited state.
Now consider the ground state. How does its energy vary as a function of
(the diamagnetic eect on the ground state)? Calculate the magnetic susceptibilityof
this state. Is the ground state, in the presence of the eldB, an eigenvector ofL
2
? of
? of? Give the form of its wave function and the corresponding probability current.
Show that the eect of the eldBis to compress the wave function about(in a ratio
[1 + ( 0)
2
]
14
) and to induce a current.
900

Index[The notation (ex.) refers to an exercise]
Absorption
and emission of photons,
collision with,
of a quantum, a photon, ,
of eld,
of several photons,
rates,
Acceptor (electron acceptor),
Acetylene (molecule),
Action,, ,
Addition
of angular momenta, ,
of spherical harmonics,
of two spins 1/2,
Adiabatic
branching of the potential,
Adjoint
matrix,
operator,
Algebra (commutators),
Allowed energy band,, ,
Ammonia (molecule),,
Amplitude
scattering amplitude,,
Angle (quantum),
Angular momentum
addition of momenta, ,
and rotations,
classical,
commutation relations,,
conservation,,,
coupling,
electromagnetic eld, ,
half-integral,
of identical particles, (ex.)
of photons,
orbital,,,
quantization,
quantum,
spin,,
standard representation,,
two coupled momenta,
Anharmonic oscillator,,
Annihilation operator,,,,
Annihilation-creation (pair), ,
Anomalous
average value, ,
dispersion,
Zeeman eect,
Anti-normal correlation function, ,
1789
Anti-resonant term,
Anti-Stokes (Raman line),,
Antibunching (photon),
Anticommutation,
eld operator,
Anticrossing of levels,,
Antisymmetric ket, state, ,
Antisymmetrizer, ,
Applications of the perturbation theory,
1231
Approximation
central eld approximation,
secular approximation,
Argument (EPR),
Atom(s),seehelium, hydrogenoid
donor,
dressed, ,
many-electron atoms, ,
mirrors for atoms,
muonic atom,
single atom uorescence,
Atomic
beam (deceleration),
orbital,, (ex.)
parameters,
Attractive bosons,
Autler-Townes
doublet,
eect,
Autoionization,
Average value (anomalous),
Azimuthal
quantum number,
Band (energy),
Bardeen-Cooper-Schrieer,
Barrier (potential barrier),,,
Basis
901
Quantum Mechanics, Volume I, Second Edition. C. Cohen-Tannoudji, B. Diu, and F. Laloë.
© 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

INDEX [The notation (ex.) refers to an exercise]
change of bases,
characteristic relations,,
continuous basis in the space of states,
99
mixed basis in the space of states,
BCHSH inequalities, ,
BCS,
broken pairs and excited pairs,
coherent length,
distribution functions,
elementary excitations,
excited states,
gap, , ,
pairs (wave function of),
phase locking, , ,
physical mechanism,
two-particle distribution,
Bell's
inequality,
theorem, ,
Benzene (molecule),,
Bessel
Bessel-Parseval relation,
spherical Bessel function,
spherical equation,
spherical function,
Biorthonormal decomposition,
Bitter,
Blackbody radiation,
Bloch
equations,, ,
theorem,
Bogolubov
excitations,
Hamiltonian,
operator method,
phonons, spectrum,
transformation,
Bogolubov-Valatin transformation, ,
1919
Bohr,
electronic magneton,
frequencies,
magneton,seefront cover pages
model,,
nuclear magneton,
radius,
Boltzmann
constant,seefront cover pages
distribution,
Born
approximation,,,
Born-Oppenheimer approximation,,
1177,
Born-von Karman conditions,
Bose-Einstein
condensation, , ,
condensation (repulsive bosons),
condensation of pairs,
distribution,,
statistics,
Bosons,
at non-zero temperature,
attractive,
attractive instability,
condensed,
in a Fock state,
paired,
Boundary conditions (periodic),
Bra,,,
Bragg reection,
Brillouin
formula,
zone,
Broadband
detector,
optical excitation,
Broadening (radiative),
Broken pairs and excited pairs (BCS),
1920
Brossel,
Bunching of bosons,
C.S.C.O.,,,,
Canonical
commutation relations,,,
ensemble,
Hamilton-Jacobi canonical equations,
214
Hamilton-Jacobi equations,
Cauchy principal part,
Center of mass,,
Center of mass frame,
Central
902

INDEX [The notation (ex.) refers to an exercise]
eld approximation,
potential,
Central potential,,
scattering,
stationary states,
Centrifugal potential,,,
Chain (von Neumann),
Chain of coupled harmonic oscillators,
Change
of bases,,,
of representation,
Characteristic equation,
Characteristic relation of an orthonormal
basis,
Charged harmonic oscillator in an elec-
tric eld,
Charged particle
in an electromagnetic eld,
Charged particle in a magnetic eld,,
321,
Chemical bond,,, ,
Chemical potential, ,
Circular quanta,,
Classical
electrodynamics,
histories,
Clebsch-Gordan coecients, ,
Closure relation,,
Coecients
Clebsch-Gordan,
Einstein, ,
Coherences (of the density matrix),
Coherent length (BCS),
Coherent state (eld),
Coherent superposition of states,,,
307
Collision,
between identical particles, , (ex.)
between identical particles in classi-
cal mechanics,
between two identical particles,
cross section,
scattering states,
total scattering cross section,
with absorption,
Combination
of atomic orbitals,
Commutation,
canonical relations,,
eld operator,
of pair eld operators,
relations,
Commutation relations
angular momentum,,
eld, ,
Commutator algebra,
Commutator(s),,,,
of functions of operators,
Compatibility of observables,
Complementarity,
Complete set of commuting observables
(C.S.C.O.),,,
Complex variables (Lagrangian),
Compton wavelength of the electron,,
1235
Condensates
relative phase,
with spins,
Condensation
BCS condensation energy,
Bose-Einstein, , ,
Condensed bosons,
Conduction band,
Conductivity (solid),
Congurations,
Conjugate momentum, , , ,
1983, ,
Conjugation (Hermitian),
Conservation
local conservation of probability,
of angular momentum,,,
of energy,
of probability,
Conservative systems,,
Constants of the motion,,
Contact term,
Contact term (Fermi), ,
Contextuality,
Continuous
spectrum,,,,
variables (in a Lagrangian),
Continuum of nal states, , ,
1380
Contractions,
903

INDEX [The notation (ex.) refers to an exercise]
Convolution product of two functions,
Cooling
Doppler,
down atoms,
evaporative,
Sisyphus,
sub-Doppler,
subrecoil,
Cooper model,
Cooper pairs,
Cooperative eects (BCS),
Correlation functions, ,
anti-normal, ,
dipole and eld,
for one-photon processes,
normal, ,
of the eld, spatial,
Correlations,
between two dipoles,
between two physical systems,
classical and quantum,
introduced by a collision,
Coulomb
eld,
gauge,
Coulomb potential
cross section,
Coupling
between angular momenta,
between two angular momenta,
between two states,
eect on the eigenvalues,
spin-orbit coupling, ,
Creation and annihilation operators,,
513,, ,
Creation operator (pair of particles), ,
1846
Critical velocity,
Cross section
and phase shifts,
scattering cross section,,,,
972
Current
metastable current in superuid,
of particles,
of probability,
probability current in hydrogen atom,
851
Cylindrical symmetry,(ex.)
Darwin term, ,
De Broglie
relation,
wavelength,seefront cover pages,,
35
Decay of a discrete state,
Deceleration of an atomic beam,
Decoherence,
Decomposition (Schmidt),
Decoupling (ne or hyperne structure),
1262,
Degeneracy
essential,,,
exchange degeneracy,
exchange degeneracy removal,
lifted by a perturbation,
rotation invariance,
systematic and accidental,
Degenerate eigenvalue,,,,
Degereracy
lifted by a perturbation,
parity,
Delta Dirac function,
potential well and barriers,85(ex.)
use in quantum mechanics,,,
280
Density
Lagrangian,
of probability,
of states,, , ,
operator,,
operator and matrix,
particle density operator,
Density functions
one and two-particle, (ex.)
Depletion (quantum),
Derivative of an operator,
Detection probability amplitude (photon),
2166
Detectors (photon),
Determinant
Slater determinant, ,
Deuterium,, (ex.)
Diagonalization
904

INDEX [The notation (ex.) refers to an exercise]
of a22matrix,
of an operator,
Diagram (dressed-atom),
Diamagnetism,
Diatomic molecules
rotation,
Diusion (momentum),
Dipole
-dipole interaction, ,
-dipole magnetic interaction,
electric dipole transition,
electric moment,
Hamiltonian,
magnetic dipole moment,
magnetic term,
trap,
Dirac,seeFermi
delta function,,,,
equation,
notation,
Direct
and exchange terms, , , ,
1646,
term, ,
Discrete
bases of the state space,
spectrum,,
Dispersion (anomalous),
Dispersion and absorption (eld),
Distribution
Boltzmann,
Bose-Einstein,
Fermi-Dirac,
function (bosons),
function (fermions),
functions, ,
functions (BCS),
Distribution law
Bose-Einstein,
Divergence (energy),
Donor atom,,
Doppler
cooling,
eect,
eect (relativistic),
free spectroscopy,
temperature,
Double
condensate,
resonance method,
spin condensate,
Doublet (Autler-Townes),
Down-conversion (parametric),
Dressed
states and energies,
Dressed-atom, ,
diagram,
strong coupling,
weak coupling,
E.P.R., (ex.)
Eckart (Wigner-Eckart theorem),seeWigner
Eect
Autler-Townes,
Mössbauer,
photoelectric,
Eective Hamiltonian,
Ehrenfest theorem,,,
Eigenresult,
Eigenstate,,
Eigenvalue,,,,
degenerate,,
equation,,
of an operator,
Eigenvector,
of an operator,
Einstein,
coecients, , ,
EPR argument,,
model,,
Planck-Einstein relations,
temperature,
Einstein-Podolsky-Rosen, ,
Elastic
scattering,
scattering (photon),
scattering, form factor, (ex.)
total cross section,
Elastically bound electron model,
Electric
conductivity of a solid,
Electric dipole
Hamiltonian,
interaction,
905

INDEX [The notation (ex.) refers to an exercise]
matrix elements,
moment,
selection rules,
transition and selection rules,
transitions,
Electric eld (quantized), ,
Electric polarisability
NH3,
Electric polarizability
of the1state in Hydrogen,
Electric quadrupole
Hamiltonian,
moment,
transitions,
Electric susceptibility
bound electron,
of an atom,
Electrical
susceptibility, (ex.)
Electrodynamics
classical,
quantum,
Electromagnetic eld
and harmonic oscillators,
and potentials,
angular momentum, ,
energy,
Lagrangian, ,
momentum, ,
polarization,
quantization,,
Electromagnetic interaction of an atom
with a wave,
Electromagnetism
elds and potentials,
Electron spin,,
Electron(s)
congurations,
gas in solids,
in solids, ,
mass and charge,seefront cover pages
Electronic
conguration,
paramagnetic resonance, (ex.)
shell,
Elements of reality,
Emergence of a relative phase, ,
Emission
of a quantum,
photon,
spontaneous, ,
stimulated (or induced),
Energy,seeConservation, Uncertainty
and momentum of the transverse elec-
tromagnetic eld,
band,
bands in solids, ,
conservation,
electromagnetic eld,
Fermi energy,
ne structure energy levels,
free energy,
levels,
levels of harmonic oscillator,
levels of hydrogen,
of a paired state,
recoil energy,
Ensemble
canonical,
grand canonical,
microcanonical,
statistical ensembles,
Entanglement
quantum, , , ,
swapping,
Entropy,
EPR, ,
elements of reality,
EPRB,
paradox/argument,
Equation of state
ideal quantum gas,
repulsive bosons,
Equation(s)
Bloch,
Hamilton-Jacobi, , ,
Lagrange, ,
Lorentz,
Maxwell,
Schrödinger,,,
von Neumann,
Essential degeneracy,,
Ethane (molecule),
Ethylene (molecule),,
906

INDEX [The notation (ex.) refers to an exercise]
Evanescent wave,,,,,
Evaporative cooling,
Even operators,
Evolution
eld operator,
of quantum systems,
of the mean value,
operator,,
operator (expansion),
operator (integral equation),
Exchange,
degeneracy,
degeneracy removal,
energy,
hole,
integral,
term, , ,
Excitations
BCS,
Bogolubov,
vacuum,
Excited states (BCS),
Exciton,
Exclusion principle (Pauli), , ,
1463,
Extensive (or intensive) variables,
Fermi
contact term,
energy, , , ,
gas,
golden rule,
level, ,
radius,
surface (modied),
,seeFermi-Dirac
Fermi level
and electric conductivity,
Fermi-Dirac
distribution, , ,
statistics,
Fermions,
in a Fock state,
paired,
Ferromagnetism,
Feynman
path,
postulates,
Fictitious spin,,
Field
absorption,
commutation relations, ,
dispersion and absorption,
intense laser,
interaction energy,
kinetic energy,
normal variables,
operator,
operator (evolution), ,
pair eld operator,
potential energy,
quantization, ,
quasi-classical state,
spatial correlation functions,
Final states continuum, ,
Fine and hyperne structure,
Fine structure
constant,seefront cover pages,
energy levels,
Hamiltonian, , ,
Helium atom,
Hydrogen,
of spectral lines,
of the states1,2et2,
Fletcher,
Fluctuations
boson occupation number,
intensity,
vacuum,,
Fluorescence (single atom),
Fluorescence triplet,
Fock
space, ,
state, , , ,
Forbidden,seeBand
energy band,,,
transition,
Forces
van der Waals,
Form factor
elastic scattering, (ex.)
Forward scattering (direct and exchange),
1874
Fourier
907

INDEX [The notation (ex.) refers to an exercise]
series and transforms,
Fragmentation (condensate), ,
Free
electrons in a box,
energy,
particle,
quantum eld (Fock space),
spherical wave,,,
spherical waves and plane waves,
Free particle
stationary states with well-dened an-
gular momentum,
stationary states with well-dened mo-
mentum,
wave packet,,,
Frequency
Bohr,
components of the eld (positive and
negative),
Rabi's frequency,
Friction (coecient),
Function
of operators,
periodic functions,
step functions,
Fundamental state,
Gap (BCS), , ,
Gauge, , , ,
Coulomb,
invariance,
Lorenz,
Gaussian
wave packet,,,
Generalized velocities,,
Geometric quantization,
Gerlach,seeStern
GHZ state, ,
Gibbs-Duhem relation,
Golden rule (Fermi),
Good quantum numbers,
Grand canonical, ,
Grand potential, , ,
Green's function,,, , ,
1789
evolution,
Greenberger-Horne-Zeilinger,
Groenewold's formula,
Gross-Pitaevskii equation, ,
Ground state,
harmonic oscillator,,
Hydrogen atom, (ex.)
Group velocity,,,
Gyromagnetic ratio,,
orbital,
spin,
H
+
2
molecular ion,(ex.),,
Hadronic atoms,
Hall eect,
Hamilton
function,
function and equations,
Hamilton-Jacobi canonical equations,,
1532, , ,
Hamiltonian,,, , , ,
1995
classical,
eective,
electric dipole, ,
electric quadrupole,
ne structure, ,
hyperne, ,
magnetic dipolar,
of a charged particle in a vector po-
tential,
of a particle in a central potential,
806,
of a particle in a scalar potential,
of a particle in a vector potential,
225,,
Hanbury Brown and Twiss,
Hanle eect, (ex.)
Hard sphere
scattering,,(ex.)
Harmonic oscillator,
in an electric eld,
in one dimension,,
in three dimensions,
in two dimensions,
innite chain of coupled oscillators,
611
quasiclassical states,
thermodynamic equilibrium,
908

INDEX [The notation (ex.) refers to an exercise]
three-dimensional,,(ex.)
two coupled oscillators,
Hartree-Fock
approximation, ,
density operator (one-particle),
equations, ,
for electrons,
mean eld, ,
potential,
thermal equilibrium, ,
time-dependent, ,
Healing length,
Heaviside step function,
Heisenberg
picture,,
relations,,,,,,,
Helicity (photon),
Helium
energy levels,
ion,
isotopes,
isotopes
3
He and
4
He, ,
solidication,
Hermite polynomials,,,
Hermitian
conjugation,
matrix,
operator,,,
Histories (classical),
Hole
creation and annihilation,
exchange,
Holes,
Hybridization of atomic orbitals,
Hydrogen,
atom,
atom in a magnetic eld,,,
862
atom, relativistic energies,
Bohr model,,
energy levels,
ne and hyperne stucture,
ionisation energy,seefront cover pages
ionization energy,
maser,
molecular ion,(ex.),,
quantum theory,
radial equation,
Stark eect,
stationary states,
stationary wave functions,
Hydrogen-like systems in solid state physics,
837
Hydrogenoid systems,
Hyperne
decoupling,
Hamiltonian, ,
Hyperne structure,seeHydrogen, muo-
nium, positronium, Zeeman ef-
fect,
Muonium,
Ideal gas, , , ,
correlations,
Identical particles, ,
Induced
emission, , ,
emission of a quantum,
emission of photons,
Inequality (Bell's),
Innite one-dimensional well,
Innite potential well,
in two dimensions,
Innitesimal unitary operator,
Insulator,
Integral
exchange integral,
scattering equation,
Intense laser elds,
Intensive (or extensive) variables,
Interaction
between magnetic dipoles,
dipole-dipole interaction, ,
electromagnetic interaction of an atom
with a wave,
eld and particles,
eld and atom,
magnetic dipole-dipole interaction,
picture,, ,
tensor interaction,
Interference
photons,
two-photon, ,
Ion H
+
2
,
909

INDEX [The notation (ex.) refers to an exercise]
Ionization
photo-ionization,
tunnel ionization,
Isotropic radiation,
Jacobi,seeHamilton
Kastler, ,
Ket,seestate,,
for identical particles,
Kuhn,seeThomas
Lagrange
equations, , ,
fonction and equations,
multipliers,
Lagrangian, ,
densities,
electromagnetic eld, ,
formulation of quantum mechanics,
339
of a charged particle in an electro-
magnetic eld,
particle in an electromagnetic eld,
323
Laguerre-Gaussian beams,
Lamb shift,, , ,
Landau levels,
Landé factor, , (ex.), ,
Laplacian,
of1,
of()
+1
,
Larmor
angular frequency,
precession,,,,,,
1071
Laser, ,
Raman laser,
saturation,
trap,
Lattices (optical),
Least action
principle of,
Legendre
associated function,
polynomial,
Length (healing),
Level
anticrossing,,
Fermi level,
Lifetime,,,
of a discrete state,
radiative,
Lifting of degeneracy by a perturbation,
1125
Light
quanta,
shifts, , , ,
Linear,seeoperator
combination of atomic orbitals,
operators,,,
response, , ,
superposition of states,
susceptibility,
Local conservation of probability,
Local realism, ,
Longitudinal
elds,
relaxation,
relaxation time,
Lorentz equations,
Lorenz (gauge),
Magnetic
dipole term,
dipole-dipole interaction,
eect of a magnetic eld on the lev-
els of the Hydrogen atom,
hyperne Hamiltonian,
interactions, ,
quantum number,
resonance,
susceptibility, ,
Magnetic dipole
Hamiltonian,
transitions and selection rules, ,
1098,
Magnetic dipoles
interactions between two dipoles,
Magnetic eld
and vector potential,
charged particle in a,,
eects on hydrogen atom,,
harmonic oscillator in a,(ex.)
Hydrogen atom in a magnetic eld,
1263,
910

INDEX [The notation (ex.) refers to an exercise]
multiplets,
quantized, ,
Magnetism (spontaneous),
Many-electron atoms,
Maser,, ,
hydrogen,
Mass correction (relativistic),
Master equation,
Matrice(s),,
diagonalization of a22matrix,
Pauli matrices,
unitary matrix,
Maxwell's equations,
Mean eld (Hartree-Fock), , ,
1725
Mean value of an observable,
evolution,
Measurement
general postulates,,
ideal von Neumann measurement,
of a spin 1/2,
of observables,
on a part of a physical system,
state after measurement,,
Mendeleev's table,
Metastable superuid ow,
Methane (molecule),
Microcanonical ensemble,
Millikan,
Minimal wave packet,,,
Mirrors for atoms,
Mixing of states, ,
Model
Cooper model,
Einstein model,
elastically bound electron,
vector model of atom,
Modes
vibrational modes,,
Modes (radiation), ,
Molecular ion,
Molecule(s)
chemical bond,,,,,
883,
rotation,
vibration,,
vibration-rotation,
Mollow,
Moment
quadrupole electric moment, (ex.)
Momentum,
conjugate,,, , ,
diusion,
electromagnetic eld, ,
mechanical momentum,
Monogamy (quantum),
Mössbauer eect, ,
Motional narrowing,
condition, , ,
Multiphoton transition, , ,
Multiplets, , ,
Multipliers (Lagrange),
Multipolar waves,
Multipole
moments,
Multipole operators
introduction, ,
parity,
Muon,,,
Muonic atom,,
Muonium,
hyperne structure,
Zeeman eect,
Narrowing (motional), ,
condition,
Natural width,,
Need for a quantum treatment, ,
Neumann
spherical function,
Neutron mass,seefront cover pages
Non-destructive detection of a photon,
2159
Non-diagonal order (BCS),
Non-locality,
Non-resonant excitation,
Non-separability,
Nonlinear
response, ,
susceptibility,
Norm
conservation,
of a state vector,,
of a wave function,,,
911

INDEX [The notation (ex.) refers to an exercise]
Normal
correlation function, ,
variables,,,,
variables (eld),
Nuclear
multipole moments,
Bohr magneton,
Nucleus
spin,
volume eect, ,
Number
occupation number, ,
photon number,
total number of particles in an ideal
gas,
Observable(s),
C.S.C.O.,,
commutation,
compatibility,
for identical particles, ,
mean value,
measurement of,,
quantization rules,
symmetric observables,
transformation by permutation,
whose commutator is},,
Occupation number, ,
operator,
Odd operators,
One-particle
Hartree-Fock density operator,
operators, , , ,
Operator(s)
adjoint operator,
annihilation operator,,,,
1597
creation and annihilation,
creation operator,,,,
derivative of an operator,
diagonalization,,
even and odd operators,
evolution operator,,
eld,
function of,
Hermitian operators,
linear operators,,,
occupation number,
one-particle operator, , , ,
1756
parity operator,
particle density operator,
permutation operators, ,
potential,
product of,
reduced to a single particle,
representation,
restriction,
restriction of,
rotation operator,
symmetric, ,
translation operator,
two-particle operator, , , ,
1756
unitary operators,
Weyl operator,
Oppenheimer,seeBorn, ,
Optical
excitation (broadband),
lattices,
pumping, ,
Orbital
angular momentum (of radiation),
atomic orbital, (ex.)
hybridization,
linear combination of atomic orbitals,
1172
quantum number,
state space,
Order parameter for pairs,
Orthonormal basis,,,,
characteristic relation,
Orthonormalization
and closure relations,,
relation,
Oscillation(s)
between two discrete states,
between two quantum states,
Rabi,
Oscillator
anharmonic,
harmonic,
strength,
Pair(s)
912

INDEX [The notation (ex.) refers to an exercise]
annihilation-creation of pairs, ,
1874,
BCS, wave function,
Cooper,
of particles (creation operator), ,
1846
pair eld (commutation),
pair eld operator,
pair wave function,
Paired
bosons,
fermions,
state energy,
states,
states (building),
Pairing term,
Paramagnetism,
Parametric down-conversion,
Parity,
degeneracy,
of a permutation operator,
of multipole operators,
operator,
Parseval
Parseval-Plancherel equality,
Parseval-Plancherel formula, ,
Partial
reection,
trace of an operator,
waves in the potential,
waves method,
Particle (current),
Particles and holes,
Partition function, , ,
Path
integral,
space-time path,
Pauli
exclusion principle, , , ,
1481
Hamiltonian, (ex.)
matrices,,
spin theory,
spinor,
Penetrating orbit,
Penrose-Onsager criterion, , ,
Peres,
Periodic
boundary conditions,
classication of elements,
functions,
potential (one-dimensional),
Permutation operators, ,
Perturbation
applications of the perturbation the-
ory,
lifting of a degeneracy,
one-dimensional harmonic oscillator,
1131
random perturbation, , ,
sinusoidal,
stationary perturbation theory,
Perturbation theory
time dependent,
Phase
locking (BCS), ,
locking (bosons), ,
relative phase between condensates,
2237,
velocity,
Phase shift (collision),, (ex.)
with imaginary part,
Phase velocity,
Phonons,,
Bogolubov phonons,
Photodetection
double, ,
single, ,
Photoelectric eect, (ex.),
Photoionization, ,
rate, ,
two-photon,
Photon,,,, , ,
absorption and emission,
angular momentum,
antibunching,
detectors,
non-destructive detection,
number,
scattering (elastic),
scattering by an atom,
vacuum,
,seeAbsorption, Emission
Picture
913

INDEX [The notation (ex.) refers to an exercise]
Heisenberg,,
interaction, ,
Pitaevskii (Gross-Pitaevskii equation), ,
1657
Plancherel,seeParseval
Planck
constant,seefront cover pages,
law ,
Planck-Einstein relations,,
Plane wave,,,,
Podolsky (EPR argument),,
Pointer states,
Polarizability
of the1state in Hydrogen,
Polarization
electromagnetic eld,
of Zeeman components,
space-dependent,
Polynomial method (harmonic oscillator),
555,
Polynomials
Hermite polynomials,,,
Position and momentum representations,
181
Positive and negative frequency compo-
nents,
Positron,
Positronium,
hyperne structure,
Zeeman eect,
Postulate (von Neumann projection),
Postulates of quantum mechanics,
Potential
adiabatic branching,
barrier,,,,
centrifugal potential,,,
Coulomb potential, cross section,
cylindrically symmetric,(ex.)
Hartree-Fock,
innite one-dimensional well,
operator,
scalar and vector potentials, ,
1960,
scattering by a,
self-consistent potential,
square potential,
square well,
step,,,,
well,,
well (arbitrary shape),
well (innite one-dimensional),
well (innite two-dimensional,
Yukawa potential,
Precession
Larmor precession,,
Thomas precession,
Preparation of a state,
Pressure (ideal quantum gas),
Principal part,
Principal quantum number,
Principle
of least action, ,
of spectral decomposition,,
of superposition,
Probability
amplitude,,,
conservation,
current,,,,,
current in hydrogen atom,
density,,
uid,
of photon absorption,
of the measurement results,,
transition probability,
Process (pair annihilation-creation), ,
1887
Product
convolution product of functions,
of matrices,
of operators,
scalar product,,,,
state (tensor product),
tensor product,
tensor product, applications,
Projection theorem,
Projector,,,,,, (ex.)
Propagator
for the Schrödinger equation,
of a particle, ,
Proper result,
Proton
mass,seefront cover pages
spin and magnetic moment, ,
Pumping,
914

INDEX [The notation (ex.) refers to an exercise]
Pure (state or case),
Quadrupolar electric moment, , (ex.)
Quanta (circular),,
Quantization
electrodynamics,
electromagnetic eld,,,
of a eld,
of angular momentum,,
of energy,,,,
of measurement results,,,
of the measurement results,
rules,,,,
Quantum
angle,
electrodynamics, , ,
entanglement, ,
monogamy,
number
orbital,
principal quantum number,
numbers (good),
resonance,
treatment needed, ,
Quasi-classical
eld states,
states,,,
states of the harmonic oscillator,
Quasi-particles, ,
Bogolubov phonons,
Quasi-particle vacuum,
Rabi
formula,,, ,
formula),
frequency,
oscillation,
Radial
equation,
equation (Hydrogen),
equation in a central potential,
integral,
quantum number,
Radiation
isotropic,
pressure,
Radiative
broadening,
cascade of the dressed atom,
Raman
eect,,, (ex.)
laser,
scattering,
scattering (stimulated),
Random perturbation, , ,
Rank (Schmidt),
Rate (photoionization), ,
Rayleigh
line,
scattering,,
Realism (local), ,
Recoil
blocking,
eect of the nucleus,
energy, ,
free atom,
suppression,
Reduced
density operator,
mass,
Reduction of the wave packet,,
Reection on a potential step,
Refractive index,
Reiche,seeThomas
Relation (Gibbs-Duhem),
Relative
motion,
particle,
phase between condensates, ,
phase between spin condensates,
Relativistic
corrections, ,
Doppler eect,
mass correction,
Relaxation,, , , , (ex.)
general equations,
longitudinal,
longitudinal relaxation time,
transverse,
transverse relaxation time,
Relay state, , ,
Renormalization,
Representation(s)
change of,
in the state space,
915

INDEX [The notation (ex.) refers to an exercise]
of operators,
position and momentum,,
Schrödinger equation,185
Repulsion between electrons,
Resonance
magnetic resonance,
quantum resonance,,
scattering resonance,,,(ex.)
two resonnaces with a sinusoidal ex-
citation,
width,
with sinusoidal perturbation,
Restriction of an operator,,
Rigid rotator,, (ex.)
Ritz theorem,
Root mean square deviation
general denition,
Rosen (EPR argument),,
Rotating frame,
Rotation(s)
and angular momentum,
invariance and degeneracy,
of diatomic molecules,
of molecules,,
operator(s),,
rotation invariance,
rotation invariance and degeneracy,
1072
Rotator
rigid rotator,, (ex.)
Rules
quantization rules,
selection rules,
Rutherford's formula,
Rydberg constant,seefront cover pages
Saturation
of linear response,
of the susceptibility,
Scalar
and vector potentials,,
interaction between two angular mo-
menta,
observable, operator,,
potential,
product,,,,,,
product of two coherent states,
Scattering
amplitude,,
by a central potential,
by a hard sphere,,(ex.)
by a potential,
cross section,,,
cross section and phase shifts,
inelastic,
integral equation,
of particles with spin,
of spin 1/2 particles, (ex.)
photon,
Raman,
Rayleigh,,
resonance,,(ex.)
resonant,
stationary scattering states,
stationary states,
stimulated Raman,
Schmidt
decomposition,
rank,
Schottky anomaly,
Schrödinger,
equation,,,,
equation in momentum representa-
tion,
equation in position representation,
183
equation, physical implications,
equation, resolution for conservative
systems,
picture,
Schwarz inequality,
Second
quantization,
harmonic generation,
Secular approximation, ,
Selection rules,,, ,
electric quadrupolar,
magnetic dipolar, ,
Self-consistent potential,
Semiconductor,,
Separability, ,
Separable density operator,
Shell (electronic),
Shift
916

INDEX [The notation (ex.) refers to an exercise]
light shift,
of a discrete state,
Singlet, ,
Sinusoidal perturbation, ,
Sisyphus
cooling,
eect,
Slater determinant, ,
Slowing down atoms,
Solids
electronic bands,
energy bands of electrons,
energy bands of electrons in solids,
381
hydrogen-like systems in solid state
physics,
Space (Fock),
Space-dependent polarization,
Space-time path,,
Spatial correlations (ideal gas),
Specic heat
of an electron gas,
of metals,
of solids,
two level system,
Spectral
decomposition principle,,,
function,
terms,
Spectroscopy (Doppler free),
Spectrum
BCS elementary excitation,
continuous,,
discrete,,
of an observable,,
Spherical
Bessel equation,
Bessel function,,
free spherical waves,
free wave,
Neumann function,
wave,
waves and plane waves,
Spherical harmonics,,
addition of,
expression for= 012,
general expression,
Spin
and magnetic moment of the proton,
1237
angular momentum,
electron,,
ctitious,
gyromagnetic ratio,,,
nuclear,
of the electron,
Pauli theory,,
quantum description,,
rotation operator,
scattering of particles with spin,
spin 1 and radiation, , ,
system of two spins,
Spin 1/2
density operator,
ensemble of,
ctitious,
interaction between two spins,
preparation and measurement,
scattering of spin 1/2 particles, (ex.)
Spin-orbit coupling, , , ,
Spin-statistics theorem,
Spinor,
rotation,
Spontaneous
emission,,, , ,
emission of photons,
magnetism of fermions,
Spreading of a wave packet,,
Square
barrier of potential,,
potential,,,,
potential well,,
spherical well,(ex.)
Standard representation (angular momen-
tum),,
Stark eect in Hydrogen atom,
State(s),seeDensity operator
density of,, , ,
Fock, , , ,
ground state,
mixing of states by a perturbation,
1121
orbital state space,
paired,
917

INDEX [The notation (ex.) refers to an exercise]
pointer states,
quasi-classical states,,,,
801
relay state, , ,
stable and unstable states,
state after measurement,
state preparation,
stationary,,,
stationary state,,
stationary states in a central poten-
tial,
unstable,
vacuum state,
vector,,
Stationary
perturbation theory,
phase condition,,
scattering states,,
states,,,,
states in a periodic potential,
states with well-dened angular mo-
mentum,,
states with well-dened momentum,
943
Statistical
entropy,
mechanics (review of),
mixture of states,,,,
Statistics
Bose-Einstein,
Fermi-Dirac,
Step
function,
potential,,,,
Stern-Gerlach experiment,
Stimulated
(or induced) emission, , ,
2081
Raman scattering,
Stokes Raman line,,
Stoner (spontaneous magnetism),
Strong coupling (dressed-atom),
Subrecoil cooling,
Sum rule (Thomas-Reiche-Kuhn),
Superuidity, ,
Superposition
of states,
principle,,
principle and physical predictions,
Surface (modied Fermi surface),
Susceptibility,seeLinear, nonlinear, ten-
sor
electric susceptibility of an atom,
electrical susceptibility,,e1223
electrical susceptibility of NH3,
magnetic susceptibility,
tensor, , (ex.)
Swapping (entanglement),
Symmetric
ket, state, ,
observables, ,
operators, , , , ,
1628, ,
Symmetrization
of observables,
postulate,
Symmetrizer, ,
System
time evolution of a quantum system,
223
two-level system,
Systematic
and accidental degeneracies,
degeneracy,
Temperature (Doppler),
Tensor
interaction,
product,,
product of operators,
product state,,
product, applications,
susceptibility tensor,
Term
direct and exchange terms, , ,
1634, ,
pairing,
spectral terms, ,
Theorem
Bell, ,
Bloch,
projection,
Ritz,
Wick, ,
918

INDEX [The notation (ex.) refers to an exercise]
Wigner-Eckart, , ,
Thermal wavelength,
Thermodynamic equilibrium,
harmonic oscillator,
ideal quantum gas,
spin 1/2,
Thermodynamic potential (minimization),
1715
Thomas precession,
Thomas-Reiche-Kuhn sum rule,
Three-dimensional harmonic oscillator,,
841,(ex.)
Three-level system, (ex.)
Three-photon transition,
Time evolution of quantum systems,
Time-correlations (uorescent photons),
2145
Time-dependent
Gross-Pitaevskii equation,
perturbation theory,
Time-energy uncertainty relation,,,
345, ,
Torsional oscillations,
Torus (ow in a),
Total
elastic scattering cross section,
reection,,
scattering cross section (collision),
Townes
Autler-Townes eect,
Trace
of an operator,
partial trace of an operator,
Transform (Wigner),
Transformation
Bogolubov,
Bogolubov-Valatin, ,
Gauge,
of observables by permutation,
Transition,seeProbability, Forbidden, Elec-
tric dipole, Magnetic dipole,
Quadrupole
electric dipole,2056
magnetic dipole transition,
probability,, , ,
probability per unit time,
probability, spin 1/2,
three-photon transition,
two-photon,
virtual,
Translation operator,,,
Transpositions,
Transverse
elds,
relaxation,
relaxation time,
Trap
dipolar,
laser,
Triplet, ,
uorescence triplet,
Tunnel
eect,,,,,,
ionization,
Two coupled harmonic oscillators,
Two-dimensional
harmonic oscillator,
innite potential well,
wave packets,
Two-level system,,,,
Two-particle operators, , , ,
1756
Two-photon
absorption, (ex.)
interference, ,
transition, (ex.),
Uncertainty
relation,,,,,,
time-energy uncertainty relation,
Uniqueness of the measurement result,
2201
Unitary
matrix,,
operator,,
transformation of operators,
Unstable states,
Vacuum
electromagnetism,,
excitations,
uctuations,
photon vacuum,
quasi-particule vacuum,
state,
Valence band,
919

INDEX [The notation (ex.) refers to an exercise]
Van der Waals forces,
Variables
intensive or extensive,
normal variables,,,,
Variational method, , , (ex.)
Vector
model,
model of the atom, ,
observable, operator,
operator,
potential,
potential of a magnetic dipole,
Velocity
critical,
generalized velocities,,
group velocity,,
phase velocity,,
Vibration(s)
modes,,
modes of a continuous system,
of molecules,,
of nuclei in a crystal,,,
of the nuclei in a molecule,
Violations of Bell's inequalities, ,
Virial theorem,,
Virtual transition,
Volume eect,,, ,
Von Neumann
chain,
equation,
ideal measurement,
reduction postulate,
statistical entropy,
Vortex in a superuid,
Water (molecule),,
Wave (evanescent),
Wave function,,,
BCS pairs, ,
Hydrogen,
norm,
pair wave functions,
particle,
Wave packet(s)
Gaussian,,
in a potential step,
in three dimensions,
minimal,,,
motion in a harmonic potential,
one-photon,
particle,
photon,
propagation,,,,
reduction,,,,
spreading,,,,(ex.)
two-dimension,
two-photons,
Wave(s)
de Broglie wavelength,,
evanescent,
free spherical waves,
multipolar,
partial waves,
plane,,,
wave function,,,,
Wave-particle duality,,
Wavelength
Compton wavelength,
de Broglie,
Weak coupling (dressed-atom),
Well
potential square well,
potential well,
Weyl
operator,
quantization,
Which path type of experiments,
Wick's theorem, ,
Wigner transform,
Wigner-Eckart theorem, , ,
Young (double slit experiment),
Yukawa potential,
Zeeman
components, polarizations,
eect,,,, , , ,
1261,
polarization of the components,
slower,
Zeeman eect
Hydrogen,
in muonium,
in positronium,
Muonium,
920

INDEX [The notation (ex.) refers to an exercise]
Zone (Brillouin zone),
921

QUANTUMMECHANICS
olumeII
Angular Momentum, Spin, and Approximation Methods
Claude Cohen-Tannoudji, Bernard Diu,
and Franck Laloë
Translated from the French bySusan Reid Hemley,
Nicole Ostrowsky, and Dan Ostrowsky
V

Authors
Prof. Dr. Claude Cohen-Tannoudji
Laboratoire Kastler Brossel (ENS)
24 rue Lhomond
75231 Paris Cedex 05
France
Prof. Dr. Bernard Diu
4 rue du Docteur Roux
91440 Boures-sur-Yvette
France
Prof. Dr. Frank Laloë
Laboratoire Kastler Brossel (ENS)
24 rue Lhomond
75231 Paris Cedex 05
France
Cover Image
© antishock/Getty Images
Second Edition
All books published by Wiley-VCH are
carefully produced. Nevertheless, authors,
editors, and publisher do not warrant the
information contained in these books,
including this book, to be free of errors.
Readers are advised to keep in mind that
statements, data, illustrations, procedural
details or other items may inadvertently be
inaccurate.
Library of Congress Card No.:
applied for
British Library Cataloguing-in-Publication
Data:
A catalogue record for this book is available
from the British Library.
Bibliographic information published by the
Deutsche Nationalbibliothek
The Deutsche Nationalbibliothek lists this
publication in the Deutsche Nationalbiblio
graﬁe; detailed bibliographic data are avail
able on the Internet at http://dnb.d-nb.de.
© 2020 WILEY-VCH Verlag GmbH &
Co. KGaA, Boschstr. 12, 69469 Weinheim,
Germany
All rights reserved (including those of trans
lation into other languages). No part of
this book may be reproduced in any form –
by photoprinting, microﬁlm, or any other
means – nor transmitted or translated into
a machine language without written per
mission from the publishers. Registered
names, trademarks, etc. used in this book,
even when not speciﬁcally marked as such,
are not to be considered unprotected by
law.
Print ISBN978-3-527-34554-0
ePDF ISBN978-3-527-82272-0
ePub ISBN978-3-527-82273-7
Cover DesignTata Consulting Services
Printing and BindingCPI Ebner & Spiegel
Printed on acid-free paper.

Directions for Use
This book is composed of chapters and their complements:
The chapterscontain the fundamental concepts. Except for a few
additions and variations, they correspond to a course given in the last
year of a typical undergraduate physics program (Volume I) or of a
graduate program (Volumes II and III). The 21 chapters arecomplete in
themselvesand can be studied independently of the complements.
The complementsfollow the corresponding chapter. Each is labelled
by a letter followed by a subscript, which gives the number of the chapter
(for example, the complements of Chapter V are, in order, AV, BV, CV,
etc.). They can be recognized immediately by the symbolthat appears
at the top of each of their pages.
The complements vary in character. Some are intended to expand the
treatment of the corresponding chapter or to provide a more detailed
discussion of certain points. Others describe concrete examples or in-
troduce various physical concepts. One of the complements (usually the
last one) is a collection of exercises.
Thedicultyof the complements varies. Some are very simple examples
or extensions of the chapter. Others are more dicult and at the grad-
uate level or close to current research. In any case, the reader should
have studied the material in the chapter before using the complements.
The complements are generally independent of one another.The student
should not try to study all the complements of a chapter at once. In
accordance with his/her aims and interests, he/she should choose a small
number of them (two or three, for example), plus a few exercises. The
other complements can be left for later study. To help with the choise,
the complements are listed at the end of each chapter in a reader's
guide, which discusses the diculty and importance of each.
Some passages within the book have been set in small type, and these
can be omitted on a rst reading.

Foreword
Foreword
Quantum mechanics is a branch of physics whose importance has continually in-
creased over the last decades. It is essential for understanding the structure and dynamics
of microscopic objects such as atoms, molecules and their interactions with electromag-
netic radiation. It is also the basis for understanding the functioning of numerous new
systems with countless practical applications. This includes lasers (in communications,
medicine, milling, etc.), atomic clocks (essential in particular for the GPS), transistors
(communications, computers), magnetic resonance imaging, energy production (solar
panels, nuclear reactors), etc. Quantum mechanics also permits understanding surpris-
ing physical properties such as superuidity or supraconductivity. There is currently a
great interest in entangled quantum states whose non-intuitive properties of nonlocality
and nonseparability permit conceiving remarkable applications in the emerging eld of
quantum information. Our civilization is increasingly impacted by technological appli-
cations based on quantum concepts. This why a particular eort should be made in the
teaching of quantum mechanics, which is the object of these three volumes.
The rst contact with quantum mechanics can be disconcerting. Our work grew
out of the authors' experiences while teaching quantum mechanics for many years. It
was conceived with the objective of easing a rst approach, and then aiding the reader
to progress to a more advance level of quantum mechanics. The rst two volumes, rst
published more than forty years ago, have been used throughout the world. They remain
however at an intermediate level. They have now been completed with a third volume
treating more advanced subjects. Throughout we have used a progressive approach to
problems, where no diculty goes untreated and each aspect of the diverse questions is
discussed in detail (often starting with a classical review).
This willingness to go further without cheating or taking shortcuts is built into
the book structure, using two distinct linked texts:chaptersandcomplements. As we
just outlined in the Directions for use, the chapters present the general ideas and
basic concepts, whereas the complements illustrate both the methods and concepts just
exposed.
Volume I presents a general introduction of the subject, followed by a second
chapter describing the basic mathematical tools used in quantum mechanics. While
this chapter can appear long and dense, the teaching experience of the authors has
shown that such a presentation is the most ecient. In the third chapter the postulates
are announced and illustrated in many of the complements. We then go on to certain
important applications of quantum mechanics, such as the harmonic oscillator, which
lead to numerous applications (molecular vibrations, phonons, etc.). Many of these are
the object of specic complements.
Volume II pursues this development, while expanding its scope at a slightly higher
level. It treats collision theory, spin, addition of angular momenta, and both time-
dependent and time-independent perturbation theory. It also presents a rst approach
to the study of identical particles. In this volume as in the previous one, each theoretical
concept is immediately illustrated by diverse applications presented in the complements.
Both volumes I and II have beneted from several recent corrections, but there have also
been additions. Chapter
perturbations, and a complement concerning relaxation has been added.
ii

Foreword
Volume III extends the two volumes at a slightly higher level. It is based on the
use of the creation and annihilation operator formalism (second quantization), which is
commonly used in quantum eld theory. We start with a study of systems of identical
particles, fermions or bosons. The properties of ideal gases in thermal equilibrium are
presented. For fermions, the Hartree-Fock method is developed in detail. It is the base
of many studies in chemistry, atomic physics and solid state physics, etc. For bosons, the
Gross-Pitaevskii equation and the Bogolubov theory are discussed. An original presen-
tation that treats the pairing eect of both fermions and bosons permits obtaining the
BCS (Bardeen-Cooper-Schrieer) and Bogolubov theories in a unied framework. The
second part of volume III treats quantum electrodynamics, its general introduction, the
study of interactions between atoms and photons, and various applications (spontaneous
emission, multiphoton transitions, optical pumping, etc.). The dressed atom method is
presented and illustrated for concrete cases. A nal chapter discusses the notion of quan-
tum entanglement and certain fundamental aspects of quantum mechanics, in particular
the Bell inequalities and their violations.
Finally note that we have not treated either the philosophical implications of quan-
tum mechanics, or the diverse interpretations of this theory, despite the great interest
of these subjects. We have in fact limited ourselves to presenting what is commonly
called the orthodox point of view. It is only in Chapter
questions concerning the foundations of quantum mechanics (nonlocality, etc.). We have
made this choice because we feel that one can address such questions more eciently
after mastering the manipulation of the quantum mechanical formalism as well as its nu-
merous applications. These subjects are addressed in the bookDo we really understand
quantum mechanics?(F. Laloë, Cambridge University Press, 2019); see also section 5 of
the bibliography of volumes I and II.
iii

Foreword
Acknowledgments:
Volumes I and II:
The teaching experience out of which this text grew were group eorts, pursued
over several years. We wish to thank all the members of the various groups and partic-
ularly Jacques Dupont-Roc and Serge Haroche, for their friendly collaboration, for the
fruitful discussions we have had in our weekly meetings and for the ideas for problems
and exercises that they have suggested. Without their enthusiasm and valuable help, we
would never have been able to undertake and carry out the writing of this book.
Nor can we forget what we owe to the physicists who introduced us to research,
Alfred Kastler and Jean Brossel for two of us and Maurice Levy for the third. It was in
the context of their laboratories that we discovered the beauty and power of quantum
mechanics. Neither have we forgotten the importance to us of the modern physics taught
at the C.E.A. by Albert Messiah, Claude Bloch and Anatole Abragam, at a time when
graduate studies were not yet incorporated into French university programs.
We wish to express our gratitude to Ms. Aucher, Baudrit, Boy, Brodschi, Emo,
Heywaerts, Lemirre, Touzeau for preparation of the mansucript.
Volume III:
We are very grateful to Nicole and Daniel Ostrowsky, who, as they translated this
Volume from French into English, proposed numerous improvements and clarications.
More recently, Carsten Henkel also made many useful suggestions during his transla-
tion of the text into German; we are very grateful for the improvements of the text
that resulted from this exchange. There are actually many colleagues and friends who
greatly contributed, each in his own way, to nalizing this book. All their complementary
remarks and suggestions have been very helpful and we are in particular thankful to:
Pierre-François Cohadon
Jean Dalibard
Sébastien Gleyzes
Markus Holzmann
Thibaut Jacqmin
Philippe Jacquier
Amaury Mouchet
Jean-Michel Raimond
Félix Werner
Some delicate aspects of Latex typography have been resolved thanks to Marco
Picco, Pierre Cladé and Jean Hare. Roger Balian, Edouard Brézin and William Mullin
have oered useful advice and suggestions. Finally, our sincere thanks go to Geneviève
Tastevin, Pierre-François Cohadon and Samuel Deléglise for their help with a number of
gures.
iv

Table of contents
Volume I
Table of contents
I WAVES AND PARTICLES. INTRODUCTION TO THE BASIC
IDEAS OF QUANTUM MECHANICS
READER'S GUIDE FOR COMPLEMENTS 33
AIOrder of magnitude of the wavelengths associated with material
particles
BIConstraints imposed by the uncertainty relations
CIHeisenberg relation and atomic parameters
DIAn experiment illustrating the Heisenberg relations
EIA simple treatment of a two-dimensional wave packet
FIThe relationship between one- and three-dimensional problems
GIOne-dimensional Gaussian wave packet: spreading of the wave packet
HIStationary states of a particle in one-dimensional square potentials
JIBehavior of a wave packet at a potential step
KIExercises
***********
II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
READER'S GUIDE FOR COMPLEMENTS 159
AIIThe Schwarz inequality
BIIReview of some useful properties of linear operators
CIIUnitary operators
DIIA more detailed study of the r and p representations
EIISome general properties of two observables,and, whose commu-
tator is equal to~ 187
FIIThe parity operator
v

Table of contents
GIIAn application of the properties of the tensor product: the two-
dimensional innite well
HIIExercises
III THE POSTULATES OF QUANTUM MECHANICS
READER'S GUIDE FOR COMPLEMENTS 267
AIIIParticle in an innite one-dimensional potential well
BIIIStudy of the probability current in some special cases
CIIIRoot mean square deviations of two conjugate observables
DIIIMeasurements bearing on only one part of a physical system
EIIIThe density operator
FIIIThe evolution operator
GIIIThe Schrödinger and Heisenberg pictures
HIIIGauge invariance
JIIIPropagator for the Schrödinger equation
KIIIUnstable states. Lifetime
LIIIExercises
MIIIBound states in a potential well of arbitrary shape
NIIIUnbound states of a particle in the presence of a potential well or
barrier
OIIIQuantum properties of a particle in a one-dimensional periodic struc-
ture 375
***********
IV APPLICATIONS OF THE POSTULATES TO SIMPLE CASES:
SPIN 1/2 AND TWO-LEVEL SYSTEMS
READER'S GUIDE FOR COMPLEMENTS 423
AIVThe Pauli matrices
BIVDiagonalization of a22Hermitian matrix
CIVFictitious spin 1/2 associated with a two-level system
vi

Table of contents
DIVSystem of two spin 1/2 particles
EIVSpin12density matrix
FIVSpin 1/2 particle in a static and a rotating magnetic elds: magnetic
resonance
GIVA simple model of the ammonia molecule
HIVEects of a coupling between a stable state and an unstable state
JIVExercises
***********
V THE ONE-DIMENSIONAL HARMONIC OSCILLATOR
READER'S GUIDE FOR COMPLEMENTS 525
AVSome examples of harmonic oscillators
BVStudy of the stationary states in the x representation. Hermite poly-
nomials
CVSolving the eigenvalue equation of the harmonic oscillator by the
polynomial method
DVStudy of the stationary states in the momentum representation
EVThe isotropic three-dimensional harmonic oscillator
FVA charged harmonic oscillator in a uniform electric eld
GVCoherent quasi-classical states of the harmonic oscillator
HVNormal vibrational modes of two coupled harmonic oscillators
JVVibrational modes of an innite linear chain of coupled harmonic
oscillators; phonons
KV Vibrational modes of a continuous physical system. Photons
LVOne-dimensional harmonic oscillator in thermodynamic equilibrium
at a temperature 647
MVExercises
***********
VI GENERAL PROPERTIES OF ANGULAR MOMENTUM IN QUAN-
TUM MECHANICS
vii

Table of contents
READER'S GUIDE FOR COMPLEMENTS 703
AVISpherical harmonics
BVIAngular momentum and rotations
CVIRotation of diatomic molecules
DVIAngular momentum of stationary states of a two-dimensional har-
monic oscillator
EVIA charged particle in a magnetic eld: Landau levels
FVIExercises
***********
VII PARTICLE IN A CENTRAL POTENTIAL, HYDROGEN ATOM
READER'S GUIDE FOR COMPLEMENTS 831
AVIIHydrogen-like systems
BVIIA soluble example of a central potential: The isotropic three-dimensional
harmonic oscillator
CVIIProbability currents associated with the stationary states of the hy-
drogen atom
DVIIThe hydrogen atom placed in a uniform magnetic eld. Paramag-
netism and diamagnetism. The Zeeman eect
EVIISome atomic orbitals. Hybrid orbitals
FVIIVibrational-rotational levels of diatomic molecules
GVIIExercises
INDEX 901
***********
viii

Table of contents
Volume II
VOLUME II
Table of contents
VIII AN ELEMENTARY APPROACH TO THE QUANTUM THEORY
OF SCATTERING BY A POTENTIAL
A Introduction
B Stationary scattering states. Calculation of the cross section
C Scattering by a central potential. Method of partial waves
READER'S GUIDE FOR COMPLEMENTS 957
AVIIIThe free particle: stationary states
with well-dened angular momentum
1 The radial equation
2 Free spherical waves
3 Relation between free spherical waves and plane waves
BVIIIPhenomenological description of collisions with absorption
1 Principle involved
2 Calculation of the cross sections
CVIIISome simple applications of scattering theory
1 The Born approximation for a Yukawa potential
2 Low energy scattering by a hard sphere
3 Exercises
***********
IX ELECTRON SPIN
A Introduction of electron spin
B Special properties of an angular momentum 1/2
C Non-relativistic description of a spin 1/2 particle
READER'S GUIDE FOR COMPLEMENTS 999
AIX Rotation operators for a spin 1/2 particle
1 Rotation operators in state space
2 Rotation of spin states
3 Rotation of two-component spinors
BIX Exercises
***********
ix

Table of contents
X ADDITION OF ANGULAR MOMENTA
A Introduction
B Addition of two spin 1/2's. Elementary method
C Addition of two arbitrary angular momenta. General method
READER'S GUIDE FOR COMPLEMENTS 1041
AX Examples of addition of angular momenta
1 Addition ofj1= 1andj2= 1. . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Addition of an integral orbital angular momentumand a spin 1/2
BX Clebsch-Gordan coecients
1 General properties of Clebsch-Gordan coecients
2 Phase conventions. Reality of Clebsch-Gordan coecients
3 Some useful relations
CX Addition of spherical harmonics
1 The functions(1; 2). . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 The functionsF(). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Expansion of a product of spherical harmonics; the integral of a product of
three spherical harmonics
DX Vector operators: the Wigner-Eckart theorem
1 Denition of vector operators; examples
2 The Wigner-Eckart theorem for vector operators
3 Application: calculation of the Landéfactor of an atomic level
EX Electric multipole moments
1 Denition of multipole moments
2 Matrix elements of electric multipole moments
FX Two angular momenta J1andJ2coupled by
an interactionJ1J2 1091
1 Classical review
2 Quantum mechanical evolution of the average valuesJ1andJ2. . . . . .
3 The special case of two spin 1/2's
4 Study of a simple model for the collision of two spin 1/2 particles
GX Exercises
***********
XI STATIONARY PERTURBATION THEORY
A Description of the method
B Perturbation of a non-degenerate level
C Perturbation of a degenerate state
x

Table of contents
READER'S GUIDE FOR COMPLEMENTS 1129
AXIA one-dimensional harmonic oscillator subjected to a perturbing
potential in,
2
,
3
1131
1 Perturbation by a linear potential
2 Perturbation by a quadratic potential
3 Perturbation by a potential in
3
. . . . . . . . . . . . . . . . . . . . . . . . .
BXIInteraction between the magnetic dipoles of two spin 1/2
particles
1 The interaction HamiltonianW. . . . . . . . . . . . . . . . . . . . . . . . . .
2 Eects of the dipole-dipole interaction on the Zeeman sublevels of two xed
particles
3 Eects of the interaction in a bound state
CXIVan der Waals forces
1 The electrostatic interaction Hamiltonian for two hydrogen atoms
2 Van der Waals forces between two hydrogen atoms in the1ground state
3 Van der Waals forces between a hydrogen atom in the1state and a hydrogen
atom in the2state
4 Interaction of a hydrogen atom in the ground state with a conducting wall
DXIThe volume eect: the inuence of the spatial extension of the nu-
cleus on the atomic levels
1 First-order energy correction
2 Application to some hydrogen-like systems
EXIThe variational method
1 Principle of the method
2 Application to a simple example
3 Discussion
FXIEnergy bands of electrons in solids: a simple model
1 A rst approach to the problem: qualitative discussion
2 A more precise study using a simple model
GXIA simple example of the chemical bond: the H
+
2
ion
1 Introduction
2 The variational calculation of the energies
3 Critique of the preceding model. Possible improvements
4 Other molecular orbitals of the H
+
2
ion
5 The origin of the chemical bond; the virial theorem
HXIExercises
***********
xi

Table of contents
XII AN APPLICATION OF PERTURBATION THEORY: THE FINE
AND HYPERFINE STRUCTURE OF THE HYDROGEN ATOM
A Introduction
B Additional terms in the Hamiltonian
C The ne structure of the= 2level
D The hyperne structure of the= 1level
E The Zeeman eect of the1ground state hyperne structure
READER'S GUIDE FOR COMPLEMENTS 1265
AXIIThe magnetic hyperne Hamiltonian
1 Interaction of the electron with the scalar and vector potentials created by
the proton
2 The detailed form of the hyperne Hamiltonian
3 Conclusion: the hyperne-structure Hamiltonian
BXIICalculation of the average values of the ne-structure Hamiltonian
in the1,2and2states
1 Calculation of1,1
2
and1
3
. . . . . . . . . . . . . . . . . . . . .
2 The average values . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 The average values . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Calculation of the coecient2associated with in the2level
CXIIThe hyperne structure and the Zeeman eect for muonium and
positronium
1 The hyperne structure of the1ground state
2 The Zeeman eect in the1ground state
DXIIThe inuence of the electronic spin on the Zeeman eect of the
hydrogen resonance line
1 Introduction
2 The Zeeman diagrams of the1and2levels
3 The Zeeman diagram of the2level
4 The Zeeman eect of the resonance line
EXIIThe Stark eect for the hydrogen atom
1 The Stark eect on the= 1level
2 The Stark eect on the= 2level
***********
XIII APPROXIMATION METHODS FOR TIME-DEPENDENT
PROBLEMS
A Statement of the problem
B Approximate solution of the Schrödinger equation
C An important special case: a sinusoidal or constant perturbation
D Random perturbation
E Long-time behavior for a two-level atom
xii

Table of contents
READER'S GUIDE FOR COMPLEMENTS 1337
AXIIIInteraction of an atom with an electromagnetic wave
1 The interaction Hamiltonian. Selection rules
2 Non-resonant excitation. Comparison with the elastically bound electron model1350
3 Resonant excitation. Absorption and induced emission
BXIIILinear and non-linear responses of a two-level system subject to a
sinusoidal perturbation
1 Description of the model
2 The approximate solution of the Bloch equations of the system
3 Discussion
4 Exercises: applications of this complement
CXIIIOscillations of a system between two discrete states under the
eect of a sinusoidal resonant perturbation
1 The method: secular approximation
2 Solution of the system of equations
3 Discussion
DXIIIDecay of a discrete state resonantly coupled to a continuum of nal
states
1 Statement of the problem
2 Description of the model
3 Short-time approximation. Relation to rst-order perturbation theory
4 Another approximate method for solving the Schrödinger equation
5 Discussion
EXIIITime-dependent random perturbation, relaxation
1 Evolution of the density operator
2 Relaxation of an ensemble of spin 1/2's
3 Conclusion
FXIIIExercises
***********
XIV SYSTEMS OF IDENTICAL PARTICLES
A Statement of the problem
B Permutation operators
C The symmetrization postulate
D Discussion
READER'S GUIDE FOR COMPLEMENTS 1457
AXIVMany-electron atoms. Electronic congurations
1 The central-eld approximation
2 Electron congurations of various elements
xiii

Table of contents
BXIVEnergy levels of the helium atom. Congurations, terms, multi-
plets
1 The central-eld approximation. Congurations
2 The eect of the inter-electron electrostatic repulsion: exchange energy, spec-
tral terms
3 Fine-structure levels; multiplets
CXIVPhysical properties of an electron gas. Application to solids
1 Free electrons enclosed in a box
2 Electrons in solids
DXIVExercises
***********
APPENDICES
I Fourier series and Fourier transforms
1 Fourier series
2 Fourier transforms
II The Dirac -function
1 Introduction; principal properties
2 The -function and the Fourier transform
3 Integral and derivatives of the-function
4 The -function in three-dimensional space
III Lagrangian and Hamiltonian in classical mechanics
1 Review of Newton's laws
2 The Lagrangian and Lagrange's equations
3 The classical Hamiltonian and the canonical equations
4 Applications of the Hamiltonian formalism
5 The principle of least action
BIBLIOGRAPHY OF VOLUMES I AND II
INDEX
***********
xiv

Table of contents
Volume III
VOLUME III
Table of contents
XV CREATION AND ANNIHILATION OPERATORS FOR IDENTI-
CAL PARTICLES
READER'S GUIDE FOR COMPLEMENTS 1617
AXVParticles and holes
BXVIdeal gas in thermal equilibrium; quantum distribution functions
CXVCondensed boson system, Gross-Pitaevskii equation
DXVTime-dependent Gross-Pitaevskii equation
EXVFermion system, Hartree-Fock approximation
FXVFermions, time-dependent Hartree-Fock approximation
GXVFermions or Bosons: Mean eld thermal equilibrium
HXVApplications of the mean eld method for non-zero temperature
***********
XVI FIELD OPERATOR
READER'S GUIDE FOR COMPLEMENTS 1767
AXVISpatial correlations in an ideal gas of bosons or fermions
BXVISpatio-temporal correlation functions, Green's functions
CXVIWick's theorem
***********
XVII PAIRED STATES OF IDENTICAL PARTICLES
READER'S GUIDE FOR COMPLEMENTS 1843
xv

Table of contents
AXVIIPair eld operator for identical particles
BXVIIAverage energy in a paired state
CXVIIFermion pairing, BCS theory
DXVIICooper pairs
EXVIICondensed repulsive bosons
***********
XVIII REVIEW OF CLASSICAL ELECTRODYNAMICS
READER'S GUIDE FOR COMPLEMENTS 1977
AXVIIILagrangian formulation of electrodynamics
***********
XIX QUANTIZATION OF ELECTROMAGNETIC RADIATION
READER'S GUIDE FOR COMPLEMENTS 2017
AXIXMomentum exchange between atoms and photons
BXIXAngular momentum of radiation
CXIXAngular momentum exchange between atoms and photons
***********
XX ABSORPTION, EMISSION AND SCATTERING OF PHOTONS
BY ATOMS
READER'S GUIDE FOR COMPLEMENTS 2095
AXXA multiphoton process: two-photon absorption
BXXPhotoionization
CXXTwo-level atom in a monochromatic eld. Dressed-atom method
DXXLight shifts: a tool for manipulating atoms and elds
EXXDetection of one- or two-photon wave packets, interference
***********
XXI QUANTUM ENTANGLEMENT, MEASUREMENTS, BELL'S IN-
EQUALITIES
xvi

Table of contents
READER'S GUIDE FOR COMPLEMENTS 2215
AXXIDensity operator and correlations; separability
BXXIGHZ states, entanglement swapping
CXXIMeasurement induced relative phase between two condensates
DXXIEmergence of a relative phase with spin condensates; macroscopic
non-locality and the EPR argument
***********
APPENDICES
IV Feynman path integral
V Lagrange multipliers
VI Brief review of Quantum Statistical Mechanics
VII Wigner transform
BIBLIOGRAPHY OF VOLUME III
INDEX
xvii

Chapter VIII
An elementary approach to the
quantum theory of scattering
by a potential
A Introduction
A-1 Importance of collision phenomena
A-2 Scattering by a potential
A-3 Denition of the scattering cross section
A-4 Organization of this chapter
B Stationary scattering states. Calculation of the cross section928
B-1 Denition of stationary scattering states
B-2 Calculation of the cross section using probability currents
B-3 Integral scattering equation
B-4 The Born approximation
C Scattering by a central potential. Method of partial waves
C-1 Principle of the method of partial waves
C-2 Stationary states of a free particle
C-3 Partial waves in the potentialV(r)
C-4 Expression of the cross section in terms of phase shifts
A. Introduction
A-1. Importance of collision phenomena
Many experiments in physics, especially in high energy physics, consist of directing
a beam of particles (1) (produced for example by an accelerator) onto a target composed
Quantum Mechanics, Volume II, Second Edition. C. Cohen-Tannoudji, B. Diu, and F. Laloë.
© 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

CHAPTER VIII SCATTERING BY A POTENTIALIncident beam
Target
Detector
Detector
particles (2)
θ
2
θ
1
particles (1)
Figure 1: Diagram of a collision experiment involving the particles (1) of an incident
beam and the particles (2) of a target. The two detectors represented in the gure measure
the number of particles scattered through angles1and2with respect to the incident
beam.
of particles (2), and studying the resulting collisions: the various particles
1
constituting
the nal state of the system that is, the state after the collision (cf.Fig.) are
detected and their characteristics (direction of emission, energy, etc.) are measured.
Obviously, the aim of such a study is to determine the interactions that occur between
the various particles entering into the collision.
The phenomena observed are sometimes very complex. For example, if particles
(1) and (2) are in fact composed of more elementary components (protons and neutrons
in the case of nuclei), the latter can, during the collision, redistribute themselves amongst
two or several nal composite particles which are dierent from the initial particles; in this
case, one speaks of rearrangement collisions. Moreover, at high energies, the relativistic
possibility of the materialization of part of the energy appears: new particles are then
created and the nal state can include a great number of them (the higher the energy of
the incident beam, the greater the number). Broadly speaking, one says that collisions
give rise toreactions, which are described most often as in chemistry:
(1) + (2)(3) + (4) + (5) + (A-1)
Amongst all the reactions possible
2
under given conditions,scatteringreactions are de-
ned as those in which the nal state and the initial state are composed of the same
particles (1) and (2). In addition, a scattering reaction is said to be elastic when none
of the particles' internal states change during the collision.
1
In practice, it is not always possible to detect all the particles emitted, and one must often be
satised with partial information about the nal system.
2
Since the processes studied occur on a quantum level, it is not generally possible to predict with cer-
tainty what nal state will result from a given collision; one merely attempts to predict the probabilities
of the various possible states.
924

A. INTRODUCTION
A-2. Scattering by a potential
We shall conne ourselves in this chapter to the study of the elastic scattering of
the incident particles (1) by the target particles (2). If the laws of classical mechanics
were applicable, solving this problem would involve determining the deviations in the
incident particles' trajectories due to the forces exerted by particles (2). For processes
occurring on an atomic or nuclear scale, it is clearly out of the question to use classical
mechanics to resolve the problem; we must study the evolution of the wave function
associated with the incident particles under the inuence of their interactions with the
target particles [which is why we speak of the scattering of particles (1) by particles
(2)]. Rather than attack this question in its most general form, we shall introduce the
following simplifying hypotheses:
()
ory considerably but should not be taken to imply that the spin of particles is
unimportant in scattering phenomena.
()
(2). The following arguments are therefore not applicable to inelastic scattering
phenomena, where part of the kinetic energy of (1) is absorbed in the nal state
by the internal degrees of freedom of (1) and (2) (cf.for example, the experiment
of Franck and Hertz). We shall conne ourselves to the case ofelastic scattering,
which does not aect the internal structure of the particles.
()
scattering processes; that is, processes during which a particular incident particle
is scattered several times before leaving the target.
()
dierent particles which make up the target. This simplication is justied when
the spread of the wave packets associated with particles (1) is small compared to
the average distance between particles (2). Therefore we shall concern ourselves
only with the elementary process of the scattering of a particle (1) of the beam by
a particle (2) of the target. This excludes a certain number of phenomena which
are nevertheless very interesting, such as coherent scattering by a crystal (Bragg
diraction) or scattering of slow neutrons by the phonons of a solid, which provide
valuable information about the structure and dynamics of crystal lattices. When
these coherence eects can be neglected, the ux of particles detected is simply
the sum of the uxes scattered by each of thetarget particles, that is,times
the ux scattered by any one of them (the exact position of the scattering particle
inside the target is unimportant since the target dimensions are much smaller than
the distance between the target and the detector).
()
by a potential energy(r1r2), which depends only on the relative position
r=r1r2of the particles. If we follow the reasoning of , Chapter, then,
in the center-of-mass reference frame
3
of the two particles (1) and (2), the problem
3
In order to interpret the results obtained in scattering experiments, it is clearly necessary to return
to the laboratory reference frame. Going from one frame of reference to another is a simple kinematic
problem that we will not consider here. See for example Messiah (1.17), vol. I. Chap. X, 7.
925

CHAPTER VIII SCATTERING BY A POTENTIAL
reduces to the study ofthe scattering of a single particle by the potential(r). The
massof this relative particle is related to the masses1and2of (1) and (2)
by the formula:
1
=
1
1
+
1
2
(A-2)
A-3. Denition of the scattering cross section
Letbe the direction of the incident particles of mass(g.). The potential
(r)is localized around the originof the coordinate system [which is in fact the center
of mass of the two real particles (1) and (2)]. We shall designate bythe ux of particles
in the incident beam, that is, the number of particles per unit time which traverse a unit
surface perpendicular toin the region wheretakes on very large negative values.
(The uxis assumed to be weak enough to allow us to neglect interactions between
dierent particles of the incident beam.)
We place a detector far from the region under the inuence of the potential and in
the direction xed by the polar anglesand, with an opening facingand subtending
the solid angle d(the detector is situated at a distance fromwhich is large compared
to the linear dimensions of the potential's zone of inuence). We can thus count the
number dof particles scattered per unit time into the solid angle dabout the direction
(). The dierential dis obviously proportional to dand to the incident ux.
We shall dene()to be the coecient of proportionality between dandd:
d= () d (A-3)
The dimensions of dandare, respectively,
1
and(
2
)
1
,()therefore has
the dimensions of a surface; it is called thedierential scattering cross sectionin the
direction(). Cross sections are frequently measured in barns and submultiples of
barns:
1 barn = 10
24
cm
2
(A-4)
The denition (A-3) can be interpreted in the following way: the number of par-
ticles per unit time which reach the detector is equal to the number of particles which
would cross a surface()dplaced perpendicular toin the incident beam.
Similarly, thetotal scattering cross sectionis dened by the formula:
= () d (A-5)
Comments:
() A-3), in which dis proportional to d, implies that only the
scattered particles are taken into consideration. The ux of these particles
reaching a given detector[of xed surface and placed in the direction
926

A. INTRODUCTION
()] is inversely proportional to the square of the distance betweenand
(this property is characteristic of a scattered ux). In practice, the incident
beam is laterally bounded [although its width remains much larger than the
extent of the zone of inuence of(r)], and the detector is placed outside
its trajectory so that it receives only the scattered particles. Of course, such
an arrangement does not permit the measurement of the cross section in
the direction= 0(the forward direction), which can only be obtained by
extrapolation from the values of()for small.
()
reaction cross sections are dened in an analogous manner.
A-4. Organization of this chapter
(r)(de-
creasing however faster than1astends toward innity). First of all, in , we
introduce the fundamental concepts of a stationary scattering state and a scattering
amplitude. We then show, in , how knowledge of the asymptotic behavior of the
wave functions associated with stationary scattering states enables us to obtain scatter-
ing cross sections. Afterwards, in , we discuss in a more precise way, using the
integral scattering equation, the existence of these stationary scattering states. Finally
(in ), we derive an approximate solution of this equation, valid for weak potentials.
This leads us to the Born approximation, in which the cross section is very simply related
to the Fourier transform of the potential.Incident beam
Region where the
potential is effective
0
V(r)
dΩ
Detector D
θ
z
Figure 2: The incident beam, whose ux of particles is, is parallel to the axis; it
is assumed to be much wider than the zone of inuence of the potential(r), which is
centered at. Far from this zone of inuence, a detectormeasures the number d
of particles scattered per unit time into the solid angle d, centered around the direction
dened by the polar anglesand. The number dis proportional toand to d; the
coecient of proportionality()is, by denition, the scattering cross section in the
direction().
927

CHAPTER VIII SCATTERING BY A POTENTIAL
For a central potential(r), the general methods described in
applicable, but the method of partial waves, set forth in , is usually considered
preferable. This method is based () on the comparison of the stationary states with
well-dened angular momentum in the presence of the potential()(which we shall
call partial waves) and their analogues in the absence of the potential (free spherical
waves). Therefore, we begin by studying, in , the essential properties of the
stationary states of a free particle, and more particularly those of free spherical waves.
Afterwards (), we show that the dierence between a partial wave in the potential
()and a free spherical wave with the same angular momentumis characterized by a
phase shift. Thus, it is only necessary to know how stationary scattering states can
be constructed from partial waves in order to obtain the expression of cross sections in
terms of phase shifts ().
B. Stationary scattering states. Calculation of the cross section
In order to describe in quantum mechanical terms the scattering of a given incident
particle by the potential(r), it is necessary to study the time evolution of the wave
packet representing the state of the particle. The characteristics of this wave packet are
assumed to be known for large negative values of the timewhen the particle is in the
negative region of theaxis, far from and not yet aected by the potential(r). It is
known that the subsequent evolution of the wave packet can be obtained immediately if
it is expressed as a superposition of stationary states. This is why we are going to begin
by studying the eigenvalue equation of the Hamiltonian:
=0+(r) (B-1)
where:
0=
P
2
2
(B-2)
describes the particle's kinetic energy.
Actually, to simplify the calculations, we are going to base our reasoning directly
on the stationary states and not on wave packets. We have already used this procedure
in Chapter, in the study of square one-dimensional potentials (
mentI). It consists of considering a stationary state to represent a probability uid
in steady ow, and studying the structure of the corresponding probability currents.
Naturally, this simplied reasoning is not rigorous: it remains to be shown that it leads
to the same results as the correct treatment of the problem, which is based on wave
packets. Assuming this will enable us to develop certain general ideas easily, without
burying them in complicated calculations
4
.
4
The proof was given in complementI, for a particular one-dimensional problem; we veried that the
same results are obtained by calculating the probability current associated with a stationary scattering
state or by studying the evolution of a wave packet describing a particle which undergoes a collision.
928

B. STATIONARY SCATTERING STATES. CALCULATION OF THE CROSS SECTION
B-1. Denition of stationary scattering states
B-1-a. Eigenvalue equation of the Hamiltonian
Schrödinger's equation describing the evolution of the particle in the potential(r)
is satised by solutions associated with a well-dened energy(stationary states):
(r) =(r) e
~
(B-3)
where(r)is a solution of the eigenvalue equation:
~
2
2
+(r)(r) =(r) (B-4)
We are going to assume that the potential(r)decreases faster than1as
approaches innity. Notice that this hypothesis excludes the Coulomb potential, which
demands special treatment; we shall not consider it here.
We shall only be concerned with solutions of (B-4) associated with a positive
energy, equal to the kinetic energy of the incident particle before it reaches the zone
of inuence of the potential. Dening:
=
~
22
2
(B-5)
(r) =
~
2
2
(r) (B-6)
enables us to write (B-4) in the form:
+
2
(r)(r) = 0 (B-7)
For each value of(that is, of the energy), equation (B-7) can be satised by an
innite number of solutions (the positive eigenvalues of the Hamiltonianare innitely
degenerate). As in square one-dimensional potential problems (cf.Chap.,
complementI), we must choose from amongst these solutions the one that corresponds
to the physical problem being studied (for example, when we wanted to determine the
probability that a particle with a given energy would cross a one-dimensional potential
barrier, we chose the stationary state which, in the region on the other side of the
barrier, was composed simply of a transmitted wave). Here, the choice proves to be more
complicated, since the particle is moving in three-dimensional space and the potential
(r)has,a priori, an arbitrary form. Therefore, we shall specify, using wave packet
properties in an intuitive way, the conditions that must be imposed on the solutions
of equation (B-7) if they are to be used in the description of a scattering process. We
shall call the eigenstates of the Hamiltonian which satisfy these conditionsstationary
scattering states, and we shall designate by
(scatt)
(r)the associated wave functions.
B-1-b. Asymptotic form of stationary scattering states. Scattering amplitude
For large negative values of, the incident particle is free [(r)is practically zero
when one is suciently far from the point], and its state is represented by a plane
wave packet. Consequently, the stationary wave function that we are looking for must
contain a term of the forme, whereis the constant which appears in equation (B-7).
929

CHAPTER VIII SCATTERING BY A POTENTIAL
When the wave packet reaches the region which is under the inuence of the potential
(r), its structure is profoundly modied and its evolution complicated. Nevertheless,
for large positive values of, it has left this region and once more takes on a simple
form: it is now split into a transmitted wave packet which continues to propagate along
in the positive direction (hence having the forme) and a scattered wave packet.
Consequently, the wave function
(scatt)
(r), representing the stationary scattering state
associated with a given energy=~
22
2, will be obtained from the superposition of
the plane waveeand a scattered wave (we are ignoring the problem of normalization).
The structure of the scattered wave obviously depends on the potential(r). Yet
its asymptotic form (valid far from the zone of inuence of the potential) is simple;
reasoning by analogy with wave optics, we see that the scattered wave must present the
following characteristics for large:
() (), its radial dependence is of the forme . It is a
divergent (or outgoing) wave which has the same energy as the incident wave. The
factor1results from the fact that there are three spatial dimensions:(+
2
)e
is not zero, while:
( +
2
)
e
= 0for 0where0is any positive distance (B-8)
(in optics, the factor1insures that the total ux of energy passing through a
sphere of radiusis independent offor large; in quantum mechanics, it is the
probability ux passing through this sphere that does not depend on).
()
depends on the direction()being considered.
Finally, the wave function
(scatt)
(r)associated with the stationary scattering state
is, by denition, the solution of equation (B-7) whose asymptotic behavior is of the form:
(scatt)
(r) e+()
e
(B-9)
In this expression, only the function(), which is called thescattering amplitude,
depends on the potential(r). It can be shown (cf.) that equation (B-7) has
indeed one and only one solution, for each value of, that satises condition (B-9).
Comments:
()
the wave packet representing the state of the incident particle, it is necessary to
expand it in terms of eigenstates of the total Hamiltonianrather than in terms
of plane waves. Therefore, let us consider a wave function of the form
5
:
(r) =
0
d()
(scatt)
(r) e
~
(B-10)
5
Actually, it is also necessary to superpose the plane waves corresponding to wave vectorskhaving
slightly dierent orientations, for the incident wave packet is limited in the directions perpendicular to
. For the sake of simplicity, we are concerning ourselves here only with the energy dispersion (which
limits the spread of the wave packet along)
930

B. STATIONARY SCATTERING STATES. CALCULATION OF THE CROSS SECTION
where:
=
~
22
2
(B-11)
and where the function(), taken to be real for the sake of simplicity, has a pro-
nounced peak at=0and practically vanishes elsewhere;(r)is a solution of
Schrödinger's equation and therefore correctly describes the time evolution of the
particle. It remains to be shown that this function indeed satises the boundary
conditions imposed by the particular physical problem being considered. Accord-
ing to (B-9), it approaches asymptotically the sum of a plane wave packet and a
scattered wave packet:
(r)
0
d() ee
~
+
0
d()()
e
e
~
(B-12)
The position of the maximum of each of these packets can be obtained from the
stationary phase condition (cf.Chap., ). A simple calculation then gives
for the plane wave packet:
() = (B-13)
with:
=
~0
(B-14)
As for the scattered wave packet, its maximum in the direction()is located at
a distance from the pointgiven by:
(;) =
0
() + (B-15)
where()is the derivative with respect toof the argument of the scattering
amplitude(). Note that formulas (B-13) and (B-15) are valid only in the
asymptotic region (that is, for large).
For large negative values of, there is no scattered wave packet, as can be seen
from (B-15). The waves of which it is composed interfere constructively only
for negative values of, and these values lie outside the domain permitted to.
Therefore, all that we nd in this region is the plane wave packet, which, according
to (B-13), is making its way towards the interaction region with a group velocity
. For large positive values of, both packets are actually present; the rst one
moves o along the positiveaxis, continuing along the path of the incident
packet, and the second one diverges in all directions. The scattering process can
thus be well described by the asymptotic condition (B-9).
()The spatial extensionof the wave packet (B-10) is related to the momentum dispersion~
by the relation:

1

(B-16)
We shall assume thatis small enough forto be much larger than the linear dimensions of
the potential's zone of inuence. Under these conditions, the wave packet moving at a velocity
towards the point(Fig.) will take a time:

1

(B-17)
931

CHAPTER VIII SCATTERING BY A POTENTIAL
to cross this zone. Let us x the time origin at the instant when the center of the incident wave
packet reaches point. Scattered waves exist only for&2, i.e. after the forward edge
of the incident wave packet has arrived at the potential's zone of inuence. For= 0, the most
distant part of the scattered wave packet is at a distance of the order of2from the point.
Let us now consider ana prioridierent problem, where we have a time-dependent potential,
obtained by multiplying(r)by a function()that increases slowly from 0 to 1 between
=2and= 0. Formuch less than2, the potential is zero and we shall assume
that the state of the particle is represented by a plane wave (extending throughout all space).
This plane wave begins to be modied only at 2, and at the instant= 0the scattered
waves look like those in the preceding case.
Thus we see that there is a certain similarity between the two dierent problems that we have
just described. On the one hand, we have scattering by a constant potential of an incident wave
packet whose amplitude at the pointincreases smoothly between the times2and zero;
on the other hand, we have scattering of a plane wave of constant amplitude by a potential that
is gradually turned on over the same time interval[20].
If 0, the wave packet (B-10) tends toward a stationary scattering state [()tends toward
( 0)]; in addition, according to (B-17),becomes innite and the turning on of the
potential associated with the function()becomes innitely slow (for this reason, it is often
said to be adiabatic). The preceding discussion, although qualitative, thus makes it possible to
describe a stationary scattering state as the result of adiabatically imposing a scattering potential
on a free plane wave. We could make this interpretation more precise by studying in a more
detailed way the evolution of the initial plane wave under the inuence of the potential()(r).Δz
ZoneυofυinΔuence
of the potential
O
υ
G
Figure 3: The incident wave packet of lengthmoves at a velocitytowards the
potential(r); it interacts with the potential during a time of the order of=
(assuming the size of the potential's zone of inuence to be negligible compared to).
B-2. Calculation of the cross section using probability currents
B-2-a. Probability uid associated with a stationary scattering state
In order to determine the cross section, one should study the scattering of an
incident wave packet by the potential(r). However, we can obtain the result much
more simply by basing our reasoning on the stationary scattering states; we consider
932

B. STATIONARY SCATTERING STATES. CALCULATION OF THE CROSS SECTION
such a state to describe aprobability uid in steady owand we calculate the cross
section from the incident and scattered currents. As we have already pointed out, this
method is analogous to the one we used in one-dimensional square barrier problems: in
those problems, the ratio between the reected (or transmitted) current and the incident
current yielded the reection (or transmission) coecient directly.
Hence we shall calculate the contributions of the incident wave and the scattered
wave to the probability current in a stationary scattering state. We recall that the
expression for the currentJ(r)associated with a wave function(r)is:
J(r) =
1
Re(r)
~
r(r) (B-18)
B-2-b. Incident current and scattered current
The incident currentJis obtained from (B-18) by replacing(r)by the plane
wavee;Jis therefore directed along theaxis in the positive direction, and its
modulus is:
J=
~
(B-19)
Since the scattered wave is expressed in spherical coordinates in formula (B-9), we
shall calculate the components of the scattered currentJalong the local axes dened
by this coordinate system. Recall that the corresponding components of the operatorr
are:
(r)=
(r)=
1
(r)=
1
sin
(B-20)
If we replace(r)in formula (B-18) by the function()e, we can easily obtain
the scattered current in the asymptotic region:
(J)=
~
1
2
()
2
(J)=
~
1
3
Re
1
()()
(J)=
~
1
3
sin
Re
1
()() (B-21)
Sinceis large,(J)and(J)are negligible compared to(J), and the scattered
current is practically radial.
B-2-c. Expression for the cross section
The incident beam is composed of independent particles, all of which are assumed
to be prepared in the same way. Sending a great number of these particles amounts to
933

CHAPTER VIII SCATTERING BY A POTENTIAL
repeating the same experiment a great number of times with one particle whose state is
always the same. If this state is
(scatt)
(r), it is clear that the incident ux(that is,
the number of particles of the incident beam that cross a unit surface perpendicular to
per unit time) is proportional to the ux of the vectorJacross this surface; that is,
according to (B-19) :
=J=
~
(B-22)
Similarly, the number dof particles that strike the opening of the detector (Fig.)
per unit time is proportional to the ux of the vectorJdacross the surface dof this
opening [the proportionality constantis the same as in (B-22)]:
d=JdS=(J)
2
d
=
~
()
2
d (B-23)
We see that dis independent ofifis suciently large.
If we substitute formulas (B-22) and (B-23) into the denition (A-3) of the dier-
ential cross section(), we obtain:
() =()
2
(B-24)
The dierential cross section is thus simply the square of the modulus of the scattering
amplitude.
B-2-d. Interference between the incident and the scattered waves
In the preceding sections, we have neglected a contribution to the current associated with
(scatt)
(r)in the asymptotic region: the one that arises from interference between the plane
waveeand the scattered wave, and which is obtained by replacing(r)in (B-18) bye
and(r)by()e , and vice versa.
Nevertheless, we can convince ourselves that these interference terms do not appear when we
are concerned with scattering in directions other than the forward direction (= 0). In order
to see this, let us go back to the description of the collision in terms of wave packets (Fig.),
and let us take into consideration the fact that in practice the wave packet always has a nite
lateral spread. Initially, the incident wave packet is moving towards the zone of inuence of(r)
(Fig.-a). After the collision (Fig.-b), we nd two wave packets: a plane one which results
from the propagation of the incident wave packet (as if there were no scattering potential) and a
scattered one moving away from the pointin all directions. The transmitted wave thus results
from the interference between these two wave packets. In general, however, we place the detector
outside the beam, so that it is not struck by transmitted particles; thus we observe only the
scattered wave packet and it is not necessary to take into consideration the interference terms
that we have just mentioned.
Yet it follows from Figure-b that interference between the plane and scattered wave packets
cannot be neglected in the forward direction, where they occupy the same region of space. The
transmitted wave packet results from this interference. It must have a smaller amplitude than the
incident packet because of conservation of total probability (that is, conservation of the number
934

B. STATIONARY SCATTERING STATES. CALCULATION OF THE CROSS SECTION+
b
a
O
D
O
Figure 4: Before the collision (g. a), the incident wave packet is moving towards the
zone of inuence of the potential. After the collision (g. b), we observe a plane wave
packet and a spherical wave packet scattered by the potential (dashed lines in the gure).
The plane and scattered waves interfere in the forward direction in a destructive way
(conservation of total probability); the detectoris placed in a lateral direction and can
only see the scattered waves.
of particles: particles scattered in all directions of space other than the forward direction leave
the beam, whose intensity is therefore attenuated after it has passed the target). It is thus the
destructive interference between the plane and forward-scattered wave packets that insures the
global conservation of the total number of particles.
B-3. Integral scattering equation
We propose to show now, in a more precise way than in , how one can
demonstrate the existence of stationary wave functions whose asymptotic behavior is of
the form (B-9). In order to do so, we shall introduce the integral scattering equation,
whose solutions are precisely these stationary scattering state wave functions.
Let us go back to the eigenvalue equation of[formula (B-7)] and put it in the
form:
( +
2
)(r) =(r)(r) (B-25)
Suppose (we shall see later that this is in fact the case) that there exists a function
(r)such that:
( +
2
)(r) =(r) (B-26)
935

CHAPTER VIII SCATTERING BY A POTENTIAL
[(r)is called the Green's function of the operator +
2
]. Then any function(r)
which satises:
(r) =0(r) +d
3
(rr)(r)(r) (B-27)
where0(r)is a solution of the homogeneous equation:
( +
2
)0(r) = 0 (B-28)
obeys the dierential equation (B-25). To show this, we apply the operator +
2
to
both sides of equation (B-27); taking (B-28) into account, we obtain:
( +
2
)(r) = ( +
2
)d
3
(rr)(r)(r) (B-29)
Assuming we can move the operator inside the integral, it will act only on the variable
r, and we shall have, according to (B-26):
( +
2
)(r) =d
3
(rr)(r)(r)
=(r)(r) (B-30)
Inversely, it can be shown that any solution of (B-25) satises (B-27)
6
. The dierential
equation (B-25) can thus be replaced by the integral equation (B-27).
We shall see that it is often easier to base our reasoning on the integral equation.
Its principal advantage derives from the fact that by choosing0(r)and(r)correctly,
one can incorporate into the equation the desired asymptotic behavior. Thus, one single
integral equation, called theintegral scattering equation, becomes the equivalent of the
dierential equation (B-25) and the asymptotic condition (B-9).
To begin with, let us consider (B-26). It implies that( +
2
)(r)must be
identically equal to zero in any region which does not include the origin [which, according
to (B-8), is the case when(r)is equal toe ]. Moreover, according to relation (61)
of Appendix,(r)must behave like14whenapproaches zero. In fact, it is
easy to show that the functions:
(r) =
1
4
e
(B-31)
are solutions of equation (B-26). We may write:
(r) = e
1
4
1
4
e
+ 2r
1
4
re (B-32)
A simple calculation then gives (cf.Appendix):
(r) =
2
(r) +(r) (B-33)
6
This can be seen intuitively if one considers(r)(r)to be the right-hand side of a dierential
equation: the general solution of (B-25) is then obtained by adding to the general solution of the
homogeneous equation a particular solution of the complete equation [second term of (B-27)].
936

B. STATIONARY SCATTERING STATES. CALCULATION OF THE CROSS SECTION
which is what we wished to prove.+and are called, respectively, outgoing and
incoming Green's functions.
The actual form of the desired asymptotic behavior (B-9) suggests the choice of
the incident plane waveefor0(r)and the choice of the outgoing Green's function
+(r)for(r). In fact, we are going to show that the integral scattering equation can
be written:
(scatt)
(r) = e+d
3
+(rr)(r)
(scatt)
(r) (B-34)
whose solutions present the asymptotic behaviour given by (B-9).
To do this, let us place ourselves at a point(positionr), very far from the
various points(positionr) of the zone of inuence of the potential
7
, whose linear
dimensions are of the order of(Fig.):
. (B-35)M
r
u
O
P
L
|r – r|
r
Figure 5: Approximate calculation of the dis-
tancerrbetween a pointvery far from
and a pointsituated in the zone of inu-
ence of the potential (the dimensions of this
zone of inuence are of the order of).
Since the angle between and is very small, the length(that is,
rr) is equal, to within a good approximation, to the projection ofon :
rr ur (B-36)
whereuis the unit vector in therdirection. It follows that, for large:
+(rr) =
1
4
e
rr
rr
1
4
e
e
ur
(B-37)
7
Recall that we have explicitly assumed that(r)decreases at innity faster than1.
937

CHAPTER VIII SCATTERING BY A POTENTIAL
Substituting this expression back into equation (B-34), we obtain the asymptotic behavior
of
(scatt)
(r):
(scatt)
(r) e
1
4
e
d
3
e
ur
(r)
(scatt)
(r) (B-38)
which is indeed of the form (B-9), since the integral is no longer a function of the distance
= but only (through the unit vectoru) of the polar anglesandwhich x the
direction of the vectorOM. Thus, by setting:
() =
1
4
d
3
e
ur
(r)
(scatt)
(r) (B-39)
we are led to an expression which is identical to (B-9).
It is therefore clear that the solutions of the integral scattering equation (B-34)
are indeed the stationary scattering states
8
.
Comment:
It is often convenient to dene theincident wave vectorkas a vector of modulus
directed along theaxis of the beam such that:
e= e
kr
(B-40)
In the same way, the vectorkwhich has the same modulusas the incident wave
vector but whose direction is xed by the anglesandis called thescattered
wave vectorin the direction():
k=u (B-41)
Finally, thescattering (or transferred) wave vectorin the direction()is the
dierence betweenkandk(Fig.):
K=kk (B-42)
B-4. The Born approximation
B-4-a. Approximate solution of the integral scattering equation
If we take (B-40) into account, we can write the integral scattering equation in the
form:
(scatt)
(r) = e
kr
+d
3
+(rr)(r)
(scatt)
(r) (B-43)
We are going to try to solve this equation by iteration.
8
In order to prove the existence of stationary scattering states rigorously, it would thus be sucient
to demonstrate that equation (B-34) admits a solution.
938

B. STATIONARY SCATTERING STATES. CALCULATION OF THE CROSS SECTIONθ
k
d
k
i
K
Figure 6: Incident wave vectork, scattered
wave vectorkand transferred wave vectorK.
A simple change of notation (r=r;r=r) permits us to write:
(scatt)
(r) = e
kr
+d
3
+(rr)(r)
(scatt)
(r) (B-44)
Inserting this expression in (B-43), we obtain:
(scatt)
(r) = e
kr
+d
3
+(rr)(r) e
kr
+d
3
d
3
+(rr)(r)+(rr)(r)
(scatt)
(r)(B-45)
The rst two terms on the right-hand side of (B-45) are known; only the third one
contains the unknown function
(scatt)
(r). This procedure can be repeated: changingr
torandrtorin (B-43) gives
(scatt)
(r), which can be reinserted in (B-45). We
then have:
(scatt)
(r) = e
kr
+d
3
+(rr)(r) e
kr
+d
3
d
3
+(rr)(r)+(rr)(r) e
kr
+d
3
d
3
d
3
+(rr)(r)+(rr)(r)
+(rr)(r)
(scatt)
(r) (B-46)
where the rst three terms are known; the unknown function
(scatt)
(r)has been pushed
back into the fourth term.
Thus we can construct, step by step, what is called theBorn expansionof the
stationary scattering wave function. Note that each term of this expansion brings in one
higher power of the potential than the preceding one. Thus, if the potential is weak, each
successive term is smaller than the preceding one. If we push the expansion far enough,
we can neglect the last term on the right-hand side and thus obtain
(scatt)
(r)entirely
in terms of known quantities.
If we substitute this expansion of
(scatt)
(r)into expression (B-39), we obtain the
Born expansion of the scattering amplitude. In particular, if we limit ourselves to rst
939

CHAPTER VIII SCATTERING BY A POTENTIALr
r
Figure 7: Schematic representation
of the Born approximation: we only
consider the incident wave and the
waves scattered by one interaction
with the potential.
order in, all we need to do is replace
(scatt)
(r)bye
kr
on the right-hand side of
(B-39). This is theBorn approximation:
()
() =
1
4
d
3
e
ur
(r) e
kr
=
1
4
d
3
e
(kk)r
(r)
=
1
4
d
3
e
Kr
(r) (B-47)
whereKis the scattering wave vector dened in (B-42). The scattering cross section,
in the Born approximation, is thus very simply related to the Fourier transform of the
potential, since, using (B-24) and (B-6), (B-47) implies:
()
() =
2
4
2
~
4
d
3
e
Kr
(r)
2
(B-48)
According to Figure, the direction and modulus of the scattering wave vectorK
depend both on the modulusofkandkand on the scattering direction(). Thus,
for a givenand, the Born cross section varies with, that is, with the energy of
the incident beam. Similarly, for a given energy,
()
varies withand. We thus see,
within the simple framework of the Born approximation, how studying the variation of
the dierential cross section in terms of the scattering direction and the incident energy
gives us information about the potential(r).
B-4-b. Interpretation of the formulas
We can give formula (B-45) a physical interpretation which brings out very clearly
the formal analogy between quantum mechanics and wave optics.
Let us consider the zone of inuence of the potential to be a scattering medium
whose density is proportional to(r). The function+(rr)[formula (B-31)] repre-
sents the amplitude at the pointrof a wave radiated by a point source situated atr.
Consequently, the rst two terms of formula (B-45) describe the total wave at the point
ras the result of the superposition of the incident wavee
kr
and an innite number of
waves coming fromsecondary sourcesinduced in the scattering medium by the incident
wave. The amplitude of each of these sources is indeed proportional to the incident wave
(e
kr
)and the density of the scattering material[(r)], evaluated at the corresponding
pointr. This interpretation, symbolized by Figure, recallsHuygens' principlein wave
optics.
940

C. SCATTERING BY A CENTRAL POTENTIAL. METHOD OF PARTIAL WAVES
Actually, formula (B-45) includes a third term. However, we can interpret in
an analogous fashion the successive terms of the Born expansion. Since the scattering
medium extends over a certain area, a given secondary source is excited not only by the
incident wave but also by scattered waves coming from other secondary sources. Figure
represents symbolically the third term of the Born expansion [cf.formula (B-46)]. If the
scattering medium has a very low density [(r)very small], we can neglect the inuence
of secondary sources on each other.
Comment:
The interpretation that we have just given for higher-order terms in the Born
expansion has nothing to do with the multiple scattering processes that can occur
inside a thick target: we are only concerned, here, with the scattering of one
particle of the beam by a single particle of the target, while multiple scattering
brings in the successive interactions of the same incident particle with several
dierent particles of the target.
C. Scattering by a central potential. Method of partial waves
C-1. Principle of the method of partial waves
In the special case of a central potential(), the orbital angular momentumLof
the particle is a constant of the motion. Therefore, there exist stationary states with well-
dened angular momentum: that is, eigenstates common to,L
2
and. We shall call
the wave functions associated with these statespartial wavesand we shall write them
(r). The corresponding eigenvalues of,L
2
and are, respectively,~
22
2,
(+ 1)~
2
and~. Their angular dependence is always given by the spherical harmonics
(); the potential()inuences only their radial dependence.
We expect that, for large, the partial waves will be very close to the common
eigenfunctions of0,L
2
and, where0is the free Hamiltonian [formula (B-2)]. This
is why we are rst going to study, in , the stationary states of a free particle, and,
in particular, those which have a well-dened angular momentum. The corresponding
wave functions
(0)
(r)arefree spherical waves: their angular dependence is, of course,
that of a spherical harmonic and we shall see that the asymptotic expression for their
radial function is the superposition of an incoming waveand an outgoing wave
with a well-determined phase dierence.
The asymptotic expression for the partial wave(r)in the potential()is
also () the superposition of an incoming wave and an outgoing wave. However, ther
r
r
Figure 8: Schematic representation
of the second-order term inin the
Born expansion: here we consider
waves which are scattered twice by
the potential.
941

CHAPTER VIII SCATTERING BY A POTENTIAL
phase dierence between these two waves is dierent from the one that characterizes the
corresponding free spherical wave: the potential()introduces a supplementaryphase
shift. This phase shift constitutes the only dierence between the asymptotic behavior
of and that of
(0)
. Consequently, for xed, the phase shiftsfor all values
ofare all we need to know to be able to calculate the cross section.
In order to carry out this calculation, we shall express ( ) the stationary
scattering state
(scatt)
(r)as a linear combination of partial waves(r)having the
same energy but dierent angular momenta. Simple physical arguments suggest that the
coecients of this linear combination should be the same as those of the free spherical
wave expansion of the plane wave e; this is in fact conrmed by an explicit calculation.
The use of partial waves thus permits us to express the scattering amplitude, and
hence the cross section, in terms of the phase shifts. This method is particularly
attractive when the range of the potential is not much longer than the wavelength asso-
ciated with the particle's motion, for, in this case, only a small number of phase shifts
are involved ( ).
C-2. Stationary states of a free particle
In classical mechanics, a free particle of massmoves along a uniform linear
trajectory. Its momentump, its energy=p
2
2and its angular momentumLLL=rp
relative to the origin of the coordinate system are constants of the motion.
In quantum mechanics, the observablesPandL=RPdo not commute. Hence
they represent incompatible quantities: it is impossible to measure the momentum and
the angular momentum of a particle simultaneously.
The quantum mechanical Hamiltonian0is written:
0=
1
2
P
2
(C-1)
0does not constitute by itself a C.S.C.O.: its eigenvalues are innitely degenerate
( 2-a). On the other hand, the four observables:
0 (C-2)
form a C.S.C.O. Their common eigenstates are stationary states of well dened momen-
tum. A free particle may also be considered as being placed in a zero central potential.
The results of Chap.
0L
2
(C-3)
form a C.S.C.O. The corresponding eigenstates are stationay states with well-dened
angular momentum (more precisely,L
2
andhave well-dened values, butand
do not).
The bases of the state space dened by the C.S.C.O.'s (C-2) and (C-3) are distinct,
sincePandLare incompatible quantities. We are going to study these two bases and
show how one can pass from one to the other.
942

C. SCATTERING BY A CENTRAL POTENTIAL. METHOD OF PARTIAL WAVES
C-2-a. Stationary states with well-dened momentum. Plane waves
We already know (cf.Chap., ) that the three observables,and
form a C.S.C.O. (for a spinless particle). Their common eigenstates give a basis for the
prepresentation:
Pp=pp (C-4)
Since0commutes with these three observables, the statespare necessarily eigenstates
of0:
0p=
p
2
2
p (C-5)
The spectrum of0is therefore continuous and includes all positive numbers and
zero. Each of these eigenvalues is innitely degenerate: to a xed positive energy
there corresponds an innite number of ketspsince there exists an innite number of
ordinary vectorspwhose modulus satises:
p=
2 (C-6)
The wave functions associated with the ketspare the plane waves (cf.Chap.,
):
rp=
1
2~
32
e
pr~
(C-7)
We shall introduce here the wave vectorkto characterize a plane wave:
k=
p
~
(C-8)
and we shall dene:
k= (~)
32
p (C-9)
The ketskare stationary states with well-dened momentum:
0k=
~
2
k
2
2
k (C-10a)
Pk=~kk (C-10b)
They are orthonormal in the extended sense:
kk=(kk) (C-11)
and form a basis in the state space:
d
3
kk= (C-12)
The associated wave functions are the plane waves; normalized in a slightly dierent
way:
rk=
1
2
32
e
kr
(C-13)
943

CHAPTER VIII SCATTERING BY A POTENTIAL
C-2-b. Stationary states with well-dened angular momentum. Free spherical waves
In order to obtain the eigenfunctions common to0,L
2
and, all we have to do
is solve the radial equation for an identically zero central potential. The detailed solution
of this problem is given in complementVIII; we shall be satised here with giving the
results.
Free spherical waves are the wave functions associated with the well-dened angular
momentum stationary states
(0)
of a free particle; they are written:
(0)
(r) =
2
2
()() (C-14)
whereis a spherical Bessel function dened by:
() = (1)
1
d
d
sin
(C-15)
The corresponding eigenvalues of0,L
2
and are, respectively,~
22
2,(+ 1)~
2
and~.
The free spherical waves (C-14) are orthonormal in the extended sense:
(0) (0)
=
2
0
()()
2
d d ()()
=( ) (C-16)
and form a basis in the state space:
0
d
=0
+
=
(0) (0)
= (C-17)
C-2-c. Physical properties of free spherical waves
. Angular dependence
The angular dependence of the free spherical wave
(0)
(r)is entirely given by
the spherical harmonic(). It is thus xed by the eigenvalues ofL
2
and(that
is, by the indicesand) and not by the energy. For example, a free(= 0)spherical
wave is always isotropic.
. Behavior in the neighborhood of the origin
Let us consider an innitesimal solid angled0about the direction(00); when
the state of the particle is
(0)
, the probability of nding the particle in this solid
angle betweenand+ dis proportional to:
22
() (00)
2
dd0 (C-18)
It can be shown (complementVIII, ) that whenapproaches zero:
()
0
(2+ 1)!!
(C-19)
944

C. SCATTERING BY A CENTRAL POTENTIAL. METHOD OF PARTIAL WAVES0 5 10
l = 4
15
ρ
2j
l
2
(ρ)
ρ
Figure 9: Graph of the function
22
()giving the radial dependence of the probability of
nding the particle in the state
(0)
. At the origin, this function behaves like
2+2
;
it remains practically zero as long as
(+ 1).
This result (which the discussion of Chapter,
that the probability (C-18) behaves like
2+2
near the origin; hence, the largeris, the
more slowly it increases.
The shape of the function
22
()is shown in Figure. We see that this function
remains small as long as:
(+ 1) (C-20)
We may thus assume that the probability (C-18) is practically zero for:
1
(+ 1) (C-21)
From a physical point of view, this result is very important for it implies that a particle
in the state
(0)
is practically unaected by what happens inside a sphere centered
atof radius:
() =
1
(+ 1) (C-22)
We shall return to this point in .
Comment:
In classical mechanics, a free particle of momentumpand angular momentumLLL
moves in a straight line whose distancefrom the pointis given (g.) by:
=
LLL
p
(C-23)
945

CHAPTER VIII SCATTERING BY A POTENTIALr
ℒ
b
O
p
Figure 10: Denition of the classi-
cal impact parameterof a particle
of momentumpand angular mo-
mentumLLLrelative to.
is called the collision parameter or impact parameter of the particle relative
to; the largerLLLis and the smaller the momentum (i.e. the energy), the larger
is. IfLLLis replaced by~
(+ 1)andpby~in (C-23), we again nd
expression (C-22) for(), which can thus be interpreted semi-classically.
. Asymptotic behavior
It can be shown (complementVIII, ) that asapproaches innity:
()
1
sin
2
(C-24)
Consequently, the asymptotic behavior of the free spherical wave
(0)
(r)is:
(0)
( )
r
2
2
()
ee
2ee2
2
(C-25)
At innity,
(0)
therefore results from thesuperposition of an incoming wave
e and an outgoing wavee , whose amplitudes dier by a phase dierence equal
to.
Comment:
Suppose that we construct a packet of free spherical waves, all corresponding to
the same values ofand. A line of reasoning analogous to that of comment
()of
negative values of, only an incoming wave packet exists; while for large positive
946

C. SCATTERING BY A CENTRAL POTENTIAL. METHOD OF PARTIAL WAVES
values of, only an outgoing wave packet exists. Therefore, a free spherical wave
may be thought of in the following manner: at rst we have an incoming wave
converging towards; it becomes distorted as it approaches this point, retraces
its steps when it is at a distance of the order of()[formula (C-22)], and gives
rise to an outgoing wave with a phase shift of.
C-2-d. Expansion of a plane wave in terms of free spherical waves
We thus have two distinct bases formed by eigenstates of0: thekbasis
associated with the plane waves and the
(0)
basis associated with the free spherical
waves. It is possible to expand any ket of one basis in terms of vectors of the other one.
Let us consider in particular the ket00, associated with a plane wave of wave
vectorkdirected along:
r00=
1
2
32
e (C-26)
00represents a state of well-dened energy and momentum (=~
22
2;pdi-
rected alongwith modulus~). Now:
e= e
cos
(C-27)
is independent of; sincecorresponds to
~
in therrepresentation, the ket
00is also an eigenvector of, with the eigenvalue zero:
00= 0 (C-28)
Using the closure relation (C-17), we can write:
00=
0
d
=0
+
=
(0) (0)
00 (C-29)
Since00and
(0)
are two eigenstates of0, they are orthogonal if the cor-
responding eigenvalues are dierent; their scalar product is therefore proportional to
( ). Similarly, they are both eigenstates ofand their scalar product is propor-
tional to0[cf.relation (C-28)]. Formula (C-29) therefore takes on the form:
00=
=0
(0)
0
(C-30)
The coecientscan be calculated explicitly (complementVIII, ). Thus we obtain:
e=
=0
4(2+ 1)()
0
()
(C-31)
A state of well-dened linear momentum is therefore formed by the superposition of
states corresponding to all possible angular momenta.
947

CHAPTER VIII SCATTERING BY A POTENTIAL
Comment:
The spherical harmonic
0
()is proportional to the Legendre polynomial(cos)(com-
plementVI, ):
0
() =
(2+ 1)4
(cos) (C-32)
Hence the expansion (C-31) is often written in the form:
e=
=0
(2+ 1)()(cos) (C-33)
C-3. Partial waves in the potentialV(r)
We are now going to study the eigenfunctions common to(the total Hamilto-
nian),L
2
and; that is, the partial waves(r).
C-3-a. Radial equation. Phase shifts
For any central potential(), the partial waves(r)are of the form:
(r) =()() =
1
()() (C-34)
where()is the solution of the radial equation:
~
2
2
d
2
d
2
+
(+ 1)~
2
2
2
+() () =
~
22
2
() (C-35)
satisfying the condition at the origin:
(0) = 0 (C-36)
It is just as if we were dealing with a one-dimensional problem, where a particle
of massis under the inuence of the potential (g.):
e() =() +
(+ 1)~
2
2
2
for0
e()innite for 0 (C-37)
For large, equation (C-35) reduces to:
d
2
d
2
+
2
() 0 (C-38)
whose general solution is:
() e+e (C-39)
Since()must satisfy condition (C-36), the constantsandcannot be arbitrary.
In the equivalent one-dimensional problem [formulas (C-37)], equation (C-36) is related
948

C. SCATTERING BY A CENTRAL POTENTIAL. METHOD OF PARTIAL WAVESV
eff
(r)
0
V(r)
r
2μr
2
l(l + 1)ħ
2
Figure 11: The eective potential
e()is the sum of the poten-
tial()and the centrifugal term
(+ 1)~
2
2
2
.
to the fact that the potential is innite for negative, and expression (C-39) represents
the superposition of an incident plane wave ecoming from the right (along the
axis on which the ctitious particle being studied moves) and a reected plane wave
epropagating from left to right. Since there can be no transmitted wave [as()
is innite on the negative part of the axis], the reected current must be equal to the
incident current. Thus we see that condition (C-36) implies that, in the asymptotic
expression (C-39):
= (C-40)
Consequently:
() ee+ ee (C-41)
which can be written in the form:
() sin ( ) (C-42)
The real phaseis completely determined by imposing continuity between (C-42)
and the solution of (C-35) which goes to zero at the origin. In the case of an identically
null potential(), we saw in thatis equal to2. It is convenient to take
this value as a point of reference, that is, to write:
() sin
2
+ (C-43)
The quantitydened in this way is called thephase shiftof the partial wave(r);
it obviously depends on, that is, on the energy.
949

CHAPTER VIII SCATTERING BY A POTENTIAL
C-3-b. Physical meaning of phase shifts
. Comparison between partial waves and free spherical waves
Taking (C-34) and (C-43) into account, we may write the expression for the asymp-
totic behavior of(r)in the form:
(r)
sin( 2 +)
()
()
ee
(
2)
ee
(
2)
2
(C-44)
We see that the partial wave(r), like a free spherical wave [formula (C-25)], results
from thesuperposition of an incoming wave and an outgoing wave.
In order to develop the comparison between partial waves and free spherical waves
in detail, we can modify the incoming wave of (C-44) so as to make it identical with
the one in (C-25). To do this, we dene a new partial wave(r)by multiplying
(r)bye(this global phase factor has no physical importance) and by choosing
the constantin such a way that:
(r) ()
ee
2
ee
2
e
2
2
(C-45)
This expression can then be interpreted in the following way (cf.the comment in
2-c-): at the outset, we have the same incoming wave as in the case of a free particle
(aside from the normalization constant
2
2
). As this incoming wave approaches
the zone of inuence of the potential, it is more and more perturbed by this potential.
When, after turning back, it is transformed into an outgoing wave,it has accumulated
a phase shift of2relative to the free outgoing wave that would have resulted if the
potential()had been identically zero. The factore
2
(which varies withand)
thus summarizes the total eect of the potential on a particle of angular momentum.
Comment:
Actually, the preceding discussion is only valid if we base our reasoning on a wave packet
formed by superposing partial waves (r)with the sameand, but slightly dif-
ferent. For large negative values of, we have only an incoming wave packet; it is
the subsequent evolution of this wave packet directed towards the potential's zone of
inuence that we have analyzed above.
We could also adopt the point of view of comment () of ; that is, we could study
the eect on a stationary free spherical wave of slowly turning on the potential().
The same type of reasoning would then demonstrate that the partial wave(r)can
be obtained from a free spherical wave
(0)
(r)by adiabatically turning on the potential
().
. Finite-range potentials
Let us suppose that the potential()has a nite range0; that is, that:
() = 0for 0 (C-46)
950

C. SCATTERING BY A CENTRAL POTENTIAL. METHOD OF PARTIAL WAVES
We pointed out earlier ( ) that the free spherical wave
(0)
scarcely
penetrates a sphere centered atof radius()[formula (C-22)]. Therefore, if we
return to the interpretation of formula (C-45) that we have just given, we see that a
potential satisfying (C-46) has virtually no eect on waves for which:
() 0 (C-47)
since the corresponding incoming wave turns back before reaching the zone of inuence
of(). Thus, for each value of the energy, there exists a critical valueof the angular
momentum, which, according to (C-22), is given approximately by:
(+ 1) 0 (C-48)
The phase shiftsare appreciable only for values ofless than or of the order of.
The shorter the range of the potential and the lower the incident energy, the
smaller the value
9
of. Therefore, it may happen that the only non-zero phase shifts
are those corresponding to the rst few partial waves: the(= 0)wave at very low
energy, followed byandwaves for slightly greater energies, etc.
C-4. Expression of the cross section in terms of phase shifts
Phase shifts characterize the modications, caused by the potential, of the asymp-
totic behavior of stationary states with well-dened angular momentum. Knowing them
should therefore allow us to determine the cross section. In order to demonstrate this,
all we must do is express the stationary scattering state
(scatt)
(r)in terms of partial
waves
10
, and calculate the scattering amplitude in this way.
C-4-a. Construction of the stationary scattering state from partial waves
We must nd a linear superposition of partial waves whose asymptotic behavior is
of the form (B-9). Since the stationary scattering state is an eigenstate of the Hamilto-
nian, the expansion of
(scatt)
(r)involves only partial waves having the same energy
~
22
2. Note also that, in the case of a central potential(), the scattering problem
we are studying is symmetrical with respect to rotation around theaxis dened by
the incident beam. Consequently, the stationary scattering wave function
(scatt)
(r)is
independent of the azimuthal angle, so that its expansion includes only partial waves
for whichis zero. Finally, we have an expression of the form:
(scatt)
(r) =
=0
0(r) (C-49)
The problem thus consists of nding the coecients.
9
is of the order of0, which is the ratio between the range0of the potential and the wavelength
of the incident particle.
10
If there exist bound states of the particle in the potential()(stationary states with negative
energy), the system of partial waves does not constitute a basis of the state space; in order to form such
a basis, it is necessary to add the wave functions of the bound states to the partial waves.
951

CHAPTER VIII SCATTERING BY A POTENTIAL
. Intuitive argument
When()is identically zero, the function
(scatt)
(r)reduces to the plane wave
e, and the partial waves become free spherical waves
(0)
(r). In this case, we already
know the expansion (C-49): it is given by formula (C-31).
For non-zero(),
(scatt)
(r)includes a diverging scattered wave as well as a plane
wave. Furthermore, we have seen that 0(r)diers from
(0)
0
(r)in its asymptotic
behavior only by the presence of the outgoing wave, which has the same radial depen-
dence as the scattered wave. We should therefore expect that the coecientsof the
expansion (C-49) will be the same as those in formula (C-31)
11
, that is:
(scatt)
(r) =
=0
4(2+ 1) 0(r) (C-50)
Comment:
We can also understand (C-50) in terms of the interpretation oered in comment () of
and the comment in . If we have a plane wave whose expansion is given by (C-31) and
we turn on the potential()adiabatically, the wave is transformed into a stationary scattering
state: the left-hand side of (C-31) must then be replaced by
(scatt)
(r). In addition, each free
spherical wave()
0
()appearing on the right-hand side of (C-31) is transformed into a
partial wave 0(r)when the potential is turned on. If we take into account the linearity of
Schrödinger's equation, we nally obtain (C-50).
. Explicit derivation
Let us now consider formula (C-50), which was suggested by a physical approach
to the problem, and let us show that it does indeed supply the desired expansion.
First of all, the right-hand side of (C-50) is a superposition of eigenstates of
having the same energy~
22
2; consequently, this superposition remains a stationary
state.
Therefore, all we must do is make sure that the asymptotic behavior of the sum (C-
50) is indeed of type (B-9). In order to do this, we use (C-45):
=0
4(2+ 1) 0(r)
=0
4(2+ 1)
0
()
1
2
ee
2ee
2e
2
(C-51)
In order to bring out the asymptotic behavior of expansion (C-31), we write:
e
2
= 1 + 2esin (C-52)
11
Note that the expansion (C-31) brings in()
0
(), that is, the free spherical wave
(0)
0
di-
vided by the normalization factor
2
2
; this is why we dened (r)[formula (C-45)] from
expression (C-25) divided by this same factor.
952

C. SCATTERING BY A CENTRAL POTENTIAL. METHOD OF PARTIAL WAVES
and, regrouping the terms that are independent of, we have:
=0
4(2+ 1) 0(r)
=0
4(2+ 1)
0
()
ee
2
ee
2
2
e1
e
2esin (C-53)
Taking (C-25) and (C-31) into consideration, we recognize, in the rst term of the right-
hand side, the asymptotic expansion of the plane wavee, and we obtain nally:
=0
4(2+ 1) 0(r) e+()
e(C-54)
with
12
:
() =
1
=0
4(2+ 1) esin
0
()
(C-55)
We have thus demonstrated that the expansion of (C-50) is correct and have found
at the same time the expression for the scattering amplitude()in terms of the phase
shifts.
C-4-b. Calculation of the cross section
The dierential scattering cross section is then given by formula (B-24):
() =()
2
=
1
2
=0
4(2+ 1) esin
0
()
2
(C-56)
from which we deduce the total scattering cross section by integrating over the angles:
=d() =
1
2
4
(2+ 1)(2+ 1) e
( )
sinsin
d
0
()
0
()(C-57)
Since the spherical harmonics are orthonormal [formula (D-23) of Chapter], we have
nally:
=
4
2
=0
(2+ 1) sin
2
(C-58)
This result shows that the terms resulting from interference between waves of dierent
angular momenta disappear from the total cross section. For any potential(), the
12
The factoris compensated bye
2= ()= (1).
953

CHAPTER VIII SCATTERING BY A POTENTIAL
contribution(2+ 1)(4
2
) sin
2
associated with a given value ofis positive and has
an upper bound, for a given energy, of(2+ 1)(4
2
).
In theory, formulas (C-56) and (C-58) necessitate knowing all the phase shifts.
Recall (cf. ) that these phase shifts can be calculated from the radial equation
if the potential()is known; this equation must be solved separately for each value of
(most of the time, moreover, this implies resorting to numerical techniques). In other
words, the method of partial waves is attractive from a practical point of view only when
there is a suciently small number of non-zero phase shifts. For a nite-range potential
(), we saw in that the phase shiftsare negligible for, the critical
valuebeing dened by formula (C-48).
When the potential()is unknown at the outset, we attempt to reproduce the ex-
perimental curves which give the dierential cross section at a xed energy by introducing
a small number of non-zero phase shifts. Furthermore, the very form of the-dependence
of the cross section often suggests the minimum number of phase shifts needed. For ex-
ample, if we limit ourselves to the-wave, formula (C-56) gives an isotropic dierential
cross section (
0
0is a constant). If the experiments imply a variation of()with,
it means that phase shifts other than that of the-wave are not equal to zero. Once
we have thereby determined, from experimental results corresponding to dierent ener-
gies, the phase shifts which do eectively contribute to the cross section, we can look
for theoretical models of potentials that produce these phase shifts and their energy
dependence.
Comment:
The dependence of cross sections on the energy=~
22
2of the incident
particle is just as interesting as the-dependence of(). In particular, in certain
cases, one observes rapid variations of the total cross sectionin the neighborhood
of certain energy values. For example, if one of the phase shiftstakes on the
value2for=0, the corresponding contribution toreaches its upper limit
and the cross section may show a sharp peak at= 0. This phenomenon
is called scattering resonance. We can compare it to the behavior described in
Chapter ) of the transmission coecient of a square one-dimensional
potential well.
References and suggestions for further reading:
Dicke and Wittke (1.14), Chap. 16; Messiah (1.17), Chap. X; Schi (1.18), Chaps. 5
and 9.
More advanced topics:
Coulomb scattering: Messiah (1.17), Chap. XI; Schi (1.18), 21; Davydov (1.20),
Chap. XI, 100.
Formal collision and S-matrix theory: Merzbacher (1.16), Chap. 19; Roman (2.3),
part II, Chap. 4; Messiah (1.17), Chap. XIX; Schweber (2.16), part 3, Chap. 11.
Description of collisions in terms of wave packets: Messiah (1.17), chap. X, 4,
5, 6; Goldberger and Watson (2.4), chaps. 3 and 4.
954

C. SCATTERING BY A CENTRAL POTENTIAL. METHOD OF PARTIAL WAVES
Determination of the potential from the phase shifts (the inverse problem): Wu
and Ohmura (2.1), G.
Applications: Davydov (1.20), Chap. XI; Sobel'man (11.12), Chap. 11; Mott and
Massey (2.5); Martin and Spearman (16.18).
Scattering by multi-particle systems in the Born approximation and space-time
correlation functions: Van Hove (2.39).
955

COMPLEMENTS OF CHAPTER VIII, READER'S GUIDE
The main purpose of Chapter
nuclear physics, where various physical applications of the theory of collisions can be found.
AVIII: THE FREE PARTICLE: STATIONARY
STATES WITH WELL-DEFINED ANGULAR MO-
MENTUM
Formal examination of stationary wave functions
for a free particle with well-dened angular
momentum. The use of theL+andLoperators
permits the introduction of spherical Bessel
functions, and the demonstration of a certain
number of their properties that were used in
of Chapter .
BVIII: PHENOMENOLOGICAL DESCRIPTION OF
COLLISIONS WITH ABSORPTION
This complement permits the extension of
the formalism of Chapter
with absorption, and establishes the optical
theorem. A phenomenological point of view is
used, whose principle is analogous to that of
ComplementIII. Not dicult if Chapter
has been well assimilated.
CVIII: SOME SIMPLE APPLICATIONS OF SCAT-
TERING THEORY
Illustration of the results of Chapter
eral specic examples. Section
for a rst reading, since it presents important
physical results in a simple manner (Rutherford's
formula). Section
example. Section
their solutions.
957

THE FREE PARTICLE: STATIONARY STATES
WITH WELL-DEFINED ANGULAR MOMENTUM
Complement AVIII
The free particle: stationary states
with well-dened angular momentum
1 The radial equation
2 Free spherical waves
2-a Recurrence relations
2-b Calculation of free spherical waves
2-c Properties
3 Relation between free spherical waves and plane waves
We introduced, in , two distinct bases of stationary states of
a free (spinless) particle whose Hamiltonian is written:
0=
P
2
2
(1)
The rst of these bases is composed of the eigenstates common to0and the three
components of the momentumP; the associated wave functions are the plane waves.
The second consists of the stationary states with well-dened angular momentum, that
is, the eigenstates common to0,L
2
and, whose principal properties we pointed out
in , c and d of Chapter . We intend to study here this second basis in more
detail. In particular, we wish to derive a certain number of results used in Chapter .
1. The radial equation
The Hamiltonian (1) commutes with the three components of the orbital angular mo-
mentumLof the particle:
[0L] =0 (2)
Consequently, we can apply the general theory developed in
particular problem. We know that the free spherical waves (eigenfunctions common to
0,L
2
and) are necessarily of the form:
(0)
(r) =
(0)
()() (3)
The radial function
(0)
()is a solution of the equation:
~
2
2
1d
2
d
2
+
(+ 1)~
2
2
2
(0)
() =
(0)
() (4)
where is the eigenvalue of0corresponding to
(0)
(r). If we set:
(0)
() =
1
(0)
() (5)
959

COMPLEMENT A VIII
the function
(0)
is given by the equation:
d
2
d
2
(+ 1)
2
+
2
~
2
(0)
() = 0 (6)
to which we must add the condition:
(0)
(0) = 0 (7)
It can be shown, rst of all, that equations (6) and (7) enable us to nd the
spectrum of the Hamiltonian0, which we already know from the study of the plane
waves [formula (C-5) of Chapter ]. To do this, note that the minimum value of the
potential (which is, in fact, identically zero) vanishes and that consequently there cannot
exist a stationary state with negative energy (cf.complementIII). Consider, therefore,
any positive value of the constantappearing in equation (6), and set:
=
1
~2 (8)
Asapproaches innity, the centrifugal term(+ 1)
2
becomes negligible compared to
the constant term of equation (6), which can thus be approximated by:
d
2
d
2
+
2(0)
() 0 (9)
Consequently, all solutions of equation (6) have an asymptotic behavior (linear combi-
nation ofeande) which is physically acceptable. Therefore, the only restriction
comes from condition (7): we know (cf.Chap., ) that there exists, for a given
value of, one and only one function (to within a constant factor) which satises (6)
and (7). For any positive, the radial equation (6) has one and only one acceptable
solution.
This means that the spectrum of0includes all positive energies. Moreover, we
see that the set of possible values ofdoes not depend on; we shall therefore omit
the indexfor the energies. As for the index, we shall identify it with the constant
dened in (8); this allows us to write:
=
~
22
2
; 0 (10)
Each of these energies is innitely degenerate. Indeed, for xed, there exists an
acceptable solution
(0)
()of the radial equation corresponding to the energyfor every
value (positive integral or zero) of. Moreover, formula (3) associates(2+1)independent
wave functions
(0)
(r)with a given radial function
(0)
(). Thus, we again nd in this
particular case the general result demonstrated in :0,L
2
and
form a C.S.C.O. inr, and the specication of the three indices,andgives
sucient information for the determination of a unique function in the corresponding
basis.
960

THE FREE PARTICLE: STATIONARY STATES
WITH WELL-DEFINED ANGULAR MOMENTUM
2. Free spherical waves
The radial functions
(0)
() =
(0)
()can be found by solving equation (6) or equa-
tion (4) directly. The latter is easily reduced (comment of below) to a dierential
equation known as the spherical Bessel equation whose solutions are well-known. In-
stead of using these results directly, we are going to see how the various eigenfunctions
common to 0,L
2
andcan be simply deduced from those which correspond to the
eigenvalue 0 ofL
2
.
2-a. Recurrence relations
Let us dene the operator:
+=+ (11)
in terms of the componentsandof the momentumP. We know thatPis a vec-
torial observable (cf.complementVI, 5-c), which implies the following commutation
relations
1
between its components and those of the angular momentumL:
[ ] = 0
[ ] =~
[ ] =~ (12)
and the equations which are deduced from these by circular permutation of the indices
,,. Using these relations, a simple algebraic calculation gives the commutators of
andL
2
with the operator+; we nd:
[ +] =~+ (13a)
[L
2
+] = 2~(+ +) + 2~
2
+ (13b)
Consider therefore any eigenfunction
(0)
(r)common to 0,L
2
and, the
corresponding eigenvalues being,(+ 1)~
2
and~. By applying the operators+
and, we can obtain the2other eigenfunctions associated with the same energy
and the same value of. Since0commutes withL, we have, for example:
0+
(0)
(r) =+0
(0)
(r) = +
(0)
(r) (14)
and+
(0)
(r)(which is not zero ifis dierent from) is an eigenfunction of0
with the same eigenvalue as
(0)
(r). Therefore:
(0)
(r)
(0)
1
(r) (15)
Let us now allow+to act on
(0)
(r). First of all, since0commutes withP,
we can repeat the preceding argument for+
(0)
. Moreover, from relation (13a):
+
(0)
(r) =+
(0)
+~+
(0)
= (+ 1)~+
(0)
(r) (16)
1
These relations can be obtained directly from the denitionL=RPand the canonical commu-
tation rules.
961

COMPLEMENT A VIII
+
(0)
is therefore an eigenfunction ofwith the eigenvalue(+ 1)~. If we use
equation (13b) in the same way, we see that the presence of the term+implies
that+
(0)
is not, in general, an eigenfunction ofL
2
; nevertheless, if=, the
contribution of this term is zero:
L
2
+
(0)
=+L
2(0)
+ 2~+
(0)
+ 2~
2
+
(0)
= [(+ 1) + 2+ 2]~
2
+
(0)
= (+ 1)(+ 2)~
2
+
(0)
(17)
Consequently,+
(0)
is a common eigenfunction of0,L
2
andwith the eigenvalues
,(+1)(+2)~
2
and(+1)~respectively. Since these three observables form a C.S.C.O.
( 1), there exists only one eigenfunction (to within a constant factor
2
) associated with
this set of eigenvalues:
+
(0)
(r)
(0)
+1+1
(r) (18)
We are going to use the recurrence relations (15) and (18) to construct the
(0)
(r)
basis from the functions
(0)
00
(r)corresponding to zero eigenvalues
3
forL
2
and.
2-b. Calculation of free spherical waves
. Solution of the radial equation for= 0
In order to determine the functions
(0)
00
(r), we return to the radial equation (6),
in which we set= 0; taking denition (10) into account, this equation can be written:
d
2
d
2
+
2(0)
0
() = 0 (19)
The solution which goes to zero at the origin [condition (7)] is of the form:
(0)
0
() =sin (20)
We choose the constantsuch that the functions
(0)
00
(r)are orthonormal in the
extended sense; that is:
d
3(0)
00
(r)
(0)
00
(r) =( ) (21)
It is easy to show (see below) that condition (21) is satised if:
=
2
(22)
2
Later ( 2-b), we shall specify the coecients that ensure the orthonormalization of the
(0)
(r)
basis (in the extended sense, sinceis a continuous index).
3
It must not be thought that the operator= allows one to step down from an arbitrary
value ofto zero. It can easily be shown, by an argument analogous to the preceding one, that:
(0)
(r)
(0)
+1(+1)
(r)
.
962

THE FREE PARTICLE: STATIONARY STATES
WITH WELL-DEFINED ANGULAR MOMENTUM
which yields (
0
0being equal to1
4):
(0)
00
(r) =
2
2
1
4
sin
(23)
Let us verify that the functions (23) satisfy the orthonormalization relation (21). To do this, it
is sucient to calculate:
d
3 (0)
00
(r)
(0)
00
(r) =
2
1
4
0
2
d
sinsin
d
=
2
0
dsinsin (24)
Replacing the sines by complex exponentials and extending the interval of integration over the
range to+, we obtain:
2
0
dsinsin=
21
4
+
de
(+)
e
( )
(25)
Sinceand are both positive,+is always dierent from zero and the contribution of
the rst term within the brackets is always zero. According to formula (34) of Appendix, the
second term yields nally:
d
3 (0)
00
(r)
(0)
00
(r) =
2
1
4
(2)( )
=( ) (26)
. Construction of the other waves by recurrence
Let us now apply the operator+dened in (11) to the function
(0)
00
(r)that we
have just found. According to relation (18):
(0)
11
(r) +
(0)
00
(r)
+
sin
(27)
In therrepresentation, which we have been using throughout,+is the dierential
operator:
+=
~
+ (28)
In formula (27), it acts on a function ofalone. Now:
+() =
~
+
d
d
()
=
~
sine
d
d
() (29)
Thus we obtain:
(0)
11
(r)sine
cos
sin
()
2
(30)
963

COMPLEMENT A VIII
We recognize the angular dependence of
1
1()[complementVI, formula (32)]; by
applying, the functions
(0)
10
(r)and
(0)
11
(r)can be calculated.
Although
(0)
11
(r)depends onand, the application of+to this function
remains very simple. The canonical commutation relations indicate immediately that:
[++] = 0 (31)
Consequently,
(0)
22
(r)is given by:
(0)
22
(r)
2
+
sin
+
+
d
d
sin
(+)+
1
d
d
sin
(+)
2
1
d
d
1d
d
sin
(32)
In general:
(0)
(r)(+)
1
d
d
sin
(33)
The angular dependence of
(0)
is contained in the factor:
(+)=(sin)e (34)
which is indeed proportional to().
Let us dene:
() = (1)
1
d
d
sin
(35)
, thus dened, is thespherical Bessel function of order. The preceding calculation
shows that
(0)
(r)is proportional to the product of()and(). We shall write
(see below the problem of normalization):
(0)
() =
2
2
() (36)
The free spherical waves are then written:
(0)
(r) =
2
2
()() (37)
They satisfy the orthonormalization relation:
d
3(0)
(r)
(0)
(r) =( ) (38)
and the closure relation:
0
d
=0
+
=
(0)
(r)
(0)
(r) =(rr) (39)
964

THE FREE PARTICLE: STATIONARY STATES
WITH WELL-DEFINED ANGULAR MOMENTUM
Let us now examine the normalization of the functions (37). To do so, let us begin by specifying
the proportionality factors of the recurrence relations (15) and (18). For the rst relation, we
already know this factor from the properties of spherical harmonics (cf.complementVI):
(0)
(r) =~
(+ 1) ( 1)
(0)
1
(r) (40)
As for relation (18), we proceed as follows. Using the explicit expression for()[formulas (4)
and (14 VI], equations (31) and (29) as well as denition (35), and taking (37)
into account, we can write this relation as:
+
(0)
(r) =
~
2+ 22+ 3
(0)
+1+1
(r) (41)
In the orthonormalization relation (38), the factors on the right-hand side arise
from the angular integration and the orthonormality of the spherical harmonics. To establish
relation (38), it is thus sucient to show that the integral:
() =d
3 (0)
(r)
(0)
(r) (42)
is equal to( ). We already know from (26) that0()has this value. Consequently, we
shall demonstrate that, if:
() =( ) (43)
then the same is true for+1(). Relation (41) permits us to write+1()in the form:
+1() =
1
~
2
2+ 3
2+ 2
d
3
+
(0)
(r) +
(0)
(r)
=
1
~
2
2+ 3
2+ 2
d
3 (0)
(r) +
(0)
(r) (44)
where = is the adjoint of+. Now:
+=
2
+
2
=P
2 2
(45)
We know that
(0)
is an eigenfunction ofP
2
. Since, in addition,is Hermitian, it results
that:
+1() =
1
~
2
2+ 3
2+ 2
~
22
() d
3 (0)
(r)
(0)
(r) (46)
We must now calculate
(0)
(r). Using the fact that()is proportional to(+),
we easily nd:
(0)
(r) =
~
2
2
cos ()+1()
=
~
1
2+ 3
(0)
+1
(r) (47)
according to formula (35) of complementVI. Putting this result into (46
+1() =
2+ 3
2+ 2
()
1
2+ 2
+1() (48)
Hypothesis (43) thus implies:
+1() =( ) (49)
which concludes the argument by recurrence.
965

COMPLEMENT A VIII
2-c. Properties
. Behavior at the origin
Whenapproaches zero, the function()behaves (see below) like:
()
0
(2+ 1)!!
(50)
Consequently,
(0)
(r)is proportional toin the neighborhood of the origin:
(0)
(r)
0
2
2
()
()
(2+ 1)!!
(51)
To demonstrate formula (50), starting from denition (35), it is sucient to expandsinin a
power series in:
sin
=
=0
(1)
2
(2+ 1)!
(52)
We then apply the operator
1
d
d
, which yields:
() = (1)
1
d
d
1
=0
(1)
2
(2+ 1)!
211
= (1)
=0
(1)
2(22)(24)[22(1)]
(2+ 1)!
22
(53)
The rstterms of the sum (= 0to1) are zero, and the(+ 1)th is written:
()
0
(1)(1)
2(22)(24)2
(2+ 1)!
(54)
which proves (50).
. Asymptotic behavior
When their argument approaches innity, the spherical Bessel functions are related
to the trigonometric functions in the following way:
()
1
sin
2
(55)
The asymptotic behavior of the free spherical waves is therefore:
(0)
(r)
2
2
()
sin( 2)
(56)
If we apply the operator
1
d
d
once to
sin
, we can write()in the form:
() = (1)
1
d
d
1
cos
2
sin
3
(57)
966

THE FREE PARTICLE: STATIONARY STATES
WITH WELL-DEFINED ANGULAR MOMENTUM
The second term inside the brackets is negligible compared to the rst term whenapproaches innity.
Moreover, when we apply
1
d
d
a second time, the dominant term still comes from the derivative of the
cosine. Thus we see that:
() (1)
1
1d
d
sin (58)
Since:
d
d
sin= (1)sin
2
(59)
the result is indeed (55).
Comment:
If we set:
= (60)
[being dened by formula (10)], the radial equation (4) becomes:
d
2
d
2
+
2d
d
+1
(+ 1)
2
() = 0 (61)
This is the spherical Bessel equation of order. It has two linearly independent
solutions, which can be distinguished, for example, by their behavior at the origin.
One of them is the spherical Bessel function(), which satises (50) and (55). For
the other, we can choose the spherical Neumann function of order, designated
as(), with the properties:
()
0
(21)!!
+1
(62a)
()
1
cos
2
(62b)
3. Relation between free spherical waves and plane waves
We already know two distinct bases of eigenstates of0: the plane waves
(0)
k
(r)are
eigenfunctions of the three components of the momentumP; the free spherical waves
(0)
(r)are eigenfunctions ofL
2
and. These two bases are dierent becausePdoes
not commute withL
2
and.
A given function of one of these bases can obviously be expanded in terms of the
other basis. For example, we shall express a plane wave
(0)
k
(r)as a linear superposition
of free spherical waves. Consider, therefore, a vectorkin ordinary space. The plane
wave
(0)
k
(r)that it characterizes is an eigenfunction of0with the eigenvalue~
2
k
2
2.
Therefore, its expansion will include only the
(0)
which correspond to this energy,
that is those for which:
=k (63)
967

COMPLEMENT A VIII
This expansion will therefore be of the form:
(0)
k
(r) =
=0
+
=
(k)
(0)
(r) (64)
the free indiceskandbeing related by equation (63). It is easy to show, using the
properties of the spherical harmonics (cf.complementVI) and the spherical Bessel
functions, that:
e
kr
= 4
=0
+
=
( )()() (65)
whereandare the polar angles that x the direction of the vectork. Ifkis directed
along, expansion (65) reduces to:
e=
=0
4(2+ 1)()
0
()
=
=0
(2+ 1)()(cos) (66)
whereis the Legendre polynomial of degree[cf.equation (57) of complementVI].
Let us rst demonstrate relation (66). To do this, let us assume that the vectorkchosen is
collinear with:
= = 0 (67)
and points in the same direction. In this case, equation (63) becomes:
= (68)
and we want to expand the function:
e= e
cos
(69)
in the
(0)
(r)basis. Since this function is independent of the angle, it is a linear combi-
nation of only those basis functions for which= 0:
e
cos
=
=0
(0)
0
(r)
=
=0
()
0
() (70)
To calculate the numbers, we can considere
cos
to be a function of the variable, with
playing the role of a parameter. Since the spherical harmonics form an orthonormal basis for
functions ofand, the coecient()can be expressed as:
() =d
0
() e
cos
(71)
Replacing
0
by its expression in terms of()[formula (25) of complementVI], we obtain:
() =
1
(2)!
d
~
()e
cos
=
1
(2)!
d ()
+
~
e
cos
(72)
968

THE FREE PARTICLE: STATIONARY STATES
WITH WELL-DEFINED ANGULAR MOMENTUM
since+is the adjoint operator of. Formula (16) of complementVIthen yields:
+
~
e
cos
= (1)e(sin)
d
d(cos)
e
cos
= (1)e(sin)()e
cos
(73)
Now(sin)eis just, to within a constant factor,()[cf.formulas (4) and (14) of
complementVI]. Consequently:
() = ()
2!
(2)!4(2+ 1)!
()
2
e
cos
(74)
It is therefore sucient to choose a particular value of, for which we know the value(),
in order to calculate. Allow, for example,to approach zero: we know that()behaves
like(), and so, in fact, does the right-hand side of equation (74). More precisely, using (50),
we nd:
1
(2+ 1)!!
=
2!
(2)!4(2+ 1)!
d ()
2
(75)
that is, sinceis normalized to 1:
=
4(2+ 1) (76)
This proves formula (66).
The general relation (65) can therefore be obtained as a consequence of the addition theorem for
spherical harmonics [formula (70) of complementVI]. Whatever the direction ofk(dened by
the polar anglesand), it is always possible, through a rotation of the system of axes, to
return to the case we have just considered. Consequently, expansion (66) remains valid, provided
is replaced bykrandcosbycos, whereis the angle betweenkandr:
e
kr
=
=0
(2+ 1)()(cos) (77)
The addition theorem for spherical harmonics permits the expression of(cos)in terms of the
angles()and( ), which yields nally formula (65).
Expansions (65) and (66) show thata state of well-dened linear momentum in-
volves all possible orbital angular momenta.
To obtain the expansion of a given function
(0)
(r)in terms of plane waves,
it is sucient to invert formula (65), using the orthonormalization relation of spherical
harmonics which are functions ofand. This yields:
d ( )e
kr
= 4 ()() (78)
Thus we nd:
(0)
(r) =
(1)
42
2
d ( ) e
kr
(79)
An eigenfunction ofL
2
andis therefore a linear superposition of all plane waves
with the same energy:a state of well-dened angular momentum involves all possible
directions of the linear momentum.
References:
Messiah (1.17), App. B, 6; Arfken (10.4), 11.6; Butkov (10.8), Chap. 9, 9;
see the subsection Special functions and tables of section 10 of the bibliography.
969

PHENOMENOLOGICAL DESCRIPTION OF COLLISIONS WITH ABSORPTION
Complement BVIII
Phenomenological description of collisions with absorption
1 Principle involved
2 Calculation of the cross sections
2-a Elastic scattering cross section
2-b Absorption cross section
2-c Total cross section. Optical theorem
In Chapter , we conned ourselves to the study of the elastic
1
scattering of
particles by a potential. But we pointed out in the introduction of that chapter that col-
lisions between particles can be inelastic and lead, under certain conditions, to numerous
other reactions (creation or destruction of particles, etc...), particularly if the energy of
the incident particles is high. When such reactions are possible, and one detects only
elastically scattered particles, one observes that certain particles of the incident beam
disappear; that is, they are not to be found either in the transmitted beam or amongst
the elastically scattered particles. These particles are said to be absorbed during the
interaction; in reality, they have taken part in reactions other than that of simple elastic
scattering. If one is interested only in the elastic scattering, one seeks to describe the
absorption globally, without going into detail about the other possible reactions. We
are going to show here that the method of partial waves provides a convenient framework
for such a phenomenological description.
1. Principle involved
We shall assume that the interactions responsible for the disappearance of the incident
particles are invariant with respect to rotation aboutThe scattering amplitude can
therefore always be decomposed into partial waves, each of which corresponds to a xed
value of the angular momentum.
In this section, we shall see how the method of partial waves can be modied to take
a possible absorption into consideration. To do this, let us return to the interpretation of
partial waves that we gave in -of Chapter . A free incoming wave penetrates
the zone of inuence of the potential and gives rise to an outgoing wave. The eect of
the potential is to multiply this outgoing wave bye
2
. Since the modulus of this factor
is 1 (the phase shiftis real), the amplitude of the outgoing wave is equal to that of the
incoming wave. Consequently (see the calculation of
incoming wave is equal to that of the outgoing wave: during the scattering, probability is
conserved, that is, the total number of particles is constant. These considerations suggest
that, in the cases where absorption phenomena occur, one can take them into account
simply by giving the phase shift an imaginary part such that:
e
2
1 (1)
1
A collision is called elastic if it changes neither the nature nor the internal state of the concerned
particles; otherwise it is called inelastic.
971

COMPLEMENT B VIII
The amplitude of the outgoing wave with angular momentumis thus smaller than that
of the incoming wave from which it arises. The fact that the outgoing probability ux
is smaller than the incoming ux expresses the disappearance of a certain number of
particles.
We are going to make this idea more explicit and deduce from it the expressions
for the scattering and absorption cross sections. However, we stress the fact that this
is a purely phenomenological method: the parameters with which we shall characterize
the absorption (modulus ofe
2
for each partial wave) mask an often very complicated
reality. Note also that if the total probability is no longer conserved it is impossible
to describe the interaction by a simple potential. A correct treatment of the set of
phenomena which can then arise during the collision would demand a more elaborate
formalism than the one developed in Chapter .
2. Calculation of the cross sections
We return to the calculations of , setting:
= e
2
(2)
Since the possibility of producing reactions other than that of elastic scattering is always
expressed by a decrease in the number of elastically scattered particles, we must have:
61 (3)
(equality corresponding to cases where only elastic scattering is possible). The asymp-
totic form of the wave function which describes the elastic scattering is therefore [cf.
formula (C-51) of Chapter ]:
(scatt)
(r)
=0
4(2+ 1)
0
()
ee
2 ee2
2
(4)
2-a. Elastic scattering cross section
The argument of
amplitude()in the form:
() =
1
=0
4(2+ 1)
0
()
1
2
(5)
From this we deduce the dierential elastic scattering cross section:
el() =
1
2
=0
4(2+ 1)
0
()
1
2
2
(6)
and the total elastic scattering cross section:
el=
2
=0
(2+ 1)1
2
(7)
972

PHENOMENOLOGICAL DESCRIPTION OF COLLISIONS WITH ABSORPTION
Comment:
According to the argument developed in 1, the absorption of the wave()reaches
a maximum when is zero, that is, when:
= 0 (8)
Formula (7) indicates however that, even in this limiting case, the contribution of
the wave()to the elastic scattering cross section is not zero
2
. In other words,
even if the interaction region is perfectly absorbing, it produces elastic scattering.
This important phenomenon is a purely quantum eect. It can be compared to
the behavior of a light wave which strikes an absorbing medium. Even if the
absorption is total (perfectly black sphere or disc), a diracted wave is observed
(concentrated into a solid angle which becomes smaller as the surface of the disc
becomes larger). Elastic scattering produced by a totally absorbing interaction is
called, for this reason,shadow scattering.
2-b. Absorption cross section
Following the same principle as in , we dene the absorption
cross sectionabs: it is the ratio between the number of particles absorbed per unit time
and the incident ux.
To calculate this cross section, it is sucient, as in , to evaluate
the total amount of probabilitywhich disappears per unit time. This probability
can be obtained from the currentJassociated with the wave function (4).is equal
to the dierence between the ux of the incoming waves across a sphere()of very large
radius0and that of the outgoing waves; it is therefore equal to minus the net ux of
the vectorJleaving this sphere. Thus:
=
()
JdS (9)
with:
J= Re
(scatt)*
(r)
~
(scatt)
(r) (10)
Only the radial componentof the current contributes to the integral (9):
=
=0
2
d (11)
with:
= Re
(scatt)*
(r)
~
(scatt)
(r) (12)
In formula (12), the derivative does not modify the angular dependence of the
various terms which compose
(scatt)
(r)[formula (4)]. Consequently, because of the
2
This contribution is zero only if= 1, that is, if the phase shift is real and equal to an integral
multiple of[this was already contained in formula (C-58) of Chapter ].
973

COMPLEMENT B VIII
orthogonality of the spherical harmonics, the cross terms between a partial wave()in
(scatt)
(r)and a dierent wave()in
(scatt)*
(r)make a zero contribution to integral (11).
We have therefore:
=
=0
=0
()2
d (13)
where
()
is the radial component of the current associated with the partial wave(). A
simple calculation gives:
()
~
(2+ 1)
22
1
2 0
()
2
(14)
that is, nally, since
0
()is normalized:
=
~
2
=0
(2+ 1)1
2
(15)
The absorption cross sectionabsis therefore equal to the probabilitydivided
by the incident current~:
abs=
2
=0
(2+ 1)1
2
(16)
It is obvious thatabsis zero if all thehave a modulus of 1; that is, according to (2), if
all the phase shifts are real. In this case, there is only elastic scattering, and the net ux
of probability leaving a sphere of large radius0is always zero. The total probability
carried by the incoming waves is entirely transferred to the outgoing waves. On the other
hand, whenis zero, the contribution of the wave()to the absorption cross section is
maximum.
Comment:
The calculation of expression (15) shows that
~
2
(2+ 1)is the amount of probability entering
per unit time, and arising from the partial wave(). If we divide this quantity by the incident
current~, we obtain a surface that can be called the incoming cross section into the partial
wave():
=
2
(2+ 1) (17)
This formula can be interpreted classically. We can consider the incident plane wave as describing
a beam of particles of uniform density, having a momentum~parallel to. What proportion of
these particles reach the scattering potential, with an angular momentum~
(+ 1)? We have
already mentioned the link between angular momentum and the impact parameter in classical
mechanics [cf.formula (C-23) of Chapter ]:
=p=~ (18)
All we must do, therefore, is to draw, in the plane passing throughand perpendicular to,
a circular ring centered at, of average radiussuch that:
~
(+ 1) =~ (19)
974

PHENOMENOLOGICAL DESCRIPTION OF COLLISIONS WITH ABSORPTION
and of widthcorresponding to= 1in formula (19) (Fig.). All the particles crossing
this surface reach the scattering potential with an angular momentum equal to~
(+ 1), to
within~. From (19) we derive:
=
1
(+ 1)
1+
1
2
(20)
if1, and consequently:
=
1
(21)
The area of the circular ring of Figure
2
2
(2+ 1) (22)
Thus we nd again, very simply,.O
b
l
Δb
l
Figure 1: The incident particles must reach the potential with the impact parameterto
withinfor their classical angular momentum to be~
(+ 1)to within~.
2-c. Total cross section. Optical theorem
When a collision can give rise to several dierent reactions or scattering phenom-
ena, the total cross sectiontotis dened as the sum of the cross sections (integrated over
all the directions of space) corresponding to all these processes. The total cross section
is thus the number of particles which, per unit time, participate in one or another of the
possible reactions, divided by the incident ux.
If, as above, we treat globally all reactions other than elastic scattering, we have
simply:
tot=el+abs (23)
975

COMPLEMENT B VIII
Formulas (7) and (16) then give:
tot=
2
2
=0
(2+ 1)(1Re) (24)
Now(1Re)is the real part of(1), which appears in the elastic scattering
amplitude [formula (5)]. Moreover, we know the value of
0
()for= 0:
0
(0) =
2+ 14
(25)
[cf.complementVI, formulas (57) and (60)]. Consequently, if we calculate from (5) the
imaginary part of the elastic scattering amplitude in the forward direction, we nd:
Im(0) =
1
=0
(2+ 1)
1Re
2
(26)
Comparing this expression to formula (24), we see that:
tot=
4
Im(0) (27)
This relation between the total cross section and the imaginary part of the elastic scat-
tering amplitude in the forward direction is valid in a very general sense; it constitutes
what is called theoptical theorem.
Comment:
The optical theorem is obviously valid in the case of purely elastic scattering
(abs= 0;tot=el). The fact that(0) i.e. the wave scattered in the forward
direction is related to the total cross section could have been predicted from
the discussion in . It is the interference in the forward
direction between the incident plane wave and the scattered wave that accounts
for the attenuation of the transmitted beam, due to the scattering of particles in
all directions of space.
References and suggestions for further reading:
Optical model: Valentin (16.1), X-3. High energy proton-proton collisions: Amaldi
(16.31).
976

SOME SIMPLE APPLICATIONS OF SCATTERING THEORY
Complement CVIII
Some simple applications of scattering theory
1 The Born approximation for a Yukawa potential
1-a Calculation of the scattering amplitude and cross section
1-b The innite-range limit
2 Low energy scattering by a hard sphere
3 Exercises
3-a Scattering of thepwave by a hard sphere
3-b Square spherical well: bound states and scattering resonances
There is no potential for which the scattering problem can be solved exactly
1
by a
simple analytical calculation. Therefore, in the examples that we are going to discuss, we
shall content ourselves with using the approximations that we introduced in Chapter .
1. The Born approximation for a Yukawa potential
Let us consider a potential of the form:
(r) =0
e
(1)
where0andare real constants, withpositive. This potential is attractive or
repulsive depending on whether0is negative or positive. The larger0, the more
intense the potential. Its range is characterized by the distance:
0=
1
(2)
since, as Figure ()is practically zero whenexceeds20or30.
The potential (1) bears the name of Yukawa, who had the idea of associating it
with nuclear forces, whose range is of the order of a fermi. To explain the origin of this
potential, Yukawa was led to predict the existence of the-meson, which was indeed
later discovered. Notice that for= 0this potential becomes the Coulomb potential,
which can thus be considered to be a Yukawa potential of innite range.
1-a. Calculation of the scattering amplitude and cross section
We assume that 0is suciently small for the Born approximation (
Chapter ) to be valid. According to formula (B-47) of Chapter , the scattering
amplitude
()
()is then given by:
()
() =
1
4
20
~
2
d
3
e
Kr
e
(3)
1
Actually, we can rigorously treat the case of the Coulomb potential; however, as we pointed out in
Chapter ), this necessitates a special method.
977

COMPLEMENT C VIIIV(r)
V
0
V
0
1
0
r
0
=
r
r
r
e
– ar
α
Figure 1: Yukawa potential and Coulomb potential. The presence of the termecauses
the Yukawa potential to approach zero much more rapidly when0= 1(range of
the potential).
whereKis the momentum transferred in the direction()dened by relation (B-42)
of Chapter .
Expression (3) involves the Fourier transform of the Yukawa potential. Since this
potential depends only on the variable, the angular integrations can easily be carried
out ( ), putting the scattering amplitude into the form:
()
() =
1
4
20
~
2
4
K
0
dsinK
e
(4)
After a simple calculation, we then nd:
()
() =
20
~
2
1
2
+K
2
(5)
Figure
K= 2sin
2
(6)
whereis the modulus of the incident wave vector andis the scattering angle.
The dierential scattering cross section is therefore written, in the Born approxi-
mation:
()
() =
4
22
0
~
4
1
2
+ 4
2
sin
2
2
2
(7)
978

SOME SIMPLE APPLICATIONS OF SCATTERING THEORY
It is independent of the azimuthal angle, as could have been foreseen from the fact that
the problem of scattering by a central potential is symmetrical with respect to rotation
about the direction of the incident beam. On the other hand, it depends, for a given
energy (that is, for xed), on the scattering angle: in particular, the cross section in
the forward direction(= 0)is larger than the cross section in the backward direction
(=). Finally,
()
(), for xed, is a decreasing function of the energy. Notice,
moreover, that the sign of0is of no importance in the scattering problem, at least in
the Born approximation.
The total scattering cross section is easily obtained by integration:
()
=d
()
() =
4
22
0
~
4
4
2
(
2
+ 4
2
)
(8)
1-b. The innite-range limit
We noted above that the Yukawa potential approaches a Coulomb potential when
tends towards zero. What happens, in this limiting case, to the formulas that we have
just established?
To obtain the Coulomb interaction potential between two particles having charges
of1and2(being the charge of the electron), we write:
= 0
0=12
2
(9)
with:
2
=
2
40
(10)
Formula (7) then gives:
()
() =
4
2
~
4
2
1
2
2
4
16
4
sin
42
=
2
1
2
2
4
16
2
sin
42
(11)
(has been replaced by its value in terms of the energy).
Expression (11) is indeed that of the Coulomb scattering cross section (Ruther-
ford's formula). Of course, the way in which we have obtained it does not constitute a
proof: the theory we have used is not applicable to the Coulomb potential. However,
it is interesting to observe that the Born approximation for the Yukawa potential gives
precisely Rutherford's formula for the limiting situation where the range of the potential
approaches innity.
Comment:
The total scattering cross section for a Coulomb potential is innite since the
corresponding integral diverges for small values of[expression (8) becomes innite
979

COMPLEMENT C VIII
whenapproaches zero]. This results from the innite range of the Coulomb
potential: even if the particle passes very far from the point, it is aected by
the potential. This suggests why the scattering cross section should be innite.
However, in reality, one never observes a rigorously pure Coulomb interaction over
an innite range. The potential created by a charged particle is always modied
by the presence, in its more or less immediate neighborhood, of other particles of
opposite charge (screening eect).
2. Low energy scattering by a hard sphere
Let us consider a central potential such that:
() = 0for 0
= for 0 (12)
In this case, we say that we are considering a hard sphere of radius0. We assume
that the energy of the incident particle is suciently small for0to be much smaller
than 1. We can then ( of chapter
phase shifts except that of thewave(= 0). The scattering amplitude()is written,
under these conditions:
() =
1
e
0()
sin0() (13)
(since
0
0= 1
4). The dierential cross section is isotropic:
() =()
2
=
1
2
sin
2
0() (14)
so that the total cross section is simply equal to:
=
4
2
sin
2
0() (15)
To calculate the phase shift0(), it is necessary to solve the radial equation
corresponding to= 0. This equation is written [cf.formula (C-35) of Chapter ]:
d
2
d
2
+
2
0() = 0for 0 (16)
which must be completed by the condition:
0(0) = 0 (17)
since the potential becomes innite for=0. The solution0()of equations (16)
and (17) is unique to within a constant factor:
0() =sin( 0)for 0
= 0 for 0 (18)
The phase shift0is, by denition, given by the asymptotic form of0():
0() sin(+0) (19)
980

SOME SIMPLE APPLICATIONS OF SCATTERING THEORY
Thus, using solution (18), we nd:
0() = 0 (20)
If we insert this value into expression (15) for the total cross section, we obtain:
=
4
2
sin
2
04
2
0 (21)
since by hypothesis0is much smaller than 1. Therefore,is independent of the energy
and equal to four times the apparent surface of the hard sphere seen by the particles of
the incident beam. A calculation based on classical mechanics would give for the cross
section the apparent surface
2
0: only the particles which bounce elastically o the hard
sphere would be deected. In quantum mechanics, however, one studies the evolution
of the wave associated with the incident particles, and the abrupt variation of()at
=0produces a phenomenon analogous to the diraction of a light wave.
Comment:
Even when the wavelength of the incident particles becomes negligible compared
to0(01), the quantum cross section does not approach
2
0. It is possible,
for very large, to sum the series which gives the total cross section in terms of
phase shifts [formula (C-58) of Chapter ]; we then nd:
2
2
0 (22)
Wave eects thus persist in the limiting case of very small wavelengths. This is
due to the fact that the potential under study is discontinuous at=0: it always
varies appreciably within an interval which is smaller than the wavelength of the
particles (cf.Chapter, ).
3. Exercises
3-a. Scattering of thepwave by a hard sphere
We wish to study the phase shift1()produced by a hard sphere on thewave
(= 1). In particular, we want to verify that it becomes negligible compared to0()at
low energy.
. 1()for 0. Show that its general
solution is of the form:
1() =
sin
cos+
cos
+ sin
whereandare constants.
. 1()implies that:
= tan1()
981

COMPLEMENT C VIII
. from the condition imposed on1()at=0.
. approaches zero,1()behaves like
2
(0)
3
, which makes it negli-
gible compared to0().
3-b. Square spherical well: bound states and scattering resonances
Consider a central potential()such that:
() = 0for 0
= 0 for 0
where0is a positive constant. Set:
0=
20~
2
We shall conne ourselves to the study of thewave(= 0).
. Bound states ( 0)
(i) Write the radial equation in the two regions0and 0, as well as the
condition at the origin. Show that, if one sets:
=
2~
2
=
2
0
2
the function0()is necessarily of the form:
0() =e for 0
=sin for 0
(ii) Write the matching conditions at=0. Deduce from them that the only
possible values forare those which satisfy the equation:
tan 0=
(iii) Discuss this equation: indicate the number ofbound states as a function
of the depth of the well (for xed0) and show, in particular, that there are no bound
states if this depth is too small.
2
This result is true in general: for any potential of nite range0, the phase shift()behaves like
(0)
2+1
at low energies.
982

SOME SIMPLE APPLICATIONS OF SCATTERING THEORY
. Scattering resonances (0)
()Again write the radial equation, this time setting:
=
2~
2
=
2
0
+
2
Show that 0()is of the form:
0() =sin(+0)for 0
=sin for 0
()Choose= 1. Show, using the continuity conditions at=0, that the
constantand the phase shift0are given by:
2
=
2
2
+
2
0
cos
2
0
0= 0+()
with:
tan() =
tan 0
()Trace the curve representing
2
as a function of. This curve clearly shows
resonances, for which
2
is maximum. What are the values ofassociated with these
resonances? What is then the value of()? Show that, if there exists such a resonance
for a small energy (01), the corresponding contribution of thewave to the total
cross section is practically maximal.
. Relation between bound states and scattering resonances
Assume that00is very close to(2+ 1)
2
, whereis an integer, and set:
00= (2+ 1)
2
+with 1
()Show that, ifis positive, there exists a bound state whose binding energy
=~
22
2is given by:
0
()Show that if, on the other hand,is negative, there exists a scattering reso-
nance at energy=~
22
2such that:
2
20
0
()Deduce from this that if the depth of the well is gradually decreased (for xed
0), the bound state which disappears when00passes through an odd multiple of2
gives rise to a low energy scattering resonance.
References and suggestions for further reading:
Messiah (1.17), Chap. IX, 10 and Chap. X, III and IV; Valentin (16.1), Annexe II.
983

Chapter IX
Electron spin
A Introduction of electron spin
A-1 Experimental evidence
A-2 Quantum description: postulates of the Pauli theory
B Special properties of an angular momentum 1/2
C Non-relativistic description of a spin 1/2 particle
C-1 Observables and state vectors
C-2 Probability calculations for a physical measurement
Until now, we have considered the electron to be a point particle possessing three
degrees of freedom associated with its three coordinates,and. Consequently, the
quantum theory that we have developed is based on the hypothesis that an electron
state, at a given time, is characterized by a wave function( )which depends only
on,andWithin this framework, we have studied a certain number of physical
systems: amongst others, the hydrogen atom (in Chapter), which is particularly
interesting because of the very precise experiments that can be performed on it. The
results obtained in Chapter
of hydrogen very accurately. They give the energy levels correctly and make it possible
to explain, using the corresponding wave functions, the selection rules (which indicate
which frequencies, out of all the Bohr frequencies that area prioripossible, appear in
the spectrum). Atoms with many electrons can be treated in an analogous fashion (by
using approximations, however, since the complexity of the Schrödinger equation, even
for the helium atom with two electrons, makes an exact analytic solution of the problem
impossible). In this case as well, agreement between theory and experiment is satisfying.
However, when atomic spectra are studied in detail, certain phenomena appear,
as we shall see, which cannot be interpreted within the framework of the theory that
we have developed. This result is not surprising. It is clear that it is necessary to
complete the preceding theory by a certain number ofrelativistic corrections: one must
take into account the modications brought in byrelativistic kinemalics(variation of
Quantum Mechanics, Volume II, Second Edition. C. Cohen-Tannoudji, B. Diu, and F. Laloë.
© 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

CHAPTER IX ELECTRON SPIN
mass with velocity, etc.) andmagnetic eectswhich we have neglected. We know that
these corrections are small ( ): nevertheless, they do exist, and can
be measured.
TheDirac equationgives a relativistic quantum mechanical description of the elec-
tron. Compared to the Schrödinger equation, it implies a profound modication in the
quantum description of the properties of the electron; in addition to the corrections al-
ready pointed out concerning its position variables, a new characteristic of the electron
appears: itsspin.In a more general context, the structure of the Lorentz group (group of
relativistic space-time transformations) reveals spin to be an intrinsic property of various
particles, on the same footing, for example, as their rest mass
1
.
Historically, electron spin was discovered experimentally before the introduction
of the Dirac equation. Furthermore, Pauli developed a theory which allowed spin to
be incorporated simply into non-relativistic quantum mechanics
2
through the addition
of several supplementary postulates. Theoretical predictions for the atomic spectra are
then obtained which are in excellent agreement with experimental results
3
.
It is Pauli's theory, which is much simpler than Dirac's, that we are going to
develop in this chapter. We shall begin, in , by describing a certain number of
experimental results, which revealed the existence of electron spin. Then we shall specify
the postulates on which Pauli's theory is based. Afterwards, we shall examine, in ,
the special properties of an angular momentum 1/2. Finally, we shall show, in , how
one can take into account simultaneously the position variables and the spin of a particle
such as the electron.
A. Introduction of electron spin
A-1. Experimental evidence
Experimental demonstrations of the existence of electron spin are numerous and
appear in various important physical phenomena. For example, the magnetic properties
of numerous substances, particularly of ferromagnetic metals, can only be explained
if spin is taken into account. Here, however, we are going to conne ourselves to a
certain number of simple phenomena observed experimentally in atomic physics: the ne
structure of spectral lines, the Zeeman eect and, nally, the behavior of silver atoms in
the Stern-Gerlach experiment.
A-1-a. Fine structure of spectral lines
The precise experimental study of atomic spectral lines (for the hydrogen atom,
for example) reveals ane structure: each line is in fact made up of several components
having nearly identical frequencies
4
but which can be clearly distinguished by a device
1
This does not mean that spin has a purely relativistic origin: it can be deduced from the structure
of the non-relativistic transformation group (the Galilean group).
2
Pauli's theory can be obtained as a limiting case of Dirac's theory when the electron's speed is small
compared to that of light.
3
We shall see, for example in Chapter
ter
for the details of the hydrogen atomic spectrum (which would be inexplicable if we limited ourselves to
the theory of Chapter).
4
For example, the resonance line of the hydrogen atom (2 1transition) is actually double: the
two components are separated by an interval of the order of10
4
eV (that is, about 10
5
times smaller
986

A. INTRODUCTION OF ELECTRON SPIN
with good resolution. This means that there exist groups of atomic levels which are very
closely spaced but distinct. In particular, the calculations of
average energies of dierent groups of levels for the hydrogen atom but do not explain
the splittings within each group.
A-1-b. Anomalous Zeeman eect
When an atom is placed in a uniform magnetic eld, each of its lines (that is, each
component of the ne structure) splits into a certain number of equidistant lines, the
interval being proportional to the magnetic eld: this is theZeeman eect. The origin of
the Zeeman eect can be easily understood by using the results of Chapters
(complementVII). The theoretical explanation is based on the fact that a magnetic
momentMis associated with the orbital angular momentumLof an electron:
M=
~
L (A-1)
where is the Bohr magneton:
=
~
2
(A-2)
However, while this theory is conrmed by experiment in certain cases (the so-called
normal Zeeman eect), it is, in other cases, incapable of accounting quantitatively for
the observed phenomena (the so-called anomalous Zeeman eect). The most striking
anomaly appears for atoms with odd atomic number(in particular, for the hydrogen
atom): their levels are divided into aneven number of Zeeman sub-levels,while, according
to the theory, this number should always be odd, being equal to(2+1)withan integer.
A-1-c. Existence of half-integral angular momenta
We are confronted with the same diculty in connection with the Stern-Gerlach
experiment, which we described in Chapter); the beam of silver atoms is split
symmetrically in two. These results suggest thathalf-integral values of j(which we saw
in a prioripossible) do indeed exist. But this poses a serious
problem, since we showed in
of a particle such as an electron could only be integral (more precisely, it is the quan-
tum numberwhich is integral). Even in atoms with several electrons, each of these
has an integral orbital angular momentum, and we shall show in Chapter
these conditions, the total orbital angular momentum of the atom is necessarily inte-
gral. The existence of half-integral angular momenta thus cannot be explained without
supplementary hypotheses.
Comment:
It is not possible to measure directly the angular momentum of the electron using
the Stern-Gerlach apparatus. Unlike silver atoms, electrons possess an electric
chargeand the force due to the interaction between their magnetic moment and
the inhomogeneous magnetic eld would be completely masked by the Lorentz
forcevB.
than the average2 1transition energy, which is equal to 10.2 eV).
987

CHAPTER IX ELECTRON SPIN
A-2. Quantum description: postulates of the Pauli theory
indexSpin!quantum description
In order to resolve the preceding diculties, Uhlenbeck and Goudsmit (1925) pro-
posed the following hypothesis: the electron spins and this gives it an intrinsic angular
momentum which is called the spin. To interpret the experimental results described
above, one must also assume that a magnetic momentMis associated
5
with this an-
gular momentumS:
M= 2
~
S (A-3)
Note that the coecient of proportionality between the angular momentum and the mag-
netic moment is twice as large in (A-3) as in (A-1): one says thatthe spin gyromagnetic
ratio is twice the orbital gyromagnetic ratio.
Pauli later stated this hypothesis more precisely and gave a quantum description
of spin which is valid in the non-relativistic limit. To the general postulates of quantum
mechanics that we set forth in Chapter
relating to spin.
Until now, we have studied the quantization oforbital variables.With the position
rand the momentumpof a particle such as the electron, we associated the observables
RandPacting in the state spacer, which is isomorphic to the spaceof wave
functions. All physical quantities are functions of the fundamental variablesrandp,
and the quantization rules enable us to associate with them observables acting inr. We
shall callrtheorbital state space.
To these orbital variables we shall addspin variableswhich satisfy the following
postulates:
(i)The spin operatorSis an angular momentum.This means ( )
that its three components are observables which satisfy the commutation relations:
[ ] =~ (A-4)
and the two formulas which are deduced by cyclic permutation of the indices,,
.
(ii) spin state space, whereS
2
and
constitute a C.S.C.O.The spaceis thus spanned by the set of eigenstates
common toS
2
and:
S
2
=(+ 1)~
2
(A-5a)
=~ (A-5b)
According to the general theory of angular momentum ( ), we
know thatmust be an integral or half-integral number, and thattakes on all
5
Actually, when one takes into account the coupling of the electron with the quantized electromag-
netic eld (quantum electrodynamics), one nds that the coecient of proportionality betweenMand
Sis not exactly2~. The dierence, which is of the order of 10
3
in relative value, is easily observable
experimentally; it is often called the anomalous magnetic moment of the electron.
988

A. INTRODUCTION OF ELECTRON SPIN
values included betweenand+diering from these two numbers by an integer
(which may be zero). A given particle is characterized bya unique value of s: this
particle is said to have a spin. The spin state spaceis therefore always of
nite dimension(2+ 1), and all spin states are eigenvectors ofS
2
with the same
eigenvalue(+ 1)~
2
.
(iii)The state spaceof the particle being consideredis the tensor product ofrand
:
=r (A-6)
Consequently ( ),all spin observables commute with all orbital
observables.
Except for the particular case where= 0, it is therefore not sucient to specify a
ket ofr(that is, a square-integrable wave function) to characterize a state of the
particle. In other words, the observables,anddo not constitute a C.S.C.O.
in the space stateof the particle (no more than do,,or any other
C.S.C.O. ofr). It is also necessary to know the spin state of the particle, that is,
to add to the C.S.C.O. ofra C.S.C.O. ofcomposed of spin observables, for
exampleS
2
and(orS
2
and). Every particle state is a linear combination of
vectors which are tensor products of a ket ofrand a ket of(see
(iv)The electron is a spin 1/2 particle(= 12) and itsintrinsic magnetic momentis
given by formula (A-3). For the electron, the spaceis therefore two-dimensional.
Comments:
(i)
particles, but their gyromagnetic ratios are dierent from that of the electron.
At the present time, we know of the existence of particles of spin 0, 1/2, 1,
3/2, 2, etc.
(ii)
like the electron, instead of being a point, has a certain spatial extension. It
would then be the rotation of the electron about its axis that would give rise
to an intrinsic angular momentum. However, it is important to note that,
in order to describe a structure that is more complex than a material point,
it would be necessary to introduce more than three position variables. If,
for example, the electron behaved like a solid body, six variables would be
required: three coordinates to locate one of its points chosen once and for
all, such as its center of gravity, and three angles to specify its orientation
in space. The theory that we are considering here is radically dierent. It
continues to treat the electron like a point (its position is xed by three
coordinates). The spin angular momentum is not derived from any position
or momentum variable
6
.Spin thus has no classical analogue.
6
If it were, moreover, it would necessarily be integral.
989

CHAPTER IX ELECTRON SPIN
B. Special properties of an angular momentum 1/2
We shall restrict ourselves from now on to the case of the electron, which is a spin 1/2
particle. From the preceding chapters, we know how to handle its orbital variables. We
are now going to study in more detail its spin degrees of freedom.
The spin state spaceis two-dimensional. We shall take as a basis the orthonor-
mal system+ of eigenkets common toS
2
andwhich satisfy the equations:
S
2
=
3
4
~
2
=
1
2
~
(B-1a)
(B-1b)
+ = 0
++= = 1
(B-2a)
(B-2b)
++ + = (B-3)
whereis the unit operator. The most general spin state is described by an arbitrary
vector of:
=+++ (B-4)
where+and are complex numbers. According to (B-1a), all the kets ofare
eigenvectors ofS
2
with the same eigenvalue3~
2
4, which causesS
2
to be proportional
to the identity operator of:
S
2
=
3
4
~
2
(B-5)
(in the right hand side of this equation, as is usually done, we have not written the unit
operatorexplicitly). SinceSis, by denition, an angular momentum, it possesses all
the general properties derived in . The action of the operators:
= (B-6)
on the basis vectors+and is given by the general formulas (C-50) of Chapter
when one sets== 12:
++= 0 + =~+ (B-7a)
+=~ = 0 (B-7b)
Any operator acting incan be represented, in the+ basis, by a22matrix.
In particular, using (B-1b) and (B-7), we nd the matrices corresponding to,and
in the form:
(S) =
~
2
(B-8)
990

C. NON-RELATIVISTIC DESCRIPTION OF A SPIN 1/2 PARTICLE
wheredesignates the set of the threePauli matrices:
=
0 1
1 0
=
0
0
=
1 0
01
(B-9)
The Pauli matrices possess the following properties, which can easily be veried
from their explicit form (B-9) (see also ComplementIV):
2
=
2
=
2
= (B-10a)
+ = 0 (B-10b)
[ ] = 2 (B-10c)
= (B-10d)
(to the last three formulas must be added those obtained through cyclic permutation of
the,,indices). It also follows from (B-9) that:
Tr= Tr= Tr= 0 (B-11a)
Det= Det= Det=1 (B-11b)
Furthermore, any22matrix can be written as a linear combination, with complex
coecients, of the three Pauli matrices and the unit matrix. This is simply due to
the fact that a22matrix has only four elements. Finally, it is easy to derive (see
ComplementIV) the following identity:
(A)(B) = (AB)+(AB) (B-12)
whereAandBare two arbitrary vectors, or two vector operators whose three components
commute with those of the spinS. IfAandBdo not commute with each other, the
identity remains valid ifAandBappear in the same order on the right-hand side as on
the left-hand side.
The operators associated with electron spin have all the properties that follow
directly from the general theory of angular momentum. They have, in addition, some
specic properties related to their particular value of(that is, of), which is the smallest
one possible (aside from zero). These specic properties can be deduced directly from
(B-8) and formulas (B-10):
2
=
2
=
2
=
~
2
4
(B-13a)
+ = 0 (B-13b)
=
2
~ (B-13c)
2
+=
2
= 0 (B-13d)
where the unit operatoris not explicitly written in the right hand side of (B-13a), as
we will do from now on for the sake of simplicity.
C. Non-relativistic description of a spin 1/2 particle
We now know how to describe separately the external (orbital) and the internal (spin)
degrees of freedom of the electron. In this section, we are going to assemble these dierent
concepts into one formalism.
991

CHAPTER IX ELECTRON SPIN
C-1. Observables and state vectors
C-1-a. State space
When all its degrees of freedom are taken into account, the quantum state of an
electron is characterized by a ket belonging to the spacewhich is the tensor product
ofrand().
We extend into, following the method described in , both
the operators originally dened inrand those which initially acted in(we shall
continue to use the same notation for these extended operators as for the operators from
which they are derived). We thus obtain a C.S.C.O. inthrough the juxtaposition of a
C.S.C.O. ofrand one of. For example, in, we can takeS
2
and(orS
2
and any
component ofS). Inr, we can choose , or , or, ifdesignates the
Hamiltonian associated with a central potential,L
2
etc. From this we deduce
various C.S.C.O. in:
S
2
(C-1a)
S
2
(C-1b)
L
2
S
2
(C-1c)
etc. Since all kets ofare eigenvectors ofS
2
with the same eigenvalue [formula (B-5)],
we can omitS
2
from the sets of observables.
We are going to use here the rst of these C.S.C.O., (C-1a). We shall take as a
basis ofthe set of vectors obtained from the tensor product of the ketsr
ofrand the ketsof:
r =r (C-2)
where the,,, components of the vectorrcan vary from to+(continuous
indices), andis equal to+or(discrete index). By denition,ris an eigenvector
common to,,,S
2
and:
r=r
r=r
r=r
S
2
r=
3
4
~
2
r
r=
~
2
r (C-3)
Each ketris unique to within a constant factor, since,,,S
2
andconstitute
a C.S.C.O. Ther system is orthonormal (in the extended sense), since the sets
rand+ are each orthonormal inrandrespectively:
r r= (rr) (C-4)
(is equal to 1 or 0 depending on whetherandare the same or dierent). Finally,
it satises a closure relation in:
d
3
rr=d
3
r+r++d
3
r r= 1 (C-5)
992

C. NON-RELATIVISTIC DESCRIPTION OF A SPIN 1/2 PARTICLE
C-1-b. r representation
. State vectors
Any stateof the spacecan be expanded on ther basis. To do this, it
suces to use the closure relation (C-5):
= d
3
rr (C-6)
Thevectorcan therefore be represented by the set of its coordinates in ther
basis, that is, by thenumbers:
r =(r) (C-7)
which depend on the three continuous indices,,(or, more succinctly,r) and on the
discrete index(+or ).In order to characterize the state of an electron completely, it
is therefore necessary to specify two functions of the space variables x, y and z:
+(r) =r+
(r) =r (C-8)
These two functions are often written in the form of atwo-component spinor, which
we shall write[](r):
[](r) =
+(r)
(r)
(C-9)
The bra associated with the ketis given by the adjoint of (C-6):
= d
3
rr (C-10)
that is, taking (C-7) into account:
= d
3
(r)r (C-11)
The bra is thus represented by the two functions
+(r)and(r), which can be
written in the form of a spinor which is the adjoint of (C-9):
[](r) =
+(r) (r) (C-12)
With this notation, the scalar product of two state vectorsand, which, according
to (C-5), is equal to:
= d
3
rr
=d
3
+(r)+(r) +(r)(r) (C-13)
993

CHAPTER IX ELECTRON SPIN
can be written in the form:
=d
3
[](r) [] (r) (C-14)
This formula is very similar to the one that permitted the calculation of the scalar product
of two kets ofrfrom the corresponding wave functions. However, it is important to
note that here the matrix multiplication of the spinors[](r)and[] (r)must precede
the spatial integration. In particular, the normalization of the vectoris expressed by:
=d
3
[](r) [] (r) =d
3
+(r)
2
+ (r)
2
= 1 (C-15)
Amongst the vectors of, some are the tensor products of a ket ofrand a ket
of(this is the case, for example, for the basis vectors). If the state vector under
consideration is of this type:
= (C-16)
with:
=d
3
(r)r r
=+++ (C-17)
the spinor associated with it takes on the simple form:
[](r) =
(r)+
(r)
=(r)
+
(C-18)
This results from the denition of the scalar product in, and we have in this case:
+(r) =r+ =r + =(r)+ (C-19a)
(r) =r =r =(r) (C-19b)
The square of the norm ofis then given by:
= = +
2
+(
2
d
3
(r)
2
(C-20)
. Operators
Let be the ket obtained from the action of the linear operatoron the ket
of. According to the results of the preceding section,and can be represented
by the two-component spinors[](r)and[](r). We are now going to show that one
can associate witha22matrix[[]]such that:
[](r) = [[]][](r) (C-21)
where the matrix elements remain in general dierential operators with respect to the
variabler.
994

C. NON-RELATIVISTIC DESCRIPTION OF A SPIN 1/2 PARTICLE
(i) Spin operators. These were initially dened in. Consequently, they act only
on theindex of the basis vectorsr, and their matrix form is the one stated in .
We shall limit ourselves to one example, say that of the operator+. Its action on a
vectorexpanded as in (C-6) gives a vector:
=~d
3
(r)r+ (C-22)
since+annihilates all ther+kets and transformsr into~r+. The compo-
nents ofin ther basis are, according to (C-22):
r+ =
+(r) =~(r)
r =(r) = 0 (C-23)
The spinor representingis therefore:
[](r) =~
(r)
0
(C-24)
This is indeed what is obtained if one performs the matrix multiplication of the spinor
[](r)by:
[[+]] =
~
2
(+) =~
0 1
0 0
(C-25)
(ii) Orbital operators. Unlike the preceding operators, they always leave unchanged
theindex of the basis vectorr: their associated22matrices are always propor-
tional to the unit matrix. On the other hand, they act on ther-dependence of the spinors
just as they act on ordinary wave functions. Consider, for example, the kets=
and = . Their components in ther basis are, respectively:
(r) =r = (r) (C-26a)
(r) =r =
~
(r) (C-26b)
The spinors[](r)and[](r)are thus obtained from[](r)by means of the22
matrices:
[[]] =
0
0
(C-27a)
[[]] =
~
0
0
(C-27b)
(iii) Mixed operators. The most general operator acting inis represented, in matrix
notation, by a22matrix whose elements are dierential operators with respect to the
rvariables. For example:
[[]] =
~
2
~
0
0
~
(C-28)
995

CHAPTER IX ELECTRON SPIN
or:
[[SP]] =
~
2
( + + ) =
~
2
2
+
(C-29)
Comments:
() r is analogous to therrepresentationr.
The matrix element of any operatorofis given by the formula:
=d
3
[](r)[[]][](r) (C-30)
where[[]]designates the22matrix that represents the operator(one
rst carries out the matrix multiplications and then integrates over all space).
This representation will only be used when it simplies the reasoning and the
calculations: as inr, the vectors and operators themselves will be used as
much as possible.
() p representation, whose basis vectors
are the eigenvectors common to the C.S.C.O. S
2
. The
denition of the scalar product inyields:
rp =rp =
1
(2~)
32
e
pr~
(C-31)
In thep representation, one associates with each vectorofa
two-component spinor:
[
](p) =
+(p)(p)
(C-32)
with:
+(p) =p+(p) =p (C-33)
According to (C-31),
+(p)and(p)are the Fourier transforms of+(r)
and(r):
(p) =p = d
3
p r r
=
1
(2~)
32
d
3
e
pr~
(r) (C-34)
The operators are still represented by22matrices, and those corresponding
to the spin operators remain the same as in ther representation.
996

C. NON-RELATIVISTIC DESCRIPTION OF A SPIN 1/2 PARTICLE
C-2. Probability calculations for a physical measurement
Using the formalism we have just described, we can apply the postulates of Chap-
ter
carrying out on an electron. We are going to give several examples.
First of all, consider the probabilistic interpretation of the components+(r)and
(r)of the state vectorwhich we assume to be normalized [formula (C-15)]. Imagine
that we are simultaneously measuring the position of the electron and the component
of its spin along. Since,,and constitute a C.S.C.O., there exists only
one state vector that corresponds to a given result:,,and~2. The probability
d
3
(r+)of the electron being found in the innitesimal volume d
3
around the point
r( )with its spin up (component alongequal to+~2) is equal to:
d
3
(r+) =r+
2
d
3
= +(r)
2
d
3
(C-35)
In the same way:
d
3
(r) =r
2
d
3
= (r)
2
d
3
(C-36)
is the probability of the electron being found in the same volume as before but with its
spin down (component alongequal to~2).
If it is the component of the spin alongthat is being measured at the same
time as the position, all we need to do is use formulas (A-20) of Chapter. The,
,andoperators also form a C.S.C.O.: to the measurement result ~2
corresponds a single state vector:
r =
1
2
[r+ r] (C-37)
The probability of the electron being found in the volume d
3
around the pointrwith
its spin in the positive direction of theaxis is then:
d
3 1
2
[r+ +r ]
2
=
1
2
+(r) +(r)
2
d
3
(C-38)
Obviously, one can measure the momentum of the electron instead of its position. One
then uses the components ofrelative to the vectorsp[cf.comment(ii)of 1], that
is, the Fourier transforms
(p)of(r). The probability d
3
(p)of the momentum
beingpto within d
3
and of the spin component alongbeing~2is given by:
d
3
(p) =p
2
d
3
=
(p)
2
d
3
(C-39)
The various measurements that we have envisaged until now are all complete in the
sense that they each relate to a C.S.C.O. For incomplete measurements, several or-
thogonal states correspond to the same result, and it is necessary to sum the squares of
the moduli of the corresponding probability amplitudes.
For example, if one does not seek to measure its spin, the probability d
3
(r)of
nding the electron in the volume d
3
in the neighborhood of the pointris equal to:
d
3
(r) = +(r)
2
+ (r)
2
d
3
(C-40)
997

CHAPTER IX ELECTRON SPIN
This is because two orthogonal state vectors,r+andr, are associated with the
result , their corresponding probability amplitudes being+(r)and(r).
Finally, let us calculate the probability+that the spin component alongis
+~2(one is not seeking to measure the orbital variables). There exist an innite number
of orthogonal states, for example all ther+with arbitraryr, which correspond to the
result of the measurement. One must therefore sum over all possible values ofrthe
squares of the moduli of the amplitudesr+ =+(r), which gives:
+=d
3
+(r)
2
(C-41)
Of course, if we are considering the component of the spin alonginstead of along
, we integrate the result (C-38) over all space. These ideas generalize those of
of Chapter, where we considered only the spin observables since the orbital variables
could be treated classically.
References and suggestions for further reading:
History of the discovery of spin and references to original articles: Jammer (4.8),
3-4.
Evidence of spin in atomic physics: Eisberg and Resnick (1.3), Chap. 8; Born
(11.4), Chap. VI; Kuhn (11.1), Chap. III, A.5, A.6 and F; see references of Chap-
ter IV relating to the Stern-Gerlach experiment.
The spin magnetic moment of the electron: Cagnac and Pebay-Peyroula (11.2).
Chap. XII; Crane (11.16).
The Dirac equation: Schi (1.18), Chap. 13; Messiah (1.17). Chap. XX: Bjorkcn
and Drell (2.6), Chaps. 1 to 4
The Lorentz group: Omnes (16.13), Chap. 4; Bacry (10.31). Chaps. 7 and 8.
Spin 1 particles: Messiah (1.17), XIII.21.
998

COMPLEMENTS OF CHAPTER IX, READER'S GUIDE
Several complements concerning the properties of spins12can be found at the end of
Chapter
AIX: ROTATION OPERATORS FOR A SPIN 1/2
PARTICLE
This complement is a continuation of Comple-
mentVI. It studies in detail the relationship
between the spin
1
2
angular momentum and the
geometric rotations of this spin. Moderately
dicult. Can be omitted upon a rst reading.
BIX: EXERCISES Exercice 4 is worked out in detail. It studies
the polarization of a beam of spin
1
2
particles
caused by their reection from a magnetized
ferromagnetic material. This method is actually
used in certain experiments.
999

ROTATION OPERATORS FOR A SPIN 1/2 PARTICLE
Complement AIX
Rotation operators for a spin 1/2 particle
1 Rotation operators in state space
1-a Total angular momentum
1-b Decomposition of rotation operators into tensor products
2 Rotation of spin states
2-a Explicit calculation of the rotation operators in. . . . . .
2-b Operator associated with a rotation through an angle of2.
2-c Relationship between the vectorial nature ofSand the behav-
ior of a spin state upon rotation
3 Rotation of two-component spinors
We are going to apply the ideas about rotation introduced in ComplementVIto
the case of a spin 1/2 particle. First, we shall study the form that rotation operators
take on in this case. We shall then examine the behavior, under rotation, of the ket
representing the particle's state and of the two-component spinor associated with it.
1. Rotation operators in state space
1-a. Total angular momentum
A spin 1/2 particle possesses an orbital angular momentumLand a spin angular
momentumS. It is natural to dene its total angular momentum as the sum of these
two angular momenta:
J=L+S (1)
This denition is clearly consistent with the general considerations discussed in Comple-
mentVI. It insures that not onlyRandP, but alsoS, be vectorial observables. Note
that, to test this, it is sucient to calculate the commutators between the components
of these observables and those ofJ;cf. VI.
1-b. Decomposition of rotation operators into tensor products
In the state space of the particle under study, the rotation operatoru()is
associated with the geometrical rotationu()through an angleabout the unit vector
u(cf.ComplementVI, ):
u() = e
~
Ju
(2)
whereJis the total angular momentum (1).
SinceLacts only inr, andSonly in(which implies, in particular, that all
components ofLcommute with all components ofS), we can writeu()in the form of
a tensor product:
u() =
(r)
u()
()
u() (3)
1001

COMPLEMENT A IX
where:
(r)
u() = e
~
Lu
(4)
and:
()
u() = e
~
Su
(5)
are the rotation operators associated withu()inrandrespectively.
Consequently, if one performs the rotationu()on a spin 1/2 particle whose
state is represented by a ket which is a tensor product:
= (6)
with:
r
(7)
its state after rotation will be:
=u()=
(r)
u()
()
u() (8)
The spin state of the particle is therefore also aected by the rotation. This is what we
are going to study in more detail in 2.
2. Rotation of spin states
We have already studied ( 3 of ComplementVI) the rotation operators
(r)
in the
spacer. Here we are interested in the operators
()
which act in the spin state space
.
2-a. Explicit calculation of the rotation operators in
As in Chapter, we set:
S=
~
2
(9)
We want to calculate the operator:
()
u() = e
~
Su
= e
2
u
(10)
To do this, let us use the denition of the exponential of an operator:
()
u() = 1
2
u+
1
2!2
2
(u)
2
++
1
!2
(u)+ (11)
Now, applying identity (B-12) of Chapter, we immediately see that:
(u)
2
=u
2
= 1 (12)
1002

ROTATION OPERATORS FOR A SPIN 1/2 PARTICLE
which leads to:
(u)=
1 if is even
u ifis odd
(13)
Consequently, if we group together the even and odd terms respectively, expansion (11)
can be written:
()
u() =1
1
2!2
2
++
(1)
(2)!2
2
+
u
2
1
3!2
3
++
(1)
(2+ 1)!2
2+1
+ (14)
that is, nally:
()
u() = cos
2
usin
2
(15)
It will be very easy to calculate the action of the operator
()
, in this form, on any spin
state.
Using this formula, we can write the rotation matrixu
(12)
()explicitly in the
+ basis, since we already know [formulas (B-9) of Chapter] the matrices
which represent the,andoperators. We nd:
(12)
u() =
cos
2
sin
2
( ) sin
2
( +) sin
2
cos
2
+ sin
2
(16)
where,andare the cartesian components of the vectoru.
2-b. Operator associated with a rotation through an angle of2
If we take2for the angle of rotation, the geometrical rotationu(2)coincides,
whatever the vectorumay be, with the identity rotation. However, if we set= 2in
formula (15), we see that:
()
u(2) =1 (17)
whereas:
()
u(0) = 1 (18)
The operator associated with a rotation through an angle of2is not the identity oper-
ator, but minus this operator. The group law is therefore conserved only locally in the
correspondance between geometrical rotations and rotation operators in[see discus-
sion in ComplementVI, comment () of ]. This is due to the half-integral value
of the spin angular momentum of the particle which we are considering.
The fact that the spin state changes sign during a rotation through an angle of2
is not disturbing, since two state vectors diering only by a global phase factor have the
1003

COMPLEMENT A IX
same physical properties. It is more important to study the way in which an observable
transforms during such a rotation. It is easy to show that:
=
()
u(2)
()
u(2) = (19)
This result is quite satisfying since a rotation through2cannot modify the measuring
device associated withConsequently, the spectrum ofmust remain the same as
that of
Comment:
We showed in ComplementVI[comment(iii) of 3-c-] that:
(r)
u(2) = 1 (20)
Consequently, in the global state space=r , as in, we have:
u(2) =
(r)
u(2)
()
u(2) =1 (21)
2-c. Relationship between the vectorial nature ofSand the behavior of a spin state
upon rotation
Consider an arbitrary spin state. We showed in Chapter
anglesandsuch that can be written (except for a global phase factor which has no
physical meaning):
= e
2
cos
2
++ e
2
sin
2
(22)
then appears as the eigenvector associated with the eigenvalue+~2of the componentSv
of the spinSalong the unit vectorvdened by the polar anglesand. Now let us perform
an arbitrary rotation on the state. Let us callvthe result of the transformation ofvby
the rotation being considered. SinceSis a vectorial observable, the stateafter the rotation
must be an eigenvector, with the eigenvalue+~2, of the componentSvofSalong the unit
vectorv(cf.ComplementVI, ):
=+= = + (23)
with:
v=v (24)
We shall be satised with verifying this for a specic case (cf.Fig.). Choose forvthe unit
vectoreof theaxis, and forvan arbitrary unit vector, with polar anglesand.vis
obtained fromv=eby a rotation through an angleabout the unit vectoru, which is xed
by the polar angles:
=
2
=+
2
(25)
Thus we must show that:
()
u()+ + (26)
The cartesian components of the vectoruare:
=sin
= cos
= 0 (27)
1004

ROTATION OPERATORS FOR A SPIN 1/2 PARTICLE
so the operator
()
u()can be written, using formula (15):
()
u() = cos
2
usin
2
= cos
2
( sin+cos) sin
2
= cos
2
1
2
+e esin
2
(28)
with:
= (29)
Now we know [cf.formulas (B-7) of Chapter] that:
++= 0
+= 2 (30)
The result of the transformation of the ket+by the operator
()
u()is therefore:
()
u()+= cos
2
++ esin
2
(31)
We recognize, to within a phase factor, the ket+[cf.formula (22)]:
()
u()+= e
2
+ (32)z
v = e
z
v
u
O
θ
y
φ
x
Figure 1: A rotation through an
angleaboutubrings the vector
v=eonto the unit vectorv, with
polar anglesand.
3. Rotation of two-component spinors
We are now prepared to study the global behavior of a spin 1/2 particle under rotation.
That is, we shall now take into account both its external and internal degrees of freedom.
1005

COMPLEMENT A IX
Consider a spin 1/2 particle whose state is represented by the ketof the state
space=r . The ket can be represented by the spinor[](r), having the
components:
(r) =r (33)
If we perform an arbitrary geometrical rotationon this particle, its state then becomes:
= (34)
where:
=
(r) ()
(35)
is the operator associated, in, with the geometrical rotation. How is the spinor,
[](r), which corresponds to the state, obtained from[](r)?
In order to answer this question, let us write the components(r)of[]:
(r) =r =r (36)
We can nd the components of(r)by inserting the closure relation relative to the
r basis betweenand:
(r) = d
3
r r r (37)
Now, since the vectors of ther basis are tensor products, the matrix elements of
the operatorin this basis can be decomposed in the following manner:
r r =r
(r)
r
()
(38)
We already know [cf.ComplementVI, formula (26)] that:
r
(r)
r=
1
rr=r(
1
r) (39)
Consequently, if we set:
()
=
(12)
(40)
formula (37) can nally be written:
(r) =
(12)
(
1
r) (41)
that is, explicitly:
+(r)
(r)
=
(12)
+ +
(12)
+
(12)
+
(12)
+(
1
r)
(
1
r)
(42)
Thus we obtain the following result: each component of the new spinor[]at the
pointris a linear combination of the two components of the original spinor[]evaluated
at the point
1
r(that is, at the point that the rotation maps intor)
1
. The coecients
of these linear combinations are the elements of the22matrix which represents
()
in the+ basis of[cf.formula (16)].
1
Note the close analogy between this behavior and that of a vector eld under rotation.
1006

ROTATION OPERATORS FOR A SPIN 1/2 PARTICLE
References and suggestions for further reading:
Feynman III (1.2), Chap. 6; Chap. 18, 18-4 and added note 1; Messiah (1.17),
App. C; Edmonds (2.21), Chap. 4.
Rotation groups and SU(2): Bacry (10.31), Chap. 6; Wigner (2.23), Chap. 15;
Meijer and Bauer (2.18), Chap. 5.
Experiments dealing with rotations of a spin 1/2: article by Werner et al. (11.18).
1007

EXERCISES
Complement BIX
Exercises
1.Consider a spin 1/2 particle. Call its spinS, its orbital angular momentumLand its
state vector. The two functions+(r)and(r)are dened by:
(r) =r
Assume that:
+(r) =()
0
0() +
1
3
0
1()
(r) =
()
3
1
1()
0
1()
where,,are the coordinates of the particle and()is a given function of
What condition must()satisfy forto be normalized?
is measured with the particle in the state. What results can be found, and
with what probabilities? Same question for, then for
A measurement ofL
2
, with the particle in the state, yielded zero. What
state describes the particle just after this measurement? Same question if the
measurement ofL
2
had given2~
2
.
2.Consider a spin 1/2 particle.PandSdesignate the observables associated with its
momentum and its spin. We choose as the basis of the state space the orthonormal basis
of eigenvectors common to,,and (whose eigenvalues are,
respectively, and~2).
We intend to solve the eigenvalue equation of the operatorwhich is dened by:
=SP
IsHermitian?
Show that there exists a basis of eigenvectors ofwhich are also eigenvectors of
,,. In the subspace spanned by the kets , where
are xed, what is the matrix representing?
What are the eigenvalues of, and what is their degree of degeneracy? Find a
system of eigenvectors common toand,,.
3. The Pauli Hamiltonian
The Hamiltonian of an electron ot mass, charge, spin~2(where
are the Pauli matrices), placed in an electromagnetic eld described by the vector po-
tentialA(r)and the scalar potential(r), is written:
=
1
2
[PA(R)]
2
+(R)
~
2
B(R)
1009

COMPLEMENT B IX
The last term represents the interaction between the spin magnetic moment
~
2
and
the magnetic eldB(R) =rA(R).
Show, using the properties of the Pauli matrices, that this Hamiltonian can also
be written in the following form (the Pauli Hamiltonian):
=
1
2
[PA(R)]
2
+(R)
4.We intend to study the reection of a monoenergetic neutron beam which is per-
pendicularly incident on a block of a ferromagnetic material. We callthe direction
of propagation of the incident beam andthe surface of the ferromagnetic material,
which lls the entire0region (see Figure). Let each incident neutron have an
energyand a massThe spin of the neutrons is= 12and their magnetic moment
is writtenM=S(is the gyromagnetic ratio andSis the spin operator).z
y
x
O
B
0
incident neutrons
Figure 1
The potential energy of the neutrons is the sum of two terms:
the rst one corresponds to the interaction with the nucleons of the substance.
Phenomenologically, it is represented by a potential(), dened by() = 0for
0,() =00for0.
the second term corresponds to the interaction of the magnetic moment of each
neutron with the internal magnetic eldB0of the material (B0is assumed to be
uniform and parallel to). Thus we have= 0for0,=0for0
(with0= 0). Throughout this exercise we shall conne ourselves to the case:
0
~0
2
0
Determine the stationary states of the particle that correspond to a positive incident
momentum and a spin which is either parallel or antiparallel to.
We assume in this question that0~02 0+~02. The incident
neutron beam is unpolarized. Calculate the degree of polarization of the reected
beam. Can you imagine an application of this eect?
1010

EXERCISES
Now consider the general case wherehas an arbitrary positive value. The spin of
the incident neutrons points in thedirection. What is the direction of the spin
of the reected particles (there are three cases, depending on the relative values of
and0~02)?
Solution of exercise 4
The Hamiltonianof the particle is:
=
P
2
2
+() + (1)
(), which acts only on the orbital variables, commutes with. Sinceis propor-
tional to, it also commutes with this operator. Furthermore,()commutes with
and, as well as with(obviously, sinceacts only on the spin variables). We can
therefore nd a basis of eigenvectors common to,,,, which can be written:
= (2)
with:
; =
; =
; =
~
2
(3)
where the ket is a solution of the eigenvalue equation:
2
2
+() +
1
2
2
+
2
~0
2
= (4)
We assume in the statement of the problem that the neutron beam is normally incident,
so we can set== 0. Let() = be the wave function associated with
; it satises the equation:
~
2
2
2
2
+()
~0
2
() = () (5)
Thus the problem is reduced to that of a classical one-dimensional square well: reec-
tion from a potential step (cf.ComplementI).
In the 0region,()is zero and the total energy(which is positive) is
greater than the potential energy. We know in this case that the wave function is a
superposition of imaginary oscillatory exponentials:
() =e+e if 0 (6)
1011

COMPLEMENT B IX
with:
=
2~
2
(7)
gives the amplitude of the wave associated with an incident particle having a spin
either parallel or antiparallel to.gives the amplitude of the wave associated with
a reected particle for the same two spin directions.
In the0region,()is equal to0and, depending on the relative values of
and0~02, the wave functions can behave like oscillatory or damped exponentials.
We shall consider three cases:
() If 0+~02, we set:
=
2~
2
0
~0
2
(8)
and the transmitted wave behaves like an oscillatory exponential:
() =e if 0 (9)
Moreover, the continuity conditions for the wave function and its derivative imply [cf.
ComplementI, relations (13) and (14)]:
=
+
=
2
+
(10)
() If, on the other hand,0~02, we must introduce the quantities:
=
2~
2
0
~0
2
(11)
and the wave in the0region is a real, damped exponential (evanescent wave):
() =e if 0 (12)
with, in this case [cf.ComplementI, equations (22) and (23)]:
=
+
;=
2
+
(13)
() Finally, in the intermediate case0~02 0+~02, we have:
+
() =+e
+
if 0 (14a)
() =e if 0 (14b)
[denitions (8) and (11) ofand+are still valid]. Depending on the spin orientation,
the wave is either a damped or an oscillatory exponential. We then have:
+
+
=
+
++
;
+
+
=
2
++
(15a)=
+
;=
2
+
(15b)
1012

EXERCISES
. When0~02 0+~02, we are in the situation of case () above.
If the projection ontoof the incident neutron spin is equal to~2, the corresponding
reection coecient is:
+=
+
+
2
=
+
++
2
= 1 (16)
On the other hand, if the projection of the spin ontois equal to~2, the reection
coecient is no longer 1, since it is given by:
=
2
=
+
2
1 (17)
Thus we see how the reected beam can be polarized since, depending on the direction
of its spin, the neutron has a dierent probability of being reected. An unpolarized
incident beam can be considered to be formed of neutrons whose spins have a probability
1/2 of being in the state+and a probability 1/2 of being in the state. Taking (16)
and (17) into account, we see that the probability that a particle of the reected beam
will have its spin in the state+is1(1 +), while for the stateit is(1 +).
Therefore, the degree of polarization of the reected beam is:
=
1
1 +
=
2
2
+
2
(18)
In practice, reection from a saturated ferromagnetic substance is actually used in
the laboratory to obtain beams of polarized neutrons. To increase the degree of polar-
ization obtained, the beam is made to fall obliquely on the surface of the ferromagnetic
mirror; thus, the theoretical results obtained here are not directly applicable. However,
the principle of the experiment is the same. The ferromagnetic substance chosen is often
cobalt. When cobalt is magnetized to saturation, one can obtain high degrees of polar-
ization(&80%). Note, furthermore, that the same neutron beam reection device
can serve as an analyzer as well as a polarizer for spin directions. This possibility
has been exploited in precision measurements of the magnetic moment of the neutron.
. Consider a neutron whose momentum, of magnitude=~, is parallel to.
Assume that the projectionof its spin is equal to~2. Its state is [cf.Chap.,
relation (A-20)]:
=
1
2
[++] (19)
with:
r=
1
(2~)
32
e
~
(20)
How can we construct a stationary state of the particle in which the incident wave has
the form (19)? We simply have to consider the state:
=
1
2
+
00
+
00
(21)
1013

COMPLEMENT B IX
which is a linear combination of two eigenkets ofdened in (2), associated with the
same eigenvalue=
2
2. The part of the ketwhich describes the reected wave
is then:
1
2
[+++ ] (22)
where+and are given, depending on the case, by (10), (13) or (15) (+and
being replaced by 1). Let us calculate, for a state such as (22), the average value
S. Since this state is a tensor product, the spin variables and the orbital variables
are not correlated. Therefore,Scan easily be obtained from the spin state vector
+++ , which gives:
=
~
2
++ +
+
2
+
2
(23a)
=
~
2
( + +)
+
2
+
2
(23b)
=
~
2
+
2 2
+
2
+
2
(23c)
Three cases can then be distinguished:
() 0+~02, we see from (10) that+and are real. Formulas (23) then
show that and are not zero but that= 0. Upon reection of the
neutron, the spin has thus undergone a rotation about. Physically, it is the
dierence between the degrees of reection of neutrons whose spin is parallel to
and those whose spin is antiparallel towhich explains why thecomponent
becomes positive.
() 0~02, equations (13) show that+and are not real: they are
two complex numbers having dierent phases but the same modulus. According
to (23), we have, in this case,= 0but = 0and = 0. Upon reection
of the neutron, the spin thus undergoes a rotation about. The physical origin
of this rotation is the following: because of the existence of the evanescent wave,
the neutron spends a certain time in the0region; the Larmor precession about
B0that it undergoes during this time accounts for the rotation of its spin.
() 0~02 0+~02,+is a complex number whileis a real
number, and their moduli are dierent. None of the spin components,,
or , is then zero. This rotation of the spin upon reection of the neutron is
explained by a combination of the eects pointed out in()and().
1014

Chapter X
Addition of angular momenta
A Introduction
A-1 Total angular momentum in classical mechanics
A-2 The importance of total angular momentum in quantum me-
chanics
B Addition of two spin 1/2's. Elementary method
B-1 Statement of the problem
B-2 The eigenvalues of and their degrees of degeneracy
B-3 Diagonalization of S
2
. . . . . . . . . . . . . . . . . . . . . . .
B-4 Results: triplet and singlet
C Addition of two arbitrary angular momenta. General method 1025
C-1 Review of the general theory of angular momentum
C-2 Statement of the problem
C-3 Eigenvalues ofJ
2
and. . . . . . . . . . . . . . . . . . . . .
C-4 Common eigenvectors ofJ
2
and . . . . . . . . . . . . . . .
A. Introduction
A-1. Total angular momentum in classical mechanics
Consider a system ofclassical particles. The total angular momentumof
this system with respect to a xed pointis the vector sum of the individual angular
momenta of theparticles with respect to this point:
=
=1
(A-1)
with:
=rp (A-2)
Quantum Mechanics, Volume II, Second Edition. C. Cohen-Tannoudji, B. Diu, and F. Laloë.
© 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

CHAPTER X ADDITION OF ANGULAR MOMENTA
The time derivative ofis equal to the moment with respect toof the external forces.
Consequently, when the external forces are zero (an isolated system) or all directed
towards the same center, the total angular momentum of the system (with respect to
any point in the rst case and with respect to the center of force in the second one) is a
constant of the motion. This is not the case for each of the individual angular momenta
if there are internal forces, that is, if the various particles of the system interact.
We shall illustrate this point with an example. Consider a system composed of
two particles, (1) and (2), subject to the same central force eld (which can be created
by a third particle assumed to be heavy enough to remain motionless at the origin). If
these two particles exert no force on each other, their angular momenta1and2with
respect to the center of forceare both constants of the motion. The only force then
acting on particle (1), for example, is directed towards; its moment with respect to this
point is therefore zero, as is
d
d
1. On the other hand, if particle (1) is also subject to a
force due to the presence of particle (2), the moment with respect toof this force is not
generally zero, and, consequently,1is no longer a constant of the motion. However, if
the interaction between the two particles obeys the principle of action and reaction, the
moment of the force exerted by (1) on (2) with respect toexactly compensates that of
the force exerted by (2) on (1): the total angular momentumis conserved over time.
Therefore, in a system of interacting particles,only the total angular momentum is
a constant of the motion: forces inside the system induce a transfer of angular momentum
from one particle to the other. Thus we see why it is useful to study the properties of
the total angular momentum.
A-2. The importance of total angular momentum in quantum mechanics
Let us treat the preceding example quantum mechanically. In the case of two
non-interacting particles, the Hamiltonian of the system is given simply, in ther1r2
representation:
0=1+2 (A-3)
with:
1=
~
2
21
1+(1)
2=
~
2
22
2+(2) (A-4)
[1and2are the masses of the two particles,()is the central potential to which
they are subject;1and2denote the Laplacian operators relative to the coordinates
of particles (1) and (2) respectively]. We know from Chapter ) that the three
components of the operatorL1associated with the angular momentum1of particle (1)
commute with 1:
[L1 1] = 0 (A-5)
Also, all observables relating to one of the particles commute with all those corresponding
to the other one ; in particular:
[L1 2] = 0 (A-6)
1016

A. INTRODUCTION
From (A-5) and (A-6), we see that the three components ofL1are constants of the
motion. An analogous argument is obviously valid forL2.
Now assume that the two particles interact, and that the corresponding potential
energy(r1r2)depends only on the distance between themr1r2
1
:
r1r2=
(1 2)
2
+ (1 2)
2
+ (1 2)
2
(A-7)
In this case, the Hamiltonian of the system is:
=1+2+(r1r2) (A-8)
where1and2are given by (A-4). According to (A-5) and (A-6), the commutator of
L1withreduces to:
[L1] = [L1(r1r2)] (A-9)
that is, for example for the component1:
[1 ] = [1(r1r2)] =
~
1
1
1
1
(A-10)
Expression (A-10) is generally not zero:L1is no longer a constant of the motion. On
the other hand, if we dene thetotal angular momentum operatorLby an expression
similar to (A-1):
L=L1+L2 (A-11)
we obtain an operator whose three components are constants of the motion. For example,
we nd:
[ ] = [1+2 ] (A-12)
According to (A-10), this commutator is equal to:
[ ] = [1+2 ]
=
~
1
1
1
1
+2
2
2
2
(A-13)
But, sincedepends only onr1r2given by (A-7), we have:
1
=
r1r2
1
=
1 2
r1r2
(A-14a)
2
=
r1r2
2
=
2 1
r1r2
(A-14b)
and analogous expressions for 1, 2, 1and 2(is the derivative
of, considered as a function of a single variable). Substituting these values into (A-13):
[ ] =
~
r1r2
1(1 2) 1(1 2) +2(2 1) 2(2 1)
= 0 (A-15)
1
The corresponding classical forces then necessarily obey the principle of action and reaction.
1017

CHAPTER X ADDITION OF ANGULAR MOMENTA
We therefore arrive at the same conclusion as in classical mechanics.
Until now we have implicitly assumed that the particles being studied had no spin.
Now let us examine another important example: that of a single particle with spin. First,
we assume that this particle is subject only to a central potential(). Its Hamiltonian
is then the one studied in . We know that the three components of
the orbital angular momentumLcommute with this Hamiltonian. In addition, since the
spin operators commute with the orbital observables, the three components of the spin
Sare also constants of the motion. But we shall see in Chapter
corrections introduce into the Hamiltonian aspin-orbit couplingterm of the form:
0=()LS (A-16)
where()is a known function of the single variable(the physical meaning of this
coupling will be explained in Chapter). When this term is taken into account,Land
Sno longer commute with the total Hamiltonian. For example
2
:
[ 0] =()[ + + ]
=()(~ ~ ) (A-17)
and, similarly:
[ 0] =()[ + + ]
=()(~ ~ ) (A-18)
However, if we set:
J=L+S (A-19)
the three components ofJare constants of the motion.To see this, we can simply add
equations (A-17) and (A-18):
[ 0] = [+ 0] = 0 (A-20)
(an analogous proof could be given for the other components ofJ). The operatorJ
dened by (A-19) is said to be the total angular momentum of a particle with spin.
In the two cases just described, we have two partial angular momentaJ1andJ2,
which commute. We know a basis of the state space composed of eigen-vectors common
toJ
2
1,1,J
2
2,2. However,J1andJ2are not constants of the motion, while the
components of the total angular momentum:
J=J1+J2 (A-21)
commute with the Hamiltonian of the system. We shall therefore try to construct, using
the preceding basis,a new basis formed by eigenvectors ofJ
2
and. The problem thus
posed in general terms is that of theaddition (or composition) of two angular momenta
J1andJ2.
The importance of this new basis, formed of eigenvectors ofJ
2
and, is easy to
understand. To determine the stationary states of the system, that is, the eigenstates of
2
To establish (A-17 A-18), one uses the fact thatL, which acts only on the angular variables
and, commutes with(), which depends only on.
1018

B. ADDITION OF TWO SPIN 1/2'S. ELEMENTARY METHOD
it is simpler to diagonalize the matrix which representsin this new basis. Since
commutes withJ
2
and, this matrix can be broken down into as many blocks as
there are eigensubspaces associated with the various sets of eigenvalues ofJ
2
and
(cf.Chap., ). Its structure is much simpler than that of the matrix which
representsin the basis of eigenvectors common toJ
2
1,1,J
2
2,2, since neither1
nor2generally commutes with
We shall leave aside for now the problem of the diagonalization of(whether exact
or approximate) in the basis of eigenstates ofJ
2
and. Rather, we shall concentrate
on the construction of this new basis from the one formed by the eigenstates ofJ
2
1,
1,J
2
2,2. A certain number of physical applications (many-electron atoms, ne and
hyperne line structure, etc.) will be considered after we have studied perturbation
theory (complements of Chapter ).
We shall begin () with an elementary treatment of a simple case, in which
the two partial angular momenta we wish to add are spin l/2's. This will allow us to
familiarize ourselves with various aspects of the problem, before we treat, in , the
addition of two arbitrary angular momenta.
B. Addition of two spin 1/2's. Elementary method
B-1. Statement of the problem
We shall consider a system of two spin 1/2 particles (electrons or silver atoms in
the ground state, for example), and we shall be concerned only with their spin degrees
of freedom. LetS1, andS2be the spin operators of the two particles.
B-1-a. State space
We have already dened the state space of such a system. Recall that it is a four-
dimensional space, obtained by taking the tensor product of the individual spin spaces
of the two particles. We know an orthonormal basis of this space, which we shall denote
by 12, that is, explicitly:
12=+++ + (B-1)
These vectors are eigenstates of the four observablesS
2
1,1,S
2
2,2(which are actually
the extensions, into the tensor product space, of operators, dened in each of the spin
spaces):
S
2
112=S
2
212=
3
4
~
2
12 (B-2a)
112=1
~
2
12 (B-2b)
212=2
~
2
12 (B-2c)
S
2
1,S
2
2,1and2constitute a C.S.C.O. (the rst two observables are actually multiples
of the identity operator, and the set of operators remains complete even if they are
omitted).
1019

CHAPTER X ADDITION OF ANGULAR MOMENTA
B-1-b. Total spin S. Commutation relations
We dene the total spinSof the system by:
S=S1+S2 (B-3)
It is simple, knowing thatS1andS2are angular momenta, to show thatSis as well.
We can calculate, for example, the commutator ofand:
[ ] = [1+2 1+2]
= [1 1] + [2 2]
=~1+~2
=~ (B-4)
The operatorS
2
can be obtained by taking the (scalar) square of equation (B-3):
S
2
= (S1+S2)
2
=S
2
1+S
2
2+ 2S1S2 (B-5)
sinceS1andS2commute. The scalar productS1S2can be expressed in terms of the
operators1,1and2,2; it is easy to show that:
S1S2=12+12+12
=
1
2
(1+2+12+) +12 (B-6)
Note that, sinceS1andS2each commute withS
2
1andS
2
2, so do the three compo-
nents ofS. In particular,S
2
andcommute withS
2
1andS
2
2:
[S
2
1] = [S
2
2] = 0 (B-7a)
[S
2
S
2
1] = [S
2
S
2
2] = 0 (B-7b)
In addition,obviously commutes with1and2:
[ 1] = [ 2] = 0 (B-8)
However,S
2
commutes with neither1nor2since, according to (B-5):
[S
2
1] = [S
2
1+S
2
2+ 2S1S21]
= 2[S1S21]
= 2[12+12 1]
= 2~(12+12) (B-9)
[this calculation is analogous to the one performed in (A-17) and (A-18)]. The com-
mutator ofS
2
with2is, of course, equal and opposite to the preceding one, so that
=1+2commutes withS
2
.
1020

B. ADDITION OF TWO SPIN 1/2'S. ELEMENTARY METHOD
B-1-c. The basis change to be performed
The basis (B-1), as we have seen, is composed of eigenvectors common to the
C.S.C.O.:
S
2
1S
2
21 2 (B-10)
Also, we have just shown that the four observables:
S
2
1S
2
2S
2
(B-11)
commute. We shall see in what follows that they also form a C.S.C.O.
Adding the two spinsS1andS2amounts to constructing the orthonormal system
of eigenvectors common to the set (B-11). This system will be dierent from (B-1), since
S
2
does not commute with1and2. We shall write the vectors of this new basis
, with the eigenvalues ofS
2
1andS
2
2(which remain the same) implicit. The vectors
therefore satisfy the equations:
S
2
1 =S
2
2 =
3
4
~
2
(B-12a)
S
2
=(+ 1)~
2
(B-12b)
=~ (B-12c)
We know thatSis an angular momentum. Consequently,must be a positive integer
or half-integer, andvaries by one-unit jumps betweenand+. The problem is
therefore to nd what valuesandcan actually have, and to express the basis vectors
in terms of those of the known basis.
In this section, we shall conne ourselves to solving this problem by the elementary
method involving the calculation and diagonalization of the44matrices representing
S
2
andin the12basis. In , we shall use another, more elegant, method,
and generalize it to the case of two arbitrary angular momenta.
B-2. The eigenvalues of and their degrees of degeneracy
The observablesS
2
1andS
2
2are easy to deal with: all vectors of the state space are
their eigenvectors, with the same eigenvalue3~
2
4. Consequently, equations (B-12a) are
automatically satised for all kets.
We have already noted [formulas (B-7) and (B-8)] thatcommutes with the
four observables of the C.S.C.O. (B-10). We should therefore expectthe basis vectors
12to be automatically eigenvectors of. We can indeed show, using (B-2b) and
(B-2c), that:
12= (1+2)12=
1
2
(1+2)~12 (B-13)
12is therefore an eigenstate ofwith the eigenvalue:
=
1
2
(1+2) (B-14)
Since1and2can each be equal to1, we see thatcan take on the values+1, 0
and1.
1021

CHAPTER X ADDITION OF ANGULAR MOMENTA
The values= 1and=1are not degenerate.Only one eigenvector corre-
sponds to each of them:++for the rst one and for the second one.On the
other hand,= 0is two-fold degenerate: two orthogonal eigenvectors are associated
with it,+ and +. Any linear combination of these two vectors is an eigenstate
ofwith the eigenvalue 0.
These results appear clearly in the matrix which representsin the12
basis. Choosing the basis vectors in the order indicated in (B-1), that matrix can be
written:
() =~
1 0 0 0
0 0 0 0
0 0 0 0
0 0 0 1
(B-15)
B-3. Diagonalization of S
2
All that remains to be done is to nd and then diagonalize the matrix which
representsS
2
in the12basis. We know in advance that it is not diagonal, since
S
2
does not commute with1and2.
B-3-a. Calculation of the matrix representing S
2
We are going to applyS
2
to each of the basis vectors. To do this, we shall use
formulas (B-5) and (B-6):
S
2
=S
2
1+S
2
2+ 212+1+2+12+ (B-16)
The four vectors12are eigenvectors ofS
2
1,S
2
2,1and2[formulas (B-2)], and the
action of the operators1and2can be derived from formulas (B-7) of Chapter.
We therefore nd:
S
2
++=
3
4
~
2
+
3
4
~
2
+++
1
2
~
2
++
= 2~
2
++ (B-17a)
S
2
+ =
3
4
~
2
+
3
4
~
2
+
1
2
~
2
+ +~
2
+
=~
2
[+ + +] (B-17b)
S
2
+=
3
4
~
2
+
3
4
~
2
+
1
2
~
2
++~
2
+
=~
2
[+++ ] (B-17c)
S
2
=
3
4
~
2
+
3
4
~
2
+
1
2
~
2
= 2~
2
(B-17d)
The matrix representingS
2
in the basis of the four vectors12, arranged in
1022

B. ADDITION OF TWO SPIN 1/2'S. ELEMENTARY METHOD
the order given in (B-1), is therefore:
(S
2
) =~
2
2 0 0 0
0 1 1 0
0 1 1 0
0 0 0 2
(B-18)
Comment:
The zeros appearing in this matrix were to be expected.S
2
commutes withand
therefore has non-zero matrix elements only between eigenvectors ofassociated
with the same eigenvalue. According to the results of 2, the only non-diagonal
elements ofS
2
which could be dierent from zero are those which relate+ to
+.
B-3-b. Eigenvalues and eigenvectors of S
2
Matrix (B-18) can be broken down into three submatrices (as shown by the dotted
lines). Two of them are one-dimensional:the vectors++and are eigenvectors
ofS
2
, as is also shown by relations (B-17a) and (B-17d
both equal to2~
2
.
We must now diagonalize the22submatrix:
(S
2
)0=~
2
1 1
1 1
(B-19)
which representsS
2
inside the two-dimensional subspace spanned by+ and +,
that is, the eigensubspace ofcorresponding to= 0. The eigenvalues~
2
of matrix
(B-19) can be obtained by solving the characteristic equation:
(1)
2
1 = 0 (B-20)
The roots of this equation are= 0and= 2. This yields the last two eigenvalues of
S
2
: 0 and2~
2
. An elementary calculation yields the corresponding eigenvectors:
1
2
[+ + +]for the eigenvalue2~
2
(B-21a)
1
2
[+ +]for the eigenvalue0 (B-21b)
(of course, they are dened only to within a global phase factor; the coecients1
2
insure their normalization).
The operatorS
2
therefore possesses two distinct eigenvalues: 0 and2~
2
. The rst
one is non-degenerate and corresponds to vector (B-21b). The second one is three-fold
degenerate, and the vectors++, and (B-21a) form an orthonormal basis in the
associated eigensubspace.
1023

CHAPTER X ADDITION OF ANGULAR MOMENTA
B-4. Results: triplet and singlet
Thus we have obtained the eigenvalues ofS
2
and, as well as a system of eigen-
vectors common to these two observables. We shall summarize these results by expressing
them in the notation of equations (B-12).
The quantum numberof (B-12b) can take on two values: 0 and 1. The rst one
is associated with a single vector, (B-21b), which is also an eigenvector ofwith the
eigenvalue 0, since it is a linear combination of+ and +; we shall therefore
denote this vector by00:
00=
1
2
[+ +] (B-22)
Three vectors which dier by their values ofare associated with the value= 1:
11=++
10=
1
2
[+ + +]
11=
(B-23)
It can easily be shown that the four vectorsgiven in (B-22) and (B-23)
form an orthonormal basis. Specication ofandsuces to dene uniquely a vector
of this basis. From this, it can be shown thatS
2
andconstitute a C.S.C.O. (which
could includeS
2
1andS
2
2, although it is not necessary here).
Therefore,when two spinl/2's(1=2= 12)are added, the numberwhich
characterizes the eigenvalues(+ 1)~
2
of the observableS
2
can be equal either to1or
to0.With each of these two values of S is associated a family of(2+ 1)orthogonal
vectors(three for= 1, one for= 0)corresponding to the(2+ 1)values of M which
are compatible with S.
Comments:
() B-23) of the three vectors1 (= 1, 0,1) constilutes
what is called atriplet; the vector00is called asingletstate.
() symmetricwith respect to exchange of the two spins,
whereas the singlet state isantisymmetric.This means that if each vector
12is replaced by the vector21, expressions (B-23) remain invariant,
while (B-22) changes sign. We shall see in Chapter
this property when the two particles whose spins are added are identical.
Furthermore, it enables us to nd the right linear combination of+ and
+which must be associated with++and (clearly symmetric)
in order to complete the triplet. The singlet state, on the other hand, is the
antisymmetric linear combination of+ and +, which is orthogonal
to the preceding one.
1024

C. ADDITION OF TWO ARBITRARY ANGULAR MOMENTA. GENERAL METHOD
C. Addition of two arbitrary angular momenta. General method
C-1. Review of the general theory of angular momentum
Consider an arbitrary system, whose state space is, and an angular momentumJ
relative to this system (Jcan be either a partial angular momentum or the total angular
momentum of the system). We showed in Chapter) that it is always possible
to construct a standard basis composed of eigenvectors common toJ
2
and
:
J
2
=(+ 1)~
2
(C-1a)
=~ (C-1b)
such that the action of the operators+and obeys the relations:
=~
(+ 1)(1) 1 (C-2)
We denote by()the vector space spanned by the set of vectors of the standard
basis which correspond to xed values ofandThere are(2+ 1)of these vectors,
and, according to (C-1) and (C-2), they can be transformed into each other byJ
2
,,+
and. The state space can be considered to be a direct sum of orthogonal subspaces
()which possess the following properties:
()()is(2+ 1)-dimensional.
()()is globally invariant under the action ofJ
2
,,, and, more generally, of
any function(J). In other words, these operators have non-zero matrix elements
only inside each of the subspaces().
() ()the matrix elements of any function(J)of the angular
momentumJare independent of
Comment:
As we pointed out in , we can give the indexa concrete physical meaning
by choosing for the standard basis the system of eigenvectors common toJ
2
,and one or
several observables which commute with the three components ofJand form a C.S.C.O. with
J
2
and. If, for example:
[J] = 0 (C-3)
and if the setJ
2
is a C.S.C.O., we require the vectors to be eigenvectors of:
= (C-4)
Relations (C-1), (C-2) and (C-4) determine the standard basis in this case. Each of
the()is an eigensubspace ofand the indexdistinguishes between the various eigenvalues
associated with each value of.
1025

CHAPTER X ADDITION OF ANGULAR MOMENTA
C-2. Statement of the problem
C-2-a. State space
Consider a physical system formed by the union of two subsystems (for example,
a two-particle system). We shall use indices 1 and 2 to label quantities relating to the
two subsystems.
We shall assume that we know, in the state space1of subsystem (1), a standard
basis11 1composed of common eigenvectors ofJ
2
1and1, whereJ1is the
angular momentum operator of subsystem (1):
J
2
111 1=1(1+ 1)~
2
11 1 (C-5a)
111 1= 1~11 1 (C-5b)
1 11 1=~
1(1+ 1) 1(11)11 11 (C-5c)
Similarly, we assume that the state space2of subsystem (2) is spanned by a standard
basis22 2:
J
2
222 2=2(2+ 1)~
2
22 2 (C-6a)
222 2= 2~22 2 (C-6b)
2 22 2=~
2(2+ 1) 2(21)22 21 (C-6c)
The state space of the global system is the tensor product of1and2:
=1 2 (C-7)
We know a basis of the global system, formed by taking the tensor product of the bases
chosen in1and2. We shall denote by12;12;1 2the vectors of this basis:
12;12;1 2=11 1 22 2 (C-8)
The spaces1and2can be considered to be the direct sums of the sub-spaces
1(11)and2(22), which possess the properties recalled in :
1= 1(11) (C-9a)
2= 2(22) (C-9b)
Consequently,is the direct sum of the subspaces(12;12)obtained by taking
the tensor product of a space1(11)and a space2(22):
= (12;12) (C-10)
with:
(12;12) =1(11) 2(22) (C-11)
The dimension of the subspace(12;12)is(21+ 1)(22+ 1). This subspace is
globally invariant under the action of any function ofJ1andJ2(J1andJ2here denote
the extensions intoof the angular momentum operators originally dened in1and2
respectively).
1026

C. ADDITION OF TWO ARBITRARY ANGULAR MOMENTA. GENERAL METHOD
C-2-b. Total angular momentum. Commutation relations
The total angular momentum of the system under consideration is dened by:
J=J1+J2 (C-12)
whereJ1andJ2, extensions of operators acting in the dierent spaces1and2, com-
mute. Of course, the components ofJ1, on the one hand, and ofJ2, on the other, satisfy
the commutation relations that characterize angular momenta. It is easy to verify that
the components ofJalso satisfy such relations [the calculation is the same as in (B-4)].
SinceJ1andJ2each commute withJ
2
1andJ
2
2, so doesJ. In particular,J
2
and
commute withJ
2
1andJ
2
2:
[J
2
1] = [J
2
2] = 0 (C-13a)
[J
2
J
2
1] = [J
2
J
2
2] = 0 (C-13b)
Furthermore,1and2obviously commute with:
[1 ] = [2 ] = 0 (C-14)
but not withJ
2
since this last operator can be written in terms ofJ1andJ2in the form:
J
2
=J
2
1+J
2
2+ 2J1J2 (C-15)
and, as in (B-9),1and2do not commute withJ1J2. We can also transform the
expression forJ
2
into:
J
2
=J
2
1+J
2
2+ 212+1+2+12+ (C-16)
C-2-c. The basis change to be performed
A vector12;12;1 2of basis (C-8) is a simultaneous eigenstate of the
observables:
J
2
1J
2
21 2 (C-17)
with the respective eigenvalues1(1+ 1)~
2
,2(2+ 1)~
2
,1~,2~.Basis(C-8)is well
adapted to the study of the individual angular momentaJ1andJ2of the two subsystems.
According to (C-13), the observables:
J
2
1J
2
2
2
(C-18)
also commute. We are going to construct an orthonormal system of common eigenvectors
of these observables:this new basis will be well adapted to the study of the total angular
momentumof the system. Note that this basis will be dierent from the preceding one,
sinceJ
2
does not commute with1and2(
Comment:
To give a physical meaning to the two indices1and2, let us assume (comment of ) that
we know, in1, a C.S.C.O.,1J
2
1
1where1commutes with the three components ofJ1,
1027

CHAPTER X ADDITION OF ANGULAR MOMENTA
and, in2, a C.S.C.O.,2J
2
2
2where2commutes with the three components ofJ2. We
can choose for a standard basis11 1the orthonormal system of eigenvectors common
to1,J
2
1
and1, and for22 2the orthonormal system of eigenvectors common to
2,J
2
2
and2. The set:
1 2;J
2
1
J
2
2
;1 2 (C-19)
then constitutes a C.S.C.O. in, whose eigenvectors are the kets (C-8). Since the observable1
commutes separately with the components ofJ1, and with those ofJ2, it also commutes with
Jand, in particular, withJ
2
and. The same is, of course, true of2. Consequently, the
observables:
1 2J
2
1
J
2
2
J
2
(C-20)
commute. We shall see that they in fact form a C.S.C.O.: the new basis we are trying to nd is
the orthonormal system of eigenvectors of this C.S.C.O.
The subspace(12;12)ofdened in (C-11) is globally invariant under the
action of any operator which is a function ofJ1andJ2, and, therefore, under the action
of any function of the total angular momentumJ. It follows that the observablesJ
2
and
, which we want to diagonalize, have non-zero matrix elements only between vectors
belonging to the same subspace(12;12). The matrices (which are, in general,
innite) representingJ
2
andin the basis (C-8) are block diagonal, that is, they can
be broken down into a series of submatrices, each of which corresponds to a particular
subspace(12;12).The problem therefore reduces to a change of basis inside each
of the subspaces(12;12), which are of nite dimension(21+ 1)(22+ 1).
Moreover, the matrix elements in the basis (C-8) of any function ofJ1andJ2are
independent of1and2This is therefore true of those ofJ
2
and. Consequently,
the problem of the diagonalization ofJ
2
and is the same inside all the subspaces
(12;12)that correspond to the same values1and2. It is for this reason that
one usually speaks ofadding angular momenta1and2without specifying the other
quantum numbers. To simplify the notation, we shall henceforth omit the indices1and
2. We shall denote by(12)the subspace(12;12)and by12;1 2,
the vectors of basis (C-8) belonging to this subspace:
(12)(12;12) (C-21a)
12;1 2 12;12;1 2 (C-21b)
SinceJis an angular momentum and(12)is globally invariant under the action
of any function ofJ, the results of Chapter ) are applicable.
Consequently,(12)is a direct sum of orthogonal subspaces(), each of which is
globally invariant under the action ofJ
2
,,+and:
(12) = () (C-22)
Thus, nally, we are left with the following double problem:
() 1and2, what are the values ofwhich appear in (C-22), andhow many
distinct subspaces()are associated with each of them?
()How can the eigenvectors ofJ
2
andbelonging to(12)be expanded on the
12;1 2basis?
1028

C. ADDITION OF TWO ARBITRARY ANGULAR MOMENTA. GENERAL METHOD

Comments:
() J1andJ2as the angular momenta of two distinct sub-
spaces. In fact, we know () that we may be adding the orbital and spin
angular momenta of the same particle. All the discussions and results of this
section are applicable to this case, with1and2simply being replaced by
rand.
()
the angular momentum so obtained to the third one, and so on until the last
one has been added.
C-3. Eigenvalues ofJ
2
and
C-3-a. Special case of two spin 1/2's
First of all, let us again take up the simple problem treated in . The spaces
1and2each contain, in this case, a single invariant subspace, and the tensor product
space, a single subspace(12), for which1=2= 12.
The results recalled in
numberassociated with the total spin. The space=(1212)must be a direct
sum of(2+1)-dimensional subspaces(). Each of these subspaces contains one and
only one eigenvector ofcorresponding to each of the values ofsuch that .
Now, we know (cf.) that the only values taken on byare 1,1and 0, the rst
two being non-degenerate and the third, two-fold degenerate. From this, the following
conclusions can be deduced directly:
() greater than 1 are excluded. For example, for= 2to be possible
there would have to exist at least one eigenvector ofof eigenvalue2~.
()= 1occurs (since= 1does) only once:= 1is not degenerate.
() = 0. The subspace characterized by= 1includes only
one vector for which= 0, and this value ofis doubly degenerate in the space
(1212).
The four-dimensional space(1212)can therefore be broken down into a sub-
space associated with= 1(which is three-dimensional) and a subspace associated with
= 0(which is one-dimensional).
Using a completely analogous argument, we shall determine the possible values of
in the general case in which1and2are arbitrary.
C-3-b. The eigenvalues of and their degrees of degeneracy
In accordance with the conclusions of , we shall consider a well-dened
subspace(12), of dimension(21+ 1)(22+ 1). We shall assume that1and2are
labeled such that:
1 2 (C-23)
1029

CHAPTER X ADDITION OF ANGULAR MOMENTA
The vectors12;1 2are already eigenstates of:
12;1 2= (1+2)12;1 2
= (1+2)~12;1 2 (C-24)
and the corresponding eigenvalues~are such that:
= 1+2 (C-25)
Consequently,takes on the following values:
1+2 1+21 1+22 (1+2) (C-26)
To nd the degree of degeneracy
12
()of these values, we can use the following
geometrical procedure. In a two-dimensional diagram, we associate with each vector
12;1 2the point whose abscissa is1and whose ordinate is2. All these
points are situated inside, or on the sides of, the rectangle whose corners are at(12),
(1 2),(1 2)and(12). Figure
basis vectors in the case in which1= 2and2= 1(the values of1and2are shown
beside each point). All points situated on the same dashed line (of slope1) correspond
to the same value of= 1+2. The number of such points is therefore equal to the
degeneracy
12
()of this value of(0, 1)
m
2
m
1
M = 3
M =  – 3
M
=  – 2
M
=  – 1
M
=  0
M
=  1
M
=  2
(– 2, 1)
(– 2, – 1) (– 1, – 1) (0, – 1) (1, – 1) (2, – 1)
(– 1, 1)
(– 2, 0) (– 1, 0)(0, 0)(1, 0)
(1, 1) (2, 1)
(2, 0)
Figure 1: Pairs of possible values(1 2)for the kets12;1 2. We have
chosen the case in which1= 2and2= 1. The points associated with a given value of
= 1+2are situated on a straight line of slope1(dashed lines).
Now consider the various values of, in decreasing order, tracing the line dened
by each of them (Fig.).=1+2is not degenerate, since the line it characterizes
1030

C. ADDITION OF TWO ARBITRARY ANGULAR MOMENTA. GENERAL METHOD– 3 – 2 – 1 0
1
2
3
ɡ
2,1
(M)
1 2 3
M
Figure 2: Value of the degree of degeneracy
12
()as a fonction of. As in Figure,
we have shown the case in which1= 2and2= 1. The degree of degeneracy
12
()
is simply obtained by counting the number of points on the corresponding dashed line of
Figure.
passes only through the upper right-hand corner, whose coordinates are(12):
12(1+2) = 1 (C-27)
=1+21is doubly degenerate, since the corresponding line contains the points
(121)and(112):
12(1+21) = 2 (C-28)
The degree of degeneracy thus increases by one whendecreases by one, until we reach
the lower right-hand corner of the rectangle(1=1 2= 2), that is, the value
=1 2. The number of points on the line is then at a maximum and is equal to:
12(1 2) = 22+ 1 (C-29)
When falls below1 2,
12
()rst remains constant and equal to its maxi-
mum value as long as the line associated withcuts across the entire width of the
rectangle, that is, until it passes through the upper left-hand corner of the rectangle
(1= 1 2=2):
12
() = 22+ 1 for(1 2) 1 2 (C-30)
Finally, forless than(1 2), the corresponding line no longer intersects with the
upper horizontal side of the rectangle, and
12()steadily decreases by one each time
decreases by one, again reaching 1 when=(1+2)(lower left-hand corner of
the rectangle). Consequently:
12
() =
12
() (C-31)
These results are summarized, for1= 2and2= 1, in Figure, which gives21()as
a function of
1031

CHAPTER X ADDITION OF ANGULAR MOMENTA
C-3-c. The eigenvalues of J
2
Note, rst of all, that the values (C-26) ofare all integral if1and2are both
integral or both half-integral, and all half-integral if one of them is integral and the other
half-integral. Consequently, the corresponding values ofwill also be all integral in the
rst case and all half-integral in the second.
Since the maximum value attained byis1+2, none of the values ofgreater
than1+2is found in(12)and therefore none appears in the direct sum (C-22).
With=1+2is associated one invariant subspace (since=1+2exists) and
only one (since=1+2is not degenerate). In this subspace(=1+2), there
is one and only one vector which corresponds to=1+21; now this value of
is two-fold degenerate in(12); therefore,=1+21also occurs, and to it
corresponds a single invariant subspace(=1+21).
More generally, we shall denote by
12()the number of subspaces()of
(12)associated with a given value of, that is, the number of dierent values of
for this value of(1and2having been xed at the beginning).
12
()and
12
()
are very simply related. Consider a particular value ofTo it corresponds one and
only one vector in each subspace()such that . Its degree of degeneracy
12()in(12)can therefore be written:
12
() =
12
(= ) +
12
(= + 1)
+
12
(= + 2) + (C-32)
Inverting, we obtain
12
()in terms of
12
():
12() =
12(=)
12(=+ 1)
=
12(=)
12(= 1) (C-33)
The results of
numberwhich actually occur in(12)and the number of invariant subspaces()
which are associated with them. First of all, we obviously have:
12
() = 0 for 1+2 (C-34)
since
12
()is zero for 1+2. Furthermore, according to (C-27) and (C-28):
12(=1+2) =
12(=1+2) = 1 (C-35a)
12(=1+21) =
12(=1+21)
12(=1+2) = 1(C-35b)
Thus, by iteration, we nd all the values of
12
():
12
(=1+22) = 1 (C-36a)
12
(=1 2) = 1 (C-36b)
and, nally, according to (C-30):
12
() = 0 for 1 2 (C-37)
Therefore, for xed1and2that is, inside a given space(12), the eigenvalues
ofJ
2
are such that
3
:
=1+2 1+21 1+22 1 2 (C-38)
3
Thus far, we have assumed1 2, but it is simple to extend the discussion to the opposite case
1 2: all we need to do is invert indices 1 and 2.
1032

C. ADDITION OF TWO ARBITRARY ANGULAR MOMENTA. GENERAL METHOD
With each of these values is associateda singleinvariant subspace(), so that the index
which appears in (C-22) is actually unnecessary. This means, in particular, that if we
x a value ofbelonging to the set (C-38) and a value ofwhich is compatible with
it, there corresponds to them one and only one vector in(12): the specication of
suces for the determination of the subspace(), in which the specication ofthen
denes one and only one vector. In other words,J
2
andform a C.S.C.O. in(12).
Comment:
It can be shown that the number of pairs()found in(12)is indeed equal
to the dimension(21+ 1)(22+ 1)of this space. This number (if, for example,
1 2) is equal to:
1+2
=1 2
(2+ 1) (C-39)
If we set:
=1 2+ (C-40)
it is easy to calculate the sum (C-39):
1+2
=1 2
(2+ 1) =
22
=0
[2(1 2+) + 1]
= [2(1 2) + 1](22+ 1) + 2
22(22+ 1)
2
= (22+ 1)(21+ 1) (C-41)
C-4. Common eigenvectors ofJ
2
and
We shall denote by the common eigenvectors ofJ
2
andbelonging to the
space(12). To be completely rigorous, we should have to recall the values of1and
2in this notation, but we shall not write them explicitly, since they are the same as in
the vectors (C-21b) of which the are linear combinations. Of course, the indices
andrefer to the eigenvalues ofJ
2
and:
J
2
=(+ 1)~
2
(C-42a)
=~ (C-42b)
and the vectors , like all those of the space(12), are eigenvectors ofJ
2
1and
J
2
2with eigenvalues1(1+ 1)~
2
and2(2+ 1)~
2
respectively.
C-4-a. Special case of two spin 1/2's
First of all, we shall show how use of the general results concerning angular mo-
menta leads us to the expression for the vectorsestablished in . It will not
be necessary to diagonalize the matrix which representsS
2
. By generalizing this method,
we shall then construct ( 4-b) the vectorsfor the case of arbitrary1and2
1033

CHAPTER X ADDITION OF ANGULAR MOMENTA
. The subspace(= 1)
The ket++is, in the state space=(1212), the only eigenvector of
associated with= 1. SinceS
2
and commute, and the value= 1is not
degenerate,++must also be an eigenvector ofS
2
( ). According
to the reasoning of , the corresponding value ofmust be 1. Therefore, we can
choose the phase of the vector= 1 = 1such that:
11=++ (C-43)
It is then easy to nd the other states of the triplet, since we know from the general
theory of angular momentum that:
11=~
1(1 + 1)1(11)10
=~
210 (C-44)
Consequently:
10=
1
~2
++ (C-45)
To calculate10explicitly in the12basis, it suces to recall that denition
(B-3) of the total spinSimplies:
=1+2 (C-46)
We then obtain:
10=
1
~2
(1+2)++
=
1
~2
[~++~+ ]
=
1
2
[+++ ] (C-47)
Finally, we can again applyto10, that is,(1+2)to expression (C-47). This
yields:
11=
1
~2
10
=
1
~2
(1+2)
1
2
[+++ ]
=
1
2~
[~ +~ ]
= (C-48)
Of course, this last result could have been obtained directly, using an argument analogous
to the one applied above to++. However, the preceding calculation has a slight
advantage: it enables us, in accordance with the general conventions set forth in
of Chapter, to x the phase factors which could appear in10and11with
respect to the one chosen for11in (C-43).
1034

C. ADDITION OF TWO ARBITRARY ANGULAR MOMENTA. GENERAL METHOD
. The state = 0 = 0
The only vector= 0 = 0of the subspace(= 0)is determined, to within
a constant factor, by the condition that it must be orthogonal to the three vectors1
that we have just constructed.
Since it is orthogonal to11=++and11= ,00must be a
linear combination of+ and +:
00=+ + + (C-49)
which will be normalized if:
0000=
2
+
2
= 1 (C-50)
We now insist that its scalar product with10[cf.(C-47)] be zero:
1
2
(+) = 0 (C-51)
The coecientsandare therefore equal in absolute value and of opposite sign. With
(C-50) taken into account, this xes them to within a phase factor:
= =
1
2
e (C-52)
whereis any real number. We shall choose= 0, which yields:
00=
1
2
[+ +] (C-53)
Thus we have calculated the four vectorswithout explicitly having had to
write the matrix that representsS
2
in the12basis.
C-4-b. General case (arbitrary 1and2)
We showed in (12)into a direct sum of
invariant subspaces()is:
(12) =(1+2)(1+21) (1 2) (C-54)
We shall now see how to determine the vectorsthat span these subspaces.
. The subspace(=1+2)
The ket12;1=1 2=2is, in(1 2), the only eigenvector of
associated with=1+2. SinceJ
2
andcommute, and the value=1+2is
not degenerate,12;1=1 2=2must also be an eigenvector ofJ
2
. According
to (C-54), the corresponding value ofcan only be1+2. We can choose the phase of
the vector:
=1+2 =1+2
1035

CHAPTER X ADDITION OF ANGULAR MOMENTA
such that:
1+2 1+2=12;12 (C-55)
Repeated application of the operatoron this expression enables us to complete
the family of vectors for which=1+2. According to the general formulas
(C-50) of Chapter:
1+2 1+2=~
2(1+2)1+2 1+21 (C-56)
We can therefore calculate the vector corresponding to=1+2and=1+21
by applying=1+2to the vector12;12:
1+2 1+21=
1
~2(1+2)
1+2 1+2
=
1
~2(1+2)
(1+2)12;12
=
1
~2(1+2)
~
2112;112
+~
2212;121 (C-57)
that is:
1+2 1+21=
1
1+2
12;112
+
2
1+2
12;121 (C-58)
Note that we obtain in this way a linear combination of the two basis vectors that
correspond to=1+21, and that this combination is directly normalized.
We then repeat the procedure: we construct1+21+22by letting
act on both sides of (C-58) (for the right-hand side, we take this operator in the form
1+2), and so on, through1+2(1+2), which is found to be equal to
12;1 2.
We therefore know how to calculate the rst[2(1+2)+1]vectors of the
basis, which correspond to=1+2and=1+21+21 (1+2)and
span the subspace(=1+2)of(12).
. The other subspaces()
Now consider the spaceS(1+2), the supplement of(1+2)in(12).
According to (C-54),S(1+2)can be broken down into:
S(1+2) =(1+21)(1+22) (1 2) (C-59)
We can therefore apply to it the same reasoning as was used in .
1036

C. ADDITION OF TWO ARBITRARY ANGULAR MOMENTA. GENERAL METHOD
InS(1+2), the degree of degeneracy
12
()of a given value ofis smaller
by one than
12
(), since(1+2)possesses one and only one vector associated with
this value of:
12
() =
12
()1 (C-60)
This means, in particular, that=1+2no longer exists inS(1+2), and that the
new maximum value =1+21is not degenerate. From this we see, as in , that
the corresponding vector must be proportional to=1+21=1+21. It
is easy to nd its expansion on the12;1 2basis, since, because of the value
of, it is surely of the form:
1+21 1+21= 12;121
+ 12;112 (C-61)
with:
2
+
2
= 1 (C-62)
to insure its normalization. It must also be orthogonal to1+21+21, which
belongs to(1+2)and whose expression is given by (C-58). The coecientsand
must therefore satisfy:
2
1+2
+
1
1+2
= 0 (C-63)
Relations (C-62) and (C-63) determineandto within a phase factor. We shall choose
andto be real and, for example,positive. With these conventions:
1+21 1+21=
1
1+2
12;121
2
1+2
12;112 (C-64)
This vector is the rst of a new family, characterized by=1+21. As in ,
we can derive the others by applyingas many times as necessary. Thus we obtain
[2(1+21) + 1]vectors corresponding to
=1+21 and =1+211+22 (1+21)
and spanning the subspace(=1+21).
Now consider the spaceS(1+21+21), the supplement of the direct sum
(1+2)(1+21)in(12)
4
:
S(1+21+21) =(1+22) (1 2) (C-65)
In the spaceS(1+2 1+21), the degeneracy of each value ofis again
decreased by one with respect to what it was inS(1+2). In particular, the maximum
4
Of course,S(1+21+21)exists only if1+22is not less than1 2
1037

CHAPTER X ADDITION OF ANGULAR MOMENTA
value is now=1+22, and it is not degenerate. The corresponding vector of
S(1+21+21)must therefore be=1+22 =1+22. To express
it in the12;1 2basis, it is sucient to note that it is a linear combination
of the three vectors12;122,12;1121,12;122. The
coecients of this combination are xed to within a phase factor by the triple condition
that it be normalized and orthogonal to1+2 1+22and1+21 1+22
(which are already known). Finally, the use ofenables us to nd the other vectors of
this third family, thus dening(1+22).
The procedure can be repeated without diculty until we have exhausted all values
ofgreater than or equal to1 2[and, consequently, according to (C-31), also all
those less than or equal to1 2]. We then know all the desiredvectors. This
method will be illustrated by two examples in ComplementX.
C-4-c. Clebsch-Gordan coecients
In each space(12), the eigenvectors ofJ
2
andare linear combinations of
vectors of the initial12;1 2basis:
=
1
1= 1
2
2= 2
12;1 2 12;1 2 (C-66)
The coecients12;1 2 of these expansions are calledClebsch-Gordan co-
ecients.
Comment:
To be completely rigorous, we should write the vectors12;1 2and
as12;12;1 2and12;12; respectively [the values of1and
2, like those of1and2, would then be the same on both sides of relations (C-
66)]. However, we shall not write1and2in the symbols which represent the
Clebsch-Gordan coecients, since we know that these coecients are independent
of1and2( ).
It is not possible to give a general expression for the Clebsch-Gordan coecients,
but the method presented in
values of1and2For practical applications, there arenumerical tablesof Clebsch-
Gordan coecients.
Actually, to determine the Clebsch-Gordan coecients uniquely, a certain number
ofphase conventionsmust be chosen. [We mentioned this fact when we wrote expressions
(C-55) and (C-64)]. Clebsch-Gordan coecients are always chosen to be real. The choice
then bears on the signs of some of them (obviously, the relative signs of the coecients
appearing in the expansion of the same vectorare xed; only the global sign of
the expansion can be chosen arbitrarily).
The results of 12;1 2 is dierent from zero only
if:
= 1+2 (C-67a)
1 2 1+2 (C-67b)
1038

C. ADDITION OF TWO ARBITRARY ANGULAR MOMENTA. GENERAL METHOD
whereis of the same type (integral or half-integral) as1+2and1 2. Condition
(C-67b) is often called the triangle rule: one must be able to form a triangle with three
line segments of lengths12and
Since the vectors also form an orthonormal basis of the space(12), the
expressions which are the inverse of (C-66) can be written:
12;1 2=
1+2
=1 2 =
12;1 2 (C-68)
As the Clebsch-Gordan coecients have all been chosen to be real, the scalar products
appearing in (C-68) are such that:
12;1 2=12;1 2 (C-69)
The Clebsch-Gordan coecients therefore enable us to express the vectors of the old
basis12;1 2, in terms of those of the new basis.
The Clebsch-Gordan coecients possess interesting properties, some of which will
be studied in ComplementX.
References and suggestions for further reading:
Messiah (1.17), Chap. XIII, V; Rose (2.19), Chap. III; Edmonds (2.21), Chaps.
3 and 6.
Relation with group theory: Meijer and Bauer (2.18), Chap. 5 5 and App. III
of that Chapter; Bacry (10.31 ),Chap. 6; Wigner (2.23), Chaps. 14 and 15.
Vectorial spherical harmonics: Edmonds (2.21), 5-10; Jackson (7.5), Chap. 16;
Berestetskii et al. (2.8), 6 and 7 ; Akhiezer and Berestetskii (2.14), 4.
1039

COMPLEMENTS OF CHAPTER X, READER'S GUIDE
AX: EXAMPLES OF ADDITION OF ANGULAR MO-
MENTA
Illustrates the results of Chapter
plest cases not treated in detail in this chapter:
two angular momenta equal to1, and an integral
angular momentum with a spin12. Easy,
recommended as an exercise illustrating methods
of addition of angular momenta.
BX: CLEBSCH-GORDAN COEFFICIENTS
CX: ADDITION OF SPHERICAL HARMONICS
Technical complements intended to demonstrate
certain useful mathematical results; can be used
as references.
BX: study of Clebsch-Gordan coecients, which
frequently appear in physical problems involving
angular momentum and rotational invariance.
CX: proof of an expression concerning the
product of spherical harmonics; useful for certain
subsequent complements and exercises.
DX: VECTOR OPERATORS: THE WIGNER-
ECKART THEOREM
EX: ELECTRIC MULTIPOLE MOMENTS
Introduction of physical concepts (vector observ-
ables, multipole moments) which play important
roles in numerous elds.
DX: study of vector operators; proof of the
Wigner-Eckart theorem, which establishes pro-
portionality rules between the matrix elements of
these operators. Rather theoretical, but recom-
mended for its numerous applications. Can be
helpful in an atomic physics course (the vector
model, calculation of Landé factors, etc.).
EX: denition and properties of electric multipole
moments of a classical or quantum mechanical
system; study of their selection rules (these
multipole moments are frequently used in atomic
and nuclear physics). Moderately dicult.
FX: EVOLUTION OF TWO ANGULAR MOMENTA
J1ANDJ2COUPLED BY AN INTERACTION J1
J2
Can be considered to be a worked exercise, treat-
ing a problem fundamental to the vector model
of the atom: the time evolution of two angular
momentaJ1andJ2coupled by an interaction
=J1J2. This dynamical point of view
completes, as it were, the results of Chapter
concerning the eigenstates of. Fairly simple.
GX: EXERCISES Exercises 7 to 10 are more dicult than the
others. Exercices 7, 8, 9 are extensions of
ComplementsXandX(concept of a standard
component and that of an irreducible tensor
operator, the Wigner-Eckart theorem). Exercise
10 takes up the problem of the various ways of
coupling three angular momenta.
1041

EXAMPLES OF ADDITION OF ANGULAR MOMENTA
Complement AX
Examples of addition of angular momenta
1 Addition of j1= 1andj2= 1. . . . . . . . . . . . . . . . . . .
1-a The subspace (= 2). . . . . . . . . . . . . . . . . . . . . .
1-b The subspace (= 1). . . . . . . . . . . . . . . . . . . . . .
1-c The vector = 0 = 0. . . . . . . . . . . . . . . . . . .
2 Addition of an integral orbital angular momentum and a
spin 1/2
2-a The subspace (=+ 12). . . . . . . . . . . . . . . . . .
2-b The subspace (= 12). . . . . . . . . . . . . . . . . .
To illustrate the general method of addition of angular momenta described in
Chapter, we shall apply it here to two examples.
1. Addition ofj1= 1andj2= 1
First consider the case in which1=2= 1. This is the case, for example, for a two-
particle system in which both orbital angular momenta are equal to 1. Since each of the
two particles is then in astate, this is said to be a
2
conguration.
The space(11)with which we are concerned has33 = 9dimensions. We
assume the basis composed of common eigenstates ofJ
2
1,J
2
2,1and2to be known:
11;1 2 with 1 2= 101 (1)
and we want to determine the basis of common eigenvectors ofJ
2
1,J
2
2,J
2
and
whereJis the total angular momentum.
According to , the possible values of the quantum number
are:
= 210 (2)
We must therefore construct three families of vectors, containing, respectively,
ve, three and one vectors of the new basis.
1-a. The subspace (= 2)
The ket= 2 = 2can be written simply:
22=11 ; 11 (3)
1043

COMPLEMENT A X
Applyingto it, we nd the vector= 2 = 1:
21=
1
2~
22
=
1
2~
(1+2)11 ; 11
=
1
2~
~
211 ; 01+~211 ; 10
=
1
2
[11 ; 10+11 ; 01] (4)
We use again to calculate= 2 = 0. After a simple calculation, we nd:
20=
1
6
[11 ; 11+ 211 ; 00+11 ;11] (5)
then:
21=
1
2
[11 ; 01+11 ;10] (6)
and, nally:
22=11;11 (7)
1-b. The subspace (= 1)
We shall now proceed to the subspace(= 1). The vector= 1 = 1must
be a linear combination of the two basis kets11 ; 10and11 ; 01(the only ones
for which= 1):
11=11 ; 10+11 ; 01 (8)
with:
2
+
2
= 1 (9)
For it to be orthogonal to the vector21, it is necessary [cf.(4)] that:
+= 0 (10)
We chooseandto be real, and choose, by convention,positive
1
. Under these
conditions:
11=
1
2
[11 ; 1011 ; 01] (11)
Application ofhere again enables us to deduce10and11. We easily nd, using
the same technique as above:
10=
1
2
[11 ; 1111 ;11] (12)
11=
1
2
[11 ; 0111 ;10] (13)
1
The component of the ket on the ket12;1=1 2= 1is always chosen to be
real and positive (cf.ComplementX, ).
1044

EXAMPLES OF ADDITION OF ANGULAR MOMENTA
It is interesting to note that expansion (12) does not contain the vector11 ; 00,
although it also corresponds to= 0. It so happens that the corresponding Clebsch-
Gordan coecient is zero:
11 ; 0010= 0 (14)
1-c. The vector = 0 = 0
We are left with the calculation of the last vector of thebasis, associated
with= = 0. This vector is a linear combination of the three basis kets for which
= 0:
00=11 ; 11+11 ; 00+11 ;11 (15)
with:
2
+
2
+
2
= 1 (16)
It must also be orthogonal to20[formula (5)] and10[formula (12)]. This gives the
two conditions:
+ 2+= 0 (17a)
= 0 (17b)
These relations imply:
= = (18)
We again choose,andreal, and agree to choosepositive (see note). We then
obtain, using (16) and (18):
00=
1
3
[11 ; 1111 ; 00+11 ;11] (19)
This completes the construction of the basis for the case1=2= 1.
Comment:
If the physical problem under study is that of a
2
conguration of a two-particle
system, the wave functions which represent the states of the initial basis are of the
form:
r1r211 ;1 2=
11(1)
21(2)
1
1
(11)
2
1
(22) (20)
wherer1(111)andr2(222)give the positions of the two particles. Since
the radial functions are independent of the quantum numbers1and 2, the
linear combinations that give the wave functions associated with the kets
are functions only of the angular dependence. For example, in ther1r2
representation, equation (19) can be written:
r1r200=
11(1)
21(2)
1
3
1
1(11)
1
1
(22)
0
1(11)
0
1(22) +
1
1
(11)
1
1(22) (21)
1045

COMPLEMENT A X
2. Addition of an integral orbital angular momentumand a spin 1/2
Now consider the addition of an orbital angular momentum (1=, an integer) and a
spin l/2 (2= 12). This problem is encountered, for example, whenever one wants to
study the total angular momentum of a spin 1/2 particle such as the electron.
The space(12)which we are considering here is2(2+ 1)-dimensional. We
already know a basis of this space
2
:
12 ; with = 1 and= (22)
formed of eigenstates of the observablesL
2
,S
2
,and, whereLandSare the orbital
angular momentum and spin under consideration. We want to construct the eigenvectors
ofJ
2
and whereJis the total angular momentum of the system:
J=L+S (23)
First of all, note that ifis zero, the solution to the problem is obvious. It is easy
to show in this case that the vectors012; 0are also eigenvectors ofJ
2
andwith
eigenvalues such as= 12and=2. On the other hand, ifis not zero, there are
two possible values of:
=+
1
2
1
2
(24)
2-a. The subspace (=+ 12)
The(2+ 2)vectors spanning the subspace(=+ 12)can be obtained
by using the general method of Chapter. We have, rst of all:
+
1
2
+
1
2
=
1
2
;+ (25)
Through the action
3
of, we obtain+
1
2
1
2
:
+
1
2
1
2
=
1
~2+ 1
+
1
2
+
1
2
=
1
~2+ 1
(+)
1
2
;+
=
1
~2+ 1
~
2
1
2
;1++~
1
2
;
=
22+ 1
1
2
;1++
1
2+ 1
1
2
; (26)
2
If we wanted to conform strictly to the notation of Chapter, we should write12, and not, in
the basis kets. But we agreed in Chapters in the spin state
space by+and .
3
To nd the numerical coecients appearing in the following equations, we can simply use the
relation:(+ 1) ( 1) = (+)( + 1).
1046

EXAMPLES OF ADDITION OF ANGULAR MOMENTA
We apply again. An analogous calculation yields:
+
1
2
3
2
=
1
2+ 121
1
2
;2+
+
2
1
2
;1 (27)
More generally, the vector+ 12 will be a linear combination of the only
two basis vectors associated with:12; 12+and12;+ 12 (
is, of course, half-integral). Comparing (25), (26) and (27), we can guess that this linear
combination should be:
+
1
2
=
1
2+ 1
++
12
1
2
;
1
2
+
+
+
12
1
2
;+
1
2
(28)
with:
=+
1
2
1
2
3
2
+
1
2
+
1
2
(29)
Reasoning by recurrence, we can show this to be true, since application ofto both
sides of (28) yields:
+
1
2
1=
1
~++
1
2
+
3
2
+
1
2
=
1
~++
1
2
+
3
2
1
2+ 1
++
12
~
+
12
+
3
2
1
2
;
3
2
+
+
++
12
~
1
2
;
1
2
+
+
12
~
++
12
+
1
2
1
2
;
1
2
=
1
2+ 1
(+
12
1
2
;
3
2
+
+
+
32
1
2
;
1
2
(30)
We indeed obtain the same expression as in (28), withchanged to 1.
1047

COMPLEMENT A X
2-b. The subspace (= 12)
We shall now try to determine the expression for the2vectors associated
with= 12. The one which corresponds to the maximum value12ofis
a normalized linear combination of12 ;1+and12 ; , and it must be
orthogonal to+ 12 12[formula (26)]. Choosing the coecient of12 ;
real and positive (cf.note), we easily nd:
1
2
1
2
=
1
2+ 12
1
2
;
1
2
;1+ (31)
The operator enables us to deduce successively all the other vectors of the
family characterized by= 12. Since there are only two basis vectors with a given
value of, and since12 is orthogonal to+ 12 , (28) leads us to expect
that:
1
2
=
1
2+ 1
++
12
1
2
;+
1
2
+
12
1
2
;
1
2
+ (32)
for:
=
1
2
3
2
+
3
2
1
2
(33)
By an argument analogous to the one in , this formula can also be proved by
recurrence.
Comments:
() 12; of a spin 1/2 particle can be represented by two-
component spinors of the form:
1
2
;+(r) =()()
1
0
(34a)
1
2
; (r) =()()
0
1
(34b)
The preceding calculations then show that the spinors associated with the
states can be written:
+
1
2
(r) =
1
2+ 1
()
++
1
2
1
2
()
+
1
2
+
1
2
()
(35a)
1
2
(r) =
1
2+ 1
()
+
1
2
1
2
()
++
1
2
+
1
2
()
(35b)
1048

EXAMPLES OF ADDITION OF ANGULAR MOMENTA
() = 1, formulas (25), (28), (31) and (32) yield:
3
2
3
2
=1
1
2
; 1+
3
2
1
2
=
23
1
1
2
; 0++
1
3
1
1
2
; 1
3
2
1
2
=
1
3
1
1
2
;1++
23
1
1
2
; 0
3
2
3
2
=1
1
2
;1 (36a)
and:
1
2
1
2
=
23
1
1
2
; 1
1
3
1
1
2
; 0+
1
2
1
2
=
1
3
1
1
2
; 0
23
1
1
2
;1+ (36b)
References and suggestions for further reading:
Addition of an angular momentumand an angular momentum= 1: see vectorial
spherical harmonics in the references of Chapter.
1049

CLEBSCH-GORDAN COEFFICIENTS
Complement BX
Clebsch-Gordan coecients
1 General properties of Clebsch-Gordan coecients
1-a Selection rules
1-b Orthogonality relations
1-c Recurrence relations
2 Phase conventions. Reality of Clebsch-Gordan coecients
2-a The coecients 12;1 2 ; phase of the ket.
2-b Other Clebsch-Gordan coecients
3 Some useful relations
3-a The signs of some coecients
3-b Changing the order of 1and2. . . . . . . . . . . . . . . . .
3-c Changing the sign of ,1and 2. . . . . . . . . . . . . .
3-d The coecients ; 00. . . . . . . . . . . . . . .
Clebsch-Gordan coecients were introduced in Chaptercf.relation (C-66)]:
they are the coecients12;1 2 involved in the expansion of the ket
on the12;1 2basis:
=
1
1= 1
2
2= 2
12;1 2 12;1 2 (1)
In this complement, we shall derive some interesting properties of Clebsch-Gordan
coecients, some of which were simply stated in Chapter.
Note that, to dene the12;1 2 completely, equation (1) is not suf-
cient. The normalized vector is xed only to within a phase factor by the
corresponding eigenvalues(+ 1)~
2
and~, and a phase convention must be chosen
in order to complete the denition. In Chapter, we used the action of theand
+operators to x the relative phase of the(2+ 1)kets associated with the
same value ofIn this complement, we shall complete this choice of phase by adopting
a convention for the phase of the kets. This will enable us to show that all the
Clebsch-Gordan coecients are then real.
However, before approaching, in 2, the problem of the choice of the phase of the
12;1 2 , we shall, in 1, study some of their most useful properties which
do not depend on this phase convention. Finally, 3 presents various relations which
will be of use in other complements.
1. General properties of Clebsch-Gordan coecients
1-a. Selection rules
Two important selection rules, which follow directly from the results of Chapter
concerning the addition of angular momenta, have already been given in that chapter [cf.
1051

COMPLEMENT B X
relations (C-67a) and (C-67b)]. We shall simply restate them here: the Clebsch-Gordan
coecient12;1 2 is necessarily zero if the following two conditions are
not simultaneously satised:
= 1+2 (2)
1 2 1+2 (3a)
Inequality (3a) is often called the triangle selection rule, since it means that a triangle
can be formed with three line segments of lengths1,2and(cf.Fig.). These three
numbers therefore play symmetrical roles here, and (3a) can also be written in the form:
1 2 +1 (3b)
or:
2 1 +2 (3c)J
j
2
j
1
Figure 1: Triangle selection rule: the coe-
cient12;1 2 can be dierent
from zero only if it is possible to form a tri-
angle with three line segments of lengths1,
2,.
Moreover, the general properties of angular momentum require that the ket
and, therefore, the coecient12;1 2 exist only iftakes on one of the
values:
= 1 2 (4a)
Similarly, it is necessary that:
1=1 11 1 (4b)
2=2 21 2 (4c)
If this is not the case, the Clebsch-Gordan coecients are not dened. However, in what
follows, it will be convenient to assume that they exist for all1,2and, but that
they are zero if at least one of conditions (4) is not satised. These relations thus play
the role of new selection rules for the Clebsch-Gordan coecients.
1052

CLEBSCH-GORDAN COEFFICIENTS
1-b. Orthogonality relations
Inserting the closure relation
1
:
1
1= 1
2
2= 2
12;1 2 12;1 2= 1 (5)
in the orthogonality relation of the kets:
= (6)
we obtain:
1
1= 1
2
2= 2
12;1 2 12;1 2 = (7a)
We shall see later [cf.relation (18b)] that the Clebsch-Gordan coecients are real, which
enables us to write this relation in the form:
1
1= 1
2
2= 2
12;1 2 12;1 2 = (7b)
Thus we obtain a rst orthogonality relation for the Clebsch-Gordan coecients. We
note, moreover, that the summation which appears in it is performed over only one index:
for the coecients of the left-hand side to be dierent from zero,1and2must be
related by (2).
Similarly, we insert the closure relation:
1+2
=1 2=
= 1 (8)
in the orthogonality relation of the kets12;1 2; we obtain:
1+2
=1 2=
12;1 2 12;
1 2=
1
1
2
2
(9a)
that is, with (18b) taken into account:
1+2
=1 2=
12;1 2 12;
1 2 =
1
1
2
2
(9b)
Again, the summation is performed over only one index: since we must have=
1+2, the summation overreduces to a single term.
1
This closure relation is valid for a given subspace(12;12)(cf.Chap., C-2).
1053

COMPLEMENT B X
1-c. Recurrence relations
In this section, we shall use the fact that the kets12;1 2form a standard
basis. Thus:
1 12;1 2=~
1(1+ 1) 1(11)12;112
2 12;1 2=~
2(2+ 1) 2(21)12;1 21 (10)
Similarly, by construction, the ketssatisfy:
=~
(+ 1) ( 1) 1 (11)
We shall therefore apply theoperator to relation (1). Since=1+2,
we obtain (if ):
(+ 1) ( 1) 1=
1
1
= 1
2
2
= 2
12;
1 2
1(1+ 1)
1
(
1
1)12;
11
2
+
2(2+ 1)
2
(
2
1)12;
1 21(12)
Multiplying this relation by the bra12;1 2, we nd:
(+ 1) ( 1)12;1 2 1
=
1(1+ 1) 1(1+ 1)12;1+ 12
+
2(2+ 1) 2(2+ 1)12;1 2+ 1 (13)
If the value ofis equal to, we have = 0, and relation (13) remains valid if
we use the convention, given above in , according to which12;1+ 12
is zero if .
Analogously, application of the operator+=1++2+to relation (1) leads to:
(+ 1) (+ 1)12;1 2 + 1
=
1(1+ 1) 1(11)12;112
+
2(2+ 1) 2(21)12;1 21 (14)
(the left-hand side of this relation is zero if=); (13) and (14) are recurrence relations
for the Clebsch-Gordan coecients.
2. Phase conventions. Reality of Clebsch-Gordan coecients
As we have seen, expressions (12) x the relative phases of the ketsassociated
with the same value ofTo complete the denition of the Clebsch-Gordan coecients
involved in (1), we must choose the phase of the various kets. To this end, we shall
begin by studying some properties of the coecients12;1 2 .
1054

CLEBSCH-GORDAN COEFFICIENTS
2-a. The coecients 12;1 2 ; phase of the ket
In the coecient12;1 21 , the maximum value of 1is1=1. According
to selection rule (2),2is then equal to 1, [whose modulus is well below2, according
to (3b)]. As1decreases from this maximum value1, one unit at a time,2increases until it
reaches its maximum value2=2[1is then equal to 2, whose modulus is well below1,
according to (3c (1+2 + 1)non-zero Clebsch-Gordan coecients
12;1 2 can exist. We are going to show that, indeed, none of them is ever zero.
If we set=in (14), we obtain:
12;112 =
2(2+ 1) 2(21)1(1+ 1) 1(11)
12;1 21 (15)
The radical on the right-hand side of this relation is never zero, nor is it innite, so long as the
Clebsch-Gordan coecients appearing there satisfy rules (4b) and (4c). Relation (15) therefore
shows that if12;1 1 were equal to zero,12;11 1+ 1 would
be zero as well, as would be all the succeeding coecients12;1 1 . Now, this is
impossible, since the ket, which is normalized, cannot be zero. Therefore, all the coecients
12;1 1 (with1 1 2) are dierent from zero.
In particular, the coecient12;1 1 , in which1takes on its max-
imum value, is not zero. To x the phase of the ket, we shall require this coecient
to satisfy the condition:
12;1 1 real and positive (16)
Relation (15) then implies by recurrence that all the coecients
12;1 1
are real [their sign being(1)
1 1
].
Comment:
The phase convention we have chosen for the ket gives the two angular
momentaJ1andJ2asymmetrical roles. It actually depends on the order in which
the quantum numbers1and2are arranged in the Clebsch-Gordan coecients:
if1and2are permuted, the phase of the ketwill be xed by the condition:
21;2 2 real and positive (17)
which is not necessarily equivalent,a priori, to (16) [(16) and (17) may dene
dierent phases for the ket]. We shall return to this point in 3-b.
2-b. Other Clebsch-Gordan coecients
Relation (13) enables us to express, in terms of the12;1 2 , all the
coecients12;1 2 1; then, by the same method, all the other coecients
12;1 2 2, etc. This relation, in which no imaginary numbers are involved,
requires that all Clebsch-Gordan coecients be real:
12;1 2 =12;1 2 (18a)
1055

COMPLEMENT B X
which can also be written:
12;1 2 = 12;1 2 (18b)
However, the signs of the12;1 2 do not obey any simple rule for
=.
3. Some useful relations
In this section, we give some useful relations, which complement those given in 1. To
prove them, we shall begin by studying the signs of a certain number of Clebsch-Gordan
coecients.
3-a. The signs of some coecients
. The coecients 12;1 21+2
Convention (16) requires the coecient12;121+21+2to be real and pos-
itive; it is, moreover, equal to 1 (cf.Chap., ). Setting==1+2in (13), we
then see that the coecients12;1 21+21+21are positive. By recurrence, it
is then easy to prove that:
12;1 21+2 0 (19)
. Coecients in which 1has its maximum value
Consider the coecient12;1 2 . In theory, the maximum value of1is
1=1. However, we then have 2= 1, which, according to (4c), is possible only if
1 2that is:
1 2 (20)
If, on the other hand:
1 2 (21)
the maximum value of1corresponds to the minimum value of2(2= 2), and is therefore
equal to1=+2.
Let us show that all Clebsch-Gordan coecients for which1has its maximum value are
non-zero and positive. To do so, we set1=1, in (13); we nd:
(+ 1) ( 1)12;1 2 1
=
2(2+ 1) 2(2+ 1)12;1 2+ 1 (22)
Using this relation, an argument by recurrence starting with (16) shows that all the coecients
12;1 1 are positive [and non-zero ifsatises (20)]. Analogously, setting
2= 2in (14), we could prove that all the coecients12;+2 2 are positive
[ifsatises (21)].
1056

CLEBSCH-GORDAN COEFFICIENTS
. The coecients 12;1 2 and12;1 2
We saw in 2-a that the sign of12;1 2 is(1)
1 1
. In particular:
the sign of12; 22 is(1)
1+2
(23)
To determine the sign of12;1 2 , we can set= in (13), whose left-
hand side then goes to zero. We therefore see that the sign of12;1 2 changes
whenever 1(or2) varies by1. Since, according to ,12;2 2 is
positive, it follows that the sign of12;1 2 is(1)
2+2
, and, in particular:
the sign of12;1 +1 is(1)
1+2
(24)
3-b. Changing the order of 1and2
With the conventions we have chosen, the phase of the ket depends on the or-
der in which the two angular momenta 1and2are arranged in the Clebsch-Gordan co-
ecients (cf.comment of 2-a). If they are taken in the order1,2, the component of
along12;1 1is positive, which means that the sign of the component along
12; 22is(1)
1+2
, as is indicated by (23). On the other hand, if we pick the
order2,1, relation (17) shows that the latter component is positive. Therefore, if we invert1
and2, the ket is multiplied by(1)
1+2
. The same is true for the kets, which
are constructed from by the action ofin such a way that the order of1and2plays
no role. Finally, the exchange of1and2leads to the relation:
21;2 1 = (1)
1+2
12;1 2 (25)
3-c. Changing the sign of ,1and 2
In Chapter (and,
therefore, the Clebsch-Gordan coecients) from the kets, by applying the operator.
We can take the opposite point of view, and start with the kets, using the operator+.
The reasoning which follows is exactly the same, and we nd for the kets the same
expansion coecients on the kets12; 1 2as for the on the12;1 2.
The only dierences that can appear are related to the phase conventions for the kets,
since the analogue of (16) then requires12; 1 +1 to be real and positive.
Now, according to (24), the sign of this coecient is, in reality,(1)
1+2
. Consequently:
12; 1 2 = (1)
1+2
12;1 2 (26)
In particular, if we set1= 2= 0, we see that the coecient12; 000is zero
when1+2 is an odd number.
3-d. The coecients ; 00
According to (3a),can be zero only if1and2are equal. We therefore substitute the
values1=2= 1= 2= 1and== 0into (13); we obtain:
;+ 1(+ 1)00= ; 00 (27)
All the coecients; 00are therefore equal in modulus. Their signs change when-
evervaries by one, and, since; 00is positive, it is given by(1). Taking into
account orthogonality relation (7b), which indicates that:
=
; 00
2
= 1 (28)
1057

COMPLEMENT B X
we nd:
; 00=
(1)
2+ 1
(29)
References
Messiah (1.17), app. C; Rose (2.19), Chap. III and app. I; Edmonds (2.21 ), Chap. 3;
Sobel'man (11.12), Chap. 4, 13.
Tables of Clebsch-Gordan coecients: Condon and Shortley (11.13), Chap. III, 14;
Bacry (10-31), app. C.
Tables of 3and 6coecients: Edmonds (2.21 ), Table 2; Rotenberg et al. (10.48).
1058

ADDITION OF SPHERICAL HARMONICS
Complement CX
Addition of spherical harmonics
1 The functions (1; 2). . . . . . . . . . . . . . . . . . . . .
2 The functions F(). . . . . . . . . . . . . . . . . . . . . . . .
3 Expansion of a product of spherical harmonics; the integral
of a product of three spherical harmonics
In this complement, we use the properties of Clebsch-Gordan coecients to prove
relations that will be of use to us later, especially in ComplementsXandXIII: the
spherical harmonic addition relations. With this aim in mind, we shall begin by intro-
ducing and studying the functions of two sets of polar angles1and2, the(1; 2).
1. The functions(1; 2)
Consider two particles (1) and (2), of state spaces
1
rand
2
rand orbital angular momenta
L1andL2. We choose for the space
1
ra standard basis, formed by the kets
11 1
,
whose wave functions are:
11 1(r1) =
11(1)
1
1
(1) (1)
(1denotes the set of polar angles11of the rst particle). Similarly, we choose for
2
ra standard basis,
22 2
. In all that follows, we shall conne the states of the
two particles to the subspaces(11)and(22), where1,1,2and2are xed;
the radial functions and
11(1)and
22(2)play no role.
The angular momentum of the total system (1) + (2) is:
J=L1+L2 (2)
According to the results of Chapter, we can construct a basis of(11)(22)of
eigenvectorscommon toJ
2
[eigenvalue(+ 1)~
2
] and(eigenvalue~). These
vectors are of the form:
=
1
1= 1
2
2= 2
12;1 2 11 1(1)
22 2(2) (3a)
the inverse change of basis being given by:
11 1(1)
22 2(2)=
1+2
=1 2=
12;1 2 (3b)
Relation (3a) shows that the angular dependence of the statesis described by the
functions:
(1; 2) =
1 2
12;1 2
1
1
(1)
2
2
(2) (4a)
1059

COMPLEMENT C X
Similarly, relation (3b) implies that:
1
1
(1)
2
2
(2) =
1+2
=1 2=
12;1 2 (1; 2) (4b)
To the observablesL1andL2correspond, for the wave functions, dierential op-
erators acting on the variables1= 11and2= 22; in particular:
1
~
1
(5a)
2
~
2
(5b)
Since, by construction, the ketis an eigenvector of=1+2, we can write:
~
1
+
2
(11;22) =~(11;22) (6)
Similarly, we have:
=~
(+ 1) ( 1)
1
(7)
which implies, with formulas (D-6) of Chapter
e
1
1
+cot1
1
+e
2
2
+cot2
2
(11;22)
=
(+ 1) ( 1)
1
(11;22) (8)
2. The functionsF()
We now introduce the functiondened by:
() () =
=
=(1= ; 2= ) (9)
is a function of a single pair of polar angles = , and can therefore characterize
the angular dependence of a wave function associated with a single particle, of state space
rand angular momentumL. In fact, we shall see thatis not a new function, but is
simply proportional to the spherical harmonic.
To demonstrate this, we shall show thatis an eigenfunction ofL
2
and
with the eigenvalues(+ 1)~
2
and~. We therefore begin by calculating the action
ofon. According to (9),depends onandby way of1= 11and
2= 22, which are both taken equal to. If we apply the dierentiation theorem
for functions of functions, we nd:
() =
~
()
=
~
1
+
2

=
=(1; 2)
1=2=
(10)
1060

ADDITION OF SPHERICAL HARMONICS
Relation (6) then yields:
() =~() (11)
which proves part of the result being sought. To calculate the action ofL
2
on, we
use the fact that:
L
2
=
1
2
(++ +) +
2
(12)
Now, by using an argument analogous to the one that enabled us to write (10) and (11),
relation (8) leads to:
() =~
(+ 1)(1)
1
() (13)
With this, (12) then yields:
L
2
() =
~
2
2
[(+ 1)(1)] + [(+ 1)(+ 1)] + 2
2
()
=(+ 1)~
2
() (14)
, which, according to (11), is an eigenfunction ofwith the eigenvalue~, is
therefore also an eigenfunction ofL
2
with the eigenvalue(+1)~
2
. SinceL
2
andform
a C.S.C.O. in the space of functions ofandalone,is necessarily proportional to the
spherical harmonic. Relation (13) enables us to show easily that the proportionality
coecient does not depend on, and we nd:
() =()() (15)
We must now calculate this proportionality coecient(). To do so, we shall
choose a particular direction in space, thedirection (= 0,indeterminate). In this
direction, all the spherical harmonicsare zero
1
, except those corresponding to= 0.
When = 0, the spherical harmonic(= 0)is given by [cf.ComplementVI,
relations (57) and (60)]:
0
(= 0) =
2+ 14
(16)
Substituting these results into (4a) and (9), we nd:
=0
(= 0) =12; 000
(21+ 1)(22+ 1)4
(17)
Furthermore, according to (15) and (16):
=0
(= 0) =()
2+ 14
(18)
We therefore have:
() =
(21+ 1)(22+ 1)4(2+ 1)
12; 000 (19)
1
Since is proportional toe, they must be zero for the value ofin thedirection to be
dened uniquely; to see this, set= 0in (66), (67) and (69 VI.
1061

COMPLEMENT C X
3. Expansion of a product of spherical harmonics; the integral of a product of
three spherical harmonics
With (9), (15) and (19) taken into account, relations (4a) and (4b
() =
(21+ 1)(22+ 1)4(2+ 1)
12; 000
1
1 2
12;1 2
1
1
()
2
2
() (20)
and:
1
1
()
2
2
() =
1+2
=1 2=
(21+ 1)(22+ 1)4(2+ 1)
12; 000
12;1 2 () (21)
This last relation (in which the summation overis actually unnecessary, since the
only non-zero terms necessarily satisfy= 1+ 2) is calledthe spherical harmonic
addition relation
2
. According to formula (26) of ComplementX, the Clebsch-Gordan
coecient12; 000is dierent from zero only if1+2 is even. The product
1
1
()
2
2
()can therefore be expanded only in terms of spherical harmonics of orders:
=1+2 1+22 1+24 1 2 (22)
In (21), the parity(1)of all the terms of the expansion on the right-hand side is thus
indeed equal to(1)
1+2
, the parity of the product which constitutes the left-hand side.
We can use the spherical harmonic addition relation to calculate the integral:
=
1
1
()
2
2
()
3
3
() d (23)
Substituting (21) into (23), we nd expressions of the type:
(;3 3) = ()
3
3
() d (24)
which, with the spherical harmonic complex conjugation relations and orthogonality
relations taken into account [cf.ComplementVI, relations (55) and (45)], are equal to:
(;3 3) = (1)
3 3
(25)
The value ofis therefore:
1
1
()
2
2
()
3
3
() d = (1)
3
(21+ 1)(22+ 1)4(23+ 1)
12; 003012;1 23 3 (26)
This integral is, consequently, dierent from zero only if:
2
In the particular case in which2= 1,2= 0[
0
1
()cos], it yields formula (35) of Comple-
mentVI.
1062

ADDITION OF SPHERICAL HARMONICS
()1+2+3= 0, as could have been predicted directly, since the integral over
in (23) is
2
0
de
(1+2+3)
=01+2+3
.
() 1,2and3
()1+2 3is even (necessary for12; 0030to be dierent from zero), that
is, if the product of the three spherical harmonics
1
1
,
2
2
and
3
3
is an even
function (obviously a necessary condition for its integral over all directions of space
to be dierent from zero).
Relation (26) expresses, for the particular case of the spherical harmonics, a more general
theorem, called the Wigner-Eckart theorem (cf.ComplementX).
1063

VECTOR OPERATORS: THE WIGNER-ECKART THEOREM
Complement DX
Vector operators: the Wigner-Eckart theorem
1 Denition of vector operators; examples
2 The Wigner-Eckart theorem for vector operators
2-a Non-zero matrix elements of V in a standard basis
2-b Proportionality between the matrix elements of J and V inside
a subspace(). . . . . . . . . . . . . . . . . . . . . . . .
2-c Calculation of the proportionality constant; the projection
theorem
3 Application: calculation of the Landé factor of an atomic
level
3-a Rotational degeneracy; multiplets
3-b Removal of the degeneracy by a magnetic eld; energy diagram1073
In ComplementVI(cf.), we dened the concept of a scalar operator: it is an
operatorwhich commutes with the angular momentumJof the system under study. An
important property of these operators was then given (cf. of that complement): in
a standard basis, , the non-zero matrix elements of a scalar
operator must satisfy the conditions=and=; in addition, these elements do
not depend
1
on, which allows us to write:
=() (1)
In particular, if the values ofandare xed, which amounts to considering the re-
striction of(cf.ComplementII, 3) to the subspace()spanned by the(2+ 1)
kets (= + 1+), we obtain a very simple(2+ 1)(2+ 1)matrix:
it is diagonal and all its elements are equal.
Now consider another scalar operatorThe matrix corresponding to it in the
subspace()possesses the same property: it is proportional to the unit matrix.
Therefore, the matrix corresponding tocan easily be obtained from the one associated
withby multiplying all the (diagonal) elements by the same constant. We therefore
see that the restrictions of two scalar operatorsandto a subspace()are always
proportional. Denoting by()the projector onto the subspace(), we can write
this result in the form
2
:
()() =()()() (2)
The aim of this complement is to study another type of operators that possesses
properties analogous to the ones just recalled: vector operators. We shall see that if
1
The proof of these properties was outlined in ComplementVI. We shall return to this point in
this complement ( 3-a) when we study the matrix elements of a scalar Hamiltonian.
2
For two given operatorsand, the proportionality coecient generally depends on the subspace
()chosen; this is why we write().
1065

COMPLEMENT D X
VandVare vectorial, their matrix elements also obey selection rules, which we shall
establish. Moreover, we shall show that the restrictions ofVandVto()are always
proportional:
()V() =()()V() (3)
These results constitute the Wigner-Eckart theorem for vector operators.
Comment:
Actually, the Wigner-Eckart theorem is much more general. For example, it en-
ables us to obtain selection rules for the matrix elements ofVbetween two kets
belonging to two dierent subspaces()and(), or to relate these elements
to the corresponding elements ofV. The Wigner-Eckart theorem can also be ap-
plied to a whole class of operators, of which scalars and vectors merely represent
special cases: the irreducible tensor operators (cf.exercise 8 of ComplementX),
which we shall not treat here.
1. Denition of vector operators; examples
In VI, we showed that an observableVis a vector if its three com-
ponents andin an orthonormal frame satisfy the following commutation
relations:
[ ] = 0 (4a)
[ ] =~ (4b)
[ ] =~ (4c)
as well as those obtained by cyclic permutation of the indices,and.
To give an idea of what this means, we shall give some examples of vector operators.
() The angular momentumJis itself a vector; replacingVbyJin formulas (4),
we simply obtain the relations that dene an angular momentum (cf.Chap.).
() For a spinless particle whose state space isr, we haveJ=L. It is then simple
to show thatRandPare vector operators. We have, for example:
[ ] = [ ] = 0
[ ] = [ ] =~ (5)
[ ] = [ ] =~
() For a particle of spinS, whose state space isr ,Jis given byJ=L+S.
In this case, the operatorsL,S,R,Pare vectors. If we take into account the fact that
all the spin operators (which act only in) commute with the orbital operators (which
act only inr), the proof of these properties follows immediately from () and ().
On the other hand, operators of the typeL
2
,LS, etc., are not vectors, but scalars
[cf.comment()of ComplementVI, 5-c]. Other vector operators could, however, be
constructed from those we have mentioned:RS, (LS)P, etc.
()Consider the system (l)+(2), formed by the union of two systems: (1), of
state space1, and (2), of state space2. IfV(1) is an operator that acts only in1, and
1066

VECTOR OPERATORS: THE WIGNER-ECKART THEOREM
if this operator is a vector [that is, satises commutation relations (4) with the angular
momentumJ1of the rst system], then the extension ofV(1) into1 2is also a vector.
For example, for a two-electron system, the operatorsL1,R1,S2, etc. are vectors.
2. The Wigner-Eckart theorem for vector operators
2-a. Non-zero matrix elements of V in a standard basis
We introduce the operators+,,+and dened by:
=
= (6)
Using relations (4), we can easily show that:
[ ] =~ (7a)
[ ] =~ (7b)
[ ] =~ (7c)
from which we can deduce the commutation relations ofand:
[++] = 0 (8a)
[+ ] = 2~ (8b)
[ +] =2~ (8c)
[ ] = 0 (8d)
Now consider the matrix elements ofVin a standard basis. We shall see that the
fact thatVis a vector implies that a large number of them are zero. First of all, we
shall show that the matrix elements are necessarily zero whenever
is dierent fromIt suces to note thatandcommute [which follows, after
cyclic permutation of the indices,and, from relation (4a)]. Therefore, the matrix
elements ofbetween two vectors corresponding to dierent eigenvalues~of
are zero (cf.Chap., D-3-a-).
For the matrix elements of we shall show that they are
dierent from zero only if =1. Equation (7c) indicates that:
= ~ (9)
Applying both sides of this relation to the ket, we obtain:
( ) = ~
= ( 1)~ (10)
This relation indicates that is an eigenvector
3
ofwith the eigenvalue
( 1)~. Since two eigenvectors of the Hermitian operatorassociated with dierent
3
It should not be concluded that is necessarily proportional to 1. In fact, the
argument we have given shows only that:
= 1
For us to be able to omit, for example, the summation over, it would be necessary forto commute
withJ
2
, which is not generally the case.
1067

COMPLEMENT D X
eigenvalues are orthogonal, it follows that the scalar product is zero
if= 1.
Summing up, the selection rules obtained for the matrix elements ofVare as
follows:
== = 0 (11a)
+== = +1 (11b)
== =1 (11c)
From these results, we can easily deduce the forms of the matrices that represent the
restrictions of the components ofVinside a subspace(). The one associated with
is diagonal, and those associated withhave matrix elements only just above and
just below the principal diagonal.
2-b. Proportionality between the matrix elements of J and V inside a subspace()
. Matrix elements of+and
Expressing the fact that the matrix element of the commutator (8a) between the
bra + 2and the ket is zero, we have:
+ 2++ = + 2++ (12)
On both sides of this relation and between the operators+and+, we insert the closure
relation:
= 1 (13)
We thus obtain the matrix elements + of+; by the very construction
of the standard basis , they are dierent from zero only if=,=and
=+ 1. The summations over,andare therefore unnecessary in this case,
and (12) can be written:
+ 2+ + 1 + 1+
= + 2+ + 1 + 1+ (14)
that is:
+ 1+
+ 1+
=
+ 2+ + 1
+ 2+ + 1
(15)
(as long as the bras and kets appearing in this relation exist, that is, as long as2
, we can show immediately that neither of the denominators can go to zero).
Writing the relation thus obtained for= + 1 2, we get:
+ 1+
+ 1+
=
+ 2+ + 1
+ 2+ + 1
=
=
+ 1+
+ 1+
=
=
+ 1
+ 1
(16)
1068

VECTOR OPERATORS: THE WIGNER-ECKART THEOREM
that is, if we call+()the common value of these ratios:
+ 1+ =+() + 1+ (17)
where+()depends onand on, but not on.
Selection rule (11b) implies that all the matrix elements + and
+ are zero if= = +1. Therefore, whateverand, we
have:
+ =+() + (18a)
This result expresses the fact that all the matrix elements of+inside()are pro-
portional to those of+.
An analogous argument can be made by taking the matrix element of the commu-
tator (8d) between the bra 2and the ket to be zero. We are thus led
to:
=() (18b)
an equation which expresses the fact that the matrix elements ofandinside()
are proportional.
. Matrix elements of
To relate the matrix elements ofto those ofwe now place relation (8c)
between the bra and the ket :
2~ = ( + +)
=~
(+ 1)(+ 1) + 1+
~
(+ 1)(1) + 1 (19)
Using (18a), we get:
=
1
2
+()
(+ 1)(+ 1) + 1+(+ 1)(1) + 1
=
~
2
+()(+ 1)(+ 1)(+ 1) +(1) (20)
that is:
=~+() (21)
Similarly, an analogous argument based on (8b) and (18b) leads to:
=~() (22)
Relations (21) and (22) show that+()and()are necessarily equal; from now
on, we shall call their common value():
() =+() =() (23)
In addition, these relations imply that:
=() (24)
1069

COMPLEMENT D X
. Generalization to an arbitrary component ofV
Any component ofVis a linear combination of+,and. Consequently,
using relation (23), we can summarize (18a), (18b) and (24) by writing:
V =() J (25)
Therefore,inside(),all the matrix elements ofVare proportional to those ofJ.
This result expresses the Wigner-Eckart theorem, for a special case. Introducing the
restrictions ofVandJto()(cf.ComplementII, 3), we can also write it:
()V() =()()J() (26)
Comment:
OperatorJcommutes with()[cf.(27)]; since, moreover:
[()]
2
=()
we can omit either one of the two projectors()on the right-hand side of (26).
2-c. Calculation of the proportionality constant; the projection theorem
Consider the operatorJV; its restriction to()is()JV(). To
transform this expression, we can use the fact that:
[J()] =0 (27)
a relation that can easily be veried by showing that the action of the commutators
[ ()]and[ ()]on any ket of the basis yields zero. Using (26),
we then get:
()JV() =J[()V()]
=()J
2
()
=()(+ 1)~
2
() (28)
The restriction to the space()of the operatorJVis therefore equal to the iden-
tity operator
4
multiplied by()(+ 1)~
2
. Therefore, ifdenotes an arbitrary
normalized state belonging to the subspace(), the average valueJV ofJV
is independent of the ketchosen, since:
JV = JV =()(+ 1)~
2
(29)
If we substitute this relation into (26), we see that
5
,inside the subspace():
V=
JV
J
2
J=
JV
(+ 1)~
2
J (30)
4
SinceJVis a scalar, the fact that its restriction is proportional to the identity operator was to
be expected.
5
We shall say that an operator relation is valid only inside a given subspace when it is actually valid
only for the restrictions to this subspace of the operators being considered. To be completely rigorous,
we should therefore have to place both sides of relation (30) between two projectors().
1070

VECTOR OPERATORS: THE WIGNER-ECKART THEOREMj
v
//
v
Figure 1: Classical interpretation of the pro-
jection theorem: since the vectorvrotates
very rapidly about the total angular momen-
tumj, only its static componentv
should
be taken into account.
This result is often called the projection theorem. Whatever the physical system being
studied, as long as we are concerned only with states belonging to the same subspace
(), we can assume that all vector operators are proportional toJ.
We can give the following classical physical interpretation of this property (cf.
of ComplementX): ifjdenotes the total angular momentum of any isolated physical
system, all the physical quantities attached to the system rotate aboutj, which is a
constant vector (cf.Fig.). In particular, for a vector quantityv, all that remains after
averaging over time is its projectionv
ontoj, that is, a vector parallel toj, given by:
v
=
jv
j
2
j (31)
a formula which is indeed analogous to (30).
Comments:
()It cannot be deduced from (30) that, in the total state space [the direct
sum of all the subspaces()],VandJare proportional. It must be
noted that the proportionality constant()(orJV) depends on
the subspace()chosen. Moreover, any vector operatorVmay possess
non-zero matrix elements between kets belonging to dierent subspaces while
the corresponding elements ofJare always zero.
()Consider a second vector operatorW. Its restriction inside()is pro-
portional toJ, and therefore also to the restriction ofV. Therefore,inside a
subspace(),all vector operators are proportional.
However, to calculate the proportionality coecient betweenVandW, we cannot simply replace
JbyWin (30) (which would give the valueVW W
2
). In the proof leading to relation
(30), we used the fact thatJcommutes with()in (28), which is not generally the case forW. To
calculate this proportionality coecient correctly, we note that, inside the subspace():
W=
JW
J
2
J (32)
1071

COMPLEMENT D X
This yields, with (30) taken into account:
V=
JV
JW
W (33)
3. Application: calculation of the Landéfactor of an atomic level
In this section, we shall use the Wigner-Eckart theorem to calculate the eect of a mag-
netic eldBon the energy levels of an atom. We shall see that this theorem considerably
simplies the calculations and enables us to predict, in a very general way, that the mag-
netic eld removes degeneracies, causing equidistant levels to appear (to rst order in
). The energy dierence of these states is proportional toand to a constant(the
Landé factor) which we shall calculate.
LetLbe the total orbital angular momentum of the electrons of an atom (the sum
of their individual orbital angular momentaL), and letSbe their total spin angular
momentum (the sum of their individual spinsS). The total internal angular momentum
of the atom (assuming the spin of the nucleus to be zero) is:
J=L+S (34)
In the absence of a magnetic eld, we call0the Hamiltonian of the atom;0
commutes
6
withJ. We shall assume that0L
2
,S
2
,J
2
andform a C.S.C.O., and
we shall call0 their common eigenvectors, of eigenvalues0,(+ 1)~
2
,
(+ 1)~
2
,(+ 1)~
2
and~, respectively.
This hypothesis is valid for a certain number of light atoms for which the angular momentum
coupling is of theLStype (cf.ComplementXIV). However, for other atoms, which have a
dierent type of coupling (for example, the rare gases other than helium), this is not the case.
Calculations based on the Wigner-Eckart theorem, similar to those presented here, can then be
performed, and the central physical ideas remain the same. For the sake of simplicity, we shall
conne ourselves here to the case in whichandare actually good quantum numbers for the
atomic state under study.
3-a. Rotational degeneracy; multiplets
Consider the ket 0 . According to the hypotheses set forth above,
commutes with 0; therefore, 0 is an eigenvector of0with the
eigenvalue0Furthermore, in accordance with the general properties of angular mo-
menta and their addition, we have:
0 =~
(+ 1) ( 1)0 1 (35)
This relation shows that, starting with a state0 , we can construct
others with the same energy: those for which . It follows that the eigenvalue
6
This general property follows from the invariance of the energy of the atom under a rotation of all
the electrons, performed about an axis passing through the origin (which is the position of the nucleus,
assumed to be motionless).0, which is invariant under rotation, therefore commutes withJ(0is a
scalar operator;cf.ComplementVI, ).
1072

VECTOR OPERATORS: THE WIGNER-ECKART THEOREM
0is necessarily at least(2+ 1)-fold degenerate. This is an essential degeneracy,
since it is related to the rotational invariance of0(an accidental degeneracy may
also be present). In atomic physics, the corresponding(2+ 1)-fold degenerate energy
level is called a multiplet. The eigensubspace associated with it, spanned by the kets
0 with= 1 , will be written(0 ).
3-b. Removal of the degeneracy by a magnetic eld; energy diagram
In the presence of a magnetic eldBparallel to, the Hamiltonian becomes (cf.
ComplementVII):
=0+1 (36)
with:
1=(+ 2) (37)
(the factor 2 beforearises from the electron spin gyromagnetic ratio). The Larmor
angular frequencyof the electron is dened in terms of its massand its charge
by:
=
2
=
~
(38)
(where=~2is the Bohr magneton).
To calculate the eect of the magnetic eld on the energy levels of the atom, we shall
consider only the matrix elements of1inside the subspace(0 )associated with
the multiplet under study. Perturbation theory, which will be explained in Chapter,
justies this procedure whenis not too large.
Inside the subspace(0 ), we have, according to the projection theorem
( 2-c):
L=
LJ
0
(+ 1)~
2
J (39a)
S=
SJ
0
(+ 1)~
2
J (39b)
whereLJ
0
andSJ
0
denote respectively the average values of the
operatorsLJandSJfor the states of the system belonging to(0 ). Now,
we can write:
LJ=L(L+S) =L
2
+
1
2
(J
2
L
2
S
2
) (40a)
as well as:
SJ=S(L+S) =S
2
+
1
2
(J
2
L
2
S
2
) (40b)
It follows that:
LJ
0
=(+ 1)~
2
+
~
2
2
[(+ 1)(+ 1)(+ 1)] (41a)
1073

COMPLEMENT D XE
E
0
M
5
2
3
2
1
2
5
=
2
J
1
2
–
–
–
3
2
5
2
Figure 2: Energy diagram showing the removal of the(2+ 1)-fold degeneracy of a
multiplet (here= 52) by a static magnetic eldB. The distance between two adjacent
levels is proportional toBand to the Landéfactor.
and:
SJ
0
=(+ 1)~
2
+
~
2
2
[(+ 1)(+ 1)(+ 1)] (41b)
Relations (41), substituted into (39) and then into (37), show that, inside the subspace
(0 ), the operator1is given by:
1= (42)
where the Landéfactor of the multiplet under consideration is equal to:
=
3
2
+
(+ 1)(+ 1)
2(+ 1)
(43)
Relation (42) implies that the eigenstates of the Hamiltonian1inside the eigen-
subspace(0 )are simply the basis vectors0 , with the eigenvalues:
1() = ~ (44)
We see that the magnetic eld completely removes the degeneracy of the multiplet.
As is shown by the diagram in Figure, a set of(2+ 1)equidistant levels appears, each
one corresponding to one of the possible values ofSuch a diagram permits general-
ization of our earlier study of the polarization and frequency of optical lines emitted by a
1074

VECTOR OPERATORS: THE WIGNER-ECKART THEOREM
ctitious atom with a single spinless electron (the normal Zeeman eect;cf.Comple-
mentVII), to the case of atoms with several electrons whose spins must be taken into
account.
References and suggestions for further reading:
Tensor operators: Schi (1.18), 28; Messiah (1.17), Chap. XIII, VI; Edmonds
(2.21), Chap. 5; Rose (2.19), Chap. 5; Meijer and Bauer (2.18), Chap. 6.
1075

ELECTRIC MULTIPOLE MOMENTS
Complement EX
Electric multipole moments
1 Denition of multipole moments
1-a Expansion of the potential on the spherical harmonics
1-b Physical interpretation of multipole operators
1-c Parity of multipole operators
1-d Another way to introduce multipole moments
2 Matrix elements of electric multipole moments
2-a General expression for the matrix elements
2-b Selection rules
2-c Physical consequences
Consider a systemScomposed ofcharged particles placed in a given electro-
static potential(r). We shall show in this complement how to calculate the interaction
energy of the systemSwith the potential(r)by introducing the electric multipole
moments ofS. First of all, we shall begin by recalling how these moments are introduced
in classical physics. Then we shall construct the corresponding quantum mechanical op-
erators, and we shall see how, in a large number of cases, their use considerably simplies
the study of the electrical properties of a quantum mechanical system. This is because
these operators possess general properties which are independent of the system being
studied, satisfying in particular certain selection rules. For example, if the state of the
systemSbeing studied has an angular momentum[i.e. is an eigenvector ofJ
2
with
the eigenvalue(+ 1)~
2
], we shall see that the average values of all multipole operators
of order higher than2are necessarily zero.
1. Denition of multipole moments
1-a. Expansion of the potential on the spherical harmonics
For the sake of simplicity, we begin by studying a systemScomposed of a single
particle, of chargeand positionr, placed in the potential(r). We shall then generalize
the results obtained to-particle systems.
. Single particle
In classical physics, the potential energy of the particle is:
(r) =(r) (1)
Since the spherical harmonics form a basis for functions ofand, we can expand(r)
in the form:
(r) =
=0=
()() (2)
1077

COMPLEMENT E X
We shall assume the charges creating the electrostatic potential to be placed outside
the region of space in which the particle being studied can be found. In this whole region,
we then have:
(r) = 0 (3)
Now, we know [cf.relation (A-15) of Chapter] that the Laplacianis related to the
dierential operatorL
2
acting on the angular variablesandby:
=
1
2
2
L
2
~
22
(4)
Also, the very denition of the spherical harmonics implies that:
L
2
() =(+ 1)~
2
() (5)
It is therefore easy to calculate the Laplacian of expansion (2). If we write, using (3),
that each of the terms thus obtained is zero, we get:
1
2
2
(+ 1)
2
() = 0 (6)
This equation has two linearly independent solutions,and
(+1)
. Since(r)is not
innite for= 0, we must choose:
() =
42+ 1
(7)
where the are coecients that depend on the potential under consideration (the
factor
4(2+ 1)is introduced for convenience, as will be seen later).
We can therefore write (2) in the form:
(r) =(r) =
=0=
(r) (8)
where the functions(r)are dened by their expressions
1
in spherical coordinates:
(r) =
42+ 1
() (9)
In quantum mechanics, the same type of expansion is possible; the potential en-
ergy operator of the particle is(R) =(R), whose matrix elements in ther
representation are (cf.ComplementII, ):
r(R)r=(r)(rr) (10)
Expansion (8) then yields:
(R) =(R) =
=0=
(11)
1
Note the dierence between the curly capitals of these classical functions and thefor the
quantum operators.
1078

ELECTRIC MULTIPOLE MOMENTS
where the operatorsare dened by:
r r= (r)(rr)
=
42+ 1
()(rr) (12)
The are called electric multipole operators.
. Generalization toparticles
Now considerclassical particles, with positionsr1,r2, ...,rand charges1,2,
...,. Their coupling energy with the external potential(r)is:
(r1r2r) =
=1
(r) (13)
The argument of the preceding section can immediately be generalized to show that:
(r1r2r) =
=0=
(r1r2r) (14)
where the coecients[which depend on the potential(r)] have the same values
as in the preceding section, and the functionsare dened by their values in polar
coordinates:
(r1r2r) =
42+ 1
=1
()( ) (15)
(and are the polar angles ofr). The multipole moments of the total system are
therefore simply the sums of the moments associated with each of the particles.
Similarly, in quantum mechanics, the coupling energy of theparticles with the
external potential is described by the operator:
(R1R2R) =
=0=
(16)
with:
r1r2r r
1r
2r
= (r1r2r)(r1r
1)(r2r
2)(rr) (17)
1-b. Physical interpretation of multipole operators
. The operator
0
0; the total charge of the system
Since
0
0is a constant(
0
0= 1
4), denition (15) implies that:
0
0=
=1
(18)
1079

COMPLEMENT E X
The operator
0
0is therefore a constant, equal to the total charge of the system.
The rst term of expansion (14) therefore gives the coupling energy of the system
with the potential(r), assuming all the particles to be situated at the origin. This
is obviously a good approximation if(r)does not vary very much in relative value over
distances comparable to those separating the various particles from(if the systemS
is centered atthis distance is of the order of the dimensions ofS). Furthermore,
there exists a special case in which expansion (14) is rigorously given by its rst term:
the case where the potential(r)is uniform, and therefore proportional to the spherical
harmonic= 0.
. The operators
1; the electric dipole moment
According to (15) and the expression for the spherical harmonics[cf.Comple-
mentVI, equations (32)], we have:
1
1=
1
2
(+)
0
1=
1
1
=
1
2
( )
(19)
These three quantities can be considered to be the components of a vector on the complex
basis of three vectorse1,e0ande1:
=
1
1
e1+
0
1e0
1
1e1 (20)
with:
e1=
1
2
(e+e);e0=e;e1=
1
2
(e e) (21)
(wheree,eandeare the unit vectors of the,and axes). The components
of this vectoron the axes are then:
1=
1
2
1
1
1
1=
1
=
2
1
1
+
1
1=
1=
0
1 = (22)
We recognize the three components of the total electric dipole moment of the systemS
with respect to the origin:
=
=1
r (23)
The operators
1are therefore actually the components of the electric dipoleD=
R.
1080

ELECTRIC MULTIPOLE MOMENTS
Relations (19) enable us, moreover, to write the= 1terms of expansion (14) in
the form:
+1
=1
1 1=
1
2
(11 11)
2
(11+11) + 10 (24)
We shall now show that the combinations of the coecients1that appear in this
expression are none other than the components of the gradient of the potential(r)at
r=0. If we take the gradient of expansion (8) of(r), the= 0term (which is constant)
disappears; the= 1term can be put into a form analogous to (24) and yields:
[r(r)]
r=0
=
1
2
(11 11)e
2
(11+11)e+10e (25)
As for the1terms of (8), they are polynomials inof degree higher than 1 (cf.
andbelow) which make no contribution to the gradient atr=0. The= 1term
of expansion (14) can therefore be written, using (23) and (25):
=1
r (r)r=0= (r=0) (26)
where:
(r) =r(r) (27)
is the electric eld at pointr. Thus we recognize (26) as the well-known expression for
the coupling energy between an electric dipole and the eld.
Comments:
()In physics, we often deal with systems whose total charge is zero (atoms, for
example).
0
0is then equal to zero, and the rst multipole operator appearing
in expansion (14) is the electric dipole moment. This expansion can often be
limited to the= 1terms [hence expression (26)], since the terms for which
2are generally much smaller (this is the case, for example, if the electric
eld varies little over distances comparable to the distances of the particles
from the origin; the2terms are, furthermore, rigorously zero in a special
case: the case in which the electric eld is uniform [cf.andbelow)].
()For a systemScomposed of two particles of opposite charge+and(an
electric dipole), the dipole momentis:
=(r1r2) (28)
Its value is related to the position of the relative particle (cf.Chap.,
) associated with the systemS; it therefore does not depend on the
choice of the originActually this is a more general property: it is simple
to show that the electric dipole moment of any electrically neutral systemS
is independent of the originchosen.
1081

COMPLEMENT E X
. The operators
2; the electric quadrupole moment
Using the explicit expression for the
2[cf.ComplementVI, relations (33)], we
could show without diculty that:
2
2
=
64
( )
2
1
2
=
62
( )
0
2=
1
2
(3
2 2
)
(29)
In this way, we obtain the ve components of the electric quadrupole moment of the
systemS. While the total charge ofSis a scalar, and its dipole moment is a vector,
it can be shown that the quadrupole moment is a second-rank tensor. In addition, an
argument similar to the one in would enable us to write the= 2terms of expansion
(14) in the form:
+2
=2
2 2=
2
r=0=1
(30)
(with,= or). These terms describe the coupling between the electric
quadrupole moment of the systemSand the gradient of the eld(r)at pointr=0.
. Generalization: the electric-pole moment
We could generalize the preceding arguments and show from the general expression
for the spherical harmonics [cf.ComplementVI, relations (26) or (30)] that:
the quantitiesare polynomials (which are homogeneous in,,) of degree.
the contribution to expansion (14) of theterms involvesth order derivatives of
the potential(r), evaluated atr=0.
Expression (14) for the potential can thus be seen to be a Taylor series expansion in
the neighborhood of the origin. As the variation of the potential(r)in the region about
Sbecomes more complicated, higher order terms must be retained in the expansion. For
example, if(r)is constant, we have seen that the= 0term is the only one involved.
If the eld(r)is uniform, the= 1terms must be added to the expansion. If it is the
gradient of the eldthat is uniform, we have2, and so forth.
1-c. Parity of multipole operators
Finally, we shall consider the parity of the. We know that the parity of
is(1)[cf.Chap., relation (D-28)]. Therefore (cf.ComplementII, ), the
electric multipole operatorhas a denite parity, equal to(1), independent of
This property will prove useful in what follows.
1082

ELECTRIC MULTIPOLE MOMENTS
1-d. Another way to introduce multipole moments
We shall consider the same system ofcharged particles as in 1-a. However,
instead of considering the interaction energy of this system with a given external potential
(r), we shall try to calculate the potential()created by these charges at a distant
point(cf.Fig.). For the sake of simplicity, we shall use classical mechanics to treat
this problem. The potential()is then:
() =
1
40
=1
r
(31)
Now, when r, it can be shown that:
1
r
=
1
=0
(cos) (32)ρ
α
1
r
1r
2
r
3
Figure 1: The potential()created at a distant point by a systemScomposed of
charged particles (of positionsr1r2) can be expressed in terms of the multipole
moments ofS.
wheredenotes the angle(r), andis theth-order Legendre polynomial. Using
the spherical harmonic addition theorem (cf.ComplementVI, ), we can write:
(cos) =
4
2+ 1
+
=
(1) ( )() (33)
(whereanddenote the polar angles of). Substituting (32) and (33) into (31), we
nally obtain:
() =
1
40
=0=
42+ 1
(1)
1
+1
() (34)
1083

COMPLEMENT E X
where(r1r2r)is dened by relation (15).
Relation (34) shows that the specication of theperfectly denes the potential
created by the particle system in regions of space outside the systemS. This potential
()can be seen to be the sum of an innite number of terms:
()The= 0term gives the contribution of the total charge of the system. This
term is isotropic (it does not depend onand) and can be written:
0() =
1
40
1
(35)
This is the1potential which would be created by the charges if they were all situated
at. It is zero if the system is globally neutral.
()The= 1term gives the contribution of the electric dipole momentof the
system. By performing transformations analogous to those in , it can be shown
that this contribution can be written:
1() =
1
40

3
(36)
This potential decreases like1
2
whenincreases.
()The= 23, ... terms give, in the same way, the contributions to the potential
()of the successive multipole moments of the system under study. Whenincreases,
each of these contributions decreases like1
+1
, and its angular dependence is described
by anth-order spherical harmonic. Moreover, we see from (34) and denition (15) that
the potential due to the multipole momentis at most of the order of magnitude of
0()(), whereis the maximum distance of the various particles of the system
Sfrom the origin. Therefore, if we are concerned with the potential at a pointsuch
that (the potential at a distant point), the1()terms decrease very rapidly
whenincreases, and we do not make a large error by retaining only the lowest values
ofin (34).
Comment:
If we wanted to calculate the magnetic eld created by a system of moving charges,
we could introduce the magnetic multipole moments of the system in an analogous
way: the magnetic dipole moment
2
, the magnetic quadrupole moment, etc. The
parities of the magnetic moments are the opposite of those of the corresponding
electric moments: the magnetic dipole moment is even, the magnetic quadrupole
moment is odd, and so on. This property arises from the fact that the electric
eld is a polar vector while the magnetic eld is an axial vector.
2. Matrix elements of electric multipole moments
We shall again consider, for the sake of simplicity, a system composed of a single spinless
particle. However, generalization to-particle systems presents no theoretical diculty.
2
There is no magnetic multipole moment of order= 0(magnetic monopole). This result is related
to the fact that the magnetic eld, whose divergence is zero according to Maxwell's equations, has a
conservative ux.
1084

ELECTRIC MULTIPOLE MOMENTS
The state spacerof the particle is spanned by an orthonormal basis,,
of common eigenvectors ofL
2
[eigenvalue(+ 1)~
2
] and(eigenvalue~). We shall
evaluate the matrix elements of a multipole operatorin such a basis.
2-a. General expression for the matrix elements
. Expansion of the matrix elements
From the general results of Chapter, we know that the wave functions associ-
ated with the states are necessarily of the form:
(r) = ()() (37)
The matrix element of the operatorcan therefore be written, using (12):
11 1 22 2=
=
0
2
d
0
sind
2
0
d
11 1
( )( )
22 2( )
=
42+ 1
0
2
d
11
()
22()
0
sind
2
0
d
1
1
()()
2
2
() (38)
Thus, in the matrix element under consideration, we have a radial integral and an angu-
lar integral. The latter, furthermore, can be simplied; using the complex conjugation
relation for spherical harmonics [cf.Chap., relation (D-29)] and relation (26) of Com-
plementX(Wigner-Eckart theorem for spherical harmonics), we can show that it can
be written:
(1)
1
0
sind
2
0
d
1
1
()()
2
2
() =
=
(2+ 1)(22+ 1)4(21+ 1)
2; 00102;2 1 1 (39)
Finally, we obtain:
11 1 22 2
=
=
1
21+ 1
11 22 2;2 1 1 (40)
where the reduced matrix element
11 22of theth-order electric multi-
pole operator is dened by:
11 22=
22+ 12; 0010
0
d
+2
11
()
22
() (41)
Relation (40) expresses, in the particular case of electric multipole operators, a general
theorem whose application in the case of vector operators has already been illustrated
(cf.ComplementX): the Wigner-Eckart theorem.
1085

COMPLEMENT E X
Comment:
We have conned ourselves here to a systemScomposed of a single spinless
particle. Nevertheless, if we consider a system ofparticles which may have spins,
we can generalize the results we have obtained. To do so, we must introduce the
total angular momentumJof the system (the sum of the orbital and spin angular
momenta of theparticles), and denote by the eigenvectors common to
J
2
and. We can then derive a relation similar to (40), in which1and2are
replaced by1and2(cf.ComplementX, exercise 8). However, the quantum
numbers1,2,1and2can then be either integral or half-integral, depending
on the physical system being considered.
. The reduced matrix element
The reduced matrix element
11 22
is independent of1and2.
It involves the radial part()of the wave functions( ). Its value therefore
depends on the basis chosen, and general properties can hardly be attributed
to it. However, it can be noted that the Clebsch-Gordan coecient2; 0010
involved in (41) is zero if1+2+is odd (cf.ComplementX, 3-c); this implies that
the reduced matrix element has the same property.
Comment:
This property is related to the(1)parity of the electric multipole operators
. For the magnetic multipole operators, we have already pointed out that their
parity is(1)
+1
; therefore it is when1+2+is even that their matrix elements
are zero.
. The angular part of the matrix element
In (40), the Clebsch-Gordan coecient2;2 1 1arises solely from the
angular integral appearing in the matrix element of[cf.(38)]. This coecient
depends only on the quantum numbers associated with the angular momenta of the
states being considered and does not involve the radial dependence()of the wave
functions. This is why it appears in the matrix elements of multipole operators whenever
one chooses a basis of eigenvectors common toL
2
and(orJ
2
andfor a system of
particles which may have spins;cf.comment of above). Now, we know that such
bases are frequently used in quantum mechanics, and, in particular, that the stationary
states of a particle in a central potential()can be chosen in this form. The radial
functions()associated with the stationary states thus depend on the potential()
chosen; this is therefore also true for the reduced matrix element
11 22
.
On the other hand, this is not the case for the angular dependence of the wave functions,
and the same Clebsch-Gordan coecient appears for all(); this is why it plays a
universal role.
1086

ELECTRIC MULTIPOLE MOMENTS
2-b. Selection rules
According to the properties of Clebsch-Gordan coecients (cf.ComplementX,
),2;2 1 1can be dierent from zero only if we have both:
1= 2+ (42)
1 2 1+2 (43)
Therefore, relation (40) implies that if at least one of these conditions is not met, the
matrix element
11 1 22 2is necessarily zero. We thus obtain selection
rules that enable us, without calculations, to simplify considerably our search for the
matrix which represents any multipole operator.
Furthermore, we saw in that the reduced matrix element of a multipole
operator obeys another selection rule:
for an electric multipole operator:
1+2+=an even number (44a)
for a magnetic multipole operator:
1+2+=an odd number (44b)
2-c. Physical consequences
. The average value of a multipole operator in a state of well-dened angular
momentum
Assume that the stateof the particle is one of the basis states
11 1
. The
average value of the operatoris then:
=
11 1 11 1
(45)
Conditions (42) and (43) are written here:
= 0 (46)
0 21 (47)
Thus we obtain the following important rules:
the average values, in a state
11 1
, of all the operatorsare zero if= 0:
= 0 if = 0 (48)
the average values, in a state
11 1
,of all operators of orderhigher than21
are zero:
= 0 if 21 (49)
1087

COMPLEMENT E X
If we now assume that the state, instead of being a state
11 1
, is any
superposition of such states, all corresponding to the same value of1, it is not dicult
to show that rule (49) remains valid [but not rule (48), since, in general, matrix elements
for which1= 2then contribute to the average value]. Relation (49) is therefore
a very general one and can be applied whenever the system is in an eigenstate ofL
2
.
Furthermore, relations (44) imply that the average value of anth-order multipole
operator can be dierent from zero only if:
for an electric multipole operator:
=an even number (50a)
for a magnetic multipole operator:
=an odd number (50b)
The preceding rules enable us to obtain, conveniently and without calculations,
some simple physical results. For example, in an= 0state (like the ground state of
the hydrogen atom), the dipole moments (electric or magnetic), quadrupole moments
(electric or magnetic), etc. are always zero. For an= 1state, only the 0th-, 1st- and
2nd-order multipole operators can be non-zero; parity rules (50) indicate that they are
the total charge and electric quadrupole of the system, as well as its magnetic dipole.
Comment:
The predictions obtained can be generalized to more complex systems (many-
electron atoms for example). If the angular momentum of such a system is
(integral or half-integral) one can show that it suces to replace, in (49),1by.
We shall apply, for example, rules (49) and (50) to the study of the electromagnetic
properties of an atomic nucleus. We know that such a nucleus is a bound system
composed of protons and neutrons, interacting through nuclear forces. If, in the
ground state
3
, the eigenvalue of the square of the angular momentum is(+1)~
2
,
the quantum numberis called the nuclear spin.
The rules we have stated indicate that:
if= 0, the electromagnetic interactions of the nucleus are characterized by
its total charge, all the other multipole moments being zero. This is the case,
for example, for
4
He nuclei (-particles),
20
Ne nuclei, etc.
if= 12, the nucleus has an electric charge and a magnetic dipole moment
[parity rule (50a) excludes an electric dipole moment]. This is the case for
the
3
He nucleus and the
1
H nucleus (i.e., the proton), as well as for all spin
1/2 particles (electrons, muons, neutrons, etc.).
if= 1, we must add the electric quadrupole moment to the charge and the
magnetic dipole moment. This is the case for
2
H (deuterium),
6
Li, etc.
This argument can be generalized to any value of. Actually, very few nuclei have
spins greater than 3 or 4.
3
In atomic physics, one generally consider the nucleus to be in its ground state: the energies involved,
although high enough to excite the electronic cloud of the atom, are much too small to excite the nucleus.
1088

ELECTRIC MULTIPOLE MOMENTS
. Matrix elements between states of dierent quantum numbers
For arbitrary1,2,1and2, the selection rules must be applied in their general
forms, (42), (43) and (44). Consider, for example, a particle of chargesubjected to a
central potential0(), whose stationary states are the states. Assume that we
then add an additional electric eld, uniform and parallel to. In the corresponding
coupling Hamiltonian, the only non-zero term is the electric dipole term (cf. 1-b-):
(R) =D
= (51)
As we saw in (22), the operatoris equal to the operator
0
1. Selection rules (42) and
(43) then indicate that:
the states coupled by the additional Hamiltonian(R)necessarily corre-
spond to the same value of
the-values of the two states necessarily dier by1[they cannot be equal, accord-
ing to (44a)]. We can predict without calculation that a large number of matrix
elements of(R)are zero. This considerably simplies, for example, the study of
the Stark eect (cf.ComplementXII), and that of the selection rules governing
the emission spectrum of atoms (cf.ComplementXIII).
References and suggestions for further reading:
Cagnac and Pebay-Peyroula (11.2), annexe IV; Valentin (16 1), Chap. VIII; Jackson
(7.5). Chaps. 4 and 16.
1089

TWO ANGULAR MOMENTA J1ANDJ2COUPLED BY
AN INTERACTION J1J2
Complement FX
Evolution of two angular momentaJ1andJ2coupled by an
interactionJ1J2
1 Classical review
1-a Equations of motion
1-b Motion of 1and 2. . . . . . . . . . . . . . . . . . . . . .
2 Quantum mechanical evolution of the average values J1
andJ2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-a Calculation of
d
d
J1and
d
d
J2. . . . . . . . . . . . . . . .
2-b Discussion
3 The special case of two spin 1/2's
3-a Stationary states of the two-spin system
3-b Calculation ofS1(). . . . . . . . . . . . . . . . . . . . . .
3-c Discussion. Polarization of the magnetic dipole transitions
4 Study of a simple model for the collision of two spin 1/2
particles
4-a Description of the model
4-b State of the system after collision
4-c Discussion. Correlation introduced by the collision
In a physical system, we must often consider the eect of a coupling between two
partial angular momentaJ1andJ2. These can, for example, be the angular momenta
of two electrons in an atom, or the orbital and spin angular momenta of an electron. In
the presence of such a coupling,J1andJ2are no longer constants of the motion; only
their sum:
J=J1+J2 (1)
commutes with the total Hamiltonian of the system.
We shall assume that the term of the Hamiltonian which introduces a coupling
betweenJ1andJ2has the simple form:
=J1J2 (2)
whereis a real constant. Such a situation is frequently encountered in atomic physics.
We shall see numerous examples in Chapter, when we use perturbation theory to
study the eect of interactions involving electron or proton spins on the hydrogen atom
spectrum. When the coupling has the form (2), classical theory predicts that the classical
angular momenta 1and 2will precess about their resultantwith an angular
velocity proportional to the constant(cf. 1 below). The vector model of the atom,
which played a very important role in the history of the development of atomic physics,
is founded on this result. In this complement, we shall show how, with the knowledge
1091

COMPLEMENT F X
of the common eigenstates ofJ
2
and, one can study the motion of the average values
J1andJ2, and again derive, at least partially, the results of the vector model of
the atom ( 2 and 3). In addition, this study will enable us to specify in simple cases
the polarization of the electro-magnetic waves emitted or absorbed in magnetic dipole
transitions. Finally, ( 4), we shall take up the case in which the two angular momenta
J1andJ2are coupled only during a collision but not permanently. This case will serve
as a simple illustration of the important concept of correlation between two systems.
1. Classical review
1-a. Equations of motion
Ifis the angle between the classical angular momenta1and 2(Fig.), the
coupling energy can be written:
= 1 2= 12cos (3)
Let0be the energy of the total system in the absence of coupling [0can represent,
for example, the sum of the rotational kinetic energies of systems (1) and (2)]. We shall
assume:
0 (4)θ

2

1
Figure 1: Two classical angular momenta
1and 2are coupled by an interaction
term= 1 2= 12cos.
Let us calculate the moment1of the forces acting on system (1). Letube a
unit vector, anddthe variation of the coupling energy when the system (1) is rotated
through an angle daboutu. We know (the theorem of virtual work) that:
1u=
d
d
(5)
1092

TWO ANGULAR MOMENTA J1ANDJ2COUPLED BY
AN INTERACTION J1J2
Starting with (3) and), we obtain, via a simple calculation:
1= 1 2 (6a)
2= 2 1 (6b)
and, consequently:
d1
d
= 1 2 (7a)
d2
d
= 2 1 (7b)
1-b. Motion of 1and 2
Adding (7a) and (7b), we obtain:
d
d
(1+ 2) =0 (8)
which shows that the total angular momentum1+ 2is indeed a constant of the
motion. Furthermore, it can easily be deduced from (7a) and (7b) that:
1
d1
d
= 2
d2
d
= 0 (9)
and:
1
d
d
2+
d
d
1 2=
d
d
(1 2) = 0 (10)
The anglebetween1and2, as well as the moduli of1and2, therefore remain
constant over time. Finally:
d
d
1= 2 1=( 1) 1= 1 (11)
Since= 1+ 2is constant, the preceding equation shows that1precesses about
with an angular velocity equal to(Fig.).
Under the eect of the coupling,1and2therefore precess about their resultant
with an angular velocity proportional toand to the coupling constant
2. Quantum mechanical evolution of the average valuesJ1andJ2
2-a. Calculation of
d
d
J1and
d
d
J2
Recall, rst of all, that ifis an observable of a quantum mechanical system of
Hamiltonianwe have (cf.Chap., ):
d
d
() =
1
~
[]() (12)
1093

COMPLEMENT F X
2

1

Figure 2: Under the eect of the coupling
= 1 2, the angular momenta1
and 2precess about their resultant,
which is a constant of the motion.
In the present case, the Hamiltonianis equal to:
=0+ (13)
where0is the sum of the energies of systems (1) and (2), andis the coupling between
J1andJ2given in (2). In the absence of such a coupling,J1andJ2are constants of the
motion (they commute with0). Therefore, in the presence of the coupling, we have
simply:
d
d
J1=
1
~
[J1]=
}
[J1J1J2] (14)
and an analogous expression for
d
d
J2. The calculation of the commutator appearing
in formula) does not present any diculty. We have, for example:
[1J1J2] = [1 12] + [1 12]
=~12 ~12
=~(J1J2) (15)
From this, we see nally that:
d
d
J1= J1J2 (16a)
d
d
J2= J2J1 (16b)
1094

TWO ANGULAR MOMENTA J1ANDJ2COUPLED BY
AN INTERACTION J1J2
2-b. Discussion
Note the close analogy between formulas (7a) and (7b) on the one hand and for-
mulas (16a) and (16b) on the other. Adding (16a 16b), we again nd thatJis a
constant of the motion, since:
d
d
J1+
d
d
J2=
d
d
J=0 (17)
However, we must recall that, in general:
J1J2=J1 J2 (18)
The motion of the average values is therefore not necessarily identical to the classical
motion. To examine this point in greater detail, we shall now study a special case: that
in whichJ1andJ2are two spin 1/2's, which we shall denote byS1andS2.
3. The special case of two spin 1/2's
The evolution of a quantum mechanical system can easily be calculated in the basis of
eigenstates of the Hamiltonian of this system. Therefore, we shall begin by determining
the stationary states of the two-spin system.
3-a. Stationary states of the two-spin system
Let:
S=S1+S2 (19)
be the total spin. Squaring both sides of (19), we obtain:
S
2
=S
2
1+S
2
2+ 2S1S2 (20)
which enables us to writein the form:
=S1S2=
2
S
2
S
2
1S
2
2=
2
S
2
3
2
~
2
(21)
(all vectors of the state space are eigenvectors ofS
2
1andS
2
2with the eigenvalue3~
2
4).
In the absence of coupling, the Hamiltonian0of the system is diagonal in the
12basis (with1=,2=) of eigenstates of1and2, as well as in the
basis (with= 0or 1, +) of eigenstates ofS
2
and. The
various vectors12or are eigenvectors of0with the same eigenvalue, which
we shall take to be the energy origin.
When we take the couplinginto account, we see from formula (21) that the
total Hamiltonian= 0+is no longer diagonal in the12basis. However,
we may write:
(0+) =
~
2
2
(+ 1)
3
2
(22)
1095

COMPLEMENT F X
The stationary states of the two-spin system therefore separate into two levels (Fig.):
the= 1level, three-fold degenerate with energy1=~
2
4, and the non-degenerate
= 0level, with energy0=3~
2
4. The splitting between the two levels is equal to
~
2
. If we set:
~
2
=~ (23)
2is the only non-zero Bohr frequency of the two-spin system.E
4
0
H
0
H
0
+ W
S = 1
S = 0
aħ
2
aħ
2
4
–
3aħ
2
Figure 3: Energy levels of a system of two spin 1/2's. On the left-hand side of the
gure, the coupling is assumed to be zero, and we obtain a single level which is four-fold
degenerate. The coupling=S1S2splits it into two distinct levels, separated by an
energy of~
2
: the triplet level (= 1, three-fold degenerate) and the singlet level (= 0,
non-degenerate).
3-b. Calculation of S1()
In order to nd the evolution ofS1(), we must rst calculate the matrices rep-
resenting1 1and1(or, more simply,1and1+=1+ 1) in the
basis of stationary states. If we use expressions (B-22) and (B-23) of Chapter, which
give the expansion of the states on the 12basis, it is possible to easily
1096

TWO ANGULAR MOMENTA J1ANDJ2COUPLED BY
AN INTERACTION J1J2
calculate the action of1or1+on the kets . We nd:
111=
~
2
11
110=
~
2
00
111=
~
2
11
100=
~
2
10
(24)
and:
1+11= 0
1+10=
~
2
11
1+11=
~
2
(10+00)
1+00=
~
2
11
(25)
From this, we can immediately derive the matrices representing1and1+in the basis
of the four statesarranged in the order11,10,11and00:
(1) =
~
2
1 0 0 0
0 0 0 1
0 01 0
0 1 0 0
(26)
(1+) =
~
2
0 1 01
0 0 1 0
0 0 0 0
0 0 1 0
(27)
Comment:
It can easily be shown that the restrictions of the1and1+matrices to the= 1subspace are
proportional respectively (with the same proportionality coecient) to the matrices representing
and+in the same subspace. This result could have been expected, in view of the Wigner-
Eckart theorem relative to vector operators (cf.ComplementX).
Let:
(0)=00+ 111+010+111 (28)
be the state of the system at the instant= 0. From this we deduce the expression for
()(to within the factor e
3~4
):
()=00+ [111+010+111] e

(29)
1097

COMPLEMENT F X
It is then easy to obtain, using (26) and (27):
1() =()1 ()
=
~
2
1
2
1
2
+ e

0+ e

0 (30)
1+() =()1+()
=
~
2
10+
01e

1+ e

1 (31)
The average values1()and 1()can be expressed in terms of1+():
1() = Re1+() (32)
1() = Im1+() (33)
Analogous calculations enable us to obtain the three components ofS2().
3-c. Discussion. Polarization of the magnetic dipole transitions
Studying the motion ofS1()does more than compare the vector model of the
atom with the predictions of quantum mechanics. It also enables us to specify the
polarization of the electromagnetic waves emitted due to the motion ofS1().
The Bohr frequency2appears in the evolution ofS1()because of the ex-
istence of non-zero matrix elements of1,1, or1between the state00and one
of the states1 (with=1, 0,+1). In (28) or (29), we shall begin by assuming
that, withnon-zero, only one of the three coecients1,0or1is dierent from
zero. The examination of the motion ofS1()in the three corresponding cases thus will
enable us to specify the polarization of the radiation associated with the three magnetic
dipole transitions:
00 10 00 11and00 11
We can always chooseto be real; we shall set:
= e (=101) (34)
Comment:
Actually, the electromagnetic waves are radiated by the magnetic momentsM1andM2associ-
ated withS1andS2(hence the name, magnetic dipole transitions).M1andM2are proportional
respectively toS1andS2. To be completely rigorous, we should then study the evolution of
M1+M2(). Here we shall assumeM1 M2. Such a situation is found, for example,
in the ground state of the hydrogen atom: the hyperne structure of this state is due to the
coupling between the spin of the electron and that of the proton (cf.Chap., ). But the
magnetic moment of the electron spin is much larger than that of the proton, so that the emis-
sion and absorption of electromagnetic waves at the hyperne transition frequency are essentially
governed by the motion of the electron spin. TakingM2into account as well would complicate
the calculations without modifying the conclusions.
1098

TWO ANGULAR MOMENTA J1ANDJ2COUPLED BY
AN INTERACTION J1J2
. The00 10transition(1= 1= 0)
If we take1= 1= 0in (30), (31), (32) and (33), we get:
1() =1() = 0
1() =~ 0cos ( 0) (35)
Furthermore, it can easily be seen that:
() = () = () = 0 (36)
S1()andS2()are then permanently of opposite direction and vibrate alongat
the frequency2(Fig.).z
O
S
2
(t)
S
1
(t)
Figure 4: If the state of the two-spin system
is a superposition of only the two stationary
states00and10,S1andS2are al-
ways of opposite direction and vibrate along
at the frequency2.
The electromagnetic waves emitted byS1therefore have a magnetic eld
1
linearly
polarized along(polarization).
We see in this example that(S1)
2
varies over time and is therefore not equal
toS
2
1(which is constant and equal to3~
2
4). This represents an important dierence
with the classical situation studied in 1, in which1maintains a constant length over
time.
1
Since these are magnetic dipole transitions, we are concerned with the magnetic eld vector of the
radiated wave. In the case of an electric dipole transition (cf.ComplementVII, ), on the other
hand, we would be concerned with the radiated electric eld.
1099

COMPLEMENT F X
. The00 11transition(0= 1= 0)
We nd in this case:
1() =
~
2
1
2
1() =
~
2
1cos( 1)
1() =
~
2
1sin( 1)
(37)
Furthermore, it can easily be veried that:
() =~1
2
() = () = 0
(38)
From this, it can be seen (Fig.) thatS1()andS2()precess counterclockwise
at an angular velocityabout their resultantS, which is parallel to. The elec-
tromagnetic waves emitted byS1()in this case therefore have a right-hand circular
polarization (+polarization).
Note that here the motion obtained for the average valuesS1andS2is the
classical motion.z
O
S

S
2
(t)
S
1
(t)
Figure 5: If the state of the two-spin sys-
tem is a superposition of only the stationary
states00and11,S1andS2precess
counterclockwise about their resultantS,
with the angular velocity.
1100

TWO ANGULAR MOMENTA J1ANDJ2COUPLED BY
AN INTERACTION J1J2z
O
S
S
2
(t)
S
1
(t)
Figure 6: If the state of the two-spin sys-
tem is a superposition of only the stationary
states00and11,S1andS2still
precess in the counterclockwise direction with
the angular velocityabout their resultant
S; however, the latter is now directed op-
posite to.
. The00 11transition(0=1= 0)
The calculations are closely analogous to those of the preceding section and lead to
the following result (Fig.):S1()andS2()precess about, again at the angular
velocity, but in the clockwise direction. It must be noted that=~ 1
2
is
now negative, so that while the direction of the precession ofS1andS2about
is dierent from what it was in the preceding case, it remains the same relative toS.
The electromagnetic waves emitted byS1are now left-hand circularly polarized (
polarization).
. General case
In the general case (any, 1,0and1), we see from (30), (31), (32) and
(33) that the components ofS1()on the three axes contain a static part and a part
modulated at the frequency2. Since these three projected motions are sinusoidal
motions of the same frequency, the tip ofS1()describes an ellipse in space. As the
sum
S1() +S2() =S
remains constant, the tip ofS2()also describes an ellipse (Fig.).
Thus we nd for the general case only part of the results of the vector model of
the atom. We do nd that, the larger the coupling constant, the more rapidlyS1()
andS2()precess aboutS. However, as we saw clearly in the special casestudied
1101

COMPLEMENT F Xz
O
S
2
(t)
S
1 (t)
S
Figure 7: Motion ofS1()andS2()in
the general case, in which the state of the
two-spin system is a superposition of the
four stationary states00,11,10and
11. The resultantSis still constant
but is not necessarily directed along.S1
andS2no longer have constant lengths,
and their tips describe ellipses.
above,S1()is not constant, and the tip ofS1()does not describe a circle in the
general case.
4. Study of a simple model for the collision of two spin 1/2 particles
4-a. Description of the model
Consider two spin 1/2 particles, whose external degrees of freedom we shall treat
classically and whose spin degrees of freedom we shall treat quantum mechanically. We
shall assume that their trajectories are rectilinear (Fig.) and that the interaction be-
tween the two spinsS1andS2is of the form=S1S2, where the coupling constant
is a rapidly decreasing function of the distanceseparating the two particles.
Sincevaries over time, so doesThe shape of the variation ofwith respect toInteraction region
(2)
(1)
Figure 8: Collision between two spin 1/2
particles (1) and (2) whose orbital variables
can be treated classically. The spin state of
each particle is represented by a large arrow.
1102

TWO ANGULAR MOMENTA J1ANDJ2COUPLED BY
AN INTERACTION J1J2t
0
a(t)
Figure 9: Shape of the variation of the cou-
pling constant()during the collision.
is shown in Figure. The maximum corresponds to the time when the distance between
the two particles is at a minimum. To simplify the reasoning, we shall replace the curve
in Figure .
The problem we have here is the following: before the collision, that is, at= ,
the spin state of the two-particle system is:
()=+ (39)
What is the state(+)of the system after the collision?
4-b. State of the system after collision
Since the Hamiltonian is zero for0, we have:
(0)=()=+
=
1
2
[10+00] (40)
The results of the preceding section concerning the eigenstates and eigenvalues of=
S1S2are applicable between times0andand enable us to calculate():
()=
1
2
10e
1~
+00e
0~
(41)
Multiplying (41) by the global phase factore
(0+1)2~
(of no physical importance),
setting1 0=~[cf.formula (23)], and returning to the12basis, we nd:
()= cos

2
+ sin

2
+ (42)
Finally, since the Hamiltonian is zero for, we have:
(+)=() (43)t
0
a(t)
a
T
Figure 10: Simplied curve used to represent
schematically the variation of the coupling
constant()during the collision.
1103

COMPLEMENT F X
Comment:
The calculation could be performed for an arbitrary function()of the type
shown in Figure. It would then be necessary to replace, in the preceding formula,
=

}
by
+
() d(cf.exercise 2 of ComplementXIII).
4-c. Discussion. Correlation introduced by the collision
If the condition:

2
=
2
+ an integer?0 (44)
is satised, we see from (42) that:
(+)= + (45)
The orientation of the two spins, in this case, is exchanged during the collision.
On the other hand, if:

2
= an integer?0 (46)
we nd that:
(+)=+ =() (47)
In this case, the collision has no eect on the orientation of the spins.
For other values of, we have:
(+)=+ + + (48)
withandsimultaneously non-zero. The state of the two-spin system has been trans-
formed by the collision into a linear superposition of the two states+ and +.
(+)is therefore no longer a tensor product,although()was one: the inter-
action of the two spins has introducedcorrelationsbetween them.
To see this, we shall analyze an experiment in which, after the collision, an observer
[observer (1)] measures1. According to formula (48) for(+), he has the proba-
bility
2
of nding+~2and
2
of nding~2[according to (42),
2
+
2
= 1].
Assume that he nds~2. Immediately after this measurement, the state of the total
system is, according to the wave packet reduction postulate,+. If, at this moment,
a second observer [observer (2)] measures2he will always nd+~2. Similarly, it can
easily be shown that if observer (1) nds the result+~2, observer (2) will then always
nd~2. Thus, the result obtained by observer (1) critically inuences the result that
observer (2) will obtain later, even if at the time of these two measurements, the particles
are extremely far apart. This apparently paradoxical result (the Einstein-Podolsky-Rosen
argument,cf.Chap. XXI, F) reects the existence of a strong correlation between the
two spins, which has appeared because of their interaction during the collision.
1104

TWO ANGULAR MOMENTA J1ANDJ2COUPLED BY
AN INTERACTION J1J2
Note, nally, that if we are concerned with only one of the two spins, it is impossible to describe
its state after the collision by a state vector, since, according to formula (48),(+)is not a
tensor product. Spin (1), for example, can be described in this case only by a density operator
(cf.ComplementIII). Let:
=(+)(+) (49)
be the density operator of the total two-spin system. According to the results of ComplementIII
(), the density operator of spin (1) can be obtained by taking the partial trace ofwith
respect to the spin variables of particle (2):
(1) = Tr2 (50)
Similarly:
(2) = Tr1 (51)
It is easy to calculate, from expression (48) for(+), the matrix representingin the four-
state basis,+++ + , arranged in this order. We nd:
=
0 0 0 0
0
2
0
0
2
0
0 0 0 0
(52)
Applying (50) and (51), we then nd:
(1) =
2
0
0
2 (53)
(2) =
2
0
0
2 (54)
Starting with expressions (53 54), we can form:
=(1)(2) (55)
whose matrix representation can be written:
=
2 2
0 0 0
0
4
0 0
0 0
4
0
0 0 0
2 2
(56)
We see thatis dierent fromreecting the existence of correlations between the two spins.
References and suggestions for further readings:
The vector model of the atom: Eisberg and Resnick (1.3), Chap. 8, 5; Cagnac
and Pebay-Peyroula (11.2), Chaps. XVI, 3B and XVII, 3E and 4C.
The Einstein-Podolsky-Rosen paradox/argument: see references of ComplementIII.
1105

EXERCISES
Complement GX
Exercises
1.Consider a deuterium atom (composed of a nucleus of spin= 1and an electron).
The electronic angular momentum isJ=L+S, whereLis the orbital angular momentum
of the electron andSis its spin. The total angular momentum of the atom isF=J+I,
whereIis the nuclear spin. The eigenvalues ofJ
2
andF
2
are(+ 1)~
2
and(+ 1)~
2
respectively.
. andfor a deuterium
atom in the 1ground state?
. 2excited state.
2.The hydrogen atom nucleus is a proton of spin= 12.
.
tum numbersandfor a hydrogen atom in the 2level?
. be the stationary states of the Hamiltonian0of the hydrogen atom
studied in .
Let be the basis obtained by addingLandSto formJ(~is
the eigenvalue of); and let be the basis obtained by addingJ
andIto formF(~is the eigenvalue of).
The magnetic moment operator of the electron is:
M=(L+ 2S)~
In each of the subspaces(= 2= 1= 12 = 12)arising from the2level
and subtended by the2+ 1vectors
= 2= 1=
1
2
=
1
2
corresponding to xed values ofandthe projection theorem (cf.ComplementX,
) enables us to write:
M= F~
Calculate the various possible values of the Landé factorscorresponding to the2
level.
3.Consider a system composed of two spin 1/2 particles whose orbital variables are
ignored. The Hamiltonian of the system is:
=11+22
where1and2are the projections of the spinsS1andS2of the two particles onto
, and1and2are real constants.
1107

COMPLEMENT G X
. = 0, is:
(0)=
1
2
[+ + +]
(with the notation of ). At timeS
2
= (S1+S2)
2
is measured.
What results can be found, and with what probabilities?
.
the evolution ofS
2
? Same question for=1+2.
4.Consider a particle()of spin 3/2 which can disintegrate into two particles,()of
spin 1/2 and()of spin 0. We place ourselves in the rest frame of(). Total angular
momentum is conserved during the disintegration.
.
two nal particles? Show that there is only one possible value if the parity of the
relative orbital state is xed. Would this result remain valid if the spin of particle
()were greater than 3/2?
. ()is initially in the spin state characterized by the eigenvalue
~of its spin component along. We know that the nal orbital state has a
denite parity. Is it possible to determine this parity by measuring the probabilities
of nding particle()either in the state+or in the state(you may use the
general formulas of ComplementX, )?
5.LetS=S1+S2+S3be the total angular momentum of three spin 1/2 particles
(whose orbital variables will be ignored). Let123be the eigenstates common to
1 2 3, of respective eigenvalues1~2,2~2,3~2. Give a basis of eigenvectors
common toS
2
and, in terms of the kets123. Do these two operators form
a C.S.C.O.? (Begin by adding two of the spins in order to obtain a partial angular
momentum, and then add it to the third one.)
6.LetS1andS2be the intrinsic angular momenta of two spin 1/2 particles,R1and
R2, their position observables, and1and 2, their masses (with=
12
1+2
, the
reduced mass). Assume that the interactionbetween the two particles is of the form:
=() +()
S1S2
~
2
where()and()depend only on the distance=R1R2between the particles.
. S=S1+S2be the total spin of the two particles.
.
1=
3
4
+
S1S2
~
2
0=
1
4
+
S1S2
~
2
are the projectors onto the total spin states= 1and= 0respectively.
1108

EXERCISES
. = 1()1+ 0()0, where1()and 0(),
are two fonctions of, to be expressed in terms of()and().
. of the relative particle in the center of mass frame;P
denotes the momentum of this relative particle. Show thatcommutes withS
2
and does not depend on. Show from this that it is possible to study separately
the eigenstates ofcorresponding to= 1and= 0.
Show that one can nd eigenstates of, with eigenvalue, of the form:
=00
0
= 0= 0+
+1
=1
1
1
= 1
where00and1are constants, and
0
and
1
are kets of the state spacer
of the relative particle (~is the eigenvalue of). Write the eigenvalue equations
satised by
0
and
1
.
.
=~
22
2be the energy of the system in the center of mass frame. We assume
in all that follows that, before the collision, one of the particles is in the+spin
state, and the other one, in thespin state. Let be the corresponding
stationary scattering state (cf.Chap. , ). Show that:
=
1
2
0
= 0= 0+
1
2
1
= 1= 0
where
0
and
1
are the stationary scattering states for a spinless particle of
mass, scattered respectively by the potentials0()and1().
. 0()and1()be the scattering amplitudes that correspond to
0
and
1
.
Calculate, in terms of0()and1(), the scattering cross section()of the two
particles in thedirection, with simultaneous ip of the two spins (the spin which
was in the+state goes into thestate, and vice versa).
.
0
and
1
be the phase shifts of thepartial waves associated respectively with
0()and 1()(cf.Chap. , ). Show that the total scattering cross
section, with simultaneous ip of the two spins, is equal to:
=
2
=0
(2+ 1) sin
2
(
1 0
)
7.We dene the standard components of a vector operatorVas the three operators:
(1)
1
=
1
2
(+)
(1)
0
=
(1)
1
=
1
2
( )
1109

COMPLEMENT G X
Using the standard components
(1)
and
(1)
of the two vector operatorsVandW,
we construct the operators:
(1) (1)
()
= 11;
(1)(1)
where the11; are the Clebsch-Gordan coecients entering into the addition
of two angular momenta 1 (these coecients can be obtained from the results of
ComplementX).
.
(1) (1)
(0)
0
is proportional to the scalar productVWof the two
vector operators.
.
(1) (1)
(1)
are proportional to the three stan-
dard components of the vector operatorVW.
.
(1) (1)
(2)
in terms of the various operators,
= = .
. V=W=R, whereRis the position observable of a particle. Show
that the ve operators
(1) (1)
(2)
are proportional to the ve components
2of the electric quadrupole moment operator of this particle [cf.formula (29)
of ComplementX].
. V=W=L, whereLis the orbital angular momentum of the particle.
Express the ve operators
(1) (1)
(2)
in terms of + . What are the
selection rules satised by these ve operators in a standard basisof
eigenstates common toL
2
and(in other words, on what conditions is the matrix
element
(1) (1)
(2)
non-zero)?
8. Irreducible tensor operators; Wigner-Eckart theorem
The2+ 1operators
()
, withan integer0and= + 1+,
are, by denition, the2+ 1components of an irreducible tensor operator of rankif
they satisfy the following commutation relations with the total angular momentumJof
the physical system:
()
=~
()
(1)
+
()
=~
(+ 1)(+ 1)
()
+1
(2)
()
=~
(+ 1)(1)
()
1
(3)
. = 0, and
that the three standard components of a vector operator (cf.exercise 7) are the
components of an irreducible tensor operator of rank= 1.
1110

EXERCISES
. be a standard basis of common eigenstates ofJ
2
and. By tak-
ing both sides of (1) to have the same matrix elements between and
, show that
()
is zero ifis not equal to
+.
. 2) and (3), show that the
(2+1)(2+1)(2+1)matrix elements
()
corresponding
to xed values of satisfy recurrence relations identical to those satised
by the(2+ 1)(2+ 1)(2+ 1)Clebsch-Gordan coecients ;
(cf.ComplementX, ) corresponding to xed values of.
.
()
= ; (4)
whereis a constant depending only on , which is usually written in
the form:
=
1
2+ 1
()
. (2+1)operators
()
satisfy relation (4) for all
and , they satisfy relations (1), (2) and (3), that is, they constitute the
(2+ 1)components of an irreducible tensor operator of rank
.
introduced in ComplementXare irreducible tensor operators of rankin the state
spacerof this particle. Show that, in addition, when the spin degrees of freedom
are taken into account, the operatorsremain irreducible tensor operators in
the state spacer (whereis the spin state space).
. in a standard basis
obtained by adding the orbital angular momentumLand the spinSof the particle
to form the total angular momentumJ=L+S[(+ 1)~
2
(+ 1)~
2
,~are
the eigenvalues ofL
2
,J
2
,respectively].
9.Let
(1)
1
be an irreducible tensor operator (exercise 8) of rank1acting in a state
space1, and
(2)
2
, an irreducible tensor operator of rank2acting in a state space
2. With
(1)
1
and
(2)
2
, we construct the operator:
()
=
(1) (2)
()
=
12
12;12
(1)
1
(2)
2
. cf.ComplementX),
show that the
()
satisfy commutation relations (1), (2) and (3) of exercise 8 with
the total angular momentumJ=J1+J2of the system. Show that the
()
are
the components of an irreducible tensor operator of rank.
1111

COMPLEMENT G X
. (1)
()()
is a scalar operator (you may use the
results of X).
10. Addition of three angular momenta
Let(1),(2),(3)be the state spaces of three systems (1), (2) and (3), of angular
momentaJ1,J2,J3. We shall writeJ=J1+J2+J3for the total angular momentum.
Let , , be the standard bases of(1),(2),(3),
respectively. To simplify the notation, we shall omit the indices, as we did in
Chapter.
We are interested in the eigenstates and eigenvalues of the total angular momentum
in the subspace( )subtended by the kets:
(1)
We want to add to form an eigenstate ofJ
2
andcharacterized by the quantum
numbersand. We shall denote by:
(); (2)
such a normalized eigenstate obtained by rst addingtoto form an angular mo-
mentum, then addingtoto form the state . One could also addand
to formand then addtoto form the normalized state, written:
() ; (3)
. ,
,, forms an orthonormal basis in( ). Same question for the system
of kets (3), corresponding to the various values of,,.
. +and, that the scalar product of kets
() ; (); does not depend on denoting such a
scalar product by() ; ();.
.
(); = () ; ();( ) ;
(4)
.
on the basis (1). Show that:
; ; =
; ;
() ; (); (5)
1112

EXERCISES
.
relations, the following relations:
; ; ;
= ; () ; (); (6)
as well as:
() ; ();=
1
2+ 1
;
; ; ; (7)
References
Exercises 8 et 9: see references of ComplementX.
Exercise 10: Edmonds (2.21), Chap. 6; Messiah (1.17), XIII-29 and App. C; Rose (2.19),
App. 1.
1113

Chapter XI
Stationary perturbation theory
A Description of the method
A-1 Statement of the problem
A-2 Approximate solution of the()eigenvalue equation
B Perturbation of a non-degenerate level
B-1 First-order corrections
B-2 Second-order corrections
C Perturbation of a degenerate state
The quantum mechanical study of conservative physical systems (that is, systems
whose Hamiltonians are not explicitly time-dependent) is based on the eigenvalue equa-
tion of the Hamiltonian operator. We have already encountered two important examples
of physical systems (the harmonic oscillator and the hydrogen atom) whose Hamiltonians
are simple enough for their eigenvalue equations to be solved exactly. However, this hap-
pens in only a very small number of problems. In general, the equation is too complicated
for us to be able to nd its solutions in an analytic form
1
. For example, we do not know
how to treat many-electron atoms, even helium, exactly. Besides, the hydrogen atom
theory explained in Chapter) takes into account only the electrostatic interac-
tion between the proton and the electron; when relativistic corrections (such as magnetic
forces) are added to this principal interaction, the equation obtained for the hydrogen
atom can no longer be solved analytically. We must then resort to solving it numerically
with a computer. There exist, however,approximation methodsthat enable us to obtain
analytically approximate solutions of the basic eigenvalue equation in certain cases. In
this chapter, we shall study one of these methods, known as stationary perturbation
theory
2
. (In Chapter , we shall describe time-dependent perturbation theory,
1
Of course, this phenomenon is not limited to the domain of quantum mechanics. In all elds of
physics, there are very few problems that can be treated completely analytically.
2
Perturbation theory also exists in classical mechanics, where it is, in principle, entirely analogous
to the one we shall describe here.
Quantum Mechanics, Volume II, Second Edition. C. Cohen-Tannoudji, B. Diu, and F. Laloë.
© 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

CHAPTER XI STATIONARY PERTURBATION THEORY
which is used to treat systems whose Hamiltonians contain explicitly time-dependent
terms.)
Stationary perturbation theory is very widely used in quantum physics, since it
reects the physicist's usual approach to problems. In studying a phenomenon or a
physical system, one begins by isolating the principal eects that are responsible for the
main features of this phenomenon or this system. When they have been understood,
one tries to explain the ner details by taking into account less important eects that
were neglected in the rst approximation. It is in treating these secondary eects that
one commonly uses perturbation theory. In Chapter, we shall see, for example, the
importance of perturbation theory in atomic physics: it will enable us to calculate the
relativistic corrections in the case of the hydrogen atom. Similarly, ComplementXIV,
which is devoted to the helium atom, indicates how perturbation theory allows us to treat
many-electron atoms. Numerous other applications of perturbation theory are given in
the complements of this chapter and the following ones.
Let us mention, nally, another often used approximation method, the variational
method, which we shall present in ComplementXI. We shall briey examine its ap-
plications in solid state physics (ComplementXI) and in molecular physics (Comple-
mentXI).
A. Description of the method
A-1. Statement of the problem
Perturbation theory is applicable when the Hamiltonianof the system being
studied can be put in the form:
=0+ (A-1)
where the eigenstates and eigenvalues of0are known, and whereis much smaller than
0. The operator0, which is time-independent, is called the unperturbed Hamilto-
nian andthe perturbation. Ifis not time-dependent, we say that we are dealing
with a stationary perturbation; this is the case we are considering in this chapter (the
case of time-dependent perturbations will be studied in Chapter ). The problem is
then to nd the modications produced in the energy levels of the system and in its
stationary states by the addition of the perturbation.
When we say thatis much smaller than0this means that the matrix elements
ofare much smaller
3
than those of0. To make this more explicit, we shall assume
thatis proportional to a real parameterwhich is dimensionless and much smaller
than 1:
=
^
with 1 (A-2)
(where
^
is an operator whose matrix elements are comparable to those of0). Per-
turbation theory consists of expanding the eigenvalues and eigenstates ofin powers of
, keeping only a nite number of terms (often only one or two) of these expansions.
We shall assume the eigenstates and eigenvalues of the unperturbed Hamiltonian
0to be known. In addition, we shall assume, thatthe unperturbed energies form a
3
More precisely, the important point is that the matrix elements ofare much smaller than the
dierences between eigenvalues of0(cf.comment of ).
1116

A. DESCRIPTION OF THE METHOD
discrete spectrum, and we shall label them by an integral index:
0
. The corresponding
eigenstates will be denoted by, the additional indexpermitting us, in the case of a
degenerate eigenvalueto distinguish between the various vectors of an orthonormal
basis of the associated eigensubspace. We therefore have:
0 =
0
(A-3)
where the set of vectorsforms an orthonormal basis of the state space:
= (A-4a)
= 1 (A-4b)
If we substitute (A-2) into (A-1), we can consider the Hamiltonian of the system
to be continuously dependent on the parametercharacterizing the intensity of the
perturbation:
() =0+
^
(A-5)
Whenis equal to zero,()is equal to the unperturbed Hamiltonian0. The eigen-
values()of()generally depend on, and Figure
their variations with respect to.
An eigenvector of()is associated with each curve of Figure. For a given value
of, these vectors form a basis of the state space [()is an observable]. Whenis
much smaller than 1, the eigenvalues()and the eigenvectors()of()remain
very close to those of0=(= 0), which they approach when 0.
The operator()may, of course, have one or several degenerate eigenvalues. For
example, in Figure, the double curve represents a doubly degenerate energy (the one
which approaches
0
4when 0), which corresponds, for all, to a two-dimensional
eigensubspace. It is also possible for several distinct eigenvalues()to approach the
same unperturbed energy
40
when 0; this happens for instance for
0
3in Figure.
In such a case, we say that the eect of the perturbation is to remove the degeneracy of
the corresponding eigenvalue of0
In the following section, we shall give an approximate solution of the eigenvalue
equation of()for 1[of course, we assume that we cannot solve this equation
exactly; otherwise it would not be necessary to resort to perturbation theory to nd the
eigenstates and eigenvalues of=()].
A-2. Approximate solution of the()eigenvalue equation
We are looking for the eigenstates()and eigenvalues()of the Hermitian
operator():
()()=()() (A-6)
4
Additional degeneracies may appear for particular non-zero values of(crossing at=1in
Figure). We shall assume here thatis small enough to avoid such a situation.
1117

CHAPTER XI STATIONARY PERTURBATION THEORY
We shall assume
5
that()and()can be expanded in powers ofin the
form:
() =0+ 1++ + (A-7a)
()=0+1++ + (A-7b)
We then substitute these two expansions, as well as denition (A-5) of(), into equa-
tion (A-6):
0+
^
=0
=
=0 =0
(A-8)
We require this equation to be satised forsmall but arbitrary. We must therefore
equate the coecients of successive powers ofon both sides. This leads to:
for 0th-order terms in:
00=00 (A-9)
for 1st-order terms:
(0 0)1+
^
10= 0 (A-10)0
E
1
0
E
2
0
E
3
0
E
4
0
E(λ)
λ
1
λ
Figure 1: Variation of the eigen-
values()of the Hamiltonian
() =0+
^
with respect to.
Each curve corresponds to an eigen-
state of(). For= 0, we obtain
the spectrum of0. We have as-
sumed here that the eigenvalues
0
3
and
0
4are doubly degenerate; ap-
plication of the perturbation
^
re-
moves the degeneracy of
0
3, but not
that of
0
4. An additional two-fold
degeneracy appears for=1.
5
This is not obvious from a mathematical point of view, the basic problem being the convergence of
the series (A-7
1118

A. DESCRIPTION OF THE METHOD
for 2nd-order terms:
(0 0)2+
^
11 20= 0 (A-11)
forth-order terms:
(0 0)+
^
1 1 2 2 0= 0 (A-12)
We shall conne ourselves here to the study of the rst three equations, that is,
we shall neglect, in expansions (A-7), terms of orders higher than 2 in.
We know that the eigenvalue equation (A-6) denes()only to within a con-
stant factor. We can therefore choose the norm of()and its phase:we shall re-
quire()to be normalized, and we shall choose its phase such that the scalar product
0()is real.To 0th order, this implies that the vector denoted by0must be
normalized:
00= 1 (A-13)
Its phase, however, remains arbitrary; we shall see in
in each particular case. To 1st order, the square of the norm of()can be written:
()()=0+10+1+(
2
)
=00+10+01+(
2
) (A-14)
(where the symbol()stands for all the terms of order higher than or equal to).
Using (A-13), we see that this expression is equal to 1 to rst order if theterm is zero.
But the choice of phase indicates that the scalar product01is real (sinceis real).
We therefore obtain:
01=10= 0 (A-15)
An analogous argument, for 2nd order in, yields:
02=20=
1
2
11 (A-16)
and, forth order:
0=0
=
1
2
11+ 22++22+11 (A-17)
When we conne ourselves to second order in, the perturbation equations are
therefore (A-9), (A-10) and (A-11). With the conventions we have set, we must add
conditions (A-13), (A-15) and (A-16).
Equation (A-9) expresses the fact that0is an eigenvector of0with the eigen-
value0.0therefore belongs to the spectrum of0This was to be expected, since each
eigenvalue of(), when 0, approaches one of the unperturbed energies. We then
1119

CHAPTER XI STATIONARY PERTURBATION THEORY
choose a particular value of0, that is, an eigenvalue
0
of0. As Figure
can exist one or several dierent energies()of()that approach
0
when 0.
Consider the set of eigenstates of()corresponding to the various eigenvalues
()that approach
0
when 0. They span a vector subspace whose dimension
clearly cannot vary discontinuously whenvaries in the neighborhood of zero. This
dimension is consequently equal to the degeneracyof
0
. In particular, if
0
is
non-degenerate, it can give rise only to a single energy(), and this energy is non-
degenerate.
To study the inuence of the perturbation, we shall consider separately the case
of non-degenerate, and degenerate levels of0.
B. Perturbation of a non-degenerate level
Consider a particular non-degenerate eigenvalue
0
of the unperturbed Hamiltonian0.
Associated with it is an eigenvectorwhich is unique to within a constant factor. We
want to determine the modications in this unperturbed energy and in the corresponding
stationary state produced by the addition of the perturbationto the Hamiltonian.
To do so, we shall use perturbation equations (A-9) through (A-12), as well as
conditions (A-13) and (A-15) through (A-17). For the eigenvalue of()that approaches
0
when 0, we have:
0=
0
(B-1)
which, according to (A-9), implies that0must be proportional to. The vectors0
and are both normalized [cf.(A-13)], and we shall choose:
0= (B-2)
Thus, when 0, we again nd the unperturbed statewith the same phase.
We call()the eigenvalue of()which, when 0, approaches the eigen-
value
0
of0. We shall assumesmall enough for this eigenvalue to remain non-
degenerate, that is, for a unique eigenvector()to correspond to it (in the case of
the= 2level of Figure, this is satised if1). We shall now calculate the rst
terms of the expansion of()and()in powers of.
B-1. First-order corrections
We shall begin by determining1and the vector1from equation (A-10) and
condition (A-15).
B-1-a. Energy correction
Projecting equation (A-10) onto the vector, we obtain:
(0 0)1+ (
^
1)0= 0 (B-3)
The rst term is zero, since=0is an eigenvector of the Hermitian operator0
with the eigenvalue
0
=0. With (B-2) taken into account, equation (B-3) then yields:
1=
^
0=
^
(B-4)
1120

B. PERTURBATION OF A NON-DEGENERATE LEVEL
In the case of a non-degenerate state
0
, the eigenvalue()ofwhich corresponds
to
0
can be written, to rst order in the perturbation=
^
:
() =
0
+ +(
2
) (B-5)
The rst-order correction to a non-degenerate energy
0
is simply equal to the average
value of the perturbation termin the unperturbed state.
B-1-b. Eigenvector correction
The projection (B-3) obviously does not exhaust all the information contained in
perturbation equation (A-10). We must now project this equation onto all the vectors of
the basis other than. We obtain, using (B-1) and (B-2):
(0
0
)1+ (
^
1) = 0 ( =) (B-6)
(since the eigenvalues
0
other than
0
can be degenerate, we must retain the degeneracy
indexhere). Since the eigenvectors of0associated with dierent eigenvalues are
orthogonal, the last term,1 , is zero. Furthermore, in the rst term, we can let
0act on the left on. (B-6) then becomes:
0 0
1+
^
= 0 (B-7)
which gives the coecients of the desired expansion of the vector1on all the unper-
turbed basis states, except:
1=
1
0 0
^
(=) (B-8)
The last coecient which we lack,1, is actually zero, according to condition (A-15),
which we have not yet used [, according to (B-2), coincides with0]:
1= 0 (B-9)
We therefore know the vector1since we know its expansion on thebasis:
1=
=
^
0 0
(B-10)
Consequently, to rst order in the perturbation=
^
, the eigenvector()
ofcorresponding to the unperturbed statecan be written:
()= +
=
0 0
+(
2
) (B-11)
The rst-order correction of the state vector is a linear superposition of all the unper-
turbed states other than: the perturbationis said to produce a mixing of the
state with the other eigenstates of0. The contribution of a given stateis
1121

CHAPTER XI STATIONARY PERTURBATION THEORY
zero if the perturbationhas no matrix element between and . In general,
the stronger the coupling induced bybetween and (characterized by the
matrix element ), and the closer the level
0
to the level
0
under study, the
greater the mixing with.
Comment:
We have assumed that the perturbationis much smaller than the unperturbed
Hamiltonian0, that is, that the matrix elements ofare much smaller than
those of0It appears here that this hypothesis is not sucient: the rst order
correction of the state vector is small only ifthe non-diagonal matrix elements of
W are much smaller than the corresponding unperturbed energy dierences.
B-2. Second-order corrections
The second-order corrections can be extracted from perturbation equation (A-11)
by the same method as above, with the addition of condition (A-16).
B-2-a. Energy correction
To calculate2, we project equation (A-11) onto the vector, using (B-1)
and (B-2):
(0
0
)2+ (
^
1)1 2 = 0 (B-12)
For the same reason as in , the rst term is zero. This is also the case for11,
since, according to (B-9),1is orthogonal to. We then get:
2=
^
1 (B-13)
that is, substituting expression (B-10) for the vector1:
2=
=
^
2
0 0
(B-14)
This result enables us to write the energy(), to second order in the perturba-
tion=
^
, in the form:
() =
0
+ +
=
2
0 0
+(
3
) (B-15)
Comment:
The second-order energy correction for the statedue to the presence of the
state has the sign of
0 0
. We can therefore say that, to second or-
der, the closer the stateto the state, and the stronger the coupling
, the more these two levels repel each other.
1122

B. PERTURBATION OF A NON-DEGENERATE LEVEL
B-2-b. Eigenvector correction
By projecting equation (A-11) onto the set of basis vectorsdierent from
, and by using conditions (A-16), we could obtain the expression for the ket2,
and therefore the eigenvector to second order. Such a calculation presents no theoretical
diculties, and we shall not give it here.
Comment:
In (B-4), the rst-order energy correction is expressed in terms of the zeroth-
order eigenvector. Similarly, in (B-13), the second-order energy correction involves
the rst-order eigenvector [which explains a certain similarity of formulas (B-10)
and (B-14)]. This is a general result: by projecting (A-12) onto, one makes
the rst term go to zero, which givesin terms of the corrections of order1,
2, ... of the eigenvector. This is why we generally retain one more term in the
energy expansion than in that of the eigenvector: for example, the energy is given
to second order and the eigenvector to rst order.
B-2-c. Upper limit of 2
If we limit the energy expansion to rst order in, we can obtain an approximate
idea of the error involved by evaluating the second-order term which is simple to obtain.
Consider expression (B-14) for2. It contains a sum (which is generally innite)
of terms whose numerators are positive or zero. We denote bythe absolute value of
the dierence between the energy
0
of the level being studied and that of the closest
level. For all, we obviously have:
0 0
> (B-16)
This gives us an upper limit for the absolute value of2:
26
1

=
^
2
(B-17)
which can be written:
26
1

=
^ ^
6
1

^
=
^
(B-18)
The operator which appears inside the brackets diers from the identity operator only
by the projector onto the state, since the basis of unperturbed states satises the
closure relation:
+
=
= 1 (B-19)
1123

CHAPTER XI STATIONARY PERTURBATION THEORY
Inequality (B-18) therefore becomes simply:
26
1

^
[1 ]
^
6
1

^2 ^
2
(B-20)
Multiplying both sides of (B-20) by
2
we obtain an upper limit for the second-
order term in the expansion of(), in the form:
2
26
1

()
2
(B-21)
whereis the root-mean-square deviation of the perturbationin the unperturbed
state. This indicates the order of magnitude of the error on the energy resulting
from taking only the rst-order correction into account.
C. Perturbation of a degenerate state
Now assume that the level
0
whose perturbation we want to study is-fold degenerate
(whereis greater than 1, but nite). We denote by
0
the corresponding eigensubspace
of0. In this case, the choice:
0=
0
(C-1)
does not suce to determine the vector0, since equation (A-9) can theoretically be
satised by any linear combination of thevectors (= 12 ). We know
only that0belongs to the eigensubspace spanned by them.
We shall see that, this time, under the action of the perturbation, the level
0
generally gives rise to several distinct sublevels. Their number,, is between 1 and
. Ifis less than, some of these sublevels are degenerate, since the total number
of orthogonal eigenvectors ofassociated with thesublevels is always equal to.
To calculate the eigenvalues and eigenstates of the total Hamiltonian, we shall limit
ourselves, as usually done, to rst order infor the energies and to zeroth order for the
eigenvectors.
To determine1and0, we can project equation (A-10) onto thebasis vectors
. Since theare eigenvectors of0with the eigenvalue
0
=0, we obtain the
relations:
^
0=1 0 (C-2)
We now insert, between the operator
^
and the vector0, the closure relation for the
basis:
^
0=1 0 (C-3)
The vector0, which belongs to the eigensubspace associated with
0
, is orthogonal to
all the basis vectorsfor whichis dierent fromConsequently, on the left-hand
1124

C. PERTURBATION OF A DEGENERATE STATE
side of (C-3), the sum over the indexreduces to a single term (=), which gives:
=1
^
0=1 0 (C-4)
We arrange the
2
numbers
^
(whereis xed and,= 12 ) in
a matrix of row indexand column index. This square matrix, which we shall
denote by(
^()
)is, so to speak, cut out of the matrix which represents
^
in the
basis:(
^()
)is the part which corresponds to
0
. Equations (C-4) then show that the
column vector of elements0(= 12 ) is an eigenvector of(
^()
)with the
eigenvalue1.
System (C-4) can, moreover, be transformed into avector equation inside
0
. All
we need to do is dene the operator
^()
, therestriction of
6^
to the subspace
0
.
^()
acts only in
0
, and it is represented in this subspace by the matrix of elements
^
, that is, by(
^()
). System (C-4) is thus equivalent to the vector equation:
^()
0=10 (C-5)
[We stress the fact that the operator
^()
is dierent from the operator
^
of which it is
the restriction: equation (C-5) is an eigenvalue equation inside
0
, and not in all space].
Therefore,to calculate the eigenvalues (to rst order) and the eigenstates (to zeroth
order) of the Hamiltonian corresponding to a degenerate unperturbed state
0
,diagonalize
the matrix(
()
),which represents the perturbation
7
,inside the eigensubspace
0
associated with
0
.
Let us examine more closely the rst-order eect of the perturbationon the
degenerate state
0
. Let
1
(= 12
(1)
) be the various distinct roots of the charac-
teristic equation of(
^()
). Since(
^()
)is Hermitian, its eigenvalues are all real, and
the sum of their degrees of degeneracy is equal to(
(1)
6). Each eigenvalue intro-
duces a dierent energy correction. Therefore, under the inuence of the perturbation
=
^
, the degenerate level splits, to rst order, into
(1)
distinct sublevels, whose
energies can be written:
() =
0
+
1
= 12
(1)
6 (C-6)
If
(1)
=, we say that, to rst order, the perturbationcompletely removes the
degeneracy of the level
0
. If
(1)
, the degeneracy, to rst order, is only partially
removed (or not at all if
(1)
= 1).
We shall now choose an eigenvalue
1
of
^()
. If this eigenvalue is non-degenerate,
the corresponding eigenvector0is uniquely determined (to within a phase factor) by
(C-5) [or by the equivalent system (C-4)]. There then exists a single eigenvalue()of
()which is equal to
0
+
1
, to rst order, and this eigenvalue is non-degenerate
8
. On
6
Ifis the projector onto the subspace
0
,
^()
can be written (ComplementII, 3):
^()
=
^
.
7
(
()
)is simply equal to(
^()
); this is why its eigenvalues yield directly the corrections1.
8
The proof of this point is analogous to the one that shows that a non-degenerate level of0gives
rise to a non-degenerate level of()(cf.end of ).
1125

CHAPTER XI STATIONARY PERTURBATION THEORY
the other hand, if the eigenvalue
1
of
^()
being considered presents a-fold degeneracy,
(C-5) indicates only that0belongs to the corresponding-dimensional subspace
(1)
.
This property of
1
can, actually, reect two very dierent situations. One could distin-
guish between them by pursuing the perturbation calculation to higher orders of, and seeing
whether the remaining degeneracy is removed. These two situations are the following:
()Suppose that there is only one exact energy()that is equal, to rst order, to
0
+
1
, and that this energy is-fold degenerate [in Figure, for example, the energy()
that approaches
0
4when 0is two-fold degenerate, for any value of]. A-dimensional
eigensubspace then corresponds to the eigenvalue(), whatever, so that the degeneracy of
the approximate eigenvalues will never be removed, to any order of.
In this case, the zeroth-order eigenvector0of()cannot be completely specied, since
the only condition imposed on0is that of belonging to a subspace which is the limit, when
0, of the-dimensional eigensubspace of()corresponding to(). This limit is none
other than the eigensubspace
(1)
of(
^
()
)associated with the eigenvalue
1
chosen.
This rst case often arises when0and possess common symmetry properties, im-
plying an essential degeneracy for(). Such a degeneracy then remains to all orders in
perturbation theory.
() It may also happen that several dierent energies()are equal, to rst order, to
0
+
1
(the dierence between these energies then appears in a calculation at second or higher
orders).
In this case, the subspace
(1)
obtained to rst order is only the direct sum of the limits,
for 0, of several eigensubspaces associated with these various energies(). In other
words, all the eigenvectors of()corresponding to these energies certainly approach kets of
(1)
, but, inversely, a particular ket of
(1)
is not necessarily the limit0of an eigenket of
().
In this situation, going to higher order terms allows one, not only to improve the accuracy
of the energies, but also to determine the zeroth-order kets0. However, in practice, the partial
information contained in equation (C-5) is often considered sucient.
Comments:
(i)
9
of the spec-
trum of0, we must diagonalize the perturbationinside each of the eigen-
subspaces
0
corresponding to these energies. It must be understood that
this problem is much simpler than the initial problem, which is the complete
diagonalization of the Hamiltonian in the entire state space. Perturbation
theory enables us to ignore completely the matrix elements ofbetween
vectors belonging to dierent subspaces
0
. Therefore, instead of having to
diagonalize a generally innite matrix, we need only diagonalize, for each of
the energies
0
in which we are interested, a matrix of smaller dimensions,
generally nite.
(ii) (
^()
)clearly depends on the basis initially chosen in this
subspace
0
(although the eigenvalues and eigenkets of
^()
obviously do not
depend on it). Therefore, before we begin the perturbation calculation, it is
advantageous to nd a basis that simplies as much as possible the form of
9
The perturbation of a non-degenerate state, studied in , can be seen as a special case of that of
a degenerate state.
1126

C. PERTURBATION OF A DEGENERATE STATE
(
()
)for this subspace, and, consequently, the search for its eigenvalues
and eigenvectors (the simplest situation is obviously the one in which this
matrix is obtained directly in a diagonal form). To nd such a basis, we
often use observables which commute both
10
with0and. Assume that
we have an observablewhich commutes with0and Since0and
commute, we can choose for the basis vectorseigenstates common to0
andFurthermore, sincecommutes with, its matrix elements are zero
between eigenvectors ofassociated with dierent eigenvalues. The matrix
(
()
)then contains numerous zeros, which facilitates its diagonalization.
(iii) cf.comment of ), the method
described in this section is valid only if the matrix elements of the pertur-
bationare much smaller than the dierences between the energy of the
level under study and those of the other levels (this conclusion would have
been evident if we had calculated higher-order corrections). However, it is
possible to extend this method to the case of a group of unperturbed levels
that are very close to each other (but distinct) and very far from all the
other levels of the system being considered. This means, of course, that the
matrix elements of the perturbationare of the same order of magnitude
as the energy dierences inside the group, but are negligible compared to the
separation between a level in the group and one outside. We can then ap-
proximately determine the inuence of the perturbationby diagonalizing
the matrix which represents=0+inside this group of levels. It is by
relying on an approximation of this type that we can, in certain cases, reduce
the study of a physical problem to that of a two-level system, such as those
described in Chapter).
References and suggestions for further reading:
For other perturbation methods, see, for example:
Brillouin-Wigner series (an expansion which is simple for all orders but which
involves the perturbed energies in the energy denominators): Ziman (2.26), 3.1.
The resolvent method (an operator method which is well suited for the calculation
of higher-order corrections): Messiah (1.17), Chap. XVI, 111; Roman (2.3), 4-5-d.
Method of Dalgarno and Lewis (which replaces the summations over the interme-
diate states by dierential equations): Borowitz (1.7). 14-5; Schi (1.18), Chap. 8,
33. Original references: (2.34), (2.35), (2.36).
The W.K.B. method, applicable to quasi-classical situations: Landau and Lifshitz
(1.19), Chap. 7; Messiah (1.17), Chap. VI, 11; Merzbacher (1.16), Chap. VII; Schi
(1.18), 34; Borowitz (1.7), Chaps. 8 and 9.
The Hartree and Hartree-Fock methods: see ComplementXV; Messiah (1.17),
Chap. XVIII, 11; Slater (11.8), Chaps. 8 and 9 (Hartree) and 17 (Hartree-Fock);
Bethe and Jackiw (1.21), Chap. 4. See also references of ComplementXIV.
10
Recall that this does not imply that0and commute.
1127

COMPLEMENTS OF CHAPTER XI, READER'S GUIDE
AXI, BXI, CXIand DXI: illustrations of stationary
perturbation theory using simple and important
examples.
AXI: A ONE-DIMENSIONAL HARMONIC OSCIL-
LATOR SUBJECTED TO A PERTURBING POTEN-
TIAL IN,
2
,
3
Study of a one-dimensional harmonic oscillator
perturbed by a potential in ,
2
,
3
. Simple,
advised for a rst reading. The last example
(perturbing potential in
3
) permits the study of
the anharmonicity in the vibration of a diatomic
molecule (a renement on the model presented in
ComplementV).
BXI: INTERACTION BETWEEN THE MAGNETIC
DIPOLES OF TWO SPIN 1/2 PARTICLES
Can be considered as a worked example, illus-
trating perturbation theory for non-degenerate
as well as degenerate states. Familiarizes the
reader with the dipole-dipole interaction between
magnetic moments of two spin 12particules.
Simple.
CXI: VAN DER WAALS FORCES Study of the long-distance forces between two
neutral atoms using perturbation theory (Van
der Waals forces). The accent is placed on the
physical interpretation of the results. A little less
simple than the two preceding complements: can
be reserved for later study.
DXI: THE VOLUME EFFECT: THE INFLUENCE OF
THE SPATIAL EXTENSION OF THE NUCLEUS ON
THE ATOMIC LEVELS
Study of the inuence of the nuclear volume
on the energy levels of hydrogen-like atoms.
Simple. Can be considered as a sequel of
ComplementVII.
EXI: THE VARIATIONAL METHOD Presentation of another approximation method,
the variational method. Important, since the
applications of the variational method are very
numerous.
1129

FXI, GXI: two important applications of the
variational method.
FXI: ENERGY BANDS OF ELECTRONS IN SOLIDS:
A SIMPLE MODEL
Introduction, using the strong-bonding approxi-
mation, of the concept of an allowed energy band
for the electrons of a solid. Essential, because of
its numerous applications. Moderetely dicult.
The accent is placed on the interpretation of the
results. The view point adopted is dierent from
that of ComplementIIIand somewhat simpler.
GXI: A SIMPLE EXAMPLE OF THE CHEMICAL
BOND: THE H
+
2
ION
Studies the phenomenon of the chemical bond for
the simplest possible case, that of the (ionized)
H
+
2
molecule. Shows how quantum mechanics
explains the attractive forces between two atoms
whose electronic wave fonctions overlap. Includes
a proof of the virial theorem. Essential from the
point of view of chemical physics. Moderately
dicult.
HXI: EXERCISES
1130

HARMONIC OSCILLATOR PERTURBED BY A POTENTIAL IN ,
2
,
3
Complement AXI
A one-dimensional harmonic oscillator subjected to a perturbing
potential in,
2
,
3
1 Perturbation by a linear potential
1-a The exact solution
1-b The perturbation expansion
2 Perturbation by a quadratic potential
3 Perturbation by a potential in
3
. . . . . . . . . . . . . . . .
3-a The anharmonic oscillator
3-b The perturbation expansion
3-c Application: the anharmonicity of the vibrations of a diatomic
molecule
In order to illustrate the general considerations of Chapter
we shall use stationary perturbation theory to study the eect of a perturbing potential
in,
2
or
3
on the energy levels of a one-dimensional harmonic oscillator (none of these
levels is degenerate,cf.Chap.).
The rst two cases (a perturbing potential inand in
2
) are exactly soluble.
Consequently, we shall be able to verify in these two examples that the perturbation
expansion coincides with the limited expansion of the exact solution with respect to the
parameter that characterizes the strength of the perturbation. The last case (a perturbing
potential in
3
) is very important in practice for the following reason. Consider a potential
()which has a minimum at= 0. To a rst approximation,()can be replaced
by the rst term (in
2
) of its Taylor series expansion, in which case we are considering
a harmonic oscillator and, therefore, an exactly soluble problem. The next term of the
expansion of(), which is proportional to
3
, then constitutes the rst correction to
this approximation. Calculation of the eect of the term in
3
, consequently, is necessary
whenever we want to study the anharmonicity of the vibrations of a physical system. It
permits us, for example, to evaluate the deviations of the vibrational spectrum of diatomic
molecules from the predictions of the (purely harmonic) model of ComplementV.
1. Perturbation by a linear potential
We shall use the notation of Chapter. Let:
0=
2
2
+
1
2
22
(1)
1131

COMPLEMENT A XI
be the Hamiltonian of a one-dimensional harmonic oscillator of eigenvectorsand
eigenvalues
1
:
0
= +
1
2
~ (2)
with= 012
We add to this Hamiltonian the perturbation:
=~
^
(3)
whereis a real dimensionless constant much smaller than 1, and
^
is given by for-
mula (B-1) of Chapter
^
is of the order of 1,~
^
is of the order of0and
plays the role of the operator
^
of Chapter). The problem consists of nding the
eigenstatesand eigenvaluesof the Hamiltonian:
=0+ (4)
1-a. The exact solution
We have already studied an example of a linear perturbation in: when the
oscillator, assumed to be charged, is placed in a uniform electric eld, we must add to
0the electrostatic Hamiltonian:
= =
~
^
(5)
whereis the charge of the oscillator. The eect of such a term on the stationary states of
the harmonic oscillator was studied in detail in ComplementV. It is therefore possible
to use the results of this complement to determine the eigenstates and eigenvalues of the
Hamiltoniangiven by (4) if we perform the substitution:
~
~
(6)
Expression (39) ofVthus yields immediately:
= +
1
2
~
2
2
~ (7)
Similarly, we see from (40) ofV(after having replacedby its expression in terms of
the creation and annihilation operatorsand):
= e
2
( )
(8)
The expansion of the exponential then yields:
=1
2
( ) +
=
+ 12
+1+
2
1+ (9)
1
To specify that we are considering the unperturbed Hamiltonian, as in Chapter, we add the
index 0 to the eigenvalue of0.
1132

HARMONIC OSCILLATOR PERTURBED BY A POTENTIAL IN ,
2
,
3
1-b. The perturbation expansion
We replace
^
by
1
2
(+)in (3) [cf.formula (B-7a) of Chapter].
We obtain:
=
~
2
+ (10)
then mixes the state only with the two states+1and 1. The only
non-zero matrix elements ofare, consequently:
+1 =
+ 12
~
1 =
2
~ (11)
According to general expression (B-15) of Chapter, we have:
=
0
+ +
=
2
0 0
+ (12)
Substituting (11) into (12) and replacing
0 0
by( )~, we immediately obtain:
=
0
+ 0
2
(+ 1)
2
~+
2
2
~+
= +
1
2
~
2
2
~+ (13)
This shows that the perturbation expansion of the eigenvalue to second order incoin-
cides
2
with the exact solution (7).
Similarly, general formula (B-11) of Chapter:
= +
=
0 0
+ (14)
yields here:
=
+ 12
+1+
2
1+ (15)
an expression which is identical to expansion (9) of the exact solution.
2. Perturbation by a quadratic potential
We now assume to have the following form:
=
1
2
~
^2
=
1
2
22
(16)
2
It can be shown that all terms of order higher than 2 in the perturbation expansion are zero.
1133

COMPLEMENT A XI
whereis a real dimensionless parameter much smaller than 1.can then be written:
=0+=
2
2
+
1
2
2
(1 +)
2
(17)
In this case, the eect of the perturbation is simply to change the spring constant of the
harmonic oscillator. If we set:
2
=
2
(1 +) (18)
we see thatis still a harmonic oscillator Hamiltonian, whose angular frequency has
become.
In this section, we shall conne ourselves to the study of the eigenvalues of.
According to (17) and (18), they can be written simply:
= +
1
2
~= +
1
2
~
1 + (19)
that is, expanding the radical:
= +
1
2
~1 +
2
2
8
+ (20)
Let us now nd result (20) by using stationary perturbation theory. Expression (16)
can also be written:
=
1
4
~ +
2
=
1
4
~
2
+
2
+ +
=
1
4
~
2
+
2
+ 2+ 1 (21)
From this, it can be seen that the only non-zero matrix elements ofassociated with
are:
=
1
2
+
1
2
~
+2 =
1
4
(+ 1)(+ 2)
12
~
2 =
1
4
(1)
12
~ (22)
When we use this result to evaluate the varions terms of (12), we nd:
=
0
+
2
+
1
2
~
2
16
(+ 1)(+ 2)
~
2
+
2
16
(1)
~
2
+
=
0
+ +
1
2
~
2
+
1
2
~
2
8
+
= +
1
2
~1 +
2
2
8
+ (23)
which indeed coincides with expansion (20).
1134

HARMONIC OSCILLATOR PERTURBED BY A POTENTIAL IN ,
2
,
3
3. Perturbation by a potential in
3
We now add to0the perturbation:
=~
^3
(24)
whereis a real dimensionless number much smaller than 1.
3-a. The anharmonic oscillator
Figure of the total potential
1
2
22
+
()in which the particle is moving. The dashed line gives the parabolic potential
1
2
22
of the unperturbed harmonic oscillalor. We have chosen0, so that the
total potential (the solid curve in the gure) increases less rapidly for0than for
0.A
E
B
x
A
x
B
mω
2
x
2
+ W(x)
x
0
1
2
Figure 1: Variation of the potential associated with an anharmonic oscillator with respect
to. We treat the dierence between the real potential (solid line) and the harmonic
potential (dashed line) of the unperturbed Hamiltonian as a perturbation (and are
the limits of the classical motion of energy).
When the problem is treated in classical mechanics, the particle with total en-
ergyis found to oscillate between two points,and(Fig.), which are no longer
symmetric with respect to. This motion, while it remains periodic, is no longer sinu-
soidal: there appears, in the Fourier expansion of(), a whole series of harmonics of the
fundamental frequency. This is why such a system is called an anharmonic oscillator
(its motion is no longer harmonic). Finally, let us point out that the period of the motion
is no longer independent of the energy, as was the case for the harmonic oscillator.
1135

COMPLEMENT A XI
3-b. The perturbation expansion
. Matrix elements of the perturbation
We replace
^
by
1
2
(+)in (24). Using relations (B-9) and (B-17) of Chapter,
we obtain, after a simple calculation:
=
~
2
32
3
+
3
+ 3 + 3(+ 1) (25)
where= was dened in Chapter B-13)].
From this can immediately be deduced the only non-zero matrix elements of
associated with:
+3 =
(+ 3)(+ 2)(+ 1)
8
1
2
~
3 =
(1)(2)
8
1
2
~
+1 = 3
+ 1
2
3
2
~
1 = 3
2
3
2
~ (26)
. Calculation of the energies
We substitute results (26) into the perturbation expansion of, see relation (12).
Since the diagonal element ofis zero, there is no rst-order correction. The four matrix
elements (26) enter, however, into the second-order correction. A simple calculation thus
yields:
= +
1
2
~
15
4
2
+
1
2
2
~
7
16
2
~+ (27)
The eect ofis therefore to lower the levels (whatever the sign of). The larger
, the greater the shift (Fig.). The dierence between two adjacent levels is equal to:
1=~1
15
2
2
(28)
It is no longer independent of, as it was for the harmonic oscillator. The energy states
are no longer equidistant and move closer together asincreases.
1136

HARMONIC OSCILLATOR PERTURBED BY A POTENTIAL IN ,
2
,
3
. Calculation of the eigenstates
Substituting relations (26) into expansion (14), we easily obtain:
= 3
+ 1
2
3
2
+1+ 3
2
3
2
1
3
(+ 3)(+ 2)(+ 1)
8
1
2
+3
+
3
(1)(2)
8
1
2
3+ (29)
Under the eect of the perturbationthe stateis therefore mixed with the states
+1, 1, +3and 3.
3-c. Application: the anharmonicity of the vibrations of a diatomic molecule
In ComplementV, we showed that a heteropolar diatomic molecule could absorb
or emit electromagnetic waves whose frequency coincides with the vibrational frequency
of the two nuclei of the molecule about their equilibrium position. If we denote by
the displacement of the two nuclei from their equilibrium position, the electric
dipole moment of the molecule can be written:
() =0+1+ (30)n – 2
n – 1
n + 1
n + 2
n
Figure 2: Energy levels of0(dashed lines) and of(solid lines). Under the eect of
the perturbation, each level of0is lowered, and the higher, the greater the shift.
1137

COMPLEMENT A XI
The vibrational frequencies of this dipole are therefore the Bohr frequencies which can
appear in the expression for(). For a harmonic oscillator, the selection rules satised
byare such that only one Bohr frequency can be involved, the frequency2(cf.
ComplementV).
When we take the perturbationinto account, the statesof the oscillator
are mixed [cf.expression (29)], andcan connect statesand for which
=1: new frequencies can thus be absorbed or emitted by the molecule.
To analyze this phenomenon more closely, we shall assume that the molecule is
initially in its vibrational ground state0(this is practically always the case at ordinary
temperaturessince, in general,~ ). By using expression (29), we can calculate,
to rst order
3
in, the matrix elements of
^
between the state0and an arbitrary
state. A simple calculation thus yields the following matrix elements (all the others
are zero to rst order in):
1
^
0=
1
2
(31a)
2
^
0=
1
2
(31b)
0
^
0=
3
2
(31c)
From this, we can nd the transition frequencies observable in the absorption
spectrum of the ground state. We naturally nd the frequency:
1=
1 0
(32a)
which appears with the greatest intensity since, according to (31a),1
^
0is of
zeroth-order in. Then, with a much smaller intensity [cf.formula (31b
frequency:
2=
2 0
(32b)
which is often called the second harmonic (although it is not rigorously equal to twice
1).
Comment:
Result (31c) means that the average value of
^
is not zero in the ground state. This can
easily be understood from Figure, since the oscillatory motion is no longer symmetric
about. Ifis negative (the case in Figure), the oscillator spends more time in the
0region than in the0region, and the average value ofmust be positive. We
thus understand the sign appearing in (31c).
The preceding calculation reveals only one new line in the absorption spectrum.
Actually, the perturbation calculation could be pursued to higher orders in, taking
into account higher order terms in expansion (30) of the dipole moment(), as well as
3
It would not be correct to keep terms of order higher than 1 in the calculation, since expansion (29)
is valid only to rst order in.
1138

HARMONIC OSCILLATOR PERTURBED BY A POTENTIAL IN ,
2
,
3
terms in
4
,
5
in the expansion of the potential in the neighborhood of= 0. All the
frequencies:
=
0
(33)
with= 345would then be present in the absorption spectrum of the molecule (with
intensities decreasing very rapidly whenincreases). This would nally give, for this
spectrum, the form shown in Figure. This is what is actually observed.0 ν
1
ν
2
ν
3
ν
4
ν
Figure 3: Form of the vibrational spectrum of a heteropolar diatomic molecule. A series of
harmonic frequencies2,3 appear in addition to the fundamental frequency
1. This results from the anharmonicity of the potential, as well as higher order terms
in the power series expansion in(the distance between the two atoms) of the molecular
dipole moment(). Note that the corresponding lines are not quite equidistant and that
their intensity decreases rapidly whenincreases.
Note that the various spectral lines of Figure
to formula (28):
10 =
1 0
=
2
1
15
2
2
(34)
2 1=
2 1
=
2
115
2
(35)
3 2=
3 2
=
2
1
45
2
2
(36)
which gives the relation:
(2 1) 1= (3 2)(2 1) =
15
4
2
(37)
Thus we see that the study of the precise positions of the lines of the absorption spectrum
makes it possible to nd the parameter.
1139

COMPLEMENT A XI
Comments:
(i) appearing in (52) of ComplementVIIcan be evaluated by using
formula (27) of the present complement. Comparing these two expressions and
replacingbyin (27), we obtain:
=
15
4
2
(38)
Now, the perturbing potential inVIIis equal to
3
, while here we have chosen
it equal to~^
3
, that is, equal to:
35
~
1
2
3
(39)
We therefore have:
=
~
35
1
2
(40)
which, substituted into (38), nally yields:
=
15
4
2
~
35
(41)
(ii) = 0, the term in
4
is
much smaller than the term in
3
but it corrects the energies to rst order, while the
term in
3
enters only in second order (cf. above). It is therefore necessary
to evaluate these two corrections simultaneously (they may be comparable) when
the spectrum of Figure
References and suggestions for further reading:
Anharmonicity of the vibrations of a diatomic molecule: Herzberg (12.4), vol. I, Chap. III,
2.
1140

INTERACTION BETWEEN THE MAGNETIC DIPOLES OF TWO SPINS 1/2
Complement BXI
Interaction between the magnetic dipoles of two spin 1/2 particles
1 The interaction Hamiltonian W. . . . . . . . . . . . . . . . .
1-a The form of the HamiltonianW. Physical interpretation
1-b An equivalent expression forW. . . . . . . . . . . . . . . . .
1-c Selection rules
2 Eects of the dipole-dipole interaction on the Zeeman sub-
levels of two xed particles
2-a Case where the two particles have dierent magnetic moments
2-b Case where the two particles have equal magnetic moments
2-c Example: the magnetic resonance spectrum of gypsum
3 Eects of the interaction in a bound state
In this complement, we intend to use stationary perturbation theory to study the
energy levels of a system of two spin 1/2 particles placed in a static eldB0and coupled
by a magnetic dipole-dipole interaction.
Such systems do exist. For example, in a gypsum monocrystal (CaSO4, 2H20),
the two protons of each crystallization water molecule occupy xed positions, and the
dipole-dipole interaction between them leads to a ne structure in the nuclear magnetic
resonance spectrum.
In the hydrogen atom, there also exists a dipole-dipole interaction between the
electron spin and the proton spin. In this case, however, the two particles are moving
relative to each other, and we shall see that the eect of the dipole-dipole interaction
vanishes due to the symmetry of the1ground state. The hyperne structure observed
in this state is thus due to other interactions (contact interaction;cf.Chap.,
and XII).
1. The interaction HamiltonianW
1-a. The form of the Hamiltonian W. Physical interpretation
LetS1andS2be the spins of particles (1) and (2), andM1andM2their corre-
sponding magnetic moments:
M1=1S1
M2=2S2 (1)
[where1and2are the gyromagnetic ratios of (1) and (2)].
We callthe interaction of the magnetic momentM2with the eld created by
M1at (2). Ifndenotes the unit vector of the line joining the two particles and, the
distance between them (Fig.),can be written:
=
0
4
12
1
3
[S1S23 (S1n) (S2n)] (2)
1141

COMPLEMENT B XI
Figure 1: Relative disposition of the magnetic momentsM1andM2of particles (1)
and (2) (is the distance between the two particles, andnis the unit vector of the
straight line between them).
The calculation which enables us to obtain expression (2) is in every way analogous to
the one that will be presented in ComplementXIand which leads to the expression for
the interaction between two electric dipoles.
1-b. An equivalent expression forW
Letandbe the polar angles ofn. If we set:
() =
0
4
12
3
(3)
we get:
=()3 [1cos+ sin(1cos+1sin)]
[2cos+ sin(2cos+2sin)]S1S2
=()31cos+
1
2
sin 1+e+1e
2cos+
1
2
sin 2+e+2e S1S2 (4)
that is:
=() [0+
0+1+ 1+2+ 2] (5)
1142

INTERACTION BETWEEN THE MAGNETIC DIPOLES OF TWO SPINS 1/2
where:
0=3 cos
2
112
0=
1
4
3 cos
2
1(1+2+1 2+)
1=
3
2
sincose(12++1+2)
1=
3
2
sincose(12+1 2)
2=
3
4
sin
2
e
2
1+2+
2=
3
4
sin
2
e
2
1 2
(6)
Each of the terms(or) appearing in (5) is, according to (6), the product
of a function ofandproportional to the second-order spherical harmonic
2
and
an operator acting only on the spin degrees of freedom [the space and spin operators
appearing in (6) are second-rank tensors;, for this reason, is often called the tensor
interaction].
1-c. Selection rules
,andare the spherical coordinates of the relative particle associated with the
system of two particles (1) and (2). The operatoracts only on these variables and on
the spin degrees of freedom of the two particles. Letbe a standard basis in the
state spacerof the relative particle, and12, the basis of eigenvectors common to
1and2in the spin state space(1= 2=). The state space in whichacts is
spanned by the 12basis, in which it is very easy, using expressions (5)
and (6), to nd the selection rules satised by the matrix elements of
. Spin degrees of freedom
0changes neither1nor2.
0ips both spins:
+ +and + +
1ips one of the two spins up:
2 +2or 1 1+
Similarly,1ips one of the two spins down:
+2 2or 1+ 1
Finally,2and 2ip both spins up and down, respectively:
++and++
1143

COMPLEMENT B XI
. Orbital degrees of freedom
When we calculate the matrix element of()between the state and
the state , the following angular integral appears:
()
2
()() d (7)
which, according to the results of ComplementX, is dierent from zero only for:
= 2+ 2 (8a)
=+ (8b)
Note that the case== 0, although not in contradiction with (8), is excluded because
we must always be able to form a triangle with,and 2, which is impossible when
== 0. We must have then:
1 (8c)
2. Eects of the dipole-dipole interaction on the Zeeman sublevels of two xed
particles
In this section, we shall assume the two particles to be xed in space. We shall therefore
quantize only the spin degrees of freedom, considering the quantities,andas given
parameters.
The two particles are placed in a static eldB0parallel to. The Zeeman
Hamiltonian0, describing the interaction of the two spin magnetic moments withB0,
can then be written:
0=11+22 (9)
with:
1= 10
2= 20 (10)
In the presence of the dipole-dipole interaction, the total Hamiltonianof the system
becomes:
=0+ (11)
We shall assume the eld0to be large enough and treatas a perturbation of0.
2-a. Case where the two particles have dierent magnetic moments
. Zeeman levels and the magnetic resonance spectrum in the absence of interaction
According to (9), we have:
012=
~
2
(11+22)12 (12)
1144

INTERACTION BETWEEN THE MAGNETIC DIPOLES OF TWO SPINS 1/2 –, +
–, –
+, –
+, +
–, +
–, –
+, –
a b
+, +
(ω
1+ω
2)
ħ
ħΩ
ħΩ
ħΩ
ħΩ
2
(ω
1–ω
2)
ħ
2
(ω
1–ω
2)–
–
–
–
ħ
2
(ω
1+ω
2)
ħ
2
Figure 2: Energy levels of two spin 1/2 particles, placed in a static eldB0parallel to
. The two Larmor angular frequencies,1= 10and2= 20, are assumed
to be dierent.
For gure a, the energy levels are calculated without taking account of the dipole-dipole
interactionbetween the two spins.
For gure b, we take this interaction into account. The levels undergo a shift whose
approximate value, to rst order in, is indicated on the right-hand side of the gure.
The solid-line arrows join the levels between which1has a non-zero matrix element,
and the shorter dashed-line arrows those for which2does.
Figurea represents the energy levels of the two-spin system in the absence of the dipole-
dipole interaction (we have assumed1 20). Since1=2, these levels are all
non-degenerate.
If we apply a radio-frequency eldB1cosparallel toOx, we obtain a series of
magnetic resonance lines. The frequencies of these resonances correspond to the vari-
ous Bohr frequencies which can appear in the evolution of11+22(the radio-
frequency eld interacts with the component alongOxof the total magnetic moment).
The solid-line (dashed-line) arrows of Figurea join levels between which1(2)has
a non-zero matrix element. Thus we see that there are two distinct Bohr angular fre-
quencies, equal to1and2(Fig.a), which correspond simply to the resonances of the
individual spins, (1) and (2).
. Modications created by the interaction
Since all the levels of Figurea are non-degenerate, the eect ofcan be obtained
to rst order by calculating the diagonal elements of12 12. It is clear from
expressions (5) and (6) that only the term0makes a non-zero contribution to this
1145

COMPLEMENT B XIa
b
ω
1
ω
1
ω
2
ω
2
4Ω 4Ω
Figure 3: The Bohr frequencies appearing in the evolution of1and 2give the
positions of the magnetic resonance lines that can be observed for the two-spin system
(the transitions corresponding to the arrows of Figure). In the absence of a dipole-
dipole interaction, two resonances are obtained, each one corresponding to one of the two
spins (g. a). The dipole-dipole interaction is expressed by a splitting of each of the two
preceding lines (g. b).
matrix element, which is then equal to:
12 12=()3 cos
2
1
12~
2
4
=12~ (13)
with:
=
~
4
()3 cos
2
1=
~0
16
12
3
3 cos
2
1 (14)
Sinceis much smaller than0, we have:
1 2 (15)
From this we can immediately deduce the level shifts to rst order in:~for++
and , and~for+ and for+(Fig.b).
What now happens to the magnetic resonance spectrum of Figurea? If we are
concerned only with lines whose intensities are of zeroth order in(that is, those that
approach the lines of Figurea when approaches zero), then to calculate the Bohr
frequencies appearing in1and 2we simply use the zeroth-order expressions
for the eigenvectors
1
. It is then the same transitions which are involved (compare the
arrows of Figuresa andb). We see, however, that the two lines which correspond to the
frequency1in the absence of coupling (solid-line arrows) now have dierent frequencies:
1
If we used higher-order expressions for the eigenvectors, we would see other lines of lower intensity
appear (they disappear when 0).
1146

INTERACTION BETWEEN THE MAGNETIC DIPOLES OF TWO SPINS 1/2
1+ 2and12. Similarly, the two lines corresponding to2(dashed-line arrows)
now have frequencies of2+ 2and22. The magnetic resonance spectrum is
therefore now composed of two doublets centered at1and2, the interval between
the two components of each doublet being equal to4(Fig.b).
Thus, the dipole-dipole interaction leads to a ne structure in the magnetic reso-
nance spectrum, for which we can give a simple physical interpretation. The magnetic
momentM1associated withS1creates a local eldbat particle (2). Since we assume
B0to be very large,S1precesses very rapidly aboutOz, so we can consider only the
1component (the local eld created by the other components oscillates too rapidly to
have a signicant eect). The local eldbtherefore has a dierent direction depending
on whether the spin is in the state+or, that is, depending on whether it points
up or down. It follows that the total eld seen by particle (2), which is the sum ofB0
andb, can take on two possible values
2
. This explains the appearance of two resonance
frequencies for the spin (2). The same argument would obviously enable us to understand
the origin of the doublet centered at1.
2-b. Case where the two particles have equal magnetic moments
. Zeeman levels and the magnetic resonance spectrum in the absence of the
interaction
Formula (12) remains valid if we choose1and2to be equal. We shall therefore
set:
1=2== 0 (16)
The energy levels are shown in Figurea. The upper level,++, and the lower level,
, of energies~and~, are non-degenerate. On the other hand, the intermediate
level, of energy 0, is two-fold degenerate: to it correspond the two eigenstates+ and
+.
The frequencies of the magnetic resonance lines can be obtained by nding the
Bohr frequencies involved in the evolution of1+2(the total magnetic moment is
now proportional to the total spinS=S1+S2). We easily obtain the four transitions
represented by the arrows in Figurea, which correspond to a single angular frequency
. This nally yields the spectrum of Figurea.
. Modications created by the interaction
The shifts of the non-degenerate levels++and can be obtained as they
were before, and are both equal to~[we must replace, however,1and2byin
expression (14) for].
Since the intermediate level is two-fold degenerate, the eect ofon this level
can now be obtained by diagonalizing the matrix that represents the restriction ofto
the subspace+ +. The calculation of the diagonal elements is performed as
above and yields:
+ + = + +=~ (17)
2
Actually, sinceB0 b, it is only the component ofbalongB0which is involved.
1147

COMPLEMENT B XI +, +
1, 1
0, 0
S, M
1, 0
+, – –, +
–, –
1, – 1
ħω
ħω ħΩ
ħΩ
– 2ħΩ
–
0
a b
Figure 4: The two spin 1/2 particles are assumed to have the same magnetic moment
and, conseqnently, the same Larmor angular frequency= 0.
In the absence of a dipole-dipole interaction, we obtain three levels, one of which is two-
fold degenerate (g. a). Under the eect of the dipole-dipole interaction (g. b), these
levels undergo shifts whose approximate values (to rst order in) are indicated on
the right-hand side of the gure. To zeroth-order in, the stationary states are the
eigenstates of the total spin. The arrows join the levels between which1+2
has a non-zero matrix element.
As for the non-diagonal element+ +, we easily see from expressions (5) and (6)
that only the term
0contributes to it:
+ +=
()
4
3 cos
2
1 +(1+2+12+)+
=()
~
2
4
3 cos
2
1=~ (18)
We are then led to the diagonalization of the matrix:
~
1 1
1 1
(19)
whose eigenvalues are2~and 0; they are respectively associated with the eigenvectors
1=
1
2
(+ ++)and2=
1
2
(+ +).
Figureb represents the energy levels of the system of two coupled spins. The
energies, to rst order inare given by the eigenstates to zeroth order.
Note that these eigenstates are none other than the eigenstatescommon to
S
2
and, whereS=S1+S2is the total spin. Since the operatorcommutes with
S
2
, it can only couple the triplet states, that is,10to11and10to11. This
1148

INTERACTION BETWEEN THE MAGNETIC DIPOLES OF TWO SPINS 1/2ω ω
6Ω
a b
Figure 5: Shape of the magnetic resonance spectrum which can be observed for a system
of two spin 1/2 particles, with the same gyromagnetic ratio, placed in a static eld0.
In the absence of a dipole-dipole interaction, we observe a single resonance (g. a). In the
presence of a dipole-dipole interaction (g. b), the preceding line splits. The separation
6between the two components of the doublet is proportional to3 cos
2
1, whereis
the angle between the static eld0and the straight line joining the two particles.
gives the two transitions represented by the arrows in Figureb, and to which correspond
the Bohr frequencies+ 3and 3. The magnetic resonance spectrum is therefore
composed of a doublet centered at, the separation between the two components of the
doublet being equal to6(Fig.b).
2-c. Example: the magnetic resonance spectrum of gypsum
The case studied in
lization water molecule in a gypsum monocrystal (CaSO4, 2H2O). These two protons
have identical magnetic moments and can be considered to occupy xed positions in the
crystal. Moreover, they are much closer to each other than to other protons (belonging to
other water molecules). Since the dipole-dipole interaction decreases very quickly when
the distance increases (1
3
law), we can neglect interactions between protons belonging
to other water molecules.
The magnetic resonance spectrum is indeed observed to contain a doublet
3
whose
separation depends on the anglebetween the eldB0and the straight line joining the
two protons. If we rotate the crystal with respect to the eldB0, this anglevaries, and
the separation between the two components of the doublet changes. Thus, by studying
the variations of this separation, we can determine the positions of the water molecules
relative to the crystal axes.
When the sample under study is not a monocrystal, but rather a powder composed
of small, randomly oriented monocrystals,takes on all possible values. We then observe
a wide band, due to the superposition of doublets having dierent separations.
3. Eects of the interaction in a bound state
We shall now assume that the two particles, (1) and (2), are not xed, but can move
with respect to each other.
3
Actually, in a gypsum monocrystal, there are two dierent orientations for the water molecules,
and, consequently, two doublets corresponding to the two possible values of.
1149

COMPLEMENT B XI
Consider, for example, the case of the hydrogen atom (a proton and an electron).
When we take only the electrostatic forces into account, the ground state of this atom
(in the center of mass frame) is described by the ket100, labeled by the quantum
numbers= 1,= 0,= 0(cf.Chap.). The proton and the electron are spin 1/2
particles. The ground state is therefore four-fold degenerate, and a possible basis in the
corresponding subspace is made up of the four vectors:
100 12 (20)
where1, and2, equal to+or, represent respectively the eigenvalues ofand(S
andI: the electron and proton spins).
What is the eect on this ground state of the dipole-dipole interaction betweenS
andI? The matrix elements ofare much smaller than the energy dierence between the
1level and the excited levels, so that it is possible to treat the eect ofby perturbation
theory. To rst order, it can be evaluated by diagonalizing the44matrix of elements
10012 10012. The calculation of these matrix elements, according to (5)
and (6), involves angular integrals of the form:
0
0()
2
()
0
0() d (21)
which are equal to zero, according to the selection rules established in
particular case, it can be shown very simply that integral (21) is equal to zero: since
0
0
is a constant, expression (21) is proportional to the scalar product of
2
and
0
0, which
is equal to zero because of the spherical harmonic orthogonality relations].
The dipole-dipole interaction does not modify the energy of the ground state to
rst order. It enters, however, into the (hyperne) structure of the excited levels with
1. We must then calculate the matrix elements
12 12, that
is, the integrals:
()
2
()() d (22)
which, according to (8c), become non-zero as soon as1.
References and suggestions for further reading:
Evidence in nuclear magnetic resonance experiments of the magnetic dipole inter-
actions between two spins in a rigid lattice: Abragam (14.1), Chap. IV, II and
Chap. VII, IA; Slichter (14.2), Chap. 3; Pake (14.6).
1150

VAN DER WAALS FORCES
Complement CXI
Van der Waals forces
1 The electrostatic interaction Hamiltonian for two hydro-
gen atoms
1-a Notation
1-b Calculation of the electrostatic interaction energy
2 Van der Waals forces between two hydrogen atoms in the
1ground state
2-a Existence of a
6
attractive potential
2-b Approximate calculation of the constant C
2-c Discussion
3 Van der Waals forces between a hydrogen atom in the 1
state and a hydrogen atom in the2state
3-a Energies of the stationary states of the two-atom system. Res-
onance eect
3-b Transfer of the excitation from one atom to the other
4 Interaction of a hydrogen atom in the ground state with a
conducting wall
The character of the forces exerted between two neutral atoms changes with the
order of magnitude of the distanceseparating these two atoms.
Consider, for example, two hydrogen atoms. Whenis of the order of atomic
dimensions (that is, of the order of the Bohr radius0), the electronic wave functions
overlap, and the two atoms attract each other, since they tend to form an H2molecule.
The potential energy of the system has a minimum
1
for a certain valueof the distance
between the atoms. The physical origin of this attraction (and therefore of the chemical
bond) lies in the fact that the electrons can oscillate between the two atoms (cf.
and ). The stationary wave functions of the two electrons are no
longer localized about only one of the nuclei; this lowers the energy of the ground state
(cf.ComplementXI).
At greater distances, the phenomena change completely. The electrons can no
longer move from one atom to the other, since the probability amplitude of such a process
decreases with the decreasing overlap of the wave functions, that is, exponentially with
the distance. The preponderant eect is then the electrostatic interaction between the
electric dipole moments of the two neutral atoms. This gives rise to a total energy which
is attractive and which decreases, not exponentially, but with1R
6
. This is the origin
of theVan der Waals forces, which we intend to study in this complement by using
stationary perturbation theory (conning ourselves, for the sake of simplicity, to the case
of two hydrogen atoms).
It should be clearly understood that the fundamental nature of Van der Waals
forces is the same as that of the forces responsible for the chemical bond: the basic
1
At very short distances, the repulsive forces between the nuclei always dominate.
1151

COMPLEMENT C XI
Figure 1: Relative position of the two hydrogen atoms.is the distance between the two
protons, which are situated atand, andnis the unit vector on the line joining them.
randrare the position vectors of the two electrons with respect to pointsand
respectively.
Hamiltonian is electrostatic in both cases. Only the variation of the energies of the
quantum stationary states of the two-atom system with respect toallows us to dene
and dierentiate these two types of forces.
Van der Waals forces play an important role in physical chemistry, especially when
the two atoms under consideration have no valence electrons (forces between rare gas
atoms, stable molecules, etc.). They are partially responsible for the dierences between
the behavior of a real gas and that of an ideal gas. Finally, as we have already said, these
are long-range forces, and are therefore involved in the stability of colloids.
We shall begin by determining the expression for the dipole-dipole interaction
Hamiltonian between two neutral hydrogen atoms ( 1). This will enable us to study the
Van der Waals forces between two atoms in the1state ( 2), or between an atom in the
2state and an atom in the1state ( 3). Finally, we shall show ( 4) that a hydrogen
atom in the1state is attracted by its electrical mirror image in a perfectly conducting
wall.
1. The electrostatic interaction Hamiltonian for two hydrogen atoms
1-a. Notation
The two protons of the two hydrogen atoms are assumed to remain motionless at
pointsand(Fig.). We shall set:
R=OBOA (1)
=R (2)
n=
R
R
(3)
1152

VAN DER WAALS FORCES
is the distance between the two atoms, andnis the unit vector on the line that joins
them. Letrbe the position vector of the electron attached to atom()with respect
to point, andr, the position vector of the electron attached to atomwith respect
to. We call:
=r (4)
=r (5)
the electric dipole moments of the two atoms (is the electron charge).
We shall assume throughout this complement that:
rr (6)
Although they are identical, the electrons of the two atoms are well separated, and their
wave functions do not overlap. It is therefore not necessary to apply the symmetrization
postulate (cf.Chap. , ).
1-b. Calculation of the electrostatic interaction energy
Atom()creates at()an electrostatic potentialwith which the charges of
()interact. This gives rise to an interaction energy.
We saw in ComplementXthatcan be calculated in terms ofnand the
multipole moments of atom(). Since()is neutral, the most important contribution
tois that of the electric dipole moment. Similarly, since()is neutral, the most
important term incomes from the interaction between the dipole momentof
()and the electric eldE=rwhich is essentially created by. This explains
the name of dipole-dipole interaction given to the dominant term of. There exist,
of course, smaller terms (dipole-quadrupoleand , quadrupole-quadrupole,
etc.), andis written:
= + + + + (7)
To calculate, we shall start with the expression for the electrostatic potential created
by at():
(R) =
1
40
R
3
(8)
which leads to:
E=rR=
40
1
3
[r3 (rn)n] (9)
and, consequently:
=E B=
e
2
3
[rr3 (rn) (rn)] (10)
We have set
2
=
2
40, and we have used expressions (4) and (5) forand . In
this complement, we shall choose theaxis parallel ton, so that (10) can be written:
=
2
3
( + 2 ) (11)
1153

COMPLEMENT C XI
In quantum mechanics, becomes the operator, which can be obtained by
replacing in (11) by the corresponding observables, , which
act in the state spacesandof the two hydrogen atoms
2
:
=
2
3
( + 2 ) (12)
2. Van der Waals forces between two hydrogen atoms in the1ground state
2-a. Existence of a
6
attractive potential
. Principle of the calculation
The Hamiltonian of the system is:
=0+0+ (13)
where0and0are the energies of atoms()and()when they are isolated.
In the absence of, the eigenstates ofare given by the equation:
(0+0) ; = (+ ) ; (14)
where the and thewere calculated in . In particular, the
ground state of0+0is
100;
100, of energy2. It is non-degenerate (we
do not take spins into account).
The problem is to evaluate the shift in this ground state due toand, in
particular, its-dependence. This shift represents, so to speak, the interaction potential
energy of the two atoms in the ground state.
Since is much smaller than0and0, we can calculate this eect by
stationary perturbation theory.
. First-order eect of the dipole-dipole interaction
Let us show that the rst-order correction:
1=
100;
100 100;
100 (15)
is zero; the energy1involves, according to expression (12) for, products of the
form
100 100 100 100(and analogous quantities in whichis
replaced by and by ). These products are zero since, in a stationary
state of the atom, the average values of the components of the position operator are zero.
Comment:
The other terms, of expansion (7) involve products of two mul-
tipole moments, one relative to () and the other one to (), at least one of which
is of order higher than 1. Their contributions are also zero to rst order: they
2
The translational external degrees of freedom of the two atoms are not quantized: for the sake of
simplicity, we assume the two protons to be innitely heavy and motionless. In (12),is therefore a
parameter and not an observable.
1154

VAN DER WAALS FORCES
are expressed in terms of average values in the ground state of multipole opera-
tors of order greater than or equal to one, and we know (cf.ComplementX,
) that such average values are zero in an= 0state (triangle rule of Clebsch-
Gordan-coecients). Therefore we must nd the second-order eect of, which
constitutes the most important energy correction.
. Second-order eect of the dipole-dipole interaction
According to the results of Chapter, the second-order energy correction can be
written:
2=
;
100;
100
2
2
(16)
where the notationmeans that the state
100;
100is excluded from the sum-
mation
3
.
Since is proportional to1
3
,2is proportional to1
6
. Furthermore, all the
energy denominators are negative, since we are starting from the ground state. Therefore,
the dipole-dipole interaction gives rise to a negative energy proportional to1
6
:
2=
6
(17)
Van der Waals forces are therefore attractive and vary with1
7
.
Finally, let us calculate the expansion of the ground state to rst order in.
We nd, according to formula (B-11) of Chapter:
0=
100;
100
+ ;
;
100;
100
2
+
(18)
Comment:
The matrix elements appearing in expressions (16) and (18) above involve the quantities
100 100(and analogous quantities in whichand
are replaced byand orand), which are dierent from zero only if= 1
and= 1. These quantites are indeed proportional to products of angular integrals
()
1
()
0
0() d ()
1
()
0
0() d
which, according to the results of ComplementX, are zero if= 1or= 1. We can
therefore, in (16) and (18), replaceandby 1.
3
This summation is performed not only over the bound states, but also over the continuous spectrum
of0+0
1155

COMPLEMENT C XI
2-b. Approximate calculation of the constant C
According to (16) and (12), the constant C appearing in (17) is given by:
= e
4
; (X + 2 )
100;
100
2
2+ +
(19)
We must have>2and>2. For bound states,=
2
is smaller than
, and the error is not signicant if we replace in (19)and by 0. For states
in the continuous spectrum,varies between 0 and+. The matrix elements of the
numerator become small, however, as soon as the size ofbecomes appreciable, since
the spatial oscillations of the wave function are then numerous in the region in which
100(r)is non-zero.
To have an idea of the order of magnitude of C, we can therefore replace all the
energy denominators of (19) by 2E. Using the closure relation and the fact that the
diagonal element ofis zero ( ), we then get:
e
4
2
100;
100(X + 2 )
2
100;
100 (20)
This expression is simple to calculate: because of the spherical symmetry of the
1state, the average values of the cross terms of the type are zero.
Furthermore, and for the same reason, the various quantities:
100
2
100 100
2
100 100
2
100
are all equal to one third of the average value ofR
2
=
2
+
2
+
2
. We nally obtain,
using the expression for the wave function100(r):
e
4
2
6
100
R
2
3
100
2
= 6
25
0 (21)
(where0is the Bohr radius) and, consequently:
2 6
2
5
0
6
=6
2
0
5
(22)
The preceding calculation is valid only if0 (no overlapping of the wave functions).
Thus we see that2is of the order of the electrostatic interaction between two charges
and, multiplied by the reduction factor(0)
5
1.
2-c. Discussion
. Dynamical interpretation of Van der Waals forces
At any given instant, the electric dipole moment (we shall say, more simply, the
dipole) of each atom has an average value of zero in the ground state
100or
100.
This does not mean that any individual measurement of a component of this dipole will
yield zero. If we make such a measurement, we generally nd a non-zero value; however,
1156

VAN DER WAALS FORCES
we have the same probability of nding the opposite value. The dipole of a hydrogen
atom in the ground state should therefore be thought of as constantly undergoing random
uctuations.
We shall begin by neglecting the inuence of one dipole on the motion of the other
one. Since the two dipoles are then uctuating randomly and independently, their mean
interaction is zero: this explains the fact thathas no rst-order eect.
However, the two dipoles are not really independent. Consider the electrostatic
eld created by dipole()at()This eld follows the uctuations of dipole(). The
dipole it induces at()is therefore correlated with dipole()so the electrostatic eld
which returns to()is no longer uncorrelated with the motion of dipole(). Thus,
although the motion of dipole()is random, its interaction with its own eld, which
is reected to it by(), does not have a average value of zero. This is the physical
interpretation of the second-order eect of.
The dynamical aspect is therefore useful for understanding the origin of Van der
Waals forces. If we were to think of the two hydrogen atoms in the ground state as two
spherical and static clouds of negative electricity (with a positive point charge at the
center of each one), we would be led to a rigorously zero interaction energy.
. Correlations between the two dipole moments
Let us show more precisely that there exists a correlation between the two dipoles.
When we take into account, the ground state of the system is no longer
100;
100, but0[cf.expression (18)]. As shown below, a simple calculation
yields:
0 0== 0 0= 0 (23)
to rst order in.
Consider, for example, the matrix element 0 0. The zeroth-order term,
100;
100 100;
100is zero, since it is equal to the average value ofin the ground
state
100. To rst order, the summation appearing in formula (18) must be included. Since
contains only products of the form , the coecients of the kets
100;
and ;
100in this summation are zero. The rst-order terms which could be dierent
from zero are therefore proportional to
;
100;
100 with= 0and= 0;
These terms are all zero sincedoes not act on
100and
100= 0for= 0.
Thus, even in the presence of an interaction, the average values of the components
of each dipole are zero. This is not surprising: in the interpretation of , the dipole
induced in()by the eld of dipole ()uctuates randomly, like this eld, and has,
consequently, an average value of zero.
Let us show, on the other hand, that the two dipoles are correlated, by evaluating
the average value of a product of two components, one relative to dipole()and the
other, to dipole(). We shall calculate0( + 2 )0, for exam-
ple, which, according to (12), is nothing more than(
32
)0 0. Using (18),
1157

COMPLEMENT C XI
we immediately nd, taking (15) and (16) into account, that:
0( + 2 )0= 22
3
2
= 0 (24)
Thus, the average values of the products, and are not zero, as would
be the products of average values , and according to (23).
This proves the existence of a correlation between the two dipoles.
. Long-range modication of Van der Waals forces
The description of above enables us to understand that the preceding
calculations are no longer valid if the two atoms are too far apart. The eld produced
by()and reected by()returns to()with a time lag due to the propagation
()()(), and we have argued as if the interactions were instantaneous.
We can see that this propagation time can no longer be neglected when it becomes
of the order of the characteristic times of the atom's evolution, that is, of the order of
2 1, where1= ( 1)~denotes a Bohr angular frequency. In other words,
the calculations performed in this complement assume that the distancebetween the
two atoms is much smaller than the wavelengths2 1of the spectrum of these atoms
(about 1 000

A).
A calculation which takes propagation eects into account gives an interaction
energy which, at large distances, decreases as1
7
. The1
6
law which we have found
therefore applies to an intermediate range of distances, neither too large (because of the
time lag) nor too small (to avoid overlapping of the wave functions).
3. Van der Waals forces between a hydrogen atom in the1state and a
hydrogen atom in the2state
3-a. Energies of the stationary states of the two-atom system. Resonance eect
The rst excited level of the unperturbed Hamiltonian0+0is eight-fold
degenerate. The associated eigensubspace is spanned by the eight states :
100;
200;
200;
100;
100;
21 with=10+1;
21
;
100with =10+1, which correspond to a situation in which one
of the two atoms is in the ground state, while the other one is in a state of the= 2
level.
According to perturbation theory for a degenerate state, to obtain the rst-order
eect of, we must diagonalize the88matrix representing the restriction of
to the eigensubspace. We shall show that the only non-zero matrix elements ofare
those which connect a state
100;
21to the state
21;
100. The operators
appearing in the expression forare odd and can therefore couple
100
only to one of the
21; an analogous argument is valid for . Finally, the
dipole-dipole interaction is invariant under a rotation of the two atoms about the
axis which joins them; thereforecommutes with + and can only join two
states for which the sum of the eigenvalues ofand is the same.
Therefore, the preceding88matrix can be broken down into four22matrices.
One of them is entirely zero (the one which concerns the 2states), and the other three
1158

VAN DER WAALS FORCES
are of the form:
0
3
3
0
(25)
where we have set:
100;
21 21;
100=
3
(26)
is a calculable constant, of the order of
22
0, which will not be specied here.
We can easily diagonalize matrix (25), obtaining the eigenvalues+
3
and
3
, associated respectively with the eigenstates:
1
2
100;
21+
21;
100
and:
1
2
100;
21 21;
100
This reveals the following important results:

3
and not as1
6
, sincenow modies the
energies to rst order. The Van der Waals forces are therefore more important than
they were between two hydrogen atoms in the1state (resonance eect between
two dierent states of the total system with the same unperturbed energy).
+
3
and
3
). There exist therefore states of the two-atom system for which there is
attraction, and others for which there is repulsion.
3-b. Transfer of the excitation from one atom to the other
The two states
100;
21and
21;
100have the same unperturbed en-
ergy and are coupled by a non-diagonal perturbation. According to the general results
of
from one level to the other with a frequency proportional to the coupling.
Therefore, if the system starts in the state
100;
21at= 0, it arrives
after a certain time (the larger, the longer the time) in the state
21;
100. The
excitation thus passes from()to(), then returns to(), and so on.
Comment:
If the two atoms are not xed but, for example, undergo collision,varies over time
and the passage of the excitation from one atom to the other is no longer periodic.
The corresponding collisions, called resonant collisions, play an important role in the
broadening of spectral lines.
1159

COMPLEMENT C XIz
A
d
O
A′
r′
A
r
A
Figure 2: To calculate the interaction energy of a hydrogen atom with a perfectly con-
ducting wall, we can assume that the electric dipole momentrof the atom interacts
with its electrical imager(is the distance between the protonand the wall).
4. Interaction of a hydrogen atom in the ground state with a conducting wall
We shall now consider a single hydrogen atom()situated at a distancefrom a wall
which is assumed to be perfectly conducting. Theaxis is taken along the perpendic-
ular to the wall passing through(Fig.). The distanceis assumed to be much larger
than the atomic dimensions. We can therefore ignore the atomic structure of the wall,
and assume that the atom interacts with its electrical image on the other side of this wall
(that is, with a symmetrical atom with opposite charges). The dipole interaction energy
between the atom and the wall can easily be obtained from expression (12) forby
making the following substitutions:
2 2
2
=
=
=
(27)
(the change of
2
to
2
is due to the sign dierence of the image charges).
Furthermore it is necessary to divide by 2 since the dipole image is ctitious,
1160

VAN DER WAALS FORCES
proportional to the electric dipole of the atom
4
. We then get:
=
2
16
3
(
2
+
2
+ 2
2
) (28)
which represents the interaction energy of the atom with the wall [acts only on the
degrees of freedom of()].
If the atom is in its ground state, the energy correction to rst order inis then:
1= 100 100 (29)
Using the spherical symmetry of the lstate, we obtain:
1=
2
16
3
4100
R
2
3
100=
22
0
4
3
(30)
We see that the atom is attracted by the wall: the attraction energy varies as 1/
3
, and,
therefore, the force of attraction varies as 1/
4
.
The fact thathas an eect even to rst order can easily be understood in terms
of the discussion of
the two dipoles, since they are images of each other.
References and suggestions for further reading:
Kittel (13.2), Chap. 3. p. 82; Davydov (1.20), Chap. XII. 124 and 125;
Langbein (12.9).
For a discussion of retardation eects, see: Power (2.11), 7.5 and 8.4 (quantum
electrodynamic approach); Landau and Lifshitz (7.12), Chap. XIII, 90 (electromag-
netic uctuation approach).
See also Derjaguin's article (12.12).
4
This12factor is easily understood if one remembers that the energy of an electrostatic system is
proportional to the integral over all space of the square of the electric eld. For the system of Fig.,
the electric eld is zero below thexOyplane.
1161

COMPLEMENT D XI
Complement DXI
The volume eect: the inuence of the spatial extension of the
nucleus on the atomic levels
1 First-order energy correction
1-a Calculation of the correction
1-b Discussion
2 Application to some hydrogen-like systems
2-a The hydrogen atom and hydrogen-like ions
2-b Muonic atoms
The energy levels and the stationary states of the hydrogen atom were studied in
Chapter
electrostatic 1Coulomb potential. Actually, this is not quite true. The proton is not
strictly a point charge; its charge lls a volume which has a certain size (of the order of
1 fermi =10
13
cm). When an electron is extremely close to the center of the proton,
it sees a potential that no longer varies as1, and which depends on the spatial
charge distribution associated with the proton. This is true, furthermore, for all atoms:
inside the volume of the nucleus, the electrostatic potential depends on how the charges
are distributed. We thus expect the atomic energy levels, which are determined by the
potential to which the electrons are subject at all points of space, to be aected by this
distribution: this is what is called the volume eect. The experimental and theoretical
study of such an eect is therefore important, since it can supply information about the
internal structure of nuclei.
In this complement, we shall give a simplied treatment of the volume eect of
hydrogen-like atoms. To have an idea of the order of magnitude of the energy shifts it
causes, we shall conne ourselves to a model in which the nucleus is represented by a
sphere of radius0, in which the chargeis uniformly distributed. In this model, the
potential created by the nucleus is (cf.ComplementV, ):
() =
2
for>0
2
200
2
3
for60
(1)
(we have set
2
=
2
40). The shape of the variation of()with respect tois
shown in Figure.
The exact solution of the Schrödinger equation for an electron subject to such a
potential poses a complicated problem. Therefore, we shall content ourselves with an
approximate solution, based on perturbation theory. In a rst approximation, we shall
consider the potential to be a Coulomb potential [which amounts to setting0= 0in (1)].
The energy levels of the hydrogen atom are then the ones found in .
1162

THE VOLUME EFFECT: THE INFLUENCE OF THE SPATIAL EXTENSION OF THE NUCLEUS ON THE
ATOMIC LEVELS0
ρ
0
V(r)
W
(
r
)
r
Figure 1: Variation with respect toof the electrostatic potential()created by the
charge distributionof the nucleus, assumed to be uniformly distributed inside a
sphere of radius0. For60, the potential is parabolic. For>0, it is a Coulomb
potential [the extension of this Coulomb potential into the60zone is represented by
the dashed line;()is the dierence between()and the Coulomb potential].
We shall treat the dierence()between the potential()written in (1) and the
Coulomb potential as a perturbation. This dierence is zero whenis greater than the
radius0of the nucleus. It is therefore reasonable that it should cause a small shift in
the atomic levels (the corresponding wave functions extend over dimensions of the order
of0 0), which justies a treatment by rst-order perturbation theory.
1163

COMPLEMENT D XI
1. First-order energy correction
1-a. Calculation of the correction
By denition,()is equal to:
() =
2
200
2
+
20
3
if0660
0 if>0
(2)
Let be the stationary states of the hydrogen-like atom in the absence of the
perturbationTo evaluate the eect ofto rst order, we must calculate the matrix
elements:
=d () ()
0
2
d () ()() (3)
In this expression, the angular integral simply gives. To simplify the radial
integral, we shall make an approximation and assume
1
that:
0 0 (4)
that is, that the 0region, in which()is not zero, is much smaller than the
spatial extent of the functions(). When 0, we then have:
() (0) (5)
The radial integral can therefore be written:
=
2
20
(0)
2
0
0
2
0
2
+
20
3 (6)
which gives:
=
2
10
2
0 (0)
2
(7)
and:
=
2
10
2
0 (0)
2
(8)
We see that the matrix representingin the subspacecorresponding to the
th level of the unperturbed Hamiltonian is diagonal. Therefore, the rst-order energy
correction associated with each statecan be written simply:
=
2
10
2
0 (0)
2
(9)
1
This is certainly the case for the hydrogen atom. In 2, we shall examine condition (4) in greater
detail.
1164

THE VOLUME EFFECT: THE INFLUENCE OF THE SPATIAL EXTENSION OF THE NUCLEUS ON THE
ATOMIC LEVELS
This correction does not depend
2
on. Furthermore, since(0)is zero unless= 0
(cf.Chap., ), only thestates (= 0states) are shifted, by a quantity which
is equal to:
0=
2
10
2
0 0(0)
2
=
2
2
5
2
0 00(0)
2
(10)
(we have used the fact that
0
0= 1
4).
1-b. Discussion
0can be written:
0=
3
10
(11)
where:
=
2
0
(12)
is the absolute value of the potential energy of the electron at a distance0from the
center of the nucleus, and:
=
4
3
3
0 00(0)
2
(13)
is the probability of nding the electron inside the nucleus.andenter into (11)
because the eect of the perturbation()is felt only inside the nucleus.
For the method which led us to (10) and (11) to be consistent, the correction 0
must be much smaller than the energy dierences between unperturbed levels. Since
is very large (an electron and a proton attract each other very strongly when they are
very close),must therefore be extremely small. Before taking up the more precise
calculation in 2, we shall evaluate the order of magnitude of these quantities. Let:
0() =
~
2
2
(14)
be the Bohr radius when the total charge of the nucleus is. Ifis not too high, the
wave functions00(r)are practically localized inside a region of space whose volume
is approximately [[0()]
3
. As for the nucleus, its volume is of the order of
3
0, so:
0
0()
3
(15)
2
This result could have been expected, since the perturbationwhich is invariant under rotation,
is a scalar (cf.ComplementVI, ).
1165

COMPLEMENT D XI
Relation (11) then yields:
0
2
0
0
0()
3
2
0()
0
0()
2
(16)
Now,
2
0()is of the order of magnitude of the binding energy()of the unper-
turbed atom. The relative value of the correction is therefore equal to:
0
()
0
0()
2
(17)
If condition (4) is met, this correction will indeed be very small. We shall now calculate
it more precisely in some special cases.
2. Application to some hydrogen-like systems
2-a. The hydrogen atom and hydrogen-like ions
For the ground state of the hydrogen atom, we have [cf.Chap., relation (C-
39a)]:
10() = 2(0)
32
e
0
(18)
[where0is obtained by setting= 1in (14)]. Formula (10) then yields:
10=
2
5
2
0
0
0
2
=
4
5
0
0
2
(19)
Now, we know that, for hydrogen:
0053

A = 53 10
11
m (20)
Furthermore, the radius0of the proton is of the order of:
0(proton)1F= 10
15
m (21)
If we substitute these numerical values into (19), we obtain:
1045 10
10
6 10
9
eV (22)
The result is therefore very small.
For a hydrogen-like ion, the nucleus has a charge of. We can then apply (10),
which amounts to replacing
2
in (19) bye
2
, and0by0() =0. We obtain:
10() =
2
5
22
0
0()
0
2
(23)
where0( )is the radius of the nucleus, composed ofnucleons (protons or neu-
trons),of which are protons. In practice, the number of nucleons of a nucleus is not very
1166

THE VOLUME EFFECT: THE INFLUENCE OF THE SPATIAL EXTENSION OF THE NUCLEUS ON THE
ATOMIC LEVELS
dierent from 2; in addition, the nuclear density saturation property is expressed by
the approximate relation:
0()A
13
Z
13
(24)
The variation of the energy correction with respect tois then given by:
E10(Z)Z
143
(25)
or:
E10(Z)
E(Z)
Z
83
(26)
E10(Z)therefore varies very rapidly with, under the eect of several concordant
factors: whenincreases,0decreases and0increases. The volume eect is therefore
signicantly larger for heavy hydrogen-like ions than for hydrogen.
Comment:
The volume eect also exists for all the other atoms. It is responsible for an
isotopic shift of the lines of the emission spectrum. For two distinct isotopes of
the same chemical element, the numberof protons of the nucleus is the same,
but the number of neutrons is dierent; the spatial distributions of the
nuclear charges are therefore not identical for the two nuclei.
Actually, for light atoms, the isotopic shift is caused principally by the nuclear
nite mass eect (cf.ComplementVII, ). On the other hand, for heavy
atoms (for which the reduced mass varies very little from one isotope to another),
the nite mass eect is small; however, the volume eect increases withand
becomes preponderant.
2-b. Muonic atoms
We have already discussed some simple properties of muonic atoms (cf.Comple-
mentV, VII, ). In particular, we have pointed out that the Bohr radius
associated with them is distinctly smaller than for ordinary atoms (this is caused by the
fact that the mass of themuon is approximately equal to 207 times that of the elec-
tron). From the qualitative discussion of , we may therefore expect an important
volume eect for muonic atoms. We shall evaluate it by choosing two limiting cases: a
light muonic atom (hydrogen) and a heavy one (lead).
. The muonic hydrogen atom
The Bohr radius is then:
0(
+
)
0
207
(27)
that is, of the order of 250 fermi. It therefore remains, in this case, distinctly greater
than0. If we replace0by0207in (19), we nd:
E10(
+
)1910
5
E(p
+
)510
2
eV (28)
Although the volume eect is much larger than for the ordinary hydrogen atom, it still
yields only a small correction to the energy levels.
1167

COMPLEMENT D XI
. The muonic lead atom
The Bohr radius of the muonic lead atom is [cf.ComplementV, relation (25)]:
0(Pb)310
15
m (29)
The muon is now very close to the lead nucleus; it is therefore practically unaected
by the repulsion of the atomic electrons which are located at distinctly greater distances.
This could lead us to believe that (10), which was proven for hydrogen-like atoms and
ions, is directly applicable to this case. Actually, this is not true, since the radius of the
lead nucleus is equal to:
0(Pb)85 F = 8510
15
m (30)
which is not small compared to0(Pb). Equation (10) would therefore lead to large
corrections (several MeV), of the same order of magnitude as the energy(Pb). This
means that, in this case, the volume eect can no longer be treated as a perturbation
(see discussion of V). To calculate the energy levels, it is necessary
to know the potential()exactly and to solve the corresponding Schrödinger equation.
The muon is therefore more inside the nucleus than outside, that is, according to
(1), in a region in which the potential is parabolic. In a rst approximation, we could con-
sider the potential to be parabolic everywhere (as is done in ComplementV) and then
treat as a perturbation the dierence which exists for>0between the real potential
and the parabolic potential. However, the extension of the wave function corresponding
to such a potential is not suciently smaller than0for such an approximation to lead
to precise results, and the only valid method consists of solving the Schrödinger equation
corresponding to the real potential.
References and suggestions for further reading:
The isotopic volume eect: Kuhn (11.1), Chap. VI, C-3; Sobel'man (11.12), Chap. 6,
24.
Muonic atoms (sometimes called mesic atoms): Cagnac and Pebay-Peyroula (11.2),
Chap. XIX, 7-C; De Benedetti (11.21); Wiegand (11.22); Weissenberg (16.19), 4-2.
1168

THE VARIATIONAL METHOD
Complement EXI
The variational method
1 Principle of the method
1-a A property of the ground state of a system
1-b Generalization: the Ritz theorem
1-c A special case where the trial functions form a subspace
2 Application to a simple example
2-a Exponential trial functions
2-b Rational wave functions
3 Discussion
The perturbation theory studied in Chapter
tion method applicable to conservative systems. We shall give a concise description here
of another of these methods, which also has numerous applications, especially in atomic
and molecular physics, nuclear physics, and solid state physics. First of all, we shall
indicate, in , the principle of the variational method. Then we shall use the simple
example of the one-dimensional harmonic oscillator to bring out its principal features
(), which we shall briey discuss in . ComplementsXIandXIapply the varia-
tional method to simple models which enable us to understand the behavior of electrons
in a solid and the origin of the chemical bond.
1. Principle of the method
Consider an arbitrary physical system whose Hamiltonianis time-independent. To
simplify the notation, we shall assume that the entire spectrum ofis discrete and
non-degenerate:
H = ;= 012 (1)
Although the Hamiltonianis known, this is not necessarily the case for its eigenvalues
and the corresponding eigenstates. The variational method is, of course, most
useful in the cases in which we do not know how to diagonalizeexactly.
1-a. A property of the ground state of a system
Choose an arbitrary ketof the state space of the system. The average value of
the Hamiltonianin the stateis such that:
H=
H
>E0 (2)
(where0is the smallest eigenvalue of), equality occuring if and only ifis an
eigenvector ofwith the eigenvalue0.
1169

COMPLEMENT E XI
To prove inequality (2), we expand the keton the basis of eigenstates of:
= (3)
We then have:
=
2
>0
2
(4)
with, of course:
=
2
(5)
which proves (2). For inequality (4) to become an equality, it is necessary and sucient
that all the coecientsbe zero, with the exception of0;is then an eigenvector
ofwith the eigenvalue0.
This property is the basis for a method of approximate determination of0. We
choose (in theory, arbitrarily, but in fact, by using physical criteria) a family of kets
()which depend on a certain number of parameters which we symbolize by. We
calculate the average value()of the Hamiltonianin these states, and we minimize
()with respect to the parameters. The minimal value so obtained constitutes
an approximation of the ground state0of the system. The kets()are calledtrial
kets, and the method itself, thevariational method.
Comment:
The preceding proof can easily be generalized to cases in which the spectrum of
is degenerate or includes a continuous part.
1-b. Generalization: the Ritz theorem
We shall show that, more generally,the average value of the Hamiltonian H is
stationary in the neighborhood of its discrete eigenvalues.
Consider the average value ofin the state:
=
(6)
as a functional of the state vector, and calculate its incrementwhen becomes
+ , where is assumed to be innitely small. To do so, it is useful to write
(6) in the form:
= (7)
and to dierentiate both sides of this relation:
+ [ + ] (8)
= +
1170

THE VARIATIONAL METHOD
that is, sinceis a number:
= [ ] + [ ] (9)
The average valuewill be stationary if:
= 0 (10)
which, according to (9), means that:
[ ] + [ ]= 0 (11)
We set:
= [ ] (12)
Relation (11) can then be written simply:
+ = 0 (13)
This last relation must be satised for any innitesimal ket. In particular, if we
choose:
= (14)
(whereis an innitely small real number), (13) becomes:
2 = 0 (15)
The norm of the ketis therefore zero, andmust consequently be zero. With
denition (12) taken into account, this means that:
= (16)
Consequently, the average valueis stationary if and only if the state vectorto
which it corresponds is an eigenvector ofand the stationary values ofare the
eigenvalues of the Hamiltonian.
The variational method can therefore be generalized and applied to the approx-
imate determination of the eigenvalues of the HamiltonianIf the function()
obtained from the trial kets()has several extrema, they give the approximate val-
ues of some of its energies(cf.exercise 10 of ComplementXI).
1-c. A special case where the trial functions form a subspace
Assume that we choose for the trial kets the set of kets belonging to a vector
subspaceof. In this case, the variational method reduces to theresolution of the
eigenvalue equation of the Hamiltonian H inside, and no longer in all of.
To see this, we simply apply the argument of , limiting it to the ketsof the
subspace. The maxima and minima of , characterized by= 0, are obtained
when is an eigenvector ofin. The corresponding eigenvalues constitute the
1171

COMPLEMENT E XI
variational method approximation for the true eigenvalues ofin. They also provide
upper bounds of these eigenvalues: we have seen that the lowest energy that is obtained
is larger than the true energy of the ground state, but it is also possible to show (cf.
MacDonald's article in the references of this complement) that the next lowest energy is
greater than the energy of the true rst excited state, etc. When the dimension ofis
increased by one unit, one obtains a new series of energies with a new energy above all
the others, which themselves decrease (or possibly remain at the same value).
We stress the fact that the restriction of the eigenvalue equation ofto a sub-
spaceof the state spacecan considerably simplify its solution. However, ifis badly
chosen, it can also yield results which are rather far from the true eigenvalues and eigen-
vectors ofin(cf.). The subspacemust therefore be chosen so as to simplify the
problem enough to make it soluble, without too greatly altering the physical reality. In
certain cases, it is possible to reduce the study of a complex system to that of a two-level
system (cf.Chap.), or at least, to that of a system of a limited number of levels.
Another important example of this procedure is the method of thelinear combination of
atomic orbitals, widely used in molecular physics. This method (cf.ComplementXI)
essentially determines the wave functions of electrons in a molecule in the form of linear
combinations of eigenfunctions associated with the various atoms which constitute the
molecule, treated as if they were isolated. It therefore limits the search for the molecular
states to a subspace chosen using physical criteria. Similarly, in ComplementXI, we
shall choose as a trial wave function for an electron in a solid a linear combination of
atomic orbitals relative to the various ions which constitute this solid.
Comment:
Note that rst-order perturbation theory ts into this special case of the variational
method:is then an eigensubspace of the unperturbed Hamiltonian0.
2. Application to a simple example
To illustrate the discussion of
obtained with the help of the variational method, we shall apply this method to the
one-dimensional harmonic oscillator, whose eigenvalues and eigenstates we know (cf.
Chap.). We shall consider the Hamiltonian:
=
~
2
2
2
2
+
1
2
22
(17)
and we shall solve its eigenvalue equation approximately by variational calculations.
2-a. Exponential trial functions
Since the Hamiltonian (17) is even, it can easily be shown that its ground state
is necessarily represented by an even wave function. To determine the characteristics of
this ground state, we shall therefore choose even trial functions. We take, for example,
the one-parameter family:
() = e
2
; 0 (18)
1172

THE VARIATIONAL METHOD
The square of the norm of the ketis equal to:
=
+
de
2
2
(19)
and we nd:
=
+
de
2~
2
2
d
2
d
2
+
1
2
22
e
2
=
~
2
2
+
1
8
2
1
+
de
2
2
(20)
so that:
() =
~
2
2
+
1
8
2
1
(21)
The derivative of the function()goes to zero for:
=0=
1
2~
(22)
and we then have:
(0) =
1
2
~ (23)
The minimum value of ()is therefore exactly equal to the energy of the ground
state of the harmonic oscillator. This result is due to the simplicity of the problem
that we are studying: the wave function of the ground state happens to be precisely
one of the functions of the trial family (18), the one which corresponds to value (22) of
the parameter. The variational method, in this case, gives the exact solution of the
problem (this illustrates the theorem proven in ).
If we want to calculate (approximately, in theory) the rst excited state1of the
Hamiltonian (17), we should choose trial functions which are orthogonal to the wave
function of the ground state. This follows from the discussion of , which shows that
the lower bound ofis1, and no longer0if the coecient0is zero. We therefore
choose the trial family of odd functions:
() =e
2
(24)
In this case:
=
+
d
2
e
2
2
(25)
and:
=
~
2
2
3+
1
2
2
3
4
+
d
2
e
2
2
(26)
which yields:
() =
3~
2
2
+
3
8
2
1
(27)
1173

COMPLEMENT E XI
This function, for the same value0as above [formula (22)], presents a minimum equal
to:
(0) =
3
2
~ (28)
Here again, we nd exactly the energy1and the associated eigenstate because the trial
family includes the correct wave function.
2-b. Rational wave functions
The calculations of
method, but they do not really allow us to judge its eectiveness as a method of approx-
imation, since the families chosen always included the exact wave function. Therefore,
we shall now choose trial functions of a totally dierent type, for example
1
:
() =
1
2
+
; a 0 (29)
A simple calculation then yields:
=
+
(x
2
+)
2
=
2
(30)
and, nally:
() =
~
2
4
1
+
1
2
2
(31)
The minimum value of this function is obtained for:
=0=
1
2
~
(32)
and is equal to:
(0) =
1
2
~ (33)
This minimum value is therefore equal to
2times the exact ground state energy~2.
To measure the error committed, we can calculate the ratio of(0)~2to the
energy quantum~:
(0)
1
2
~
~
=
212
20% (34)
3. Discussion
The example of
without signicant error, starting with arbitrarily chosen trial kets. This is one of the
1
Our choice here is dictated by the fact that we want the necessary integrals to be analytically
calculable. Of course, in most real cases, one resorts to numerical integration.
1174

THE VARIATIONAL METHOD
principal advantages of the variational method. Since the exact eigenvalue is a minimum
of the average value, it is not surprising thatdoes not vary very much near this
minimum.
On the other hand, as the same reasoning shows, the approximate state can be
rather dierent from the true eigenstate. Thus, in the example of , the wave function
1(
2
+0)[where0is given by formula (32)] decreases too rapidly for small values of
and much too slowly whenbecomes large. Table I gives quantitative support for this
qualitative assertion. It gives, for various values of
2
, the values of the exact normalized
eigenfunction:
0() = (20)
14
e
0
2
[where0was dened in (22)] and of the approximate normalized eigenfunction:
2
(0)
34
0
() =
2
(0)
34
2
+0
=
2
2
20
14 1
1 + 220
2
(35)
0
2
14
e
0
2
2
(2
2)
14
1+220
2
0 0.893 1.034
1/2 0.696 0.605
1 0.329 0.270
3/2 0.094 0.140
2 0.016 0.083
5/2 0.002 0.055
3 0.000 1 0.039
Table I
It is therefore necessary to be very careful when physical properties other than
the energy of the system are calculated using the approximate state obtained from the
variational method. The validity of the result obtained varies enormously depending
on the physical quantity under consideration. In the particular problem which we are
studying here, we nd, for example, that the approximate average value
2
of the operator
2
is not very dierent from the exact value:
0
2
0
0 0
=
1
2
~
(36)
which is to be compared with~2. On the other hand, the average value of
4
is
innite for the wave function (35), while it is, of course, nite for the real wave function.
2
The average value ofis automatically zero, as is correct since we have chosen even trial functions.
1175

COMPLEMENT E XI
More generally, Table I shows that the approximation will be very poor for all properties
that depend strongly on the behavior of the wave function for&2
0.
The drawback we have just mentioned is all the more serious as it is very dicult,
if not impossible, to evaluate the error in a variational calculation if we do not know
the exact solution of the problem (and, of course, if we use the variational method, it is
because we do not know this exact solution).
The variational method is therefore a very exible approximation method, which
can be adapted to very diverse situations and which gives great scope to physical intu-
ition in the choice of trial kets. It gives good values for the energy rather easily, but
the approximate state vectors may present certain completely unpredictable erroneous
features, and we cannot check these errors. This method is particularly valuable when
physical arguments give us an idea of the qualitative or semi-quantitative form of the
solutions.
References and suggestions for further reading:
The Hartree-Fock method, often used in physics, is an application of the variational
method. See references of Chapter XVof Volume III.
The variational method is of fundamental importance in molecular physics. See
references of ComplementXI.
For a simple presentation of the use of variational principles in physics, see Feyn-
man II (7.2), Chap. 19.
J.K.L. MacDonald,Physical Reviewvol. 143, pages 830 à 833 (1933).
1176

ENERGY BANDS OF ELECTRONS IN SOLIDS: A SIMPLE MODEL
Complement FXI
Energy bands of electrons in solids: a simple model
1 A rst approach to the problem: qualitative discussion
2 A more precise study using a simple model
2-a Calculation of the energies and stationary states
2-b Discussion
A crystal is composed of atoms evenly distributed in space so as to form a three-
dimensional periodic lattice. The theoretical study of the properties of a crystal, which
brings into play an extremely large number of particles (nuclei and electrons), poses a
problem which is so complicated that it is out of the question to treat it rigorously. We
must therefore resort to approximations.
The rst of these is of the same type as the Born-Oppenheimer approximation
(which we encountered in V). It consists of considering, rst of all,
the positions of the nuclei as xed, which enables us to study the stationary states of the
electrons subjected to the potential created by the nuclei. The motion of the nuclei is not
treated until later, using the knowledge of the electronic energies
1
. In this complement,
we shall concern ourselves only with the rst step of this calculation, and we shall assume
the nuclei to be motionless at the nodes of the crystalline lattice.
This problem still remains extremely complicated. It is necessary to calculate the
energies of a system of electrons subjected to a periodic potential and interacting with
each other. We then make a second approximation: we assume that each electron, at a
positionr, is subjected to the inuence of a potential(r)which takes into account
the attraction exerted by the nuclei and the average eect of the repulsion of all the other
electrons
2
. The problem is thus reduced to one involving independent particles, moving
in a potential that has the periodicity of the crystalline lattice.
The physical characteristics of a crystal therefore depend, in a rst approximation,
on the behavior of independent electrons subjected to a periodic potential. We could be
led to think that each electron remains bound to a given nucleus, as happens in isolated
atoms. We shall see that, in reality, the situation is completely dierent. Even if an
electron is initially in the neighborhood of a particular nucleus, it can move into the
zone of attraction of an adjacent nucleus by the tunnel eect, then into another, and so
on. Actually, the stationary states of the electrons are not localized in the neighborhood
of any nucleus, but are completely delocalized: the probability density associated with
them is uniformly distributed over all the nuclei
3
. Thus, the properties of an electron
1
Recall that the study of the motion of the nuclei leads to the introduction of the normal vibrational
modes of the crystal: the phonons (cf.ComplementV).
2
This approximation is of the same type as the central eld approximation for isolated atoms (cf.
ComplementXIV, ).
3
This phenomenon is analogous to the one we encountered in the study of the ammonia molecule (cf.
ComplementIV). There, since the nitrogen atom can move from one side of the plane of the hydrogen
atoms to the other, by the tunnel eect, the stationary states give an equal probability of nding it in
each of the two corresponding positions.
1177

COMPLEMENT F XI
placed in a periodic potential resemble those of an electron free to move throughout
the crystal more than they do those of an electron bound to a particular atom. Such a
phenomenon could not exist in classical mechanics: the direction of a particle traveling
through a crystal would change constantly under the inuence of the potential variations
(for example, upon skirting an ion). In quantum mechanics, the interference of the waves
scattered by the dierent nuclei permit the propagation of an electron inside the crystal.
In , we shall study very qualitatively how the energy levels of isolated atoms
are modied when they are brought gradually closer together to form a linear chain.
Then, in , still conning ourselves, for simplicity, to the case of a linear chain, we shall
calculate the energies and wave functions of stationary states a little more precisely. We
shall perform the calculation in the tight bonding approximation: when the electron
is in one site, it can move to one of two neighboring sites via the tunnel eect. The tight
bonding approximation is equivalent to assuming that the probability of its tunneling
is small. We shall, in this way, establish a certain number of results (the delocalization
of stationary states, the appearance of allowed and forbidden energy bands, the form of
Bloch functions) which remain valid in more realistic models (three-dimensional crystals,
bonds of arbitrary strength).
The perturbation approach that we shall adopt here constructs the stationary
states of the electrons from atomic wave functions localized about the various ions. It has
the advantage of showing how atomic levels change gradually to energy bands in a solid.
Note, however, that the existence of energy bands can be directly established from the
periodic nature of the structure in which the electron is placed (see, for example, Com-
plementIII, in which we study quantization of the energy levels in a one-dimensional
periodic potential).
Finally, we stress the fact that we are concerned here only with the properties of the
individual stationary states of the electrons. To construct the stationary state of a system
ofelectrons from these individual states, it is necessary to apply the symmetrization
postulate (cf.Chap. ), since we are dealing with a system of identical particles.
We shall treat this problem again in ComplementXIV, when we shall describe the
spectacular consequences of Pauli's exclusion principle on the physical behavior of the
electrons in a solid. Many other examples of the eects of the symmetrization will be
discussed in Chapters XV to XVII.
1. A rst approach to the problem: qualitative discussion
Let us go back to the example of the ionized H
+
2
molecule, studied in
of Chapter. Consider, therefore, two protons1and2whose positions are xed,
and an electron which is subject to their electrostatic attraction. This electron sees a
potential(r), which has the form indicated in Figure. In terms of the distance
between1and2(considered as a parameter) what are the possible energies and the
corresponding stationary states?
We shall begin by considering the limiting case in which0(where0is the
Bohr radius of the hydrogen atom). The ground state is then two-fold degenerate: the
electron can form a hydrogen atom either with1or with2; it is practically unaected
by the attraction of the other proton, which is very far away. In other words, the
coupling between the states1and2considered in Chapter
the neighborhood of1or2;cf.Fig. ) is then negligible, so that1
1178

ENERGY BANDS OF ELECTRONS IN SOLIDS: A SIMPLE MODELR
2
R
V(x)
x2
+
0
–
Figure 1: The potential seen by the electron as it moves along theaxis dened by the
two protons in the ionized
+
2
molecule. We obtain two wells separated by a barrier. If,
at any instant, the electron is localized in one of the two wells, it can move into the other
well via the tunnel eect.
and2are practically stationary states.
If we now choose a value ofcomparable to0it is no longer possible to neglect
the attraction of one or the other of the protons. If, at= 0, the electron is localized
in the neighborhood of one of them, and even if its energy is lower than the height
of the potential barrier situated between1and2(cf.Fig.), it can move to the
other proton by the tunnel eect. In Chapter
the states1and 2, and we showed that it produces an oscillation of the system
between these two states (the dynamical aspect). We have also seen (the static aspect)
that this coupling removes the degeneracy of the ground state and that the corresponding
stationary states are delocalized (for these states, the probability of nding the electron
in the neighborhood of1or2is the same). Figure
with respect toof the possible energies of the system
4
.
Two eects appear when we decrease the distancebetween1and2. On the
one hand, an=energy value gives rise to two distinct energies whendecreases
(when the distanceis xed at a given value0, the stronger the coupling between the
states1and2, the greater the dierence between these two energies). On the other
hand, the stationary states are delocalized.
It is easy to imagine what will happen if the electron is subject to the inuence, not
of two, but of three identical attractive particles (protons or positive ions), arranged, for
example, in a straight line at intervals ofWhen is very large, the energy levels are
triply degenerate, and the stationary states of the electron can be chosen to be localized
in the neighborhood of any one of the xed particles.
Ifis decreased, each energy gives rise to three generally distinct energies and, in a
4
A detailed study of the
+
2
ion is presented in ComplementXI.
1179

COMPLEMENT F XIE
0 R
0
R
E
I
Δ
Figure 2: Variation of the energy of stationary states of the electron in terms of the
distancebetween the two protons of the H
+
2
ion. Whenis large, there are two prac-
tically degenerate states, of energy. Whendecreases, this degeneracy is removed.
The smaller, the greater the splitting.
stationary state, the probabilities of nding the electron in the three wells are comparable.
Moreover, if, at the initial instant, the electron is localized in the right-hand well, for
example, it moves into the other wells during its subsequent evolution
5
.
The same ideas remain valid for a chain composed of an arbitrary numberof
ions which attract an electron. The potential seen by the electron is then composed of
regularly spaced identical wells (in the limit in which, it is a periodic potential).
When the distancebetween the ions is large, the energy levels are-fold degenerate.
This degeneracy disappears if the ions are moved closer together: each level gives rise
to distinct levels, which are distributed, as shown in Figure, in an energy interval of
width. What now happens if the value ofis very large? In each of the intervals
, the possible energies are so close that they practically form a continuum: allowed
energy bands are thus obtained, separated by forbidden bands. Each allowed band
containslevels (actually2if the electron spin is taken into account). The stronger
the coupling causing the electron to pass from one potential well to the next one, the
greater the band width. (Consequently, we expect the lowest energy bands to be the
narrowest since the tunnel eect which is responsible for this passage is less probable
when the energy is smaller). The stationary states of the electron are all delocalized.
The analogue here of Figure IIIis Figure, which represents the
energy levels and gives an idea of the spatial extension of the associated wave functions.
Finally, note that if, at= 0, the electron is localized at one end of the chain, it
propagates along the chain during its subsequent evolution.
5
See exercise 8 of ComplementIV.
1180

ENERGY BANDS OF ELECTRONS IN SOLIDS: A SIMPLE MODELE
R
0
R
EE′
Δ′
Δ
0
Figure 3: Energy levels of an electron subject to the action ofregularly spaced identical
ions. Whenis very large, the wave functions are localized about the various ions, and
the energy levels are the atomic levels,-fold degenerate (the electron can form an atom
with any one of theions). In the gure, two of these levels are shown, of energies
and . Whendecreases, the electron can pass from one ion to another by the
tunnel eect, and the degeneracy of the levels is removed. The smaller, the greater the
splitting. For the value0offound in a crystal, each of the two original atomic levels
is therefore broken down intovery close levels. Ifis very large, these levels are so
close that they yield energy bands, of widthsand, separated by a forbidden band.
2. A more precise study using a simple model
2-a. Calculation of the energies and stationary states
To complete the qualitative considerations of the preceding section, we shall dis-
cuss the problem more precisely, using a simple model. We shall perform calculations
analogous to those of , but adapted to the case in which the system
under consideration contains an innite number of ions (instead of two), regularly spaced
1181

COMPLEMENT F XIΔ
∆
V(x)
x
Figure 4: Energy levels for a potential composed of several regularly spaced wells. Two
bands are shown in this gure, one of widthand the other of width. The deeper the
band, the more narrow it is, since crossing the barrier by the tunnel eect is then more
dicult.
in a linear chain.
. Description of the model; simplifying hypotheses
Consider, therefore, an innite linear chain of regularly spaced positive ions. As
in Chapter, we shall assume that the electron, when it is bound to a given ion, has
only one possible state: we shall denote bythe state of the electron when it forms
an atom with theth ion of the chain. For the sake of simplicity, we shall neglect the
mutual overlap of the wave functions()associated with neighboring atoms, and we
shall assume the basis to be orthonormal:
= (1)
Moreover, we shall conne ourselves to the subspace of the state space spanned by the
kets. It is obvious that by restricting the state space accessible to the electron in
this way, we are making an approximation. This can be justied by using the variational
method (cf.ComplementXI): by diagonalizing the Hamiltonian, not in the total
space, but in the one spanned by the, it can be shown that we obtain a good
approximation for the true energies of the electron.
We shall now write the matrix representing the Hamiltonianin the basis.
Since the ions all play equivalent roles, the matrix elementsare necessarily
all equal to the same energy0. In addition to these diagonal elements,also has non-
diagonal elements (coupling between the various states, which expresses
the possibility for an electron to move from one ion to another). This coupling is obviously
very weak for distant ions; this is why we shall take into account only the matrix elements
1, which we shall choose equal to a real constant. Under these conditions,
1182

ENERGY BANDS OF ELECTRONS IN SOLIDS: A SIMPLE MODEL
the (innite) matrix that representscan be written:
() =
.
.
.
0 0 0
0 0
0 0
0 0 0
.
.
.
(2)
To nd the possible energies and the corresponding stationary states, we must diagonalize
this matrix.
. Possible energies; the concept of an energy band
Letbe an eigenvector of; we shall write it in the form:
=
+
=
(3)
Using (2), the eigenvalue equation:
= (4)
projected onto, yields:
0 +1 1= (5)
When takes on all positive or negative integral values, we obtain an innite
system of coupled linear equations which, in certain ways, recall the coupled equations
(5) of ComplementV. As in that complement, we shall look for simple solutions of the
form:
= e (6)
whereis the distance between two adjacent ions, andis a constant whose dimensions
are those of an inverse length. We requireto belong to the rst Brillouin zone, that
is, to satisfy:
6 + (7)
This is always possible, because two values ofdiering by2give all the coecients
the same value. Substituting (6) into (5), we obtain:
0e e
(+1)
+ e
(1)
=e (8)
that is, dividing by e:
=() =02cos (9)
If this condition is satised, the ketgiven by (3) and (6) is an eigenket of; its
energy depends on the parameter, as is indicated by (9).
Figure with respect to. It shows that the possible
energies are situated in the interval[02 0+ 2]. We therefore obtain an allowed
energy band, whose width 4is proportional to the strength of the coupling.
1183

COMPLEMENT F XI– π/l + π/l
k
E
0
– 2A
E
0
+ 2A
E(k)
0
Figure 5: Possible energies of the
electron in terms of the parameter
(varies within the rst Brillouin
zone). An energy band therefore
appears, with a width 4which is
proportional to the coupling between
neighboring atoms.
. Stationary states; Bloch functions
Let us calculate the wave function() = associated with the stationary
state of energy(). Relations (3) and (6) lead to:
=
+
=
e (10a)
that is:
() =
+
=
e () (10b)
where:
() = (11)
is the wave function associated with the state. Since the statecan be obtained
from the state0by a translation ofwe have:
() =0( ) (12)
so that (10b) can be written:
() =
+
=
e 0( ) (13)
1184

ENERGY BANDS OF ELECTRONS IN SOLIDS: A SIMPLE MODEL
We now calculate(+):
(+) =
+
=
e 0[(1)]
= e
+
=
e
(1)
0[(1)]
= e () (14)
To express this remarkable property simply, we set:
() = e () (15)
The function()so dened then satises:
(+) =() (16)
Therefore, the wave function()is the product of eand a periodic function which
has the periodof the lattice. A function of type (15) is called aBloch function. Note
that, ifis any integer:
(+)
2
= ()
2
(17)
a result which demonstrates the delocalization of the electron: the probability density of
nding the electron at any point on the-axis is a periodic function of
Comment:
Expressions (15) and (16) have been proven here for a simple model. Actually, this result
is more general and can be proven directly from the symmetries of the Hamiltonian
(Bloch's theorem). To show this, let us call()the unitary operator associated with a
translationalong (cf.ComplementII, ). Since the system is invariant under
any translation that leaves the ion chain unchanged, we must have:
[()] = 0 (18a)
We can therefore construct a basis of eigenvectors common to the operator()and.
Now, equation (14) is simply the one that denes the eigenfunctions of()[since
this operator is unitary, its eigenvalues can always be written in the form e, where
satises condition (7);cf.ComplementII, ]. It is then simple to get, as before,
(15) and (16) from (14).
Note that, for any, we have, in general:
[()]= 0 (18b)
unlike the situation of a free particle (or one subject to the inuence of a constant
potential). For a free particle, sincecommutes with all operators()(that is, with
the momentum ;cf.ComplementII, ), the stationary wave functions are of the
form:
()e (19)
This means that the function()appearing in (15) is necessarily a constant, which is a
more restrictive condition. In the problem studied in this complement, the commutator of
relation (18b) vanishes only for certain values of, which implies less restrictive conditions
for the wave function.
1185

COMPLEMENT F XI
. Periodic boundary conditions
To each value ofin the interval[ +]corresponds an eigenstateof,
with the coecientsappearing in expansion (3) ofgiven by equation (6). We thus
obtain an innite continuum of stationary states. This is due to the fact that we have
considered a linear chain containing an innite number of ions. What happens when we
consider a nite linear chain, of length, composed of a large numberof ions?
The qualitative considerations of levels in the
band (2if spin is taken into account). The exact determination of the corresponding
stationary states is a dicult problem, since it is necessary to take into account the
boundary conditions at the ends of the chain. It is clear, however, that the behavior of
electrons suciently far from the ends are little aected by the edge eects
6
. This is
why one generally prefers, in solid state physics, to substitute new boundary conditions
for the real boundary conditions; despite their articial character, these new conditions
lead to much simpler calculations, while conserving the most important properties nec-
essary for the comprehension of the physical eects (other than the edge eects).
These new boundary conditions, called periodic boundary conditions, or Born-
Von Karman conditions (B.V.K. conditions), require the wave function to take on the
same value at both ends of the chain. We can also imagine that we are placing an innite
number of identical chains, all of lengthend to end. We then require the wave function
of the electron to be periodic, with a periodEquations (5) remain valid, as does their
solution (6), but the periodicity of the wave function now implies:
e= 1 (20)
Consequently, the only possible values ofare of the form:
=
2
(21)
whereis a positive or negative integer or zero. Let us now verify that the B.V.K.
conditions give the correct result for the number of stationary states contained in the
band. To do so, we must calculate the number of allowed valuesincluded in the rst
Brillouin zone dened in (7). We obtain this number by dividing the width2of this
zone by the interval2 between two adjacent values ofwhich indeed gives us:
2
2
== 1 (22)
We should also show that thestationary states obtained with the B.V.K. con-
ditions are distributed in the allowed band with the same density
7
()as the true sta-
tionary states (associated with the real boundary conditions). As the density of states
()plays a very important role in the comprehension of the physical properties of
a solid (we shall discuss this point in ComplementXIV), it is important for the new
boundary conditions to leave it unchanged. That the B.V.K. conditions give the correct
density of states will be proven in ComplementXIV() for the simple example of a
free electron gas enclosed in a rigid box. In this case, the true stationary states can be
calculated and compared with those obtained by using the periodic boundary conditions
on the walls of the box (see also III).
6
For a three-dimensional crystal, this amounts to establishing a distinction between bulk eects
and surface eects.
7
() dis the number of distinct stationary states with energies included betweenand+ d.
1186

ENERGY BANDS OF ELECTRONS IN SOLIDS: A SIMPLE MODEL
2-b. Discussion
Starting with a discrete non-degenerate level for an isolated atom (for example,
the ground level) we have obtained a series of possible energies, grouped in an allowed
band of width 4for the chain of ions being considered. If we had started with another
level of the atom (for example, the rst excited level), we would have obtained another
energy band, and so on. Each atomic level yields one energy band, as Figure
and there appears a series of allowed bands, separated by forbidden bands.
Relation (6) shows that, for a stationary state, the probability amplitude of nd-
ing the electron in the stateis an oscillating function of, whose modulus does
not depend onThis recalls the properties of phonons, the normal vibrational modes
of an innite number of coupled oscillators for which all the oscillators participate in
the collective vibration with the same amplitude, but with a certain phase shift (cf.
ComplementV).Allowed bands
E
Figure 6: Allowed bands and forbidden bands on the energy axis.
How can we obtain states in which the electron is not completely delocalized? For
a free electron, we saw in Chapter
free wave packet :
^
() =
1
2
d^() e
[ ()~]
(23)
The maximum of this wave packet propagates at the group velocity (cf.Chap., ):
^
=
1
~
d
d
=0
=
~0
(24)
[where0is the value offor which the function^()presents a peak]. Here, we must
superpose wave functions of type (15), and the corresponding ket can be written:
()=
1
2
d()e
()~
(25)
where()is a function ofwhich has the form of a peak about=0. We shall
calculate the probability amplitude of nding the electron in the state. Using (10a)
and (1), we can write:
()=
1
2
d() e
[ ()~]
(26)
1187

COMPLEMENT F XI
Replacingbyin this relation, we obtain a function of:
() =
1
2
d() e
[ ()~]
(27)
Only the values at the points= 0,, etc... of this function are really signicant
and yield the desired probability amplitudes.
Relation (27) is entirely analogous to (23). By applying (24), it can be shown that
()takes on signicant values only in a limited domain of the-axis whose center
moves at the velocity:
=
1
~
d()
d
=0
(28)
It follows that the probability amplitude()is large only for certain values of:
therefore, the electron is no longer delocalized, but moves in the crystal at the velocity
given by (28).
Equation (9) enables us to calculate this velocity explicitly:
=
2
~
sin0 (29)
This function is shown in Figure. It is zero when0= 0, that is, when the energy is0
k
V
G
(k)
+ π/l
2Al
–
h
– π/l
2Al
+
h
Figure 7: Group velocity of the electron as
a function of the parameterThis velocity
goes to zero, not only for= 0(as for the
free electron), but also for= (the
edges of the rst Brillouin zone).
minimal; this is also a property of the free electron. However, when0takes on non-zero
values, important departures from the behavior of a free electron occur. For example, as
soon as0 2, the group velocity is no longer an increasing function of the energy.
It even goes to zero when0= (at the borders of the rst Brillouin zone). This
indicates that an electron cannot move in the crystal if its energy is too close to the
maximum value 0+ 2appearing in Figure. The optical analogy of this situation
is Bragg reection. X rays whose wavelength is equal to the unit edge of the crystalline
lattice cannot propagate in it: interference of the waves scattered by each of the ions
lead to total reection.
References and suggestions for further reading:
Feynman III (1.2), Chap. 13; Mott and Jones (13.7), Chap. II, 4; references of
section 13 of the bibliography.
1188

A SIMPLE EXAMPLE OF THE CHEMICAL BOND: THE H
+
2
ION
Complement GXI
A simple example of the chemical bond: the H
+
2ion
1 Introduction
1-a General method
1-b Notation
1-c Principle of the exact calculation
2 The variational calculation of the energies
2-a Choice of the trial kets
2-b The eigenvalue equation of the Hamiltonianin the trial ket
vector subspace. . . . . . . . . . . . . . . . . . . . . . . .
2-c Overlap, Coulomb and resonance integrals
2-d Bonding and antibonding states
3 Critique of the preceding model. Possible improvements
3-a Results for small. . . . . . . . . . . . . . . . . . . . . . . .
3-b Results for large. . . . . . . . . . . . . . . . . . . . . . . .
4 Other molecular orbitals of the H
+
2
ion
4-a Symmetries and quantum numbers. Spectroscopic notation
4-b Molecular orbitals constructed from the 2atomic orbitals
5 The origin of the chemical bond; the virial theorem
5-a Statement of the problem
5-b Some useful theorems
5-c The virial theorem applied to molecules
5-d Discussion
1. Introduction
In this complement, we intend to show how quantum mechanics enables us to understand
the existence and properties of thechemical bond, which is responsible for the formation
of various molecules from isolated atoms. Our aim is to explain the basic nature of these
phenomena and not, of course, to enter into details which could only be covered in a
specialized book on molecular physics. This is why we shall study the simplest existing
molecule, the H
+
2
ion, which is composed of two protons and a single electron. We
have already discussed certain aspects of this problem, in Chapter ) and in
exercise 5 of ComplementI; we shall consider it here in a more realistic and systematic
fashion.
1-a. General method
When the two protons are very far from each other, the electron forms a hydrogen
atom with one of them, and the other one remains isolated, in the form of an H
+
ion.
If the two protons are brought closer together, the electron will be able to jump from
1189

COMPLEMENT G XIr
1
r
2
P
2
P
1
R
M
Figure 1: We call1the distance between the electron()and proton1,2the distance
between the electron and proton2andthe internuclear distance12.
one to the other. This radically modies the situation (cf. Chap., ). We shall
therefore study the variation of the energies of the stationary states of the system with
respect to the distance between the two protons. We shall see that the energy of the
ground state reaches a minimum for a certain value of this distance, which explains the
stability of the H
+
2
molecule.
In order to treat the problem exactly, it would be necessary to write the Hamilto-
nian of the three-particle system and solve its eigenvalue equation. However, it is possible
to simplify this problem considerably by using theBorn-Oppenheimer approximation(cf.
ComplementV, ). Since the motion of the electron in the molecule is considerably
more rapid than that of the protons, the latter can be neglected in a rst approxima-
tion. The problem is then reduced to the resolution of the eigenvalue equation of the
Hamiltonian of the electron subject to the attraction of two protons which are assumed
to be xed. In other words, the distancebetween the two protons is treated, not like a
quantum mechanical variable, but like aparameter,on which the electronic Hamiltonian
and total energy of the system depend.
In the case of the H
+
2
ion, it so happens that the equation simplied in this way
is exactly soluble for all values ofHowever, this is not true for other, more complex,
molecules. Thevariational method, described in ComplementXI, must then be used.
Although we are conning ourselves here to the study of the H
+
2
ion, we shall use the
variational method, since it can be generalized to the case of other molecules.
1-b. Notation
We shall callthe distance between the two protons, situated at1and2, and
1and2the distances of the electron to each of the two protons (Fig.). We shall relate
these distances to a natural atomic unit, the Bohr radius0(cf.Chap., ), by
setting:
= 0
1=10 2=20 (1)
The normalized wave function associated with the ground state 1of the hydrogen
1190

A SIMPLE EXAMPLE OF THE CHEMICAL BOND: THE H
+
2
ION
atom formed around proton1can be written:
1=
1
3
0
e
1
(2)
Similarly, we express the energies in terms of the natural unit=
2
20;is the
ionization energy of the hydrogen atom.
It will sometimes be convenient in what follows to use a system of elliptic coordi-
nates, in which a pointof space (here, the electron) is dened by:
=
1+2
=
1+2
=
1 2
=
1 2
(3)
and the anglewhich xes the orientation of the12plane about the12axis
(this angle also enters into the system of polar coordinates whoseaxis coincides with
12). If we xand, and ifvaries between 0 and2, the pointdescribes a circle
about the12axis. If(or) andare xed,describes an ellipse (or a hyperbola)
of foci1and2when(or) varies. It can easily be shown that the volume element
in this coordinate system is:
d
3
=
3
8
(
2 2
)ddd (4)
To do so, we simply calculate the Jacobianof the transformation:
= (5)
We see immediately that, if12is chosen as theaxis, with the originin the middle of
12:
2
1=
2
+
2
+
2
2
2
2=
2
+
2
++
2
2
tg=
(6)
We can then nd:
=
11
+
2
=
1
1
+
2
=
12
=
112
=
12
=
12
=
12
=
1/2
1
+
+/2
2
=
+/2
12
=
1/2
1
+/2
2
=
+/2
12
=
2
+
2
=
2
+
2
= 0 (7)
1191

COMPLEMENT G XI
The Jacobiancan therefore be written:
=
1
(12)
2
+ 2
2
2
+
2 2
+
2
0
=
1
(12)
2
2
(
2 2
) (8)
Since:
2 2
=
412
2
(9)
we get, nally:
=
8
3
(
2 2
)
(10)
1-c. Principle of the exact calculation
In the Born-Oppenheimer approximation, the equation to be solved in order to
nd the energy levels of the electron in the Coulomb eld of the two xed protons can
be written:
~
2
2

2
1
2
2
+
2
(r) =(r) (11)
If we go into the elliptical coordinates dened in (3), we can separate the variables,
and. Solving the equations so obtained, we nd a discrete spectrum of possible
energies for each value of. We shall not perform this calculation here, but shall merely
represent (the solid-line curve in Figure) the variation of the ground state energy with
respect toThis will enable us to compare the results we shall obtain by the variational
method with the values given by the exact solution of equation (11).
2. The variational calculation of the energies
2-a. Choice of the trial kets
Assumeto be much larger than0. If we are concerned with values of1of the
order of0, we have, practically:
2
2
2
for 2 0 (12)
The Hamiltonian:
=
P
2
2
2
1
2
2
+
2
(13)
is then very close to that of a hydrogen atom centered at proton1. Analogous con-
clusions are, of course, obtained formuch larger than0, and2of the order of0.
1192

A SIMPLE EXAMPLE OF THE CHEMICAL BOND: THE H
+
2
ION1 2
0
E
– E
I
3 4
R
a
0
=ρ
Figure 2: Variation of the energyof the molecular ion H
+
2
with respect to the distance
between the two protons.
. solid line: the exact total energy of the ground state (the stability of the H
+
2
ion is due
to the existence of a minimum in this curve).
. dotted line: the diagonal matrix element11=22of the Hamiltonian(the variation
of this matrix element cannot explain the chemical bond).
. dashed line: the results of the simple variational calculation of
antibonding states (though approximate, this calculation explains the stability of the H
+
2
ion).
. triangles: the results of the more elaborate variational calculation of
atomic orbitals of adjustable radius considerably improves the accuracy, especially at
small distances).
Therefore, when the two protons are very far apart, the eigenfunctions of the Hamiltonian
(13) are practically the stationary wave functions of hydrogen atoms.
This is, of course, no longer true when0is not negligible compared to. We
see, however, that it is convenient, for all, to choose a family of trial kets constructed
from atomic states centered at each of the two protons. This choice constitutes the
application to the special case of the H
+
2
ion of a general method known as themethod
of linear combination of atomic orbitals.More precisely, we shall call1and2the
1193

COMPLEMENT G XI
kets which describe the 1states of the two hydrogen atoms:
r 1=
1
3
0
e
1
r 2=
1
3
0
e
2
(14)
We shall choose as trial kets all the kets belonging to the vector subspacespanned by
these two kets, that is, the set of ketssuch that:
=11+22 (15)
The variational method (ComplementXI) consists of nding the stationary values of:
=
(16)
within this subspace. Since this is a vector subspace, the average valueis minimal or
maximal when is an eigenvector ofinside this subspace, and the corresponding
eigenvalue constitutes an approximation of a true eigenvalue ofin the total state space.
2-b. The eigenvalue equation of the Hamiltonian in the trial ket vector subspace
The resolution of the eigenvalue equation ofwithin the subspaceis slightly
complicated by the fact that1and2are not orthogonal.
Any vectorofis of the form (15). For it to be an eigenvector ofinwith
the eigenvalue, it is necessary and sucient that:
= = 12 (17)
that is:
2
=1
=
2
=1
(18)
We set:
=
= (19)
We must solve a system of two linear homogeneous equations:
(11 11)1+ (12 12)2= 0
(21 21)1+ (22 22)2= 0 (20)
This system has a non-zero solution only if:
11 11 12 12
21 21 22 22
= 0 (21)
The possible eigenvaluesare therefore the roots of a second-degree equation.
1194

A SIMPLE EXAMPLE OF THE CHEMICAL BOND: THE H
+
2
ION
2-c. Overlap, Coulomb and resonance integrals
1and2are normalized; consequently:
11=22= 1 (22)
On the other hand,1and 2are not orthogonal. Since the wave-functions (14)
associated with these two kets are real, we have:
12=21= (23)
with:
= 1 2=d
3
1(r)2(r) (24)
is called anoverlap integral, since it receives contributions only from points of space
at which the atomic wave functions1and2are both dierent from zero (such points
exist if the two atomic orbitals partially overlap). A simple calculation gives:
=e1 ++
1
3
2
(25)
To nd this result, we can use elliptic coordinates (3), since:
1=
+
2
2=
2
(26)
According to expression (14) for the wave functions and the one for the volume element, (4), we
must calculate:
=
1
3
0
+
1
d
+1
1
d
2
0
d
33
0
8
2 2
e
=
3
2
+
1
d
21
3
e (27)
which easily yields (25).
By symmetry:
11=22 (28)
According to expression (13) for the Hamiltonian, we obtain:
11= 1
P
2
2
2
1
1 1
2
2
1+
2
11 (29)
Now, 1is a normalized eigenket of
P
2
2
2
1
. The rst term of (29) is therefore equal
to the energyof the ground state of the hydrogen atom, and the third term is equal
to
2
; we thus have:
11= +
2
(30)
1195

COMPLEMENT G XI
with:
= 1
2
2
1=d
3
2
2
[1(r)]
2
(31)
is called aCoulomb integral.It describes (to within a change of sign) the electro-
static interaction between the proton2and the charge distribution associated with the
electron when it is in the 1atomic state around the proton1. We nd:
=
2
1e
2
(1 +) (32)
To nd this result, we use elliptic coordinates again:
=
2
0
1
3
0
33
0
8
2 2
ddd
2
e
(+)
=
2
+
1
d
+1
1
d(+)e
(+)
(33)
Elementary integrations then lead to result (32).
In formula (30),can be considered to be a modication of the repulsive energy
2
of the two protons: when the electron is in the state1, the corresponding charge
distribution screens the proton1. Since1(r)
2
is spherically symmetric about1,
if the proton2was far enough from it this charge distribution would appear to2like
a negative point chargesituated at its center1, (so that the charge of the proton1
would be totally cancelled). This does not actually happen unlessis much larger than
0:
lim
2
= 0 (34)
For nite, the screening eect can only be partial, and we must have:
2
0 (35)
The variation of the energy
2
with respect tois shown in Figure
line. It is clear that the variation of11(or22) with respect tocannot explain the
chemical bond, since this curve has no minimum.
Finally, let us calculate12and21. Since the wave functions1(r)and2(r)
are real, we have:
12=21 (36)
Expression (13) for the Hamiltonian gives:
12= 1
P
2
2
2
2
2+
2
12 1
2
1
2 (37)
that is, according to denition (24) of:
12= +
2
(38)
1196

A SIMPLE EXAMPLE OF THE CHEMICAL BOND: THE H
+
2
ION
with:
= 1
2
1
2=d
3
1(r)
2
1
2(r) (39)
We shall calltheresonance integral
1
. It is equal to:
= 2e(1 +) (40)
The use of elliptic coordinates enables us to writein the form:
=
2
0
1
3
0
33
0
8
2 2
ddd
2e
(+)
=
2
+
1
d2e
(41)
The fact that12is dierent from zero expresses the possibility of the electron
jumping from the neighborhood of one of the protons to that of the other one. If,
at some time, the electron is in the state1(or2), it oscillates in time between
the two sites, under the inuence of12. This non-diagonal matrix element is therefore
responsible for the phenomenon ofquantum resonance, which we described qualitatively
in ).
To sum up, the parameters which are functions ofand are involved in equation
(21) for the approximate energiesare:
11=22= 1 12=21=
11=22= +
2
12=21= +
2
(42)
where,andare given by (25), (32) and (40), and are plotted in Figure. Note
that the non-diagonal elements of determinant (21) take on signicant values only if the
orbitals1(r)and2(r)partially overlap, since the product1(r)2(r)appears in
denition (39) ofas well as in that of
2-d. Bonding and antibonding states
. Calculation of the approximate energies
We set:
=
=
= (43)
1
Certain authors callan exchange integral. We prefer to restrict the use of this term to another
type of integral which is encountered in many-particle systems (ComplementXIV, ).
1197

COMPLEMENT G XI0 1 2 3 4 5 6
0.5
1
S
S
S
C
C–
R
e
2
A
C
A
E
I
C, A
=
2a
0
e
2
a
0
e
2
ρ=
a
0
R
Figure 3: Variation of(the overlap integral),(the Coulomb integral) and(the
resonance integral) with respect to= 0. When ,andapproach zero
exponentially, whiledecreases only with
2
(the screened interaction
2
of
the proton1with the atom centered at2also decreases exponentially, however).
Equation (21) can then be written:
1 +
2
1 +
2
1 +
2
1 +
2
= 0 (44)
or:
++ 1
2
2
=++ 1
2
2
(45)
This gives the following two values for:
+=1 +
2
+
1
(46a)
=1 +
2
+
1 +
(46b)
+and both approach1whenapproaches innity. This means that the two
approximate energiesapproach , the ground state energy of an isolated hydrogen
1198

A SIMPLE EXAMPLE OF THE CHEMICAL BOND: THE H
+
2
ION
atom, as expected (). Furthermore, it is convenient to choose this value as the energy
origin, that is, to set:
=() () =+ (47)
Using (25), (32) and (40), the approximate energies+and can be written:
=
2
2e(1 +)
2
1e
2
(1 +)
1e(1 ++
2
3)
(48)
The variation of with respect tois shown in dashed lines in Figure. We see
that has a negative minimum for a certain value of the distancebetween the two
protons. Although this is an approximation (cf.Fig.), it explains the existence of the
chemical bond.
As we have already pointed out, the variation with respect toof the diagonal
elements11and22of determinant (21) has no minimum (dotted-line curve of Fig-
ure). The minimum of therefore is due to the non-diagonal elements12and
12. This shows that the phenomenon of the chemical bond appears only if the electronic
orbitals of the two atoms participating in the bond overlap suciently.
. Eigenstates ofinside the subspace
The eigenstate corresponding tois called abonding state, and the one corre-
sponding to+, anantibonding state, since+always remains greater than the energy
of the system formed by a hydrogen atom in the ground state and an innitely
distant proton.
According to (45):
++ 1
2
= ++ 1
2
(49)
System (20) then gives:
1 2= 0 (50)
The bonding and antibonding states are therefore symmetric and antisymmetric linear
combinations of the kets1and2. To normalize them, it must be recalled that1
and2are not orthogonal (their scalar product is equal to). We therefore obtain:
+=
1
2(1)
[1 2] (51a)
=
1
2(1 +)
[1+ 2] (51b)
Note that thebonding state, associated with,is symmetricunder exchange of
1and2, while the antibonding state is antisymmetric.
1199

COMPLEMENT G XI
Comment:
It could have been expected that the eigenstates ofinside the subspacewould
be symmetric and antisymmetric combinations of1and2: for given positions
of the two protons, there is symmetry with respect to the bisecting plane of12,
andremains unchanged if the roles of the two protons are exchanged.
The bonding and antibonding states are approximate stationary states of the sys-
tem under study. We pointed out in ComplementXIthat the variational method can
give a valid approximation for the energies but gives a more debatable result for the
eigenfunctions. It is instructive, however, to have an idea of the mechanism of the chem-
ical bond, to represent graphically the wave functions associated with the bonding and
antibonding states, which are often called bonding and antibondingmolecular orbitals.
To do so, we can, for example, trace the surfaces of equal(the locus of points in space
for which the modulusof the wave function has a given value). Ifis real, we indicate
by a + (or) sign the regions in which it is positive (or negative). This is what is done
in Figure +and (the surfaces of equalare surfaces of revolution about the
12axis, and Figure 12). The
dierence between the bonding orbital and the antibonding orbital is striking. In the
rst one, the electronic cloud streches out to include both protons, while in the second
one, the position probability of the electron is zero in the bisecting plane of12.
Comment:
We can calculate the average value of the potential energy in the state, which,
if we use (51b), (31) et (39), is equal to:
=
2
2
1
2
2
=
2
1
1 +
1
2
1
1+ 1
2
1
2
+1
2
2
1+ 1
2
2
2
=
2
1
1 +
(2 + 2+) (52)
Subtracting this from (46b
=
P
2
2
=
=
1
1 +
(1 +) (53)
We shall discuss later () to what extent (52) and (53) give good approximations
for the kinetic and potential energies.
1200

A SIMPLE EXAMPLE OF THE CHEMICAL BOND: THE H
+
2
IONP
1
a b
P
1
++ –
P
2
P
2
Figure 4: Schematic drawings of the bonding molecular orbital (g. a) and the antibonding
molecular orbital (g. b) for the H
+
2
ion. We have shown the cross section in a plane
containing12of a family of surfaces for which the modulusof the wave function
has a constant given value. These are surfaces of revolution about12(we have shown
4 surfaces, corresponding to 4 dierent values of). The+andsigns indicated in
the gure are those of the wave function (which is real) in the corresponding regions.
The dashed line is the trace of the bisecting plane of12, which is a nodal plane for the
antibonding orbit.
3. Critique of the preceding model. Possible improvements
3-a. Results for small
What happens to the energy of the bonding state and the corresponding wave function
when 0?
We see from Figure ,andapproach, respectively, 1,2and2when
0. If we subtract the repulsion term
2
of the two protons, to obtain the electronic
energy, we nd:
2
0
3 (54)
In addition, since1approaches2, reduces to1(the ground state1of the hydrogen
atom).
This result is obviously incorrect. When= 0, we have the equivalent
2
of a helium
ion He
+
. The electronic energy of the ground state of H
+
2
must coincide, for= 0, with that
of the ground state of He
+
. Since the helium nucleus is a= 2nucleus, this energy is (cf.
ComplementVII):
2
=4 (55)
and not3. Furthermore, the wave function(r)should not approach1(r) = (
3
0)
12
e
1
,
but rather(
3
0
3
)
12
e
1
with= 2(the Bohr orbit is twice as small). This enables us
to understand why the disagreement between the exact result and that of
2
In addition to the two protons, the helium nucleus of course contains one or two neutrons.
1201

COMPLEMENT G XI
important for small values of(Fig.): this calculation uses atomic orbitals which are too
spread out when the two protons are too close to each other.
A possible improvement therefore consists of enlarging the family of trial kets because of
these physical arguments and using kets of the form:
=11()+22() (56)
where 1()and 2()are associated with1atomic orbitals of radius0centered at1
and2. The ground state still corresponds, for reasons of symmetry, to1=2. We consider
like a variational parameter in seeking, for each value ofthe value ofwhich minimizes the
energy.
The calculation can be performed completely in elliptic coordinates. We nd (cf.Fig.)
that the optimal value ofdecreases from= 2for= 0to= 1for , as it should.
The curve obtained for is much closer to the exact curve (cf.Fig.). Table I
gives the values of the abscissa and ordinate of the minimum of obtained from the various
models considered in this complement. It can be seen from this table that the energies found by
the variational method are always greater than the exact energy of the ground state; in addition,
we see that enlarging the family of trial kets improves the results for the energy.
3-b. Results for large
When , we see from (48) that+and exponentially approach the same
value . Actually, this limit should not be obtained so rapidly. To see this, we shall use a
perturbation approach, as in ComplementXI, (Van der Waals forces) orXII(the Stark eect
of the hydrogen atom). Let us evaluate the perturbation of the energy of a hydrogen atom (in
the 1state), situated at2, produced by the presence of a proton1situated at a distance
much greater than0( 1). In the neighborhood of2, the proton1creates an electric eld
E, which varies like1
2
. This eld polarizes the hydrogen atom and causes an electric dipole
momentD, proportional toE, to appear. The electronic wave function is distorted, and the
barycenter of the electronic charge distribution moves closer to1(Fig.).EandDare both
proportional to 1/
2
and have the same sign. The electrostatic interaction between the proton
1and the atom situated at2must therefore lower
3
. Consequently, the asymptotic behavior
of+and must vary, not exponentially, but as
4
(whereis a positive constant)
the energy by an amount which, likeED, varies as 1/
4
.
It is actually possible to nd this result by the variational method. Instead of linearly
superposing1orbitals centered at1and2, we shall superpose hybrid orbitals1, and2,
which are not spherically symmetric about1and2. The hybrid orbital2is obtained, for
example, by linearly superposing a1orbital and a 2orbital, both centered
4
at2:
2(r) =
2
1(r) +
2
1(r) (57)
and has a form analogous to the one shown in Figure. Now, consider determinant (21). The
non-diagonal elements12= 1 2and12= 12still approach zero exponentially
when . This is because the product1(r)2(r)appears in the corresponding integrals;
even though distorted, the orbitals1(r)and2(r)still remain localized in the neighborhoods
of1and2respectively, and their overlap goes to zero exponentially when. The two
eigenvalues+and therefore both approach11= 22when , since determinant
(21) becomes diagonal.
Now, what does 22represent? As we have seen (cf.), it is the energy of a
hydrogen atom placed at2and perturbed by the proton1. The calculation of
3
More precisely, the energy is lowered by
1
2
ED(cf.ComplementXII, ).
4
The symmetry axis of the2orbital is chosen along the straight line joining the two protons.
1202

A SIMPLE EXAMPLE OF THE CHEMICAL BOND: THE H
+
2
ION1
1
2
2
3 4 5
ρ = R/a
0
Z
0
Figure 5: For each value of the internuclear distance, we have calculated the value of
which minimizes the energy. For= 0, we have the equivalent of a He
+
ion, and
we indeed nd= 2. For 0, we have essentially an isolated hydrogen atom,
which gives= 1. Between these two extremes,is a decreasing function of. The
corresponding optimal energies are represented by triangles in Figure.
any polarization of the1electronic orbital due to the eect of the electric eld created by
1, and this is why we found an energy correction decreasing exponentially whenincreases.
However, if, as we are doing here, we take into account the polarization of the electronic orbital,
we nd a correction in
4
. The fact that, in (57), we consider only the mixing with the 2
orbital causes the value ofgiven by the variational calculation to be approximate (whereas the
perturbation calculation of the polarization involves all the excited states,cf.ComplementXII).
The two curves+and therefore do approach each other exponentially, since the
dierence between+and involves only the non-diagonal elements12and12, and their
common value for largeapproaches zero proportionally to1
4
(Fig.).
The preceding discussion also suggests using polarized orbitals like the one in (57), not
only for largebut also for all other values ofas well. We would thus enlarge the familyD
E
P
2
P
1
Figure 6: Under the eect of the electric eldEcreated by the proton1, the electronic
cloud of the hydrogen atom centered at2becomes distorted, and this atom acquires an
electric dipole momentD. An interaction energy results which decreases with 1/
4
when
increases.
1203

COMPLEMENT G XIR
ΔE
–
ΔE
+
Figure 7: When , the energies of the bonding and antibonding states approach
each other exponentially. However, they approach their limiting value less rapidly (like
1
4
).
of trial kets and consequently improve the accuracy. In expression (57), we then consideras
a variation parameter, like the parameterthat denes the Bohr radius0associated with
the1and 2orbitals. To make the method even more exible, we choose dierent parameters
andfor1and2. For each value ofwe then minimize the average value ofin the
state1+ 2(which, for reasons of symmetry, is still the ground state), and we determine
the optimal values of,,. The agreement with the exact solution then becomes excellent
(cf.Table I).
4. Other molecular orbitals of the H
+
2
ion
In the preceding sections, we obtained by the variational method a bonding and an
antibonding molecular orbitals. They were obtained from the ground state1of each of
Distance d'équilibre
des deux protons
(abscisse du minimum
de)
Profondeur du puits de
potentiel
(valeur du minimum de
)
Méthode variationnelle du
(orbitales1avec= 1) 2,50 0 1,76 eV
Méthode variationnelle du ??
(orbitales1avecvariable) 2,00 0 2,35 eV
Méthode variationnelle du
(orbitales hybrides avec
variables)
2,000 2,73 eV
Valeurs exactes 2,00 0 2,79 eV
Tableau I
1204

A SIMPLE EXAMPLE OF THE CHEMICAL BOND: THE H
+
2
ION
the two hydrogen atoms, formed about the two protons. Of course, we chose the1state
because it was clear that this would be the best choice for obtaining an approximation of
the ground state of a system of two protons and one electron. We can obviously envisage,
with the method of linear combination of atomic orbitals (), using excited states
of the hydrogen atom to obtain other molecular orbitals of higher energies. The main
interest of these excited orbitals will be to give us an idea of the phenomena which can
come into play in molecules which are more complex than the H
+
2
ion. For example, to
understand the properties of a diatomic molecule containing several electrons, we can,
in a rst approximation, treat these electrons individually, as if they did not interact
with each other. We thus determine the various possible stationary states for an isolated
electron placed in the Coulomb eld of the nuclei, and then place the electrons of the
molecule in these states, taking the Pauli principle into account (Chap. , ) and
lling the lowest energy states rst (this procedure is analogous to the one described
for many-electron atoms in ComplementXIV). In this section, we shall indicate the
principal properties of the excited molecular orbitals of the H
+
2
ion, while keeping in
mind the possibilities of generalization to more complex molecules.
4-a. Symmetries and quantum numbers. Spectroscopic notation
()The potentialcreated by the two protons is symmetric with respect to revolution
about the12axis, which we shall choose as theaxis. This means that
and, consequently, the Hamiltonianof the electron, do not depend on the angular
variablewhich xes the orientation aboutof the 12plane containing the
axis and the point. It follows thatcommutes with the component
of the orbital angular momentum of the electron [in therrepresentation,
becomes the dierential operator
~
, which commutes with any-independent
operator]. We can then nd a system of eigenstates ofthat are also eigenstates
of, and class them according to the eigenvalues~of.
()The potentialis also invariant under reection through any plane containing
12, that is, theaxis. Under such a reection, an eigenstate ofof eigenvalue
~is transformed into an eigenstate ofof eigenvalue~(the reection changes
the sense of revolution of the electron about). Because of the invariance of
the energy of a stationary state depends only on.
In spectroscopic notation, we label each molecular orbital with a Greek letter in-
dicating the value of, as follows:
= 0
= 1 (58)
= 2
(note the analogy with atomic spectroscopic notation:,,recall,,). For
example, since the ground state1of the hydrogen atom has a zero orbital angular
momentum, the two orbitals studied in the preceding sections areorbitals (it can
be shown that this is also true for the exact stationary wave functions, and not
only for the approximate states obtained by the variational method).
1205

COMPLEMENT G XI
This notation does not use the fact that the two protons of the H
+
2
ion have equal
charges. The,,classication of molecular orbitals therefore remains valid for
a heteropolar diatomic molecule.
()In the H
+
2
ion (and, more generally, in homopolar diatomic molecules), the potential
is invariant under reection through the middleof12. We can therefore
choose eigenfunctions of the Hamiltonianin such a way that they have a denite
parity with respect to the point. For an even orbital, we add to the Greek letter
which characterizes, an index(from the German gerade); this index is
(ungerade) for odd orbitals. Thus, the bonding orbital obtained above from the
1atomic states is aorbital, while the corresponding antibonding orbital is.
()Finally, we can use the invariance ofunder reection through the bisecting plane
of12to choose stationary wave functions which have a denite parity in this
operation, that is, a parity dened with respect to the change in sign of the variable
. Functions which are odd under this reection are labeled with an asterisk. They
are necessarily zero at all points of the bisecting plane of12, like the orbital
shown in Figureb; these are antibonding orbitals.
Comment:
Reection through the bisecting plane of12can be obtained by performing a
reection throughfollowed by a rotation ofabout. The parity () is
therefore not independent of the preceding symmetries (the states will have
an asterisk for oddand none for even; the situation is reversed for the
states). However, it is convenient to consider this parity, since it enables us to
determine the antibonding orbitals immediately.
4-b. Molecular orbitals constructed from the 2atomic orbitals
If we start with the excited state2of the hydrogen atom, arguments analogous
to those of the preceding sections will give a bonding(2)orbital and an antibonding
(2)orbital, with forms similar to those in Figure. We shall therefore concern
ourselves instead with molecular orbitals obtained from the excited atomic states2.
. Orbitals constructed from2states
We shall denote by
1
2and
2
2the atomic states2(cf.ComplementVII,
), centered at1and2respectively. The form of the corresponding orbitals is
shown in Figure
By a variational calculation analogous to the one in , we can construct, starting
with these two atomic states, two approximate eigenstates of the Hamiltonian (13). The
symmetries recalled in
states can be written:
1
2+
2
2 (59a)
1
2
2
2 (59b)
1206

A SIMPLE EXAMPLE OF THE CHEMICAL BOND: THE H
+
2
ION– + – +P
1
P
2
z
Figure 8: Schematic representation of the2atomic orbitals centered at1and2(the
axis is chosen along12) and used as a basis for constructing the excited molecular
orbitals(2)and(2)shown in Figure
The shape of the two molecular orbitals so obtained can easily be deduced from Figure;
they are shown in Figure.
The two atomic states2are eigenstates ofwith the eigenvalue zero; the same
is therefore also true of the two states (59). The molecular orbital associated with (59a)
is even and is written(2); the one corresponding to (59b) is odd under a reection
throughas well as under a reection through the bisecting plane of12, and we shall
therefore denote it by(2).– – – ++ –+
P
1
P
2
P
2
a b
σ
g
(2p
z
) σ
u
*
(2p
z
)
P
1
Figure 9: Schematic representation of the excited molecular orbitals: the bonding orbital
(2)(g. a) and the antibonding orbital(2)(g. b). As in Figure, we have
drawn the cross section in a plane containing12of a constant modulussurface.
This is a surface of revolution about12. The sign shown is that of the (real) wave
function. The dashed-line curves are the cross sections in the plane of the gure of the
nodal surfaces (= 0).
1207

COMPLEMENT G XI
Remark:
As we mentioned in the introduction of this complement, the major interest of the excited
orbitals we study here is their application to molecules more complex than H
+
2
. For such
molecules, the orbitals2and2have dierent energies, so that it is legitimate to consider
them separately. If, however, we specically study the hydrogen molecular H
+
2
ion, the
sates2and2are then degenerate, so that they are immediately mixed by the electric
eld of a nearby proton. In this case, there is no reason to study the orbitals2et2
separately; it is more appropriate to introduce hybrid orbitals similar to those discussed
in VII.
. Orbitals constructed from2or2states
We shall now start with the atomic states
1
2and
2
2, with which are associ-
ated the real wave functions (cf.ComplementVII, ) shown in Figure
the surfaces of equalwhose cross sections in theplane are given in Figure
are surfaces of revolution, not about, but about axes parallel toand passing
through1and2). Recall that the atomic orbital2is obtained by the linear combi-
nation of eigenstates ofcorresponding to= 1and=1. The molecular orbitals
constructed from these atomic orbitals therefore have= 1; they areorbitals.+
–
P
1
+
–
P
2
zO
x
Figure 10: Schematic representation of the atomic orbitals2centered at1and2(the
axis is chosen along12) and used as a basis for constructing the excited molecular
orbitals(2)and(2)shown in Figure. For each orbital, the surface of equal
, whose cross section in theplane is shown, is a surface of revolution, no longer
about, but about a straight line parallel toand passing either through1or2.
Here again, the approximate molecular states produced from the atomic states2
are the symmetric and antisymmetric linear combinations:
1
2+
2
2 (60a)
1
2
2
2 (60b)
The form of these molecular orbitals can easily be qualitatively deduced from Figure.
The surfaces of equalare not surfaces of revolution about, but are simply sym-
metric with respect to theplane. Their cross sections in this plane are shown in
1208

A SIMPLE EXAMPLE OF THE CHEMICAL BOND: THE H
+
2
ION–
–+
–
+
+
P
2
a b
P
1
P
2
P
1
π
u
(2p
x
) π
g
*
(2p
x
)
Figure 11: Schematic representation of the excited molecular orbitals: the bonding orbital
(2)(g. a) and the antibonding orbital(2)(g. b). For each of these two
orbitals, we have shown the cross section in theplane of a surface on whichhas
a given constant value. This surface is no longer a surface of revolution but is simply
symmetric with respect to theplane. The meaning of the signs and the dashed lines
is the same as in Figures,,,.
Figure. We see immediately in this gure that the orbital associated with state (60a)
is odd with respect to the middleof12but even with respect to the bisecting plane
of12; it will therefore be denoted by(2). On the other hand, the orbital corre-
sponding to (60b) is even with respect to pointand odd with respect to the bisecting
plane of12: it is an antibonding orbital, denoted by(2). We stress the fact that
theseorbitals haveplanes of symmetry,not axes of revolution like theorbitals.
Of course, the molecular orbitals produced by the atomic states2can be deduced
from the preceding ones by a rotation of2about12.
orbitals analogous to the preceding ones are involved in the double or triple
bonds of atoms such as carbon (cf.ComplementVII, ).
Comment:
We saw earlier () that the energy separation of the bonding and antibonding
levels is due to the overlap of the atomic wave functions. Now, for the same
distancethe overlap of the
1
2and
2
2orbitals, which point towards each
other, is larger than that of
1
2and
2
2, whose axes are parallel (Fig.).
We see that the energy dierence between(2)and(2)is larger than that
between(2)and(2)[or(2)and(2)]. The hierarchy of the
corresponding levels is indicated in Figure.
1209

COMPLEMENT G XI2p
z
, 2p
x
, 2p
y
π
g
*
2p
x
π
g
*
2p
y
σ
u
*
2p
z
σ
g
2p
z
π
u
2p
x
π
u
2p
y
Figure 12: The energies of the various excited molecular orbitals constructed from the
atomic orbitals2,2and2centered at1and2(theaxis is chosen along
12). By symmetry, the molecular orbitals produced by the2atomic orbitals are
degenerate with those produced by the2atomic orbitals. The dierence between the
bonding and antibonding molecular orbitals(2)and(2)is, however, smaller
than the corresponding dierence between the(2)and(2)molecular orbitals.
This is due to the larger overlap of the two2atomic orbitals.
5. The origin of the chemical bond; the virial theorem
5-a. Statement of the problem
When the distancebetween the protons decreases, their electrostatic repulsion
2
increases. The fact that the total energy()of the bonding state decreases
(whendecreases from a very large value) and then passes through a minimum therefore
means that the electronic energy begins by decreasing faster than
2
increases (of
course, since this term diverges when0, it is the repulsion between the protons
which counts at short distances). We can then ask the following question: does the
lowering of the electronic energy, which makes the chemical bond possible, arise from a
lowering of the electronic potential energy or from a lowering of the kinetic energy or
from both?
We have already calculated, in (52) and (53), approximate expressions for the
(total) potential and kinetic energies. We might then consider studying the variation of
these expressions with respect toSuch a method, however, would have to be used
with caution, since, as we have already pointed out, the eigenfunctions supplied by a
variational calculation are much less precise than the energies. We shall discuss this
point in greater detail in below.
Actually, it is possible to answer this question rigorously, thanks to the virial
theorem, which provides exact relations between()and the average kinetic and
potential energies. Therefore, in this section, we shall prove this theorem and discuss its
physical consequences. The results obtained, furthermore, are completely general and
1210

A SIMPLE EXAMPLE OF THE CHEMICAL BOND: THE H
+
2
ION
can be applied, not only to the molecular ion H
+
2
, but also to all other molecules. Before
considering the virial theorem itself, we shall begin by establishing some results which
we shall need later.
5-b. Some useful theorems
. Euler's theorem
Recall that a function(1,2, ...,) of several variables1,2, ...is said to
be homogeneous of degreeif it is multiplied bywhen all the variables are multiplied
by:
(1 2 ) =(12 ) (61)
For example, the potential of a three-dimensional harmonic oscillator:
( ) =
1
2
2
(
2
+
2
+
2
) (62)
is homogeneous of degree 2. Similarly, the electrostatic interaction energy of two particles:
=
( )
2
+ ( )
2
+ ( )
2
(63)
is homogeneous of degree1.
Euler's theorem indicates that any functionwhich is homogeneous of degree
satises the identity:
=1
=(1 ) (64)
To prove this, we calculate the derivatives with respect toof both sides of (61). The
left-hand side yields:
(1 )() =(1 ) (65)
and the right-hand side yields:
1
(1 ) (66)
If we set (65) equal to (66), with= 1, we obtain (64).
Euler's theorem can very easily be veried in examples (62) and (63).
. The Hellman-Feynman theorem
Let()be a Hermitian operator which depends on a real parameter, and()
a normalized eigenvector of()of eigenvalue():
()()=()() (67)
()()= 1 (68)
1211

COMPLEMENT G XI
The Hellmann-Feynman theorem indicates that:
d
d
() =()
d
d
()() (69)
This relation can be proven as follows. According to (67) and (68), we have:
() =()()() (70)
If we dierentiate this relation with respect to, we obtain:
d
d
() =()
d
d
()()
+
d
d
()()()+()()
d
d
() (71)
that is, using (67) and the adjoint relation [()is Hermitian, hence()is real]:
d
d
() =()
d
d
()()
+()
d
d
()()+()
d
d
() (72)
On the right-hand side, the expression inside curly brackets is the derivative of()(),
which is zero since()is normalized; we therefore nd (69).
. Average value of the commutator [,] in an eigenstate of
Letbe a normalized eigenvector of the Hermitian operatorof eigenvalue.
For any operator:
[]= 0 (73)
since, as= and = :
( )= = 0 (74)
5-c. The virial theorem applied to molecules
. The potential energy of the system
Consider an arbitrary molecule composed ofnuclei andelectrons. We shall
denote byr(= 12 )the classical positions of the nuclei, and byrandp(=
12 )the classical positions and momenta of the electrons. The components of these
vectors will be written,,, etc.
We shall use the Born-Oppenheimer approximation, considering theras given
classical parameters. In the quantum mechanical calculation, only therandpbecome
operators,RandP. We must therefore solve the eigenvalue equation:
(r
1 r)(r
1 r)=(r
1 r)(r
1 r) (75)
1212

A SIMPLE EXAMPLE OF THE CHEMICAL BOND: THE H
+
2
ION
of a Hamiltonianwhich depends on the parametersr
1 rand which acts in the
state space of the electrons. The expression forcan be written:
=+(r
1 r) (76)
whereis the kinetic energy operator of the electrons:
=
=1
1
2
(P)
2
(77)
and(r
1 r)is the operator obtained by replacing therby the operatorsRin
the expression for the classical potential energy. The latter is the sum of the repulsion
energy between the electrons, the attraction energybetween the electrons and
the nuclei, and the repulsion energybetween the nuclei, so that:
(r
1 r) =+ (r
1 r) +(r
1 r) (78)
Actually, sincedepends only on therand does not involve theR, is a
number and not an operator acting in the state space of the electrons. The only eect
of is therefore to shift all the energies equally, since equation (75) is equivalent to:
(r
1 r)(r
1 r)=(r
1 r)(r
1 r) (79)
where:
(r
1 r) =+ + (r
1 r) = (r
1 r) (80)
and where the electronic energyis related to the total energyby:
(r
1 r) =(r
1 r) (r
1 r) (81)
We can apply Euler's theorem to the classical potential energy, since it is a ho-
mogeneous function of degree1of thesetof electronic and nuclear coordinates. Since
the operatorsRall commute with each other, we get the relation between the quantum
mechanical operators:
=1
rr +
=1
Rr= (82)
whererandrdenote the operators obtained by substitution of theRfor ther
in the gradients with respect torandrin the classical expression for the potential
energy. Relation (82) will serve as the foundation of our proof of the virial theorem.
. Proof of the virial theorem
We apply (73) to the special case in which:
=
=1
RP (83)
1213

COMPLEMENT G XI
To do so, we nd the commutator ofwith:
=1
RP=
=1
[ ]+[ ]
=~
=1
(P)
2
+Rr (84)
(we have used the commutation relations of a function of the momentum with the posi-
tion, or vice versa;cf.ComplementII, ). The rst term inside the curly brackets
is proportional to the kinetic energy. According to (82), the second term is equal to:
=1
rr (85)
Consequently, relation (73) yields:
2 + +
=1
rr = 0 (86)
that is, since the Hamiltoniandepends on the parametersronly through:
2 + =
=1
rr (87)
The componentsrhere play a role analogous to that of the parameterin (69). Ap-
plication of the Hellmann-Feynman theorem to the right-hand side of equation (87) then
gives:
2 + =
=1
rr(r
1 r r) (88)
Furthermore, we obviously have:
+ =(r
1 r) (89)
We can then easily nd from (88) and (89):
=
=1
rr
= 2+
=1
rr
(90)
Thus, we obtain a very simple result: the virial theorem applied to molecules. It enables
us to calculate the average kinetic and potential energies if we know the variation of the
total energy with respect to the positions of the nuclei.
1214

A SIMPLE EXAMPLE OF THE CHEMICAL BOND: THE H
+
2
ION
Comment:
The total electronic energyand the electronic potential energyare also
related by:
= 2+
=1
rr (91)
This relation can be proven by substituting (81) and the explicit expression forin
terms of therinto the second relation of (90). However, it is simpler to note that the
electronic potential energy= +, like the total potential energy, is a homoge-
neous function of degree1of the coordinates of the system of particles. Consequently,
the preceding arguments apply toas well as toand we can simultaneously replace
byandbyin both relations (90).
. A special case: the diatomic molecule
When the number of nuclei is equal to two, the energies depend only on the
internuclear distanceThis further simplies the expression for the virial theorem,
which becomes:
=
d
d
= 2+
d
d
(92)
Sincedepends on the nuclear coordinates only throughwe have:
=
d
d
(93)
and, consequently:
=12
=
d
d
=12
(94)
Now, the distancebetween the nuclei is a homogeneous function of degree 1 of the coordinates
of the nuclei. Application of Euler's theorem to this function enables us to replace the double
summation appearing on the right-hand side of (94) by, and we nally obtain:
=12
rr =
d
d
(95)
When this result is substituted into (90), it gives relations (92).
In (92) as in (90), we can replacebyandby
5-d. Discussion
. The chemical bond is due to a lowering of the electronic potential energy
Let be the value of the total energyof the system when the various nuclei
are innitely far apart. If it is possible to form a stable molecule by moving the nuclei
1215

COMPLEMENT G XI
closer together, there must exist a certain relative arrangement of these nuclei for which
the total energypasses through a minimum0 . For the corresponding values
ofr, we then have:
r =0 (96)
Relations (90) then indicate that, for this equilibrium position, the kinetic and potential
energies are equal to:
0= 0
0= 20 (97)
Furthermore, when the nuclei are innitely far from each other, the system is composed
of a certain number of atoms or ions without mutual interactions (the energy no longer
depends on ther). For each of these subsystems, the virial theorem indicates that
= ,= 2and, for the system as a whole, we must therefore also have:
=
= 2 (98)
Subtracting (98) from (97) then gives:
0 =(0 )0
0 = 2(0 )0 (99)
The formation of a stable molecule is therefore always accompanied by an increase in the
kinetic energy of the electrons and a decrease in the total potential energy. The electronic
potential energy must, furthermore, decrease even more since the average value(the
repulsion between the nuclei), which is zero at innity, is always positive. It is therefore
a lowering of the potential energy of the electrons+ that is responsible for the
chemical bond. At equilibrium, this lowering must outweigh the increase inand
.
. The special case of the H
+
2
ion
()Application of the virial theorem to the approximate variational energy.
We return to the study of the variation ofand for the H
+
2
ion. We shall begin
by examining the predictions of the variational model of
expressions (52) and (53). From the second of these relations, we deduce that:
= =
1
1 +
(2 ) (100)
Sinceis always greater than2(cf.Fig.), this calculation would tend to indicate
thatis always negative. This appears, moreover, in Figure, where the dashed lines
represent the variations of the approximate expressions (52) and (53). In particular, we see
that, according to the variational calculation,is negative at equilibrium( 25)and
is positive. These results are both incorrect, according to (99). We see here the limits of a
variational calculation, which gives an acceptable value for the total energy+, but not for
and separately. These latter average values depend too strongly on the wave function.
1216

A SIMPLE EXAMPLE OF THE CHEMICAL BOND: THE H
+
2
IONE
– E
I
E
I
00
1 2 3 4 5
R
a
0
e
2
a
0
e
2
a
0
–
T
e

V
Figure 13: The electronic kinetic energyand the potential energyof the H
+
2
ion as functions of= 0(for purposes of comparison, we have also shown the total
energy= + ).
. solid lines: the exact values (the chemical bond is due to the fact thatdecreases a
little faster thanincreases).
. long dashes: the average values calculated from the bonding wave function given by the
simple variational method of .
. short dashes: the values obtained by the application of the virial theorem to the energy
given by the same variational calculation.
The virial theorem enables us, without having to resort to the rigorous calculation men-
tioned in , to obtain a much better approximation forand . All we need to do is
apply the exact relations (92) to the energycalculated by the variational method. We should
expect an acceptable result, since the variational approximation is now used only to supply the
total energyThe values thus obtained for and are represented by short dashed
lines in Figure. For purposes of comparison, we have shown in solid lines the exact values of
and (obtained by application of the virial theorem to the solid-line curve of Figure).
First of all, we see that for= 25, the curve in short dashed lines indicates, as expected, that
is positive andis negative. In addition, the general shape of these curves reproduces
rather well that of the solid-line curves. As long as&15, the virial theorem applied to the
variational energy does give values which are very close to reality. This represents a considerable
improvement over the direct calculation of the average values in the approximate states.
() Behavior ofand
The solid-line curves of Figure 4and
+when 0. Indeed, when= 0, we have the equivalent of a He
+
ion for which the
1217

COMPLEMENT G XI
electronic kinetic energy is4. The divergence ofis due to the term =
2
, which
becomes innite when 0(the electronic potential energy=
2
remains nite
and approaches8, which is indeed its value in the He
+
ion).
The behavior for largedeserves a more detailed discussion. We have seen above ()
that the energyof the ground state behaves, for 0, like:
4
(101)
whereis a constant which is proportional to the polarizability of the hydrogen atom. By
substituting this result into formulas (92), we obtain:
3
4
2+
2
4
(102)
When decreases from a very large value,begins by decreasing with1
4
from its asymp-
totic value, andbegins by increasing from2. These variations then change sign (this
must be so since 0is larger than and 0is smaller than ): ascontinues
to decrease (cf.Fig.), passes through a minimum and then increases until it reaches
its value4for= 0. As for the potential energy, it passes through a maximum, then
decreases, passes through a minimum, and then approaches innity when0. How can we
interpret these variations?P
1
P
2
z
a
0
a
0
E
I
–
–
Figure 14: Variation of the potential energyof the electron subjected to the simulta-
neous attraction of the two protons1and2as one moves along the line12. In the
bonding state, the wave function is concentrated in the region between1and2, and
the electron benets simultaneously from the attraction of both protons.
As we have noted several times, the non-diagonal elements12and 21of determinant
(21) approach zero exponentially when . We can therefore argue only in terms of11
or22in discussing the variation of the energy of the H
+
2
ion at large internuclear distances.
The problem is then reduced to the study of the perturbation of a hydrogen atom centered at
2by the electric eld of the proton1. This eld tends to distort the electronic orbital by
1218

A SIMPLE EXAMPLE OF THE CHEMICAL BOND: THE H
+
2
ION
stretching it in the1direction (cf.Fig.). Consequently, the wave function extends into a
larger volume. According to Heisenberg's uncertainty relations, this allows the kinetic energy
to decrease; this can explain the behavior offor large
Arguing in terms of22, we can also explain the asymptotic behavior of. The
discussion of 0, the polarization of the hydrogen atom situated at
2makes its interaction energy
2
1
+
2
with1slightly negative (proportional to1
4
).
If is positive, it is because the potential energy
2
2
of the atom at2increases more
rapidly, when1is brought closer to2, than
2
1
+
2
decreases. This increase in
2
2
is due to the fact that the attraction of1moves the electron slightly away from2and carries
it into regions of space in which the potential created by2is less negative.
For 0(the equilibrium position of the H
+
2
ion), the wave function of the
bonding state is highly localized in the region between the two protons. The decrease
in (despite the increase in
2
) is due to the fact that the electron is in a region
of space in which it benets simultaneously from the attraction of both protons. This
lowers its potential energy (cf.Fig.). This combined attraction of the two protons
also leads to a decrease in the spatial extension of the electronic wave function, which is
concentrated in the intermediate region. This is why, forclose to0, increases
whendecreases.
References and suggestions for further reading (H
+
2
ion, H2molecule, nature of the chemical
bond, etc.):
Pauling (12.2); Pauling and Wilson (1.9), Chaps. XII and XIII; Levine (12.3),
Chaps. 13 and 14; Karplus and Porter (12.1), Chap. 5, 6; Slater (1.6), Chaps.
8 and 9; Eyring et al (12.5), Chaps. XI and XII; Coulson (12.6), Chap. IV; Wahl
(12.13).
1219

EXERCISES
Complement HXI
Exercises
1.A particle of massis placed in an innite one-dimensional well of width:
() = 0for0
() = + everywhere else
It is subject to a perturbationof the form:
() = 0
2
where0is a real constant with the dimensions of an energy.
. 0, the modications induced by()to the energy
levels of the particle.
. =
2 ~
2
, show that the
possible values of the energy are given by one of the two equations sin(2) = 0
or tan(2) =~
2
0(as in exercise 2 of ComplementI, watch out for
the discontinuity of the derivative of the wave function at=2).
Discuss the results obtained with respect to the sign and size of0. In the limit0 0,
show that one obtains the results of the preceding question.
2.Consider a particle of massplaced in an innite two-dimensional potential well of
width(cf.ComplementII) :
() = 0if0 and 0
() = + everywhere else
This particle is also subject to a perturbationdescribed by the potential:
() =0for0
2
and 0
2
() = 0everywhere else.
. 0, the perturbed energy of the ground state.
.
zeroth order in0.
3.A particle of mass, constrained to move in theplane, has a Hamiltonian:
0=
2
2
+
2
2
+
1
2
2
(
2
+
2
)
(a two-dimensional harmonic oscillator, of angular frequency). We want to study the
eect on this particle of a perturbationgiven by:
=11+22
1221

COMPLEMENT H XI
where1and2are constants, and the expressions for1and2are:
1=
2
2=~
2
~
2
2
(is the component alongof the orbital angular momentum of the particle).
In the perturbation calculations, consider only the corrections to rst order for the
energies and to zeroth order for the state vectors.
. 0, their degrees of degeneracy
and the associated eigenvectors.
In what follows, consider only the second excited state of0of energy3~and
which is three-fold degenerate.
. 1and2to the eigensub-
space of the eigenvalue3~of0.
. 2= 0and11.
Calculate, using perturbation theory, the eect of the term11on the second
excited state of0.
. with the limited expansion of the exact solution,
to be found with the help of the methods described in ComplementV(normal
vibrational modes of two coupled harmonic oscillators).
. 2 11. Considering the results of questionto be a new unper-
turbed situation, calculate the eect of the term22.
. 1= 0and21.
Using perturbation theory, nd the eect of the term22on the second excited
state of0.
. with the exact solution, which can be found
from the discussions of ComplementVI.
. 1 21. Considering the results of questionto be a
new unperturbed situation, calculate the eect of the term11.
4.Consider a particleof massconstrained to move in theplane in a circle
centered atwith xed radius(a two-dimensional rotator). The only variable of
the system is the angle= ( ), and the quantum state of the particle is dened
by the wave function()(which represents the probability amplitude of nding the
particle at the point of the circle xed by the angle). At each point of the circle,()
can take on only one value, so that:
(+ 2) =()
()is normalized if:
2
0
()
2
d= 1
1222

EXERCISES
. =
~
d
d
. IsHermitian? Calculate the eigenvalues and
normalized eigenfunctions ofWhat is the physical meaning of?
.
0=
2
2
2
Calculate the eigenvalues and eigenfunctions of0. Are the energies degenerate?
. = 0, the wave function of the particle iscos
2
(whereis a normalization
coecient). Discuss the localization of the particle on the circle at a subsequent
time
. and that it interacts with a uniform elec-
tric eldparallel to. We must therefore add to the Hamiltonian0the
perturbation:
= cos
Calculate the new wave function of the ground state to rst order in. Determine
the proportionality coecient(the linear suceptibility) between the electric dipole
parallel toacquired by the particle and the eld.
. 3 CH3, a rotation of one CH3group relative
to the other about the straight line joining the two carbon atoms.
To a rst approximation, this rotation is free, and the Hamiltonian0introduced
indescribes the rotational kinetic energy of one of the CH3groups relative to the
other (2
2
must, however, be replaced by, whereis the moment of inertia of
the CH3group with respect to the rotational axis andis a constant). To take
account of the electrostatic interaction energy between the two CH3groups, we
add to0a term of the form:
=cos 3
whereis a real constant.
Give a physical justication for the-dependence ofCalculate the energy and
wave function of the new ground state (to rst order infor the wave function and
to second order for the energy). Give a physical interpretation of the result.
5.Consider a system of angular momentumJ. We conne ourselves in this exercise
to a three-dimensional subspace, spanned by the three kets+ 1,0 1, common
eigenstates ofJ
2
(eigenvalue2~
2
) and(eigenvalues+~0~). The Hamiltonian0
of the system is:
0= +
~
2
whereandare two positive constants, which have the dimensions of an angular
frequency.
1223

COMPLEMENT H XI
. is there
degeneracy?
. B0is applied in a directionuwith polar anglesand. The
interaction withB0of the magnetic moment of the system:
M=J
(: the gyromagnetic ratio, assumed to be negative) is described by the Hamilto-
nian:
=0
where0= B0is the Larmor angular frequency in the eldB0, andis the
component ofJin theudirection:
=cos+sincos+sinsin
Write the matrix that representsin the basis of the three eigenstates of0.
. =and that theudirection is parallel to. We also have0 .
Calculate the energies and eigenstates of the system, to rst order in0for the
energies and to zeroth order for the eigenstates.
. = 2and that we again have0 , the direction ofunow being
arbitrary.
In the+ 10 1basis, what is the expansion of the ground state0of
0+, to rst order in0?
Calculate the average valueMof the magnetic momentMof the system in the
state0. AreMandB0parallel?
Show that one can write:
=
with= . Calculate the coecients(the components of the suscepti-
bility tensor).
6.Consider a system formed by an electron spinSand two nuclear spinsI1andI2(S
is, for example, the spin of the unpaired electron of a paramagnetic diatomic molecule,
andI1andI2are the spins of the two nuclei of this molecule).
Assume thatS,I1,I2are all spin 1/2's. The state space of the three-spin system
is spanned by the eight orthonormal kets12, common eigenvectors of,1,
2, with respective eigenvalues~2,1~2,2~2(with= 1= 2=). For
example, the ket+ +corresponds to the eigenvalues+~2for,~2for1,
and+~2for2.
1224

EXERCISES
We begin by neglecting any coupling of the three spins. We assume, however, that
they are placed in a uniform magnetic eldBparallel to. Since the gyromagnetic
ratios ofI1andI2are equal, the Hamiltonian0of the system can be written:
0= + 1+ 2
whereandare real, positive constants, proportional toB. Assume 2.
What are the possible energies of the three-spin system and their degrees of degen-
eracy? Draw the energy diagram.
We now take coupling of the spins into account by adding the Hamiltonian:
=SI1+SI2
whereis a real, positive constant (the direct coupling ofI1andI2is negligible).
What conditions must be satised by,1,2,,
1,
2forSI1to have a
non-zero matrix element between 12and
12? Same question for
SI2.
Assume that:
~
2
~~
so thatcan be treated like a perturbation with respect to0. To rst order in
, what are the eigenvalues of the total Hamiltonian= 0+? To zeroth
order in, what are the eigenstates of? Draw the energy diagram.
Using the approximation of the preceding question, determine the Bohr frequencies
which can appear in the evolution ofwhen the couplingof the spins is taken
into account.
In an E.P.R. (Electronic Paramagnetic Resonance) experiment, the frequencies of
the observed resonance lines are equal to the preceding Bohr frequencies. What is
the shape of the E.P.R. spectrum observed for the three-spin system? How can the
coupling constantbe determined from this spectrum?
Now assume that the magnetic eldBis zero, so that == 0. The Hamiltonian
then reduces to
LetI=I1+I2be the total nuclear spin. What are the eigenvalues ofI
2
and
their degrees of degeneracy? Show thathas no matrix elements between
eigenstates ofI
2
of dierent eigenvalues.
LetJ=S+Ibe the total spin. What are the eigenvalues ofJ
2
and their
degrees of degeneracy? Determine the energy eigenvalues of the three-spin sys-
tem and their degrees of degeneracy. Does the setJ
2
form a C.S.C.O.?
Same question forI
2
J
2
.
7.Consider a nucleus of spin= 32, whose state space is spanned by the four vectors
(= +32,+12,12,32), common eigenvectors ofI
2
(eigenvalue15~
2
4) and
(eigenvalue~).
1225

COMPLEMENT H XI
This nucleus is placed at the coordinate origin in a non-uniform electric eld derived
from a potential( ). The directions of the axes are chosen such that, at the origin:
2
=
2
=
2
= 0
Recall thatsatises Laplace's equation:
= 0
We shall assume that the interaction Hamiltonian between the electric eld gradi-
ent at the origin and the electric quadrupole moment of the nucleus can be written:
0=
2(21)
1
~
2
2
+
2
+
2
whereis the electron charge,is a constant with the dimensions of a surface and
proportional to the quadrupole moment of the nucleus, and:
=
2
2
0
; =
2
2
0
; =
2
2
0
(the index 0 indicates that the derivatives are evaluated at the origin).
Show that, ifis symmetric with respect to revolution about,0has the form:
0=[3
2
(+ 1)]
whereis a constant to be specied. What are the eigenvalues of0, their degrees
of degeneracy and the corresponding eigenstates?
Show that, in the general case,0can be written:
0=[3
2
(+ 1)] +(
2
++
2
)
whereandare constants, to be expressed in terms ofand
What is the matrix which represents0in the basis? Show that it can be
broken down into two22submatrices. Determine the eigenvalues of0and their
degrees of degeneracy, as well as the corresponding eigenstates.
In addition to its quadrupole moment, the nucleus has a magnetic momentM=I
(: the gyromagnetic ratio). Onto the electrostatic eld is superposed a magnetic
eldB0, of arbitrary directionu. We set0= B0.
What term must be added to0in order to take into account the coupling
betweenMandB0? Calculate the energies of the system to rst order in0
AssumeB0to be parallel toand weak enough for the eigenstates found in
and the energies to rst order in0found into be good approximations.
What are the Bohr frequencies which can appear in the evolution of? Deduce
from them the shape of the nuclear magnetic resonance spectrum which can be
observed with a radiofrequency eld oscillating along.
1226

EXERCISES
8.A particle of massis placed in an innite one-dimensional potential well of width
:
() = 0for0
() = + elsewhere
Assume that this particle, of charge, is subject to a uniform electric eld, with the
corresponding perturbationbeing:
=
2
. 1and2be the corrections to rst- and second-order infor the ground state
energy.
Show that1is zero. Give the expression for2in the form of a series, whose terms
are to be calculated in terms of ~(the integrals given at the end of the
exercise can be used).
By nding upper bounds for the terms of the series for2, give an upper bound for
2(cf. ). Similarly, give a lower bound for2, obtained by
retaining only the principal term of the series.
With what accuracy do the two preceding bounds enable us to bracket the exact
value of the shiftin the ground state to second order in?
We now want to calculate the shiftby using the variational method. Choose
as a trial function:
() =
2
sin1 +
2
whereis the variational parameter. Explain this choice of trial functions.
Calculate the average energy()of the ground state to second order in
[assuming the expansion of()to second order into be sucient]. Determine
the optimal value of. Find the resultvargiven by the variational method for
the shift in the ground state to second order in.
By comparingvarwith the results of, evaluate the accuracy of the variational
method applied to this example.
We give the integrals:
2
02
sinsin
2
d=
16
2
1
(14
2
)
2
= 123
2
02
2
sin
2
d=
2
2
1
6
1
2
2
02
sincosd=
2
For all the numerical calculations, take
2
= 987.
1227

COMPLEMENT H XI
9.We want to calculate the ground state energy of the hydrogen atom by the variational
method, choosing as trial functions the spherically symmetrical functions(r)whose
-dependcnce is given by:
() =1
for
() = 0for
is a normalization constant andis the variational parameter.
.
the state. Express the average value of the kinetic energy in terms ofr, so
as to avoid the delta functions which appear in(sinceris discontinuous).
. 0of. Compare0with the Bohr radius0.
.
exact value1.
10.We intend to apply the variational melhod to the determination of the energies of a
particle of massin an innite potential well:
() = 0
() = everywhere else
We begin by approximating, in the interval[+], the wave function of the
ground state by the simplest even polynomial which goes to zero at=:
() =
2 2
for
() = 0everywhere else
(a variational family reduced to a single trial function).
Calculate the average value of the Hamiltonianin this state. Compare the result
obtained with the true value.
Enlarge the family of trial functions by choosing an even fourth-degree polynomial
which goes to zero at=:
() =
2 2 2 2
for
() = 0everywhere else
(a variational family depending on the real parameter).
()Show that the average value ofin the state()is:
() =
~
2
2
2
33
2
42+ 105
2
2
12+ 42
1228

EXERCISES
()Show that the values ofwhich minimize or maximize()are given by
the roots of the equation:
13
2
98+ 21 = 0
()Show that one of the roots of this equation gives, when substituted into
(), a value of the ground state energy that is much more precise than the
one obtained in.
()What other eigenvalue is approximated when the second root of the equa-
tion obtained in b-is used? Could this have been expected? Evaluate the
precision of this determination.
.
excited state wave function is(
2 2
).
What approximate value is then obtained for the energy of this state?
1229

Chapter XII
An application of perturbation
theory: the ne and hyperne
structure of hydrogen
A Introduction
B Additional terms in the Hamiltonian
B-1 The ne-structure Hamiltonian
B-2 Magnetic interactions related to proton spin: the hyperne
Hamiltonian
C The ne structure of the = 2level
C-1 Statement of the problem
C-2 Matrix representation of the ne-structure Hamiltonian
inside the= 2level
C-3 Results: the ne structure of the= 2level
D The hyperne structure of the = 1level
D-1 Statement of the problem
D-2 Matrix representation of in the1level
D-3 The hyperne structure of the1level
E The Zeeman eect of the 1ground state hyperne structure1251
E-1 Statement of the problem
E-2 The weak-eld Zeeman eect
E-3 The strong-eld Zeeman eect
E-4 The intermediate-eld Zeeman eect
Quantum Mechanics, Volume II, Second Edition. C. Cohen-Tannoudji, B. Diu, and F. Laloë.
© 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

CHAPTER XII THE FINE AND HYPERFINE STRUCTURE OF THE HYDROGEN ATOM
A. Introduction
The most important forces inside atoms are Coulomb electrostatic forces. We took them
into account in Chapter
0=
P
2
2
+() (A-1)
The rst term represents the kinetic energy of the atom in the center of mass frame (
is the reduced mass). The second term:
() =
2
40
1
=
2
(A-2)
represents the electrostatic interaction energy between the electron and the proton (is
the electron charge). In , we calculated in detail the eigenstates and
eigenvalues of0.
Actually, expression (A-1) is only approximate: it does not take any relativistic
eects into account. In particular, all the magnetic eects related to the electron spin
are ignored. Moreover, we have not introduced the proton spin and the corresponding
magnetic interactions. The error is, in reality, very small, since the hydrogen atom is a
weakly relativistic system (recall that, in the Bohr model, the velocityin the rst orbit
= 1satises=
2
~= 11371). In addition, the magnetic moment of the
proton is very small.
However, the considerable accuracy of spectroscopic experiments makes it possible
to observe eects that cannot be explained in terms of the Hamiltonian (A-1). Therefore,
we shall take into account the corrections we have just mentioned by writing the complete
hydrogen atom Hamiltonian in the form:
=0+ (A-3)
where0is given by (A-1) and whererepresents all the terms neglected thus far. Since
is much smaller than0, it is possible to calculate its eects by using the perturbation
theory presented in Chapter. This is what we propose to do in this chapter. We shall
show thatis responsible for a ne structure, as well as for a hyperne structure
of the various energy levels calculated in Chapter. Furthermore, these structures can
be measured experimentally with very great accuracy (the hyperne structure of the1
ground state of the hydrogen atom is currently known with 12 signicant gures; the
ratio between certain atomic frequencies has been measured with 18 digits!). We shall
also consider, in this chapter and its complements, the inuence of an external static
magnetic or electric eld on the various levels of the hydrogen atom (the Zeeman eect
and the Stark eect).
This chapter actually has two goals. On the one hand, we want to use a concrete
and realistic case to illustrate the general stationary perturbation theory discussed in
the preceding chapter. On the other hand, this study, which bears on one of the most
fundamental systems of physics (the hydrogen atom), brings out certain concepts which
are basic to atomic physics. For example,
various relativistic and magnetic corrections. This chapter, while not indispensable for
the study of the last two chapters, presents concepts fundamental to atomic physics.
1232

B. ADDITIONAL TERMS IN THE HAMILTONIAN
B. Additional terms in the Hamiltonian
The rst problem to be solved obviously consists of nding the expression for
B-1. The ne-structure Hamiltonian
B-1-a. The Dirac equation in the weakly relativistic domain
In Chapter, we mentioned that the spin appears naturally when we try to estab-
lish an equation for the electron which satises both the postulates of special relativity
and those of quantum mechanics. Such an equation exists: it is theDirac equation, which
makes it possible to account for numerous phenomena (electron spin, the ne structure
of hydrogen, etc.) and to predict the existence of positrons.
The most rigorous way of obtaining the expression for the relativistic corrections
[appearing in the termof (A-3)] therefore consists of rst writing the Dirac equation
for an electron placed in the potential()created by the proton (considered to be
innitely heavy and motionless at the coordinate origin). One then looks for its limiting
form when the system is weakly relativistic, as is the case for the hydrogen atom. We then
recognize that the description of the electron state must include a two-component spinor
(cf.Chap., ). The spin operators,,, introduced in Chapter
appear naturally. Finally, we obtain an expression such as (A-3) for the Hamiltonian,
in whichappears in the form of a power series expansion inwhich we can evaluate.
It is out of the question here to study the Dirac equation, or to establish its form
in the weakly relativistic domain. We shall conne ourselves to giving the rst terms of
the power series expansion inofand their interpretation.
=
2
+
P
2
2
+()
0
P
4
8
32
+
1
2
22
1d()
d
LS
+
~
2
8
22
()
+(B-1)
We recognize in (B-1) the rest-mass energy
2
of the electron (the rst term) and the
non-relativistic Hamiltonian0(the second and third terms)
1
. The following terms are
called ne structure terms.
Comment:
Note that it is possible to solve the Dirac equation exactly for an electron placed in
a Coulomb potential. We thus obtain the energy levels of the hydrogen atom without
having to make a limited power series expansion inof the eigenstates and eigenvalues
of. The perturbation point of view we are adopting here is, however, very useful in
bringing out the form and physical meaning of the various interactions which exist inside
an atom. This will later permit a generalization to the case of many-electron atoms (for
which we do not know how to write the equivalent of the Dirac equation).
1
Expression (B-1) was obtained by assuming the proton to be innitely heavy. This is why it is the
mass of the electron that appears, and not, as in (A-1), the reduced massof the atom. As far as
0is concerned, the proton nite mass eect is taken into account by replacingby. However, we
shall neglect this eect in the subsequent terms of, which are already corrections. It would, moreover,
be dicult to evaluate, since the relativistic description of a system of two interacting particles poses
serious problems [it is not sucient to replacebyin the last terms of (B-1)].
1233

CHAPTER XII THE FINE AND HYPERFINE STRUCTURE OF THE HYDROGEN ATOM
B-1-b. Interpretation of the various terms of the ne-structure Hamiltonian
. Variation of the mass with the velocity (term)
()The physical origin
The physical origin of theterm is very simple. If we start with the relativistic
expression for the energy of a classical particle of rest-massand momentump:
=
p
2
+
22
(B-2)
and perform a limited expansion ofin powers ofp , we obtain:
=
2
+
p
2
2
p
4
8
32
+ (B-3)
In addition to the rest-mass energy (
2
) and the non-relativistic kinetic energy (p
2
2),
we nd the termp
4
8
32
, which appears in (B-1). This term represents the rst en-
ergy correction, due to the relativistic variation of the mass with the velocity.
()Order of magnitude
To evaluate the size of this correction, we shall calculate the order of magnitude
of the ratio 0:
0
p
4
8
32
p
2
2
=
p
2
4
22
=
1
4
2
2
1
137
2
(B-4)
since we have already mentioned that, for the hydrogen atom,. Since010eV,
we see that 10
3
eV.
. Spin-orbit coupling (Wterm)
()The physical origin
The electron moves at a velocityv=p in the electrostatic eldEcreated by
the proton. Special relativity indicates that there then appears, in the electron frame, a
magnetic eldBgiven by:
B=
1
2
vE (B-5)
to rst order in. Since the electron possesses an intrinsic magnetic momentM=
S, it interacts with this eldB. The corresponding interaction energy can be
written:
=MB (B-6)
Let us expressmore explicitly. The electrostatic eldEappearing in (B-5) is equal
to
1
d()
d
r
, where() =
2
is the electrostatic energy of the electron. From this,
we get:
B=
1
2
1d()
d
p
r (B-7)
1234

B. ADDITIONAL TERMS IN THE HAMILTONIAN
In the corresponding quantum mechanical operator, there appears:
PR=L (B-8)
Finally, we obtain:
=
1
22
1d()
d
LS=
2
22
1
3
LS (B-9)
Thus we nd, to within the factor
2
1/2, the spin-orbit termwhich appears in (B-1).
This term then represents the interaction of the magnetic moment of the electron spin
with the magnetic eld seen by the electron because of its motion in the electrostatic
eld of the proton.
()Order of magnitude
SinceLandSare of the order of~, we have:
2
22
~
2
3
(B-10)
Let us compare with0, which is of the order of
2
:
0
2
~
2
2232
=
~
2
222
(B-11)
is of the order of the Bohr radius,0=~
2 2
. Consequently:
0
4
~
22
=
2
=
1
137
2
(B-12)
. The Darwin term
()The physical origin
In the Dirac equation, the interaction between the electron and the Coulomb eld
of the nucleus is local; it only depends on the value of the eld at the electron position
r. However, the non-relativistic approximation (the series expansion in) leads, for
the two-component spinor which describes the electron state, to an equation in which
the interaction between the electron and the eld has become non-local. The electron is
then aected by all the values taken on by the eld in a domain centered at the pointr,
and whose size is of the order of the Compton wavelength~ of the electron. This is
the origin of the correction represented by the Darwin term.
To understand this more precisely, assume that the potential energy of the electron,
instead of being equal to(r), is given by an expression of the form:
d
3
()(r+) (B-13)
2
It can be shown that the factor 1/2 is due to the fact that the motion of the electron about the
proton is not rectilinear. The electron spin therefore rotates with respect to the laboratory reference
frame (Thomas precession: see Jackson (7.5) section 11-8, Omnes (16.13) chap. 4 2, or Bacry (10.31)
Chap. 7 5-d).
1235

CHAPTER XII THE FINE AND HYPERFINE STRUCTURE OF THE HYDROGEN ATOM
where()is a function whose integral is equal to 1, which only depends on, and which
takes on signicant values only inside a volume of the order of(~ )
3
, centered at=0.
If we neglect the variation of(r)over a distance of the order of~ , we can replace
(r+)by(r)in (B-13) and take(r)outside the integral, which is then equal to 1. (B-13)
reduces, in this case, to(r).
A better approximation consists of replacing, in (B-13),(r+)by its Taylor series
expansion in the neighborhood of=0. The zeroth-order term gives(r). The rst-order
term is zero because of the spherical symmetry of(). The second-order term involves the
second derivatives of the potential energy(r)at the pointrand quadratic functions of the
components of, weighted by()and integrated overd
3
. This leads to a result of the order
of
(~ )
2
(r)
It is therefore easy to accept the idea that this second-order term should be the Darwin term.
()Order of magnitude
Replacing()by
2
, we can write the Darwin term in the form:
2
~
2
8
22

1
=
2
~
2
2
22
(R) (B-14)
(we have used the expression for the Laplacian of 1/given by formula (61) of Ap-
pendix).
When we take the average value of (B-14) in an atomic state, we nd a contribution
equal to:
2
~
2
2
22
(0)
2
where(0)is the value of the wave function at the origin. The Darwin term therefore
aects only theelectrons, which are the only ones for which(0)= 0(cf.Chap.,
). The order of magnitude of(0)
2
can be obtained by taking the integral of
the square of the modulus of the wave function over a volume of the order of
3
0(where
0is the Bohr radius) to be equal to 1. Thus we obtain:
(0)
21
3
0
=
36
~
6
(B-15)
which gives the order of magnitude of the Darwin term:
2
~
2
2
22
(0)
2 2
8
~
44
=
24
(B-16)
Since0
22
, we again see that:
0
2
=
1
137
2
(B-17)
Thus, all the ne structure terms are about 10
4
times smaller than the non-relativistic
Hamiltonian of Chapter.
1236

B. ADDITIONAL TERMS IN THE HAMILTONIAN
B-2. Magnetic interactions related to proton spin: the hyperne Hamiltonian
B-2-a. Proton spin and magnetic moment
Thus far, we have considered the proton to be a physical point of massand
charge=. Actually, the proton, like the electron, is a spin 1/2 particle. We shall
denote byIthe corresponding spin observable.
With the spinIof the proton is associated a magnetic momentM. However, the
gyromagnetic ratio is dierent from that of the electron:
M= I~ (B-18)
whereis thenuclear Bohr magneton:
=
~
2
(B-19)
and the factor, for the proton, is equal to:5585. Because of the presence of
(the proton mass) in the denominator of (B-19),is close to 2 000 times smaller
than the Bohr magneton(recall that=~2). Although the angular momenta
of the proton and the electron are the same, nuclear magnetism, because of the mass
dierence, is much less important than electronic magnetism. The magnetic interactions
due to the proton spinIare therefore very weak.
B-2-b. The magnetic hyperne Hamiltonian
The electron moves, therefore, not only in the electrostatic eld of the proton,
but also in the magnetic eld created byM. When we introduce the correspond-
ing vector potential into the Schrödinger equation
3
, we nd that we must add to the
Hamiltonian (B-1) an additional series of terms for which the expression is (cf.Comple-
mentXII):
=
0
4
3
LM+
1
3
[3(Mn)(Mn)MM]
+
8
3
MM(R) (B-20)
Mis the spin magnetic moment of the electron, andnis the unit vector of the straight
line joining the proton to the electron (Fig.).
We shall see thatintroduces energy shifts which are small compared to those
created by. This is whyis called the hyperne structure Hamiltonian.
B-2-c. Interpretation of the various terms of
The rst term ofrepresents the interaction of the nuclear magnetic moment
Mwith the magnetic eld(04)L
3
created at the proton by the rotation of
the electronic charge.
The second term represents the dipole-dipole interaction between the electronic
and nuclear magnetic moments: the interaction of the magnetic moment of the electron
spin with the magnetic eld created byM(cf.ComplementXI) or vice versa.
3
Since the hyperne interactions are very small corrective terms, they can be found using the non-
relativistic Schrödinger equation.
1237

CHAPTER XII THE FINE AND HYPERFINE STRUCTURE OF THE HYDROGEN ATOMM
I
M
s
p
e
n
Figure 1: Relative disposition of the
magnetic momentsMandMof
the proton and the electron;nis the
unit vector on the line joining the
two particles.
Finally, the last term, also called Fermi's contact term, arises from the singularity
at= 0of the eld created by the magnetic moment of the proton. In reality, the proton
is not a point. It can be shown (cf.ComplementXII) that the magnetic eld inside
the proton does not have the same form as the one created outside byM(and which
enters into the dipole-dipole interaction). The contact term describes the interaction of
the magnetic moment of the electron spin with the magnetic eld inside the proton (the
delta function expresses the fact that this contact term exists, as its name indicates,
only when the wave functions of the electron and proton overlap).
B-2-d. Orders of magnitude
It can easily be shown that the order of magnitude of the rst two terms of
is:
2
~
2
3
0
4
=
2
~
2
2
1
3
(B-21)
By using (B-10), we see that these terms are about 2 000 times smaller than.
As for the last term of (B-20), it is also 2 000 times smaller than the Darwin term,
which also contains a(R)function.
C. The ne structure of the= 2level
C-1. Statement of the problem
C-1-a. Degeneracy of the = 2level
We saw in Chapter
quantum number. The2(= 2= 0)and2(= 2= 1)states therefore have
the same energy, equal to:
4
=
1
8
22
If the spins are ignored, the2subshell is composed of a single state, and the2subshell
of three distinct states which dier by their eigenvalue~of the componentof the
orbital angular momentumL(= 101). Because of the existence of electron and
proton spins, the degeneracy of the= 2level is higher than the value calculated in
1238

C. THE FINE STRUCTURE OF THE = 2LEVEL
Chapter. The componentsandof the two spins can each take on two values:
=12,=12. One possible orthonormal basis in the= 2level is given by
the kets:
= 2 ;= 0 ;= 0 ;=
1
2
;=
1
2
(C-1)
(2subshell, of dimension 4) and:
= 2 ;= 1 ;=10+1 ; =
1
2
;=
1
2
(C-2)
(2subshell, of dimension 12).
The= 2shell then has a total degeneracy equal to 16.
According to the results of Chapter), in order to calculate the eect of
a perturbationon the= 2level, it is necessary to diagonalize the1616matrix
representing the restriction ofto this level. The eigenvalues of this matrix are the rst
order corrections to the energy, and the corresponding eigenstates are the eigenstates of
the Hamiltonian to zeroth order.
C-1-b. The perturbation Hamiltonian
In all of this section, we shall assume that no external eld is applied to the
atom. The dierencebetween the exact Hamiltonianand the Hamiltonian0of
Chapter) contains ne structure terms, indicated in
= + + (C-3)
and hyperne structure terms, introduced in . We thus have:
= + (C-4)
Sinceis close to 2 000 times larger than(cf. ), we must obviously begin
by studying the eect of, before considering that of, on the= 2level. We shall
see that the= 16degeneracy of this level is partially removed by. The structure
which appears in this way is called the ne structure.
may then remove the remaining degeneracy of the ne structure levels and
cause a hyperne structure to appear inside each of these levels.
In this section (), we shall conne ourselves to the study of the ne structure
of the= 2level. The calculations can easily be generalized to other levels.
C-2. Matrix representation of the ne-structure Hamiltonianinside the= 2level
C-2-a. General properties
The properties of, as we shall see, enable us to show that the1616matrix
which represents it in the= 2level can be broken down into a series of square sub-
matrices of smaller dimensions. This will considerably simplify the determination of the
eigenvalues and eigenvectors of this matrix.
1239

CHAPTER XII THE FINE AND HYPERFINE STRUCTURE OF THE HYDROGEN ATOM
. does not act on the spin variables of the proton
We see from (B-1) that the ne structure terms do not depend onI. It follows
that the proton spin can be ignored in the study of the ne structure (afterwards, we
multiply by 2 all the degrees of degeneracy obtained). The dimension of the matrix to
be diagonalized therefore falls from 16 to 8.
. does not connect the2and2subshells
Let us rst prove thatL
2
commutes with . The operatorL
2
commutes with
the various components ofL, with(L
2
acts only on the angular variables), withP
2
[cf.
formula (A-16) of Chapter], and withS(L
2
does not act on the spin variables).L
2
therefore commutes with (which is proportional toP
4
), with (which depends
only on,L,S), and with(which depends only on).
The2and2states are eigenstates ofL
2
with dierent eigenvalues (0 and2~
2
).
Therefore,, which commutes withL
2
, has no matrix elements between a2state and
a2state. The88matrix representinginside the= 2level can be broken down,
consequently, into a22matrix relative to the2state and a66matrix relative to
the2state:
()=2=
2 2
2
2
0
0
Comment:
The preceding property can also be considered to be a consequence of the fact that
is even. Under a reection,Rchanges toR(=Rremains unchanged),
PtoP,LtoL, andStoS. It is then easy to see thatremains invariant.
therefore has no matrix elements between the2and2states, which are of
opposite parity (cf.ComplementII).
C-2-b. Matrix representation of in the2subshell
The dimension 2 of the2subspace is the result of the two possible values=
12of(since we are ignoringfor the moment).
and do not depend onS. The matrices which represent these two oper-
ators in the2subspace are therefore multiples of the unit matrix, with proportionality
coecients equal, respectively, to the purely orbital matrix elements:
= 2 ;= 0 ;= 0
P
4
8
32
= 2 ;= 0 ;= 0
1240

C. THE FINE STRUCTURE OF THE = 2LEVEL
and:
= 2 ;= 0 ;= 0
~
2
8
32
()= 2 ;= 0 ;= 0
Since we know the eigenfunctions of0, the calculation of these matrix elements presents
no theoretical diculty. We nd (cf.ComplementXII):
2
=
13
128
24
(C-5)
2
=
1
16
24
(C-6)
Finally, calculation of the matrix elements ofinvolves angular matrix ele-
ments of the form= 0 = 0 = 0 = 0, which are zero because of the
value= 0of the quantum number. Therefore:
2
= 0 (C-7)
Thus, under the eect of the ne structure terms, the2subshell is shifted as a
whole with respect to the position calculated in Chapter
5
24
128.
C-2-c. Matrix representation of in the2subshell
. and terms
The and terms commute with the various components ofL, sinceLacts
only on the angular variables and commutes withandP
2
(which depends on these
variables only throughL
2
;cf.chapter).Ltherefore commutes with and .
Consequently, and are scalar operators with respect to the orbital variables
(cf.ComplementVI, ). Since and do not act on the spin variables,
it follows that the matrices which representand inside the2subspace are
multiples of the unit matrix. The calculation of the proportionality coecient is given
in ComplementXIIand leads to:
2
=
7
384
24
(C-8)
2
= 0 (C-9)
The result (C-9) is due to the fact thatis proportional to(R)and can therefore
have a non-zero average value only in anstate (for1, the wave function is zero at
the origin).
. term
We must calculate the various matrix elements:
= 2 ;= 1 ;=
1
2
; ; ()LS= 2 ;= 1 ;=
1
2
; ;
(C-10)
1241

CHAPTER XII THE FINE AND HYPERFINE STRUCTURE OF THE HYDROGEN ATOM
with:
() =
2
2
22
1
3
(C-11)
If we use therrepresentation, we can separate the radial part of matrix element
(C-10) from the angular and spin parts. Thus we obtain:
2 = 1 ;=
1
2
;; LS= 1 ;=
1
2
; ; (C-12)
where2is a number, equal to the radial integral:
2=
2
2
22
0
1
3
21()
22
d (C-13)
Since we know the radial function21()of the2state, we can calculate2. We nd
(cf.ComplementXII):
2=
1
48~
2
24
(C-14)
The radial variables have therefore disappeared. According to (C-12), the problem
is reduced to the diagonalization of the operator2LS, which acts only on the angular
and spin variables.
To represent the operator2LSby a matrix, several dierent bases can be
chosen:
rst of all, the basis:
= 1;=
1
2
;; (C-15)
which we have used thus far and which is constructed from common eigenstates ofL
2
,
S
2
,,;
or, introducing the total angular momentum:
J=L+S (C-16)
the basis:
= 1;=
1
2
;; (C-17)
constructed from the eigenstates common toL
2
,S
2
,J
2
,. According to the results of
chapter, since= 1and= 12,can take on two values:= 1 + 12 = 32and
= 112 = 12. Furthermore, we know how to go from one basis to the other, thanks
to the Clebsch-Gordan coecients [formulas (36) of ComplementX].
We shall now show that the second basis (C-17) is better adapted than the rst
one to the problem which interests us here, since2LSis diagonal in the basis (C-17).
To see this, we square both sides of (C-16). We nd (LandScommute):
J
2
= (L+S)
2
=L
2
+S
2
+ 2LS (C-18)
1242

C. THE FINE STRUCTURE OF THE = 2LEVEL
which gives:
2LS=
1
2
2J
2
L
2
S
2
(C-19)
Each of the basis vectors (C-17) is an eigenstate ofL
2
,S
2
,J
2
; we thus have:
2LS= 1;=
1
2
;; =
1
2
2~
2
(+ 1)2
3
4
= 1;=
1
2
;;
(C-20)
We see from (C-20) that the eigenvalues of2LSdepend only onand not on
; they are equal to:
1
2
2
3
4
2
3
4
~
2
= 2~
2
=
1
48
24
(C-21)
for= 12, and:
1
2
2
15
4
2
3
4
~
2
= +
1
2
2~
2
=
1
96
24
(C-22)
for= 32.
The six-fold degeneracy of the2level is therefore partially removed by. We
obtain a four-fold degenerate level corresponding to= 32, and a two-fold degenerate
level corresponding to= 12. The(2+ 1)-fold degeneracy of eachstate is an
essential degeneracy related to the rotation invariance of.
Comments:
()In the2subspace(= 0= 12),can take on a single value,= 0+12 =
12.
()In the2subspace, and are represented by multiples of the unit
matrix. This property remains valid in any basis since the unit matrix is invariant
under a change of basis. The choice of basis (C-17), required by theterm, is
therefore also adapted to theand terms.
C-3. Results: the ne structure of the= 2level
C-3-a. Spectroscopic notation
In addition to the quantum numbers,(and), the preceding discussion in-
troduced the quantum numberon which the energy correction due to the spin-orbit
coupling term depends.
For the2level,= 12; for the2level,= 12or= 32. The level associated
with a set of values,,,is generally denoted by adding an indexto the symbol
representing the()subshell in spectroscopic notation (cf.Chap., ):
(C-23)
wherestands for the letterfor= 0,for= 1,for= 2,for= 3... Thus, the
= 2level of the hydrogen atom gives rise to the2
12,2
12and2
32levels.
1243

CHAPTER XII THE FINE AND HYPERFINE STRUCTURE OF THE HYDROGEN ATOM
C-3-b. Positions of the2
12,2
12and2
32levels
By regrouping the results of 2, we can now calculate the positions of the2
12,
2
12and2
32levels with respect to the unperturbed energy of the= 2level
calculated in Chapter
22
8.
According to the results of 2.b, the2
12level is lowered by a quantity equal to:
5
128
24
(C-24)
According to the results of 2.c, the2
12level is lowered by a quantity equal to:
7
384
1
48
24
=
5
128
24
(C-25)
Thus we see that the2
12and2
12levels have the same energy. According to the
theory presented here, this degeneracy must be considered to beaccidental, as opposed
to the essential(2+ 1)-fold degeneracy of eachlevel.
Finally, the2
32level is lowered by a quantity:
7
384
+
1
96
24
=
1
128
24
(C-26)
The preceding results are shown in Figure.
Comments:
()Only the spin-orbit coupling is responsible for the separation between the2
12
and2
32levels, sinceand shift the entire2level as a whole.
()The hydrogen atom can go from the2state to the1state by emitting a
Lyman photon (= 1 216

A). The material presented in this chapter shows
that, because of the spin-orbit coupling, the Lymanline actually contains two
neighboring lines
4
,2
12 1
12and2
32 1
12, separated by an energy
dierence equal to:
4
128
24
=
1
32
24
When they are observed with a sucient resolution, the lines of the hydrogen
spectrum therefore present a ne structure.
()We see in Figure have the same en-
ergy. This result is not merely true to rst order in: it remains valid to all
orders. The exact solution of the Dirac equation gives, for the energy of a level
characterized by the quantum numbers , the value:
=
2
1 +
2
1
2
+
(+ 12)
2 2
2
12
(C-27)
4
In the ground state,= 0and= 12, socan take on a single value= 12. therefore does
not remove the degeneracy of the1state, and there is only one ne structure level, the1
12level.
This is a special case, since the ground state is the only one for whichis necessarily zero. This is why
we have chosen here to study the excited= 2level.
1244

C. THE FINE STRUCTURE OF THE = 2LEVEL2s
1/2
m
e
c
2
α
4
2p
1/2
2p
3/2
128
5
m
e
c
2
α
4
n = 2
128
1
Figure 2: Fine structure of the= 2level of the hydrogen atom. Under the eect of
the ne structure Hamiltonian, the= 2level splits into three ne structure levels,
written2
12,2
12and2
32. We have indicated the algebraic values of the shifts,
calculated to rst order in. The shifts are the same for the2
12and2
12levels (a
result which remains valid, moreover, to all orders in). When we take into account
the quantum mechanical nature of the electromagnetic eld, we nd that the degeneracy
between the2
12and2
12levels is removed (the Lamb shift; see Figure).
We see that the energy depends only onand, and not on.
If we make a limited expansion of formula (C-27) in powers of, we obtain:
=
2
1
2
22
1
2
2
2
4
+ 12)
3
4
4
+ (C-28)
The rst term is the rest-mass-energy of the electron. The second term follows
from the theory of Chapter. The third term gives the correction to rst order
incalculated in this chapter.
()Even in the absence of an external eld and incident photons, a uctuating
electromagnetic eld must be considered to exist in space (cf.ComplementV,
). This phenomenon is related to the quantum mechanical nature of the
electromagnetic eld, which we have not taken into consideration here. The cou-
pling of the atom with these uctuations of the electromagnetic eld removes the
degeneracy between the2
12and2
12levels. The2
12level is raised with re-
spect to the2
12level by a quantity called the Lamb shift, which is of the order
of 1 060 MHz (Fig., page ).
The theoretical and experimental study of this phenomenon, which was
discovered in 1949, has been the object of a great deal of research, leading to the
development of modern quantum electrodynamics.
1245

CHAPTER XII THE FINE AND HYPERFINE STRUCTURE OF THE HYDROGEN ATOM
D. The hyperne structure of the= 1level
It would now seem logical to study the eect ofinside the ne structure levels2
12,
2
12and2
32, in order to see if the interactions related to the proton spinIcause a
hyperne structure to appear in each of these levels. However, sincedoes not remove
the degeneracy of the ground state1, it is simpler to study the eect ofon this
state. The results obtained in this special case can easily be generalized to the2
12,
2
12and2
32levels.
D-1. Statement of the problem
D-1-a. The degeneracy of the 1level
For the1level, there is no orbital degeneracy (= 0). On the other hand, the
andcomponents ofSandIcan still take on two values:=12and =12.
The degeneracy of the1level is therefore equal to 4, and a possible basis in this level is
given by the vectors:
= 1;= 0;= 0;=
1
2
;=
1
2
(D-1)
D-1-b. The 1level has no ne structure
We shall show that theterm does not remove the degeneracy of the1level.
The and terms do not act onand, and are represented in the1
subspace by multiples of the unit matrix. We nd (cf.ComplementXII):
1
=
5
8
24
(D-2)
1
=
1
2
24
(D-3)
Finally, calculation of the matrix elements of theterm involves the angular ma-
trix elements= 0 = 0 = 0 = 0, which are obviously zero(= 0);
therefore:
1
= 0 (D-4)
In conclusion,merely shifts the1level as a whole by a quantity equal to:
5
8
+
1
2
24
=
1
8
24
(D-5)
without splitting the level. This result could have been foreseen: since= 0and= 12,
can take on a single value,= 12, and the1level therefore gives rise to only one
ne structure level,2
12.
Since the Hamiltoniandoes not split the1level, we can now consider the
eect of the term. To do so, we must rst calculate the matrix which represents
in the1level.
1246

D. THE HYPERFINE STRUCTURE OF THE = 1LEVEL
D-2. Matrix representation of in the1level
D-2-a. Terms other than the contact term
Let us show that the rst two terms of[formula (B-20)] make no contribution.
Calculation of the contribution from the rst term,
0
4
3LM, leads to the
angular matrix elements= 0;= 0L= 0 = 0, which are obviously zero
(= 0).
Similarly, it can be shown (cf.ComplementXI, ) that the matrix elements of
the second term (the dipole-dipole interaction) are zero because of the spherical symmetry
of the1state.
D-2-b. The contact term
The matrix elements of the last term of (B-20), that is, of the contact term, are of
the form:
= 1;= 0;= 0;;
20
3
MM(R)= 1;= 0;= 0;;
(D-6)
If we go into therrepresentation, we can separate the orbital and spin parts
of this matrix element and put it in the form:
; IS ; (D-7)
whereis a number given by:
=
2
30
2
= 1;= 0;= 0(R)= 1;= 0;= 0
=
2
30
2
1
4
10(0)
2
=
4
3
24
1 +
3
1
~
2
(D-8)
We have used the expressions relatingMandMtoSandI[cf.B-18], as well as the
expression for the radial function10()given in
5
.
The orbital variables have therefore completely disappeared, and we are left with
a problem of two spin 1/2's,IandS, coupled by an interaction of the form:
IS (D-9)
whereis a constant.
5
The factor(1 + )
3
in (D-8) arises from the fact that it is the reduced masswhich enters
into10(0). It so happens that, for the contact term, it is correct to take the nuclear nite mass eect
into account in this way.
1247

CHAPTER XII THE FINE AND HYPERFINE STRUCTURE OF THE HYDROGEN ATOM
D-2-c. Eigenstates and eigenvalues of the contact term
To represent the operatorIS, we have thus far considered only the basis:
=
1
2
;=
1
2
;; (D-10)
formed by the eigenvectors common toS
2
,I
2
,,. We can also, by introducing the
total angular momentum
6
:
F=S+I (D-11)
use the basis:
=
1
2
;=
1
2
;; (D-12)
formed by the eigenstates common toS
2
,I
2
,F
2
,. Since== 12,can take on
only the two values= 0and= 1. We can easily pass from one basis to the other by
means of (B-22) and (B-23) of Chapter.
The basis is better adapted than the basis to the study of
the operatorIS, as this operator is represented in thebasis by a diagonal
matrix (for the sake of simplicity, we do not explicitly write= 12and= 12). This
is true, since we obtain, from (D-11):
IS=
2
F
2
I
2
S
2
(D-13)
It follows that the statesare eigenstates ofIS:
IS =
~
2
2
[(+ 1)(+ 1)(+ 1)] (D-14)
We see from (D-14) that the eigenvalues depend only on, and not on. They are
equal to:
~
2
2
2
3
4
3
4
=
~
2
4
(D-15)
for= 1, and:
~
2
2
0
3
4
3
4
=
3~
2
4
(D-16)
for= 0.
6
The total angular momentum is actuallyF=L+S+I, that is,F=J+I. However, for the ground
state, the orbital angular momentum is zero, soFreduces to (D-11).
1248

D. THE HYPERFINE STRUCTURE OF THE = 1LEVEL
The four-fold degeneracy of the1level is therefore partially removed by.
We obtain a three-fold degenerate= 1level and a non-degenerate= 0level. The
(2+ 1)-fold degeneracy of the= 1level is essential and is related to the invariance
of under a rotation of the total system.
D-3. The hyperne structure of the1level
D-3-a. Positions of the levels
Under the eect of, the energy of the1level is lowered by a quantity
24
8
with respect to the value
22
2calculated in Chapter. then splits the1
12
level into two hyperne levels, separated by an energy~
2
(Fig.).~
2
is often called
the hyperne structure of the ground state.1s
1s
1/2
m
e
c
2
α
4
8
1
1
4
3
4
+
–
ħ
2
ħ
2
ħ
2
F = 1
F = 0
Figure 3: The hyperne structure of the= 1level of the hydrogen atom. Under the
eect of, the= 1level undergoes a global shift equal to
24
8;can take
on a single value,= 12. When the hyperne couplingis taken into account, the
1
12level splits into two hyperne levels,= 1and= 0. The hyperne transition
= 1 = 0(the 21 cm line studied in radioastronomy) has a frequency which is
known experimentally to twelve signicant gures (thanks to the hydrogen maser).
1249

CHAPTER XII THE FINE AND HYPERFINE STRUCTURE OF THE HYDROGEN ATOM2p
3/2
ΔE
2s
1/2
2p
1/2
F = 2
F = 1
F = 1
F = 0
F = 1
F = 0
Figure 4: The hyperne structure of the= 2level of the hydrogen atom. The separation
Sbetween the two levels2
12and2
12is the Lamb shift, which is about ten times
smaller than the ne structure splittingseparating the two levels2
12and2
32
(S 1 0578MHz: 10 9691MHz). When the hyperne coupling is taken
into account, each level splits into two hyperne sublevels (the corresponding value of
the quantum numberis indicated on the right-hand side of the gure). The hyperne
splittings are equal to 23.7 MHz for the2
32level, 177.56 MHz for the2
12level and
59.19 MHz for the2
12level (for the sake of clarity, the gure is not drawn to scale).
Comment:
It could be found, similarly, thatsplits each of the ne structure levels2
12,
2
12and2
32into a series of hyperne levels, corresponding to all the values of
separated by one unit and included between+and . For the2
12
and2
12levels, we have= 12. Therefore,takes on the two values= 1
and= 0. For the2
32level,= 32, and, consequently, we have= 2and
= 1(cf.Fig.).
D-3-b. Importance of the hyperne structure of the1level
The hyperne structure of the ground state of the hydrogen atom is currently the
physical quantity which is known experimentally to the highest number of signicant
1250

E. THE ZEEMAN EFFECT OF THE 1GROUND STATE HYPERFINE STRUCTURE
gures. Expressed in Hz, it is equal to
7
:
~
2
= 1 420 405 7517670001 Hz (D-17)
Such a high degree of experimental accuracy was made possible by the development
of the hydrogen maser in 1963. The principle of such a device is, very schematically,
the following: hydrogen atoms, previously sorted (by a magnetic selection of the Stern-
Gerlach type) so as to choose those in the upper hyperne level= 1, are stored in a glass
cell (the arrangement is similar to the one shown in Figure IV). This
constitutes an amplifying medium for the hyperne frequency[(= 1)(= 0)].
If the cell is placed in a cavity tuned to the hyperne frequency, and if the losses of the
cavity are small enough for the gain to be greater than the losses, the system becomes
unstable and can oscillate: we obtain an atomic oscillator (a maser). The frequency of
the oscillator is very stable and of great spectral purity. Its measurement gives directly
the value of the hyperne splitting, expressed in Hz.
Note, nally, that hydrogen atoms in interstellar space are detected in radioastron-
omy by the radiation they emit spontaneously when they fall from the= 1hyperne
level to the= 0hyperne level of the ground state (this transition corresponds to a
wave length of 21 cm). Most of the information we possess about interstellar hydrogen
clouds is supplied by the study of this 21 cm line.
E. The Zeeman eect of the1ground state hyperne structure
E-1. Statement of the problem
E-1-a. The Zeeman Hamiltonian
We now assume the atom to be placed in a static uniform magnetic eldB0parallel
to. This eld interacts with the various magnetic moments present in the atom: the
orbital and spin magnetic moments of the electron,M=
2
LandM=S, and
the magnetic moment of the nucleus,M=
2
I[cf.expression (B-18)].
The Zeeman Hamiltonian which describes the interaction energy of the atom
with the eldB0can then be written:
=B0(M+M+M)
=0(+ 2) + (E-1)
where0(the Larmor angular frequency in the eldB0) andare dened by:
0=
2
0 (E-2)
=
2
0 (E-3)
Since , we clearly have:
0 (E-4)
7
The calculations presented in this chapter are obviously completely incapable of predicting all these
signicant gures. Moreover, even the most advanced theories cannot, at the present time, explain more
than the rst ve or six gures of (D-17).
1251

CHAPTER XII THE FINE AND HYPERFINE STRUCTURE OF THE HYDROGEN ATOM
Comment:
Rigorously,contains another term, which is quadratic in0(the diamagnetic
term). This term does not act on the electronic and nuclear spin variables and
merely shifts the1level as a whole, without modifying its Zeeman diagram, which
we shall study later. Moreover, it is much smaller than (E-1). Recall that a detailed
study of the eect of the diamagnetic term is presented in ComplementVII.
E-1-b. The perturbation seen by the 1level
In this section, we propose to study the eect ofon the1ground state of
the hydrogen atom (the case of the= 2level is slightly more complicated since, in a
zero magnetic eld, this level possesses both a ne and a hyperne structure, while the
= 1level has only a hyperne structure; the principle of the calculation is nevertheless
the same). Even with the strongest magnetic elds that can be produced in the labo-
ratory,is much smaller than the distance between the1level and the other levels;
consequently, its eect can be treated by perturbation theory.
The eect of a magnetic eld on an atomic energy level is called the Zeeman
eect. When0is plotted on the-axis and the energies of the various sublevels it
creates are plotted on the-axis, aZeeman diagramis obtained.
If0is suciently strong, the Zeeman Hamiltoniancan be of the same order
of magnitude as the hyperne Hamiltonian
8
, or even larger. On the other hand, if
0is very weak, . Therefore, in general it is not possible to establish the
relative importance ofand . To obtain the energies of the various sublevels,
(+ )must be diagonalized inside the= 1level.
We showed in to the= 1level could be put
in the formIS. Using expression (E-1) for, we see that we must also calculate
matrix elements of the form:
= 1;= 0;= 0;; 0(+ 2) + = 1;= 0;= 0;;
(E-5)
The contribution of0is zero, sinceand are zero. Since20+ acts
only on the spin variables, we can, for this term, separate the orbital part of the matrix
element:
= 1;= 0;= 0= 1;= 0;= 0= 1 (E-6)
from the spin part.
In conclusion, therefore, we must, ignoring the quantum numbers, diago-
nalize the operator:
IS+ 20+ (E-7)
which acts only on the spin degrees of freedom. To do so, we can use either the
basis or the basis.
According to (E-4), the last term of (E-7) is much smaller than the second one. To
simplify the discussion, we shall neglect the termfrom now on (it would be possible,
8
Recall thatshifts the1level as a whole: it therefore also shifts the Zeeman diagram as a whole.
1252

E. THE ZEEMAN EFFECT OF THE 1GROUND STATE HYPERFINE STRUCTURE
however, to take it into account
9
). The perturbation seen by the1level can therefore
be written, nally:
IS+ 20 (E-8)
E-1-c. Dierent domains of eld strength
By varying0, we can continuously modify the magnitude of the Zeeman term
20. We shall consider three dierent eld strengths, determined by the respective
orders of magnitude of the hyperne term and the Zeeman term:
()~0 ~
2
: weak elds
()~0 ~
2
: strong elds
()~0 ~
2
: intermediate elds
We shall later see that it is possible to diagonalize operator (E-8) exactly. However,
in order to give a particularly simple example of perturbation theory, we shall use a
slightly dierent method in cases()and(). In case(), we shall treat20like a
perturbation with respect toIS. On the other hand, in case(), we shall treatIS
like a perturbation with respect to20. The exact diagonalization of the set of two
operators, indispensable in case(), will allow us to check the preceding results.
E-2. The weak-eld Zeeman eect
The eigenstates and eigenvalues ofIShave already been determined ().
We therefore obtain two dierent levels: the three-fold degenerate level,
= 1;=10+1
of energy~
2
4, and the non-degenerate level,= 0;= 0, of energy3~
2
4.
Since we are treating20like a perturbation with respect toIS, we must now
separately diagonalize the two matrices representing20in the two levels,= 1and
= 0, corresponding to two distinct eigenvalues ofIS.
E-2-a. Matrix representation of in the basis
Since we shall need it later, we shall begin by writing the matrix which represents
in the basis (for the problem which concerns us here, it would suce to
write the two submatrices corresponding to the= 1and= 0subspaces).
By using formulas (B-22) and (B-23) of Chapter, we easily obtain:
= 1;= 1=
~
2
= 1;= 1
= 1;= 0=
~
2
= 0;= 0
= 1;=1=
~
2
= 1;=1
= 0;= 0=
~
2
= 1;= 0
(E-9)
9
This is what we do in ComplementXII, in which we study the hydrogen-like systems (muonium,
positronium) for which it is not possible to neglect the magnetic moment of one of the two particles.
1253

CHAPTER XII THE FINE AND HYPERFINE STRUCTURE OF THE HYDROGEN ATOM
which gives the following expression for the matrix representingin the basis
(the basis vectors are arranged in the order11,10,11,00):
() =
}
2
1 0 0 0
0 0 0 1
0 0 -1 00 1 0 0
(E-10)
Comment:
It is instructive to compare the preceding matrix with the one which representsin the
same basis:
() =}
1 0 0 0
0 0 0 0
0 0 -1 0
0 0 0 0
(E-11)
We see, rst of all, that the two matrices are not proportional: the()matrix is diagonal,
while the()one is not.
However, if we conne ourselves to the restrictions of the two matrices in the
= 1subspace [limited by the darker line in expressions (E-10) and (E-11)], we see
that they are proportional. Denoting by1the projector onto the= 1subspace (cf.
ComplementII), we have:
1 1=
1
2
1 1 (E-12)
It would be simple to show that the same relation exists betweenand on the one
hand, andand, on the other.
We have thus found a special case of the Wigner-Eckart theorem (ComplementX),
according to which, in a given eigensubspace of the total angular momentum, all the ma-
trices which represent vector operators are proportional. It is clear from this example that
this proportionality exists only for the restrictions of operators to a given eigensubspace
of the total angular momentum, and not for the operators themselves.
Moreover, the proportionality coecient 1/2 which appears in (E-12) can be ob-
tained immediately from the projection theorem. According to formula (30) of Comple-
mentX, this coecient is equal to:
SF=1
F
2
=1
=
(+ 1) +(+ 1)(+ 1)
2(+ 1)
(E-13)
Since== 12, (E-13) is indeed equal to 1/2.
E-2-b. Weak-eld eigenstates and eigenvalues
According to the results of a, the matrix which represents20in the= 1
level can be written:
~00 0
00 0
0 0 ~0
(E-14)
1254

E. THE ZEEMAN EFFECT OF THE 1GROUND STATE HYPERFINE STRUCTURE
In the= 0level, this matrix reduces to a number, equal to 0.
Since these two matrices are diagonal, we can immediately nd the weak-eld
eigenstates (to zeroth order in0) and the eigenvalues (to rst order in0):
Eigenstates Eigenvalues
= 1;= 1
~
2
4
+~0
= 1;= 0
~
2
4
+ 0
= 1;=1
~
2
4
~0
= 0;= 0 3
~
2
4
+ 0
(E-15)
In Figure, we have plotted~0on the-axis and the energies of the four Zeeman
sublevels on the-axis (Zeeman diagram). In a zero eld, we have the two hyperne levels,
= 1and= 0. When the eld0is turned on, the= 0 = 0sublevel, which
is not degenerate, starts horizontally; as for the= 1level, its three-fold degeneracy is
completely removed: three equidistant sublevels are obtained, varying linearly with~0
with slopes of+1,0,1respectively.E
m
F
+ 1
– 1
0
0
0
F

= 0
F

= 1
ħω
0
ħ
2
Figure 5: The weak-eld Zeeman diagram of
the1ground state of the hydrogen atom.
The hyperne= 1level splits into three
equidistant levels, each of which corresponds
to a well-dened value of the quantum num-
ber. The= 0level does not undergo
any shift to rst order in0.
The preceding treatment is valid as long as the dierence~0between two adjacent
Zeeman sublevels of the= 1level remains much smaller than the zero-eld dierence
between the= 1and= 0levels (the hyperne structure).
1255

CHAPTER XII THE FINE AND HYPERFINE STRUCTURE OF THE HYDROGEN ATOM
Comment:
The Wigner-Eckart theorem, mentioned above, makes it possible to show that, in a
given levelof the total angular momentum, the Zeeman Hamiltonian0(+ 2)is
represented by a matrix proportional to. Thus, we can write, denoting the projector
onto thelevel by:
[0(+ 2)]= 0 (E-16)
is called theLandé factorof thestate. In the case which concerns us here,=1= 1.
E-2-c. The Bohr frequencies involved in the evolution of and ; comparison with the vector
model of the atom
In this section, we shall determine the dierent Bohr frequencies which appear in the
evolution ofFandS, and show that certain aspects of the results obtained recall those
found by using the vector model of the atom (cf.ComplementX).
First of all, we shall briey review the predictions of the vector model of the atom (in
which the various angular momenta are treated like classical vectors) as far as the hyperne
coupling betweenIandSis concerned. In a zero eld,F=I+Sis a constant of the motion.
IandSprecess about their resultantFwith an angular velocity proportional to the coupling
constantbetweenIandS. If the system is, in addition, placed in a weak static eldB0
parallel to, onto the rapid precessional motion ofIandSaboutFis superposed a slow
precessional motion ofFabout (Larmor precession; Fig.).
is therefore a constant of the motion, whilehas a static part (the projection onto
of the component ofSparallel toF), and a part which is modulated by the hyperne precession
frequency (the projection ontoof the component ofSperpendicular toF, which precesses
aboutF).
Let us compare these semi-classical results with those of the quantum theory presented
earlier in this section. To do so, we must consider the time evolution of the average values
et . According to the discussion of , the average value()
of a physical quantitycontains a series of components which oscillate at the various Bohr
frequencies( )of the system. Also, a given Bohr frequency appears in()only if the
matrix element ofbetween the states corresponding to the two energies is dierent from zero.
In the problem which concerns us here, the eigenstates of the weak-eld Hamiltonian are the
states. Now consider the two matrices (E-10) and (E-11) which representandin
this basis. Sincehas only diagonal matrix elements, no Bohr frequency dierent from zero
can appear in (): is therefore static. On the other hand,has, not only diagonal
matrix elements (with which is associated a static component of), but also a non-diagonal
element between the= 1;= 0and= 0;= 0states, whose energy dierence is
~
2
, according to Table (E-15) (or Figure). It follows thathas, in addition to a static
component, a component modulated at the angular frequency~. This result recalls the one
obtained using the vector model of the atom
10
.
10
A parallel could also be established between the evolution of, , , , and that of the
projections of the vectorsFandSof Figure and . However, the motion ofFandS
does not coincide perfectly with that of the classical angular momenta. In particular, the modulus of
Sis not necessarily constant (in quantum mechanics,S
2
=S
2
); see discussion of ComplementX.
1256

E. THE ZEEMAN EFFECT OF THE 1GROUND STATE HYPERFINE STRUCTUREz
B
0
F
I
S
Figure 6: The motion ofS,IandFin the vector model of the atom.S,Iprecess rapidly
aboutFunder the eect of the hyperne coupling. In a weak eld,Fslowly precesses
aboutB0(Larmor precession).
Comment:
A relation can be established between perturbation theory and the vector model of the
atom. The inuence of a weak eld0on the= 1and= 0levels can be obtained
by retaining in the Zeeman Hamiltonian20only the matrix elements in the= 1
and= 0levels, forgetting the matrix element ofbetween= 1;= 0and
= 0;= 0. Proceeding in this way, we also forget the modulated component of
, which is proportional to this matrix element. We therefore keep only the component
ofSparallel toF.
Now, this is precisely what we do in the vector model of the atom when we want to
evaluate the interaction energy with the eldB0. In a weak eld,Fdoes precess about
B0much more slowly thanSdoes aboutF. The interaction ofB0with the component
ofSperpendicular toFtherefore has no eect, on the average; only the projection ofS
ontoFcounts. This is how, for example, the Landé factor is calculated.
E-3. The strong-eld Zeeman eect
We must now start by diagonalizing the Zeeman term.
1257

CHAPTER XII THE FINE AND HYPERFINE STRUCTURE OF THE HYDROGEN ATOM
E-3-a. Eigenstates and eigenvalues of the Zeeman term
This term is diagonal in the basis:
20 = 2~0 (E-17)
Since=12, the eigenvalues are equal to~0. Each of them is therefore two-fold
degenerate, because of the two possible values of. We therefore have
11
:
20+ = +~0+
20 =~0
(E-18)
E-3-b. Eects of the hyperne term considered as a perturbation
The corrections to rst order incan be obtained by diagonalizing the restrictions
of the operatorISto the two subspaces+ and corresponding to the
two dierent eigenvalues of20.
First of all, notice that, in each of these two subspaces, the two basis vectors++
and+ (or+and ) are also eigenvectors of, but do not correspond to
the same value of = +. Since the operatorIS=
2
2 2 2
commutes with, it has no matrix elements between the two states++and+,
or+and . The two matrices representingISin the two subspaces+
and are then diagonal, and their eigenvalues are simply the diagonal elements
; IS ; which can also be written, using the relation:
IS= +
1
2
(++ +) (E-19)
in the form:
IS
= =~
2
(E-20)
Finally, in a strong eld, the eigenstates (to zeroth order in) and the eigenvalues
(to rst order in) are:
Eigenstates Eigenvalues
++ ~0+
~
2
4
+ ~0
~
2
4
+ ~0
~
2
4
~0+
~
2
4
(E-21)
11
To simplify the notation, we shall often write instead of , where and are
equal to+or, depending on the signs ofand .
1258

E. THE ZEEMAN EFFECT OF THE 1GROUND STATE HYPERFINE STRUCTURE
In Figure, the solid-line curves on the right-hand side (for~0 ~
2
) represent
the strong-eld energy levels: we obtain two parallel straight lines of slope+1, separated
by an energy~
2
2, and two parallel straight lines of slope1, also separated by~
2
2.
The perturbation treatments presented in this section and the preceding one therefore
give the strong-eld asymptotes and the tangents at the origin of the energy levels.
Comment:
The strong-eld splitting~
2
2of the two states,++and+ or+and
, can be interpreted in the following way. We have seen that only the term–
– –
–+
+ +
+
F

= 1
E
ε
Sε
l
F

= 0
0
ħω
0
"ħ
2
Figure 7: The strong-eld Zeeman diagram of the1ground state of the hydrogen atom.
For each orientation of the electronic spin (= +or=), we obtain two par-
allel straight lines separated by an energy~
2
2, each one corresponding to a dierent
orientation of the proton spin (= +or=).
1259

CHAPTER XII THE FINE AND HYPERFINE STRUCTURE OF THE HYDROGEN ATOM
of expression (E-19) forISis involved in a strong eld, when the hyperne
coupling is treated like a perturbation of the Zeeman term. The total strong-eld
Hamiltonian (E-8) can therefore be written:
20+ = 2 0+
2
(E-22)
It is as if the electronic spin saw, in addition to the external eldB0, a smaller
internal eld, arising from the hyperne coupling betweenIandSand having
two possible values, depending on whether the nuclear spin points up or down.
This eld adds to or substracts fromB0and is responsible for the energy dierence
between++and+ or between+and .
E-3-c. The Bohr frequencies involved in the evolution of
In a strong eld, the Zeeman coupling ofSwithB0is more important than the
hyperne coupling ofSwithI. If we start by neglecting this hyperne coupling, the
vector model of the atom predicts thatSwill precess (very rapidly sinceB0is large)
about thedirection ofB0(Iremains motionless, since we have assumedto be
negligible).z
B
0
S
I
Figure 8: The motion ofSin the vector
model of the atom. In a strong eld,Spre-
cesses rapidly aboutB0(here we are neglect-
ing both the Zeeman coupling betweenIand
B0and the hyperne coupling betweenIand
S, so thatIremains motionless).
1260

E. THE ZEEMAN EFFECT OF THE 1GROUND STATE HYPERFINE STRUCTURE
Expression (E-19) for the hyperne coupling remains valid for classical vectors.
Because of the very rapid precession ofS, the terms+and oscillate very fast and
have, on the average, no eect, so that only the termcounts. The eect of the
hyperne coupling is therefore to add a small eld parallel toand proportional to
(cf.comment of the preceding section), which accelerates or slows down the precession
ofSabout, depending on the sign of. The vector model of the atom thus predicts
thatwill be static in a strong eld.
We shall show that quantum theory gives an analogous result for the average value
of the observableIn a strong eld, the well-dened energy states are, as we have
seen, the states . Now, in this basis, the operatorhas only diagonal matrix
elements. No non-zero Bohr frequency can therefore appear in, which, consequently,
is a static quantity
12
, unlike its weak-eld counterpart (cf. ).
E-4. The intermediate-eld Zeeman eect
E-4-a. The matrix which represents the total perturbation in the basis
The states are eigenstates of the operatorIS. The matrix which
represents this operator in the basis is therefore diagonal. The diagonal ele-
ments corresponding to= 1are equal to~
2
4, and those corresponding to= 0, to
3~
2
4. Furthermore, we have already written, in (E-10), the matrix representation
ofin the same basis. It is now very simple to write the matrix which represents the
total perturbation (E-8). Arranging the basis vectors in the order11,11,10,
00, we thus obtain:
~
2
4
+~0 0 0 0
0
~
2
4
~00 0
0 0
~
2
4
~0
0 0 ~0
3~
2
4
(E-23)
Comment:
and commute;20can therefore have non-zero matrix elements only
between two states with the same. Thus, we could have predicted all the
zeros of matrix (E-23).
12
The study of and presents no diculty. We nd two Bohr angular frequencies: one,
0+~2, slightly larger than0, and the other one,0 ~2, slightly smaller. They correspond to
the two possible orientations of the internal eld, produced by, which adds to the external eld0.
Similarly, we nd thatIprecesses about the internal eld produced by.
1261

CHAPTER XII THE FINE AND HYPERFINE STRUCTURE OF THE HYDROGEN ATOM
E-4-b. Energy values in an arbitrary eld
Matrix (E-23) can be broken into two11matrices and one22matrix. The
two11matrices immediately yield two eigenvalues:
1=
~
2
4
+~0
2=
~
2
4
~0
(E-24)
corresponding respectively to the state11(that is, the state++) and to the state
11(that is, the state). In Figure, the two straight lines of slopes ,+1and
1passing through the point whose ordinate is+~
2
4for a zero eld (for which the
perturbation theory treatment gave only the initial and asymptotic behavior) therefore
represent, for any0, two of the Zeeman sublevels.
The eigenvalue equation of the remaining22matrix can be written:
~
2
4
3~
2
4
~
22
0= 0 (E-25)
The two roots of this equation can easily be found:
3=
~
2
4
+
~
22
2
+~
22
0
(E-26)
4=
~
2
4
~
22
2
+~
22
0
(E-27)
When~0varies, the two points of abscissas~0and ordinates3and4follow the two
branches of a hyperbola (Fig.). The asymptotes of this hyperbola are the two straight
lines whose equation is=(~
2
4)~0, obtained in 3 above. The two turning
points of the hyperbola have abscissas of0= 0and ordinates of(~
2
4)~
2
2,
that is,~
2
4and3~
2
4. The tangents at both these points are horizontal. This is
in agreement with the results of 2 for the states= 1;= 0and= 0;= 0.
The preceding results are summarized in Figure, which is the Zeeman diagram
of the1ground state.
E-4-c. Partial hyperne decoupling
In a weak eld, the well-dened energy states are the states; in a strong
eld, the states ; in an intermediate eld, the eigenstates of matrix (E-23), which
are intermediate between the states and the states .
One thus moves continuously from a strong coupling betweenIandS(coupled
bases) to a total decoupling (uncoupled bases) via a partial coupling.
Comment:
An analogous phenomenon exists for the Zeeman ne structure eect. If, for
simplicity, we neglect, we know () that, in a zero eld, the eigenstates
of the Hamiltonianare the states corresponding to a strong coupling
1262

E. THE ZEEMAN EFFECT OF THE 1GROUND STATE HYPERFINE STRUCTUREE
+ 1
– 1
0
0
0
m
F
(F = 1)
(F = 0)
•
2
/4
–3•
2
/4
ħω
0
Figure 9: The Zeeman diagram (for an arbitrary eld) of the1ground state of the
hydrogen atom: remains a good quantum number for any value of the eld. We
obtain two straight lines, of opposite slopes, corresponding to the values,+1and1, of
, as well as a hyperbola whose two branches are associated with the two= 0levels.
Figures
levels shown in this diagram.
betweenLandS(the spin-orbit coupling). This property remains valid as long
as . If, on the other hand,0is strong enough to make ,
we nd that the eigenstates ofare the states corresponding to a total
decoupling ofLandS. The intermediate zone ( ) corresponds to a partial
1263

CHAPTER XII THE FINE AND HYPERFINE STRUCTURE OF THE HYDROGEN ATOM
coupling ofLandS. See, for example, ComplementXII, in which we study the
Zeeman eect of the2level (without takinginto account).
References and suggestions for further reading:
The hydrogen atom spectrum: Series (11.7), Bethe and Salpeter (11.10).
The Dirac equation: the subsection Relativistic quantum mechanics of section 2
of the bibliography and Messiah (1.17), Chap. XX, especially V and IV-27.
The ne structure of the= 2level and the Lamb shift: Lamb and Retherford
(3.11); Frisch (3.13); Series (11.7), Chaps. VI, VII and VIII.
The hyperne structure of the ground state: Crampton et al. (3.12).
The Zeeman eect and the vector model of the atom: Cagnac and Pebay-Peyroula
(11.2), Chap. XVII, 3E and 4C; Born (11.4), Chap. 6, 2.
Interstellar hydrogen: Roberts (11.17); Encrenaz (12.11), Chap. IV.
1264

COMPLEMENTS OF CHAPTER XII, READER'S GUIDE
AXII: THE MAGNETIC HYPERFINE HAMILTO-
NIAN
Derivation of the expression for the hyperne
Hamiltonian used in Chapter . Gives the
physical interpretation of the various terms
appearing in the Hamiltonian in particular the
contact term. Rather dicult.
BXII: CALCULATION OF THE AVERAGE VALUES
OF THE FINE-STRUCTURE HAMILTONIAN IN THE
1,2AND2STATES
The detailed calculations of certain radial
integrals appearing in the expression obtained
in Chapter
conceptually dicult.
CXII: THE HYPERFINE STRUCTURE AND THE
ZEEMAN EFFECT FOR MUONIUM AND POSITRO-
NIUM
Extension of the study of
Chapter
systems, muonium and positronium, already
presented in ComplementVII. Brief description
of experimental methods for studying these two
systems. Simple if the calculations of
of Chapter
DXII: THE INFLUENCE OF THE ELECTRON SPIN
ON THE ZEEMAN EFFECT OF THE HYDROGEN
RESONANCE LINE
Study of the eect of the electronic spin on
the frequencies and polarizations of the Zeeman
components of the resonance line of hydrogen. Im-
proves the results obtained in ComplementVII,
in which the electron spin was ignored (uses
certain results of that complement). Moderately
dicult.
EXII: THE STARK EFFECT OF THE HYDROGEN
ATOM
Study of the eect of a static electric eld on
the ground state (= 1) and of the rst excited
state (= 2) of the hydrogen atom (Stark eect).
Shows the importance for the Stark eect of the
existence of a degeneracy between two states of
dierent parities. Rather simple.
1265

THE MAGNETIC HYPERFINE HAMILTONIAN
Complement AXII
The magnetic hyperne Hamiltonian
1 Interaction of the electron with the scalar and vector po-
tentials created by the proton
2 The detailed form of the hyperne Hamiltonian
2-a Coupling of the magnetic moment of the proton with the or-
bital angular momentum of the electron
2-b Coupling with the electron spin
3 Conclusion: the hyperne-structure Hamiltonian
The aim of this complement is to justify the expression for the hyperne Hamil-
tonian given in Chapter B-20)]. As in that chapter, we shall conne our
reasoning to the hydrogen atom, which is composed of a single electron and a proton,
although most of the ideas remain valid for any atom. We have already said that the
origin of the hyperne Hamiltonian is the coupling between the electron and the elec-
tromagnetic eld created by the proton. We shall therefore callA(r)and(r)the
vector and scalar potentials associated with this electromagnetic eld. We shall begin
by considering the Hamiltonian of an electron subjected to these potentials.
1. Interaction of the electron with the scalar and vector potentials created by
the proton
LetRandPbe the position and momentum of the electron,S, its spin;, its mass;
and, its charge;=~2is the Bohr magneton.
The Hamiltonianof the electron in the eld of the proton can be written:
=
1
2
[PA(R)]
2
+ (R)2
S
~
rA(R) (1)
This operator is obtained from expression (B-46) of Chapter
a spinless particle) by adding to it the coupling energy between the magnetic moment
2S~associated with the spin and the magnetic eldrA(r).
We shall begin by studying the terms which, in (1), arise from the scalar potential
(r). According to ComplementX, this potential results from the superposition of
several contributions, each of them associated with one of the electric multipole moments
of the nucleus. For an arbitrary nucleus, we must consider:
()The total chargeof the nucleus (the moment of order= 0), which yields
a potential energy:
0(r) =0(r) =
2
40
(2)
1267

COMPLEMENT A XII
(with, for the proton,= 1). Now, the Hamiltonian which we chose in Chapter
the study of the hydrogen atom is precisely:
0=
P
2
2
+0(R) (3)
0(R)has therefore already been taken into account in the Hamiltonian0.
()The electric quadrupole moment(= 2)of the nucleus. The corresponding
potential adds to the potential0and yields a term of the hyperne Hamiltonian, called
the electric quadrupole term. The results of ComplementXenable us to write this term
without diculty. In the case of the hydrogen atom, it is zero, since the proton, which is
a spin 1/2 particle, has no electric quadrupole moment (cf. of ComplementX).
()The electric multipole moments of order= 4, 6, etc... which are theoretically
involved as long as2; for the proton, they are all zero.
Thus, for the hydrogen atom, potential (2) is really the potential seen by the
electron
1
. There is no need to add any corrections to it (by hydrogen atom, we mean
the electron-proton system, excluding isotopes such as deuterium: since the deuterium
nucleus has a spin= 1, we would have to take into account an electric quadrupole
hyperne Hamiltonian see comment () at the end of this complement).
Now let us consider the terms arising from the vector potentialA(r)in (1). We
denote byMthe magnetic dipole moment of the proton (which, for the same reason as
above, cannot have magnetic multipole moments of order1). We have:
A(r) =
0
4
M r
3
(4)
The hyperne Hamiltonian can now be obtained by retaining in (1) the terms which
are linear inA:
=
2
[PA(R) +A(R)P]2
S
~
rA(R) (5)
and by replacingAby expression (4) (sincealready makes a very small correction
to the energy levels of0, it is perfectly legitimate to ignore the second-order term, in
A
2
). This is what we shall do in the following section.
2. The detailed form of the hyperne Hamiltonian
2-a. Coupling of the magnetic moment of the proton with the orbital angular
momentum of the electron
First of all, we shall calculate the rst term of (5). Using (4), we have:
PA(R) +A(R)P=
0
4
P(M R)
1
3
+
1
3
(M R)P (6)
1
We are concerned here only with the potential outside the nucleus, where the multipole moment
expansion is possible. Inside the nucleus, we know that the potential does not have form (2). This causes
a shift in the atomic levels called the volume eect. This eect was studied in ComplementXI, and
we shall not take it into account here.
1268

THE MAGNETIC HYPERFINE HAMILTONIAN
We can apply the rules for a mixed vector product to vector operators as long as we do
not change the order of two non-commuting operators. The components ofMcommute
withRandP, so we have:
(M R)P= (RP)M=LM (7)
where:
L=RP (8)
is the orbital angular momentum of the electron. It can easily be shown that:
L
1
3
=0 (9)
(any function ofRis a scalar operator), so that:
1
3
(M R)P=
LM
3
(10)
Similarly:
P(M R)
1
3
=M(PR)
1
3
=
ML
3
(11)
since:
PR=L (12)
Thus, the rst term of (5) makes a contributionto which is equal to:
=
0
42
2
ML
3
=
0
4
2
M(L~)
3
(13)
This term corresponds to the coupling between the nuclear magnetic momentMand
the magnetic eld:
B=
0
4
L
3
created by the current loop associated with the rotation of the electron (cf.Fig.).
Comment:
The presence of the 1/
3
term in (13) might lead us to believe that there is a
singularity at the origin, and that certain matrix elements ofare innite. Ac-
tually, this is not the case. Consider the matrix element ,
where and are the stationary states of the hydrogen atom found
in Chapter. In therrepresentation, we have:
r = (r) =()() (14)
1269

COMPLEMENT A XIIL
B
L
M
I
v
q
Figure 1: Relative disposition of the mag-
netic momentMof the proton and the eld
Bcreated by the current loop associated
with the motion of the electron of charge
and velocityv(Bis antiparallel to the or-
bital angular momentumLof the electron).
with [cf.Chap., relation (A-28)]:
()
0
(15)
With the presence of the
2
dterm in the integration volume element taken into
account, the function to be integrated overbehaves at the origin like
++23
=
+ 1
. Furthermore, the presence of the Hermitian operatorLin (13) means that
the matrix element is zero whenoris zero. We then
have+ 2, and
+ 1
remains nite at the origin.
2-b. Coupling with the electron spin
We shall see that, for the last term of (5), the problems related to the singularity
at the origin of the vector potential (4) are important. This is why, in studying this
term, we shall choose a proton of nite size, letting its radius approach zero at the end
of the calculations. Furthermore, from a physical point of view, we now know that the
proton does possess a certain spatial extension and that its magnetism is spread over a
certain volume. However, the dimensions of the proton are much smaller than the Bohr
radius0. This justies our treating the proton as a point particle in the nal stage of
the calculation.
. The magnetic eld associated with the proton
Consider the proton to be a particle of radius0(Fig.), placed at the origin. The
distribution of magnetism inside the proton creates, at a distant point, a eldBwhich
can be calculated by attributing to the proton a magnetic momentMwhich we shall
choose parallel to. For 0, we obtain the components ofBfrom the curl of the
1270

THE MAGNETIC HYPERFINE HAMILTONIANB
i
M
I B
z
x
y
Figure 2: The magnetic eld created by the proton. Outside the proton, the eld is that
of a dipole. Inside, the eld depends on the exact distribution of the magnetism of the
proton, but we can, in a rst approximation, consider it to be uniform. The contact term
corresponds to the interaction between the spin magnetic moment of the electron and this
uniform eldBinside the proton.
vector potential written in (4):
=
0
4
3
5
=
0
4
3
5
=
0
4
3
2 2
5
(16)
Expressions (16), moreover, remain valid even ifis not very large compared to0. We
have already emphasized that the proton, since it is a spin 1/2 particle, has no magnetic
multipole moments of order1. The eld outside the proton is therefore a pure dipole
eld.
Inside the proton, the magnetic eld depends on the exact magnetic distribution.
We shall assume this eldBto be uniform (by symmetry, it must then be parallel to
Mand, therefore, to)
2
.
To calculate the eldBinside the proton, we shall write the equation stating that
the ux of the magnetic eld through a closed surface, bounded by theplane and
the upper hemisphere centered atand of innite radius, is zero. Since, as,B
decreases as1
3
, the ux through this hemisphere is zero. Therefore, if(0)denotes
the ux through a disk centered atof radius0in the plane, and(0), the ux
2
The following argument can be generalized to cases whereBvaries in a more complicated fashion
(see comment () at the end of this complement).
1271

COMPLEMENT A XII
through the rest of theplane, we have:
(0) + (0) = 0 (17)
Relations (16) enable us to calculate(0)easily, and we get:
(0) = 2
+
0
d
0
4
1
3
=
0
4
2
0
(18)
As for the ux(0)ofB, it is equal to:
(0) =
2
0 (19)
so that (17) and (18) yield:
=
0
4
2
3
0
(20)
Thus, we know the values of the eld created by the proton at all points in space. We
can now calculate the part ofrelated to the electron spinS.
. The magnetic dipole term
If we substitute (16) into the term2
S
~
rA, we obtain the operator:
dip
=
0
4
2
~
3
+ +
53
(21)
that is, taking into account the fact thatMis, by hypothesis, parallel to:
dip
=
0
4
2
~
1
3
SM 3
(SR)(MR)
2
(22)
This is the expression for the Hamiltonian of the dipole-dipole interaction between two
magnetic momentsMandM= 2S~(cf.ComplementXI, ).
Actually, expression (16) for the magnetic eld created by the proton is valid only
for 0, and (22) should be applied only to the part of the wave functions which
satises this condition. However, when we let0approach zero, expression (22) gives no
singularity at the origin; it is therefore valid in all space.
Consider the matrix element:
dip
(we are adding here the indicesandto the states considered above in order to label
the eigenvalues~2and~2of) and, in particular, the radial integral which corresponds
to it. At the origin, the function ofto be integrated behaves like
++23
=
+ 1
. Now,
according to condition (8-c) of ComplementXI, the non-zero matrix elements are obtained for
+ 2. There is therefore no divergence at the origin. In the limit where0 0, the
integral overbecomes an integral from 0 to innity, and expression (22) is valid in all space.
1272

THE MAGNETIC HYPERFINE HAMILTONIAN
. The contact term
We shall now substitute (20) into the last term of (5), so as to obtain the con-
tribution of the internal eld of the proton to. We then obtain an operator,
which we shall call the contact term, and whose matrix elements in the
representation are:
=
0
4
2
~
2
3
0 0
d
3
(r) (r)(23)
Let0approach zero. The integration volume,4
3
03, also approaches zero, and the
right-hand side of (23) becomes:
0
4
2
~
8
3
(r=0) (r=0) (24)
The contact term is therefore given by:
=
0
4
8
3
M
2S
~
(R) (25)
Although the volume containing an internal magnetic eld (20) approaches zero when
0 0, the value ofremains nite, since this internal eld approaches innity as
1/
3
0.
Comments:
()In (25), the function(R)of the operatorRis simply the projector:
(R) =r=0r=0 (26)
()The matrix element written in (23) is dierent from zero only if== 0.
This is a necessary condition for(r=0)and (r=0)to be non-zero
(cf.Chap., ). The contact term therefore exists only for thestates.
()In order to study, in , the coupling betweenMand the orbital angular
momentum of the electron, we assumed expression (4) forA(r)to be valid in
all space. This amounts to ignoring the fact that the eldBactually has the
form (20) inside the proton. We might wonder if this procedure is correct, or if
there is not also an orbital contact term in.
Actually, this is not the case. The term inPA+APwould lead, for
the eldB, to an operator proportional to:
BL=
0
4
M
2
3
0
(27)
Let us calculate the matrix element of such an operator in therepresen-
tation. The presence of the operatorrequires, as above,1. The radial
function to be integrated between 0 and0then behaves at the origin like
++2
and therefore goes to zero at least as rapidly as
4
. Despite the presence of the
1/
3
0term in (27), the integral between= 0and=0therefore goes to zero in
the limit where0 0.
1273

COMPLEMENT A XII
3. Conclusion: the hyperne-structure Hamiltonian
Now, let us take the sum of the operators,
dip
and . We use the fact that the
magnetic dipole momentMof the proton is proportional to its angular momentumI:
M=
I
~
(28)
(cf. ). We obtain:
=
0
4
2
~
2
IL
3
+ 3
(IR)(SR)
5
IS
3
+
8
3
IS(R) (29)
This operator acts both in the state space of the electron and in the state space of the
proton. It can be seen that this is indeed the operator introduced in Chaptercf.
(B-20)].
Comments:
()We will now discuss the generalization of formula (29) to the case of an atom
having a nuclear spin12.
First of all, if= 1, we have already seen that the nucleus can have an
electric quadrupole moment which adds a contribution to the potential0(r)
given by (2). An electric quadrupole hyperne term is therefore present in
the hyperne Hamiltonian, in addition to the magnetic dipole term (29).
Since an electrical interaction does not directly aect the electron spin, this
quadrupole term only acts on the orbital variables of the electrons.
If now 1, other nuclear electric or magnetic multipole moments can
exist, increasing in number asincreases. The electric moments give rise
to hyperne terms acting only on the orbital electron variables, while the
magnetic terms act on both the orbital and the spin variables. For elevated
values of, the hyperne Hamiltonian has therefore a complex structure. In
practice however, for the great majority of cases, one can limit the hyperne
Hamiltonian to magnetic dipole and electric quadrupole terms. This is due
to the fact that the multipole nuclear moments of an order superior to 2
make extremely small contributions to the hyperne atomic structures. These
contributions are therefore dicult to observe experimentally. This arises
essentially from the extremely small size of the nuclei compared to the spatial
extent0of the electronic wave functions.
()The simplifying hypothesis which we have made concerning the eldB(r)created
by the proton (a perfectly uniform eld within a sphere, a dipole eld outside)
is not essential. The form (25) of the magnetic dipole Hamiltonian remains valid
whenever the nuclear magnetism has an arbitrary repartition, giving rise to more
complicated internal eldsB(r)(assuming however that the spatial extent of the
nucleus is negligible compared to0;cf.the following comment). The argument is
actually a direct generalization of that given in this complement. Consider a sphere
centered at the origin, containing the nucleus and having a radius0.
1274

THE MAGNETIC HYPERFINE HAMILTONIAN
If=
1
2
, the eld outsidehas the form (16) and, sinceis very small
compared to0, its contribution leads to the terms (13) and (22). As for the
contribution of the eldB(r)inside, it depends only on the value at the origin
of the electronic wave functions and on the integral ofB(r)inside. Since the
ux ofB(r)across all closed surfaces is zero, the integral inof each component
ofB(r)can be transformed into an integral outside of, whereB(r)has the
form (16). A simple calculation will again give exactly expression (25) which is
therefore independent of the simplifying hypothesis that we have made.
If
1
2
, the nuclear contribution to the electromagnetic eld outside of
gives rise to the multipole hyperne Hamiltonian which we have discussed in
comment()above. On the other hand, one can easily show that the contribution
of the eld insidedoes not give rise to any new term: only the magnetic dipole
possesses a contact term.
()In all of the above, we have totally neglected the dimensions of the nucleus compared
to those of the electronic wave functions (we have taken the limit000). This
is obviously not always realistic, in particular for heavy atoms whose nuclei have a
relatively large spatial extension. If one studies these volume eects (keeping for
example several of the lower order terms in00), a series of new terms appears
in the electron-nucleus interaction Hamiltonian. We have already encountered this
type of eect in ComplementXIwhere we studied the eects of the radial distribu-
tion of the nuclear charge (nuclear multipole moments of order= 0). Analogous
phenomena occur concerning the spatial distribution of nuclear magnetism and lead
to modications of dierent terms of the hyperne Hamiltonian (29). In particu-
lar, a new term must be added to the contact term (25) when the electronic wave
functions vary signicantly within the nucleus. This new term is neither simply
proportional to(R), nor to the total magnetic moment of the nucleus. It depends
on the spatial distribution of the nuclear magnetism. From a practical point of
view, such a term is interesting since, using precise measurements of the hyperne
structure of heavy atoms, it permits obtaining information concerning the variation
of the magnetism within the corresponding nuclei.
References and suggestions for further reading:
The hyperne Hamiltonian including the electric quadrupole interaction term: Abragam
(14.1), Chap. VI; Kuhn (11.1), Chap. VI, B; Sobel'man (11.12), Chap. 6.
1275

COMPLEMENT B XII
Complement BXII
Calculation of the average values of the ne-structure Hamiltonian in
the1,2and2states
1 Calculation of 1,1
2
and1
3
. . . . . . . . . . . . .
2 The average values . . . . . . . . . . . . . . . . . . . .
3 The average values . . . . . . . . . . . . . . . . . . . . .
4 Calculation of the coecient 2associated with in
the2level
For the hydrogen atom, the ne-structure Hamiltonianis the sum of three
terms:
= + + (1)
studied in .
The aim of this complement is to give the calculation of the average values of these
three operators for the1,2and2states of the hydrogen atom, a calculation which
was omitted in Chapter
average values of1,1
2
and1
3
in these states.
1. Calculation of1,1
2
and1
3
The wave function associated with a stationary state of the hydrogen atom is (cf.
Chap., ):
(r) = ()() (2)
()is a spherical harmonic. The expressions for the radial functions()corre-
sponding to the1,2,2states are:
10() = 2(0)
32
e
0
20() = 2(20)
32
1
20
e
20
21() = (20)
32
(3)
12
0
e
20
(3)
where0is the Bohr radius:
0= 40
~
2
2
=
~
2
2
(4)
The are normalized with respect toand, so that the average valueof the
th power (whereis a positive or negative integer) of the operatorassociated with
1276

AVERAGE VALUES OF THE FINE-STRUCTURE HAMILTONIAN IN THE 1,2AND2STATES
=rin the state can be written
1
:
=
0
+2
()
2
d (5)
It therefore does not depend on. If (3) is substituted into (5), there appear integrals
of the form:
() =
0
e
r0
d (6)
whereandare integers. We shall assume here that0, that is, 2. An
integration by parts then yields directly:
() =
0
e
r0
0
+
0
0
1
e
r0
d
=
0
(1) (7)
Since, furthermore:
(0) =
0
e
r0
d=
0
(8)
we obtain, by recurrence:
() =!
0
+1
(9)
Now, let us apply this result to the average values to be determined. We obtain:
1 1=
4
3
00
e
2r0
d
=
4
3
0
(12) =
1
0
(10a)
1 2=
4
8
3
00
1
20
2
e
r0
d
=
1
2
3
0
(11)
1
0
(21) +
1
4
2
0
(31)
=
1
40
(10b)
1 2=
1
8
3
0
1
3
00
2
e
r0
d
=
1
24
5
0
(31) =
1
40
(10c)
1
Of course, this average value exists only for values ofwhich make integral (5) convergent.
1277

COMPLEMENT B XII
Similarly:
1
2
1=
4
3
0
(02) =
2
2
0
(11a)
1
2
2=
1
2
3
0
(01)
1
0
(11) +
1
4
2
0
(21)=
1
4
2
0
(11b)
1
2
2=
1
24
5
0
(21) =
1
12
2
0
(11c)
It is clear that the expression for the average value of1
3
is meaningless for the1and
2states [since integral (5) is divergent]. For the2state, it is equal to:
1
3
2=
1
24
5
0
(11) =
1
24
3
0
(12)
2. The average values
Let:
0=
P
2
2
+ (13)
be the Hamiltonian of the electron subjected to the Coulomb potential. We have:
P
4
= 4
2
[0 ]
2
(14a)
with:
=
2
(14b)
so that:
=
1
2
2
[0 ]
2
(15)
We shall take the average values of both sides of this expression in a state.
Since0andare Hermitian operators, we obtain:
=
1
2
2
()
2
+ 2
2
1 +
4
1
2
(16)
In this expression, we have set:
=
2
=
1
2
2
2 2
(17)
where:
=
2
~
(18)
is the ne-structure constant.
1278

AVERAGE VALUES OF THE FINE-STRUCTURE HAMILTONIAN IN THE 1,2AND2STATES
If we apply relation (16) to the case of the1state, we obtain, using (10a) and
(11a):
1=
1
2
2
1
4
424 2 2
2
0
+ 2
4
2
0
(19)
that is, since, according to (4) and (18),
2
0=
2 2
:
1=
1
2
4 2
1
4
1 + 2=
5
8
4 2
(20)
The same type of calculation, for the2state, leads to:
2=
1
2
4 2
1
8
2
2
1
8
1
4
+
1
4
=
13
128
4 2
(21)
and, for the2state, to:
2=
1
2
4 2
1
8
2
2
1
8
1
4
+
1
12
=
7
384
4 2
(22)
3. The average values
With (14b) and the fact that(1) =4(r)taken into account, the average value
of in the state can be written [see also formula (B-14) of Chapter]:
=
~
2
8
22
4
2
(r=0)
2
(23)
This expression goes to zero if(r=0) = 0, that is, if= 0. Therefore:
2= 0 (24a)
For the1and2levels, we obtain, using (2), (23) and the fact that
0
0= 1
4:
1=
~
2
8
22
2
10(0)
2
=
1
2
4 2
(24b)
as well as:
2=
~
2
8
22
2
20(0)
2
=
1
16
4 2
(24c)
4. Calculation of the coecient2associated with in the2level
In of Chapter, we dened the coecient:
2=
2
2
22
0
21()
2
d (25)
1279

COMPLEMENT B XII
According to (3):
2=
2
2
22
1
24
5
0
(11) (26)
Relation (9) then yields:
2=
2
2
22
1
24
3
0
=
1
48~
2
4 2
(27)
References:
Several radial integrals for hydrogen-like atoms are given in Bethe and Salpeter
(11.10).
1280

THE HYPERFINE STRUCTURE AND THE ZEEMAN EFFECT FOR MUONIUM AND POSITRONIUM
Complement CXII
The hyperne structure and the Zeeman eect for muonium and
positronium
1 The hyperne structure of the 1ground state
2 The Zeeman eect in the 1ground state
2-a The Zeeman Hamiltonian
2-b Stationary state energies
2-c The Zeeman diagram for muonium
2-d The Zeeman diagram for positronium
In ComplementVII, we studied some hydrogen-like systems, composed, like the
hydrogen atom, of two oppositely charged particles electrostatically attracted to each
other. Of all these systems, two are particularly interesting: muonium (composed of
an electron,, and a positive muon,
+
) and positronium (composed of an electron,
, and a positron,
+
). Their importance lies in the fact that the various particles
which come into play (the electron, the positron and the muon) are not directly aected
by strong interactions (while the proton is). The theoretical and experimental study of
muonium and positronium therefore permits a very direct test of the validity of quantum
electrodynamics.
Actually, a very precise information we now possess about these two systems comes
from the study of the hyperne structure of their1ground state [the optical lines joining
the1ground state to the various excited states have been observed for positronium;cf.
Ref. (11.25)]. This hyperne structure is the result, as in the case of the hydrogen atom,
of magnetic interactions between the spins of the two particles. We shall describe some
interesting features of the hyperne structure and the Zeeman eect for muonium and
positronium in this complement.
1. The hyperne structure of the1ground state
LetS1be the electron spin andS2, the spin of the other particle (the muon or the
positron, which are both spin l/2 particles). The degeneracy of the1ground state is
then, as for hydrogen, four-fold.
We can use stationary perturbation theory to study the eect on the1ground
state of the magnetic interactions betweenS1andS2. The calculation is analogous to
the one in . We are left with a problem of two spin 1/2's coupled by
an interaction of the form:
S1S2 (1)
whereis a constant which depends on the system under study. We shall denote by
,,the three values ofwhich correspond respectively to hydrogen, muonium
and positronium.
1281

COMPLEMENT C XII
It is easy to see that:
(2)
since the smaller the mass of particle (2), the larger its magnetic moment. Now the
positron is about 200 times lighter than the muon, which is close to 10 times lighter than
the proton.
Comment:
The theory of Chapter
structure of hydrogen, muonium and positronium. In particular, the hyperne Hamilto-
nian given in
particles (1) and (2). For example, the fact that the electron and the positron are an-
tiparticles of each other (they can annihilate to produce photons) is responsible for an
additional coupling between them which has no equivalent for hydrogen and muonium.
In addition, a series of corrections (relativistic, radiative, recoil eects, etc.) must be
taken into account. These are complicated to calculate and must be treated by quantum
electrodynamics. Finally, for hydrogen, nuclear corrections are also involved which are
related to the structure and polarizability of the proton. However, it can be shown that
the form (1) of the coupling betweenS1andS2remains valid, the constantbeing given
by an expression which is much more complicated than formula (D-8) of Chapter.
The hydrogen-like systems studied in this complement are important precisely because
they enable us to compare the theoretical value ofwith experimental results.
The eigenstates ofS1S2are the states, whereand are the
quantum numbers related to the total angular momentum:
F=S1+S2 (3)
As in the case of the hydrogen atom,can take on two values,= 1and= 0. The
two levels,= 1and= 0, have energies equal to~
2
4and3~
2
4, respectively.
Their separation~
2
gives the hyperne structure of the1ground state. Expressed in
MHz, this interval is equal to:
~
2
= 4 4633170021 MHz (4)
for muonium, and:
~
2
= 203 40312 MHz (5)
for positronium.
2. The Zeeman eect in the1ground state
2-a. The Zeeman Hamiltonian
If we apply a static eldB0parallel to, we must add, to the hyperne Hamil-
tonian (1), the Zeeman Hamiltonian which describes the coupling ofB0to the magnetic
1282

THE HYPERFINE STRUCTURE AND THE ZEEMAN EFFECT FOR MUONIUM AND POSITRONIUM
moments:
M1=1S1 (6)
and:
M2=2S2 (7)
of the two spins, with gyromagnetic ratios1and2. If we set:
1= 10 (8)
2= 20 (9)
this Zeeman Hamiltonian can be written:
11+22 (10)
In the case of hydrogen, the magnetic moment of the proton is much smaller than
that of the electron. We used this property in
coupling of the proton, compared to that of the electron
1
. Such an approximation is less
justied for muonium, since the magnetic moment of the muon is larger than that of
the proton. We shall therefore take both terms of (10) into account. For positronium,
furthermore, they are equally important: the electron and positron have equal masses
and opposite charges, so that:
1= 2 (positronium) (11)
or:
1= 2 (positronium) (12)
2-b. Stationary state energies
When 0is not zero, it is necessary, in order to nd the stationary state energies,
to diagonalize the matrix representing the total Hamiltonian:
S1S2+11+22 (13)
in an arbitrary orthonormal basis, for example, thebasis. A calculation which
is analogous to the one in
four basis vectors are arranged in the order11111000):
~
2
4
+
~
2
(1+2)
0
0
~
2
4
~
2
(1+2)
0 0
0 0
0 0
0 0
~
2
4
~
2
(1 2)
~2
(1 2)
3~
2
4
(14)
1
Recall that the gyromagnetic ratio of the electron spin is1= 2 ~(: the Bohr magneton).
Thus, if we set0= 0~(the Larmor angular frequency), the constant1dened by (8) is equal
to20(this is, furthermore, the notation used in ; to obtain the results of that section,
it therefore suces, in this complement, to replace1by20and2by 0).
1283

COMPLEMENT C XII
Matrix (14) can be broken down into two11submatrices and a22submatrix.
Two eigenvalues are therefore obvious:
1=
~
2
4
+
~
2
(1+2) (15)
2=
~
2
4
~
2
(1+2) (16)
They correspond, respectively, to the states11and11, which, moreover, coincide
with the states++and of the12basis of common eigenstates of1and
2. The other two eigenvalues can be obtained by diagonalizing the remaining22
submatrix. They are equal to:
3=
~
2
4
+
~
22
2
+
~
2
4
(1 2)
2
(17)
4=
~
2
4
~
22
2
+
~
2
4
(1 2)
2
(18)
In a weak eld, they correspond to the states10and00, respectively, and, in a
strong eld, to the states+ and+.
2-c. The Zeeman diagram for muonium
The only dierences with the results of
here, we are taking the Zeeman coupling of particle (2) into account. These dierences
appear only in a suciently strong eld.
Let us therefore consider the form taken on by the energies3and4when
~(1 2) ~
2
. In this case:1 2
3
~
2
4
+
~
2
(1 2) (19)
4
~
2
4
~
2
(1 2) (20)
Now, compare (19) with (15) and (20) with (16). We see that, in a strong eld, the
energy levels are no longer represented by pairs of parallel lines, as was the case in
of Chapter. The slopes of the asymptotes of the1and3levels are, respectively,
~
2
(1+2)and
~
2
(1 2); those of the2and4levels,
~
2
(1+2)and
~
2
(1 2).
Since the two particles (1) and (2) have opposite charges,1and2have opposite signs.
Consequently, in a suciently strong eld, the3level (which then corresponds to the
+ state) moves above the1level (the++state), since its slope,
~
2
(1 2)is
greater than
~
2
(1+2).
The distance between the1and 3levels therefore varies in the following way with
respect to0(cf.Fig.): starting from 0, it increases to a maximum for the value of0which
makes the derivative of:
1 3=
~
2
2
+(0) (21)
1284

THE HYPERFINE STRUCTURE AND THE ZEEMAN EFFECT FOR MUONIUM AND POSITRONIUME
F = 1
F = 0
0
| –, +
| –, –
| +, –
| +, +
B
0
E
4
E
2
E
3
E
1
Figure 1: The Zeeman diagram for the1ground state of muonium. Since we are not
neglecting here the Zeeman coupling between the magnetic moment of the muon and the
static eldB0, the two straight lines (which correspond, in a strong eld, to the same
electron spin orientation but dierent muon spin orientations) are no longer parallel,
as was the case for hydrogen (in the Zeeman diagram of Figure , the
Larmor angular frequencyof the proton was neglected). For the same value of the
static eld0, the splitting between the1and3levels is maximal and that between the
2and4levels is minimal. The arrows represent the transitions studied experimentally
for this value of the eldB0.
1285

COMPLEMENT C XII
equal to zero, with:
(0) =
~
2
(1+2)0
~
22
2
+
~
22
0
4
(1 2)
2
(22)
The distance then goes to zero again, and nally increases without bound. As for the distance
between the2and 4levels, it starts with the value~
2
, decreases to a minimum for the
value of0which makes the derivative of:
2 4=
~
2
2
(0) (23)
equal to zero and then increases without bound.
Since it is the same function(0)that appears in (21) and (23), we can show that, for
the same value of0[the one which makes the derivative of(0)go to zero], the distances
between the1and3levels and between the2and4levels are either maximal or minimal.
This property was recently used to improve the accuracy of experimental determinations of the
hyperne structure of muonium.
By stopping polarized muons (for example, in the+state) in a rare gas target, one
can prepare, in a strong eld, muonium atoms which will be found preferentially in the++
and +states. If we then apply simultaneously two radio frequency elds whose frequencies
are close to(1 3)~and(2 4)~, we induce resonant transitions from++to+
and from +to (arrows in Figure). It is these transitions which are detected
experimentally, since they correspond to a ip of the muon spin which is revealed by a change in
the anisotropy of the positrons emitted during the-decay of the muons. If we are operating in
a eld0such that the derivative of(0)is zero, the inhomogeneities of the static eld, which
may exist from one point to another of the cell containing the rare gas, are not troublesome,
since the resonant frequencies of muonium,(1 3)and(2 4), are not aected, to
rst order, by a variation of0[ref. (11.24)].
Comment:
For the ground state of the hydrogen atom, we obtain a Zeeman diagram analogous
to the one in Figure
the proton spin and the eldB0.
2-d. The Zeeman diagram for positronium
If we set1= 2(this property is a direct consequence of the fact that the
positron is the antiparticle of the electron) in (15) and (16), we see that the1and2
levels are independent of0:
1=2=
~
2
4
(24)
On the other hand, we obtain from (17) and (18):
3=
~
2
4
+
~
22
2
+~
22
1
2
0
(25)
4=
~
2
4
~
22
2
+~
22
1
2
0
(26)
1286

THE HYPERFINE STRUCTURE AND THE ZEEMAN EFFECT FOR MUONIUM AND POSITRONIUME
E
3
E
1
E
2
E
4
B
0
0
F = 1
F = 0
Figure 2: The Zeeman diagram for the1ground state of positronium. As in the cases
of hydrogen and muonium, this diagram is composed of one hyperbola and two straight
lines. However, since the gyromagnetic ratios of the electron and positron are equal
and opposite, the two straight lines have a zero slope and, consequently, are superposed
(in the two corresponding states, with energy1and2, the total magnetic moment
is zero, since the electron and positron spins are parallel). The arrow represents the
experimentally studied transition.
1287

COMPLEMENT C XII
The Zeeman diagram for positronium therefore has the form shown in Figure. It is
composed of two superposed straight lines parallel to the0axis and one hyperbola.
Actually, positronium is not stable. It decays by emitting photons. In a zero eld,
it can be shown by symmetry considerations that the= 0state (the singlet spin state,
or parapositronium) decays by emitting two photons. Its half-life is of the order of0
125 10
10
s. On the other hand, the= 1state (the triplet spin state, or orthopositronium)
can decay only by emitting three photons (since the two-photon transition is forbidden). This
process is much less probable, and the half-life of the triplet is much longer, on the order of
114 10
7
s.
When a static eld is applied, the1and2levels retain the same lifetimes since the
corresponding eigenstates do not depend on0. On the other hand, the10state is mixed
with the00state, and vice versa. Calculations analogous to those of ComplementIVshow
that the lifetime of the3level is reduced relative to its zero-eld value1(that of the4level
is increased relative to the value0). The positronium atoms in the3state then have a certain
probability of decaying by emission of two photons.
This inequality of the lifetimes of the three states of energies1,2,3when 0is
non-zero is the basis of the methods for determining the hyperne structure of positronium.
Formation of positronium atoms by positron capture by an electron generally populates the
four states of energies1,2,3,4equally. In a non-zero eld0, the two states1and2
decay less rapidly than the3state, so that in the stationary state, they are more populated. If
we then apply a radiofrequency eld oscillating at the frequency(3 1)= (3 2), we
induce resonant transitions from the1and2states to the3state (the arrow of Figure).
This increases the decay rate via two-photon emission, which permits the detection of resonance
when (with xed0) we vary the frequency of the oscillating eld. Determination of3 1
for a given value of0then allows us to nd the constantby using (24) and (25).
In a zero eld, resonant transitions could also be induced between the unequally populated
= 1and= 0levels. However, the corresponding resonant frequency, given by (5), is high
and not easily produced experimentally. This is why one generally prefers to use the low
frequency transition represented by the arrow of Figure.
References and suggestions for further reading:
See the subsection Exotic atoms of section 11 of the bibliography.
The annihilation of positronium is discussed in Feynman III (1.2), 18-3.
1288

ZEEMAN EFFECT OF THE HYDROGEN LYMAN LINE
Complement DXII
The inuence of the electronic spin on the Zeeman eect of the
hydrogen resonance line
1 Introduction
2 The Zeeman diagrams of the 1and2levels
3 The Zeeman diagram of the 2level
4 The Zeeman eect of the resonance line
4-a Statement of the problem
4-b The weak-eld Zeeman components
4-c The strong-eld Zeeman components
1. Introduction
The conclusions of ComplementVIIrelative to the Zeeman eect for the resonance line
of the hydrogen atom spectrum (the1 2transition) must be modied to take into
account the electron spin and the associated magnetic interactions. This is what we shall
do in this complement, using the results obtained in Chapter.
To simplify the discussion, we shall neglect eects related to nuclear spin (which
are much smaller than those related to the electron spin). Therefore, we shall not take
the hyperne coupling(chap., ) into account, choosing the Hamiltonian
in the form:
=0+ + (1)
0is the electrostatic Hamiltonian studied in Chapter),, the sum of the
ne structure terms (chap., ):
= + + (2)
and , the Zeeman Hamiltonian (chap., ) describing the interaction of the
atom with a magnetic eldB0parallel to:
=0(+ 2) (3)
where the Larmor angular frequency0is given by:
0=
2
0 (4)
[we shall neglectrelative to0; see formula (E-4) of Chapter].
We shall determine the eigenvalues and eigenvectors ofby using a method anal-
ogous to that of : we shall treatand like perturbations of0.
Although they have the same unperturbed energy, the2and2levels can be studied
1289

COMPLEMENT D XII
separately since they are connected neither by(chap., ) nor by.
In this complement, the magnetic eldB0will be called weak or strong, depending on
whether is small or large compared to. Note that the magnetic elds considered
here to be weak are those for whichis small compared tobut large compared
to which we have neglected. These weak elds are therefore much stronger than
those treated in .
Once the eigenstates and eigenvalues ofhave been obtained, it is possible to
study the evolution of the average values of the three components of the electric dipole
moment of the atom. Since an analogous calculation was performed in detail in Com-
plementVII, we shall not repeat it. We shall merely indicate, for weak elds and for
strong elds, the frequencies and polarization states of the various Zeeman components
of the resonance line of hydrogen (the Lymanline).
2. The Zeeman diagrams of the1and2levels
We saw in shifts the1level as a whole and gives rise
to only one ne-structure level,1
12. The same is true for the2level, which becomes
2
12. In each of these two levels, we can choose a basis:
;= 0;= 0;=
1
2
;=
1
2
(5)
of eigenvectors common to0,L
2
,,,(the notation is identical to that of Chap-
ter; sincedoes not act on the proton spin, we shall ignorein all that follows).
The vectors (5) are obviously eigenvectors ofwith eigenvalues2~0. Thus,
each1
12or2
12level splits, in a eld0, into two Zeeman sublevels of energies:
(;= 0;= 0;) =(
12) + 2~0 (6)
where(
12)is the zero-eld energy of the
12level, calculated in
D-1-b . The Zeeman diagram of the1
12level (as well as the one for
the2
12level) is therefore composed of two straight lines of slopes+1 and 1 (Fig.),
corresponding, respectively, to the two possible orientations of the spin relative toB0
(= +12and =12).
Comparison of Figure
are doing here, the eects related to nuclear spin amounts to considering eldsB0which
are so large that . We are then in the asymptotic region of the diagram of
Figure , where we can ignore the splitting of the energy levels due to
the proton spin and hyperne coupling.
3. The Zeeman diagram of the2level
In the six-dimensional2subspace, we can choose one of the two bases:
= 2;= 1;; (7)
or:
= 2;= 1;; (8)
1290

ZEEMAN EFFECT OF THE HYDROGEN LYMAN LINEE
E(l s
1/2
)
0
m
S
= +
1
2
m
S
= –
1
2
ħω
0
Figure 1: The Zeeman diagram of the1
12
level when the hyperne couplingis ne-
glected. The ordinate of the point at which
the two levels=12cross is the en-
ergy of the1
12level (i.e., the eigenvalue
of0, corrected for the global shift
produced by the ne-structure Hamiltonian
). Figure
of the modications of this diagram produced
by .
adapted, respectively, to the individual angular momentaLandSand to the total
angular momentumJ=L+S[cf.(36a) and (36b) of ComplementX].
The terms and which appear in expression (2) forshift the2level as
a whole. Therefore, to study the Zeeman diagram of the2level, we simply diagonalize
the66matrix which represents + in either one of the two bases, (7) or
(8). Actually, sinceand =2LSboth commute with = +, this
66matrix can be broken down into as many submatrices as there are distinct values
of. Thus, there appear two one-dimensional submatrices (corresponding respectively
to = +32and =32) and two two-dimensional submatrices (corresponding
respectively to= +12and =12). The calculation of the eigenvalues and
associated eigenvectors (which is very much like that of ) presents
no diculties and leads to the Zeeman diagram shown in Figure. This diagram is
composed of two straight lines and four hyperbolic branches.
In a zero eld, the energies depend only on. We obtain the two ne-structure
levels,2
32and2
12, already studied in , whose energies are equal
to:
(2
32) =
~
(2) +
1
2
2~
2
(9)
(2
12) =
~
(2) 2~
2
(10)
~
(2)is the2level energy(2)corrected for the global shift due toand
[cf.expressions (C-8) and (C-9) of Chapter].2is the constant which appears in
the restriction2LSof to the2level [cf.expression (C-13) of Chapter].
In weak magnetic elds ( ), the slope of the energy levels can also be
obtained by treatinglike a perturbation of. It is then necessary to diagonalize the
1291

COMPLEMENT D XIIħω
0
E
E(2p
3/2
)
E(2p
1/2
)
0
E(2p)
Figure 2: The Zeeman diagram of the2level when the hyperne couplingis ne-
glected. In a zero eld, we nd the ne-structure levels,2
12and2
32. The Zeeman
diagram is composed of two straight lines and two hyperbolas (for which the asymptotes
are shown in dashed lines). The hyperne couplingwould signicantly modify this
diagram only in the neighborhood of0= 0.
~
(2)is the2level energy (the eigenvalue
4of0) corrected for the global shift produced by+ .
44and22matrices representingin the2
32and2
12levels. Calculations anal-
ogous to those of
proportional to those which represent0in the same subspaces. The proportionality
coecients, called Landé factors (cf.ComplementX, ), are equal, respectively,
1292

ZEEMAN EFFECT OF THE HYDROGEN LYMAN LINE
to
1
:
(2
32) =
4
3
(11)
(2
12) =
2
3
(12)
In weak elds, each ne-structure level therefore splits into2+ 1equidistant Zeeman
sublevels. The eigenstates are the states of the coupled basis, (8), corresponding to
the eigenvalues:
( ) =(2) + (2)~0 (13)
where the(2)are given by expressions (9) and (10).
In strong elds ( ), we can, on the other hand, treat=2LS
like a perturbation of, which is diagonal in basis (7). As in ,
it can easily be shown that only the diagonal elements of2LSare involved when the
corrections are calculated to rst order in. Thus, we nd that in strong elds, the
eigenstates are the states of the decoupled basis, (7), and the corresponding eigenvalues
are:
( ) =
~
(2) + (+ 2)~0+ ~
2
2 (14)
Formula (14) gives the asymptotes of the diagram of Figure.
As the magnetic eld0increases, we pass continuously from basis (8) to basis (7).
The magnetic eld gradually decouples the orbital angular momentum and the spin. This
situation is the analogue of the one studied in , in which the angular
momentaSandIwere coupled or decoupled, depending on the relative importance of
the hyperne and Zeeman terms.
4. The Zeeman eect of the resonance line
4-a. Statement of the problem
Arguments of the same type as those of VII(see, in particu-
lar, the comment at the end of that complement) show that the optical transition between
a2Zeeman sublevel and a1Zeeman sublevel is possible only if the matrix element
of the electric dipole operatorRbetween these two states is dierent from zero
2
. In
addition, depending on whether it is the(+),( )oroperator which has
a non-zero matrix element between the two Zeeman sublevels under consideration, the
polarization state of the emitted light is
+
,or. Therefore, we use the previously
determined eigenvectors and eigenvalues ofin order to obtain the frequencies of the
various Zeeman components of the hydrogen resonance line and their polarization states.
Comment:
1
These Landé factors can be calculated directly from formula (43) of ComplementX.
2
The electric dipole, since it is an odd operator, has no matrix elements between the1and2states,
which are both even. This is why we are ignoring the2states here.
1293

COMPLEMENT D XIIE(2p
3/2
)
E
4
3/2 3/2
3/2 1/2
3/2 1/2–
–3/2 3/2
3
J m
J
E(2p
1/2
)
E(1s
1/2
)
ħω
0
2 1/2 1/2
1/2
1/2 1/2
1/2 – 1/2
1/2–3
ħω
0
2ħω
0
σ
+
σ
+
σ
+
σ
–
σ
–
σ
–
πππ π
Figure 3: The disposition, in a weak eld, of the Zeeman sublevels arising from the ne-
structure levels,1
12,2
12,2
32(whose zero-eld energies are marked on the vertical
energy scale). On the right-hand side of the gure are indicated the splittings between
adjacent Zeeman sublevels (for greater clarity, these splittings have been exaggerated with
respect to the ne-structure splitting which separates the2
12and2
32levels), as well
as the values of the quantum numbersand associated with each sublevel. The arrows
indicate the Zeeman components of the resonance line, each of which has a well-dened
polarization,
+
,or.
1294

ZEEMAN EFFECT OF THE HYDROGEN LYMAN LINE
The(+),( )and operators act only on the orbital part of the wave
function and causeto vary, respectively, by+1,1 and 0 (cf.ComplementVII,
);is not aected. Since= + is a good quantum number (whatever
the strength of the eld0), = +1transitions have a
+
polarization; =1
transitions, apolarization; and = 0transitions, apolarization.
4-b. The weak-eld Zeeman components
Figure
from the1
12,2
12and2
32levels, obtained from expressions (6), (13), (11) and (12).
The vertical arrows indicate the various Zeeman components of the resonance line. The
polarization is
+
,or, depending on whether = +11or 0.
Figure
relative to the zero-eld positions of the lines. The result diers notably from that of
ComplementVII(see Figure
perpendicular toB0, we had three equidistant components of polarization
+
,,,
separated by a frequency dierence02.σ
–
σ
+
σ
+
σ
+
v
σ
–
σ
–
π π
2π
a
b
ω
0
ξ
2p
ħ
π π
4π
3
Figure 4: Frequencies of the various Zeeman components of the hydrogen resonance line.
a) In a zero eld: two lines are observed, separated by the ne-structure splitting32~4
(2is the spin-orbit coupling constant of the2level) and corresponding respectively
to the transitions2
32 1
12(the line on the right-hand side of the gure) and
2
121
12(the line on the left-hand side).
b) In a weak eld0: each line splits into a series of Zeeman components whose polar-
izations are indicated;02is the Larmor frequency in the eld0.
4-c. The strong-eld Zeeman components
Figure
1and2levels [see expressions (6) and (14)]. To rst order in, the degeneracy
between the states=1 = 12and = 1 =12is not removed. The
vertical arrows indicate the Zeeman components of the resonance line. The polarization
is
+
,or, depending on whether = +11or 0 (recall that in an electric
dipole transition, the quantum numberis not aected).
1295

COMPLEMENT D XIIE m
L
m
S
E(1s
1/2
)
0
1
2
–
E(2p)
0
1
2
– 1
1
2
–
0
1
2
–
1
1
2
–
0
1
2
1
1
2
–1
1
2
– ħω
0
ħω
0
2ħω
0
ħω
0
– ħω
0
– 2ħω
0
+
ħ
2
ħ
2
+
2
2
ħ
2
2
ξ
2p
ξ
2p
ξ
2p
0 –
σ
–
σ
–
σ
+
σ
+
π π
Figure 5: The disposition, in a strong eld (decoupled ne structure), of the Zeeman
sublevels arising from the1and2levels. On the right-hand side of the gure are
indicated the values of the quantum numbersand associated with each Zeeman
sublevel, as well as the corresponding energy, given relative to(1
12)or
~
(2). The
vertical arrows indicate the Zeeman components of the resonance line.
The corresponding optical spectrum is shown in Figure. The twotransitions
have the same frequency (cf.Fig.). On the other hand, there is a small splitting,
~22, between the frequencies of the two
+
transitions and between those of the two
transitions. The mean distance between the
+
doublet and theline (or between
theline and thedoublet) is equal to02. The spectrum of Figure
1296

ZEEMAN EFFECT OF THE HYDROGEN LYMAN LINEω
0
2π
2π 2π
ω
0
2π
σ
–
σ
–
σ
+
σ
+
π π
ħξ
2p
ħξ
2p
Figure 6: The strong-eld positions of the Zeeman components of the hydrogen resonance
line. Aside from the splitting of the
+
andlines, this spectrum is identical to the one
obtained in ComplementVII, where the eects related to electron spin were ignored.
similar to that of Figure VII. Furthermore, the splitting of the
+
and lines, due to the existence of the electron spin, is easy to understand.
In strong elds,LandSare decoupled. Since the1 2transition is an electric
dipole transition, only the orbital angular momentumLof the electron is aected by the
optical transition. An argument analogous to the one in
that the magnetic interactions related to the spin can be described by an internal eld
which adds to the external eldB0and whose sign changes, depending on whether the
spin points up or down. It is this internal eld that causes the splitting of the
+
and
lines (theline is not aected, since its quantum numberis zero).
References and suggestions for further reading:
Cagnac and Pebay-Peyroula (11.2), Chaps. XI and XVII (especially 5-A of that
chapter); White (11.5), Chap. X; Kuhn (11.1), Chap. III, F; Sobel'man (11.12),
Chap. 8, 29.
1297

COMPLEMENT E XII
Complement EXII
The Stark eect for the hydrogen atom
1 The Stark eect on the = 1level
1-a The shift of the1state is quadratic in. . . . . . . . . . .
1-b Polarizability of the1state
2 The Stark eect on the = 2level
Consider a hydrogen atom placed in a uniform static electric eldparallel to.
To the Hamiltonian studied in Chapter ,
which describes the interaction energy of the electric dipole momentRof the atom with
the eld. can be written:
= R= (1)
Even for the strongest electric elds that can be produced in the laboratory, we
always have 0. On the other hand, ifis strong enough,can have the same
order of magnitude asand or be even larger. To simplify the discussion, we
shall assume throughout this complement thatis strong enough for the eect of
to be much larger than that ofor . We shall therefore calculate directly, using
perturbation theory, the eect ofon the eigenstates of0found in Chapter
next step, which we shall not consider here, would consist of evaluating the eect of,
and then of, on the eigenstates of0+ ).
Since both0and do not act on the spin variables, we shall ignore the quantum
numbers and.
1. The Stark eect on the = 1level
1-a. The shift of the1state is quadratic in
According to perturbation theory, the eect of the electric eld can be obtained to
rst order by calculating the matrix element:
= 1= 0 = 0 = 1= 0 = 0
Since the operatoris odd, and since the ground state has a well-dened parity (it is
even), the preceding matrix element is zero.
There is therefore no eect which is linear in, and we must go on to the next
term of the perturbation series:
2=
22
=1
100
2
1
(2)
where=
2
is the eigenvalue of0associated with the eigenstate (cf.
Chap., ). The preceding sum is certainly not zero, since there exist states
1298

THE STARK EFFECT FOR THE HYDROGEN ATOM
whose parity is opposite to that of100. We conclude that, to lowest order in, the
Stark shift of the1ground state is quadratic. Since1 is always negative, the
ground state is lowered.
1-b. Polarizability of the1state
We have already mentioned that, for reasons of parity, the average values of the
components of the operatorRare zero in the state100(the unperturbed ground
state).
In the presence of an electric eldparallel to, the ground state is no longer
100, but rather (according to the results of ):
0=100
=1
100
1
+ (3)
This shows that the average value of the electric dipole momentRin the perturbed
ground state is, to rst order in,0R0. Using expression (3) for0, we then
obtain:
0R0=
2
=1
100R 100+100 R100
1
(4)
Thus, we see that the electric eldcauses an induced dipole moment to appear,
proportional to. It can easily be shown, by using the spherical harmonic orthogonality
relation
1
, that0 0and 0 0are zero, and that the only non-zero average
value is:
0 0=2
2
= 1
100
2
1
(5)
In other words, the induced dipole moment is parallel to the applied eld.This is not
surprising, given the spherical symmetry of the1state. The coecient of proportionality
between the induced dipole moment and the eld is called the linear electric suscepti-
bility. We see that quantum mechanics permits the calculation of this susceptibility for
the1state:
1=2
2
= 1
100
2
1
(6)
1
This relation implies that100 is dierent from zero only if= 1,= 0(the argument
is the same as the one given for21 200in the beginning of 2 below). Consequently, in (2),
(3), (4), (5), (6), the summation is actually carried out only over(it includes, furthermore, the states
of the positive energy continuum).
1299

COMPLEMENT E XII
2. The Stark eect on the = 2level
The eect ofon the= 2level can be obtained to rst order by diagonalizing the
restriction ofto the subspace spanned by the four states of the200;21 =
10+1basis.
The200state is even; the three21 states are odd. Sinceis odd, the
matrix element200 200and the nine matrix elements21 21 are
zero (cf.ComplementII). On the other hand, since the200and21states have
opposite parities,21 200can be dierent from zero.
Let us show that actually only210 200is non-zero.is proportional
to=cosand, therefore, to
0
1(). The angular integral which enters into the
matrix elements21 200is therefore of the form:
1()
0
1()
0
0() d
Since
0
0is a constant, this integral is proportional to the scalar product of
0
1and
1
and is therefore dierent from zero only if= 0. Moreover, since
0
1,21()and20()
are real, the corresponding matrix element ofis real. We shall set:
210 200= (7)
without concerning ourselves with the exact value of[which could be calculated without
diculty since we know the wave functions210(r)and200(r)].
The matrix which representsin the= 2level, therefore, has the following
form (the basis vectors are arranged in the order211,211,210,200):
0 0 0 00 0 0 00 0 00 0 0
(8)
We can immediately deduce the corrections to rst order inand the eigenstates to
zeroth order:
Eigenstates Corrections
211 0
211 0
1
2
(210+200)
1
2
(210200)
(9)
Thus, we see that the degeneracy of the= 2level is partially removed and that the
energy shifts arelinear,and not quadratic, in. The appearance of a linear Stark eect
is a typical result of the existence of two levels of opposite parities and the same energy,
here the2and2levels. This situation exists only in the case of hydrogen (because of
the-fold degeneracy of the= 1shells).
1300

THE STARK EFFECT FOR THE HYDROGEN ATOM
Comment:
The states of the= 2level are not stable. Nevertheless, the lifetime of the
2state is considerably longer than that of the2states, since the atom passes
easily from2to1by spontaneous emission of a Lymanphoton (lifetime of
the order of 10
9
s), while decay from the2state requires the emission of two
photons (lifetime of the order of a second). For this reason, the2states are said
to be unstable and the2state, metastable.
Since the Stark Hamiltonianhas a non-zero matrix element between2
and2, any electric eld (static or oscillating) mixes the metastable2state with
the unstable2state, greatly reducing the2state's lifetime. This phenomenon
is called metastability quenching (see also ComplementIV, in which we study
the eect of a coupling between two states of dierent lifetimes).
References and suggestions for further reading:
The Stark eect in atoms: Kuhn (11.1), Chap. III, A-6 and G. Ruark and Urey
(11.9), Chap. V, 12 and 13; Sobel'man (11.12), Chap. 8, 28.
The summation over the intermediate states which appears in (2) and (6) can be
calculated exactly by the method of Dalgarno and Lewis; see Borowitz (1.7), 14-5;
Schi (1.18), 33. Original references: (2.34), (2.35), (2.36).
Quenching of metastability: see Lamb and Retherford (3.11), App. II; Sobel'man
(11.12), Chap. 8, 28.5.
1301

Chapter XIII
Approximation methods for
time-dependent problems
A Statement of the problem
B Approximate solution of the Schrödinger equation
B-1 The Schrödinger equation in the representation
B-2 Perturbation equations
B-3 Solution to rst order in. . . . . . . . . . . . . . . . . . . .
C An important special case: a sinusoidal or constant per-
turbation
C-1 Application of the general equations
C-2 Sinusoidal perturbation coupling two discrete states: the res-
onance phenomenon
C-3 Coupling with the states of the continuous spectrum
D Random perturbation
D-1 Statistical properties of the perturbation
D-2 Perturbative computation of the transition probability
D-3 Validity of the perturbation treatment
E Long-time behavior for a two-level atom
E-1 Sinusoidal perturbation
E-2 Random perturbation
E-3 Broadband optical excitation of an atom
A. Statement of the problem
Consider a physical system with Hamiltonian0. The eigenvalues and eigenvectors of
0will be denoted byand :
0 = (A-1)
Quantum Mechanics, Volume II, Second Edition. C. Cohen-Tannoudji, B. Diu, and F. Laloë.
© 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

CHAPTER XIII APPROXIMATION METHODS FOR TIME-DEPENDENT PROBLEMS
For the sake of simplicity, we shall consider the spectrum of0to be discrete and non-
degenerate; the formulas obtained can easily be generalized (see, for example, ). We
assume that0is not explicitly time-dependent, so that its eigenstates are stationary
states.
At= 0, a perturbation is applied to the system. Its Hamiltonian then becomes:
() =0+() (A-2)
with:
() =
^
() (A-3)
whereis a real dimensionless parameter much smaller than 1 and
^
()is an observable
(which can be explicitly time-dependent) of the same order of magnitude as0, and zero
for0.
The system is assumed to be initially in the stationary state, an eigenstate
of0of eigenvalue. Starting at= 0when the perturbation is applied, the system
evolves: the stateis no longer, in general, an eigenstate of the perturbed Hamiltonian.
We propose, in this chapter, to calculate the probabilityP()of nding the system in
another eigenstateof0at time. In other words, we want to study the transitions
that can be induced by the perturbation()between the stationary states of the
unperturbed system.
The treatment is very simple. Between the times 0 and, the system evolves in
accordance with the Schrödinger equation:
~
d
d
()= 0+
^
()() (A-4)
The solution()of this rst-order dierential equation corresponding to the initial
condition:
(= 0)= (A-5)
is unique. The desired probabilityP()can be written:
P() = ()
2
(A-6)
The whole problem therefore consists of nding the solution()of (A-4) that
corresponds to the initial condition (A-5). However, such a problem is not generally
rigorously soluble. This is why we resort to approximation methods. We shall show
in this chapter how, ifis suciently small, the solution()can be found in the
form of a power series expansion in. Thus, we shall calculate()explicitly to rst
order in, as well as the corresponding probability (). The general formulas obtained
will then be applied () to the study of an important special case, the one in which
the perturbation is a sinusoidal function of time or a constant (the interaction of an
atom with an electro-magnetic wave, which falls into this category, is treated in detail in
ComplementXIII). This is an example of theresonancephenomenon. Two situations
will be considered: the one in which the spectrum of0is discrete, and the one in
which the initial stateis coupled to a continuum of nal states. In the latter case,
we shall prove an important formula known as Fermi's golden rule. In
1304

B. APPROXIMATE SOLUTION OF THE SCHRÖDINGER EQUATION
consider another important case in which the perturbation uctuates randomly; it is
then characterized by its time-dependent correlation function, and will be treated with
a perturbative calculation that is valid for short times. We will then show in
to extend the valitidy of this calculation to long times, within a general approximation
called motional narrowing approximation.
Comment:
The situation treated in
of the general problem discussed in this chapter. Recall that, in Chapter, we
discussed a two-level system (the states1and2), initially in the state1,
subjected, beginning at time= 0, to a constant perturbation. The probability
P12()can then be calculated exactly, leading toRabi's formula.
The problem we are taking up here is much more general. We shall consider
a system with an arbitrary number of levels (sometimes, as in , with a
continuum of states) and a perturbation()which is an arbitrary function of the
time. This explains why, in general, we can obtain only an approximate solution.
B. Approximate solution of the Schrödinger equation
B-1. The Schrödinger equation in the representation
The probabilityP()explicitly involves the eigenstatesand of0. It
is therefore reasonable to choose therepresentation.
B-1-a. The system of dierential equations for the components of the state vector
Let()be the components of the ket()in the basis:
()= () (B-1)
with:
() = () (B-2)
^
()denotes the matrix elements of the observable
^
()in the same basis:
^
()=
^
() (B-3)
Recall that0is represented in thebasis by a diagonal matrix:
0 = (B-4)
We shall project both sides of Schrödinger equation (A-4) onto. To do so, we
insert the closure relation:
= 1 (B-5)
1305

CHAPTER XIII APPROXIMATION METHODS FOR TIME-DEPENDENT PROBLEMS
and use relations (B-2), (B-3) and (B-4). We obtain:
~
d
d
() = () +
^
()() (B-6)
The set of equations (B-6), written for the various values of, constitutes a system of
coupled linear dierential equations of rst order in, which enables us, in theory, to
determine the components()of(). The coupling between these equations arises
solely from the existence of the perturbation
^
(), which, by its non-diagonal matrix
elements
^
(), relates the evolution of()to that of all the other coecients().
B-1-b. Changing functions
When
^
()is zero, equations (B-6) are no longer coupled, and their solution is
very simple. It can be written:
() =e
~
(B-7)
whereis a constant which depends on the initial conditions.
Now, if
^
()is not zero, while remaining much smaller than0because of the
condition 1, we expect the solution()of equations (B-6) to be very close to
solution (B-7). In other words, if we perform the change of functions:
() =()e
~
(B-8)
we can predict that the()will be slowly varying functions of time.
We substitute (B-8) into equation (B-6); we obtain:
~e
~
d
d
() + () e
~
= () e
~
+
^
()() e
~
(B-9)
We now multiply both sides of this relation by e
+ ~
, and introduce the Bohr angular
frequency:
=
~
(B-10)
related to the pair of statesand. We obtain:
~
d
d
() = e
^
()() (B-11)
B-2. Perturbation equations
The system of equations (B-11) is rigorously equivalent to Schrödinger equation
(A-4). In general, we do not know how to nd its exact solution. This is why we shall
use the fact thatis much smaller than 1 to try to determine this solution in the form
1306

B. APPROXIMATE SOLUTION OF THE SCHRÖDINGER EQUATION
of a power series expansion in(which we can hope to be rapidly convergent ifis
suciently small):
() =
(0)
() +
(1)
() +
2(2)
() + (B-12)
If we substitute this expansion into (B-11), and if we set equal the coecients of
on both sides of the equation, we nd:
()for= 0:
~
d
d
(0)
() = 0 (B-13)
since the right-hand side of (B-11) has a common factor. Relation (B-13) expresses the
fact that
(0)
does not depend on. Thus, ifis zero,()reduces to a constant [cf.
(B-7)].
()for= 0:
~
d
d
()
() =e
^
()
(1)
() (B-14)
We see that, with the zeroth-order solution determined by (B-13) and the initial condi-
tions, recurrence relation (B-14) enables us to obtain the rst-order solution (= 1). It
then furnishes the second-order solution (= 2) in terms of the rst-order one and, by
recurrence, the solution to any orderin terms of the one to order1.
B-3. Solution to rst order in
B-3-a. The state of the system at time
For 0, the system is assumed to be in the state. Of all the coecients
(), only()is dierent from zero (and, furthermore, independent ofsince
^
is
then zero). At time= 0,
^
()may become discontinuous in passing from a zero value
to the value
^
(0). However, since
^
()remains nite, the solution of the Schrödinger
equation is continuous at= 0. It follows that:
(= 0) = (B-15)
and this relation is valid for all. Consequently, the coecients of expansion (B-12)
must satisfy:
(0)
(= 0) = (B-16)
()
(= 0) = 0 if 1 (B-17)
Equation (B-13) then immediately yields, for all positive:
(0)
() = (B-18)
which completely determines the zeroth-order solution.
1307

CHAPTER XIII APPROXIMATION METHODS FOR TIME-DEPENDENT PROBLEMS
This result now permits us to write (B-14), for= 1, in the form:
~
d
d
(1)
() =e
^
()
= e
^
() (B-19)
an equation which can be integrated without diculty. Taking into account initial con-
dition (B-17), we nd:
(1)
() =
1
~
0
e
^
() d (B-20)
If we now substitute (B-18) and (B-20) into (B-8) and then into (B-1), we obtain
the state()of the system at time, calculated to rst order in.
B-3-b. The transition probabilityP()
According to (A-6) and denition (B-2) of(), the transition probabilityP()
is equal to()
2
, that is, since()and()have the same modulus [cf.(B-8)]:
P() =()
2
(B-21)
where:
() =
(0)
() +
(1)
() + (B-22)
can be calculated from the formulas established in the preceding section.
From now on, we shall assume that the statesand are dierent. We shall
therefore be concerned only with the transitions induced by
^
()between two distinct
stationary states of0. We then have
(0)
() = 0, and, consequently:
P() =
2(1)
()
2
(B-23)
Using (B-20) and replacing
^
()by()[cf.(A-3)], we nally obtain:
P() =
1
~
2
0
e () d
2 (B-24)
Consider the function
~
(), which is zero for0and , and equal to
()for0 (cf.Fig.).
~
()is the matrix element of the perturbation
seen by the system between the time= 0and the measurement time, when we
try to determine if the system is in the state. Result (B-24) shows thatP()is
proportional to the square of the modulus of the Fourier transform of the perturbation
actually seen,
~
(). This Fourier transform is evaluated at an angular frequency
equal to the Bohr angular frequency associated with the transition under consideration.
Note also that the transition probabilityP()is zero to rst order if the matrix
element()is zero for all.
1308

C. AN IMPORTANT SPECIAL CASE: A SINUSOIDAL OR CONSTANT PERTURBATION
Comment:
We have not discussed the validity conditions of the approximation to rst order
in. Comparison of (B-11) with (B-19) shows that this approximation simply
amounts to replacing, on the right-hand side of (B-11), the coecients()by
their values(0)at time= 0. It is therefore clear that, so long asremains small
enough for(0)not to dier very much from(), the approximation remains
valid. On the other hand, whenbecomes large, there is no reason why the
corrections of order 2, 3, etc. inshould be negligible.
C. An important special case: a sinusoidal or constant perturbation
C-1. Application of the general equations
Now assume that()has one of the two simple forms:
^
() =
^
sin (C-1a)
^
() =
^
cos (C-1b)
where
^
is a time-independent observable and, a constant angular frequency. Such
a situation is often encountered in physics. For example, in ComplementsXIIIand
BXIII, we consider the perturbation of a physical system by an electromagnetic wave
of angular frequency;P()then represents the probability, induced by the incident
monochromatic radiation, of a transition between the initial stateand the nal state
.
With the particular form (C-1a) of
^
(), the matrix elements
^
()take on the
form:
^
() =
^
sin=
^
2
(e e) (C-2)0
W
fi
(t)
~
t t
Figure 1: The variation of the func-
tion
~
()with respect to. This
function coincides with()in
the interval0 , and goes to
zero outside this interval. It is the
Fourier transform of
~
()that
enters into the transition probabil-
ityP()to lowest order.
1309

CHAPTER XIII APPROXIMATION METHODS FOR TIME-DEPENDENT PROBLEMS
where
^
is a time-independent complex number. Let us now calculate the state vector
of the system to rst order in. If we substitute (C-2) into general formula (B-20), we
obtain:
(1)
() =
^
2~
0
e
(+)
e
( )
d (C-3)
The integral which appears on the right-hand side of this relation can easily be calculated
and yields:
(1)
() =
^
2~
1e
(+)
+
1e
( )
(C-4)
Therefore, in the special case we are treating, general equation (B-24) becomes:
P(;) =
2(1)
()
2
=
2
4~
2
1e
(+)
+
1e
( )
2
(C-5a)
(we have added the variablein the probabilityP, since the latter depends on the
frequency of the perturbation).
If we choose the special form (C-1b) for
^
()instead of (C-1a), a calculation
analogous to the preceding one yields:
P(;) =
2
4~
2
1e
(+)
+
+
1e
( )
2
(C-5b)
The operator
^
cosbecomes time-independent if we choose= 0. The transition
probabilityP()induced by a constant perturbationcan therefore be obtained by
replacingby 0 in (C-5b):
P() =
2
~
22
1e
2
=
2
~
2
( ) (C-6)
with:
( ) =
sin(2)
(2)
2
(C-7)
In order to study the physical content of equations (C-5b) and (C-6), we shall
rst consider the case in whichand are two discrete levels (), and then
that in which belongs to a continuum of nal states ( ). In the rst case,
P(;)[orP()] really represents a transition probability which can be measured,
while, in the second case, we are actually dealing with a probability density (the truly
measurable quantities then involve a summation over a set of nal states). From a
physical point of view, there is a distinct dierence between these two cases. We shall
see in ComplementsXIIIandXIIIthat, over a suciently long time interval, the system
1310

C. AN IMPORTANT SPECIAL CASE: A SINUSOIDAL OR CONSTANT PERTURBATIONE
f
E
i
a b
E
i
φ
f

φ
i

φ
i

φ
f

E
f
Figure 2: The relative disposition of the energiesand associated with the states
and . If (g. a), the transition occurs through absorption
of an energy quantum~. If, on the other hand, (g. b), the
transition occurs through induced emission of an energy quantum~.
oscillates between the statesand in the rst case, while it leaves the state
irreversibly in the second case.
In , in order to concentrate on the resonance phenomenon, we shall choose a
sinusoidal perturbation, but the results obtained can easily be transposed to the case of a
constant perturbation. On the other hand, we shall use this latter case for the discussion
of .
C-2. Sinusoidal perturbation coupling two discrete states: the resonance phenomenon
C-2-a. Resonant nature of the transition probability
When the timeis xed, the transition probabilityP(;)is a function only of
the variable. We shall see that this function has a maximum for:
(C-8a)
or:
(C-8b)
A resonance phenomenon therefore occurs when the angular frequency of the pertur-
bation coincides with the Bohr angular frequency associated with the pair of states
and . If we agree to choose0, relations (C-8) give the resonance conditions cor-
responding respectively to the cases0and 0. In the rst case (cf.Fig.-a),
the system goes from the lower energy levelto the higher levelby the resonant
absorption of an energy quantum~. In the second case (cf.Fig.-b), the resonant
perturbation stimulates the passage of the system from the higher levelto the lower
level(accompanied by the induced emission of an energy quantum~). Throughout
this section, we shall assume thatis positive (the situation of Figure-a). The case
in whichis negative could be treated analogously.
1311

CHAPTER XIII APPROXIMATION METHODS FOR TIME-DEPENDENT PROBLEMS
To reveal the resonant nature of the transition probability, we note that both
expressions (C-5a) and (C-5b) forP(;)involve the square of the modulus of a sum
of two complex terms. The rst of these terms is proportional to:
+=
1e
(+)
+
=e
(+)2
sin [(+)2]
(+)2
(C-9a)
and the second one, to:
=
1e
( )
=e
( )2
sin [( )2]
( )2
(C-9b)
The denominator of theterm goes to zero for= , and that of the+term,
for= . Consequently, forclose to, we expect only theterm to be
important; this is why it is called the resonant term, while the+term is called the
anti-resonant term (+would become resonant if, for negative,were close to
).
Let us then consider the case in which:
(C-10)
neglecting the anti-resonant term+(the validity of this approximation will be discussed
in C-9b) into account, we then obtain:
P(;) =
2
4~
2
( ) (C-11)
with:
( ) =
sin [( )2]
( )2
2
(C-12)
Figure P(;)with respect to, for a given time. It
clearly shows the resonant nature of the transition probability. This probability presents
a maximum for= , when it is equal to
22
4~
2
. As we move away from,
it decreases, going to zero for= 2. When continues to increase, it
oscillates between the value
2
~
2
( )
2
and zero (diraction pattern).
C-2-b. The resonance width and the time-energy uncertainty relation
The resonance widthcan be approximately dened as the distance between
the rst two zeros ofP(;)on each side of= . It is inside this interval that
the transition probability takes on its largest values [the rst secondary maximumP,
attained when( )2 = 32, is equal to
22
9
2
~
2
, that is, less than 5% of
the transition probability at resonance]. We then have:

4
(C-13)
The larger the time, the smaller this width.
Result (C-13) presents a certain analogy with the time-energy uncertainty relation
(cf.Chap., ). Assume that we want to measure the energy dierence=
1312

C. AN IMPORTANT SPECIAL CASE: A SINUSOIDAL OR CONSTANT PERTURBATION0 ω
fi
ω
ω=
t
4π
4ħ
2
W
fi
2
t
2
"
if
(t;ω)
Figure 3: Variation, with respect to, of the rst-order transition probabilityP(;)
associated with a sinusoidal perturbation of angular frequency;is xed. When
, a resonance appears whose intensity is proportional to
2
and whose width is inversely
proportional to.
~by applying a sinusoidal perturbation of angular frequencyto the system and
varyingso as to detect the resonance. If the perturbation acts during a time, the
uncertaintyon the value will be, according to (C-13), of the order of:
=~
~
(C-14)
Therefore, the productcannot be smaller than~. This recalls the time-energy
uncertainty relation, althoughhere is not a time interval characteristic of the free
evolution of the system, but is externally imposed.
C-2-c. Validity of the perturbation treatment
Now let us examine the limits of validity of the calculations leading to result (C-11).
We shall rst discuss the resonant approximation, which consists of neglecting the anti-
resonant term+, and then the rst-order approximation in the perturbation expansion
of the state vector.
. Discussion of the resonant approximation
Using the hypothesis , we have neglected+relative to. We shall
therefore compare the moduli of+and.
1313

CHAPTER XIII APPROXIMATION METHODS FOR TIME-DEPENDENT PROBLEMS
The shape of the function()
2
is shown in Figure. Since +()
2
=
()
2
,+()
2
can be obtained by plotting the curve which is symmetric with
respect to the preceding one relative to the vertical axis= 0. If these two curves, of
width, are centered at points whose separation is much larger than, it is clear
that, in the neighborhood of=, the modulus of+is negligible compared to that
of. The resonant approximation is therefore justied on the condition
1
that:
2 (C-15)
that is, using (C-13):
1
1
(C-16)
Result (C-11) is therefore valid only if the sinusoidal perturbation acts during a time
which is large compared to1. The physical meaning of such a condition is clear:
during the interval[0], the perturbation must perform numerous oscillations to appear
to the system as a sinusoidal perturbation. If, on the other hand,were small compared
to1, the perturbation would not have enough time to oscillate and would be equivalent
to a perturbation varying linearly in time [in the case (C-1a)] or constant [in the case
(C-1b)].
Comment:
For a constant perturbation, condition (C-16) can never be satised, sinceis
zero. However, it is not dicult to adapt the calculations of
case. We have already obtained [in (C-6)] the transition probabilityP()for
a constant perturbation by directly setting= 0in (C-5b). Note that the two
terms+and are then equal, which shows that if (C-16) is not satised, the
anti-resonant term is not negligible.
The variation of the probabilityP()with respect to the energy dierence~
(with the timexed) is shown in Figure. This probability is maximal when
= 0, which corresponds to what we found in
frequency is zero, the perturbation is resonant when= 0(degenerate lev-
els). More generally, the considerations of
resonance can be transposed to this case.
. Limits of the rst-order calculation
We have already noted (cf.comment at the end of ) that the rst-order
approximation can cease to be valid when the timebecomes too large. This can indeed
be seen from expression (C-11), which, at resonance, can be written:
P(;=) =
2
4~
2
2
(C-17)
1
Note that if condition (C-15) is not satised, the resonant and anti-resonant terms interfere: it is
not correct to simply add+
2
and
2
.
1314

C. AN IMPORTANT SPECIAL CASE: A SINUSOIDAL OR CONSTANT PERTURBATION
This function becomes innite when , which is absurd, since a probability can
never be greater than 1.
In practice, for the rst-order approximation to be valid at resonance, the proba-
bility in (C-17) must be much smaller than 1, that is
2
:
~
(C-18)
To show precisely why this inequality is related to the validity of the rst-order approximation,
it would be necessary to calculate the higher-order corrections from (B-14) and to examine
under what conditions they are negligible. We would then see that, although inequality (C-18)
is necessary, it is not rigorously sucient. For example, in the terms of second or higher order,0 ω
fi

if
(t)
!ω≃
t
4π
ħ
2
W
fi
2
t
2
Figure 4: Variation of the transition probabilityP()associated with a constant pertur-
bation with respect to= ( )~, for xed. A resonance appears, centered about
= 0(conservation of energy), with the same width as the resonance of Figure, but
an intensity four times greater (because of the constructive interference of the resonant
and anti-resonant terms, which, for a constant perturbation, are equal).
2
For this theory to be meaningful, it is obviously necessary for conditions (C-16) and (C-18) to be
compatible. That is, we must have:
1
~
This inequality means that the energy dierence =~ is much larger than the matrix
element of()between and .
1315

CHAPTER XIII APPROXIMATION METHODS FOR TIME-DEPENDENT PROBLEMS
there appear matrix elements
^
of
^
other than
^
, on which certain conditions must be
imposed for the corresponding corrections to be small.
Note that the problem of calculating the transition probability whendoes not
satisfy (C-18) is taken up in ComplementXIII, in which an approximation of a dierent
type is used (the secular approximation).
C-3. Coupling with the states of the continuous spectrum
If the energybelongs to a continuous part of the spectrum of0, that is, if the
nal states are labeled by continuous indices, we cannot measure the probability of nding
the system in awell-denedstate at time. The postulates of Chapter
that in this case the quantity()
2
which we determined above (approximately) is
a probability density. The physical predictions for a given measurement then involve an
integration of this probability density over a certain group of nal states (which depends
on the measurement to be made). We shall consider what happens to the results of the
preceding sections in this case.
C-3-a. Integration over a continuum of nal states: density of states
. Example
To understand how this integration is performed over the nal states, we shall rst
consider a concrete example.
We shall discuss the problem of the scattering of a spinless particle of mass
by a potential(r)(cf.Chap. ). The state()of the particle at timecan be
expanded on the statespof well-dened momentapand energies:
=
p
2
2
(C-19)
The corresponding wave functions are the plane waves:
rp=
1
2~
32
e
pr~
(C-20)
The probability density associated with a measurement of the momentum isp()
2
[()is assumed to be normalized].
The detector used in the experiment (see, for example, Figure )
gives a signal when the particle is scattered with the momentump. Of course, this
detector always has a nite angular aperture, and its energy selectivity is not perfect: it
emits a signal whenever the momentumpof the particle points within a solid angle
aboutpand its energy is included in the intervalcentered at=p
2
2. If
denotes the domain ofp-space dened by these conditions, the probability of obtaining
a signal from the detector is therefore:
P(p) =
p
d
3
p()
2
(C-21)
To use the results of the preceding sections, we shall have to perform a change of variables
which results in an integral over the energies. This does not present any diculties, since
1316

C. AN IMPORTANT SPECIAL CASE: A SINUSOIDAL OR CONSTANT PERTURBATION
we can write:
d
3
=
2
dd (C-22)
and replace the variableby the energy, to which it is related by (C-19). We thus
obtain:
d
3
=()dd (C-23)
where the function(), called thedensity of nal states, can be written, according to
(C-19), (C-22) and (C-23):
() =
2
d
d
=
2
=
2 (C-24)
(C-21) then becomes:
P(p) =
;
d d ()p()
2
(C-25)
. The general case
Assume that, in a particular problem, certain eigenstates of0are labeled by a
continuous set of indices, symbolized by, such that the orthonormalization relation can
be written:
=( ) (C-26)
The system is described at timeby the normalized ket(). We want to calculate the
probabilityP()of nding the system, in a measurement, in a given group of nal
states. We characterize this group of states by a domainof values of the parameters
, centered at, and we assume that their energies form a continuum. The postulates
of quantum mechanics then yield:
P() = d ()
2
(C-27)
As in the example of above, we shall change variables, and introduce the density
of nal states. Instead of characterizing these states by the parameters, we shall use
the energyand a set of other parameters(which are necessary when0alone does
not constitute a C.S.C.O.). We can then express din terms of dand d:
d=()dd (C-28)
in which the density of nal states
3
()appears. If we denote byand the
range of values of the parametersanddened by, we obtain:
P() =
;
dd() ()
2
(C-29)
where the notationhas been replaced by in order to point up the- and
-dependence of the probability density()
2
.
3
In the general case, the density of statesdepends on bothand. However, it often happens
(cf.example of above) thatdepends only on.
1317

CHAPTER XIII APPROXIMATION METHODS FOR TIME-DEPENDENT PROBLEMS
C-3-b. Fermi's golden rule
In expression (C-29),()is the normalized state vector of the system at time.
As in
of0[therefore belongs to the discrete spectrum of0, since the initial state
of the system must, like(), be normalizable]. In (C-29), we shall replace the notation
P()byP( )in order to remember that the system starts from the state
.
The calculations of
perturbation () remain valid when the nal state of the system belongs
to the continuous spectrum of0. If we assumeto be constant, we can therefore use
(C-6) to nd the probability density()
2
to rst order in. We then get:
()
2
=
1
~
2
2
~
(C-30)
whereandare the energies of the statesand respectively, andis the
function dened by (C-7). We get forP( ), nally:
P( ) =
1
~
2
;
dd()
2
~
(C-31)
The function
~
varies rapidly about= (cf.Fig.). Ifis suciently
large, this function can be approximated, to within a constant factor, by the function
( ), since, according to (11) and (20) of Appendix, we have:
lim
~
=
2~
= 2~( ) (C-32)
On the other hand, the function()
2
generally varies much more slowly
with. We shall assume here thatis suciently large for the variation of this function
over an energy interval of width4~centered at=to be negligible
4
. We can then
in (C-31) replace
~
by its limit (C-32). This enables us to perform the integral
overimmediately. If, in addition,is very small, integration overis unnecessary,
and we nally get:
when the energybelongs to the domain:
P( ) =
2
~
=
2
( =)(C-33a)
when the energydoes not belong to this domain:
P( ) = 0 (C-33b)
As we saw in the comment of , a constant perturbation can induce tran-
sitions only between states of equal energies. The system must have the same energy (to
within2~) in the initial and nal states. This is why, if the domainexcludes the
energy, the transition probability is zero.
4
()
2
must vary slowly enough to enable the nding of values ofthat satisfy the
stated condition but remain small enough for the perturbation treatment ofto be valid. Here, we
also assume that 4~.
1318

C. AN IMPORTANT SPECIAL CASE: A SINUSOIDAL OR CONSTANT PERTURBATION
The probability (C-33a) increases linearly with time. Consequently,the transition
probability per unit time,( ), dened by:
( ) =
d
d
P( ) (C-34)
is time-independent. We introduce the transition probability density per unit time and
per unit interval of the variable:
( ) =
( )
(C-35)
It is equal to:
( ) =
2
~
=
2
( =)
(C-36)
This important result is known asFermi's golden rule.
Comments:
()Assume that is a sinusoidal perturbation of the form (C-1a) or (C-1b
which couples a state (to a continuum of states with energies
close to+~. Starting with (C-11), we can carry out the same procedure
as above, which yields:
( ) =
2~
=+~
2
( =+~)(C-37)
()Let us return to the problem of the scattering of a particle by a potentialwhose
matrix elements in therrepresentation are given by:
rr=(r)(rr) (C-38)
Now assume that the initial state of the system is a well-dened momentum state:
(= 0)=p (C-39)
and let us calculate the scattering probability of an incident particle of momentum
pinto the states of momentumpgrouped about a given valuep(withp=p).
(C-36) gives the scattering probability(pp)per unit time and per unit solid
angle aboutp=p:
(pp) =
2
~
p p
2
(=) (C-40)
Taking into account (C-20), (C-38) and expression (C-24) for(), we then get:
(pp) =
2
~2
1
2~
6
d
3
e
(pp)r~
(r)
2
(C-41)
On the right-hand side of this relation, we recognize the Fourier transform of the
potential(r), evaluated for the value ofpequal topp.
1319

CHAPTER XIII APPROXIMATION METHODS FOR TIME-DEPENDENT PROBLEMS
Note that the initial statepchosen here is not normalizable, and it cannot
represent the physical state of a particle. However, although the norm ofpis
innite, the right-hand side of (C-41) maintains a nite value. Intuitively, we can
therefore expect to obtain a correct physical result from this relation. If we divide
the probability obtained by the probability current:
=
1
2~
3
~
=
1
2~
3
2
(C-42)
associated, according to (C-20), with the statep, we obtain:
(pp)
=
2
4
2
~
4
d
3
e
(pp)r~
(r)
2
(C-43)
which is the expression for the scattering cross section in the Born approximation
( ).
Although it is not rigorous, the preceding treatment enables us to show that
the scattering cross sections of the Born approximation can also be obtained by a
time-dependent approach, using Fermi's golden rule.
D. Random perturbation
Another interesting case occurs when the perturbation applied to the system uctuates
in a random fashion. Consider for example an atom (a) having a spin magnetic moment,
and moving in a gas of particles (b) which also have magnetic moments. As atom (a)
undergoes a series of random collisions with particles (b), it is subjected to a magnetic
eld that varies randomly from one collision to another. The resulting interactions can
change the orientation of the atom's magnetic moment. This type of situation is treated
here (). We shall go back to the calculation of B assuming that the matrix element
^
()of the perturbation is a random function of time. Our aim is to study the transi-
tion probabilityP()for going from the stateto the stateafter a time, and
determine how it diers from the result found in the previous section.
D-1. Statistical properties of the perturbation
Here we consider the evolution of a single quantum system, atom (a) in the example
described above, and study its evolution averaged over time. We thus need to consider
the properties of statistical averages over time
5
of the perturbation(). We note
()the average value of the matrix element(), and assume it is equal to zero:() = 0 (D-1)
This means that()uctuates between values that can be opposite. Since the matrix
elements and are two complex conjugate numbers, their product is necessarily
positive or zero, hence having in general a non-zero average value:
()()0 (D-2)
5
The point of view of ComplementXIIIis more directly related to most experimental situations:
we study an ensemble ofindividual quantum systems described by their density operator. The two
approaches are nevertheless equivalent since, in statistical mechanics, averaging over a Gibbs ensemble
is equivalent to averaging a single system over a long time.
1320

D. RANDOM PERTURBATION
It will be useful in what follows to also consider the average value of such a product taken
at dierent instantsand+, called the correlation function():
(+)() =()= 0 (D-3)
Thedependence of()characterizes the time during which the perturbation keeps
a memory of its value:()is non-zero as long as(+)remains correlated with
(). The correlation function()goes to0when the time dierenceis longer
than a characteristic timecalled the correlation time :
()( ) =() 0 if (D-4)
We shall consider the case whereis very short compared to all the other evolution times
of the system. For instance, in the example mentioned above of an atom (a) diusing
in a gas of particles (b), the correlation time is of the order of the duration of a single
collision, generally (much) shorter than10
10
s.
We assume the random perturbation to be stationary, meaning that the correla-
tion functions depend only on the dierencebetween the two instants+and.
Consequently, we can also write:
()( ) =() (D-5)
Using complex conjugation, relation (D-3) can be written:
(+)() =()(+) =() (D-6)
Comparing with (D-5) yields:
() =() (D-7)
Changing the sign of the variabletransforms the function()into its complex
conjugate; in particular,(0)is real.
In the following computations, il will be useful to introduce the Fourier transform
~
()of the function():
~
() =
1
2
+
de () (D-8)
leading to its inverse relation:
() =
1
2
+
de
+~
() (D-9)
Relation (D-7) implies that
~
()is a real function.
D-2. Perturbative computation of the transition probability
As in (B-8), we perform the change of functions that transform()into(),
which eliminates the variations of the coecients due to0alone (this amounts to using
the interaction picture,cf.exercise 15 of ComplementIII). We assume that(0) = 1
1321

CHAPTER XIII APPROXIMATION METHODS FOR TIME-DEPENDENT PROBLEMS
and(0) = 0. We then look for the probability amplitude()for the system, starting
at time= 0from the state, to be found at timein the state. Equation (B-20)
is now written:
(1)
() =
1
~
0
e ()d (D-10)
The probability of nding the system in the stateat timeis obtained by multiplying
(D-10) by its complex conjugate. Since= and = , we get:
[
(1)
()][
(1)
()] =
1
~
2
0
e ()d
0
e ()d (D-11)
The transition probabilityP()is the average of (D-11) over the various values of the
random perturbation. This leads to:
P() =
[
(1)
()][
(1)
()] =
1
~
2
0
d
0
de
( )
()()(D-12)
Setting:
= (D-13)
and using (D-5) enable us to write (D-12) as:
P() =
1
~
2
0
d de () (D-14)
(the change of sign coming from d=dis accounted for by interchanging the inte-
gration limits).
We assume in what follows that:
(D-15)
The integral over dis taken over a time interval from0to, very large compared to.
Its value will not be signicantly modied if we shorten that interval at both end by a
few. Ifis of the order of a few units (= 2or= 3for instance), we can write:
P()
1
~
2
d de () (D-16)
In the integral over d, the upper limit is ; this upper limit may be extended to
innity since()goes to zero when , and hence the additional contribution to
the integral is zero. In the same way, the negative lower limitcan be replaced by
, since the condition ensures that the function to be integrated is zero
in the additional integration domain. The integral over dbecomes independent of, so
that the integral over dis easily computed and leads to:
d= (2) (D-17)
1322

D. RANDOM PERTURBATION
We then get:
P() (D-18)
where the constantis dened by:
=
1
~
2
+
() e d=
2~
2
~
() (D-19)
This result involves the Fourier transform
~
()of the correlation function()
dened by relation (D-8), taken at the (angular) frequency= of the transition
between the initial stateand the nal state. As already mentioned, relation
(D-7) shows that the constantis real.
The transition probability fromto after a timeis thus proportional to
that time. This means that when the perturbation is random, one can dene (at least
for short times
6
where the perturbative treatment to lowest order is valid) a transition
probability per unit time fromto . It is proportional to the Fourier transform
of the correlation function of the perturbation, computed at the angular frequency.
This is a very dierent result from the one obtained in (C-11) and (C-12) for a sinusoidal
perturbation. In that case, the transition probability increased as
2
for short times, and
then oscillated as a function of time.
D-3. Validity of the perturbation treatment
Result (D-18), obtained by a perturbative treatment, is valid as long as the tran-
sition probability remains small, that is if:
1

(D-20)
On the other hand, to establish (D-18) we assumed in (D-15) thatwas much larger
than. The two conditions (D-20) and (D-15) are compatible only if:
1

(D-21)
The calculation we just presented implies the existence of two very dierent time scales:
the evolution time of the system, of the order of1, often called the relaxation time;
the correlation time,, which is much shorter and characterizes the memory of the
uctuations of the random perturbation.
Equation (D-21) simply expresses the fact that, during this correlation time, the
system barely evolves. Using forrelation (D-19), this inequality can be written solely
with parameters concerning the perturbation. This inequality is often called the mo-
tional narrowing condition, for reasons that will be explained in of Complement
EXIII.
6
We shall see in the next section under which conditions this result remains valid for long times.
1323

CHAPTER XIII APPROXIMATION METHODS FOR TIME-DEPENDENT PROBLEMS
E. Long-time behavior for a two-level atom
Until now, we have limited ourselves in this chapter to perturbative calculations of the
transition probability after a time. We found that it increases as
2
for a sinusoidal
perturbation, but linearly asfor a random perturbation with a short memory. As a
probability cannot become larger than one, such approximations are only valid for small
values of. In the last part of this chapter, we shall present treatments that permit us
to study and compare the long-time behaviors of a system subjected to these two types
of perturbations. For the sake of simplicity, we shall limit our study to the case of a
two-level system.
E-1. Sinusoidal perturbation
We already studied in ComplementIVthe long-time behavior of a two-level sys-
tem subjected to a sinusoidal perturbation. We computed the exact evolution of a spin
12in the special case where the Hamiltonianobeys relation (14) ofIV:
() =
}
2
0 1e
1e 0
(E-1)
(this matrix is written using the basis+ of the eigenvectors of thespin compo-
nent). The diagonal of this matrix yields the matrix elements of the Hamiltonian0; this
Hamiltonian comes from the coupling of the spin with a static magnetic eldB0, parallel
to theaxis. The perturbation Hamiltonian()corresponds to the non-diagonal
parts of the matrix; it comes from the coupling of the spin with a radiofrequency eld
rotating around theaxis at the angular frequency. We showed in ComplementIV
that the quantum evolution of a spin12was identical to the classical evolution of a
magnetic dipole with a proportional angular momentum. This led to a useful image for
the evolution of a spin in a magnetic eld, composed of a constant and a rotating eld.
Now we saw in ComplementIVthat any two-level system is perfectly isomorphic
to a spin12. The statesand are associated with the spin states+and, and
the Hamiltonian0leads to two non-perturbed energies=}02and=}02;
this means that0= . We assume that the perturbationthat couples the two
statesand is the analog of the action of a magnetic eldB1rotating in the
plane at the frequency; it is thus responsible for the non-diagonal matrix elements
of (E-1), with:
() =
}1
2
e
() =
}1
2
e (E-2)
(the number1is supposed to be real; if this is not the case, a change of the relative
phase ofand allows this condition to be fullled). We can then directly trans-
pose the results of ComplementIV, with no additional computations. Relation (27) of
ComplementIVshows that the transition probability is given by Rabi's formula:
P() =
(1)
2
(1)
2
+ ( )
2
sin
2
(1)
2
+ ( )
22
(E-3)
1324

E. LONG-TIME BEHAVIOR FOR A TWO-LEVEL ATOM
Since the quantum and classical evolution coincide in the present situation, they
can be simply interpreted in terms of the classical precession of a magnetic moment
around an eective eld. At resonance, the eective eld is located in theplane,
for instance along theaxis. The spin, initially parallel to, precesses around,
hence tracing large circles in the plane. Relation (E-3) shows that the probability
for the spin to reverse its initial orientation is written:
P() = sin
2 1
2
(E-4)
This probability oscillates between0and1with a precession angular frequency1=
2 }, called Rabi's frequency. This type of long lasting oscillations could not have
been obtained
7
by a perturbation treatment.
For a non-resonant perturbation, the eective eld has a component along the
axis. As it precesses, the magnetic moment now follows a cone; the larger the discrepancy
betweenand the resonant frequency, the smaller the cone's aperture becomes (Figure
of ComplementIV). We must now use the complete relation (E-3) which, also, predicts
a sinusoidal oscillation. It should be noted that, if 2}, we nd again
the result of equations (C-11) and (C-12) of Chapter , which thus provide a good
approximation in this case.
Comment:
The previous results assume that the perturbation can be reduced to a single
rotating eld. Replacing in (E-1) the exponentialsby the sinusoidal functions
sinorcos, it introduces two rotating elds with opposite frequencies; they
both act simultaneously on the system, leading to a more complex situation. The
results remain, however, valid as long as the perturbation is weak enough (meaning
1 0) andnot too far from one of the two resonances (0or 0).
E-2. Random perturbation
As for the spin12case considered above, we assume here that the perturbation
does not have diagonal elements:
= = 0 (E-5)
(ComplementXIIIpresents a more general calculation, where this hypothesis is no longer
necessary).
In , we assumed that the system was initially in the state, hence only
(= 0)was dierent from zero. This rules out the possibility of any superposition of
states at the initial moment. To remove this restriction, we now assume that the system
is, at instant, in any superposition of the statesand :
()=()e
}
+()e
}
(E-6)
We then consider a later instant,+, and compute the evolution of the system between
the timesand+ , to second order in.
7
One must sum an innite number of perturbative terms to reconstruct a sine squared.
1325

CHAPTER XIII APPROXIMATION METHODS FOR TIME-DEPENDENT PROBLEMS
E-2-a. State vector of the system at time + to second order in
To zeroth-order of the perturbation, relation (B-13) shows that neither(+ )
nor(+ )depend on:
(0)
(+ ) =() ;
(0)
(+ ) =() (E-7)
To rst order, we use relation (B-14) with= 1; hypothesis (E-5) means thatis only
coupled to, and vice versa. Integrating over time, we get:
(1)
(+ ) =
1
~
()
+
e () d
=
1
~
()

0
e
(+)
(+) d (E-8)
where we have set= ; we also get a similar relation where the indicesandare
interchanged:
(1)
(+ ) =
1
~
()

0
e
(+)
(+) d (E-9)
The term (E-8) describes an atom that was at timein the stateand is found at
time+in the state; the term (E-9) describes the inverse process.
To second order, we again use relation (B-14), this time with= 2; after integra-
tion, it leads to:
(2)
(+ ) =
1
~
+
e ()
(1)
() d
=
1
~

0
e
(+)
(+)
(1)
(+) d (E-10)
We now change the integration variableto, and insert relation (E-9) after replacing
by; this leads to:
(2)
(+ ) =
1
~
2
()

0
e
(+)
(+) d
0
e
(+)
(+) d
=
1
~
2
()

0
e (+) d
0
e (+) d (E-11)
This perturbative term describes an atom that was at timein the state, then made
a transition to the stateat time+(included betweenand+), then came
back to the stateat time+(included betweenand+ ). Here also we can
interchange the indicesandto get the probability amplitude of the inverse process.
E-2-b. Average occupation probabilities to second order
For given values of the random variablesand , the probability of nding
the system in the stateis(+ )
2
. To second order in, this squared modulus
contains the following terms:
1326

E. LONG-TIME BEHAVIOR FOR A TWO-LEVEL ATOM
the squared modulus of
(0)
(+ ), which is of zeroth order.
a rst-order term containing the product of
(0)
(+)and the complex conjugate
of
(1)
(+ ), or the opposite. Due to condition (E-5), this term, linear in, averages
out to zero over all the possible values ofand . It will not be taken into account.
the squared modulus of
(1)
(+ ), which is of order2.
nally, twice the real part of the product of
(0)
(+)and the complex conjugate
of
(2)
(+ ), which is also of order2.
We thus get:
(+ )
2
(0)
(+ )
2
+
(1)
(+ )
2
+2Re[
(0)
(+ )][
(2)
(+ )](E-12)
This expression is rewritten below in a slightly dierent form. The rst and third term
are regrouped in a rst line, while the second term is rewritten in the second line. Note
that the rst term is rewritten using (E-7), while for the third term we use the complex
conjugate of the second line of (E-11), obtained by replacingby (and vice versa)
as well asby (and vice versa). This leads to:
(+ )
2
()
2
1
2
~
2
Re

0
e (+) d
0
e (+) d
+ ()
2
1
~
2

0
e (+) d

0
e (+) d (E-13)
We now average this probability over the various values of the random variables
and . We get in (E-13) the product of two matrix elements ofand of the
amplitude squared of()and(). Rigorously speaking, these quantities are not
mutually independent, since the system's state at timeis determined by the values of
the perturbation at an earlier time. This correlation actually lasts over a time much
shorter than, of the order of the correlation timeof the functionsand
cf. relation (D-4); therefore, a very short timeafter the instant, and are no
longer correlated with the values of()
2
. It is then justied to compute separately
the two averages:
()
2
Double integral()
2Double integral (E-14)
The averages
()
2
and()
2
of the populations of the statesand
are simply the diagonal elements of the density operator (ComplementXIII). Noting
~()this operator, we get:
~() =
()
2
~() =
()
2
(E-15)
1327

CHAPTER XIII APPROXIMATION METHODS FOR TIME-DEPENDENT PROBLEMS
The variation rate of~()between the timesand+ is:
~()

=
(+ )
2
()
2

(E-16)
Relation (E-13) then yields:
~()

=~() [1+
1] + ~()2 (E-17)
where1and2are the averages of double integrals:
1=
1
~
2
1

0
d
0
de
( )
(+)(+)
2=
1
~
2
1

0
d

0
de
( )
(+)(+) (E-18)
The computation of the average values of these two double integrals is similar to
the one performed in . It is carried out in detail below and leads to:
2= (E-19)
1=

2
+ (E-20)
In these relations,and are expressed in terms of the Fourier transform
~
()of
(), which was introduced in (D-8). The constantwas given in (D-19):
=
2~
2
~
() =
1
~
2
+
() e d (E-21)
and is dened by:
=
1
2~
2
+
d
1
~
() (E-22)
wheremeans the principal part (Appendix, 1-b).
Computation of 1and2:
The double integral appearing in2has already been encountered in (D-12), while com-
puting the value ofP(); we must simply replaceby. Its value is given in (D-18),
which becomes here. According to the denition (E-18) of2, we must then divide
this result by, which leads to relation (E-19).
The computation of1is similar to that of2, except that the upper limit of the integral
over disinstead of. In the rst line of (E-18), we can make the change of variables
=to transform the integrations according to:

0
d
0
d

0
d
0
d=

0
d
0
d (E-23)
1328

E. LONG-TIME BEHAVIOR FOR A TWO-LEVEL ATOM
As we did in (D-16), when we can replace the lower limit of the integral over d
by a few, without any signicant change of the result. The upper limit of the integral
over dis then longer than a fewand can be extended to+. The integral over d
then reduces to a simple factor, which cancels that same factor in the denominator
of (E-18). This leads to:
1
1
~
2
+
0
de
()( ) (E-24)
or else, using relation (D-9) to introduce the Fourier transform of():
1
1
2~
2
+
d
~
()
+
0
de
( )
(E-25)
The integral overleads to:
+
0
de
( )
(E-26)
To make this integral convergent, we introduce an innitesimal (positive) factorand
write:
+
0
de
( +)
=
1
( +)
=
+
(E-27)
In the limit0we get, taking into account relation (12) of Appendix:
+
0
de
( +)
=( ) +
1
(E-28)
Inserting this result in (E-25) we nd (E-20), whereand are given by (E-21) and
(E-22).
E-2-c. Time evolution of the populations
According to (E-20), we can write1+
1= . Inserting this result into (E-17)
and using (E-19), we get:
~()

=~() + ~()
~()

= +~()~() (E-29)
The interpretation of these two equations is straightforward: at any timethe system
goes from to , and from to, with a probability per unit time that is
constant and equal to. If, at time, the two populations ofand are dierent,
they will both tend exponentially towards the same value12, without ever coming back
to their initial values. This long-time irreversibility is clearly very dierent from what
we obtained for a two-level system subjected to a sinusoidal perturbation. We no longer
observe the oscillating and reversible behavior, similar to the Rabi precession of the spin
12associated with the two-level system ().
One may wonder how a prediction valid for long times can be obtained while using
perturbation calculations limited to second order in: expressions such as (E-11) and
1329

CHAPTER XIII APPROXIMATION METHODS FOR TIME-DEPENDENT PROBLEMS
(E-13) are certainly no longer valid for large values of. This is due to the random
character of the perturbation, which has a correlation timemuch shorter than the
evolution time
1
. At any time(even very distant from the initial time= 0) the
system has little memory of its past evolution. Between the instantand+ , its
evolution only depends on what occurred beforeduring the time interval[ ].
When is very short compared to, the system barely evolves during the timeto
+ , and a perturbation treatment can be applied. This amounts to dividing the time
into time intervals of width, very long compared to, but nevertheless very short
compared to the characteristic evolution time
1
.
E-2-d. Time evolution of the coherences
In addition to the populations (E-15) of the two levels, we must also consider
the coherences existing between them. It is the non-diagonal matrix elements of the
density operator:
~(+ ) =
[(+ )] [(+ )] (E-30)
that characterize the existence of coherent linear superpositions of the two levels. Up to
second order in, such a non-diagonal element includes zeroth order, rst order and
second order terms. The zeroth order term is a constant, since we dened in (E-6) the
coecients[()]and[()]in the interaction representation (in the usual represen-
tation, this term would correspond to the free evolution of the coherence at the Bohr
frequency). The rst order terms cancel out since we assumed that the average values
of the perturbation matrix elements are zero. To second order in, these coherences
are obtained by rst replacing, in (E-30),[(+ )]and[(+ )]by their series
expansion in power of. One must then take the average over the various values of the
random perturbation. This leads to:
~(+ )~() =
[
(0)
(+ )] [
(2)
(+ )]
+
[
(2)
(+ )] [
(0)
(+ )]
+
[
(1)
(+ )] [
(1)
(+ )]+ (E-31)
(i) Let us consider the rst two lines of this relation; we will show in (ii) below
why the third line can be left out. The zeroth order coecients,[
(0)
(+ )]and
[
(0)
(+ )], remain equal to their initial values, written[()]and[()]. Using the
second line of (E-11), we can write:
[
(2)
(+ )] [
(0)
(+ )]
=
1
~
2
()()

0
e (+) d
0
e (+) d (E-32)
whose average value yields the second line of (E-31).
The rst line of (E-31) is obtained by interchangingandin the second line of
1330

E. LONG-TIME BEHAVIOR FOR A TWO-LEVEL ATOM
(E-11), taking its complex conjugate, and multiplying the result by(). This leads to:
[
(0)
(+ )] [
(2)
(+ )]
=
1
~
2
()()

0
e (+)d
0
e (+) d (E-33)
This expression is identical to (E-32) since= and = .
We now average over the various values ofand . As we did before, we
can average independently()()and the double integral of (E-32). The computation
of the average value of this double integral has already been performed in
yields
1, where1is given in (E-20). This result must be doubled since the two
terms (E-32) and (E-33) are equal and add up. This nally leads to:
~(+ ) = ~()[12
1] = ~()[1(2)] (E-34)
or else:
~()

=
~(+ )~()

=(2) ~() (E-35)
Let us go back to the initial components()and()of the state vector. Relation
(B-8) shows that the elements of the density matrix constructed with these components
are:
() =
[()] [()]=e
( )}
[()] [()]=e ~() (E-36)
which leads to:
d
d
() = () +e
d
d
~() (E-37)
Now for short enough values of, the derivative of~()is given by (E-35). Using this
relation in (E-37), we get:
d
d
() =[ +( 2)]() (E-38)
This means that the coherence betweenand is damped at a rate, and that
its evolution frequency is shifted by2.
(ii) The third line of (E-31) is proportional to
(+ )(+ ), and is therefore
responsible for a coupling between~(+ )and~(+ ): the rate of variation of
~(+ )is a priori dependent on~(+ ). However, if the energy dierence
~ is suciently large, the unperturbed evolution frequencies of these nondiagonal
elements are very dierent, so that the eect of this coupling by the perturbation remains
negligible (it actually disappears within the secular approximation). Moreover, if the
statistical distribution of the random perturbation has a rotational symmetry around
theaxis
8
, this third line is equal to zero; this is demonstrated in a more general case
9
in ComplementXIII.
8
The axis is dened for the spin12associated with the two-level system.
9
The diagonal elements ofare generally not equal to zero. This means that the coherences can
also be coupled to the populations. In ComplementXIII,is supposed to be invariant with respect
to a rotation around any axis.
1331

CHAPTER XIII APPROXIMATION METHODS FOR TIME-DEPENDENT PROBLEMS
E-2-e. Energy shifts
The previous computation shows that the two states are shifted by the perturba-
tion, but it does not give the value of the shift of each state; relation
predicts that the dierence of the shifts~( )of the two states must be equal to
2~. We will now prove that the shifts are opposite,= and= +. The
most convenient demonstration uses the theory of the dressed atom, which will be
presented in ComplementXX. A more elementary demonstration is given below.
Imagine that there exists a third, so called spectator, state, which is not coupled
to the perturbation, so that its energy is not shifted by the perturbation. We assume
that there is a coherence~()between and , and study how it is modied by
the perturbation acting on. The computation of~(+ )is quite similar to that
of~(+ ), except that(+ )remains equal to(), up to any order insince
the stateis not coupled to the perturbation. Replacingbyin (E-31), the only
non-zero term is on the second line, equal to
(2)
(+ )(). The computation proceeds
as for
(2)
(+ )()and leads to the same result as (E-32) whereis replaced by.
Averaging overyields the same result as (E-34) whereis again replaced by; the
factor2in front of
1is no longer there, since the term of the rst line of (E-31) no
longer comes into play to double the value of the second line. We nally get:
~()

=
~(+ )~()

=(

2
) ~() (E-39)
The coherence between and is damped at a rate2and its evolution frequency,
equal to= in the absence of the perturbation, is changed by. Since the
state is not coupled to the perturbation, the statemust be shifted by=~.
As for the state, it is shifted by= +~, since relation (E-38) indicates that the
dierence between the two shifts ofand must be equal to2~.
Let us focus on the sign of, given by relation (E-22). We saw in
~
()
is real; we assume this function to be positive in an angular frequency domain centered
around=, and zero everywhere else. Relation (E-22) shows that:
0 ; 0 (E-40)
For , we have 0; when , relation (E-38) shows that the shift
decreases the energy dierence between the two statesand . For , we
have 0; it is now when 0that the shift decreases the energy dierence.
In both cases, and ifhas the same sign as, the energy dierence is decreased when
; in the opposite case, the energy levels get further from each other.
E-3. Broadband optical excitation of an atom
We now apply the previous results to the excitation of a two-level atom by broad-
band radiation. The radiation is described by an incoherent superposition of monochro-
matic elds with frequenciesspreading over an interval of width, and with random
phases. Consequently, the coupling between the atom and the radiation is a random
perturbation. The larger, the shorter the perturbation correlation time, as we shall
see below. We can directly use the results of
of the radiation by the atom, as well as the energy shifts due to the coupling between
the atom and the radiation
10
.
10
This problem is also studied in XIIIusing a dierent method. In that
complement, we shall sum the transition probabilities associated with each of the monochromatic waves
1332

E. LONG-TIME BEHAVIOR FOR A TWO-LEVEL ATOM
E-3-a. Correlation functions of the interaction Hamiltonian
The matrix element ()of the perturbationassociated with the atom-
radiation coupling can be written as:
() = ()= () (E-41)
whereis the electric dipole moment of the atom and()the electric eld of the
incident radiation
11
; we have set:
= (E-42)
and will assume, for the sake of simplicity, thatis real (this can be obtained by a change
of the relative phase ofand ). The correlation function of the perturbation is
then proportional to that of the electric eld:
()( ) =
2
()( ) (E-43)
This eld can be expanded on its Fourier components:
() =
1
2
+
de
~
() (E-44)
with, since the eld is real:
~
() =
~
() (E-45)
We assume that the phases of thecomponents are independent of each other, and
completely random. This leads to:
~
()
~
() =()( ) (E-46)
where()describes the spectral distribution of the incident radiation; the function()
is supposed to have a width.
Taking (E-45) and (E-46) into account, we can now write the correlation function
(E-43) as:
()( ) =
2
2
d dee
()
~
()
~
()
=
2
2
d()e (E-47)
We rst note that this function only depends on the dierence of times: the perturbation
is a stationary random function. Secondly, if the spectral distribution()of the inci-
dent radiation is a bell shaped curve of width, the time correlation function of the
perturbation decreases with a characteristic timeinversely proportional to this width:
1

(E-48)
This means that, if we assume that the atom is excited by a radiation with a broad
enough spectrum, the correlation timewill be short enough to fulll the conditions
necessary for applying the results of .
present in the incident radiation.
11
To simplify the equations, we shall ignore the vectorial nature of()et.
1333

CHAPTER XIII APPROXIMATION METHODS FOR TIME-DEPENDENT PROBLEMS
E-3-b. Absorption rates and light shifts
Relation (E-47) shows that:
() =
2
2
() (E-49)
Using (D-19) for the value of, we get:
=
2
}
2
() (E-50)
As we saw above,yields the transition rate per unit time between the states
and . This rate is proportional to(), i.e. to the Fourier transform of the time
correlation function of the perturbation, calculated at the frequencyof the transition.
This result is entirely dierent from the result obtained with a monochromatic radiation.
In this latter case, and at resonance, one expects a Rabi oscillation between the two
statesand .
When considering the excitation probability of an atom in its ground state, the
statesand are, respectively, the lower and upper states of the transition. This
means that:
=~ 0 (E-51)
The angular frequencyappearing in (E-50) is then negative: in an absorption pro-
cess, it is the Fourier components with negative frequenciesthat come into play (note
however that for an electric eld incosorsin, the positive and negative frequency
components have the same intensity, and the distinction is no longer essential). Further-
more, the previous calculations are still valid when0, a case that corresponds
to stimulated emission, or induced emission (see ). During this
process, the radiation stimulates the transition of the atom from an excited state to its
ground state. This treatment justies the introduction by Einstein ( )
of the coecientsand describing the absorption and stimulated emission in the
presence of black body radiation (which is broadband).
We can also evaluate the atomic energy shifts due to the presence of the radiation.
The results of
the statesand by the respective values~and+~. The light shift
is proportional (with a positive proportionality constant) to the following integral over
:
d()
1
(E-52)
These light shifts are proportional to the light intensity since they depend linearly on
(). Their sign depends on the detuning between the central frequencyof the incident
radiation and the frequencyof the atomic transition. As we have seen in ,
ifis larger that, meaning that the incident radiation is detuned towards the
blue, the energies of the two levels get closer under the eect of the radiation. We get
the opposite conclusion if the incident radiation is detuned towards the red. These light
shifts will be further discussed using the dressed atom approach in ComplementXX.
We will show that they also exist for an atom excited by monochromatic radiation, and
that they are useful tools for manipulating atoms and their motion.
1334

E. LONG-TIME BEHAVIOR FOR A TWO-LEVEL ATOM
References and suggestions for further reading:
Perturbation series expansion of the evolution operator: Messiah (1.17), Chap. XVII,
1 and 2.
Sudden or adiabatic modication of the Hamiltonian: Messiah (1.17), Chap. XVII,
II ; Schi (1.18), Chap. 8, 35.
Diagramatic representation of a perturbation series (Feynman diagrams) : Ziman
(2.26), Chap. 3 ; Mandl (2.9), Chaps. 12 to 14 ; Bjorkén and Drell (2.10), Chaps. 16 and
17.
1335

COMPLEMENTS OF CHAPTER XIII, READER'S GUIDE
AXIII: INTERACTION OF AN ATOM WITH AN
ELECTROMAGNETIC WAVE
Illustration of the general considerations of
of Chapter , using the important example
of an atom interacting with a sinusoidal electro-
magnetic wave. Introduces fundamental concepts
such as: spectral line selection rules, absorption
and induced emission of radiation, oscillator
strength... Although moderately dicult, can be
recommended for a rst reading, because of the
importance of the concepts introduced.
BXIII: LINEAR AND NON-LINEAR RESPONSE OF
A TWO-LEVEL SYSTEM SUBJECTED TO A SINU-
SOIDAL PERTURBATION
A simple model for the study of some non-linear
eects that appear in the interaction of an
electromagnetic wave with an atomic system
(saturation eects, multiple-quanta transitions,
etc.). More dicult thanXIII(graduate level);
should therefore be reserved for a subsequent
study.
CXIII: OSCILLATIONS OF A SYSTEM BETWEEN
TWO DISCRETE STATES UNDER THE EFFECT OF
A RESONANT PERTURBATION
Study of the behavior, over a long time interval, of
a system that has discrete energy levels, subjected
to a resonant perturbation. Completes, in greater
detail, the results of , which
are valid only for short times. Relatively simple.
DXIII: DECAY OF A DISCRETE STATE RESO-
NANTLY COUPLED TO A CONTINUUM OF FINAL
STATES
Study of the behavior, over a long time in-
terval, of a discrete state resonantly coupled
to a continuum of nal states. Completes
the results obtained in
(Fermi golden rule), which were established
only for short time intervals. Proves that the
probability of nding the particle in the discrete
level decreases exponentially, and justies the
concept of lifetimes introduced phenomeno-
logically in Complement III. Important for
its numerous physical applications; graduate level.
1337

EXIII: TIME-DEPENDENT RANDOM PERTURBA-
TION, RELAXATION
This complement provides a more detailed and
precise view of the study of
the eects of a random perturbation. The
motional narrowing condition is assumed to
be valid, which means that the memory time
of the perturbation is much shorter than the
time it takes for the perturbation to have a
signicant eect. This complement rst part,
this complement provides de general equations of
evolution of the density matrix. In a second part,
the theory is applied to an ensemble of spins12
coupled to a random isotropic perturbation. This
complement is important because of its numerous
applications: magnetic resonance, optics, etc.
FXIII: EXERCISES Exercise 10 can be done at the end of Comple-
ment XIII; it is a step by step study of the
eects of the external degrees of freedom of a
quantum mechanical system on the frequencies
of the electromagnetic radiation it can absorb
(Doppler eect, recoil energy, Mössbauer eect).
Some exercises (especially 8 and 9) are more di-
cult than others, but treat important phenomena.
1338

INTERACTION OF AN ATOM WITH AN ELECTROMAGNETIC WAVE
Complement AXIII
Interaction of an atom with an electromagnetic wave
1 The interaction Hamiltonian. Selection rules
1-a Fields and potentials associated with a plane electromagnetic
wave
1-b The interaction Hamiltonian at the low-intensity limit
1-c The electric dipole Hamiltonian
1-d The magnetic dipole and electric quadrupole Hamiltonians
2 Non-resonant excitation. Comparison with the elastically
bound electron model
2-a Classical model of the elastically bound electron
2-b Quantum mechanical calculation of the induced dipole moment1351
2-c Discussion. Oscillator strength
3 Resonant excitation. Absorption and induced emission
3-a Transition probability associated with a monochromatic wave
3-b Broad-line excitation. Transition probability per unit time
In , we studied the special case of a sinusoidally time-dependent
perturbation:() =sin. We encountered the resonance phenomenon which oc-
curs whenis close to one of the Bohr angular frequencies= ( )~of the
physical system under consideration.
A particularly important application of this theory is the treatment of an atom
interacting with a monochromatic wave. In this complement, we will use this example to
illustrate the general considerations of Chapter
concepts of atomic physics such as spectral line selection rules, absorption and induced
emission of radiation, oscillator strength, etc...
As in Chapter , we shall conne ourselves to rst-order perturbation calcula-
tions. Some higher-order eects in the interaction of an atom with an electromagnetic
wave (non-linear eects) will be taken up in ComplementXIII.
We shall begin () by analyzing the structure of the interaction Hamiltonian
between an atom and the electromagnetic eld. This will permit us to isolate the electric
dipole, magnetic dipole and electric quadrupole terms, and to study the corresponding
selection rules. Then we shall calculate the electric dipole moment induced by a non-
resonant incident wave () and compare the results obtained with those of the model of
the elastically bound electron. Finally, we shall study () the processes of absorption
and induced emission of radiation which appear in the resonant excitation of an atom.
Comment:
In all complements of Chapter
tromagnetic eld is treated classically, as a time-dependent perturbation acting on the atom.
In Chapter XX and its complements, a more elaborate study will be given with a full quan-
tum treatment of both the electromagnetic eld and the atom; the interaction hamiltonian is
1339

COMPLEMENT A XIII
then time-independent. This permits the description of physical eects such as the spontaneous
emission of photons by atoms in excited states, which does not appear when the eld is treated
classically.
1. The interaction Hamiltonian. Selection rules
1-a. Fields and potentials associated with a plane electromagnetic wave
Consider a plane electromagnetic wave
1
, of wave vectork(parallel to) and
angular frequency=. The electric eld of the wave is parallel toand the
magnetic eld, to(Fig.).E
z
y
O
x
k
B
Figure 1: The electric eld E and
magnetic eld B of a plane wave of
wave vectork.
For such a wave, it is always possible, with a suitable choice of gauge (cf.Ap-
pendix, ), to make the scalar potential(r)zero. The vector potential
A(r)is then given by the real expression:
A(r) =0ee
( )
+
0ee
( )
(1)
where0is a complex constant whose argument depends on the choice of the time origin.
We then have:
E(r) =
A(r) = 0ee
( )
0ee
( )
(2)
B(r) =rA(r) = 0ee
( )
0ee
( )
(3)
We shall choose the time origin such that the constant0is pure imaginary, and we set:
0=
2
(4a)
0=
2
(4b)
1
For the sake of simplicity, we shall conne ourselves here to the case of a plane wave. The results
obtained in this complement, however, can be generalized to an any electromagnetic eld.
1340

INTERACTION OF AN ATOM WITH AN ELECTROMAGNETIC WAVE
whereandare two real quantities such that:
== (5)
We then obtain:
E(r) =ecos( ) (6)
B(r) =ecos( ) (7)
andare therefore the amplitudes of the electric and magnetic elds of the plane wave
considered.
Finally, we shall calculate the Poynting vector
2
Gassociated with this plane wave:
G=0
2
EB (8)
ReplacingEandBin (8) by their expressions (6) and (7), and taking the time-average
value over a large number of periods, we obtain, using (5):
G=0
2
2
e (9)
1-b. The interaction Hamiltonian at the low-intensity limit
The preceding wave interacts with an atomic electron (of massand charge
) situated at a distancefromand bound to this pointby a central potential
()(created by a nucleus assumed to be motionless at). The quantum mechanical
Hamiltonian of this electron can be written:
=
1
2
[PA(R)]
2
+()SB(R) (10)
The last term of (10) represents the interaction of the spin magnetic moment of the
electron with the oscillating magnetic eld of the plane wave.A(R)andB(R)are
the operators obtained by replacing, in the classical expressions (1) and (3),,,by
the observables,,.
In expanding the square that appears on the right-hand side of (10), we should,
in theory, remember thatPdoes not generally commute with a function ofR. Such
a precaution is, however, unnecessary in the present case, since, asAis parallel to
[formula (1)], only thecomponent enters into the double product; nowcommutes
with thecomponent ofR, which is the only one to appear in expression (1) forA(R).
We can then take:
=0+() (11)
where:
0=
P
2
2
+() (12)
2
Recall that the energy ux across a surface element dperpendicular to the unit vectornis equal
toGnd.
1341

COMPLEMENT A XIII
is the atomic Hamiltonian, and:
() =
PA(R)SB(R) +
2
2
[A(R)]
2
(13)
is the interaction Hamiltonian with the incident plane wave [the matrix elements of()
approach zero when0approaches zero].
The rst two terms on the right-hand side of (13) depend linearly on0, and
the third one depends on it quadratically. With ordinary light sources, the intensity is
suciently low that the eect of the
2
0term can be neglected compared to that of the
0term. We shall therefore write:
() () +() (14)
with:
() =
PA(R) (15)
() =
SB(R) (16)
We shall evaluate the relative orders of magnitude of the matrix elements of()
and ()between two bound states of the electron. Those ofSare of the order of~,
andBis of the order of0[cf.formula (3)]. Thus:
()
()
~00
=
~
(17)
According to the uncertainty relations,~is, at most, of the order of atomic dimensions
(characterized by the Bohr radius,005

A).is equal to2, whereis the
wavelength associated with the incident wave. In the spectral domains used in atomic
physics (the optical or Hertzian domains),is much greater than0, so that:
()
()
0
1 (18)
1-c. The electric dipole Hamiltonian
. The electric dipole approximation. Interpretation
Using expression (1) forA(R), we can put()in the form:
() =
[0ee+
0ee] (19)
We now expand the exponential ein powers of:
e = 1
1
2
22
+ (20)
Sinceis of the order of atomic dimensions, we have, as above:
0
1 (21)
1342

INTERACTION OF AN ATOM WITH AN ELECTROMAGNETIC WAVE
We therefore obtain a good approximation forby retaining only the rst term of
expansion (20). Let be the operator obtained by replacing eby 1 on the
right-hand side of (19). Using (4-a), we get:
() =
sin (22)
()is called the electric dipole Hamiltonian. The electric dipole approximation,
which is based on conditions (18) and (21), therefore consists of neglecting()relative
to()and identifying()with ():
() () (23)
Let us show that, if we replace()by (), the electron oscillates as if it were
subjected to auniformsinusoidal electric eldecos, whose amplitude is that of the
electric eld of the incident plane wave evaluated at the point. Physically, this means
that the wave function of the bound electron is too localized aboutfor the electron
to feel the spatial variation of the electric eld of the incident plane wave. We shall
therefore calculate the evolution ofR(). Ehrenfest's theorem (cf.Chap., )
leads to:
d
d
R=
1
~
[R 0+ ]=
P
+esin
d
d
P=
1
~
[P 0+ ]=r()
(24)
EliminatingPfrom these two equations, we obtain, after a simple calculation:
d
2
d
2
R=r()+ecos (25)
which is indeed the predicted result: the center of the wave packet associated with the
electron moves like a particle of massand charge, subjected to both the central force
of the atomic bond [the rst term on the right-hand side of (25)] and the inuence of a
uniform electric eld [the second term of (25)].
Comment:
Expression (22) for the electric dipole interaction Hamiltonian seems rather unusual for
a particle of chargeinteracting with a uniform electric eldE=ecos. We would
tend to write the interaction Hamiltonian in the form:
() =DE= cos (26)
whereD=Ris the electric dipole moment associated with the electron.
Actually, expressions (22) and (26) are equivalent. We shall show that we can go
from one to the other by a gauge transformation (which does not modify the physical
content of quantum mechanics;cf.ComplementIII). The gauge used to obtain (22) is:
A(r) =
esin( ) (27a)
(r) = 0 (27b)
1343

COMPLEMENT A XIII
[to write (27a), we have replaced0by2in (1);cf.formula (4a)]. Now consider
the gauge transformation associated with the function:
(r) =
sin (28)
It leads to a new gaugeA characterized by:
A=A+r=e
[sin( ) + sin] (29a)
=
= cos (29b)
The electric dipole approximation amounts to replacingby 0 everywhere. We then
see that in this approximation:
A e
[sin() + sin] = 0 (30)
If, in addition, we neglect, as we did above, the magnetic interaction terms related to the
spin, we obtain, for the system's Hamiltonian:
=
1
2
(P A)
2
+() +(R)
=
P
2
2
+() +(R)
= 0+() (31)
where0is the atomic Hamiltonian given by (12), and:
() =(R) = cos= () (32)
is the usual form (26) of the electric dipole interaction Hamiltonian.
Recall that the state of the system is no longer described by the same ket when we go
from gauge (27) to gauge (29) (cf.ComplementIII). The replacement of ()by
()must therefore be accompanied by a change of state vector, the physical content,
of course, remaining the same.
In the rest of this complement, we shall continue to use gauge (27).
. The matrix elements of the electric dipole Hamiltonian
Later, we shall need the expressions for the matrix elements ofbetween
and , eigenstates of0of eigenvaluesand. According to (22), these matrix
elements can be written:
()=
sin (33)
It is simple to replace the matrix elements ofby those ofon the right-hand
side of (33). Insofar as we are neglecting all magnetic eects in the atomic Hamiltonian
[cf.expression (12) for0], we can write:
[ 0] =~
0
=~ (34)
1344

INTERACTION OF AN ATOM WITH AN ELECTROMAGNETIC WAVE
which yields:
[ 0]= 0 0
=( ) =
~
(35)
Introducing the Bohr angular frequency= ( )~, we then get:
= (36)
and, consequently:
()=
sin (37)
The matrix elements of()are therefore proportional to those of.
Comment:
It is the matrix element ofwhich appears in (37) because we have chosen the
electric eldE(r)parallel to. In practice, we may be led to choose a frame
related, not to the light polarization, but to the symmetry of the states
and . For example, if the atoms are placed in a uniform magnetic eldB0,
the most convenient quantization axis for the study of their stationary states
is obviously parallel toB0. The polarization of the electric eldE(r)can then
be arbitrary relative to. In this case, we must replace the matrix element of
in (37) by that of a linear combination of,and.
. Electric dipole transition selection rules
If the matrix element ofbetween the statesand is dierent from
zero, that is, if is non-zero
3
, the transition is said to be an
electric dipole transition. To study the transitions induced betweenand by the
incident wave, we can then replace()by (). If, on the other hand, the matrix
element of ()between and is zero, we must pursue the expansion of()
further, and the corresponding transition is either a magnetic dipole transition or an
electric quadrupole transition, etc...
4
(see following sections). Since()is much
larger than the subsequent terms of the power series expansion of()in0, electric
dipole transitions will be, by far, the most intense. In fact, most optical lines emitted by
atoms correspond to electric dipole transitions.
Let:
(r) = ()()
(r) = ()()
(38)
3
Actually, it suces for one of the three numbers , or to be dierent
from zero (cf.comment of above).
4
It may happen that all the terms of the expansion have zero matrix elements. The transition is
then said to be forbidden to all orders (it can be shown that this is always the case ifand both
have zero angular momenta).
1345

COMPLEMENT A XIII
be the wave functions associated withand . Since:
=cos=
43
0
1() (39)
the matrix element ofbetween and is proportional to the angular integral:
d ()
0
1() () (40)
According to the results of ComplementX, this integral is dierent from zero only if:
= 1 (41)
and:
= (42)
Actually, it would suce to choose another polarization of the electric eld (for example,
parallel toor; see comment of ) to have:
= 1 (43)
From (41), (42) and (43), we obtain the electric dipole transition selection rules:
= =1 (44a)
= =10+1 (44b)
Comments:
()is an odd operator. It can connect only two states of dierent parities.
Since the parities ofand are those ofand,= must be
odd, as is compatible with (44a).
()If there exists a spin-orbit coupling()LSbetweenLandS(cf.Chap.,
1-b-), the stationary states of the electron are labeled by the quantum numbers
,,, (withJ=L+S). The electric dipole transition selection rules can
be obtained by looking for the non-zero matrix elements ofRin the
basis. By using the expansions of these basis vectors on the kets (cf.
ComplementX, ), we nd, starting with (44a) and (44b), the selection rules:
= 01 (44a)
=1 (44b)
= 01 (44c)
Note that a= 0transition is not forbidden [unless== 0;cf.note on
the preceding page]. This is due to the fact thatis not related to the parity of
the level.
Finally, we point out that selection rules (44a, , ) can be generalized to
many-electron atoms.
1346

INTERACTION OF AN ATOM WITH AN ELECTROMAGNETIC WAVE
1-d. The magnetic dipole and electric quadrupole Hamiltonians
. Higher-order terms in the interaction Hamiltonian
The interaction Hamiltonian given by (14) can be written in the form:
() =() +() = () + [() ()] +() (45)
Thus far, we have studied(). As we have seen, the ratio of() ()and
()to ()is of the order of0.
To calculate() (), we simply replacee bye 1 +
in (19), which yields:
() () =
[ 0e
0e]+ (46)
or, using (4b):
() () =
cos + (47)
If we writein the form:
=
1
2
( ) +
1
2
( + ) =
1
2
+
1
2
( + ) (48)
we obtain, nally:
() () =
2
cos
2
cos[+ ] + (49)
In the expression for()[formulas (16) and (3)], it is entirely justied to replace
e by 1. We thus obtain a term of order0relative to(), that is, of the same
order of magnitude as() ():
() =
cos+ (50)
Substituting (49) and (50) into (45) and grouping the terms dierently, we obtain:
() = () + () + () + (51)
with:
=
2
(+ 2)cos (52)
=
2
( + )cos (53)
[we have replacedby in (53)]. and (which are,a priori, of the same
order of magnitude) are, respectively, the magnetic dipole and electric quadrupole Hamil-
tonians.
1347

COMPLEMENT A XIII
. Magnetic dipole transitions
The transitions induced byare called magnetic dipole transitions.rep-
resents the interaction of the total magnetic moment of the electron with the oscillating
magnetic eld of the incident wave.
The magnetic dipole transition selection rules can be obtained by considering the
conditions which must be met byand in order for to have a non-zero
matrix element between these two states. Since neithernorchanges the quantum
number, we must have, rst of all,= 0.changes the eigenvalueofby1,
which gives =1.changes the eigenvaluesofby1, so that =1.
Note, furthermore, that if the magnetic eld of the incident wave were parallel to,
we would have = 0and = 0. Grouping these results, we obtain the magnetic
dipole transition selection rules:
= 0
=10
=10
(54)
Comment:
In the presence of a spin-orbit coupling, the eigenstates of0are labeled by the
quantum numbersand. Sinceanddo not commute withJ
2
, can
connect states with the samebut dierent. By using the addition formulas
for an angular momentumand an angular momentum 1/2 (cf.ComplementX,
), it can easily be shown that selection rules (54) become:
= 0
=10
=10
(55)
Note that the hyperne transition= 0 = 1of the ground state of the
hydrogen atom (cf.Chap., ) is a magnetic dipole transition, since the
components ofShave non-zero matrix elements between the= 1states and the
= 0 = 0state.
. Electric quadrupole transitions
Using (34), we can write:
+ = + =
~
[ 0] + [0]
=
~
( 0 0) (56)
from which we obtain, as in (36):
()=
2
cos (57)
The matrix element of ()is therefore proportional to that of, which is
a component of the electric quadrupole moment of the atom (cf.ComplementX). In
1348

INTERACTION OF AN ATOM WITH AN ELECTROMAGNETIC WAVE
addition, the following quantity appears in (57):
== (58)
which, according to (2), is of the order of. The operator ()can therefore
be interpreted as the interaction of the electric quadrupole moment of the atom with the
gradient
5
of the electric eld of the plane wave.
To obtain the electric quadrupole transition selection rules, we simply note that,
in therrepresentation,is a linear superposition of
21
2()and
2 1
2
().
Therefore, in the matrix element there appear angular integrals:
d ()
1
2
()() (59)
which, according to the results of ComplementX, are dierent from zero only if= 0,
2and=1. This last relation becomes=21, 0 when we consider an
arbitrary polarization of the incident wave (cf.comment of 1-c-), and the electric
quadrupole transition selection rules can be written, nally:
= 02
= 012
(60)
Comments:
() and are even operators and can therefore connect only states of
the same parity, which is compatible with (54) and (60). For a given tran-
sition, and are never in competition with. This facilitates
the observation of magnetic dipole and electric quadrupole transitions.
Most of the transitions that occur in the microwave or radio-frequency
domain in particular, magnetic resonance transitions (cf.ComplementIV)
are magnetic dipole transitions.
()For a= 0,= 01transition, the two operatorsand
simultaneously have non-zero matrix elements. However, it is possible to
nd experimental conditions under which only magnetic dipole transitions
are induced. All we need to do is place the atom, not in the path of a plane
wave, but inside a cavity or radiofrequency loops, at a point whereBis large
but the gradient ofEis negligible.
()For a= 2transition, cannot be in competition with, and we
have a pure quadrupole transition. As an example of a quadrupole transition,
we can mention the green line of atomic oxygen (5577

A), which appears in
the aurora borealis spectrum.
()If we pursued the expansion offurther, we would nd electric octupole
and magnetic quadrupole terms, etc.
In the rest of this complement, we shall conne ourselves to electric dipole tran-
sitions. In the next Complement,XIII, on the other hand, we shall consider a
magnetic dipole transition.
5
It is normal for the electric eld gradient to appear, since()was obtained by expanding the
potentials in a Taylor series in the neighborhood of
1349

COMPLEMENT A XIII
2. Non-resonant excitation. Comparison with the elastically bound electron
model
In this section, we shall assume that the atom, initially in the ground state0, is excited
by a non-resonant plane wave:coincides with none of the Bohr angular frequencies
associated with transitions from0.
Under the eect of this excitation, the atom acquires an electric dipole moment
D()which oscillates at the angular frequency(forced oscillation) and is proportional
towhenis small (linear response). We shall use perturbation theory to calculate
this induced dipole moment, and we shall show that the results obtained are very close
to those found with the classical model of the elastically bound electron.
This model has played a very important role in the study of the optical properties
of materials. It enables us to calculate the polarization induced by the incident wave
in a material. This polarization, which depends linearly on the eld, behaves like a
source term in Maxwell's equations. When we solve these equations, we nd plane waves
propagating in the material at a velocity dierent from. This allows us to calculate the
refractive index of the material in terms of various characteristics of elastically bound
electrons (natural frequencies, number per unit volume, etc.). Thus, we see that it is
very important to compare the predictions of this model (which we shall review in a)
with those of quantum mechanics.
2-a. Classical model of the elastically bound electron
. Equation of motion
Consider an electron subjected to a restoring force directed towards the point
and proportional to the displacement. In the classical Hamiltonian corresponding to (12),
we then have:
() =
1
2
2
0
2
(61)
where0is the electron's natural angular frequency.
If we make the same approximations, using the classical interaction Hamiltonian, as
those which enabled us to obtain expression (22) for()(the electric dipole approxi-
mation) in quantum mechanics, a calculation similar to that of [cf.equation (25)]
yields the equation of motion:
d
2
d
2
+
2
0=cos (62)
This is the equation of a harmonic oscillator subject to a sinusoidal force.
. General solution
The general solution of (62) can be written:
=cos(0 ) +
(
2
0
2
)
cos (63)
whereandare real constants which depend on the initial conditions. The rst term
of (63),cos(0 ), represents the general solution of the homogeneous equation (the
1350

INTERACTION OF AN ATOM WITH AN ELECTROMAGNETIC WAVE
electron's free motion). The second term is a particular solution of the equation (forced
motion of the electron).
We have not, thus far, taken damping into account. Without going into detailed
calculations, we shall cite the eects of weak damping: after a certain time, it causes
the natural motion to disappear and very slightly modies the forced motion (provided
that we are suciently far from resonance:0 1). We shall therefore retain
only the second term of (63):
=
cos
(
2
0
2
)
(64)
Comment:
Far from resonance, the exact damping mechanism is of little importance, provided that
it is weak. We shall not, therefore, take up the problem of the exact description of this
damping, either in quantum or in classical mechanics. We shall merely use the fact that
it exists to ignore the free motion of the electron.
It would be dierent for a resonant excitation: the induced dipole moment would then
depend critically on the exact damping mechanism (spontaneous emission, thermal relax-
ation, etc.). This is why we shall not try to calculateD()in
excitation). We shall be concerned only with calculating the transition probabilities.
In ComplementXIII, we shall study a specic model of a system placed in an electro-
magnetic wave and at the same time subject to dissipative processes (Bloch equations
of a system of spins). We shall then be able to calculate the induced dipole moment for
any exciting frequency.
. Susceptibility
Let= be the electric dipole moment of the system. According to (64), we
have:
==
2
(
2
0
2
)
cos=cos (65)
where the susceptibilityis given by:
=
2
(
2
0
2
)
(66)
2-b. Quantum mechanical calculation of the induced dipole moment
We shall begin by calculating, to rst order in, the state vector()of the
atom at time. We shall choose for the interaction Hamiltonian, the electric dipole
Hamiltonian given by (22). In addition, we shall assume that:
(= 0)= 0 (67)
1351

COMPLEMENT A XIII
We apply the results of , replacingby
and
by0. This leads to
6
:
()= e
0~
0+
=0
(1)
() e
~
(68)
or, using (C-4) of Chapter ()by the global phase factor e
0~
,
which has no physical importance:
()= 0+
=0
2~
0
e
0
e
0+
e
0
e
0
(69)
From this, we nd()and () =() (). In the calculation of this
average value, we retain only the terms linear in, and we neglect all those that oscillate
at angular frequencies0(the natural motion, which would disappear if we took
weak damping into account). Finally, replacing 0by its expression in terms of
0[cf.equation (36)], we nd:
() =
2
2
~
cos
0 0
2
2
0
2
(70)
2-c. Discussion. Oscillator strength
. The concept of oscillator strength
We set:
0=
2 0 0
2
~
(71)
0is a real dimensionless number, characteristic of the0 transition and called
the oscillator strength
7
of this transition. If0is the ground state,0is positive, like
0.
Oscillator strengths satisfy the following sum rule (the Thomas-Reiche-Kuhn sum
rule):
0= 1 (72)
This can be shown as follows. Using (36), we can write:
0=
1
~
0 0
1
~
0 0 (73)
The summation overcan be performed by using the closure relation relative to the
basis, and we get:
0=
1
~
0( )0= 00= 1 (74)
6
Since is odd,0 ()0is zero, so
(1)
0
() = 0.
7
The operatorenters into (71 . It would,
however, be possible to give a general denition of the oscillator strength, independent of the polarization
of the incident wave.
1352

INTERACTION OF AN ATOM WITH AN ELECTROMAGNETIC WAVE
. The quantum mechanical justication for the elastically bound electron model
We substitute denition (71) into (70) and multiply the expression so obtained by
the numberof atoms contained in a volume whose linear dimensions are much smaller
than the wavelengthof the radiation. The total electric dipole moment induced in this
volume can then be written:
() = 0
2
(
2
0
2
)
cos (75)
Comparing (75) and (), we see that it is like havingclassical oscillators [since
0=according to (72)] whose natural angular frequencies are not all the same
since they are equal to the various Bohr angular frequencies of the atom associated
with the transitions from0. According to (75), the proportion of oscillators with the
angular frequency0is0.
Thus, for a non-resonant wave, we have justied the classical model of the elasti-
cally bound electron. Quantum mechanics gives the frequencies of the various oscillators
and the proportion of oscillators that have a given frequency. This result shows the
importance of the concept of oscillator strength and enables us to understanda poste-
rioriwhy the elastically bound electron model was so useful in the study of the optical
properties of materials.
3. Resonant excitation. Absorption and induced emission
3-a. Transition probability associated with a monochromatic wave
Consider an atom initially in the stateplaced in an electromagnetic wave
whose angular frequency is close to a Bohr angular frequency.
The results of
to the calculation of the transition probabilityP(;). We nd, using expression (37)
(thus making the electric dipole approximation):
P(;) =
2
4~
2
2
22
( ) (76)
where:
( ) =
sin[( )2]
( )2
2
(77)
We have already discussed the resonant nature ofP(;)in Chapter . At
resonance,P(;)is proportional to
2
, that is, to the incident ux of electromagnetic
energy [cf.formula (9)].
Comments:
()If instead of using the gauge (27), leading to the matrix element (37), we had
used the gauge (29) leading to the Hamiltonian (32), the factor( )
2
in
(76) would be missing. The fact that the results are dierent is not at all
surprising. The statesand , and consequentlyP(;), do not have
the same physical meaning in the two gauges.
1353

COMPLEMENT A XIII
()However, as , the diraction function( )tends towards
( ), and the factor( )
2
approaches unity. This leads to the same
probability densityP(;)in the two gauges. This result can be easily
understood if we consider the incident electromagnetic wave to be a quasi-
monochromatic wave packet of very large but nite spatial extent, rather
than a plane wave extending to innity. When theEeld seen by
the atom tends towards zero and the gauge transformation associated with
the functiondened in (28) tends towards unity. Consequentlyand
each represent the same physical states in the two gauges.
()Obviously, it is also possible to consider the transition probability between
two well-dened energy states of the atomic system for a nite time inter-
val. In this case, the eigenstatesand of the atomic Hamiltonian0
written in (12) only represent states of well-dened atomic energy (kinetic
plus potential) in the gauge (29) whereAis zero [see (30)] andp
2
2rep-
resents the kinetic energy. The same physical states would be represented
in gauge (27) by the statesexp[ (r)~]andexp[ (r)~]
respectively. For nite, calculations are therefore simpler in the gauge (29).
Since in the rest of this complement we replace( )by( )[see
(79)], we will be considering the limitfor which the above diculties
disappear.
3-b. Broad-line excitation. Transition probability per unit time
In practice, the radiation which strikes the atom is very often non-monochromatic.
We shall denote by()dthe incident ux of electromagnetic energy per unit surface
within the interval[+ d]. The variation of()with respect tois shown in
Figure.is the excitation line width. Ifis innite, we say that we are dealing with
a white spectrum.
The dierent monochromatic waves which constitute the incident radiation are
generally incoherent: they have no well-dened phase relation. The total transition
probability
Pcan therefore be obtained by summing the transition probabilities asso-
ciated with each of these monochromatic waves. We must, consequently, replace
2
by
2()d 0in (76) [formula (9)] and integrate over. This gives:
P() =
2
20~
2
2
d
2
()( ) (78)
We can then proceed as in
appears in (78). Compared to a function ofwhose width is much larger than4, the
function( )(see Figure ) behaves like( ). Ifis large
enough to make4 (: excitation line width) while remaining small enough for
the perturbation treatment to be valid, we can, in (78), assume that:
( )2( ) (79)
which yields:
P() =
2
0~
2
2
() (80)
1354

INTERACTION OF AN ATOM WITH AN ELECTROMAGNETIC WAVEω
fi
ℐ(ω)
ω
∆
Figure 2: The spectral distribution of the incident ux of electromagnetic energy per unit
surface.is the width of this spectral distribution.
We can write (80) in the form:
P() = () (81)
where:
=
4
2
~
2
(82)
andis the ne-structure constant:
=
2
40
1
~
=
2
~
1
137
(83)
This result shows that
P()increases linearly with time. Thetransition proba-
bility per unit timeis therefore equal to:
= () (84)
is proportional to the value of the incident intensity for the resonance frequency,
to the ne-structure constant, and to the square of the modulus of the matrix element
of, which is related [by (71)] to the oscillator strength of the transition.
In this complement, we have considered the case of radiation propagating along
a given direction with a well-dened polarization state. By averaging the coecients
over all propagation directions and over all possible polarization states, we could
introduce coecients, analogous to the coecients, dening the transition prob-
abilities per unit time for an atom placed in isotropic radiation. The coecients
(and) are none other than the coecients introduced by Einstein to describe the
absorption (and induced emission). Thus, we see how quantum mechanics enables us to
calculate these coecients.
1355

COMPLEMENT A XIII
Comment:
A third coecient,, was introduced by Einstein to describe the spontaneous emission
of a photon, which occurs when the atom falls back from the upper stateto the
lower state. The theory presented in this complement does not explain spontaneous
emission. In the absence of incident radiation, the interaction Hamiltonian is zero, and
the eigenstates of0(which is then the total Hamiltonian) are stationary states.
Actually, the preceding model is insucient, since it treats asymmetrically the atomic
system (which is quantized) and the electromagnetic eld (which is considered classi-
cally). When we quantize both systems, we nd, even in the absence of incident photons,
that the coupling between the atom and the electromagnetic eld continues to have ob-
servable eects (a simple interpretation of these eects is given in ComplementV). The
eigenstates of0are no longer stationary states, since0is no longer the Hamiltonian
of the total system, and we can indeed calculate the probability per unit time of sponta-
neous emission of a photon (cf. Chap. XX, C-3) . Quantum mechanics therefore also
enables us to obtain the Einstein coecient.
References and suggestions for further reading:
See, for example: Schi (1.18), Chap. 11; Bethe and Jackiw (1.21), Part II,
Chaps. 10 and 11; Bohm (5.1), Chap. 18, 12 to 44.
For the elastically bound electron model: Berkeley 3 (7.1), supplementary topic 9;
Feynman I (6.3), Chap. 31 and Feynman II (7.2), Chap. 32.
For Einstein coecients: the original article (1.31), Cagnac and Pebay-Peyroula
(11.2), Chap. III and Chap. XIX, 4.
For the exact denition of oscillator strength: Sobel'man (11.12), Chap. 9, 31.
For atomic multipole radiation and its selection rules: Sobel'man (11.12), Chap. 9,
32.
1356

LINEAR AND NON-LINEAR RESPONSES OF A TWO-LEVEL SYSTEM
Complement BXIII
Linear and non-linear responses of a two-level system
subject to a sinusoidal perturbation
1 Description of the model
1-a Bloch equations for a system of spin 1/2's interacting with a
radiofrequency eld
1-b Some exactly and approximately soluble cases
1-c Response of the atomic system
2 The approximate solution of the Bloch equations of the
system
2-a Perturbation equations
2-b The Fourier series expansion of the solution
2-c The general structure of the solution
3 Discussion
3-a Zeroth-order solution: competition between pumping and re-
laxation
3-b First-order solution: the linear response
3-c Second-order solution: absorption and induced emission
3-d Third-order solution: saturation eects and multiple-quanta
transitions
4 Exercises: applications of this complement
In the preceding complement, we applied rst-order time-dependent perturbation
theory to the treatment of some eects produced by the interaction of an atomic system
and an electromagnetic wave: appearance of an induced dipole moment, induced emission
and absorption processes, etc.
We shall now consider a simple example, in which it is possible to pursue the per-
turbation calculations to higher orders without too many complications. This will allow
us to demonstrate some interesting non-linear eects: saturation eects, non-linear sus-
ceptibility, the absorption and induced emission of several photons, etc. In addition, the
model we shall describe takes into account (phenomenologically) the dissipative coupling
of the atomic system with its surroundings (the relaxation process). This will enable us
to complete the results related to the linear response obtained in the preceding com-
plement. For example, we shall calculate the atom's induced dipole moment, not only
far from resonance, but also at resonance.
Some of the eects we are going to describe are objects of a great deal of research.
Their study necessitates very strong electromagnetic elds, which cab be obtained only
with lasers. New branches of research have thus appeared with lasers: quantum electron-
ics, non-linear optics, etc. The calculation methods described in this complement (for a
very simple model) are applicable to these problems.
1357

COMPLEMENT B XIII
Comment:
The Comment at the end of the introduction of ComplementXIIIapplies to the present com-
plement as well: here, we limit ourselves to a semi-classical treatment, where the atomic system
is treated quantum mechanically but the electromagnetic eld classically. A full quantum treat-
ment of both systems will be given in Chapter XX; see in particular ComplementXX, which
describes the dressed atom method and non-perturbative calculations.
1. Description of the model
1-a. Bloch equations for a system of spin 1/2's interacting with a radiofrequency eld
We shall return to the system described in IV: a system
of spin 1/2's placed in a static eldB0parallel to, interacting with an oscillating
radiofrequency eld and subject to pumping and relaxation processes.
If()is the total magnetization of the spin system contained in the cell (Fig.
of ComplementIV), we showed in ComplementIVthat:
d
d
() =
0
1
() + ()B() (1)
The rst term on the right-hand side describes the preparation, or the pumping of the
system:spins enter the cell per unit time, each one with an elementary magnetization
0parallel to. The second term arises from relaxation processes, characterized by the
average timerequired for a spin either to leave the cell or have its direction changed
by collision with the walls. Finally, the last term of (1) corresponds to the precession of
the spins about the total magnetic eld:
B() =0e+B1() (2)
B()is the sum of a static eld0eparallel toand a radiofrequency eldB1()of
angular frequency.
Comments:
()The transitions which we shall study in this complement (which connect the
two states+and of each spin 1/2) are magnetic dipole transitions.
()One could question our using expression (1) relative to average values rather
than the Schrödinger equation. We do so because we are studying a statistical
ensemble of spins coupled to a thermal reservoir (via collisions with the cell
walls). We cannot describe this ensemble in terms of a state vector: we must
use a density operator (see ComplementIII). The equation of motion of
this operator is called a master equation and we can show that it is exactly
equivalent to (1) (see ComplementIV, , and ComplementIV,
where we show that the average value of the magnetization determines the
density matrix of an ensemble of spin 1/2's).
It turns out that the master equation satised by the density operator
and the Schrödinger equation studied in
structure as (1): a linear dierential equation, with constant or sinusoidally
varying coecients. The approximation methods we describe in this chapter
are, therefore, applicable to any of these equations.
1358

LINEAR AND NON-LINEAR RESPONSES OF A TWO-LEVEL SYSTEM
1-b. Some exactly and approximately soluble cases
If the radiofrequency eldB1()is rotating, that is, if:
B1() =1(ecos+esin) (3)
equation (1) can be solved exactly [changing to the frame which is rotating withB1
transforms (1) into a time-independent linear dierential system]. The exact solution of
(1) corresponding to such a situation is given in IV.
Here, we shall assumeB1to be linearly polarized along:
B1() =1ecos (4)
In this case, it is not possible
1
to nd an exact analytic solution of equation (1) (there is
no transformation equivalent to changing to the rotating frame). We shall see, however,
that a solution can be found in the form of a power series expansion inB1.
Comment:
The calculations we shall present here for spin 1/2's can also be applied to other
situations in which we can conne ourselves to two levels of the system and ignore
all others. We know (cf.ComplementIV) that we can associate a ctitious spin
1/2 with any two-level system. The problem considered here is therefore that of
an arbitrary two-level system subject to a sinusoidal perturbation.
1-c. Response of the atomic system
The set of terms which, in,,, depend on1constitute the response
of the atom to the electromagnetic perturbation. They represent the magnetic dipole
moment induced in the spin system by the radiofrequency eld. We shall see that such a
dipole moment is not necessarily proportional to1; the terms in1represent the linear
response, and the others (terms in
2
1,
3
1, ...), the non-linear response. In addition, we
shall see that the induced dipole moment does not oscillate only at the angular frequency
, but also at its various harmonics(= 0, 2, 3, 4, ...).
It is easy to see why we should be interested in calculating the response of the atomic
system. Such a calculation is useful for the theory of the propagation of an electromagnetic
wave in a material, or for the theory of atomic oscillators, masers or lasers.
Consider an electromagnetic eld. Because of the coupling between this eld and the
atomic system, a polarization appears in the material, due to the atomic dipole moments (arrow
directed towards the right in Figure). This polarization acts like a source term in Maxwell's
equations and contributes to the creation of the electromagnetic eld (arrow directed towards
the left in Figure). When we close the loop, that is, when we take the eld so created
to be equal to the one with which we started, we obtain the wave propagation equations in
the material (refractive index) or the oscillator equations (in the absence of external elds, an
1
A linearly polarized eld can be obtained as a superposition of a left and a right circular components.
It would be possible to nd an exact solution for each of these components taken separately. However,
equation (1) is not linear, in the sense that a solution corresponding to (4) cannot be obtained by
superposing two exact solutions, one of which corresponds to (3) and the other one to the eld rotating
in the opposite direction [in the term Bthat appears on the right-hand side of (1),depends
onB1].
1359

COMPLEMENT B XIII
electromagnetic eld may appear in the material, if there is sucient amplication: the system
becomes unstable and can oscillate spontaneously). In this complement, we shall be concerned
only with the rst step of the calculation (the atomic response).Atomic
dipole
moments
Response of the atomic system
Maxwell’s equations
Electromagnetic
field
Figure 1: Schematic representation of the calculations to be performed in studying the
propagation of an electromagnetic wave in a material (or the operation of an atomic
oscillator, a laser or a maser). We begin by calculating the dipole moments induced in
the material by a given electromagnetic eld (the response of the atomic system). The
corresponding polarization acts like a source term in Maxwell's equations and contributes
to the creation of the electromagnetic eld. We then take the eld obtained to be equal to
the one with which we started.
2. The approximate solution of the Bloch equations of the system
2-a. Perturbation equations
As in ComplementIV, we set:
0= 0
1= 1
(5)
(6)
~0represents the energy dierence of the spin states+and (Fig.). Substituting
(4) into (2), and (2) into (1), we obtain, after a simple calculation:
d
d
= 0+
1
2
cos( +) (7a)
d
d
=+ 0 1cos (7b)
with:
= (8)
1360

LINEAR AND NON-LINEAR RESPONSES OF A TWO-LEVEL SYSTEM +
–
ħω
Figure 2: Energy levels of a spin 1/2 in a static magnetic eldB0;0is the Larmor
angular frequency in the eldB0.
Note that the source term0exists only in the equation of motion of, since
0is parallel to, and the pumping is said to be longitudinal
2
. We also point out
that the relaxation time can be dierent for the longitudinal components () and the
transverse components () of the magnetization. For the sake of simplicity, we shall
choose a single relaxation time here.
Equations (7a) and (7b), called the Bloch equations, cannot be solved exactly.
We shall therefore determine their solution in the form of a power series expansion in1:
=
(0)
+1
(1)
+
2
1
(2)
++
1
()
+ (9a)
=
(0)
+1
(1)
+
2
1
(2)
++
1
()
+ (9b)
Substituting (9a) and (9b) into (7a) and (7b), and setting equal the coecients of terms
in
1, we obtain the following perturbation equations:
n=0 :
d
d
(0)
= 0
1
(0)
(10a)
d
d
(0)
=
1
(0)
0
(0)
(10b)
= 0 :
d
d
()
=
1
()
+
2
cos[
(1) (1)
+] (11a)
d
d
()
=
1
()
0
()
cos
(1)
(11b)
2
In certain experiments, the pumping is transverse (0is perpendicular toB0). See exercise 1 at
the end of this complement.
1361

COMPLEMENT B XIII
2-b. The Fourier series expansion of the solution
Since the only time-dependent terms on the right-hand side of (10) and (11) are
sinusoidal, the steady-state solution of (10) and (11) is periodic, of period2. We can
expand it in a Fourier series:
()
=
+
=
()
e (12a)
()
=
+
=
()
e (12b)
()
and
()
represent theFourier components of theth-order solution.
By taking
()
real and
()
+and
()
as complex conjugates of each other,
we obtain the following reality conditions:
()
=
()
(13a)
()
=
()
(13b)
Substituting (12a) and (12b) into (10) and (11), and setting equal to zero the
coecient of each exponentiale, we nd:
= 0 :
(0)
0
= 0
(0)
= 0 if = 0
(0)
= 0 for any
(14)
= 0 :
+
1
()
=
4
(1)
+1
+
(1)
1
(1)
+1 +
(1)
1 +
(15a)
( 0) +
1
()
=
2
(1)
+1
+
(1)
1
(15b)
These algebraic equations can be solved immediately:
()
=
4 +
1
(1)
+1
+
(1)
1
(1)
+1 +
(1)
1 +
(16a)
()
=
2( 0) +
1
(1)
+1
+
(1)
1
(16b)
1362

LINEAR AND NON-LINEAR RESPONSES OF A TWO-LEVEL SYSTEM
Thus, expressions (16) give theth-order solution explicitly in terms of the(1)th-
order solution. Since the zeroth-order solution is known [cf.equations (14)], the problem
is, in theory, entirely solved.
2-c. The general structure of the solution
It is possible to arrange the various terms of the expansion of the solution in a
double-entry table in which the perturbation orderlabels the columns and the degree
of the harmonicbeing considered labels the rows. To zeroth-order, only
(0)
0
is
dierent from zero. By iteration, using (16), we can deduce the other non-zero higher-
order terms (table I), thus obtaining a tree-like structure. The following properties can
be found directly by recurrence, using (16):
()At even perturbation orders, only the longitudinal magnetization is modied; at
odd orders, only the transverse magnetization.
()At even perturbation orders, only the even harmonics are involved; at odd orders,
only the odd harmonics.
()For each value of, the values ofto be retained are,2, ...+ 2,
Table I:Double-entry table indicating theFourier components of the magnetization
that are non-zero to theth perturbation order in1.
1363

COMPLEMENT B XIII
Comment:
This structure is valid only for a particular polarization of the radiofrequency eld
B1()(perpendicular toB0). Analogous tables could be constructed for other
radiofrequency polarizations.
3. Discussion
We shall now interpret the results of this calculation, through third order.
3-a. Zeroth-order solution: competition between pumping and relaxation
According to (14), the only non-zero zeroth-order component is:
(0)
0
= 0 (17)
In the absence of radiofrequency elds, there is only a static longitudinal magnetization
(= 0). Since, is proportional to the population dierence of the states+and
shown in Figurecf.ComplementIV), it can also be said that the pumping populates
these two states unequally.
The larger the number of particles entering the cell (the more ecient the pumping)
and the longer(the slower the relaxation), the larger
(0)
0
. The zeroth-order solution
(17) therefore describes the dynamic equilibrium resulting from competition between the
pumping and relaxation processes.
From now on, in order to simplify the notation, we shall set:
0=
(0)
0
(18a)
=
1
(18b)
3-b. First-order solution: the linear response
To rst order, only the transverse magnetizationis dierent from zero. Since
+= , it suces to study+.
. Motion of the transverse magnetization
According to Table I, for= 1, we have=1. Setting= 1and=1in
(16b), using (18), we get:
(1)
1 +=
0
2
1
0 +
(19a)
(1)
1+=
0
2
1
0++
(19b)
Substituting these expressions into (12b) and then into (9b), we obtain+to rst
order in1:
+=1
0
2
e
0 +
+
e
0++
(20)
1364

LINEAR AND NON-LINEAR RESPONSES OF A TWO-LEVEL SYSTEM
The point representing+describes the same motion in the complex plane as the
projectionofin the plane perpendicular toB0. According to (20), this motion
results from the superposition of two circular motions with the same angular velocity,
one of them right circular (theeterm) and the other left circular (thee term).
The resulting motion, in the general case, is therefore elliptical.
. Existence of two resonances
The right circular motion has a maximum amplitude when0=, and the left
circular motion, when0= . therefore presents two resonances (while for a
rotating eld, there was a single resonance; see ComplementIV). The interpretation
of this phenomenon is as follows: the linear radiofrequency eld can be broken down
into a left and a right circular eld, each of which induces a resonance; since the rotation
directions are opposed, the static eldsB0for which these resonances appear are opposed.
. Linear susceptibility
Near a resonance (0 , for example), we can neglect the non-resonant term in
(20). We then get:
+
0
1
0
2
e
0 +
(21)
+is therefore proportional to the rotating radiofrequency eld component in the di-
rection corresponding to the resonance,1e2in this case.
The ratio of+to this component is called the linear susceptibility():
() = 0
1
0 +
(22)
()is a complex susceptibility because of the existence of a phase dierence between
and the rotating component of the radiofrequency eld responsible for the resonance.
The square of the modulus of()has the classical resonant form in the neigh-
borhood of=0(Fig.), over an interval of width:
= 2=
2
(23)
The longer the relaxation time, the sharper the resonance curve. From now on, we
shall assume that the two resonances0=and0= are completely separated, i.e.
that:
= 1 (24)
The phase dierence varies from 0 towhen we pass through resonance. It is
equal to2at resonance: it is whenand the rotating component are out of phase
by2that the work of the couple exerted by the eld onis maximal. The sign of
this work depends on the sign of0, that is, on that of0: it depends on whether the
spin states of the entering particles are+or. In one case (spins entering in the
lower level), the work is furnished by the eld, and energy is transferred from the eld to
the spins (absorption). In the opposite case (particles entering in the higher level), the
work is negative, and energy is transferred from the spins to the eld (induced emission).
The latter situation occurs in atomic ampliers and oscillators (masers and lasers).
1365

COMPLEMENT B XIII2
(ω)
2
0 ω
0
ω
T
R
Figure 3: Variation of the square of the modulus()
2
of the linear susceptibilite of the
spin system, with respect to. A resonance appears, of width2, centered at=0.
3-c. Second-order solution: absorption and induced emission
To second order, according to Table I, only
(2)
0
and
(2)
2
are non-zero. First,
we shall study
(2)
0
, that is, the static population dierence of the states+and
to second order. We shall then consider
(2)
2
, that is, the generation of the second
harmonic.
. Variation of the population dierence of the two states of the system
(2)
0 0corrects the zeroth-order result obtained for
(0)
0 0. According to (16a) and
(13b):
(2)
0
=
4
(1)
1
+
(1)
1
(1)
1 +
(1)
1+
=
4
(1)
1++
(1)
1 +
(1)
1 +
(1)
1+ (25)
which, according to rst-order solutions (19a) and (19b), yields:
(2)
0
=
0
4
1
( 0)
2
+
2
+
1
(+0)
2
+
2
(26)
Grouping the static terms(= 0)through second order in (9a), we get:
(static) =01
2
1
4
1
( 0)
2
+
2
+
1
(+0)
2
+
2
+ (27)
Figure 0.
1366

LINEAR AND NON-LINEAR RESPONSES OF A TWO-LEVEL SYSTEMℳ
z
(static)
ℳ
0
4
2
T
R
ℳ
0
ω
1
2
T
R
2
– ω ω ω
00
2
T
R
Figure 4: Variation of the static longitudinal magnetization with respect to0. To second
order in the perturbation treatment, there appear two resonances of width2, centered
at0=and0= . The calculation is valid only if the relative intensity of the
resonances is small, that is, if1 1.
The population dierence is therefore alwaysdecreased, to second order, relative to
its value in the absence of radiofrequency, and the decrease is proportional to theintensity
of the radiofrequency eld. This is simple to understand: under the eect of the incident
eld, transitions are induced from+to (induced emission) or fromto+
(absorption); whatever the sign of the initial population dierence, the transitions from
the more populated state are the more numerous, so that they decrease the population
dierence.
Comment:
The maximum value of
2
1
(2)
0
is0
2
14
2
= 0
2
1
2
4(the resonance
amplitude which appears as a dip in Figure). For the perturbation expansion to
make sense, it is therefore necessary that:
1 1 (28)
1367

COMPLEMENT B XIII
. Generation of the second harmonic
According to (16a), (13b), (19a) and (19b):
(2)
2
=
1
4(2 )
(1)
1+
(1)
1 +
=
0
8(2 )
1
0+
1
0 +
(29)
(2)
2
describes a vibration of the magnetic dipole alongat the angular fre-
quency2. The system can therefore radiate a wave of angular frequency2, polarized
(as far as the magnetic eld is concerned) linearly along.
Thus, we see that an atomic system is not generally a linear system. It can double
the excitation frequency, triple it (as we shall see later), etc. The same type of phe-
nomenon exists in optics for very high intensities (non-linear optics): a red laser beam
(produced, for example, by a ruby laser) falling on a material such as a quartz crystal
can give rise to an ultraviolet light beam (doubled frequency).
Comment:
It will prove useful to compare
(2)
0
and
(2)
2
in the neighborhood of0=.
According to (29), for 0, we have:
(2)
2
0
160
(30)
Similarly, (26) indicates that:
(2)
0
0
4
2
(31)
Therefore, for 0:
(2)
2
(2)
0

40
=
1
40
1 (32)
according to (24).
3-d. Third-order solution: saturation eects and multiple-quanta transitions
To third order, Table I shows that only
(3)
1
and
(3)
3
are non-zero; it suces
to study
(3)
+.
(3)
1 +corrects to third order the right circular motion of, found to rst order
and analyzed in
(3)
1 +corresponds to a saturation eect
in the susceptibility of the system.
(3)
3 +represents a new component of the motion of, of angular frequency
3of the motion of (generation of the third harmonic). Moreover, the resonant
nature of
(3)
3 +in the neighborhood of0= 3can be interpreted as resulting from
the simultaneous absorption of three radiofrequency photons, a process which conserves
both the total energy and the total angular momentum.
1368

LINEAR AND NON-LINEAR RESPONSES OF A TWO-LEVEL SYSTEM
. Saturation of the susceptibility of the system
According to (16b):
(3)
1 +=
1
2
1
0 +
(2)
2
+
(2)
0
(33)
Since we are interested in the correction to the right circular motion discussed in 3.b,
which is resonant at=0, we shall place ourselves in the neighborhood of0=. We
can then, according to the comment in the preceding section [cf.formula (32], neglect
(2)
2
compared to
(2)
0
. Thus we obtain, using expression (26) for
(2)
0
(neglecting
the term whose resonance peak is at0=):
(3)
1 +
0
8
1
0 +
1
( 0)
2
+
2
(34)
If we regroup results (34) and (19a), we nd the expression for the right circular
motion of+at the frequency2, to third order in1:
+(right circular)=
1
0
2
e
0 +
1
2
1
4
1
( 0)
2
+
2
(35)
Comparing (35) and (21), we see that the susceptibility of the system goes from
value (22) to the value:
() = 0
1
0 +
1
2
1
4
1
(0)
2
+
2
(36)
It is therefore multiplied by a factor smaller than one; the greater the intensity of the
radiofrequency eld and the nearer we are to resonance, the smaller the factor. The
system is then said to be saturated. The
2
1term of (36) is called the non-linear
susceptibility.
The physical meaning of this saturation is very clear. A weak electromagnetic eld
induces in the atomic system a dipole moment which is proportional to it. If the eld
amplitude is increased, the dipole cannot continue to increase proportionally to the eld.
The absorption and emission transitions induced by the eld decrease the population
dierence of the atomic states involved. Consequently, the atomic system responds less
and less to the eld. Furthermore, we see that the term in brackets in (36) is none other
than the term that expresses the decrease in the population dierence to second order
[cf.formula (27), in which the term resonant at0= was neglected].
Comment:
The saturation terms play a very important role in all maser or laser theories. Consider
Figure
(arrow directed to the right), the induced dipole moment is proportional to the eld. If the
material amplies (and if the losses of the electromagnetic cavity are suciently small),
the reaction of the dipole on the eld (arrow directed to the left) tends to increase the
1369

COMPLEMENT B XIII
eld by a quantity proportional to it. Thus, we obtain for the eld a linear dierential
equation which leads to a solution which increases linearly with time.
It is the saturation terms that prevent this unlimited increase. They lead to an
equation whose solution remains bounded and approaches a limit which is the steady-
state laser eld in the cavity. Physically, these saturation terms express the fact that the
atomic system cannot furnish the eld with an energy greater than that corresponding
to the population dierence initially introduced by the pumping.
. Three-photon transitions
According to (16b) and (29):
(3)
3 +=
1
2
1
03+
(2)
2
=
0
16
1
03+
1
2
1
0+
1
0 +
(37)
With respect to the term
(3)
3 +, we could make the same comment as for
(2)
2
:
the atomic system produces harmonics of the excitation frequency (here, the third har-
monic).
The dierence with the discussion of the preceding section relative to
(2)
2
is the
appearance of a resonance centered at0= 3[due to the rst resonant denominator
of (37)].
We can give a particle interpretation of the0=resonance discussed in the
preceding sections: the spin goes from the stateto the state+by absorbing a
photon (or emitting it, depending on the relative positions of the+and states).
There is resonance when the energy~of the photon is equal to the energy~0of the
atomic transition. We could give an analogous particle interpretation of the0= 3
resonance. Since~0= 3~, the transition necessarily involves three photons, since the
total energy must be conserved.
We may wonder why no resonance has appeared to second order for~0= 2~
(two-photon transition). The reason is that the total angular momentum must also be
conserved during the transition. The linear radiofrequency eld is, as we have already
said, a superposition of two elds rotating in opposite directions. With each of these
rotating elds are associated photons of a dierent type. For the right circular eld, it is
+
photons, transporting an angular momentum+~relative to. For the left circular
eld, it isphotons, transporting an angular momentum~. To go from thestate
to the+state, the spin must absorb an angular momentum+~relative to(the
dierence between the two eigenvalues of). It can do so by absorbing a
+
photon;
if0=, there is also conservation of the total energy, which explains the appearance
of the0=resonance. The system can also acquire an angular momentum+~by
absorbing three photons (Fig.): two
+
photons and one photon. Therefore, if
0= 3, both energy and total angular momentum can be conserved, which explains
the0= 3resonance. On the other hand, two photons can never give the atom an
angular momentum+~: either both photons are
+
and they carry 2~, or they are both
and they carry2~, or one is
+
and one isand they carry no total angular
momentum.
1370

LINEAR AND NON-LINEAR RESPONSES OF A TWO-LEVEL SYSTEMħω
ħω
ħω
ħω
0
σ
–
σ
+
σ
+
–
+
Figure 5: The spin can go from thestate to the+state by absorbing three photons
of energy~. The total energy is conserved if~0= 3~. The angular momentum is
conserved if two photons have a
+
polarization (each carries an angular momentum+~
relative to) and the third has apolarization (it carries an angular momentum
~).
These arguments can easily be generalized and enable us to show that resonances
appear when0=, 3, 5, 7, ...,(2+ 1), ..., corresponding to the absorption of an
odd number of photons. Furthermore, we see from formula (16b) that
(2+1)
2+1 +gives
rise to a resonance peak for0= (2+ 1). Nothing analogous occurs at even orders
since, according to Table I, we must then use equation (16a).
Comments:
()If the eldB1is rotating, there is only one type of photon,
+
or. The
same argument shows that asingle resonancecan then occur, at0=if the
photons are
+
and at0= if they are. This enables us to understand
why the calculations are much simpler for a rotating eld and lead to an exact
solution. It is instructive to apply the method of this complement to the case
of a rotating eld and to show that the perturbation series can be summed
to give the solution found directly in ComplementIV.
()Consider a system having two levels of dierent parities, subject to the in-
uence of an oscillating electric eld. The interaction Hamiltonian then has
the same structure as the one we are studying in this complement:has
only non-diagonal elements. Similarly, the electric dipole Hamiltonian, since
it is odd, can have no diagonal elements. In the second case, the calculations
1371

COMPLEMENT B XIII
are very similar to the preceding ones and lead to analogous conclusions: res-
onances are found for0=, 3, 5, ... The interpretation of the odd
nature of the spectrum is then as follows: the electric dipole photons have a
negative parity, and the system must absorb an odd number of them in order
to move from one level to another of dierent parity.
()For the spin 1/2 case, assume that the linear radiofrequency eld is neither
parallel nor perpendicular toB0(Fig.).B1can then be broken down into
a component parallel toB0,B
1, with which are associatedphotons (with
zero angular momentum relative to), and a componentB1, with which,
as we have seen,
+
and photons are associated. In this case, the atom
can increase its angular momentum relative toby+~, and move from
to+, by absorbing two photons, one
+
and the other. It can be shown,
by applying the method of this complement, that for this polarization of the
radiofrequency, a complete (even and odd) spectrum of resonances appears:
0=,2, 3, 4, ...B
0
B
1
B
1 //
B
1 ⊥
Figure 6: The static magnetic eldB0and
the radiofrequency eldB1, in the case in
whichB1is neither parallel nor perpendicu-
lar toB0.B
1andB1are the components
ofB1parallel and perpendicular toB0.
4. Exercises: applications of this complement
EXERCISE 1
In equations (1), set1= 0(no radiofrequency) and choose0parallel to
(transverse pumping).
Calculate the steady-state values of, and . Show that and
undergo resonant variations when the static eld is swept about zero (the Hanle eect).
Give a physical interpretation of these resonances (pumping in competition with Larmor
precession) and show that they permit the measurement of the product.
1372

LINEAR AND NON-LINEAR RESPONSES OF A TWO-LEVEL SYSTEM
EXERCISE 2
Consider a spin system subjected to the same static eldB0and to the same
pumping and relaxation processes as in this complement. These spins are also subjected
to two linear radiofrequency elds, the rst one of angular frequencyand amplitude
1, parallel to, and the second one of angular frequencyand amplitude
1, parallel
to.
Using the general methods described in this complement, calculate the magneti-
zationof the spin system to second order in1= 1and
1=
1(terms in
2
1,
2
1,11). We x0= 0and1. Assume0 , and letvary. Show that,
to this perturbation order, two resonances appear, one at=0 and the other at
=0+.
Give a physical interpretation of these two resonances (the rst one corresponds
to a two-photon absorption, and the second, to a Raman eect).
References and suggestions for further reading:
See section 15 of the bibliography.
Semiclassical theories of masers and lasers: Lamb (15.4) and (15.2), Sargent et al.
(15.5), Chap. VIII, IX and X.
Non-linear optics: Baldwin (15.19), Bloembergen (15.21), Giordmaine (15.22).
Iterative solution of the master equation: Bloembergen (15.21), Chap. 2, 3, 4
and 5 and Appendix III.
Multiphoton processes in R. F. range, Hanle eect: Brossel's lectures in (15.2).
1373

COMPLEMENT C XIII
Complement CXIII
Oscillations of a system between two discrete states under the eect
of a sinusoidal resonant perturbation
1 The method: secular approximation
2 Solution of the system of equations
3 Discussion
The approximation method used to calculate the eect of a resonant perturbation
in Chapter cf.condition (C-18)
of this chapter] thatmust satisfy:
~
(1)
Suppose we want to study the behavior of a system subjected to a resonant perturbation
over a considerable time [for which condition (1) is not satised]. Since the rst-order
solution is then insucient, we could try to calculate a certain number of higher-order
terms to obtain a better expression forP(;):
P(;) =
(1)
() +
2(2)
() +
3(3)
() +
2
(2)
Such a method would lead to unnecessarily long calculations.
We shall see here that it is possible to solve the problem more elegantly and rapidly
by tting the approximation method to the resonant nature of the perturbation. The
resonance condition implies that only the two discrete statesand are
eectively coupled by(). Since the system, at the initial instant, is in the state
[(0) = 1], the probability amplitude()of nding it in the stateat timecan be
appreciable. On the other hand, all the coecients()(with=,) remain much
smaller than 1 since they do not satisfy the resonance condition. This is the basis of the
method we shall use.
1. The method: secular approximation
In Chapter , we replaced all the components()on the right-hand side of (B-11)
by their values(0)at time= 0. Here, we shall do the same thing for the components
for which=,. However, we shall explicitly keep()and(). Thus, in order to
determine()and(), we are led to the system of equations [the perturbation having
the form (C-1a) of Chap. ]:
~
d
d
() =
1
2
e e () +e
( )
e
(+ )
()
~
d
d
() =
1
2
e
(+ )
e
( )
() +e e () (3)
1374

OSCILLATIONS OF A SYSTEM BETWEEN TWO DISCRETE STATES
On the right-hand side of these equations, certain coecients of()and()are
proportional toe
( )
, so they oscillate slowly in time when.
On the other hand, the coecients proportional either toe or toe
(+ )
oscillate much more rapidly. Here, we shall use thesecular approximation, which consists
in neglecting the second type of terms. The remaining ones, called secular terms, are
then those whose coecients reduce to constants for= . When integrated over
time, they make signicant contributions to the variations of the components()and
(). On the other hand, the contribution of the other terms is negligible, since their
variation is too rapid (the integration ofe

causes a factor1to appear, and the
average value ofe

over a large number of periods is practically zero).
Comment:
For the preceding argument to be valid, it is necessary for the temporal variation
of a terme ()to be due principally to the exponential, and not to the
component (). Sinceis very close to, this means that()must
not signicantly vary over a time interval of the order of1 . This is indeed
true with the assumptions we have made, that is, with 0. The variations
of()and()(which are constants if= 0) are due to the presence of
the perturbation, and are appreciable for times of the order of~ [this
can be veried directly from formulas (8), obtained below]. Since by hypothesis
~ , this time is much greater than1 .
In conclusion, the secular approximation leads to the system of equations:
d
d
() =
1
2~
e
( )
() (4a)
d
d
() =
1
2~
e
( )
() (4b)
whose solution, very close to that of system (3), is easier to calculate, as we shall see in
the next section.
2. Solution of the system of equations
We shall begin by considering the case for which= . Dierentiating (4a) and
substituting (4b) into the result, we obtain:
d
2
d
2
() =
1
4~
2
2
() (5)
Since the system is in the stateat time= 0, the initial conditions are:
(0) = 1 (6a)
(0) = 0 (6b)
1375

COMPLEMENT C XIII
which, according to (4), gives:
d
d
(0) = 0 (7a)
d
d
(0) =
2~
(7b)
The solution of (5) that satises (6a) and (7a) can be written:
() = cos
2~
(8a)
We can then calculatefrom (4a):
() = esin
2~
(8b)
where is the argument of. The probabilityP(;=)of nding the system
in the stateat timeis therefore, in this case, equal to:
P(;=) = sin
2
2~
(9)
Whenis dierent from(while remaining close to the resonance value), the
dierential system (4) is still exactly soluble. In fact, it is completely analogous to the one
we obtained in ComplementIV[cf.equation (15)] in studying the magnetic resonance
of a spin 1/2. The same type of calculation as in that complement leads to the analogue
of relation (27) (Rabi's formula), which can be written here:
P(;) =
2
2
+~
2
( )
2
sin
2
2
~
2
+ ( )
2
2
(10)
[when=, this expression does reduce to (9)].
3. Discussion
The discussion of the result obtained in (10) is the same as that of the magnetic resonance
of a spin 1/2 (cf.ComplementIV, ). The probabilityP(;)is an oscillating
function of time; for certain values of,P(;) = 0, and the system has gone back
into the initial state.
Furthermore, equation (10) measures the magnitude of the resonance phenomenon.
When = , however small the perturbation is, it can cause the system to move
completely from the stateto the state
1
. On the other hand, if the perturbation
is not resonant, the probabilityP(;)always remains less than 1.
Finally, it is interesting to compare the result obtained in this complement with
the one obtained using the rst-order theory in Chapter . First of all, note that,
1
The magnitude of the perturbation, characterized by, enters, at resonance, only into the time
taken by the system to move from to . The smaller , the longer the time.
1376

OSCILLATIONS OF A SYSTEM BETWEEN TWO DISCRETE STATES
for all values of, the probabilityP(;)obtained in (10) is included between 0 and
1. The approximation method used here therefore enables us to avoid the diculties
encountered in Chapter cf. ). When we letapproach zero in (9), we
get (C-17) of this chapter. Thus, rst-order perturbation theory is indeed valid for
suciently small (cf.comment of ). It amounts to replacing the sinusoid which
representsP(;)as a function of time by a parabola.
1377

COMPLEMENT D XIII
Complement DXIII
Decay of a discrete state resonantly coupled
to a continuum of nal states
1 Statement of the problem
2 Description of the model
2-a Assumptions about the unperturbed Hamiltonian 0. . . . .
2-b Assumptions about the coupling . . . . . . . . . . . . . .
2-c Results of rst-order perturbation theory
2-d Integrodierential equation equivalent to the Schrödinger equa-
tion
3 Short-time approximation. Relation to rst-order pertur-
bation theory
4 Another approximate method for solving the Schrödinger
equation
5 Discussion
5-a Lifetime of the discrete state
5-b Shift of the discrete state due to the coupling with the continuum1387
5-c Energy distribution of the nal states
1. Statement of the problem
In , we showed that the coupling induced by a constant perturbation
between an initial discrete state of energyand a continuum of nal states (some of
which have an energy equal to) causes the system to go from the initial state to
this continuum of nal states. More precisely, the probability of nding the system in
a well-dened group of states of the continuum at timeincreases linearly with time.
Consequently, the probabilityP()of nding the system in the initial stateat time
must decrease linearly over time from the valueP(0) = 1. It is clear that this result
is valid only over short times, since extrapolation of the linear decrease ofP()to long
times would lead to negative values ofP(), which would be absurd for a probability.
This raises the problem of determining the long-time behavior of the system.
We encountered an analogous problem when we studied the resonant transitions
induced by a sinusoidal perturbation between two discrete statesand . First-
order perturbation theory predicts a decrease proportional to
2
ofP()from the initial
valueP(0) = 1. The method presented in ComplementXIIIshows that the system
actually oscillates between the statesand . The decrease with
2
found in
of Chapter
We might expect an analogous result in the problem with which we are concerned
here (oscillations of the system between the discrete state and the continuum). We shall
show that this is not the case:the physical system leaves the stateirreversibly. We
nd an exponential decreasee

forP()(for which the perturbation treatment gives
1378

DECAY OF A DISCRETE STATE RESONANTLY COUPLED TO A CONTINUUM OF FINAL STATES
only the short-time behavior1). Thus, thecontinuousnature of the set of nal
states causes the reversibility found in ComplementXIIIto disappear; it is responsible
for adecayof the initial state, which thus acquires anite lifetime(unstable state;cf.
ComplementIII).
The situation envisaged in the present complement is very frequently encountered
in physics. For example, a system, initially in a discrete state, can split, under the eect
of an internal coupling (described, consequently, by a time-independent Hamiltonian
), into two distinct parts whose energies (kinetic in the case of material particles and
electromagnetic in the case of photons) can have, theoretically, any value; this gives
the set of nal states a continuous nature. Thus, in-decay, a nucleus initially in a
discrete state is transformed (via the tunnel eect) into a system composed of an-
particle and another nucleus. A many-electron atominitially in a conguration (cf.
ComplementsXIVandXIV) in which several electrons are excited can, under the eect
of electrostatic interactions between electrons, give rise to a system formed of an ion
+
and a free electron (the energy of the initial conguration must, of course, be greater
than the simple ionization limit of): this is the autoionization phenomenon. We can
also cite thespontaneous emissionof a photon by an excited atomic (or nuclear) state:
the interaction of the atom with the quantized electromagnetic eld couples the discrete
initial state (the excited atom in the absence of photons) with a continuum of nal states
(the atom in a lower state in the presence of a photon of arbitrary direction, polarization
and energy). Finally, we can mention the photoelectric eect, in which a perturbation,
now sinusoidal, couples a discrete state of an atomto a continuum of nal states (the
ion
+
and the photoelectron).
These few examples of unstable states taken from various domains of physics are
sucient to underline the importance of the problem we are treating in this complement.
2. Description of the model
2-a. Assumptions about the unperturbed Hamiltonian 0
To simplify the calculations as much as possible, we shall make the following as-
sumptions about the spectrum of the unperturbed Hamiltonian0. This spectrum
includes:
()a discrete stateof (non-degenerate) energy:
0 = (1)
()a set of statesforming a continuum:
0= (2)
can take on a continuous innity of values, distributed over a portion of the real axis
including. We shall assume, for example, thatvaries from 0 to+:
0 (3)
Each stateis characterized by its energyand a set of other parameters which
we shall denote by(as in of Chapter ).can therefore also be written
1379

COMPLEMENT D XIII
in the form . We have [cf.formula (C-28) of Chap. ]:
d=() dd (4)
where()is the density of nal states.
The eigenstates of0satisfy the following relations (orthogonality and closure
relations):
= 1
= 0
=( )
(5a)
(5b)
(5c)
+d = 1 (6)
2-b. Assumptions about the coupling
We shall assume thatis not explicitly time-dependent and has no diagonal
elements:
= 0
= 0 (7)
(if these diagonal elements were not zero, we could always add them to those of0,
which would simply amount to changing the unperturbed energies). Similarly, we shall
assume thatcannot couple two states of the continuum:
= 0 (8)
The only non-zero matrix elements ofare then those connecting the discrete state
with the states of the continuum. It is these matrix elements,, that are
responsible for the decay of the state.
The preceding assumptions are not too restrictive. In particular, condition (8) is very
often satised in the physical problems alluded to at the end of 1. The advantage of this
model is that it enables us to investigate the physics of the decay phenomenon without too
many complicated calculations. The essential physical conclusions would not be modied by
using a more elaborate model.
Before taking up the new method for solving the Schrödinger equation which we are
describing in this complement, we shall indicate the results of the rst-order perturbation
theory of Chapter
2-c. Results of rst-order perturbation theory
The discussion of
formula (C-36)] the probability of nding the physical system at time(initially in
the state) in a nal state of arbitrary energy belonging to a group of nal states
characterized by the intervalaround the value.
1380

DECAY OF A DISCRETE STATE RESONANTLY COUPLED TO A CONTINUUM OF FINAL STATES
Here, we shall concern ourselves with the probability of nding the system in
any of the nal states: neithernoris specied. We must therefore integrate
expression (C-36) of Chapter , which gives the probability density
[the integration over the energy was already performed in (C-36)]. Thus, we introduce
the constant:
=
2
~
d =
2
( =) (9)
The desired probability is then equal to. With the assumptions of , it represents
the probability of the system having left the stateat time. If we callP()the
probability that the system is still in this state at time, we have:
P() = 1 (10)
In the discussion of the following sections, it is important to recall the validity
conditions for (10):
()Expression (10) results from a rst-order perturbation theory which is valid only if
P()diers only slightly from its initial valueP(0) = 1. We then must have:
1

(11)
()Furthermore, (10) is valid only for suciently long times.
To state the second condition more precisely, and to see, in particular, if it is
compatible with (11), we return to expression (C-31) of Chapter andare no
longer constrained to vary only inside the intervalsand ). Instead of proceeding
as we did in Chapter , we shall integrate the probability density appearing in (C-31),
rst overand then over. The following integral then appears:
1
~
2
0
d
~
() (12)
where(), which results from the rst integration over, is given by:
() =d
2
() (13)
~
is the diraction function dened by (C-7) of Chapter , centered at
=and of width4~.
Let~be the width of():~represents the order of magnitude of the
variation needed for()to change signicantly (cf.Fig.). As soon asis suciently
large that:
1

(14)
~
behaves like a delta function with respect to(). Using relation (C-32)
1381

COMPLEMENT D XIIIEE
i0
–
K(E)
F
E E
i
t,
4πћ
ћ?
ћ
t
Figure 1: Variation of the functions()and
~
with respect to. The
respective widths of the two curves are of the order of~and4~. For suciently
large,
~
behaves like a delta function with respect to().
of Chapter , we can then write (12) in the form:
2
~
d( )() =
2
~
(=) = (15)
since by comparing (9) and (13), it can easily be seen that:
2
~
(=) = (16)
Again we nd that the linear decrease appearing in (10) is valid only ifis large
enough to satisfy (14).
Conditions (11) and (14), obviously, are compatible only if:
(17)
We have thus given a quantitative form to the condition stated in the note of Chap.
on page . In the rest of this complement, we shall assume that inequality (17) is
satised.
2-d. Integrodierential equation equivalent to the Schrödinger equation
It is easy to adapt expressions (B-11) of Chapter
here.
The state of the system at timecan be expanded on the basis:
()=() e
~
+d() e
~
(18)
1382

DECAY OF A DISCRETE STATE RESONANTLY COUPLED TO A CONTINUUM OF FINAL STATES
When we substitute state vector (18) into the Schrödinger equation, using the assump-
tions stated in , we obtain, after a calculation that is analogous to the one
in , the following equations of motion:
~
d
d
() =de
( )~
()
~
d
d
() = e
( )~
()
(19)
(20)
The problem consists of using these rigorous equations to predict the behavior of the
system after a long time, taking into account the initial conditions:
(0) = 1
(0) = 0
(21a)
(21b)
The simplifying assumptions which we made forimply that
d
d
()depends
only on(), and
d
d
(), only on(). Consequently, we can integrate equation
(20), taking initial condition (21b) into account. Substituting the value obtained in this
way for()into (19), we obtain the following equation describing the evolution of
():
d
d
() =
1
~
2
d
0
de
( )( )~ 2
() (22)
By using (4) and performing the integration over, we obtain, according to (13):
d
d
() =
1
~
2
0
d
0
d() e
( )( )~
() (23)
Thus, we have been able to obtain an equation involving only. However, it must
be noted that this equation is no longer a dierential equation, but an integrodierential
equation: the time derivative
d
d
()depends on the entire history of the system
between the times 0 and.
Equation (23) is rigorously equivalent to the Schrödinger equation, but we do not
know how to solve it exactly. In the following sections, we shall describe two approximate
methods for solving this equation. One of them ( 3) is equivalent to the rst-order theory
of Chapter ; the other one ( 4) enables us to study the long-time behavior of the
system more satisfactorily.
3. Short-time approximation. Relation to rst-order perturbation theory
Ifis not too large, that is, if()is not too dierent from(0) = 1, we can replace
()by(0) = 1on the right-hand side of (23). This right-hand side then reduces to a
double integral, overand, whose integration presents no diculties:
1
~
2
0
d
0
d() e
( )( )~
(24)
1383

COMPLEMENT D XIII
We shall perform this calculation explicitly, since it allows us to introduce two constants
[one of which is, dened by (9)] which play an important role in the more elaborate
method described in 4.
We shall begin by integrating over' in (24). According to relation (47) of Ap-
pendix, the limit of this integral foris the Fourier transform of the Heaviside
step function. More precisely:
Lim
0
e
( )~
d=~( ) +
1
(25)
(we have set =).
Actually, it is not necessary to letapproach innity in order to use (25) in the
calculation of (24). It suces for~to be much smaller than the width~of(),
that is, forto be much greater than1. We again nd the validity condition (14). If
this condition is satised, we can use (25) to write (24) in the form:
~
(=)
~
0
()
d (26)
The rst term of (26) is, according to (16), simply2. We shall set:
=
0
()
d (27)
Therefore, the double integral (24) is equal to:

2~
(28)
When()is replaced by(0) = 1in (23), this equation then becomes [as soon
as (14) is satised]:
d
d
() =

2~
(29)
The solution of (29), using the initial condition (21a), is very simple:
() = 1

2
+
~
(30)
Obviously, this result is valid only if()diers slightly from 1, that is if:
1

~
(31)
This is the other validity condition, (11), for rst-order perturbation theory.
Using (30), we can easily calculate the probabilityP() =()
2
that the system
is still in the stateat time. If we neglect terms in
2
and
2
, we obtain:
P() = 1 (32)
All the results obtained in Chapter 23) when
()is replaced by(0). This equation has also enabled us to introduce the parameter
, whose physical signicance will be discussed later [note thatdoes not appear in
the treatment of Chapter
the probability()
2
, and not with that of the probability amplitude()].
1384

DECAY OF A DISCRETE STATE RESONANTLY COUPLED TO A CONTINUUM OF FINAL STATES
4. Another approximate method for solving the Schrödinger equation
A better approximation consists of replacing()by()rather than by(0)in (23).
To see this, we shall begin by performing the integral overwhich appears on the
right-hand side of the rigorous equation, (23). We obtain a function ofand :
( ) =
1
~
2
0
d() e
( )( )~
(33)
which is clearly dierent from zero only ifis very small. In (33), we are integrating
overthe product of(), which varies slowly with(cf.Fig.), and an exponential
whose period with respect to the variableis2~( ). If we choose values ofand
such that this period is much smaller than the width~of(), the product of these
two functions undergoes numerous oscillations whenis varied, and its integral over
is negligible. Consequently, the modulus of( )is large for 0and becomes
negligible as soon as 1. This property means that, for all, the only values
of()to enter signicantly into the right-hand side of (23) are those which correspond
tovery close to(more precisely, 1). Indeed, once the integration over
has been performed, this right-hand side becomes:
0
( )() d (34)
and we see that the presence of( )practically eliminates the contribution of
()as soon as 1.
Thus, the derivative
d
d
()has only a very short memory of the previous values of
()between 0 and. Actually, it depends only on the values ofat times immediately
before, andthis is true for all. This property enables us to transform the integrodif-
ferential equation (23) into a dierential equation. If()varies very little over a time
interval of the order of1, we make only a small error by replacing()by()in
(34). This yields:
()
0
( )d=

2
+
~
() (35)
[to write the right-hand side of (35), we used the fact that the integral overof( )
is simply, according to (33), the double integral (24) evaluated in 3 above].
Now, according to the results of
characteristic of the evolution of()is of the order of1or~. The validity
conditions for (35) are then:

~
(36)
which we have already assumed to be fullled [cf.(17)].
To a good approximation, and for all, equation (23) can therefore be written:
d
d
() =

2
+
~
() (37)
1385

COMPLEMENT D XIII
whose solution, using (21a), is obvious:
() = e
2
e (38)
It can easily be shown that the limited expansion of (38) gives (30) to rst order in
and.
Comment:
No upper bound has been imposed on the time. On the other hand, the integral
0
( )dwhich appears in (35) is equal to(2+ ~)only if1,
as we saw in
the same limitations as perturbation theory; however, it has the great advantage
of being valid for long times.
If we now substitute expression (38) for()into equation (20), we obtain a
very simple equation which enables us to determine the probability amplitude()
associated with the state:
() =
1
~
0
e
2
e
( )~
d (39)
that is:
() =
~
1e
2
e
( )~
1~
( ) +

2
(40)
Equations (38) and (40), respectively, describe the decay of the initial state and
the lling of the nal states. Now let us study in greater detail the physical content
of these two equations.
5. Discussion
5-a. Lifetime of the discrete state
According to (38), we have:
P() =()
2
= e

(41)
P()therefore decreasesirreversiblyfromP(0) = 1and approaches zero as
(Fig.). The discrete initial state is said to have anite lifetime, whereis the time
constant of the exponential of Figure
=
1

(42)
This irreversible behavior contrasts sharply with the oscillations of the system (Rabi's
formula) between two discrete states when it is subject to a resonant perturbation cou-
pling these two states.
1386

DECAY OF A DISCRETE STATE RESONANTLY COUPLED TO A CONTINUUM OF FINAL STATES1
1
e
0

ii
(t)
1
Γ
τ =
t
Figure 2: Variation with respect to time of the probability of nding the system in the
discrete stateat time. We obtain an exponential decrease,e, for which Fermi's
golden rule gives the tangent at the origin (this tangent is represented by a dashed line).
5-b. Shift of the discrete state due to the coupling with the continuum
If we go from()to()[cf.formula (B-8) of Chapter ], we obtain, from
(38):
() = e
2
e
(+)~
(43)
Recall that, in the absence of the coupling, we would have:
() = e
~
(44)
In addition to the exponential decrease,e
2
, the coupling with the continuum is
therefore responsible for a shift in the discrete state energy, which goes fromto
+. This is the interpretation of the quantityintroduced in 3.
Let us analyze expression (27) formore closely. Substituting denition (13) of
()into (27), we get:
=
0
d
d()
2
(45)
or, if we use (4) and replace by:
= d
2
(46)
The contribution to this integral of a particular stateof the continuum, for
which=, is:
2
(47)
1387

COMPLEMENT D XIII
We recognize (47) as a familiar expression in stationary perturbation theory [cf.formula
(B-14) of Chapter]. (47) represents the energy shift of the statedue to the coupling
with the state, to second order in.is simply the sum of the shifts due to the
various statesof the continuum. We might imagine that a problem would appear
for the statesfor which=. Actually, the presence in (46) of the principal
partimplies that the contribution of the statessituated immediately above
compensates that of the states situated immediately below.
Summing up:
()The coupling of with the statesof the same energy is responsible for
the nite lifetime of[the function( )of formula (25) enters into the
expression for].
()The coupling ofwith the statesof dierent energies is responsible for an
energy shift of the state. This shift can be calculated by stationary perturbation
theory (this was not obvious in advance).
Comment:
In the particular case of the spontaneous emission of a photon by an atom,represents
the shift of the atomic level under study due to the coupling with the continuum of nal
states (an atom in another discrete state, in the presence of a photon). The dierence
between the shiftsof the212and212states of the hydrogen atom is the Lamb
shift [cf.ComplementV, and Chapter, , comment (iv)].
5-c. Energy distribution of the nal states
Once the discrete state has decayed, that is, when1, the nal state of
the system belongs to the continuum of states. It is interesting to study the energy
distribution of the possible nal states. For example, in the spontaneous emission of a
photon by an atom, this energy distribution is that of the photon emitted when the atom
falls back from the excited level to a lower level (the natural width of spectral lines).
When 1, the exponential which appears in the numerator of (40) is practi-
cally zero. We then have:
()
2
1
2
1
( )
2
+~
2

2
4
(48)
()
2
actually represents a probability density. The probability of nding the system,
after the decay, in a group of nal states characterized by the intervalsdandd
aboutand can be calculated directly from (48):
dP( ) =
2
( )
1
( )
2
+~
2

2
4
dd (49)
Let us examine the-dependence of the probability density:
dP( )
dd
1388

DECAY OF A DISCRETE STATE RESONANTLY COUPLED TO A CONTINUUM OF FINAL STATES0
d(β
f
, E
f
, t)
dβ
f
dE
f
E
i
+ δE
ħΓ =
E
f
τ
ħ
Figure 3: Form of the energy distribution of the nal states attained by the system after
the decay of the discrete state. We obtain a Lorentzian distribution centered at+
(the energy of the discrete state corrected by the shiftdue to the coupling with the
continuum). The shorter the lifetimeof the discrete state, the wider the distribution
(time-energy uncertainty relation).
Since
2
( )remains practically constant whenvaries over an
interval of the order of~, the variation of the probability density with respect tois
essentially determined by the function:
1
( )
2
+~
2

2
4
(50)
and has, consequently, the form shown in Figure. The energy distribution of the nal
states has a maximum for=+, that is, when the nal state energy is equal
to that of the initial state, corrected by the shift. The form of the distribution
is that of a Lorentz curve of width~, called the natural width of the state. An
energy dispersion of the nal states therefore appears. The larger~(that is, the shorter
the lifetime= 1of the discrete state), the greater the dispersion. More precisely:
=~ =
~
(51)
Note again the analogy between (51) and the time-energy uncertainty relation. In
the presence of the coupling, the statecan be observed only during a nite time,
of the order of its lifetime. When we want to determine its energy by measuring that
of the nal state of the system, the uncertaintyof the result cannot be much less
than~.
References:
The original article: Weisskopf and Wigner (2.33).
1389

COMPLEMENT E XIII
Complement EXIII
Time-dependent random perturbation, relaxation
1 Evolution of the density operator
1-a Coupling Hamiltonian, correlation times
1-b Evolution of a single system
1-c Evolution of the ensemble of systems
1-d General equations for the relaxation
2 Relaxation of an ensemble of spin 1/2's
2-a Characterization of the operators, isotropy of the perturbation
2-b Longitudinal relaxation
2-c Transverse relaxation
3 Conclusion
This complement examines the problem studied in , both in a
more precise and general way. Rather than studying a single system, we shall study an
ensemble of individual quantum systems subjected to an external random perturbation.
This type of situation often occurs in magnetic resonance experiments where one mea-
sures the global magnetization of an ensemble of spins each carrying a small magnetic
moment, as for example the nuclear spins of atoms in a gas. As the atoms move, they un-
dergo collisions with impurities contained in the gas or on the walls of the container. As
mentioned in Chapter , if these impurities carry a magnetic moment, such collisions
may change the directions of the nuclear spins of the colliding atoms. The corresponding
perturbation lasts for a very short time (the collision time), and is of a random nature
since the magnetic moment of the impurities can have any direction. The gas of atomic
spins is thus subjected to a sum of random perturbations that rapidly change their val-
ues (and signs), hence having a very short correlation time. Another classic example of
random perturbation is an experiment where an ensemble of atoms is illuminated by a
light source. Several reasons give the interaction between the atoms and the incident
electromagnetic eld a random character. First of all, most light sources produce elds
that have rapid frequency and phase uctuations. This means that the eld itself must
be characterized in a stochastic manner, with a short coherence time. Furthermore, even
if the light source is an almost perfectly monochromatic laser, the atoms' motion is ran-
dom. Because of the Doppler eect, the atoms will be coupled, in their own reference
frame, to a eld having a random frequency. Studying the propagation of a light beam
in an atomic gas thus involves the study of a large number of individual atoms, each sub-
jected to a dierent and random perturbation. Many examples exist of similar situations
involving rapidly uctuating perturbations.
This complement examines how the eect of such a random perturbation on an
ensemble of individual systems must be treated using quantum mechanics. In a more
general framework than the one used in , we will show that the
coupling with the random perturbation produces a so-called relaxation phenomenon
in the global system, very dierent from the evolution in the absence of the random
1390

TIME-DEPENDENT RANDOM PERTURBATION, RELAXATION
event. We saw that the coupling with a constant or perfectly sinusoidal perturbation
produces oscillations in the physical system, at Bohr frequencies that are modied by the
interaction. This complement will describe a totally dierent behavior, an exponential
evolution with real exponents, leading to irreversible evolutions. An example of such
behavior could be the relaxation of a system towards thermal equilibrium, very dierent
than an oscillation.
We rst study () the evolution of the density operator characterizing the ensem-
ble of systems. This leads to a general relaxation equation valid whenever the correlation
times are very short. In the following section () we apply this general equation to
an important specic case, an ensemble of spin12's coupled to statistically isotropic
perturbations. This will enable us to explain the important concepts of longitudinal
relaxation and transverse relaxation, which play a central role in many magnetic
resonance experiments.
1. Evolution of the density operator
Consider an ensemble ofindividual systems labeled by the index= 1,2, ...,.
Each system is described by a density operator (ComplementsIIIandIV) noted().
Statistically, the ensemble of thesystems is described by the following density operator
():
() =
1
=1
() (1)
Each individual system evolves under the eect of an operator(), the sum of two
Hamiltonians:
() =0+ () (2)
The rst,0, is the Hamiltonian common to all the individual systems, corresponding for
example to the coupling of their spins with an external static magnetic eld. We assume
that this Hamiltonian does not depend on time. The second,(), is the coupling
Hamiltonian with the random perturbation. It depends not only on the time but also on
the indexof the individual system.
We note the eigenvectors of0:
0 = (3)
having the energies=}; we set:
= (4)
The matrix elements of the coupling Hamiltonian are written, in this basis:
()
() = () (5)
1391

COMPLEMENT E XIII
1-a. Coupling Hamiltonian, correlation times
Consider the ensemble of thefunctions
()
()obtained whenvaries from
1to: they aredierent realizations of the same matrix element. Choosing
randomly
1
introduces a random function of time,(). Changing the values ofand
, and callingthe dimension of the state space, we can dene
2
random functions.
These random functions can be considered as the matrix elements of an operator()
that is also a random function of time. In other words,()is the operator obtained
by choosing randomly the value oflabeling theoperators(). The statistical
correlation properties of this random operator (or of its matrix elements) play an essential
role in what follows.
The ensemble formed by thesystems can be considered as an ensemble of dier-
ent possible realizations of the same individual system, called in statistical mechanics the
Gibbs ensemble. It is equivalent to take an average over this ensemble at a given time
or over a single system taken at a large number of dierent times (ergodic hypothesis).
This average will be symbolized by placing a horizontal line over the letter.
We rst assume that the average value of the perturbation is zero:
() = 0 that is:() =() = 0for any, (6)
This hypothesis is not restrictive since, if the average value of
()is any constant
operator, that operator can be added to the Hamiltonian0, hence keeping the average
value of the perturbation equal to zero. As()and ()are complex conjugates,
their product is always a positive number whose average is, a priori, not zero. The
amplitude of the perturbation is then dened by the average value of the products:
() () =
0
(7)
As we assume the random function to be stationary, this mean square is independent of
time. In a more general way, one can dene a series of cross-correlation coecients, also
time-independent:
() () =
0
(8)
To characterize a function random in time, we also need to consider averages of
products taken at dierent times. As we did in D of Chap. , we introduce the
correlation functions between timeand+:
(+) () = () (9)
Since the random functions are stationary by hypothesis, the functiononly depends
on the dierence between the times+and, and we can also write:
() (0) =() (10)
We know that, by denition, the function()starts from a positive value for
= 0. As the delaystarts increasing, the correlation between()and (0)
1
We assume the number to be very large. As an example, a millimeter cube of gas, at standard
temperature and pressure, contains roughly10
16
atoms.
1392

TIME-DEPENDENT RANDOM PERTURBATION, RELAXATION
rapidly decreases. The function()tends towards zero, with a characteristic time
called the correlation time and noted:
(+) ()0if (11)
For instance, if the perturbation is induced by the collision between an atom and an
impurity, it will clearly lose any memory of its value between one collision and the next,
or even right after a single collision. As collision times are often very short, there are
many examples whereis a very short time. This analysis can be generalized to cross-
correlation coecients like the one described in (8).
One often uses a model where()is a decreasing exponential of the delay:
(+) () =
0
(12)
To simplify the notation, we only took into account a single correlation time, indepen-
dent ofand; the generalization to several correlation times is straightforward. Similar
relations as the ones we just wrote can be obtained for cross-correlation coecients like
the one described in (8). This leads to a whole series of correlation times depending on
numerous indices. In the collision example discussed above, all these times are of the
same order of magnitude as the very short collision time; a natural approximation is to
assume that they are comparable, and to callthe longest amongst all these times.
1-b. Evolution of a single system
We now study the evolution of a single quantum system (the value ofis xed).
It will be treated in the interaction picture (exercise 15 of ComplementIII), which we
now briey review.
. Interaction picture
The evolution of each density operator()obeys the usual von Neumann equa-
tion, with a commutator on the right-hand side:
}
d
d
() = [0+ ()()] (13)
In this right-hand side, the term containing0may lead to a rapid evolution. The term
containing()is assumed to be smaller, hence leading to a slower evolution that can
be treated using approximations.
() To start with, let us assume that the coupling Hamiltonian()is zero. The
evolution of()is only due to0. It is useful to express it as a function of the
evolution operator0()between the timesand(ComplementIII) associated
with the non-perturbed Hamiltonian0:
0() =
0( )}
(14)
This is a unitary operator:
0
()0() = 1for anyor (15)
1393

COMPLEMENT E XIII
When ()is zero, we simply have:
() =0()()
0
() (16)
as we now show. Taking the derivative of this relation with respect to, the derivation
of0()introduces a term 0()}; the derivation of
0
()introduces a term
+()0}. Both terms together reconstruct the0term of the commutator on the
right-hand side of (13), which shows that equation (13) is veried by solution (16). As
for the initial condition for=, it is also veried since the unitary operator0()
then becomes the unit operator.
We can also use0()to perform an inverse unitary transformation on()
and dene the modied density operator()as:
() =
0
()()0() (17)
Inserting relation (16) in this denition, we obtain relation (15) twice, both on the left
and on the right of(). All the evolution operators thus disappear, and we get:
() =() (18)
This shows that()does not depend on time as long as the coupling()remains
equal to zero.
() Even when ()is no longer zero, it is still useful to apply the unitary
transformation (17) to the density operator. This operation is generally referred to as the
passage to the interaction picture. In this picture, the evolution of the density operator
()is only due to the presence of the interaction(). According to our assumptions,
this evolution is much slower than the evolution(), which is also governed by0.
This property considerably facilitates the use of approximations and will be used in this
complement.
Let us take the time derivative of (17), starting with the derivative of the two
unitary operators on the left and on the right of(), followed by the derivative of the
operator itself. This yields:
}
d
d
() = 0() +()0+
0
()}
d
d
()0()
=[0()] +
0
() [0+ ()()]0() (19)
Now for any operator, the fact that0is a unitary operator allows transforming the
following commutator according to:
0
() [()]0() =
0
() 0()
0
()()0() (20)
(to check this, one can simply expand both commutators and use the relation00
= 1).
The right-hand side now contains(). If= 0+ (), since0commutes with
0, we get:
0
() [0+ ()()]0()
= [0()] +
0
()()0()() (21)
1394

TIME-DEPENDENT RANDOM PERTURBATION, RELAXATION
The right-hand side contains the unitary transform()of(), obtained by the
same unitary transformation that led from()to():
() =
0
()()0() (22)
Inserting this expression in the right-hand side of (19), the commutators containing0
on both sides cancel out. We nally get the simple relation:
}
d
d
() = ()() (23)
This evolution equation only contains operators in the interaction picture. The hamil-
tonian0is no longer explicitly present (but is implicitly contained in the unitary
transformation that leads to the interaction picture). It is easy to verify that()does
not evolve in the absence of perturbation.
. Approximate calculation of the evolution
Integrating over time equation (23) yields:
() =() +
1
}
d ()() (24)
which, inserted in the same equation, leads to:
}
d
d
() = ()()+
1
}
d () ()() (25)
This evolution equation for(), which now contains a double commutator, is exact.
As()appears in the integral, it is an integro-dierential equation.
This equation can be transformed into a simple dierential equation, using the
following approximation. If the eect of the perturbationremains limited during
the time interval fromto, a good approximation of the evolution of()in that
interval is to replace()by its value for any time chosen in that interval. Choosing
for example the time, yields the dierential equation:
}
d
d
() = ()()+
1
}
d () ()() (26)
The delaycan be introduced explicitly by performing the change of integration variable:
= (27)
Dividing both sides by}, we obtain:
d
d
() =
1
}
()()
1
}
2
0
d () ( )() (28)
1395

COMPLEMENT E XIII
1-c. Evolution of the ensemble of systems
To obtain the evolution equation for the density operator()describing the en-
semble of thesystems, we rst transform that operator as in (17) to use the interaction
picture:
() =
0
()()0() (29)
The initial density operator can easily be retrieved using the inverse unitary transforma-
tion. Denition (1) of()shows that its evolution is obtained by summing relation
(28) over the index, and dividing the result by, the total number of systems. In other
words, it means that we have to take the ensemble average of both sides of (28). This
operation is dicult to carry out without making some hypotheses about the character-
istics of the random functions that come into play. We shall assume that the evolution
of()occurs with time constants that are much longer than the correlation time.
We shall explain below ( ) what this implies in terms of the parameters dening
the interactions, hence verifying that the computation is consistent.
With each timewe can associate a previous timesuch that is very large
compared to the correlation time, while remaining small compared to the characteristic
evolution time of the density operator in the interaction picture. We shall then use
relations (9) and (11) that characterize the random perturbation. The rst commutator
on the right-hand side of the evolution equation (28) involves two operators at dierent
times and which are therefore not correlated:()depends on values of the perturbation
at times earlier than, whereas()is the value of the perturbation at a time later
thanby a time larger than. Taking the average over all the values ofthen shows
that this rst term cancels out since we assumed in (6) that the average values of the
matrix elements of the perturbation are zero. As for the following integral, it contains the
average value overof the product()( )(), in that order or any other
order. Contributions to this integral only come from values of the delayof the order
of the correlation time; if ,()depends neither on( )nor on(),
which allows factoring an average value that is equal to zero. This has two consequences.
First()is not correlated with the two terms in, so that we can compute its
average separately and replace()by(). The second consequence is that we can
replace the integral upper boundby innity without changing signicantly its value.
This leads to:
d
d
() =
1
}
2
0
d
()[( )()] (30)
where, as before, the bar on top of the operators stands for the ensemble average (this
average only concerns the perturbation, not the density operator).
An additional simplication comes from the fact that we assumedto be
short compared to the evolution time of(). It is therefore a good approximation to
replace()by()in the right-hand side of this equation. This nally leads to the
relaxation equation of the density operator in the interaction representation:
d
d
() =
1
}
2
0
d
()[( )()]
(31)
Using (29), we obtain the corresponding equation in the usual representation.
1396

TIME-DEPENDENT RANDOM PERTURBATION, RELAXATION
1-d. General equations for the relaxation
We now use the previous results to compute the evolution of the density matrix.
. Evolution of the matrix elements of the density operator
Relation (31) can be written in the basis of the eigenvectorsof the Hamiltonian
0, in order to directly obtain the coupled evolution equations of the dierent matrix
elements of the density operator. Since:
() = e
( )
() (32)
where is dened in (4), we get:
d
d
() =
1
}
2
0
d
e
( )
e
() ( ) ()
e
( )( )
e
() ( ) ()
e
( )( )
e
( ) () ()
+ e
( )
e
( ) () () (33)
The random functions associated withare stationary, as seen from relation (9). This
allows adding an arbitrary time to the two variables they contain, in the right-hand side
of the previous equation. We can thus replacebyand by0.
We now leave the interaction picture and come back to the usual picture (labora-
tory picture) using the unitary transformation (17), written in thebasis:
() = e
( )
() (34)
This relation leads to:
d
d
() =( ) () + e
( )
d
d
() (35)
The general relaxation equations are then written:
d
d
() =( ) ()
1
}
2
0
d e
() (0) ()
e
() (0) ()
e
(0) () ()
+ e ()
(0) () (36)
Notingthe dimension of the state space, the previous relations (33) or (36) yield
2
dierential equations that govern the time evolution of the matrix elements of()
or of(). These dierential equations are coupled with each other; their coecients are
time integrals of correlation functions of the perturbation, which are supposedly known
for a given physical problem.
1397

COMPLEMENT E XIII
. Short memory approximation
In view of the approximations we used, let us nd under which conditions our
calculations are consistent. The general validity condition is that there exists, for each
time, a previous timesuch that the intervalobeys two conditions: it must be
simultaneously very long compared to the correlation time and very short compared to
the evolution time in the interaction picture.
We can evaluate this evolution time by using an approximate expression of relation
(33). We introduced in (7) the mean square of the matrix element(). Let us call
2
the order of magnitude of such a mean square for the various values ofand. The
coecients that multiply()on the right-hand side of (33) can be replaced by this
factor
2
, integrated over d. Taking (12) into account, this integral introduces a factor
. The coecients can thus be approximated by:
2
}
2
(37)
With this approximation, the evolution equation (33) yields an evolution time of the
order of}
2
/
2
. Our computations are consistent if this time is much larger than,
that is if:
2
()
2
}
2
(38)
In other words, our computations are valid if the correlation (memory) time of the
perturbation is short compared to the characteristic time of its intensity,}. This
means that the perturbation will frequently change its value (and sign) before it can
signicantly change the system. This validity condition is often called the motional
narrowing condition, for a reason explained in .
Relations (33) or (36) are sets of rst order dierential equations. They describe
the exponential relaxation of all the populations towards a situation where they all be-
come equal. Carrying out calculations with these equations is not particularly dicult.
However, it leads to the writing of complicated equations, in particular due to the large
number of indices involved. In the general case, the populations() are
not only coupled to each other, but also to non-diagonal elements() with
=. All the matrix elements of the density operator can a priori be coupled to each
other. One must then use additional approximations to select the terms essential for
determining the relaxation properties.
In this complement, we shall only consider a simple particular case, that still allows
us to develop a large number of physical concepts: the study of an ensemble of spin 1/2's
undergoing an statistically isotropic perturbation.
2. Relaxation of an ensemble of spin 1/2's
Consider an ensemble of spin 1/2's, contained for example in a sample measured in a
magnetic resonance experiment, such as the one mentioned in the introduction. The
evolution equations then take on a simple form, easy to interpret. There are only two
levels, which will be noted+and. Their energy dierence is:
}(+ ) =}0 (39)
1398

TIME-DEPENDENT RANDOM PERTURBATION, RELAXATION
It is useful to characterize the density operator of the spins by the average value of their
angular momentum, which amounts to expanding this density operator on Pauli matrices
( IV).
2-a. Characterization of the operators, isotropy of the perturbation
All the operators appearing in the previous equations now act in a2-dimensional
space. They are represented by matrices that can be expanded on the three Pauli ma-
trices,and, as well as on the identity matrix, as shown in relation (22) of
ComplementIV.
. Transformation of the operators
We set:
() =
1
2
[1 +M()] (40)
wherestands for the vector operator whose components are the three Pauli matrices.
The components of the vectorM()are three real numbers that play the role of param-
eters dening(). As we now show, the vectorM()is simply the mean value of
over the whole sample, whose total magnetization is thus proportional toM(). Relation
(11) of ComplementIVindicates that:
= + (41)
where is equal to zero if two indices are equal, equal to+1if the series of indices
is an even permutation of the three axes,and, and equal to1if the
permutation is odd. It follows that the trace of a product of Pauli matrices is zero, unless
two of the matrices are identical (in which case the trace equals2). Consequently, the
average value of the operatoris given by:
() =Tr ()=
1
2
Tr[]
2
()
= () (42)
The operator0is written in a form
2
similar to that of relation (12) in Comple-
mentIV, which studies a magnetic resonance experiment:
0=
}0
2
(43)
This operator corresponds to the eect of a magnetic eld parallel to theaxis, which
induces a rotation of the spins around that axis at the angular frequency0. The
unitary operator0()dened in (14) is now a rotation operator of the spins (Com-
plementIV) through an angle0( ); as for the adjoint operator
0
(), it is also
a rotation operator, but through an opposite angle.
2
We did not give the operators0or ()a component on the identity operator, since it would
not alter the commutators where these operators come into play.
1399

COMPLEMENT E XIII
The interaction operator can be written in a similar way:
() =
1
2
h() (44)
whereh()is a random vector function that characterizes the perturbation acting on the
spins. In the same way as()was dened () as a random choice amongst the
possible outcomes of the individual system labeled by the index,h()characterizes
the statistical properties of the three components of the local eld acting on each spin at
time. Contrary to the eld associated with0, this local eld is random and can point
in any direction, not necessarily parallel to theaxis. The three components= 1,2,
3of this vector are noted(). For ensemble averages, we assume that
() = 0and,
as in (9) and (12), we shall write the correlation functions in the following way:
(+)() =() (45)
where the()are rapidly decreasing functions ofover times of the order of. The
coecients()are auto-correlation functions of the various components ofh(), the
()for=are cross-correlation functions pertaining to two dierent components.
. Isotropy
We introduce an additional hypothesis, and assume that the perturbation aect-
ing the spins is statistically isotropic: the correlation functions of the components of
h()have no preferred direction. This means that the correlation functions()are
identical for the three axes,and (corresponding to= 1,2and3respec-
tively), whatever the value of. In other words, the ensemble of the()form a33
matrix, which is rotation invariant, hence necessarily proportional to the unit matrix.
Consequently, not only the auto-correlation coecients are equal to each other, but also
the cross-correlation coecientsfor=must be equal to zero. Added to the
stationarity of the perturbation, this hypothesis leads to:
(+)() = () (46)
One frequently models the decrease of()withby a simple exponential, with a time
constant. This leads to:
(+)() =[(0)]
2
(47)
2-b. Longitudinal relaxation
When = =+, the rst term on the right-hand side of equation (36)
in( )cancels out (no evolution of the populations due to0). In the following
terms, we must replace()and(0)by their expression given by (44). This leads to
products of matrix elements of two Pauli matricesand, multiplied by the statistical
average (46) of the perturbation. Because of the factorwe must choose the same Pauli
matrix in both operators()and(0).
1400

TIME-DEPENDENT RANDOM PERTURBATION, RELAXATION
. Calculation of the relaxation time
Let us start by choosing twice the matrix. Since this matrix does not couple
the states+and, we must have= =+in relation (36). As a result, the
sum of the last4terms on the right-hand side is zero.
We then choose twice the matrix, which only has non-diagonal elements, all
equal to1. In the second line of the right-hand side of (36), when== +, we must
necessarily have= and =+, whereas in the fth line, the opposite is true
( =+and = ); for the third and fourth lines we have= = .
This yields the following term:
1
4}
2
0
d
()(+)
0
+ ()+
0
()
0
()+
0
+ ()+ (48)
or:
1
2}
2
0
d
()(+) cos0[ () + ()+] (49)
We nally choose twice the matrix, which has the same structure, with two matrix
elements equal to+and, so that their product is also equal to unity. This term is
the same as theterm, except that we must replaceby.
We nally obtain:
d
d
+ ()+=
1
21
() + ()+ (50)
with:
1
1
=
1
}
2
0
d
()(+) +()(+)cos0 (51)
The time1is called the longitudinal relaxation time. Its properties are discussed
below.
The calculation of the evolution of the other diagonal element()is
practically the same and yields:
d
d
()=
1
21
+ ()+ () (52)
Now using (42) we can write the evolution of thecomponent of the magnetization:
d
d
() =
d
d
()=
d
d
+ ()+ () (53)
Taking the dierence between (50) and (52) leads to:
d
d
() =
1
1
() (54)
This equation shows that the longitudinal (parallel to the static magnetic eld) compo-
nent ofM()decreases exponentially with a time constant1, and tends toward zero
when . The relaxation rate11depends on the sum of correlation functions
of both transverse (perpendicular to) components of the perturbation. This was to
be expected since it is the operatorsandthat can induce transitions of the spins
between their levels+and.
1401

COMPLEMENT E XIII
. Role of the spectral density
The dependence in0of the relaxation probability11can be interpreted in
view of the results of Chapter ), where we studied a sinusoidal perturbation
coupling two levels+and. We showed that the closer the perturbation frequency
is to the Bohr frequency0associated with the energy dierence between the two
levels, the more eective the perturbation. In our present case, the perturbation is not
a sinusoid but a random function. We thus expect the probability amplitude of the
transition to involve the0Fourier component of the perturbation that acts between the
instants0and .
To further examine this idea, we introduce the Fourier transform()of the
correlation function, called the spectral density:
() =
1
2
+
d
()( )()( ) =
1
2
+
d () (55)
As the random function is stationary, we have:
()( ) =(+)() =()(+) (56)
This shows that the correlation function is an even function. As the Fourier transform
of an even and real function is also even and real (Appendix), we can write:
() =() (57)
Relation (51) can be rewritten as:
1
1
=
1
22}
2
0
d
+
d [() +()]
0
+
0
(58)
Taking (57
ablesanddoes not change the function to be integrated. It merely transforms the
rst integral over dto an integral between0and , while the limits of the second
become+and . Since two sign changes cancel each other, we can reverse the limits
in each of the two integrals. This leads to:
1
1
=
1
22}
2
0
d
+
d [() +()]
0
+
0
=
1
42}
2
d
+
d [() +()]
0
+
0
(59)
Note that the nal form (second line) of this relation was obtained by adding the rst
line of this relation to relation (58), and dividing by two. The integral over dleads to:
d
(+0)
+
( 0)
= 2[(+0) +( 0)] (60)
1402

TIME-DEPENDENT RANDOM PERTURBATION, RELAXATION
Taking again (57) into account, the two terms in(+0)and( 0)yield the same
contribution, and we obtain:
1
1
=
2
(0) +(0)
}
2
(61)
The transition probability is proportional to the sum of the spectral densities of the two
perturbations responsible for the transitions. As expected, it is the resonant components
of the perturbation that induce the transitions between the states+and.
. Exponential correlation function
The correlation functions of these components are often modeled by a simple ex-
ponential, as in (47). In that case, the integral over dof (51) is easy to compute:
0
d
()(+)
0
+
0
=[(0)]
2
0
d
0
+
0
=[(0)]
2 1
01
+
1
01
= [(0)]
2 2
1 + (0)
2
(62)
Adding the term corresponding to the eect of thecomponent, we get:
1
1
=
1
}
2
[(0)]
2
+[(0)]
2
1 + (0)
2
(63)
The longitudinal relaxation rate varies as a Lorentzian function of0, plotted in Fig..
The relaxation rate is maximum when0= 0(zero static eld), and is equal to:
1
1(0= 0)
=
1
}
2
[(0)]
2
+[(0)]
2
(64)
With our present notation, the motional narrowing condition (38) is written:
[(0)]
2
()
2
}
2
(65)
This leads to:
1
1
1
(66)
We thus verify that1, the characteristic evolution time of(), is very long compared
to the correlation time, hence proving the consistency of the approximations we have
used.
2-c. Transverse relaxation
We now study the evolution of the non-diagonal elements of the density matrix
between the states+and. Let us rst show that the matrix element+ (),
or its complex conjugate ()+, characterizes the transverse components()
1403

COMPLEMENT E XIII
Figure 1: Plot of the longitudinal relaxation rate11as a function of the energy dier-
ence}0between the energy levels (left-hand side of the gure), or as a function of the
correlation time (right-hand side of the gure). This rate is proportional to the power
spectrum of the perturbation at the frequency0, and hence follows a Lorentzian function
when plotted as a function of0 cf. relation (63). In the regime where01, the
power spectrum of the perturbation decreases as1
2
0: the relaxation rate can be greatly
reduced by increasing0. If we now keep0xed and increase the correlation time,
we rst get a linear variation of the relaxation probability, proportional to the time
during which the perturbation acts in a coherent way. The probability then reaches a
maximum for0= 1, followed by a decrease in1as the Fourier components of the
perturbation at the frequency0become weaker and weaker.
and ()ofM(). Using the expressions of the Pauli matrices cf.for example
relations (2) of ComplementIV we can compute the dierence :
=
0 0
2 0
(67)
which leads to:
= ( )()= 2+ () (68)
We saw in M()were equal to the average values of
the corresponding Pauli matrices. It thus follows:
() ()
2
=+ () (69)
The real part of the non-diagonal matrix element+ ()directly yields()2,
whereas its imaginary part yields the opposite of()2.
To avoid taking into account the evolution of this matrix element due to0, which
introduces the rst term in( )on the right-hand side of (36), we shall use the
interaction picture. The evolution of+()is then given by (33). In this relation,
we replace the interaction operators()and(0)by their expression (44). As before,
the statistical isotropy of the perturbation leads us to only keep the terms where the
same component ofh()appears in the two operators. We shall rst examine the case
where this component is either(), or(); the case where this component is()
will be examined later.
1404

TIME-DEPENDENT RANDOM PERTURBATION, RELAXATION
. Eect of the transverse components of the perturbation
We successively take into account the components on theand axes.
() Thecomponent of the perturbation introduces in (33) the matrix elements
of, which are all non-diagonal and equal to one. Each matrix element ofchanges
a ket+into a bra, or vice versa. The Pauli matrix simply introduces a coecient
equal to1.
In the rst term on the right-hand side of (33), when= +, we have=
and hence= +; this term couples+()to itself. As for the exponential, it
introduces
0
. The same result is obtained for the fourth term: since=, we have
= +and=, and the exponential again introduces the term
0
. The sum of
these two terms yields:
1
2}
2
0
d
0
()( )+ () (70)
The second and third terms on the right-hand side of (33) are dierent, since if
= +we have=, whereas if=we have= +, which introduces on the right-
hand side the matrix element()+. This means there is a term that couples two
complex conjugate matrix elements:
20( )
2}
2
0
d
0
()( ) ()+ (71)
() Thecomponent of the perturbation introduces in (33) the matrix elements
of, also non-diagonal but now equal to. For the rst and fourth term on the right-
hand side of (33), this introduces in the previous calculation a factor() () = 1, which
does not change anything. As for the second and third term, the factor equals()
2
=1,
which changes the sign of the result. Since the isotropy requires the correlation functions
of()and of()to be equal, we obtain the opposite of (71), and both terms cancel
out.
() We nally get:
1
2}
2
0
d
0
()( ) +()( ) + () (72)
Expanding
0
intocos (0) +sin (0), and taking (51) into account, we get:
1
21
+ + () (73)
where the coecientis dened by:
=
1
2}
2
0
dsin (0)
()( ) +()( ) (74)
The physical signicance of this coecient is discussed below.
1405

COMPLEMENT E XIII
. Eect of the longitudinal component of the perturbation
The component of the perturbation does not change the spin state. As a result,
the rst line in (33) contains=== +and=; the exponentials and the matrix
elements ofare all equal to unity. In the fourth line,= +and===,
the exponentials are again equal to unity, as is the product of two matrix elements of
(each equal to1), and the nal result is the same. As for the second and third lines,
we have== +and==, so that the exponentials are equal to1, whereas
the product of the matrix elements ofis now equal to1; these two terms double the
two preceding terms. Taking into account the factor12of (44), we nally obtain the
contribution:
1
}
2
0
d
()( )+ () (75)
that leads to the coecient of transverse relaxation:
1
2
=
1
}
2
0
d
()( ) (76)
. Discussion, role of the spectral density
Grouping together both contributions (73) and (75), we can write the complete
evolution of the non-diagonal element as:
d
d
+ ()=
1
21
+
1
2
++ () (77)
Leaving the interaction picture to go back to the density operator()in the usual
laboratory picture, we must add to this evolution the rst term appearing on the right-
hand side of (36); this yields:
d
d
+ ()=
1
21
+
1
2
+(0+ ) + () (78)
() Damping
In either picture, the non-diagonal element of the density matrix is damped with
a time constant2given by:
1
2
=
1
21
+
1
2
(79)
which is the sum of two contributions.
The rst is directly related to the longitudinal relaxation process. This process
changes the distribution of the populations between the two levels+and, hence de-
stroying the coherence between these two levels. This rate of destruction of the coherence
is the same as the one aecting the populations in (50), but half the one aecting()
in (54). Note that it is only the transverse components of the perturbation that play a
role in this contribution, since they are the ones that can induce transitions between the
two levels.
1406

TIME-DEPENDENT RANDOM PERTURBATION, RELAXATION
The second contribution comes only from the longitudinal component of the
perturbation. This uctuating component directly modies the energy dierence}0
between the two levels+and, and hence the precession velocity of the spins' transverse
component. When the dierent spins have dierent precession velocities around the
axis, their transverse components spread out and their vector sum diminishes. This leads
to a decrease of the transverse component of the global spin of the system.
() Frequency shift
Relation (78) shows that the term inis equivalent to a change in the precession
frequency0of the spins. In addition to the damping associated with the relaxation, the
perturbation introduces a shift in the evolution frequency of the non-diagonal elements.
The same calculation as the one leading to (58) yields for the frequency shift:
=
1
42}
2
0
d
+
d [() +()]
0 0
(80)
This expression contains an integral over d:
1
0
d
(+0) ( 0)
=
1
+0
1
0
[(+0)( 0)] (81)
As the functions()and ()are even, the terms containing the delta functions
cancel out, whereas the terms containing the principal parts double each other. This
leads to:
=
2
1
}
2
+
d
1
0
[() +()] (82)
Note that, contrary to the longitudinal relaxation characterized by the time1, it is not
the power spectrum of the perturbation at the resonant frequency=0that plays a
role. Only non-resonant frequencies contribute to the shift.
. Exponential correlation functions
When the correlation function is modeled by an exponential as in relation (47),
equalities (74) and (76) become:
1
2
=
1
}
2
[(0)]
2
(83)
and:
=
1
2}
2
[(0)]
2
+ [(0)]
2 0
1 + (0)
2
(84)
It is interesting to discuss the eect of the uctuations of the perturbation on the re-
laxation time
2. Imagine rst that the ensemble of spins is placed in a magnetic eld
along, which does not change in time but has a dierent value for each spin. We
1407

COMPLEMENT E XIII
notethe root mean square of the corresponding uctuation of the Hamiltonian. If the
spins are initially oriented in the same direction, their transverse orientations will start
spreading aroundsince each spin has a dierent precession velocity. The average
value of the global transverse orientation of the sample will go to zero over a time of the
order}. This means that the transverse orientation diminishes at a rate of the order
of}, which depends linearly on the perturbation amplitude. Now in the presence
of time dependent uctuations of the perturbation, result (83) predicts a totally dierent
behavior. The relaxation rate1
2is of the order of
2
}
2
and hence varies as the
square of the perturbation amplitude. This evolution rate in the presence of uctuations
is thus }times the evolution rate in the static case. As the factor}1
see the motional narrowing condition (38) the quadratic relaxation is much slower
than the relaxation in the absence of uctuation. This eect is even stronger whenis
shorter, which shows that it is the rapidly changing uctuations that are responsible for
the decrease of the relaxation rate.
Now the width of the magnetic resonance lines is an increasing function of the
transverse relaxation rate
3
. Consequently, the shorter the correlation time, the narrower
the lines. It often happens that the perturbation uctuations come from the spins' motion
in the sample, in which case the more rapid the motion, the narrower the magnetic
resonance lines. This explains the origin of the expression motional narrowing.
On the other hand, comparing relations (63) and (84) shows that the relaxation
probability and the frequency shift have a very dierent dependence on0. The relax-
ation probability follows a Lorentzian function, with a maximum for0= 0, whereas
the frequency shift is maximum for0= 1. This dierence comes from the fact that,
as we discussed at the end of , it is not the resonant but rather the non-resonant
frequencies of the spectral density that determine the frequency shift.
3. Conclusion
As mentioned in the introduction, there are many situations where an ensemble of indi-
vidual quantum systems is subjected to a random perturbation with a correlation time
very short compared to the other characteristic times of the problem. In a more gen-
eral way than in , we examined in this complement how, in
the limit whereis too short for the perturbation to have an eect during that time,
the perturbation no longer induces a Rabi type oscillation. This led us to introduce a
transition probability between the levels, leading to an exponential (and not oscillating)
evolution of the populations. Note that in ComplementXIII, we also obtained, with the
Fermi golden rule, a transition probability. In that case, it was the summation over the
energies of all the nal states that transformed the oscillation into a real and damped
exponential. In the present complement, it is the random character of the perturbation
that has a similar eect, even though the nal state is unique and has a perfectly well
dened energy. Another result we obtained concerns the existence of a frequency shift
induced by the random perturbation. In the case of an optical excitation such as the one
considered in , they are called light shifts, and have numerous
applications in atomic physics (ComplementXX).
3
Figure IVshows the variation of these lines, assuming there exists only one
longitudinal and transverse relaxation rate1.
1408

EXERCISES
Complement FXIII
Exercises
1.Consider a one-dimensional harmonic oscillator of mass, angular frequency
0and charge. Let and = (+ 12)~0be the eigenstates and eigenvalues of
its Hamiltonian0.
For0, the oscillator is in the ground state0. At= 0, it is subjected to an
electric eld pulse of duration. The corresponding perturbation can be written:
() =
for0
0 for0 and
is the eld amplitude andis the position observable. LetP0be the probability of
nding the oscillator in the stateafter the pulse.
. CalculateP01by using rst-order time-dependent perturbation theory. How
doesP01vary with, for xed0?
. Show that, to obtainP02, the time-dependent perturbation theory calculation
must be pursued at least to second order. CalculateP02to this perturbation order.
. Give the exact expressions forP01andP02in which the translation operator
used in ComplementVappears explicitly. By making a limited power series expansion
inof these expressions, nd the results of the preceding questions.
2.Consider two spin 1/2's,S1andS2, coupled by an interaction of the form
()S1S2;()is a function of time which approaches zero whenapproaches innity,
and takes on non-negligible values (on the order of0) only inside an interval, whose
width is of the order of, about= 0.
. At= , the system is in the state+ (an eigenstate of1and2with
the eigenvalues+~2and~2). Calculate, without approximations, the state of the
system at= +. Show that the probabilityP(+ +)of nding, at= +,
the system in the state+depends only on the integral
+
() d.
. CalculateP(+ +)by using rst-order time-dependent perturbation
theory. Discuss the validity conditions for such an approximation by comparing the
results obtained with those of the preceding question.
. Now assume that the two spins are also interacting with a static magnetic eld
B0parallel to. The corresponding Zeeman Hamiltonian can be written:
0= 0(11+22)
where1and2are the gyromagnetic ratios of the two spins, assumed to be dierent.
Assume that() =0e
22
. CalculateP(+ +)by rst-order time-
dependent perturbation theory. Considering0andas xed, discuss the variation of
P(+ +)with respect to0.
3. Two-photon transitions between non-equidistant levels
Consider an atomic level of angular momentum= 1, subject to static electric
and magnetic elds, both parallel to. It can be shown that three non-equidistant
1409

COMPLEMENT F XIII
energy levels are then obtained. The eigenstatesof(=10+1), of energies
, correspond to them. We set1 0=~0,0 1=~
0(0=
0).
The atom is also subjected to a radiofrequency eld rotating at the angular fre-
quencyin the plane. The corresponding perturbation()can be written:
() =
1
2
(+e+e)
where1is a constant proportional to the amplitude of the rotating eld.
. We set (notation identical to that of Chapter ):
()=
+1
=1
()e
~
Write the system of dierential equations satised by the().
. Assume that, at time= 0, the system is in the state1. Show that if we
want to calculate1()by time-dependent perturbation theory, the calculation must be
pursued to second order. Calculate1()to this perturbation order.
. For xed, how does the probabilityP1+1() =1()
2
of nding the system
in the state1at timevary with respect to? Show that a resonance appears, not
only for=0and=
0, but also for= (0+
0)2. Give a particle interpretation
of this resonance.
4.Returning to exercise 5 of ComplementXIand using its notation, assume that
the eldB0is oscillating at angular frequency, and can be writtenB0() =B0cos.
Assume that= 2and thatis not equal to any Bohr angular frequency of the system
(non-resonant excitation).
Introduce the susceptibility tensor, of components(), dened by:
() =Re ()0e
with= . Using a method analogous to the one in 2 of ComplementXIII,
calculate(). Setting= 0, nd the results of exercise 5 of ComplementXI.
5. The Autler-Townes eect
Consider a three-level system:1,2, and3, of energies1,2and3.
Assume 3 2 1and3 2 2 1.
This system interacts with a magnetic eld oscillating at the angular frequency.
The states2and 3are assumed to have the same parity, which is the opposite of
that of1, so that the interaction Hamiltonian()with the oscillating magnetic eld
can connect2and 3to1. Assume that, in the basis of the three states1,
2,3, arranged in that order,()is represented by the matrix:
0 0 0
0 0 1sin
01sin 0
where1is a constant proportional to the amplitude of the oscillating eld.
1410

EXERCISES
. Set (notation identical to that of Chapter ):
()=
3
=1
()e
~
Write the system of dierential equations satised by the().
. Assume thatis very close to32= (3 2)~. Making approximations
analogous to those used in ComplementXIII, integrate the preceding system, with the
initial conditions:
1(0) =2(0) =
1
2
3(0) = 0
(neglect, on the right-hand side of the dierential equations, the terms whose coecients,
e
(+32)
, vary very rapidly, and keep only those whose coecients are constant or vary
very slowly, ase
( 32)
).
. The component alongof the electric dipole moment of the system is
represented, in the basis of the three states1,2,3, arranged in that order, by
the matrix:
00
0 0
0 0 0
whereis a real constant (is an odd operator and can connect only states of dierent
parities).
Calculate () =() (), using the vector()calculated in
Show that the time evolution of()is given by a superposition of sinusoidal
terms. Determine the frequenciesand relative intensitiesof these terms.
These are the frequencies that can be absorbed by the atom when it is placed in
an oscillating electric eld parallel to. Describe the modications of this absorption
spectrum when, forxed and equal to32,1is increased from zero. Show that the
presence of the magnetic eld oscillating at the frequency322splits the electric dipole
absorption line at the frequency212, and that the separation of the two components
of the doublet is proportional to the oscillating magnetic eld amplitude (the Autler-
Townes doublet).
What happens when, for1xed, 32is varied?
6. Elastic scattering by a particle in a bound state. Form factor
Consider a particle()in a bound state0described by the wave function0(r)
localized about a point. Towards this particle()is directed a beam of particles(), of
mass, momentum~k, energy=~
2
k
2
2and wave function
1
(2)
32
e
kr
. Each
particle()of the beam interacts with particle(). The corresponding potential energy,
, depends only on the relative positionrrof the two particles.
. Calculate the matrix element:
:0;:k(RR):0;:k
1411

COMPLEMENT F XIII
of(R R)between two states in which particle()is in the same state0and
particle()goes from the statekto the statek. The expression for this matrix
element should include the Fourier transform
(k)of the potential(rr):
(rr) =
1
(2)
32(k) e
k(rr)
d
3
. Consider the scattering processes in which, under the eect of the interaction
, particle()is scattered in a certain direction, with particle()remaining in the same
quantum state0after the scattering process (elastic scattering).
Using a method analogous to the one in Chapter cf.comment()of ],
calculate, in the Born approximation, the elastic scattering cross section of particle()
by particle()in the state0.
Show that this cross section can be obtained by multiplying the cross section
for scattering by the potential(r)(in the Born approximation) by a factor which
characterizes the state0, called the form factor.
7. A simple model of the photoelectric eect
Consider, in a one-dimensional problem, a particle of mass, placed in a potential
of the form() = (), whereis a real positive constant.
Recall (cf.exercises 2 and 3 of ComplementI) that, in such a potential, there is
a single bound state, of negative energy0=
2
2~
2
, associated with a normalized
wave function0() =
~
2
e
~
2
. For each positive value of the energy=
~
22
2, on the other hand, there are two stationary wave functions, corresponding,
respectively, to an incident particle coming from the left or from the right. The expression
for the rst eigenfunction, for example, is:
() =
1
2
e
1
1 +~
2
e for 0
1
2
~
2
1 +~
2
e for 0
. Show that the()satisfy the orthonormalization relation (in the extended
sense):
=( )
The following relation [cf.formula (47) of Appendix II] can be used:
0
ed=
0
ed= Lim
0
1
+
=()
1Calculate the density of states()for a positive energy.
1412

EXERCISES
. Calculate the matrix element 0of the position observablebetween
the bound state0and the positive energy statewhose wave function was given
above.
. The particle, assumed to be charged (charge) interacts with an electric eld
oscillating at the angular frequency. The corresponding perturbation is:
() = sin
whereis a constant.
The particle is initially in the bound state0. Assume that~ 0. Calculate,
using the results of C-37)], the transition
probabilityper unit time to an arbitrary positive energy state (the photoelectric or
photoionization eect). How doesvary withand?
8. Disorientation of an atomic level due to collisions with rare gas atoms
Consider a motionless atomat the origin of a coordinate frame(Fig.).
This atomis in a level of angular momentum= 1, to which correspond the three
orthonormal kets(=10+1), eigenstates ofof eigenvalues~.
A second atom, in a level of zero angular momentum, is in uniform rectilinear
motion in theplane: it is travelling at the velocityalong a straight line parallel to
and situated at a distancefrom this axis (is the impact parameter). The time
origin is chosen at the time whenarrives at pointof theaxis (=). At
time, atomis therefore at point, where =. Callthe angle between
and .
The preceding model, which treats the external degrees of freedom of the two atoms
classically, permits the simple calculation of the eect on the internal degrees of freedom
of atom(which are treated quantum mechanically) of a collision with atom(which
is, for example, a rare gas atom in the ground state). It can be shown that, because
of the Van der Waals forces (cf.ComplementXI) between the two atoms, atomisz
O
M
H
θ
x
y
Figure 1
1413

COMPLEMENT F XIII
subject to a perturbationacting on its internal degrees of freedom, and given by:
=
6
2
whereis a constant,is the distance between the two atoms, andis the component
of the angular momentumJof atomon the axis joining the two atoms.
. Expressin terms of = . Introduce the dimensionless
parameter= .
. Assume that there is no external magnetic eld, so that the three states+ 1,
0,1of atomhave the same energy.
Before the collision, that is, at= , atomis in the state1. Using rst-
order time-dependent perturbation theory, calculate the probabilityP1+1of nding,
after the collision (that is, at= +), atomin the state+ 1. Discuss the variation
ofP1+1with respect toand. Similarly, calculateP10.
. Now assume that there is a static eldB0parallel to, so that the three
stateshave an additional energy~0(the Zeeman eect), where0is the Larmor
angular frequency in the eldB0.
. With ordinary magnetic elds (B010
2
gauss),010
9
rad.sec
1
;
is of the order of 5

A, and, of the order of510
2
m.sec
1
. Show that, under these
conditions, the results of questionremain valid.
Without going into detailed calculations, explain what happens for much
higher values of0. Starting with what value of0(whereandhave the values
indicated in) will the results ofno longer be valid?
. Without going into detailed calculations, explain how to calculate the disori-
entation probabilitiesP1+1andP10for an atomplaced in a gas of atomsin
thermodynamic equilibrium at the temperature, containing a numberof atoms per
unit volume suciently small that only binary collisions need be considered.
N.B. We give:
+
d(1 +
2
)
4
= 516
9. Transition probability per unit time under the eect of a random
perturbation. Simple relaxation model
This exercice uses the results of ComplementXIII. We consider a system ofspin
1/2 particles, with gyromagnetic ratio, placed in a static eldB0(set0= 0).
These particles are enclosed in a spherical cell of radius. Each of them bounces
constantly back and forth between the walls. The mean time between two collisions of
the same particle with the wall is called the ight time. During this time, the particle
sees only the eldB0. In a collision with the wall, each particle remains adsorbed on
the surface during a mean time( ), during which it sees, in addition toB0,
a constant microscopic magnetic eldb, due to the paramagnetic impurities contained
in the wall. The direction ofbvaries randomly from one collision to another; the mean
amplitude ofbis denoted by0.
. What is the correlation time of the perturbation seen by the spins? Give the
physical justication for the following form, to be chosen for the correlation function of
the components of the microscopic eldb:
()( ) =
1
3
2
0e
1414

EXERCISES
with analogous expressions for the components alongand, all the cross terms
()( )... being zero.
. Let be the component along theaxis dened by the eldB0of the
macroscopic magnetization of theparticles. Show that, under the eect of the collisions
with the wall,relaxes with a time constant1:
d
d
=
1
(1is called the longitudinal relaxation time). Calculate1in terms of,0,,,0.
. Show that studying the variation of1with0permits the experimental de-
termination of the mean adsorption time.
. We have at our disposition several cells, of dierent radii, constructed from
the same material. By measuring1, how can we determine experimentally the mean
amplitude0of the microscopic eld at the wall?
10. Absorption of radiation by a many-particle system forming a bound
state. The Doppler eect. Recoil energy. The Mössbauer eect
In ComplementXIII, we consider the absorption of radiation by a charged particle
attracted by a xed center(the hydrogen atom model for which the nucleus is innitely
heavy). In this exercise, we treat a more realistic situation, in which the incident radiation
is absorbed by a system of several particles of nite masses interacting with each other and
forming a bound state. Thus, we are studying the eect on the absorption phenomenon
of the degrees of freedom of the center of mass of the system.
I-Absorption of radiation by a free hydrogen atom. The doppler eect.
Recoil energy
LetR1andP1,R2andP2be the position and momentum observables of two
particles, (1) and (2), of masses1and2and opposite charges1and2(a hydrogen
atom). LetRandP,RandPbe the position and momentum observables of the
relative particle and of the center of mass (cf.Chap., ).= 1+ 2is the
total mass, and= 12(1+2)is the reduced mass. The Hamiltonian0of the
system can be written:
0= + (1)
where:
=
1
2
P
2
(2)
is the translational kinetic energy of the atom, assumed to be free (external degrees of
freedom), and where(which depends only onRandP) describes the internal energy
of the atom (internal degrees of freedom). We denote byKthe eigenstates of,
with eigenvalues~
2
K
2
2. We concern ourselves with only two eigenstates of,
and , of energiesand( ). We set:
=~0 (3)
. What energy must be furnished to the atom to move it from the stateK;
(the atom in the statewith a total momentum~K) to the stateK;?
1415

COMPLEMENT F XIII
. This atom interacts with a plane electromagnetic wave of wave vectorkand
angular frequency=, polarized along the unit vectoreperpendicular tok. The
corresponding vector potentialA(r)is:
A(r) =0ee
(kr )
+c.c. (4)
where0is a constant. The principal term of the interaction Hamiltonian between this
plane wave and the two-particle system can be written (cf.ComplementXIII, ):
() =
2
=1
PA(R) (5)
Express()in terms ofR,P,R,P,,and(setting1= 2=), and
show that, in the electric dipole approximation which consists of neglectingkR(but
notkR) compared to 1, we have:
() =e+ e (6)
where:
=
0
ePe
kR
(7)
. Show that the matrix element ofbetween the stateK;and the state
K;is dierent from zero only if there exists a certain relation betweenK,k,K(to
be specied). Interpret this relation in terms of the total momentum conservation during
the absorption of an incident photon by the atom.
. Show from this that if the atom in the stateK;is placed in the plane wave
(4), resonance occurs when the energy~of the photons associated with the incident
wave diers from the energy~0of the atomic transition by a quantity
which is to be expressed in terms of~,0,K,k,,(sinceis a corrective term, we
can replaceby0in the expression for). Show thatis the sum of two terms: one
of which,1, depends onKand on the angle betweenKandk(the Doppler eect); and
the other,2, is independent ofK. Give a physical interpretation of1and2(showing
that2is the recoil kinetic energy of the atom when, having been initially motionless, it
absorbs a resonant photon).
Show that2is negligible compared to1when~0is of the order of 10 eV (the
domain of atomic physics). Choose, for, a mass of the order of that of the proton
2
10
9
eV), and, forK, a value corresponding to a thermal velocity at= 300K.
Would this still be true if~0were of the order of 10
5
eV (the domain of nuclear physics)?
II-Recoilless absorption of radiation by a nucleus vibrating about its
equilibrium position in a crystal. The Mössbauer eect
The system under consideration is now a nucleus of massvibrating at the
angular frequencyabout its equilibrium position in a crystalline lattice (the Einstein
model;cf.ComplementV, 2). We again denote byRandPthe position and
momentum of the center of mass of this nucleus. The vibrational energy of the nucleus
is described by the Hamiltonian:
=
1
2
P
2
+
1
2

2
(
2
+
2
+
2
) (8)
1416

EXERCISES
which is that of a three-dimensional isotropic harmonic oscillator. Denote by
the eigenstate ofof eigenvalue(+++ 32)~. In addition to these external
degrees of freedom, the nucleus possesses internal degrees of freedom with which are
associated observables which all commute withRandP. Letbe the Hamiltonian
that describes the internal energy of the nucleus. As above, we concern ourselves with
two eigenstates of, and , of energiesand, and we set~0= .
Since~0falls into the-ray domain, we have, of course:
0 (9)
. What energy must be furnished to the nucleus to allow it to go from the state
000;(the nucleus in the vibrational state dened by the quantum numbers= 0,
= 0,= 0and the internal state) to the state00;?
. This nucleus is placed in an electromagnetic wave of the type dened by (4),
whose wave vectorkis parallel to. It can be shown that, in the electric dipole approx-
imation, the interaction Hamiltonian of the nucleus with this plane wave (responsible for
the absorption of the-rays) can be written as in (6), with:
=0() e (10)
where()is an operator which acts on the internal degrees of freedom and consequently
commutes withRandP. Set() = ().
The nucleus is initially in the state000;. Show that, under the inuence
of the incident plane wave, a resonance appears whenever~coincides with one of the
energies calculated in, with the intensity of the corresponding resonance proportional to
()
2
00e 000
2
, where the value ofis to be specied. Show, furthermore,
that condition (9) allows us to replaceby0=0in the expression for the intensity
of the resonance.
. We set:
(0) = e
0
0
2
(11)
where the states are the eigenstates of a one-dimensional harmonic oscillator of
position, massand angular frequency.
. Calculate(0)in terms of~,,,0,(see also exercise 7 of Comple-
mentV). Set=
~
22
0
2
~.
Hint: establish a recurrence relation betweene
0
0and 1e
0
0,
and express all the(0)in terms of0(0), which is to be calculated directly from the
wave function of the harmonic oscillator ground state. Show that the(0)are given
by a Poisson distribution.
. Verify that
=0
(0) = 1
. Show that
=0
~ (0) =~
22
02
2
.
. Assume that~ ~
22
02
2
, i.e. that the vibrational energy of the nucleus
is much greater than the recoil energy (very rigid crystalline bonds). Show that the
absorption spectrum of the nucleus is essentially composed of a single line of angular
frequency0. This line is called the recoilless absorption line. Justify this name. Why
does the Doppler eect disappear?
1417

COMPLEMENT F XIII
. Now assume that~ ~
22
02
2
(very weak crystalline bonds). Show that
the absorption spectrum of the nucleus is composed of a very large number of equidistant
lines whose barycenter (obtained by weighting the abscissa of each line by its relative
intensity) coincides with the position of the absorption line of the free and initially
motionless nucleus. What is the order of magnitude of the width of this spectrum (the
dispersion of the lines about their barycenter)? Show that one obtains the results of the
rst part in the limit 0.
REFERENCES
Exercise 3:
References: see Brossel's lectures in (15.2).
Exercise 5:
References: see Townes and Schawlow (12.10), Chap. 10, 9.
Exercise 6:
References: see Wilson (16.34).
Exercise 9:
References: see Abragam (14.1), Chap. VIII; Slichter (14.2), Chap. 5.
Exercise 10:
References: see De Benedetti (16.23); Valentin (16.1), annex XV.
1418

Chapter XIV
Systems of identical particles
A Statement of the problem
A-1 Identical particles: denition
A-2 Identical particles in classical mechanics
A-3 Identical particles in quantum mechanics: the diculties of
applying the general postulates
B Permutation operators
B-1 Two-particle systems
B-2 Systems containing an arbitrary number of particles
C The symmetrization postulate
C-1 Statement of the postulate
C-2 Removal of exchange degeneracy
C-3 Construction of physical kets
C-4 Application of the other postulates
D Discussion
D-1 Dierences between bosons and fermions. Pauli's exclusion
principle
D-2 The consequences of particle indistinguishability on the cal-
culation of physical predictions
In Chapter, we stated the postulates of non-relativistic quantum mechanics, and
in Chapter, we concentrated on those which concern spin degrees of freedom. Here,
we shall see () that, in reality, these postulates are not sucient when we are dealing
with systems containing several identical particles since, in this case, their application
leads to ambiguities in the physical predictions. To eliminate these ambiguities, it is
necessary to introduce a new postulate, concerning the quantum mechanical description
of systems of identical particles. We shall state this postulate in
physical implications in . Before we do so, however, we shall (in ) dene and study
permutation operators, which considerably facilitate the reasoning and the calculations.
Quantum Mechanics, Volume II, Second Edition. C. Cohen-Tannoudji, B. Diu, and F. Laloë.
© 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

CHAPTER XIV SYSTEMS OF IDENTICAL PARTICLES
A. Statement of the problem
A-1. Identical particles: denition
Two particles are said to be identical if all their intrinsic properties (mass, spin,
charge, etc.) are exactly the same: no experiment can distinguish one from the other.
Thus, all the electrons in the universe are identical, as are all the protons and all the
hydrogen atoms. On the other hand, an electron and a positron are not identical, since,
although they have the same mass and the same spin, they have dierent electrical
charges.
An important consequence can be deduced from this denition: when a physical
system contains two identical particles, there is no change in its properties or its evolution
if the roles of these two particles are exchanged.
Comment:
Note that this denition is independent of the experimental conditions. Even if,
in a given experiment, the charges of the particles are not measured, an electron
and a positron can never be treated like identical particles.
A-2. Identical particles in classical mechanics
In classical mechanics, the presence of identical particles in a system poses no
particular problems. This special case is treated just like the general case. Each particle
moves along a well-dened trajectory, which enables us to distinguish it from the others
and follow it throughout the evolution of the system.
To treat this point in greater detail, we shall consider a system of two identical
particles. At the initial time0, the physical state of the system is dened by specifying
the position and velocity of each of the two particles; we denote these initial data by
r0v0andr
0v
0. To describe this physical state and calculate its evolution, we
number the two particles:r1()andv1()denote the position and velocity of particle
(1) at time, andr2()andv2(), those of particle (2). This numbering has no physical
foundation, as it would if we were dealing with two particles having dierent natures.
It follows that the initial physical state which we have just dened may, in theory, be
described by two dierent mathematical states as we can set, either:
r1(0) =r0r2(0) =r
0
v1(0) =v0v2(0) =v
0 (A-1)
or:
r1(0) =r
0r2(0) =r0
v1(0) =v
0v2(0) =v0 (A-2)
Now, let us consider the evolution of the system. Suppose that the solution of the
equations of motion dened by initial conditions (A-1) can be written:
r1() =r()r2() =r() (A-3)
1420

A. STATEMENT OF THE PROBLEM
wherer()andr()are two vector functions. The fact that the two particles are iden-
tical implies that the system is not changed if they exchange roles. Consequently, the
Lagrangian(r1v1;r2v2)and the classical Hamiltonian(r1p1;r2p2)are invariant
under exchange of indices 1 and 2. It follows that the solution of the equations of motion
corresponding to the initial state (A-2) is:
r1() =r()r2() =r() (A-4)
where the functionsr()andr()are the same as in (A-3).
The two possible mathematical descriptions of the physical state under considera-
tion are therefore perfectly equivalent, since they lead to the same physical predictions.
The particle which started fromr0v0at0is atr()with the velocityv() = drdat
time, and the one which started fromr
0v
0is atr()with the velocityv() = drd
(Fig.). Under these conditions, all we need to do is choose, at the initial time, either
one of the two possible mathematical states and ignore the existence of the other one.
Thus, we treat the system as if the two particles were actually of dierent natures. The
numbers (1) and (2), with which we label them arbitrarily at0, then act like intrinsic
properties to distinguish the two particles. Since we can follow each particle step-by-step
along its trajectory (arrows in Figure), we can determine the locations of the particle
numbered (1) and the one numbered (2) at any time.r
0
, v
0 r(t), v(t)
r(t), v(t)r
0
, v
0
Initial state State at the instant l
Figure 1: Position and velocity of each of the two particles at the initial time0and at
time.
A-3. Identical particles in quantum mechanics: the diculties of applying the general
postulates
A-3-a. Qualitative discussion of a rst simple example
It is immediately apparent that the situation is radically dierent in quantum
mechanics, since the particles no longer have denite trajectories. Even if, at0, the
wave packets associated with two identical particles are completely separated in space,
their subsequent evolution may mix them. We then lose track of the particles; when
we detect one particle in a region of space in which both of them have a non-zero position
probability, we have no way of knowing if the particle detected is the one numbered (1)
or the one numbered (2). Except in special cases for example, when the two wave
packets never overlap the numbering of the two particles becomes ambiguous when
their positions are measured, since, as we shall see, there exist several distinct paths
taking the system from its initial state to the state found in the measurement.
1421

CHAPTER XIV SYSTEMS OF IDENTICAL PARTICLES
To investigate this point in greater detail, consider a concrete example; a collision
between two identical particles in their center of mass frame (Fig.). Before the collision,
we have two completely separate wave packets, directed towards each other (Fig.a). We
can agree, for example, to denote by (1) the particle on the left and by (2), the one on the
right. During the collision (Fig.b), the two wave packets overlap. After the collision,
the region of space in which the probability density of the two particles is non-zero
1
looks like a spherical shell whose radius increases over time (Fig.c). Suppose that a
detector placed in the direction which makes an anglewith the initial velocity of wave
packet (l) detects a particle. It is then certain (because momentum is conserved in the
collision) that the other particle is moving away in the opposite direction. However, it is
impossible to know if the particle detected atis the one initially numbered (1) or the
one numbered (2). Thus, there are two dierent paths that could have led the system
from the initial state shown in Figurea to the nal state found in the measurement.
These two paths are represented schematically in Figuresa andb. Nothing enables us
to determine which one was actually followed.
Figure 2: Collision between two identical particles in the center of mass frame: schematic
representation of the probability density of the two particles. Before the collision (g. a),
the two wave packets are clearly separated and can be labeled. During the collision (g. b),
the two wave packets overlap. After the collision (g. c), the probability density is non-
zero in a region shaped like a spherical shell whose radius increases over time. Because
the two particles are identical, it is impossible, when a particle is detected at, to know
with which wave packet, (1) or (2), it was associated before the collision.
A fundamental diculty then arises in quantum mechanics when using the postu-
lates of Chapter. In order to calculate the probability of a given measurement result
it is necessary to know the nal state vectors associated with this result. Here, there are
two, which correspond respectively to Figuresa andb. These two kets are distinct
(and, furthermore, orthogonal). Nevertheless, they are associated with a single physical
1
The two-particle wave function depends on six variables (the components of the two particles coor-
dinatesrandr) and is not easily represented in 3 dimensions. Figure
grey regions are those to which bothrandrmust belong for the wave function to take on signicant
values.
1422

A. STATEMENT OF THE PROBLEM(1)
(1)
(1)
(1)
(2)
(2)
D D
a b
(2)
(2)
Figure 3: Schematic representation of two types of paths which the system could have
followed in going from the initial state to the state found in the measurement. Because
the two particles are identical, we cannot determine the path that was actually followed.
state since it is impossible to imagine a more complete measurement that would permit
distinguishing between them. Under these conditions, should one calculate the probabil-
ity using path 3a, path 3b or both? In the latter case, should one take the sum of the
probabilities associated with each path, or the sum of their probability amplitudes (and
in this case, with what sign)? These dierent possibilities lead, as we shall verify later,
to dierent predictions.
The answer to the preceding questions will be given in
symmetrization postulate. Before going on, we shall study another example that will aid
us in understanding the diculties related to the indistinguishability of two particles.
A-3-b. Origin of the diculties: Exchange degeneracy
In the preceding example, we considered two wave packets which, initially, did
not overlap; this enabled us to label each of them arbitrarily with a number, (1) or
(2). Ambiguities appeared, however, when we tried to determine the mathematical state
(or ket) associated with a given result of a position measurement. Actually, the same
diculty arises in the choice of the mathematical ket used to describe the initial physical
state. This type of diculty is related to the concept of exchange degeneracy which
we shall introduce in this section. To simplify the reasoning, we shall rst consider a
dierent example, so as to conne ourselves to a nite-dimensional space. Then, we shall
generalize the concept of exchange degeneracy, showing that it can be generalized to all
quantum mechanical systems containing identical particles.
. Exchange degeneracy for a system of two spin 1/2 particles
Let us consider a system composed of two identical spin 1/2 particles, conning
ourselves to the study of its spin degrees of freedom. As in , we shall distinguish
between the physical state of the system and its mathematical description (a ket in state
space).
It would seem natural to suppose that, if we made a complete measurement of each
of the two spins, we would then know the physical state of the total system perfectly.
Here, we shall assume that the component alongof one of them is equal to+~/2 and
that of the other one, ~/2 (this is the equivalent for the two spins of the specication
1423

CHAPTER XIV SYSTEMS OF IDENTICAL PARTICLES
ofr0v0andr
0v
0in ).
To describe the system mathematically, we number the particles:S1andS2denote
the two spin observables, and12(where1and2can be equal to+or) is the
orthonormal basis of the state space formed by the common eigenkets of1(eigenvalue
1~2) and2(eigenvalue2~2).
Just as in classical mechanics, two dierent mathematical states could be asso-
ciated with the same physical state. Either one of the two orthogonal kets:
1= + 2= (A-5a)
1= 2= + (A-5b)
can,a priori, describe the physical state considered here.
These two kets span a two-dimensional subspace whose normalized vectors are of
the form:
+ + + (A-6)
with:
2
+
2
= 1 (A-7)
By the superposition principle, all mathematical kets (A-6) can represent the same phys-
ical state as (A-5a) or (A-5b) (one spin pointing up and the other one pointing down).
This is called exchange degeneracy.
Exchange degeneracy creates fundamental diculties, since application of the pos-
tulates of Chapter A-6) can lead to physical predictions that
depend on the ket chosen. Let us determine, for example, the probability of nding the
components of the two spins alongboth equal to+~2. With this measurement result
is associated a single ket of the state space. According to formula (A-20) of Chapter,
this ket can be written:
1
2
[1= ++1=]
1
2
[2= ++2=]
=
1
2
+++ +++ + (A-8)
Consequently, the desired probability, for the vector (A-6), is equal to:
1
2
(+)
2
(A-9)
This probability does depend on the coecientsand. It is not possible, therefore, to
describe the physical state under consideration by the set of kets (A-6) or by any one of
them chosen arbitrarily. The exchange degeneracy must be removed. That is, we must
indicate unambiguously which of the kets (A-6) is to be used.
Comment:
In this example, exchange degeneracy appears only in the initial state, since we
chose the same value for the components of the two spins in the nal state. In the
general case (for example, if the measurement result corresponds to two dierent
eigenvalues of), exchange degeneracy appears in both the initial and the nal
state.
1424

B. PERMUTATION OPERATORS
. Generalization
The diculties related to exchange degeneracy arise in the study of all systems
containing an arbitrary numberof identical particles (1).
Consider, for example, a three-particle system. With each of the three particles,
taken separately, are associated a state space and observables acting in this space. Thus,
we are led to number the particles:(1),(2)and(3)will denote the three one-particle
state spaces, and the corresponding observables will be labeled by the same indices. The
state space of the three-particle system is the tensor product:
=(1)(2)(3) (A-10)
Now, consider an observable(1), initially dened in(1). We shall assume
that(1)alone constitutes a C.S.C.O. in(1)[or that(1)actually denotes several
observables which form a C.S.C.O.]. The fact that the three particles are identical implies
that the observables(2)and(3)exist and that they constitute C.S.C.O.'s in(2)and
(3)respectively.(1),(2)and(3)have the same spectrum,;= 12 . Using
the bases that dene these three observables in(1),(2)and(3), we can construct,
by taking the tensor product, an orthonormal basis of, which we shall denote by:
1 :; 2 :; 3 :; = 12 (A-11)
The kets1 :; 2 :; 3 :are common eigenvectors of the extensions of(1),(2)
and(3)in, with respective eigenvalues,et.
Since the three particles are identical, we cannot measure(1)or(2)or(3),
since the numbering has no physical signicance. However, we can measure the physical
quantityfor each of the three particles. Suppose that such a measurement has resulted
in three dierent eigenvalues,,and. Exchange degeneracy then appears, since
the state of the system after this measurement can,a priori, be represented by any one
of the kets of the subspace ofspanned by the six basis vectors:
1 :; 2 :; 3 : 1 :; 2 :; 3 : 1 :; 2 :; 3 :
1 :; 2 :; 3 : 1 :; 2 :; 3 : 1 :; 2 :; 3 : (A-12)
Therefore,a complete measurement on each of the particles does not permit the determi-
nation of a unique ketof the state space of the system.
Comment:
The indeterminacy due to exchange degeneracy is, of course, less important if
two of the eigenvalues found in the measurement are equal. This indeterminacy
disappears in the special case in which the three results are identical.
B. Permutation operators
Before stating the additional postulate that enables us to remove the indeterminacy
related to exchange degeneracy, we shall study certain operators, dened in the total state
space of the system under consideration, which actually permute the various particles of
the system. The use of these permutation operators will simplify the calculations and
reasoning in .
1425

CHAPTER XIV SYSTEMS OF IDENTICAL PARTICLES
B-1. Two-particle systems
B-1-a. Denition of the permutation operator 21
Consider a system composed of two particles with the same spin. Here it is not
necessary for these two particles to be identical; it is sucient that their individual state
spaces be isomorphic. Therefore, to avoid the problems that arise when the two particles
are identical, we shall assume that they are not: the numbers (1) and (2) with which
they are labeled indicate their natures. For example, (1) will denote a proton and (2),
an electron.
We choose a basis, , in the state space(1)of particle (1). Since the two
particles have the same spin,(2)is isomorphic to(1), and it can be spanned by the
same basis. By taking the tensor product, we construct, in the state spaceof the
system, the basis:
1 :; 2 : (B-1)
Since the order of the vectors is of no importance in a tensor product, we have:
2 :; 1 : 1 :; 2 : (B-2)
However, note that:
1 :; 2 :=1 :; 2 :if= (B-3)
The permutation operator21is then dened as the linear operator whose action
on the basis vectors is given by:
211 :; 2 :=2 :; 1 :=1 :; 2 : (B-4)
Its action on any ket ofcan easily be obtained by expanding this ket
2
on the basis
(B-1).
Comment:
If we choose the basis formed by the common eigenstates of the position observableR
and the spin component, (B-4) can be written:
211 :r; 2 :r =1 :r; 2 :r (B-5)
Any ket of the state spacecan be represented by a set of(2+ 1)
2
functions of six
variables:
= d
3
d
3
(rr)1 :r; 2 :r (B-6)
with:
(rr) =1 :r; 2 :r (B-7)
2
It can easily be shown that the operator21so dened does not depend on the basis chosen.
1426

B. PERMUTATION OPERATORS
We then have:
21=
3 3
(rr)1 :r; 2 :r (B-8)
By changing the names of the dummy variables:
rr (B-9)
we transform formula (B-8) into:
21=
3 3
(rr)1 :r; 2 :r (B-10)
Consequently, the functions:
(rr) =1 :r; 2 :r 21 (B-11)
which represent the ket= 21can be obtained from the functions (B-7) which
represent the ketby inverting(r)and(r):
(rr) = (rr) (B-12)
B-1-b. Properties of 21
We see directly from denition (B-4) that:
(21)
2
= 1 (B-13)
The operator21is its own inverse.
It can easily be shown that21is Hermitian:
21
=21 (B-14)
The matrix elements of21in the1 :; 2 :basis are:
1 :; 2 : 211 :; 2 :=1 :; 2 :1 :; 2 :
= (B-15)
Those of
21
are, by denition:
1 :; 2 :
21
1 :; 2 := (1 :; 2 : 211 :; 2 :)
= (1 :; 2 :1 :; 2 :)
= (B-16)
Each of the matrix elements of
21
is therefore equal to the corresponding matrix element
of21. This leads to relation (B-14).
It follows from (B-13) and (B-14) that21is alsounitary:
2121=2121
= 1 (B-17)
1427

CHAPTER XIV SYSTEMS OF IDENTICAL PARTICLES
B-1-c. Symmetric and antisymmetric kets. Symmetrizer and antisymmetrizer
According to relation (B-14), the eigenvalues of21must be real. Since, accord-
ing to (B-13), their squares are equal to 1, these eigenvalues are simply+1 and1.
The eigenvectors of21associated with the eigenvalue+1 are calledsymmetric, those
corresponding to the eigenvalue1,antisymmetric:
21 = = symmetric
21 = = antisymmetric (B-18)
Now consider the two operators:
=
1
2
(1 +21) (B-19a)
=
1
2
(1 21) (B-19b)
These operators areprojectors, since (B-13) implies that:
2
= (B-20a)
2
= (B-20b)
and, in addition, (B-14) enables us to show that:
= (B-21a)
= (B-21b)
andare projectors onto orthogonal subspaces, since, according to (B-13):
= = 0 (B-22)
These subspaces are supplementary, since denitions (B-19) yield:
+= 1 (B-23)
Ifis an arbitrary ket of the state space, is a symmetric ket and,
an antisymmetric ket, as it is easy to see, using (B-13) again, that:
21 =
21 = (B-24)
For this reason,andare called, respectively, asymmetrizerand anantisymmetrizer.
Comment:
The same symmetric ket is obtained by applyingto21or toitself:
21= (B-25)
For the antisymmetrizer, we have, similarly:
21= (B-26)
1428

B. PERMUTATION OPERATORS
B-1-d. Transformation of observables by permutation
Consider an observable(1), initially dened in(1)and then extended into.
It is always possible to construct thebasis in(1)from eigenvectors of(1)(the
corresponding eigenvalues will be written). Let us now calculate the action of the
operator21(1)
21
on an arbitrary basis ket of:
21(1)
21
1 :; 2 :=21(1)1 :; 2 :
= 211 :; 2 :
=1 :; 2 : (B-27)
We would obtain the same result by applying the observable(2)directly to the basis
ket chosen. Consequently:
21(1)
21
=(2) (B-28)
The same reasoning shows that:
21(2)
21
=(1) (B-29)
In, there are also observables, such as(1) +(2)or(1)(2), which involve both
indices simultaneously. We obviously have:
21[(1) +(2)]
21
=(2) +(1) (B-30)
Similarly, using (B-17), we nd:
21(1)(2)
21
=21(1)
2121(2)
21
=(2)(1) (B-31)
These results can be generalized to all observables inwhich can be expressed in terms
of observables of the type of(1)and(2), to be denoted by(12):
21(12)
21
=(21) (B-32)
(21)is the observable obtained from(12)by exchanging indices 1 and 2 throughout.
An observable(12)is said to besymmetricif:
(21) =(12) (B-33)
According to (B-32), all symmetric observables satisfy:
21(12) =(12)21 (B-34)
that is:
[(12)21] = 0 (B-35)
Symmetric observables commute with the permutation operator.
1429

CHAPTER XIV SYSTEMS OF IDENTICAL PARTICLES
B-2. Systems containing an arbitrary number of particles
In the state space of a system composed ofparticles with the same spin (tem-
porarily assumed to be of dierent natures),!permutation operators can be dened
(one of which is the identity operator). Ifis greater than 2, the properties of these
operators are more complex than those of21. To have an idea of the changes involved
whenis greater than 2, we shall briey study the case in which= 3.
B-2-a. Denition of the permutation operators
Consider, therefore, a system of three particles that are not necessarily identical,
but have the same spin. As in , we construct a basis of the state space of the
system by taking a tensor product:
1 :; 2 :; 3 : (B-36)
In this case, there exist six permutation operators, which we shall denote by:
123 312 231 132 213 321 (B-37)
By denition, the operator(where,,is an arbitrary permutation of the numbers
1, 2, 3) is the linear operator whose action on the basis vectors obeys:
1 :; 2 :; 3 :=:;:;: (B-38)
For example:
2311 :; 2 :; 3 :=2 :; 3 :; 1 :
=1 :: 2 :; 3 : (B-39)
123therefore coincides with the identity operator. The action ofon any ket of the
state space can easily be obtained by expanding this ket on the basis (B-36).
The!permutation operators associated with a system ofparticles with the
same spin could be dened analogously.
B-2-b. Properties
. The set of permutation operators constitutes a group
This can easily be shown for the operators (B-37):
()123is the identity operator.
()The product of two permutation operators is also a permutation operator. We can show,
for example, that:
312132=321 (B-40)
To do so, we apply the left-hand side to an arbitrary basis ket:
3121321 :; 2 :; 3 :
=3121 :; 3 :; 2 :
=3121 :; 2 :; 3 :
=3 :; 1 :; 2 :
=1 :; 2 :; 3 : (B-41)
1430

B. PERMUTATION OPERATORS
The action of321eectively leads to the same result:
3211 :; 2 :; 3 :=3 :; 2 :; 1 :
=1 :; 2 :; 3 : (B-42)
()Each permutation operator has an inverse, which is also a permutation operator. Rea-
soning as in(), we can easily show that:
1
123=123;
1
312=231;
1
231=312
1
132=132;
1
213=213;
1
321=321 (B-43)
Note thatthe permutation operators do not commute with each other.For example:
132312=213 (B-44)
which, compared to (B-40), shows that the commutator of132and312is not zero.
. Transpositions. Parity of a permutation operator
Atranspositionis a permutation which simply exchanges the roles of two of the
particles, without touching the others. Of the operators (B-37), the last three are trans-
position operators
3
. Transposition operators are Hermitian, and each of them is the same
as its inverse, so that they are also unitary [the proofs of these properties are identical
to those for (B-14), (B-13) and (B-17)].
Any permutation operator can be broken down into a product of transposition op-
erators. For example, the second operator (B-37) can be written:
312=132213=321132=213321=132213(132)
2
= (B-45)
This decomposition is not unique. However, for a given permutation, it can be shown that
the parity of the number of transpositions into which it can be broken down is always the
same: it is called theparity of the permutation. Thus, the rst three operators (B-37) are
even, and the last three, odd. For any, there are always as many even permutations
as odd ones.
. Permutation operators are unitary
Permutation operators, which are products of transposition operators, all of which
are unitary, are therefore also unitary. However, they are not necessarily Hermitian, since
transposition operators do not generally commute with each other.
Finally, note that the adjoint of a given permutation operator has the same parity
as that of the operator, since it is equal to the product of the same transposition operators,
taken in the opposite order.
B-2-c. Completely symmetric or antisymmetric kets. Symmetrizer and antisymmetrizer
Since the permutation operators do not commute for2, it is not possible to
construct a basis formed by common eigenvectors of these operators. Nevertheless, we
shall see that there exist certain kets which are simultaneously eigenvectors of all the
permutation operators.
3
Of course, for= 2, the only possible permutation is a transposition.
1431

CHAPTER XIV SYSTEMS OF IDENTICAL PARTICLES
We shall denote byan arbitrary permutation operator associated with a system
ofparticles with the same spin;represents an arbitrary permutation of the rst
integers. A ketsuch that:
= (B-46)
for any permutation, is said to becompletely symmetric. Similarly, acompletely
antisymmetricket satises, by denition
4
:
= (B-47)
where:
= +1ifis an even permutation
=1ifis an odd permutation (B-48)
The set of completely symmetric kets constitutes a vector subspaceof the state space
; the set of completely antisymmetric kets, a subspace.
Now consider the two operators:
=
1
!
(B-49)
=
1
!
(B-50)
where the summations are performed over the!permutations of the rstintegers,
andis dened by (B-48). We shall show thatandare the projectors ontoand
respectively. For this reason, they are called asymmetrizerand anantisymmetrizer.
andare Hermitian operators:
= (B-51)
= (B-52)
The adjointof a given permutation operator is, as we saw above (cf. ), another
permutation operator, of the same parity (which coincides, furthermore, with
1
). Taking
the adjoints of the right-hand sides of the denitions ofandtherefore amounts simply
to changing the order of the terms in the summations (since the set of the
1
is again the
permutation group).
Also, if
0
is an arbitrary permutation operator, we have:
0
=
0
= (B-53a)
0
=
0
=
0
(B-53b)
This is due to the fact that
0is also a permutation operator:
0= (B-54)
4
According to the property stated in , this denition can also be based solely on the
transposition operators: any transposition operator leaves a completely symmetric ket invariant and
transforms a completely antisymmetric ket into its opposite.
1432

B. PERMUTATION OPERATORS
such that:
=
0 (B-55)
If, for
0xed, we choose successively forall the permutations of the group, we see that
theare each identical to one and only one of these permutations (in, of course, a dierent
order). Consequently:
0=
1
!
0=
1
!
= (B-56a)
0=
1
!
0=
1
!
0 =
0 (B-56b)
Similarly, we could prove analogous relations in whichandare multiplied by
0from the
right.
From (B-53), we see that:
2
=
2
= (B-57)
and, moreover:
= = 0 (B-58)
This is because:
2
=
1
!
=
1
!
=
2
=
1
!
=
1
!
2
= (B-59)
as each summation includes!terms; furthermore:
=
1
!
=
1
!
= 0 (B-60)
since half theare equal to+1 and half equal to1 (cf. ).
andare thereforeprojectors. They project respectively ontoandsince,
according to (B-53), their action on any ketof the state space yields a completely
symmetric or completely antisymmetric ket:
0
= (B-61a)
0
=
0
(B-61b)
Comments:
()The completely symmetric ket constructed by the action ofon , where
is an arbitrary permutation, is the same as that obtained from, since
expressions (B-53) indicate that:
= (B-62)
1433

CHAPTER XIV SYSTEMS OF IDENTICAL PARTICLES
As for the corresponding completely antisymmetric kets, they dier at most
by their signs:
= (B-63)
()For 2, the symmetrizer and antisymmetrizer are not projectors onto
supplementary subspaces. For example, when= 3, it is easy to obtain
[by using the fact that the rst three permutations (B-37) are even and the
others odd] the relation:
+=
1
3
(123+231+312)= 1 (B-64)
In other words, the state space is not the direct sum of the subspaceof
completely symmetric kets and the subspaceof completely antisymmetric
kets.
B-2-d. Transformation of observables by permutation
We have indicated ( ) that any permutation operator of an-particle
system can be broken down into a product of transposition operators analogous to the
operator21studied in . For these transposition operators, we can use the argu-
ments of
they are multiplied from the left by an arbitrary permutation operatorand from the
right by.
In particular, the observables(12 )which are completely symmetric un-
der exchange of the indices 1, 2, . . . ,, commute with all the transposition operators,
and, therefore, with all the permutation operators:
[(12 )] = 0 (B-65)
C. The symmetrization postulate
C-1. Statement of the postulate
When a system includes several identical particles, only certain kets of its state space
can describe its physical states. Physical kets are, depending on the nature of the
identical particles, either completely symmetric or completely antisymmetric with re-
spect to permutation of these particles. Those particles for which the physical kets are
symmetric are calledbosons, and those for which they are antisymmetric,fermions.
The symmetrization postulate thus limits the state space for a system of identical
particles. This space is no longer, as it was in the case of particles of dierent natures, the
tensor productof the individual state spaces of the particles constituting the system.
It is only a subspace of, namelyor, depending on whether the particles are
bosons or fermions.
From the point of view of this postulate, particles existing in nature are divided
into two categories. All currently known particles obey the followingempirical rule
5
:
particles of half-integral spin (electrons, positrons, protons, neutrons, muons, etc.) are
fermions, and particles of integral spin (photons, mesons, etc.) are bosons.
5
The spin-statistics theorem, proven in quantum eld theory, makes it possible to consider this
1434

C. THE SYMMETRIZATION POSTULATE
Comment:
Once this rule has been veried for the particles which are called elementary, it holds for
all other particles as well, inasmuch as they are composed of these elementary particles.
Consider a system of many identical composite particles. Permuting two of them is
equivalent to simultaneously permuting all the particles composing the rst one with the
corresponding particles (necessarily identical to the aforementioned ones) of the second
one. This permutation must leave the ket describing the state of the system unchanged
if the composite particles being studied are formed only of elementary bosons or if each
of them contains an even number of fermions (no sign change, or an even number of sign
changes); in this case, the particles are bosons. On the other hand, composite particles
containing an odd number of fermions are themselves fermions (an odd number of sign
changes in the permutation). Now, the spin of these composite particles is necessarily
integral in the rst case and half-integral in the second one (Chap., ). They
therefore obey the rule just stated. For example, atomic nuclei are known to be composed
of neutrons and protons, which are fermions (spin 1/2). Consequently, nuclei whose
mass number(the total number of nucleons) is even are bosons, and those whose mass
number is odd are fermions. Thus, the nucleus of the
3
He isotope of helium is a fermion,
and that of the
4
He isotope, a boson.
C-2. Removal of exchange degeneracy
We shall begin by examining how this new postulate removes the exchange degen-
eracy and the corresponding diculties.
The discussion of be a ket
which can mathematically describe a well-dened physical state of a system containing
identical particles. For any permutation operator, can describe this physical
state as well as. The same is true for any ket belonging to the subspacespanned
byand all its permutations. Depending on the ketchosen, the dimension of
can vary between 1 and!. If this dimension is greater than 1, several mathematical
kets correspond to the same physical state: there is then an exchange degeneracy.
The new postulate which we have introduced considerably restricts the class of
mathematical kets able to describe a physical state: these kets must belong tofor
bosons, or tofor fermions. We shall be able to say that the diculties related to
exchange degeneracy are eliminated if we can show thatcontainsa singleket of
ora singleket of.
To do so, we shall use the relations= or= , proven in (B-53). We
obtain:
= (C-1a)
= (C-1b)
These relations express the fact that the projections ontoandof the various kets
which spanand, consequently, of all the kets of, are collinear. The symmetrization
postulate thus unambiguously indicates (to within a constant factor)theket ofwhich
rule to be a consequence of very general hypotheses. However, these hypotheses may not all be correct,
and discovery of a boson of half-integral spin or a fermion of integral spin remains possible. It is not
inconceivable that, for certain particles, the physical kets might have more complex symmetry properties
than those envisaged here.
1435

CHAPTER XIV SYSTEMS OF IDENTICAL PARTICLES
must be associated with the physical state considered:for bosons and for
fermions. This ket will be called thephysical ket.
Comment:
It is possible for all the kets ofto have a zero projection onto(or). In
this case, the symmetrization postulate excludes the corresponding physical state.
Later ( ), we shall see examples of such a situation when dealing
with fermions.
C-3. Construction of physical kets
C-3-a. The construction rule
The discussion of the preceding section leads directly to the following rule for the
construction of theuniqueket (the physical ket) corresponding to a given physical state
of a system ofidentical particles:
()Number the particles arbitrarily, and construct the ketcorresponding to the
physical state considered and to the numbers given to the particles.
()Applyorto, depending on whether the identical particles are bosons or
fermions.
()Normalize the ket so obtained.
We shall describe some simple examples to illustrate this rule.
C-3-b. Application to systems of two identical particles
Consider a system composed of two identical particles. Suppose that one of them
is known to be in the individual state characterized by the normalized ket, and the
other one, in the individual state characterized by the normalized ket.
First of all, we shall envisage the case in which the two kets,and, are
distinct. The preceding rule is applied in the following way:
()We label with the number 1, for example, the particle in the state, and with
the number 2, the one in the state. This yields:
=1 :; 2 : (C-2)
()We symmetrize if the particles are bosons:
=
1
2
[1 :; 2 :+1 :; 2 :] (C-3a)
We antisymmetrizeif the particles are fermions:
=
1
2
[1 :; 2 : 1 :; 2 :] (C-3b)
1436

C. THE SYMMETRIZATION POSTULATE
()The kets (C-3a) and (C-3b), in general, are not normalized. If we assumeand
to be orthogonal, the normalization constant is very simple to calculate. All we
have to do to normalizeor is replace the factor12appearing in formulas
(C-3) by1
2. The normalized physical ket, in this case, can therefore be written:
;=
1
2
[1 :; 2 :+1 :; 2 :] (C-4)
with= +1for bosons and1for fermions.
We shall now assume that the two individual states,and, are identical:
= (C-5)
(C-2) then becomes:
=1 :; 2 : (C-6)
is already symmetric. If the two particles are bosons, (C-6) is then the physical ket
associated with the state in which the two bosons are in the same individual state.
If, on the other hand, the two particles are fermions, we see that:
=
1
2
1 :; 2 : 1 :; 2 := 0 (C-7)
Consequently, there exists no ket ofable to describe the physical state in which
two fermions are in the same individual state. Such a physical state is therefore
excluded by the symmetrization postulate. We have thus established, for a special case,
a fundamental result known as Pauli's exclusion principle:two identical fermions
cannot be in the same individual state. This result has some very important physical
consequences which we shall discuss in .
C-3-c. Generalization to an arbitrary number of particles
These ideas can be generalized to an arbitrary numberof particles. To see how
this can be done, we shall rst treat the case= 3.
Consider a physical state of the system dened by specifying the three individual
normalized states,and. The statewhich enters into the rule of a can be
chosen in the form:
=1 :; 2 :; 3 : (C-8)
We shall discuss the cases of bosons and fermions separately.
. The case of bosons
The application oftogives:
=
1
3!
=
1
6
1 :; 2 :; 3 :+1 :; 2 :; 3 :+1 :; 2 :; 3 :
+1 :; 2 :; 3 :+1 :; 2 :; 3 :+1 :; 2 :; 3 : (C-9)
1437

CHAPTER XIV SYSTEMS OF IDENTICAL PARTICLES
It then suces to normalize the ket (C-9).
First of all, let us assume that the three kets,and are orthogonal. The
six kets appearing on the right-hand side of (C-9) are then also orthogonal. To normalize
(C-9), all we must do is replace the factor 1/6 by1
6.
If the two statesandcoincide, while remaining orthogonal to, only three
distinct kets now appear on the right-hand side of (C-9). It can easily be shown that the
normalized physical ket can then be written:
;;=
1
3
1 :; 2 :; 3 :
+1 :; 2 :; 3 :+1 :; 2 :; 3 : (C-10)
Finally, if the three states,,are the same, the ket:
=1 :; 2 :; 3 : (C-11)
is already symmetric and normalized.
. The case of fermions
The application oftoleads to:
=
1
3!
1 :; 2 :; 3 : (C-12)
The signs of the various terms of the sum (C-12) are determined by the same rule as
those of a33determinant. This is why it is convenient to writein the form of a
Slater determinant:
=
1
3!
1 :1 :1 :
2 :2 :2 :
3 :3 :3 :
(C-13)
is zero if two of the individual states,or coincide, since the de-
terminant (C-13) then has two identical columns. We obtain Pauli's exclusion principle,
already mentioned in : the same quantum mechanical state cannot be simultane-
ously occupied by several identical fermions.
Finally, note that if the three states,, are orthogonal, the six kets
appearing on the right-hand side of (C-12) are orthogonal. All we must then do to
normalize is replace the factor13!appearing in (C-12) or (C-13) by1
3!.
If, now, the system being considered contains more than three identical particles,
the situation actually remains similar to the one just described. It can be shown that,
foridentical bosons, it is always possible to construct the physical statefrom
arbitrary individual states,, . . . On the other hand, for fermions, the physical ket
can be written in the form of anSlater determinant; this excludes the case
in which two individual states coincide (the ketis then zero). This shows, and we
shall return to this in detail in , how dierent the consequences of the new postulate
can be for fermion and boson systems.
1438

C. THE SYMMETRIZATION POSTULATE
C-3-d. Construction of a basis in the physical state space
Consider a system ofidentical particles. Starting with a basis,, in the
state space of a single particle, we can construct the basis:
1 :; 2 :; ;:
in the tensor product space. However, since the physical state space of the system is
not, but rather one of the subspacesor, the problem arises of how to determine
a basis in this physical state space.
By application of(or) to the various kets of the basis:
1 :; 2 :; ;:
we can obtain a set of vectors spanning(or). Letbe an arbitrary ket of,
for example (the case in whichbelongs tocan be treated in the same way).,
which belongs to, can be expanded in the form:
= 1 :; 2 :; : (C-14)
Since, by hypothesis, belongs to, we have =, and we simply apply the
operatorto both sides of (C-14) to show thatcan be expressed in the form of a
linear combination of the various kets1 :; 2 :; ;:.
However, it must be noted that the various kets1 :; 2 :; ;:
are not independent. Let us permute the roles of the various particles in one of the kets
1 :; 2 :; ;:of the initial basis (before symmetrization). On this new
ket, application oforleads, according to (B-62) and (B-63), to the same ket of
or(possibly with a change of sign).
Thus, we are led to introduce the concept of anoccupation number: by denition,
for the ket1 :; 2 :; ;:, the occupation numberof the individ-
ual stateis equal to the number of times the stateappears in the sequence
, that is, the number of particles in the state(we have, obviously,
=). Two dierent kets1 :; 2 :; ;:for which the occupation
numbers are equal can be obtained from each other by the action of a permutation op-
erator. Consequently, after the action of the symmetrizer(or the antisymmetrizer),
they give the same physical state, which we shall denote by12 :
12
= 1 :1; 2 :1; 1;1;
1particles
in the state |1
1+ 1 :2; ;1+2:2;
2particles
in the state2
(C-15)
For fermions,would be replaced byin (C-15) (is a factor which permits the normal-
ization of the state obtained in this way
6
). We shall not study the states12
in detail here; we shall conne ourselves to giving some of their important properties:
6
A simple calculation yields:=
!1!2!for bosons and!for fermions.
1439

CHAPTER XIV SYSTEMS OF IDENTICAL PARTICLES
()The scalar product of two kets12 and
12 is dierent
from zero only if all the occupation numbers are equal (=for all).
By using (C-15) and denitions (B-49) and (B-50) ofand, we can obtain the
expansion of the two kets under consideration on the orthonormal basis,1 :; 2 :
; ;: . It is then easy to see that, if the occupation numbers are not all
equal, these two kets cannot simultaneously have non-zero components on the same basis
vector.
()If the particles under study are bosons, the kets12 , in which the
various occupation numbersare arbitrary (with, of course=) form an
orthonormal basis of the physical state space.
Let us show that, for bosons, the kets12 dened by (C-15
zero. To do so, we replaceby its denition (B-49). There then appear, on the right-hand
side of ( ), various orthogonal kets1 :; 2 :; ;:, all with positive
coecients. The ket12 cannot, therefore, be zero.
The 12 form a basis insince these kets span, are all non-zero,
and are orthogonal to each other.
()If the particles under study are fermions, a basis of the physical state spaceis
obtained by choosing the set of kets12 in which all the occupation
numbers are equal either to 1 or to 0 (again with=).
The preceding proof is not applicable to fermions because of the minus signs which
appear before the odd permutations in denition (B-50) of. Furthermore, we saw in
c that two identical fermions cannot occupy the same individual quantum state: if any
one of the occupation numbers is greater than 1, the vector dened by (C-15) is zero. On
the other hand, it is never zero if all the occupation numbers are equal to one or zero;
this is because two particles are then never in the same individual quantum state, so that
the kets1 :; 2 :; ;:and1 :; 2 :; ;:are always
distinct and orthogonal. Relation (C-15) therefore denes a non-zero physical ket in this
case. The rest of the proof is the same as for bosons.
C-4. Application of the other postulates
It remains for us to show how the general postulates of Chapter
in light of the symmetrization postulate introduced in , and to verify that no
contradictions arise. More precisely, we shall see how measurement processes can be
described with kets belonging only to eitheror, and we shall show that the time
evolution process does not take the ket()associated with the state of the system
out of this subspace. Thus, all the quantum mechanical formalism can be applied inside
eitheror.
C-4-a. Measurement postulates
. Probability of nding the system in a given physical state
Consider a measurement performed on a system of identical particles. The ket
()describing the quantum state of the system before the measurement must, ac-
cording to the symmetrization postulate, belong toor to, depending on whether
the system is formed of bosons or fermions. To apply the postulates of Chapter
1440

C. THE SYMMETRIZATION POSTULATE
concerning measurements, we must take the scalar product of()with the ket
corresponding to the physical state of the system after the measurement. This ket
is to be constructed by applying the rule given in . The probability amplitude
()can therefore be expressed in terms of two vectors, both belonging either to
or to. In , we shall discuss a certain number of examples of such calculations.
If the measurement envisaged is a complete measurement (yielding, for example,
the positions and spin componentsfor all the particles), the physical ketis unique
(to within a constant factor). On the other hand, if the measurement is incomplete (for
example, a measurement of the spins only, or a measurement bearing on a single particle),
several orthogonal physical kets are obtained, and the corresponding probabilities must
then be summed.
. Physical observables: invariance ofand
In certain cases, it is possible to specify the measurement performed on the system
of identical particles by giving the explicit expression of the corresponding observable in
terms ofR1,P1,S1,R2,P2,S2, etc.
We shall give some concrete examples of observables which can be measured in a
three-particle system:
Position of the center of massR, total momentumPand total angular momentum
L:
R=
1
3
(R1+R2+R3) (C-16)
P=P1+P2+P3 (C-17)
L=L1+L2+L3 (C-18)
Electrostatic repulsion energy:
=
2
40
1
R1R2
+
1
R2R3
+
1
R3R1
(C-19)
Total spin:
S=S1+S2+S3 (C-20)
etc.
It is clear from these expressions that the observables associated with the physical
quantities considered involve the various particles symmetrically. This important prop-
erty follows directly from the fact that the particles are identical. In (C-16), for example,
R1,R2andR3have the same coecient, since the three particles have the same mass.
It is the equality of the charges which is at the basis of the symmetric form of (C-19).
In general, since no physical properties are modied when the roles of theidentical
particles are permuted, theseparticles must play a symmetric role
7
in any actually
measurable observable. Mathematically, the corresponding observable, which we shall
call aphysicalobservable, must be invariant under all permutations of theidentical
7
Note that this reasoning is valid for fermions as well as for bosons.
1441

CHAPTER XIV SYSTEMS OF IDENTICAL PARTICLES
particles. It must therefore commute with all the permutation operatorsof the
particles (cf. ):
[ ] = 0for all (C-21)
For a system of two identical particles, for example, the observableR1R2(the vector
dierence of the positions of the two particles), which is not invariant under the eect
of the permutation21(R1R2changes sign) is not a physical observable; indeed, a
measurement ofR1R2assumes that particle (1) can be distinguished from particle
(2). On the other hand, we can measure the distance between the two particles, that is,
(R1R2)
2
, which is symmetric.
Relation (C-21) implies thatand are both invariant under the action of a
physical observable. Let us show that, ifbelongs to, also belongs to
(the same proof also applies, of course, to). The fact thatbelongs tomeans
that:
= (C-22)
Now let us calculate . According to (C-21) and (C-22), we have:
= = (C-23)
Since the permutationis arbitrary, (C-23) expresses the fact thatis completely
antisymmetric and therefore belongs to.
All operations normally performed on an observable in particular, the determi-
nation of eigenvalues and eigenvectors can therefore be applied toentirely within one
of the subspaces,or. Only the eigenkets ofbelonging to the physical subspace,
and the corresponding eigenvalues, are retained.
Comments:
()All the eigenvalues ofwhich exist in the total spaceare not necessarily
found if we restrict ourselves to the subspace(or). The eect of the
symmetrization postulate on the spectrum of a symmetric observablemay
therefore be to abolish certain eigenvalues. On the other hand, it adds no
new eigenvalues to this spectrum, since, because of the global invariance of
(or) under the action of, any eigenvector ofin(or) is also
an eigenvector ofinwith the same eigenvalue.
()Consider the problem of writing mathematically, in terms of the observablesR1,
P1,S1, etc., the observables corresponding to the dierent types of measurement
envisaged in . This problem is not always simple. For example, for a system of
three identical particles, we shall try to write the observables corresponding to the
simultaneous measurement of the three positions in terms ofR1,R2andR3. We
can resolve this problem by considering several physical observables chosen such
that we can, using the results obtained by measuring them, unambiguously deduce
the position of each particle (without, of course, being able to associate a numbered
particle with each position). For example, we can choose the set:
1+2+3 12+23+31 123
1442

D. DISCUSSION
(and the corresponding observables for theandcoordinates). However, this
point of view is essentially formal. Rather than trying to write the expressions for
the observables in all cases, it is simpler to follow the method used in , in which
we conned ourselves to using the physical eigenkets of the measurement.
C-4-b. Time-evolution postulates
The Hamiltonian of a system of identical particles must be a physical observable.
We shall write, for example, the Hamiltonian describing the motion of the two electrons
of the helium atom about the nucleus, assumed to be motionless
8
:
(12) =
P
2
1
2
+
P
2
2
2
2
2
1
2
2
2
+
2
R1R2
(C-24)
The rst two terms represent the kinetic energy of the system; they are symmetric
because the two masses are equal. The next two terms are due to the attraction of the
nucleus (whose charge is twice that of the proton). The electrons are obviously equally
aected by this attraction. Finally, the last term describes the mutual interaction of
the electrons. It is also symmetric, since neither of the two electrons is in a privileged
position. It is clear that this argument can be generalized to any system of identical
particles. Consequently, all the permutation operators commute with the Hamiltonian
of the system:
[ ] = 0 (C-25)
Under these conditions, if the ket(0)describing the state of the system at a
given time0is a physical ket, the same must be true of the ket()obtained from
(0)by solving the Schrödinger equation. According to this equation:
(+ d)=1 +
d
~
() (C-26)
Now, applyingand using relation (C-25):
(+ d)= (1 +
d
~
) () (C-27)
If()is an eigenvector of,(+ d)is also an eigenvector of, with the same
eigenvalue. Since(0), by hypothesis, is a completely symmetric or completely anti-
symmetric ket, this property is conserved over time.
The symmetrization postulate is therefore also compatible with the postulate that
gives the time evolution of physical systems: the Schrödinger equation does not remove
the ket()fromor.
D. Discussion
In this nal section, we shall examine the consequences of the symmetrization postulate
on the physical properties of systems of identical particles. First of all, we shall indicate
8
Here, we shall consider only the most important terms of this Hamiltonian. See ComplementXIV
for a more detailed study of the helium atom.
1443

CHAPTER XIV SYSTEMS OF IDENTICAL PARTICLES
the fundamental dierences introduced by Pauli's exclusion principle between systems of
identical fermions and systems of identical bosons. Then, we shall discuss the implications
of the symmetrization postulate concerning the calculation of the probabilities associated
with the various physical processes.
D-1. Dierences between bosons and fermions. Pauli's exclusion principle
In the statement of the symmetrization postulate, the dierence between bosons
and fermions may appear insignicant. Actually, this simple sign dierence in the sym-
metry of the physical ket has extremely important consequences. As we saw in , the
symmetrization postulate does not restrict the individual states accessible to a system
of identical bosons. On the other hand, it requires fermions to obey Pauli's exclusion
principle: two identical fermions cannot occupy the same individual quantum state.
The exclusion principle was formulated initially in order to explain the properties
of many-electron atoms ( XIV). It can now be seen
to be more than a principle applicable only to electrons: it is a consequence of the
symmetrization postulate, valid for all systems of identical fermions. Predictions based
on this principle, which are often spectacular, have always been conrmed experimentally.
We shall give some examples of them.
D-1-a. Ground state of a system of independent identical particles
The Hamiltonian of a system of identical particles (bosons or fermions) is always
symmetric with respect to permutations of these particles ( ). Consider such a
system in which the various particles are independent, that is, do not interact with each
other (at least in a rst approximation). The corresponding Hamiltonian is then a sum
of one-particle operators of the form:
(12 ) =(1) +(2) ++() (D-1)
(1)is a function only of the observables associated with the particle numbered (1); the
fact that the particles are identical [which implies a symmetric Hamiltonian(12 )]
requires this functionto be the same in theterms of expression (D-1). In order to
determine the eigenstates and eigenvalues of the total Hamiltonian(12 ), we
simply calculate those of the individual Hamiltonian()in the state space()of one
of the particles:
() = ; () (D-2)
For the sake of simplicity, we shall assume that the spectrum of()is discrete and
non-degenerate.
If we are considering a system of identical bosons, the physical eigenvectors of the
Hamiltonian(12 )can be obtained by symmetrizing the tensor products of
arbitrary individual states:
()
12
= 1 :
1; 2 :
2; ;: (D-3)
where the corresponding energy is the sum of theindividual energies:
12
=
1
+
2
++ (D-4)
1444

D. DISCUSSION
[it can easily be shown that each of the kets appearing on the right-hand side of (D-3) is
an eigenket ofwith the eigenvalue (D-4); this is also true of their sum]. In particular,
if1is the smallest eigenvalue of(), and1is the associated eigenstate, the ground
state of the system is obtained when theidentical bosons are all in the state1. The
energy of this ground state is therefore:
111= 1 (D-5)
and its state vector is:
()
12
=1 :1; 2 :1; ;:1 (D-6)
Now, suppose that theidentical particles considered are fermions. It is no
longer possible for theseparticles all to be in the individual state1. To obtain the
ground state of the system, Pauli's exclusion principle must be taken into account. If the
individual energiesare arranged in increasing order:
1 2 1 +1 (D-7)
the ground state of the system ofidentical fermions has an energy of:
12 =1+2++ (D-8)
and it is described by the normalized physical ket:
()
12
=
1
!
1 :11 :21 :3
2 :12 :22 :3
.
.
.
:1 :2 :3
(D-9)
The highest individual energyfound in the ground state is called theFermi energy
of the system.
Pauli's exclusion principle thus plays a role of primary importance in all domains
of physics in which many-electron systems are involved, such as atomic and molecu-
lar physics (cf.ComplementsXIVandXIV) and solid state physics (cf.Comple-
mentXIV), and in all those in which many-proton and many-neutron systems are in-
volved, such as nuclear physics
9
.
Comment:
In most cases, the individual energiesare actually degenerate. Each of them
can then enter into a sum such as (D-8) a number of times equal to its degree of
degeneracy.
9
The ket representing the state of a nucleus must be antisymmetric both with respect to the set of
protons and with respect to the set of neutrons.
1445

CHAPTER XIV SYSTEMS OF IDENTICAL PARTICLES
D-1-b. Quantum statistics
The object of statistical mechanics is to study systems composed of a very large
number of particles (in numerous cases, the mutual interactions between these particles
are weak enough to be neglected in a rst approximation). Since we do not know the
microscopic state of the system exactly, we content ourselves with describing it globally
by its macroscopic properties (pressure, temperature, density, etc.). A particular macro-
scopic state corresponds to a whole set of microscopic states. We then use probabilities:
the statistical weight of a macroscopic state is proportional to the number of distinct mi-
croscopic states that correspond to it, and the system, at thermodynamic equilibrium, is
in its most probable macroscopic state (with any constraints that may be imposed taken
into account). To study the macroscopic properties of the system, it is therefore essential
to determine how many dierent microscopic states possess certain characteristics and,
in particular, a given energy.
In classical statistical mechanics (Maxwell-Boltzmann statistics), theparticles
of the system are treated as if they were of dierent natures, even if they are actually
identical. Such a microscopic state is dened by specifying the individual state of each
of theparticles. Two microscopic states are considered to be distinct when these
individual states are the same but the permutation of the particles is dierent.
In quantum statistical mechanics, the symmetrization postulate must be taken into
account. A microscopic state of a system of identical particles is characterized by the
enumeration of theindividual states which form it, the order of these states being
of no importance since their tensor product must be symmetrized or anti-symmetrized.
The numbering of the microscopic states therefore does not lead to the same result as
in classical statistical mechanics. In addition, Pauli's principle radically dierentiates
systems of identical bosons and systems of identical fermions: the number of particles
occupying a given individual state cannot exceed one for fermions, while it can take on
any value for bosons (cf. C-3). Dierent statistical properties result: bosons obey
Bose-Einstein statisticsand fermions,Fermi-Dirac statistics. This is the origin of the
terms bosons and fermions.
The physical properties of systems of identical fermions and systems of identical
bosons are very dierent. This subject will be discussed in more detail in the rst three
chapters of Volume III. The dierences can be observed, for example, at low temperatures,
when the particles tend to accumulate in the individual states of lowest energy. Identical
bosons may then exhibit a phenomenon calledBose-Einstein condensationof particles
(ComplementsXVandXV); by contrast identical fermions, subject to the restrictions
of Pauli's principle, build a Fermi sphere (ComplementXIV) and can undergo only a
pair condensation (Chapter ). Bose-Einstein condensation is at the origin of the
remarkable properties (superuidity) of the
4
He isotope of helium, in particular the
superuid properties (ComplementXV) of its liquid at low temperatures (a few K) .
The
3
He isotope, which is a fermion (cf.Comment of ), has very dierent properties
and is superuid only at much lower temperatures because of pair condensation.
D-2. The consequences of particle indistinguishability on the calculation of physical
predictions
In quantum mechanics, all the predictions concerning the properties of a system
are expressed in terms of probability amplitudes (scalar products of two state vectors)
1446

D. DISCUSSION
or matrix elements of an operator. It is then not surprising that the symmetrization
or antisymmetrization of state vectors causes specialinterference eectsto appear in
systems of identical particles. First, we shall specify these eects, and then we shall
see how they disappear under certain conditions (the particles of the system, although
identical, then behave as if they were of dierent natures). To simplify the discussion,
we shall conne ourselves to systems containing only two identical particles.
D-2-a. Interferences between direct and exchange processes
. Predictions concerning a measurement on a system of identical particles: the
direct term and the exchange term
Consider a system of two identical particles, one of which is known to be in the
individual stateand the other, in the individual state. We shall assumeand
to be orthogonal, so that the state of the system is described by the normalized
physical ket [cf.formula (C-4)]:
;=
1
2
[1 +21]1 :; 2 : (D-10)
where:
= +1if the particles are bosons
=1if the particles are fermions (D-11)
With the system in this state, suppose that we want to measure on each of the two
particles the same physical quantitywith which the observables(1)and(2)are
associated. For the sake of simplicity, we shall assume that the spectrum ofis entirely
discrete and non-degenerate:
= (D-12)
What is the probability of nding certain given values in this measurement (for one
of the particles andfor the other one)? We shall begin by assumingandto be
dierent, so that the corresponding eigenvectorsand are orthogonal. Under
these conditions, the normalized physical ket dened by the result of this measurement
can be written:
; =
1
2
[1 +21]1 :; 2 : (D-13)
which gives the probability amplitude associated with this result:
; ;=
1
2
1 :; 2 :(1 +
21
)(1 +21)1 :; 2 : (D-14)
Using properties (B-13) and (B-14) of the operator21, we can write:
1
2
(1 +
21
)(1 +21) = 1 +21 (D-15)
(D-14) then becomes:
; ;=1 :; 2 :(1 +21)1 :; 2 : (D-16)
1447

CHAPTER XIV SYSTEMS OF IDENTICAL PARTICLES
Letting1 +21act on the bra, we obtain:
; ;=1 :; 2 :(1 :; 2 :
+1 :; 2 :1 :; 2 :
=1 :1 :2 :2 :
+1 :1 :2 :2 :
= + (D-17)
The numbering has disappeared from the probability amplitude, which is now expressed
directly in terms of the scalar products . Also, the probability amplitude
appears either as a sum (for bosons) or a dierence (for fermions) of two terms, with
which we can associate the diagrams of Figuresa andb.u
n φ
χ
φ
χ u
n

u
n
u
n

a b
Figure 4: Schematic representation of the direct term and the exchange term associated
with a measurement performed on a system of two identical particles. Before the measure-
ment, one of the particles is known to be in the stateand the other one, in the state
. The measurement result obtained corresponds to a situation in which one particle
is in the stateand the other one, in the state. Two probability amplitudes are
associated with such a measurement; they are represented schematically by gures a and
b. These amplitudes interfere with a+sign for bosons and with a sign for fermions.
We can interpret result (D-17) in the following way. The two ketsand
associated with the initial state can be connected to the two brasand associated
with the nal state by two dierent paths, represented schematically by Figuresa
andb. With each of these paths is associated a probability amplitude, or
, andthese two amplitudes interfere with a+sign for bosons and asign
for fermions. Thus, we obtain the answer to the question posed in
desired probability(;)is equal to the square of the modulus of (D-17):
(;) = +
2
(D-18)
One of the two terms on the right-hand side of (D-17), the one which corresponds, for
example, to path 4-a, is often called thedirect term. The other term is called theexchange
term.
1448

D. DISCUSSION
Comment:
Let us examine what happens if the two particles, instead of being identical, are
of dierent natures. We shall then choose as the initial state of the system the
tensor product ket:
=1 :; 2 : (D-19)
Now, consider a measurement instrument which, although the two particles, (1)
and (2), are not identical, is not able to distinguish between them. If it yields the
resultsand, we do not know ifis associated with particle (l) or particle
(2) (for example, for a system composed of a muonand an electron, the
measurement device may be sensitive only to the charge of the particles, giving
no information about their masses). The two eigenstates1 :; 2 : and
1 : ; 2 : (which, in this case, represent dierent physical states) then
correspond to the same measurement result. Since they are orthogonal, we must
add the corresponding probabilities, which gives:
(;) =1 :; 2 :1 :; 2 :
2
+1 :; 2 :1 :; 2 :
2
=
2 2
+
2 2
(D-20)
Comparison of (D-18) with (D-20) clearly reveals the signicant dierence in the
physical predictions of quantum mechanics depending on whether the particles
under consideration are identical or not.
Now consider the case in which the two statesand are the same. When
the two particles are fermions, the corresponding physical state is excluded by Pauli's
principle, and the probability(;)is zero. On the other hand, if the two particles
are bosons, we have:
; =1 :; 2 : (D-21)
and, consequently:
; ;=
1
2
1 :; 2 :(1 +21)1 :; 2 :
=
2 (D-22)
which gives:
(;) = 2
2
(D-23)
Comments:
()Let us compare this result with the one which would be obtained in the case,
already considered above, in which the two particles are dierent. We must
1449

CHAPTER XIV SYSTEMS OF IDENTICAL PARTICLESφ
χ
ω
u
n

u
n

u
n

φ
χ
ω
u
n

u
n

u
n

φ
χ
ω
u
n

u
n

u
n

φ
χ
ω
u
n

u
n

u
n

φ
χ
ω
u
n

u
n

u
n

φ
χ
ω
u
n

u
n

u
n

+
ε ε ε
+ +
Figure 5: Schematic representation of the six probability amplitudes associated with a
system of three identical particles. Before the measurement, one particle is known to be
in the state, another, in the state, and the last one, in the state. The result
obtained corresponds to a situation in which one particle is in the state, another, in
the state, and the last one, in the state. The six amplitudes interfere with a
sign which is shown beneath each one (= +1for bosons,1 for fermions).
then replace;by1 :; 2 :and ; by1 :; 2 :, which
gives the value for the probability amplitude:
(D-24)
and, consequently:
(;) =
2
(D-25)
()For a system containingidentical particles, there are, in general,!distinct
exchange terms which add (or subtract) in the probability amplitude. For example,
consider a system of three identical particles in the individual states,and,
and the probability of nding, in a measurement, the results,and . The
possible paths are then shown in Figure. There are six such paths (all dierent
if the three eigenvalues,,and are dierent). Some always contribute to
the probability amplitude with a+sign, others with ansign (+for bosons and
for fermions).
. Example: elastic collision of two identical particles
To understand the physical meaning of the exchange term, let us examine a con-
crete example (already alluded to in ): that of the elastic collision of two identical
1450

D. DISCUSSION
particles in their center of mass frame
10
. Unlike the situation in above, here we must
take into account the evolution of the system between the initial time when it is in the
stateand the timewhen the measurement is performed. However, as we shall see,
this evolution does not change the problem radically, and the exchange term enters the
problem as before.
In the initial state of the system (Fig.a), the two particles are moving towards
each other with opposite momenta. We choose theaxis along the direction of these
momenta, and we denote their modulus by. One of the particles thus possesses the
momentumez, and the other one, the momentume(whereeis the unit vector of
theaxis). We shall write the physical ketrepresenting this initial state in the
form:
=
1
2
(1 +21)1 :e; 2 :e (D-26)
describes the state of the system at0, before the collision.a
Initial state
O Oz
z
n
b
Final state
Figure 6: Collision between two identical particles in the center of mass frame: the
momenta of the two particles in the initial state (g. a) and in the nal state found in
the measurement (g. b) are represented. For the sake of simplicity, we ignore the spin
of the particles.
The Schrödinger equation which governs the time evolution of the system is lin-
ear. Consequently, there exists a linear operator(), which is a function of the
Hamiltonian, such that the state vector at timeis given by:
()=(0) (D-27)
(ComplementIII). In particular, after the collision, the state of the system at time1
is represented by the physical ket:
(1)=(10) (D-28)
10
We shall give a simplied treatment of this problem, intended only to illustrate the relation between
the direct term and the exchange term. In particular, we ignore the spin of the two particles. However,
the calculations of this section remain valid in the case in which the interactions are not spin-dependent
and the two particles are initially in the same spin state.
1451

CHAPTER XIV SYSTEMS OF IDENTICAL PARTICLES
Note that, since the Hamiltonianis symmetric, the evolution operatorcommutes
with the permutation operator:
[()21] = 0 (D-29)
Now, let us calculate the probability amplitude of the result envisaged in , in
which the particles are detected in the two opposite directions of theOnaxis, of unit
vectorn(Fig.b). We denote the physical ket associated with this nal state by:
=
1
2
(1 +21)1 :n; 2 :n (D-30)
The desired probability amplitude can therefore be written:
(1)= (10)
=
1
2
1 :n; 2 :n(1 +
21
)
(10)(1 +21)1 :e; 2 :e (D-31)
According to relation (D-29) and the properties of the operator21, we nally obtain:
=(0)
=1 :n; 2 :n(1 +
21
)(10)1 :e; 2 :e
=1 :n; 2 :n(10)1 :e; 2 :e
+1 :n; 2 :n(10)1 :e; 2 :e (D-32)
The direct term corresponds, for example, to the process shown in Figurea, and the
exchange term is then represented by Figureb. Again, the probability amplitudes
associated with these two processes must be added or subtracted. This causes an inter-
ference term to appear when the square of the modulus of expression (D-32) is taken.
Note also that this expression is simply multiplied byifnis changed to n, so that the
corresponding probability is invariant under this change.
D-2-b. Situations in which the symmetrization postulate can be ignored
If the application of the symmetrization postulate were always indispensable, it
would be impossible to study the properties of a system containing a restricted number
of particles, because it would be necessary to take into account all the particles in the
universe which are identical to those in the system. We shall see in this section that
this is not the case. In fact, under certain special conditions, identical particles behave
as if they were actually dierent, and it is not necessary to take the symmetrization
postulate into account in order to obtain correct physical predictions. It seems natural
to expect, considering the results of , that such a situation would arise whenever
the exchange terms introduced by the symmetrization postulate are zero. We shall give
two examples.
. Identical particles situated in two distinct regions of space
Consider two identical particles, one of which is in the individual stateand the
other, in the state. To simplify the notation, we shall ignore their spin. Suppose that
1452

D. DISCUSSION
the domain of the wave functions representing the ketsand are well separated in
space:
(r) =r= 0 ifr
(r) =r= 0 ifr
(D-33)
where the domainsanddo not overlap. The situation is analogous to the classical
mechanical one (): as long as the domainsanddo not overlap, each of the par-
ticles can be tracked; we therefore expect application of the symmetrization postulate
to be unnecessary.
In this case, we can envisage measuring an observable related to one of the two
particles. All we need is a measurement device placed so that it cannot record what
happens in the domain, or in the domain. If it iswhich is excluded in this way,
the measurement will only concern the particle in, an vice versa.
Now, imagine a measurement concerning the two particles simultaneously, but per-
formed with two distinct measurement devices, one of which is not sensitive to phenomena
occurring in, and the other, to those in. How can the probability of obtaining a
given result be calculated? Letandbe the individual states associated respectively
with the results of the two measurement devices. Since the two particles are identical,
the symmetrization postulate must, in theory, be taken into account. In the probability
amplitude associated with the measurement result, the direct term is then,
and the exchange term is . Now, the spatial disposition of the measurement
devices implies that:
(r) =r= 0 ifr
(r) =r= 0 ifr (D-34)
According to (D-33) and (D-34), the wave functions(r)and(r)do not overlap; neither
do(r)and(r), so that:
= = 0 (D-35)n n
zz
a b
Figure 7: Collision between two identical particles in the center of mass frame: schematic
representation of the physical processes corresponding to the direct term and the exchange
term. The scattering amplitudes associated with these two processes interfere with a plus
sign for bosons and a minus sign for fermions.
1453

CHAPTER XIV SYSTEMS OF IDENTICAL PARTICLES
The exchange term is therefore zero. Consequently, it is unnecessary, in this situation, to
use the symmetrization postulate. We obtain the desired result directly by reasoning as if
the particles were of dierent natures, labeling, for example, the one in the domainwith
the number 1, and the one situated inwith the number 2. Before the measurement, the
state of the system is then described by the ket1 :; 2 :, and with the measurement
result envisaged is associated the ket1 :; 2 :. Their scalar product gives the
probability amplitude .
This argument shows that the existence of identical particles does not prevent the
separate study of restricted systems, composed of a small number of particles.
Comment:
In the initial state chosen, the two particles are situated in two distinct regions of space.
In addition, we have dened the state of the system by specifying two individual states.
We might wonder if, after the system has evolved, it is still possible to study one of the
two particles and ignore the other one. For this to be the case, it is necessary, not only
that the two particles remain in two distinct regions of space, but also that they do not
interact. Whether the particles are identical or not, an interaction always introduces
correlations between them, and it is no longer possible to describe each of them by a
state vector.a b
OO
z
z
n
Figure 8: Collision between two identical spin 1/2 particles in the center of mass frame:
a schematic representation of the momenta and spins of the two particles in the initial
state (g. a) and in the nal state found in the measurement (g. b). If the interactions
between the two particles are spin-independent, the orientation of the spins does not
change during the collision. When the two particles are not in the same spin state before
the collision (the case of the gure), it is possible to determine the path followed by the
system in arriving at a given nal state. For example, the only scattering process which
leads to the nal state of gure b and which has a non-zero amplitude is of the type shown
in Figurea.
1454

D. DISCUSSION
. Particles which can be identied by the direction of their spins
Consider an elastic collision between two identical spin 1/2 particles (electrons, for
example), assuming that spin-dependent interactions can be neglected, so that the spin
states of the two particles are conserved during the collision. If these spin states are
initially orthogonal, they enable us to distinguish between the two particles at all times,
as if they were not identical; consequently, the symmetrization postulate should again
have no eect here.
We can show this, using the calculation of . The initial physical ket will
be, for example (Fig.a):
=
1
2
(1 21)1 :e+ ; 2 :e (D-36)
(where the symbol+oradded after each momentum indicates the sign of the spin
component along a particular axis). The nal state we are considering (Fig.b) will be
described by:
=
1
2
(1 21)1 :n+ ; 2 :n (D-37)
Under these conditions, only the rst term of (D-32) is dierent from zero, since the
second one can be written:
1 :n; 2 :n+(10)1 :e+ ; 2 :e (D-38)
This is the matrix element of a spin-independent operator (by hypothesis) between two
kets whose spin states are orthogonal; it is therefore zero. Consequently, we would obtain
the same result if we treated the two particles directly as if they were dierent, that is, if
we did not antisymmetrize the initial and nal kets and if we associated index 1 with the
spin state+and index 2 with the spin state. Of course, this is no longer possible if
the evolution operator, that is, the Hamiltonianof the system, is spin-dependent.
References and suggestions for further reading:
The importance of interference between direct and exchange terms is stressed in
Feynman III (1.2), 3.4 and Chap. 4.
Quantum statistics: Reif (8.4). Kittel (8.2).
Permutation groups: Messiah (1.17), app. D, IV; Wigner (2.23), Chap. 13; Bacry
(10.31), 41 and 42.
The eect of the symmetrization postulate on molecular spectra: Herzberg (12.4),
Vol. I, Chap. III, 2f.
An article giving a popularized version: Gamow (1.27).
1455

COMPLEMENTS OF CHAPTER XIV, READER'S GUIDE
AXIV: MANY-ELECTRON ATOMS;
ELECTRONIC CONFIGURATIONS
Simple study of many-electron atoms in the
central-eld approximation. Discusses the
consequences of the Pauli exclusion principle
and introduces the concept of an electronic
conguration. Remains qualitative.
BXIV: ENERGY LEVELS OF THE HELIUM ATOM:
CONFIGURATIONS, TERMS, MULTIPLETS
Study, in the case of the helium atom, of the
eect of the electronic repulsion between electrons
and of the magnetic interactions. Introduces
the concepts of terms and multiplets. Can be
reserved for later study.
CXIV: PHYSICAL PROPERTIES OF AN ELECTRON
GAS. APPLICATION TO SOLIDS
Study of the ground stante of a gas of free
electrons enclosed in a box. Introduces the
concept of Fermi energy and periodic boundary
conditions. Generalization to electrons in solids
and qualitative discussion of the relation between
electrical conductivity and the position of the
Fermi level. Moderately dicult. The physical
discussions are emphasized. Can be considered to
be a sequel ofXI.
DXIV: EXERCISES
1457

MANY-ELECTRON ATOMS. ELECTRONIC CONFIGURATIONS
Complement AXIV
Many-electron atoms. Electronic congurations
1 The central-eld approximation
1-a Diculties related to electron interactions
1-b Principle of the method
1-c Energy levels of the atom
2 Electron congurations of various elements
The energy levels of the hydrogen atom were studied in detail in Chapter.
Such a study is considerably simplied by the fact that the hydrogen atom possesses a
single electron, so that Pauli's principle is not relevant. In addition, by using the center
of mass frame, we can reduce the problem to the calculation of the energy levels of a
single particle (the relative particle) subjected to a central potential.
In this complement, we shall consider many-electron atoms, for which these simpli-
cations cannot be made. In the center-of-mass frame, we must solve a problem involving
several non-independent particles. This is a complex problem and we shall give only an
approximate solution, using the central-eld approximation (which will be outlined, with-
out going into details of the calculations). In addition Pauli's principle, as we shall show,
plays an important role.
1. The central-eld approximation
Consider a-electron atom. Since the mass of its nucleus is much larger (several thousand
times) than that of the electrons, the center-of-mass of the atom practically coincides
with the nucleus, which we shall therefore assume to be motionless at the coordinate
origin
1
. The Hamiltonian describing the motion of the electrons, neglecting relativistic
corrections and, in particular, spin-dependent terms, can be written:
=
=1
P
2
2
=1
2
+
2
RR
(1)
We have numbered the electrons arbitrarily from 1 to, and we have set:
2
=
2
40
(2)
whereis the electron charge. The rst term of the Hamiltonian (1) represents the total
kinetic energy of the system ofelectrons. The second one arises from the attraction
exerted on each of them by the nucleus, which bears a positive charge equal to.
The last one describes the mutual repulsion of the electrons [note that the summation is
carried out over the(1)2dierent ways of pairing the-electrons].
The Hamiltonian (1) is too complicated for us to solve its eigenvalue equation
exactly, even in the simplest case, that of helium(= 2).
1
Making this approximation amounts to neglecting the nuclear nite mass eect.
1459

COMPLEMENT A XIV
1-a. Diculties related to electron interactions
In the absence of the mutual interaction term
2
RR
in, the electrons
would be independent. It would then be easy to determine the energies of the atom. We
would simply sum the energies of theelectrons placed individually in the Coulomb
potential
2
, and the theory presented in Chapter
diately. As for the eigenstates of the atom, they could be obtained by antisymmetrizing
the tensor product of the stationary states of the various electrons.
It is then the presence of the mutual interaction term that makes it dicult to
solve the problem exactly. We might try to treat this term by perturbation theory.
However, a rough evaluation of its relative magnitude shows that this would not yield a
good approximation. We expect the distanceRRbetween two electrons to be, on
the average, roughly the distanceof an electron from the nucleus. The ratioof the
third term of formula (1) to the second one is therefore approximately equal to:
1
2
(1)
2
(3)
varies between 1/4 for= 2and 1/2 formuch larger than 1. Consequently, the
perturbation treatment of the mutual interaction term would yield, at most, more or less
satisfactory results for helium (= 2), but it is out of the question to apply it to other
atoms (is already equal to 1/3 for= 3). A more elaborate approximation method
must therefore be found.
1-b. Principle of the method
To understand the concept of a central eld, we shall use a semi-classical argument.
Consider a particular electron(). In a rst approximation, the existence of the1
other electrons aects it only because their charge distribution partially compensates the
electrostatic attraction of the nucleus. In this approximation, the electron()can be
considered to move in a potential that depends only on its positionrand takes into
account the average eect of the repulsion of the other electrons. We choose a potential
()that depends only on the modulus ofrand call it the central potential of
the atom under consideration. Of course, this can only be an approximation: since the
motion of the electron()actually inuences that of the(1)other electrons, it is
not possible to ignore the correlations which exist between them. Moreover, when the
electron()is in the immediate vicinity of another electron(), the repulsion exerted by
the latter becomes preponderant, and the corresponding force is not central. However,
the idea of an average potential appears more valid in quantum mechanics, where we
consider the delocalization of the electrons as distributing their charges throughout an
extended region of space.
These considerations lead us to write the Hamiltonian (1) in the form:
=
=1
P
2
2
+()+ (4)
with:
=
=1
2
+
2
RR
=1
() (5)
1460

MANY-ELECTRON ATOMS. ELECTRONIC CONFIGURATIONS
If the central potential()is suitably chosen,should play the role of a
small correction in the Hamiltonian. The central-eld approximation then consists of
neglecting this correction, that is of choosing the approximate Hamiltonian:
0=
=1
P
2
2
+() (6)
will then be treated like a perturbation of0(cf.ComplementXIV, ). The diag-
onalization of0leads to a problem ofindependent particles: to obtain the eigenstates
of0, we simply determine those of the one-electron Hamiltonian:
P
2
2
+() (7)
Denitions (4) and (5) do not, of course, determine the central potential(),
since we always have= 0+, for all(). However, in order to treatlike
a perturbation,()must be wisely chosen. We shall not take up the problem of
the existence and determination of such an optimal potential here. This is a complex
problem. The potential()to which a given electron is subjected depends on the
spatial distribution of the(1)other electrons, and this distribution, in turn, depends
on the potential(), since the wave functions of the(1)electrons must also be
calculated from(). We must therefore arrive at a coherent solution (one generally
says self-consistent), for which the wave functions determined from()give a charge
distribution which reconstitutes this same potential().
1-c. Energy levels of the atom
While the exact determination of the potential()requires rather long calcula-
tions, the short- and long-distance behavior of this potential is simple to predict. We
expect, for small, the electron()under consideration to be inside the charge distri-
bution created by the other electrons, so that it sees only the attractive potential of
the nucleus. On the other hand, for large, that is, outside the cloud formed by the
(1)electrons treated globally, it is as if we had a single point charge situated at the
coordinate origin and equal to the sum of the charges of the nucleus and the cloud [the
(1)electrons screen the eld of the nucleus]. Consequently (Fig.):
()w
2
for large
()w
2
for small (8)
For intermediate values of, the variation of()can be more or less complicated,
depending on the atom under consideration.
Although these considerations are qualitative, they give an idea of the spectrum
of the one-electron Hamiltonian (7). Since()is not simply proportional to1, the
accidental degeneracy found for the hydrogen atom (Chap., ) is no longer
observed. The eigenvalues of the Hamiltonian (7) depend on the two quantum numbers
and[however, they remain independent of, since()is central]., of course,
characterizes the eigenvalue of the operator
2
, andis, by denition (as for the hydrogen
1461

COMPLEMENT A XIV0
V
c
(r)
e
2
Ze
2
r
r
r
Figure 1: Variation of the central potential()with respect to. The dashed-line
curves represent the behavior of this potential at short distances(
2
)and at long
distances(
2
).
atom), the sum of the azimuthal quantum number, and the radial quantum number
introduced in solving the radial equation corresponding to;andare therefore integer
and satisfy:
066 1 (9)
Obviously, for a given value of, the energiesincrease with:
if (10)
For xed, the energy is lower when the corresponding eigenstate is more penetrating,
that is, when the probability density of the electron in the vicinity of the nucleus is larger
[according to (8), the screening eect is then smaller]. The energiesassociated with
the same value ofcan therefore be arranged in order of increasing angular momenta:
0 1 1 (11)
It so happens that the hierarchy of states is approximately the same for all atoms, al-
though the absolute values of the corresponding energies obviously vary with. Figure
1462

MANY-ELECTRON ATOMS. ELECTRONIC CONFIGURATIONS
indicates this hierarchy, as well as the2(2+1)-fold degeneracy of each state (the factor 2
comes from the electron spin). The various states are represented in spectroscopic nota-
tion (cf.Chap., ). Those shown inside the same brackets are very close to
each other, and may even, in certain atoms, practically coincide (we stress the fact that
Figure
with respect to each other; no attempt is made to establish an even moderately realistic
energy scale).
Note the great dierence between the energy spectrum shown and that of the
hydrogen atom (cf.Chap., Fig.). As we have already pointed out, the energy
depends here on the orbital quantum number, and, in addition, the order of the states
is dierent. For example, Figure 4shell has a slightly lower energy
than that of the3shell. This is explained, as mentioned above, by the fact that the4
wave function is more penetrating. Analogous inversions occur for the= 4and= 5
shells, etc. This demonstrates the importance of inter-electron repulsion.
2. Electron congurations of various elements
In the central-eld approximation, the eigenstates of the total Hamiltonian0of the
atom are Slater determinants, constructed from the individual electron states associated
with the energy statesthat we have just described. This is therefore the situation
envisaged in : the ground state of the atom is obtained when
theelectrons occupy the lowest individual states compatible with Pauli's principle.
The maximum number of electrons that can have a given energyis equal to the
2(2+ 1)-fold degeneracy of this energy level. The set of individual states associated
with the same energyis called ashell. The list of occupied shells with the number
of electrons found in each is called theelectronic conguration. The notation used will
be specied below in a certain number of examples. The concept of a conguration also
plays an important role in the chemical properties of atoms. Knowledge of the wave
functions of the various electrons and of the corresponding energies makes it possible to
interpret the number, stability, and geometry of the chemical bonds which can be formed
by this atom (cf.ComplementXII).
To determine the electronic conguration of a given atom in its ground state, we
simply ll the various shells successively, in the order indicated in Figure
of course, with the1level), until theelectrons are exhausted. This is what we shall
do, in a rapid review of Mendeleev's table.
In the ground state of the hydrogen atom, the single electron of this atom occupies
the1level. The electronic conguration of the next element (helium,= 2) is:
He : 1
2
(12)
which means that the two electrons occupy the two orthogonal states of the1shell
(same spatial wave function, orthogonal spin states). Then comes lithium (= 3),
whose electronic conguration is:
Li : 1
2
2 (13)
The1shell can accept only two electrons, so the third one must go into the level directly
above it, that is, according to Figure, into the2shell. This shell can accept a second
1463

COMPLEMENT A XIVE
etc...
5f
6d
7s
6p
6s
5d
5p
5s
4d
4p
3d
3p
3s
2p
2s
1s
4s
4f
(14)
(10)
(2)
(6)
(2)
(10)
(6)
(2)
(10)
(6)
Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn
K, Ca
Al, Si, P, S, Cl, A
Na, Mg
B, C, N, O, F, Ne
Li, Be
H, He
(2)
(10)
(6)
(2)
(6)
(2)
(2)
n = 1n = 2 n = 3 n = 4 n = 5 n = 6 n = 7
(14)
Figure 2: Schematic representation of the hierarchy of energy levels (electronic shells) in
a central potential of the type shown in Figure. For each value of, the energy increases
with. The degeneracy of each level is indicated in parentheses. The levels that appear
inside the same bracket are very close to each other, and their relative disposition can
vary from one atom to another. On the right-hand side of the gure, we have indicated
the chemical symbols of the atoms for which the electronic shell appearing on the same
line is the outermost shell occupied in the ground state conguration.
1464

MANY-ELECTRON ATOMS. ELECTRONIC CONFIGURATIONS
electron, which gives beryllium(= 4)the electronic conguration:
Be : 1
2
2
2
(14)
For 4, the2shell (cf.Fig.) is the rst to be gradually lled, and so on. As the
numberof electrons increases, higher and higher electronic shells are brought in (on the
right-hand side of Figure, we have shown, opposite each of the lowest shells, the symbols
of the atoms for which this shell is the outermost). Thus, we obtain the congurations
of the ground state for all the atoms. This explains Mendeleev's classication. However,
it must be noted that levels that are very close to each other (those grouped in brackets
in Figure) may be lled in a very irregular fashion. For example, although Figure
gives the 4shell a lower energy than that of the 3shell, chromium (= 24) has ve
3electrons although the 4shell is incomplete. Similar irregularities arise for copper
(= 29), niobium(= 41), etc.
Comments:
()The electronic congurations which we have analyzed characterize the ground
state of various atoms in the central-eld approximation. The lowest excited
states of the Hamiltonian0are obtained when one of the electrons moves
to an individual energy level which is higher than the last shell occupied in
the ground state. We shall see, for example, in ComplementXIV, that the
rst excited conguration of the helium atom is:
12 (15)
()A single non-zero Slater determinant is associated with an electronic congu-
ration ending with a complete shell, since there are then as many orthogonal
individual states as there are electrons. Thus, the ground state of the rare
gases (. . . ,
2
,
6
) is non-degenerate, as is that of the alkaline-earths (. . . ,
2
). On the other hand, when the number of external electrons is smaller
than the degree of degeneracy of the outermost shell, the ground state of
the atom is degenerate. For the alkalines (), the degree of degener-
acy is equal to 2; for carbon (1
2
,2
2
,2
2
), it is equal to
2
6= 15, since
two individual states can be chosen arbitrarily from the six orthogonal states
constituting the2shell.
()It can be shown that, for a complete shell, the total angular momentum is
zero, as are the total orbital angular momentum and the total spin (the sums,
respectively, of the orbital angular momenta and the spins of the electrons
occupying this shell). Consequently, the angular momentum of an atom
2
is due only to its outer electrons. Thus, the total angular momentum of a
helium atom in its ground state is zero, and that of an alkali metal is equal
to 1/2 (a single external electron of zero orbital angular momentum and spin
1/2).
2
The angular momentum being discussed here is that of the electronic cloud of the atom. The nucleus
also possesses an angular momentum which should be added to this one.
1465

COMPLEMENT A XIV
References and suggestions for further reading:
Pauling and Wilson (1.9), Chap. IX; Levine (12.3), Chap. 11, 1, 2 and 3; Kuhn
(11.1), Chap. IV, A and B; Schi (1.18), 47; Slater (1.6), Chap. 6; Landau and
Lifshitz (1.19), 68, 69 and 70. See also references of Chap. XI (Hartree and Hartree-
Fock methods).
The shell model in nuclear physics: Valentin (16.1), Chap. VI; Preston (16.4),
Chap. 7; Deshalit and Feshbach (16.6), Chap. IV and V. See also articles by Mayer
(16.20); Peierls (16.21) and Baranger (16.22).
1466

ENERGY LEVELS OF THE HELIUM ATOM. CONFIGURATIONS, TERMS, MULTIPLETS
Complement BXIV
Energy levels of the helium atom. Congurations, terms, multiplets
1 The central-eld approximation. Congurations
1-a The electrostatic Hamiltonian
1-b The ground state conguration and rst excited congurations
1-c Degeneracy of the congurations
2 The eect of the inter-electron electrostatic repulsion: ex-
change energy, spectral terms
2-a Choice of a basis of(;)adapted to the symmetries of
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-b Spectral terms. Spectroscopic notation
2-c Discussion
3 Fine-structure levels; multiplets
In the preceding complement, we studied many-electron atoms in the central-eld
approximation in which the electrons are independent. This enabled us to introduce
the concept of a conguration. We shall evaluate the corrections that must be made to
this approximation, taking into account the inter-electron electrostatic repulsion more
precisely. In order to simplify the reasoning, we shall conne ourselves to the simplest
many-electron atom, the helium atom. We shall show that, under the eect of the inter-
electron electrostatic repulsion, the congurations of this atom () split into spectral
terms (), which give rise to ne-structure multiplets () when smaller terms in the
atomic Hamiltonian (magnetic interactions) are taken into account. The concepts we
shall bring out in this treatment can be generalized to more complex atoms.
1. The central-eld approximation. Congurations
1-a. The electrostatic Hamiltonian
As in the preceding complement, we shall take into account only the electrostatic
forces at rst, writing the Hamiltonian of the helium atom [formula (C-24) of Chap-
ter ] in the form:
=0+ (1)
where:
0=
P
2
1
2
+
P
2
2
2
+(1) +(2) (2)
and:
=
2
2
1
2
2
2
+
2
R1R2
(1) (2) (3)
1467

COMPLEMENT B XIV1s,2p
1s,2s
1s
2
Figure 1: The ground state conguration
and rst excited congurations of the helium
atom (the energies are not shown to scale).
The central potential()is chosen so as to makea small correction of0.
When is neglected, the electrons can be considered to be independent (although
their average electrostatic repulsion is partially taken into account by the potential).
The energy levels of0then dene the electronic congurations we shall study in this
section. We shall then examine the eect ofby using stationary perturbation theory
in 2.
1-b. The ground state conguration and rst excited congurations
According to the discussion of ComplementXIV(), the congurations of the
helium atom are specied by the quantum numbers,and,of the two electrons
(placed in the central potential). The corresponding energycan be written:
= + (4)
Thus (Fig.), the ground state conguration, written1
2
, is obtained when the two
electrons are in the1shell; the rst excited conguration,12, when one electron
is in the1shell and the other one is in the2shell. Similarly, the second excited
conguration is the1,2conguration.
The excited congurations of the helium atom are of the form1,. Actually, there
also exist doubly excited congurations of the type,(with, 1). But, for helium,
their energy is greater than the ionization energyof the atom (the limit of the energy of
the conguration1, when ). Most of the corresponding states, therefore, are very
unstable: they tend to dissociate rapidly into an ion and an electron and are called autoionizing
states. However, there exist levels belonging to doubly excited congurations which are not
autoionizing, but which decay by emitting photons. Some of the corresponding spectral lines
have been observed experimentally.
1-c. Degeneracy of the congurations
Sinceis central and not spin-dependent, the energy of a conguration does not
depend on the magnetic quantum numbersand(66,66) or on
the spin quantum numbersand(=,=) associated with the two electrons.
Most of the congurations, therefore, are degenerate; it is this degeneracy we shall now
calculate.
A state belonging to a conguration is dened by specifying the four quantum
numbers ( ) and ( ) of each electron. Since the electrons are identical
1468

ENERGY LEVELS OF THE HELIUM ATOM. CONFIGURATIONS, TERMS, MULTIPLETS
particles, the symmetrization postulate must be taken into account. The physical ket
associated with this state can, according to the results of , be
written in the form:
; =
1
2
(1 21)1 : ; 2 : (5)
Pauli's principle excludes the states of the system for which the two electrons would be
in the same individual quantum state (=,=,=,=). According to the
discussion of , the set of physical kets (5) for which,,,
are xed and which are not null (that is, not excluded by Pauli's principle) constitute an
orthonormal basis in the subspace(;)ofassociated with the conguration
,.
To evaluate the degeneracy of a conguration, we shall distinguish between
two cases:
() The two electrons are not in the same shell (we do not have=and=).
The individual states of the two electrons can never coincide, and,,,can
independently take on any value. The degeneracy of the conguration, consequently, is
equal to:
2(2+ 1)2(2+ 1) = 4(2+ 1)(2+ 1) (6)
The1,2and1,2congurations enter into this category; their degeneracies are equal
to 4 and 12 respectively.
() The two electrons are in the same shell (=and=).
In this case, the states for which= and=must be excluded. Since
the number of distinct individual quantum states is equal to2(2+ 1), the degree of
degeneracy of the
2
conguration is equal to the number of pairs that can be formed
from these individual states (cf. ), that is:
2
2(2+1)
= (2+ 1)(4+ 1) (7)
Thus, the1
2
conguration, which enters into this category, is not degenerate. It
is useful to expand the Slater determinant corresponding to this conguration. If, in (5),
we set== 1,=== = 0,= +,=, we obtain, writing the spatial
part as a common factor:
1
2
=1 : 100; 2 : 100
1
2
(1 : +; 2 : 1 :; 2 : +) (8)
In the spin part of (8), we recognize the expression for the singlet state= 0,= 0,
whereand are the quantum numbers related to the total spinS=S1+S2(cf.
Chap., ). Thus, although the Hamiltonian0does not depend on the spins,
the constraints introduced by the symmetrization postulate require the total spin of the
ground state to have the value= 0.
2. The eect of the inter-electron electrostatic repulsion: exchange energy,
spectral terms
We shall now study the eect ofby using stationary perturbation theory. To do
so, we must diagonalize the restriction ofinside the subspace(;)associated
1469

COMPLEMENT B XIV
with the, conguration. The eigenvalues of the corresponding matrix give the
corrections of the conguration energyto rst order in; the associated eigenstates
are the zero-order eigenstates.
To calculate the matrix which representsinside(;), we can choose any
basis, in particular, the basis of kets (5). Actually, it is to our advantage to use a basis
well adapted to the symmetries of. We shall see that we can choose a basis in which
the restriction ofis already diagonal.
2-a. Choice of a basis of (;)adapted to the symmetries of
. Total orbital momentumLand total spinS
does not commute with the individual orbital angular momentaL1andL2of
each electron. However, we have already shown (cf.Chap., ) that, ifLdenotes
the total orbital angular momentum:
L=L1+L2 (9)
we have:
[L] = [
2
12
L] = 0 (10)
Therefore,Lis a constant of the motion
1
. Moreover, sincedoes not act in the spin
state space, this is also true for the total spinS:
[S] =0 (11)
Now, consider the set of the four operators,L
2
,S
2
,,. They commute with
each other and with. We shall show that they constitute a C.S.C.O. in the subspace
(;)of. This will enable us in
restriction ofin this subspace.
To do this, we shall return to the space, the tensor product of the state spaces
(1)and(2)relative to the two electrons, assumed to be numbered arbitrarily. The
subspace(;)ofassociated with the,conguration can be obtained
2
by
antisymmetrizing the various kets of the subspace(1) (2)of. If we choose the
basis1 : 2 : in this subspace, we obtain the basis of physical
kets (5) by antisymmetrization.
However, we know from the results of Chapter (1)
(2)another basis composed of common eigenvectors ofL
2
,,S
2
,and entirely
dened by the specication of the corresponding eigenvalues. We shall write this basis:
1 :; 2 :; (12)
with:
=+ + 1
= 10
(13)
1
This result is related to the fact that, under a rotation involving both electrons, the distance between
them,12, is invariant. However, it changes if only one of the two electrons is rotated. This is why
commutes with neitherL1norL2.
2
We could also start with the subspace(1) (2)[cf.comment (i) of ,
p. ].
1470

ENERGY LEVELS OF THE HELIUM ATOM. CONFIGURATIONS, TERMS, MULTIPLETS
SinceL
2
,,S
2
,are all symmetric operators (they commute with21), the vectors
(12) remain, after antisymmetrization, eigenvectors ofL
2
,,S
2
,with the same
eigenvalues (some of them may, of course, have a zero projection onto, in which
case the corresponding physical states are excluded by Pauli's principle; see below).
The non-zero kets obtained by antisymmetrization of (12) are therefore orthogonal, since
they correspond to dierent eigenvalues of at least one of the four observables under
consideration. Since they span(;), they constitute an orthonormal basis of this
subspace, which we shall write:
;; ; (14)
with:
;; ;
=(1 21)1 :; 2 :; (15)
whereis a normalization constant.L
2
,,S
2
,therefore form a C.S.C.O. inside
(;).
Now, we shall introduce the permutation operator
()
21
in the spin state space:
()
21
1 :; 2 :=1 :; 2 : (16)
We showed in cf.comment()] that:
()
21
= (1)
+1
(17)
Furthermore, if
(0)
21
is the permutation operator in the state space of the orbital variables,
we have:
21=
(0)
21
()
21
(18)
Using (17) and (18), we can, nally, put (15) in the form:
;; ;
=[1(1)
+1 (0)
21
]1 :; 2 :; (19)
. Constraints imposed by the symmetrization postulate
We have seen that the dimension of the space(;)is not always equal to
4(2+ 1)(2+ 1), that is, to the dimension of(1) (2). Certain kets of(1)
(2)can therefore have a zero projection onto(;). It is interesting to study
the consequences for the basis (14) of this constraint imposed by the symmetrization
postulate.
First of all, assume that the two electrons do not occupy the same shell. It is then
easy to see that the orbital part of (19) is a sum or a dierence of two orthogonal kets
and, consequently, is never zero
3
. Since the same is true of , we see that all the
3
The normalization constantis then equal to
12
1471

COMPLEMENT B XIV
possible values ofand[cf.formula (13)] are allowed. For example, for the1,2
conguration, we can have= 0,= 0and= 1,= 0; for the1,2conguration,
we can have= 0,= 1and= 1,= 1, etc.
If we now assume that the two electrons occupy the same shell, we have=and
=, and certain of the kets (19) can be zero. Let us write1 :; 2 :; in
the form:
1 :; 2 :;
= ; 1 : ; 2 : (20)
According to relation (25) of ComplementX:
; = (1) ; (21)
By using (20), we then get:
(0)
21
1 :; 2 :; = (1)1 :; 2 :; (22)
Substituting this result into (19), we obtain
4
:
;; ; =
0 if+is odd
1 :; 2 :; if+is even
(23)
Therefore,andcannot be arbitrary:+must be even. In particular, for the
1
2
conguration, we must have= 0, so= 1is excluded. This is a result found
previously.
Finally, note that the symmetrization postulate introduces a close correlation be-
tween the symmetry of the orbital part and that of the spin part of the physical ket (19).
Since the total ket must be antisymmetric, and the spin part, depending on the value of
, is symmetric(= 1)or antisymmetric(= 0), the orbital part must be antisymmet-
ric when= 1and symmetric when= 0. We shall see the importance of this point
later.
2-b. Spectral terms. Spectroscopic notation
commutes with the four observablesL
2
,,S
2
,, which form a C.S.C.O.
inside(;). It follows that the restriction ofinside(;)is diagonal in
the basis:
;; ; and has eigenvalues of:
() = ;; ; ;; ; (24)
This energy depends neither onnor on, since relations (10) and (11) imply that
commutes not only withandbut also withand:is therefore a scalar
4
The normalization constant is then 1/2.
1472

ENERGY LEVELS OF THE HELIUM ATOM. CONFIGURATIONS, TERMS, MULTIPLETS
operator in both the orbital state space and the spin state space (cf.ComplementVI,
).
Inside each conguration, we thus obtain energy levels(;) +
(), labeled by their values ofand. Each of them is(2+ 1)(2+ 1)-fold
degenerate. Such levels are calledspectral termsand denoted in the following way. With
each value ofis associated, in spectroscopic notation (Chap., ) a letter
of the alphabet; we write the corresponding capital letter and add, at the upper left, a
number equal to2+ 1. For example, the1
2
conguration leads to a single spectral
term, written
1
(the
3
, as we have seen, is forbidden by Pauli's principle). The1,2
conguration produces two terms,
1
(non-degenerate) and
3
(three-fold degenerate);
the1,2conguration, two terms,
1
(degeneracy 3) and
3
(degeneracy 9). For a
more complicated conguration such as, for example,2
2
, we obtain (cf. 2-a-) the
spectral terms
1
,
1
and
3
(+must be even), etc.
Under the eect of the electrostatic repulsion, the degeneracy of each congura-
tion is therefore partially removed (the1
2
conguration, which is non-degenerate, is
simply shifted). We shall study this eect in greater detail in the simple example of the
12conguration. We shall try to understand why the two terms
1
and
3
resulting
from this conguration, and whose total spin values are dierent, have dierent energies
although the original Hamiltonian is purely electrostatic.
2-c. Discussion
. Energies of the spectral terms arising from the1,2conguration
In the1,2conguration,=== 0. It is then easy to obtain from (20):
1 := 1= 0; 2 := 2= 0;= = 0
=1 := 1== 0; 2 := 2== 0 (25)
a vector that we shall write, more simply,1 : 1; 2 : 2. If
3
and
1
0
denote the states corresponding to the two spectral terms
3
and
1
arising from the1,
2conguration, we obtain, substituting (25) into (19):
3
=
1
2
[(1
(0)
21
1 : 1; 2 : 2] = 1 (26a)
1
0 =
1
2
[(1 +
(0)
21
1 : 1; 2 : 2] = 0 = 0 (26b)
Sincedoes not act on the spin variables, the eigenvalues given by (24) can be written:
(
3
) =
1
2
1 : 1; 2 : 2(1
(0)
21
)(1
(0)
21
)1 : 1; 2 : 2 (27a)
(
1
) =
1
2
1 : 1; 2 : 2(1 +
(0)
21
)(1 +
(0)
21
)1 : 1; 2 : 2 (27b)
(we have used the fact that
(0)
21
is Hermitian). Moreover,
(0)
21
commutes with, and
the square of
(0)
21
is the identity operator. Therefore:
(1
(0)
21
)(1
(0)
21
) = (1
(0)
21
)
2
= 2(1
(0)
21
) (28)
1473

COMPLEMENT B XIV
Finally, we obtain:
(
3
) = (29a)
(
1
) =+ (29b)
with:
= 1 : 1; 2 : 21 : 1; 2 : 2 (30)
= 1 : 1; 2 : 2
(0)
21
1 : 1; 2 : 2=1 : 2; 2 : 11 : 1; 2 : 2 (31)
therefore represents an overall shift of the energy of the two terms and does not
contribute to their separation.is more interesting, as it introduces an energy dierence
between the
3
and
1
terms (cf.Fig.). We shall therefore study it in a little more
detail.1
S
3
S
2J ≃ 0.8 eV
2s1s
K
Figure 2: The relative position of the spectral terms
1
and
3
arising from the1,2
conguration of the helium atom.represents an overall shift of the conguration. The
removal of the degeneracy is proportional to the exchange integral
. The exchange integral
When we substitute expression (3) forinto (31), there appear terms of the form:
1 : 2; 2 : 1(1)1 : 1; 2 : 2
=1 : 2(1)1 : 1 2 : 12 : 2(32)
Now, the scalar product of the two orthogonal states,2 : 1and2 : 2is zero.
Expression (32) is then equal to zero. The same type of reasoning shows that the terms
that arise from the operators(2),2
2
1,2
2
2are also zero, since each of
these operators acts only in the single-electron spaces while the state of the two electrons
is dierent in the ket and bra of (31). Finally, there remains:
=1 : 2; 2 : 1
2
R1R2
1 : 1; 2 : 2 (33)
1474

ENERGY LEVELS OF THE HELIUM ATOM. CONFIGURATIONS, TERMS, MULTIPLETS
therefore involves only the electrostatic repulsion between the electrons.
Let (r)be the wave functions associated with the states (the
stationary states of an electron in the central potential):
(r) =r (34)
In therrepresentation, the calculation offrom (33) yields:
=d
3
1d
3
2200(r1)
100(r2)
2
r1r2
100(r1)200(r2) (35)
This integral is called the exchange integral. We shall not calculate it explicitly here;
we point out, however, that it is positive.
. The physical origin of the energy dierence between the two spectral terms
We see from expressions (26) and (27) that the origin of the energy separation of
the
3
and
1
terms lies in the symmetry dierences of the orbital parts of these terms.
As we emphasized at the end of 2-a, a triplet term(= 1)must have an orbital part
which is antisymmetric under exchange of the two electrons ; hence thesign before
(0)
21
in (26a) and (27a). On the other hand, a singlet term(= 0)must have a symmetric
orbital part [+sign in (26b 27b)].
This explains the relative position of the
3
and
1
terms shown in Figure. For
the singlet term, the orbital wave function is symmetric with respect to exchange of
the two electrons, which then have a non-zero probability of being at the same point
in space. This is why the electrostatic repulsion, which gives an energy of
2
12which
is large when the electrons are near each other, signicantly increases the singlet state
energy. On the other hand, for the triplet state, the orbital function is antisymmetric
with respect to exchange of the two electrons, which then have a zero probability of
being at the same point in space. The average value of the electrostatic repulsion is then
smaller. Therefore, the energy dierence between the singlet and triplet states arises
from the fact that the correlations between the orbital variables of the two electrons
depend, because of the symmetrization postulate, on the value of the total spin.
. Analysis of the role played by the symmetrization postulate
At this point in the discussion, it might be thought that the degeneracy of a
conguration is removed by the symmetrization postulate. We now show
5
that this is
not the case. This postulate merely xes the value of the total spin of the terms arising
from a given conguration (because of the inter-electron electrostatic repulsion).
To see this, imagine for a moment that we do not need to apply the symmetrization
postulate. Suppose, for example, that the two electrons are replaced by two particles
(ctitious, of course) of the same mass, the same charge and the same spin as the elec-
trons but with another intrinsic property that permits us to distinguish between them
[without, however, changing the Hamiltonianof the problem, which is still given by
formula (1)]. Sinceis not spin-dependent and we do not have to apply the symmetriza-
tion postulate, we can ignore the spins completely until the end of the calculations, and
then multiply the degeneracies obtained by 4. The energy level of0corresponding to
5
See also comment (i) of of Chapter , p. .
1475

COMPLEMENT B XIV
the1,2conguration is two-fold degenerate from the orbital point of view, because
two orthogonal states1 : 1; 2 : 2and1 : 2; 2 : 1correspond to it (they are
dierent physical states since the two particles are of dierent natures). To study the
eect of, we must diagonalizein the two-dimensional space spanned by these two
kets. The corresponding matrix can be written:
(36)
whereandare given by (30) and (31) [the two diagonal elements of (36) are equal
becauseis invariant under permutation of the two particles]. Matrix (36) can be
diagonalized immediately. The eigenvalues found are+and , associated
respectively with the symmetric and antisymmetric linear combinations of the two kets
1 : 1; 2 : 2and1 : 2; 2 : 1. The fact that these orbital eigenstates have
well-dened symmetries relative to exchange of the two particles has nothing to do with
Pauli's principle. It arises only from the fact thatcommutes with
(0)
21
(common
eigenstates ofand
(0)
21
can therefore be found).
When the two particles are not identical, we obtain the same arrangement of levels
and the same orbital symmetry as before. On the other hand, the degeneracy of the
levels is obviously dierent: the lower level, with energy, can have a total spin of
either= 0or= 1, as can the upper level.
If we return to the real helium atom, we now see very clearly the role played by
Pauli's principle. It is not responsible for the splitting of the initial level1,2into
the two energy levels+and , since this splitting would also appear for two
particles of dierent natures. Similarly, the symmetric or antisymmetric character of the
orbital part of the eigenvectors is related to the invariance of the electrostatic interaction
under permutation of the two electrons. Pauli's principle merely forbids the lower state
to have a total spin= 0and the upper state to have a total spin= 1, since the
corresponding states would be globally symmetric, which is unacceptable for fermions.
. The eective spin-dependent Hamiltonian
We replaceby the operator:
=+S1S2 (37)
whereS1andS2denote the two electron spins. We also have:
=
3~
2
4
+
2
S
2
(38)
so that the eigenstates ofare the triplet states, with the eigenvalue+~
2
4, and
the singlet state, with the eigenvalue3~
2
4. Therefore, if we set:
=
2
=
2
~
2
(39)
1476

ENERGY LEVELS OF THE HELIUM ATOM. CONFIGURATIONS, TERMS, MULTIPLETS
we obtain, by diagonalizing, the same eigenstates and eigenvalues we found above
6
.
We can then consider that it is as if the perturbation responsible for the appearance
of the terms were(the eective Hamiltonian), which is of the same form as the
magnetic interaction between two spins. However, one should not conclude that the
coupling energy between the electrons, which is responsible for the appearance of the
two terms, is of magnetic origin: two magnetic moments equal to that of the electron
and placed at a distance of the order of 1

Afrom each other would have an interaction
energy much smaller than. However, because of the very simple form of, this
eective Hamiltonian is often used instead of
An analogous situation arises in the study of ferromagnetic materials. In these
substances, the electron spins tend to align themselves parallel to each other. Since
the spin state is then completely symmetric, Pauli's principle requires the orbital state
to be completely antisymmetric. For the same reasons as for the helium atom, the
electronic repulsion energy is then minimal. When we study such phenomena, we often
use eective Hamiltonians of the same type as (37). However, it must be noted that the
physical interaction which is at the origin of the coupling is again electrostatic and not
magnetic.
Comments:
() 1,2conguration can be treated in the same way. We then have
= 1, so that= +1, 0 or1. As for the1,2conguration, the
shells occupied by the two electrons are dierent, so that the two terms
3
and
1
exist simultaneously. The rst one is nine-fold degenerate, and the
second, three-fold. It can be shown, as above, that the
3
term has an energy
lower than that of the
1
term, and the dierence between the two energies
is proportional to an exchange integral which is analogous to the one written
in (35). We would proceed in the same way for all other congurations of the
type1,.
()We have treatedlike a perturbation of0. For this approach to be coherent, the
energy shifts associated with[for example, the exchange integral written in (35)]
must be much smaller than the energy dierences between congurations. Actually,
this is not the case. For the1,2and1,2congurations, for example, while the
energy dierence(
1 3
)in the1,2conguration is of the order of 0.8 eV,
the minimum distance between levels is[(12)
3
(12)
1
]035eV. We
might therefore believe that it is not valid to treatlike a perturbation of0.
However, the approach we have given is correct. This is due to the fact that, for all
congurations of the type1,, we have=. Therefore, which according
to (10) commutes withL, has zero matrix elements between the states of the1,
2conguration and those of the1,2conguration, since they correspond to
dierent values of. The operator couples a1, conguration only to
congurations with distinctly higher energies, of the1, type with=
(only the values ofare dierent) or of the, type, withand dierent
from 1 (the angular momentaandcan be added to give).
6
We must, obviously, keep only the eigenvectors ofthat belong to.
1477

COMPLEMENT B XIV
3. Fine-structure levels; multiplets
Thus far, we have taken into account in the Hamiltonian only interactions of purely elec-
trostatic origin; we have neglected all eects of relativistic and magnetic origin. Actually,
such eects exist, and we have already studied them in the case of the hydrogen atom
(cf.Chap., ), where they arise from the variation of the electron mass with the
velocity, from theLSspin-orbit coupling, and from the Darwin term. For helium, the
situation is more complicated because of the simultaneous presence of two electrons. For
example, there is a spin-spin magnetic coupling term in the Hamiltonian (cf.Comple-
mentXI) which acts in both the spin state space and the orbital state space of the two
electrons
7
. Nevertheless, a great simplication arises from the fact that the energy dier-
ences associated with these couplings of relativistic and magnetic origin are much weaker
than those which exist between two dierent spectral terms. This enables us to treat the
corresponding Hamiltonian (the ne-structure Hamiltonian) like a perturbation.
The detailed study of the ne structure levels of helium falls outside the domain of
this complement. We shall conne ourselves to describing the symmetries of the problem
and indicating how to distinguish between the dierent energy levels. We shall use the
fact that the ne-structure Hamiltonianis invariant under a simultaneous rotation
of all the orbital and spin variables. This means (cf.ComplementVI, 6) that, ifJ
denotes the total angular momentum of the electrons:
J=L+S (40)
we have:
[ J] =0 (41)
On the other hand, the ne-structure Hamiltonian changes if the rotation acts only on
the orbital variables or only on the spins:
[ L] =[ S]=0 (42)
These properties can easily be seen for the operators()LS, for example, or for
the dipole-dipole magnetic interaction Hamiltonian (cf.ComplementXI).
The state space associated with a term is spanned by the ensemble of states
;; ; written in (19), whereandare xed, and where:
6 6+
6 6+
(43)
In this subspace, it can be shown thatJ
2
and form a C.S.C.O. which, according
to (41), commutes with . The eigenvectors common toJ
2
[eigenvalue
(+ 1)~
2
] and(eigenvalue~) are therefore necessarily eigenvectors of, with
an eigenvalue that depends onbut not on(this last property arises from the fact
7
See for example 19.6 in Sobel'man (11.12) for an explicit expression of the dierent terms of the
ne structure Hamiltonian (Breit Hamiltonian).
1478

ENERGY LEVELS OF THE HELIUM ATOM. CONFIGURATIONS, TERMS, MULTIPLETS1
P
3
P
1
P
1
3
P
0
3
P
1
3
P
2
0.25 eV
1.2 10
–4
eV
1 10
–5
eV
1s2p
Figure 3: The relative position of the spectral terms and multiplets arising from the12
conguration of the helium atom (the splitting of the three multiplets
3
0,
3
1,
3
2has
been greatly exaggerated in order to make the gure clearer).
that commutes with+and). According to the general theory of addition of
angular momenta, the possible values ofare:
=+ + 1+ 2 (44)
The eect ofis therefore a partial removal of the degeneracy. For each term,
there appear as many distinct levels as there are dierent values of, according to
relation (44). Each of these levels is (2+ 1)-fold degenerate and is called a multiplet.
The usual spectroscopic notation consists of denoting a multiplet by adding a right lower
index equal to the value ofto the symbol representing the term from which it arises.
For example, the ground state of the helium atom gives a single multiplet,
1
0. Similarly,
each of the terms
1
and
3
of the1,2conguration leads to a single multiplet:
1
0
and
3
1, respectively. On the other hand, the
3
term arising from1, 2yields three
multiplets,
3
2,
3
1and
3
0(cf.Fig.), and so on. We point out that the measurement
and theoretical calculation of the ne structure of the
3
level of the1,2conguration
is of great fundamental interest, since it can lead to the very precise knowledge of the
ne structure constant,=
2
~.
1479

COMPLEMENT B XIV
Comments:
()For many atoms, the ne-structure Hamiltonian is essentially given by:
=1
()LS (45)
whereR,LandSdenote the positions, angular momenta and spins of each
of theelectrons. It can then be shown, using the Wigner-Eckart theorem (cf.
ComplementX), that the energy of themultiplet is proportional to(+ 1)
(+ 1)(+ 1). This result is sometimes called the Landé interval rule.
For helium, the
3
1and
3
2levels arising from the1,2conguration are much
closer than would be predicted by this rule. This arises from the importance of the
dipole-dipole magnetic coupling of the spins of the two electrons.
()In this complement, we have neglected the hyperne eects related to nu-
clear spin (cf.Chap., ). Such eects actually exist only for the
3
He
isotope, whose nucleus has a spin= 12(the nucleus of the
4
He isotope has
a zero spin). Each multiplet of electronic angular momentumsplits, in the
case of
3
He, into two hyperne levels of total angular momentum= 12,
(2+ 1)-fold degenerate (unless, of course,= 0).
References and suggestions for further reading:
Kuhn (11.1), Chap. III-B; Slater (11.8), Chap. 18; Bethe and Salpeter (11.10).
Multiplet theory and the Pauli principle: Landau and Lifshitz (1.19), 64 and
65; Slater (1.6), Chap. 7 and (11.8), Chap. 13; Kuhn (11.1), Chap. V, A; Sobel'man
(11.12), Chap. 2, 5.3.
1480

PHYSICAL PROPERTIES OF AN ELECTRON GAS. APPLICATION TO SOLIDS
Complement CXIV
Physical properties of an electron gas. Application to solids
1 Free electrons enclosed in a box
1-a Ground state of an electron gas; Fermi energy . . . . . .
1-b Importance of the electrons with energies close to. . . .
1-c Periodic boundary conditions
2 Electrons in solids
2-a Allowed bands
2-b Position of the Fermi level and electric conductivity
In ComplementsXIVandXIV, we studied, taking the symmetrization postulate
into account, the energy levels of a small number of independent electrons placed in
a central potential (the shell model of many-electron atoms). Now, we shall consider
systems composed of a much larger number of electrons, and we shall show that Pauli's
exclusion principle has an equally spectacular eect on their behavior.
To simplify the discussion, we shall neglect interactions between electrons. More-
over, we shall assume, at rst ( 1), that they are subjected to no external potential
other than the one that restricts them to a given volume and which exists only in the
immediate vicinity of the boundary (a free-electron gas enclosed in a box). We shall
introduce the important concept of theFermi energy, which depends only on the
number of electrons per unit volume. We shall also show that the physical properties of
the electron gas (specic heat, magnetic susceptibility, ...) are essentially determined by
the electrons whose energy is close to.
A free-electron model describes the principal properties of certain metals rather
well. However, the electrons of a solid are actually subjected to the periodic potential
created by the ions of the crystal. We know that the energy levels of each electron are then
grouped into allowed energy bands, separated by forbidden bands (cf. ComplementsXI
andIII). We shall show qualitatively in 2 that the electric conductivity of a solid is
essentially determined by the position of the Fermi level of the electron system relative
to the allowed energy bands. Depending on this position, the solid is an insulator or a
conductor.
1. Free electrons enclosed in a box
1-a. Ground state of an electron gas; Fermi energy
Consider a system ofelectrons, whose mutual interactions we shall neglect, and
which, furthermore, are subjected to no external potential. Theseelectrons, however,
are enclosed in a box, which, for simplicity, we shall choose to be a cube with edges of
length
If the electrons cannot pass through the walls of the box, it is because the walls
constitute practically innite potential barriers. Since the potential energy of the elec-
trons is zero inside the box, the problem is reduced to that of the three-dimensional
1481

COMPLEMENT C XIV
innite square well (cf.ComplementsIIandI). The stationary states of a particle in
such a well are described by the wave functions:
(r) =
2
32
sinsinsin (1a)
= 123 (1b)
[expression (1a) is valid for06 6, since the wave function is zero outside this
region]. The energy associated with is equal to:
=
2
}
2
2
2
(
2
+
2
+
2
) (2)
Of course, the electron spin must be taken into account: each of the wave functions
(1) describes the spatial part of two distinct stationary states which dier by their spin
orientation; these two states correspond to the same energy, since the Hamiltonian of the
problem is spin-independent.
The set of these stationary states constitutes a discrete basis, enabling us to con-
struct any state of an electron enclosed in this box (that is, whose wave function goes
to zero at the walls). Note that, by increasing the dimensions of the box, we can make
the interval between two consecutive individual energies as small as we wish, since this
interval is inversely proportional to
2
. Ifis suciently large, therefore, we cannot,
in practice, distinguish between the discrete spectrum (2) and a continuous spectrum
containing all the positive values of the energy.
The ground state of the system of theindependent electrons can be obtained
by antisymmetrizing the tensor product of theindividual states associated with the
lowest energies compatible with Pauli's principle. Ifis small, it is thus simple to ll
the rst individual levels (2) and to nd the ground state of the system, as well as its
degree of degeneracy and the antisymmetrized kets that correspond to it. However, when
is much larger than 1 (in a macroscopic solid,is of the order of10
23
), this method
cannot be used in practice, and we must follow a more global reasoning.
We shall begin by evaluating the number()of individual stationary states whose
energies are lower than a given value. To do so, we shall write expression (2) for the
possible energies in the form:
=
~
2
2
k
2
(3)
with:
(k )=
(k )=(k )=
(4)
According to (1), a vectork corresponds to each function (r). Con-
versely, to each of these vectors, there corresponds one and only one function.
The number of states()can then be obtained by multiplying by 2 the number of vec-
torsk whose modulus is smaller than
2 ~
2
(the factor 2 arises, of course,
1482

PHYSICAL PROPERTIES OF AN ELECTRON GAS. APPLICATION TO SOLIDS
from the existence of electron spin). The tips of the vectorsk dividek-space into
elementary cubes of edge(see Figure, in which, for simplicity, a two-dimensional
rather than a three-dimensional space is shown). Each of these tips is common to eight
neighboring cubes, and each cube has eight corners. Consequently, if the elementary
cubes are suciently small (that is, ifis suciently large), there can be considered to
be one vectork per volume element()
3
ofk-space.0
π/L
π/L
(k)
x
(k)
y
Figure 1: Tips of the vectorsk characterizing the stationary wave functions in a
two-dimensional innite square well.
The valueof the energy which we have chosen denes, ink-space, a sphere
centered at the origin, of radius
2 ~
2
. Only one-eigth of the volume of this sphere
is involved, since the components ofkare positive [cf.(1b) and (4)]. If we divide it
by the volume element()
3
associated with each stationary state, and if we take into
account the factor 2 due to the spin, we obtain:
() = 2
1
8
4
3
2
~
2
32
1
()
3
=
3
3
2
2
~
2
32
(5)
This result enables us to calculate immediately the maximal individual energy of
an electron in the ground state of the system, that is, theFermi energyof the electron
gas. This energysatises:
() = (6)
which gives:
=
~
2
2
3
2
3
23
(7)
1483

COMPLEMENT C XIV
Note that, as might be expected, the Fermi energy depends only on the number
3
of
electrons per unit volume. At absolute zero, all the individual states of energy less than
are occupied, and all those whose energies are greater thanare empty. We shall
see in
We can also deduce thedensity of states()from (5); by denition,()dis
the number of states whose energies are included betweenand+ d. This density
of states, as we shall see later, is of considerable physical importance. It can be obtained
simply by dierentiating()with respect to:
() =
d()
d
=
3
2
2
2
~
2
32
12
(8)
()therefore varies like
. At absolute zero, the number of electrons with a given
energy betweenand+ d(less than, of course) is equal to()d. By using
the value (7) of the Fermi energy, we can put()in the form:
() =
3
2
12
32
(9)
Comment:
It can be seen from (5) that the dimensions of the box are involved only through the
intermediary of the volume element()
3
associated, ink-space, with each stationary
state. If, instead of choosing a cubic box of edge, we had considered a parallelepiped
of edges1,2,3, we would have obtained a volume element of
3
123: only the
volume123of the box, therefore, enters into the density of states. This result can
be shown to remain valid, whatever the exact form of the box, provided it is suciently
large.
1-b. Importance of the electrons with energies close to
The results obtained in the preceding section make it possible to understand the
physical properties of a free electron gas. We shall give two simple examples here, that
of the specic heat and that of the magnetic susceptibility of the system. We shall
conne ourselves, however, to semi-quantitative arguments which simply illustrate the
fundamental importance of Pauli's exclusion principle.
. Specic heat
At absolute zero, the electron gas is in its ground state: all the individual levels of
energy less thanare occupied, and all the others are empty. Taking into account the
form (8) of the density of states(), we can represent the situation schematically as in
Figurea: the number() dof electrons with an energy betweenand+ dis
() dfor and zero for . What happens if the temperatureis low
but not strictly zero?
If the electrons obeyed classical mechanics, each of them, in going from absolute
zero to the temperature, would gain an energy of the order of(whereis the
1484

PHYSICAL PROPERTIES OF AN ELECTRON GAS. APPLICATION TO SOLIDSa
0
E
F
E
0
≃ kT
E
F
E
v(E) v(E)
b
Figure 2: Variation of()with respect to[()dis the number of electrons with
energy betweenand+ d]. At absolute zero, all the levels whose energies are less
than the Fermi energyare occupied (g. a). At a slightly higher temperature, the
transition between empty and occupied levels occurs over an energy interval of a few
(g. b).
Boltzmann constant). The total energy per unit volume of the electron gas would then
be approximately:
()
3
(10)
This would lead to a specic heat at constant volume that is independent of
the temperature.
In reality, the physical phenomena are totally dierent, since Pauli's principle
prevents most of the electrons from gaining energy. For an electron whose initial energy
is much less than(more precisely, if ), the states to which it could
go if its energy increased byare already occupied and are therefore forbidden to it.
Only electrons having an initial energyclose to( ) can heat up, as
shown by Figureb. The number of these electrons is approximately:
()=
3
2
(11)
[according to (9)]. Since the energy of each one increases by about, the total energy
per unit volume can be written:
()
3
(12)
instead of the classical expression (10). Consequently, the constant volume specic heat
is proportional to the absolute temperature:
=
3
(13)
1485

COMPLEMENT C XIV
For a metal, to which the free-electron model can be applied,is typically on the order
of a few eV. Sinceis about 0.03 eV at ordinary temperatures, we see that in this case
the factor introduced by Pauli's principle is of the order of 1/100.
Comments:
()In order to calculate the specic heat of the electron gas quantitatively, we must
know the probability()for an individual state of energyto be occupied
when the system is at thermodynamic equilibrium at the temperature. The
number()dof electrons whose energies are included betweenand+ d
is then:
() d=()() d (14)
It is shown in statistical mechanics (ComplementXV, ) that, for fermions,
the function()can be written:
() =
1
e
( )
+ 1
(15)
whereis thechemical potential(Appendix), also called theFermi levelof the
system. This is theFermi-Dirac distribution. The Fermi level is determined by the
condition that the total number of electrons must be equal to:
+
0
()d
e
( )
+ 1
= (16)
depends on the temperature, but it can be shown that it varies very slowly for
small. The shape of the function()is shown in Figure. At absolute zero,
(0)is equal to 1 for and to 0 for (step function). At non-zero
temperatures,()has the form of a rounded step (the energy interval over
which it varies is of the order of a fewas long as ).
For a free electron gas, it is clear that the Fermi levelat absolute zero coincides
with the Fermi energy calculated in . According to (14) and the form
that()takes for= 0(Fig.),then characterizes, like, the highest
individual energy.
On the other hand, for a system with a discrete spectrum of energies(1,2,
,), the Fermi levelobtained from formula (16) does not coincide with
the highest individual energyin the ground state at absolute zero. In this
case, the density of states is composed of a series of delta functions centered at
1 2 Consequently, at absolute zero,can take on any value between
and +1, since, according to (14), all these possibilities lead to the same
value of(). We choose to deneat absolute zero as the limit of()as
approaches zero. Since at non-zero temperatures the levelempties a little, and
+1begins to ll, the limit of()is found to be a value betweenand +1
(halfway between these two values if the two statesand +1have the same
degree of degeneracy).
Similarly, for a system containing a series of allowed energy bands separated by
forbidden bands (electrons of a solid;cf.ComplementXI), the Fermi levelis
in a forbidden band when the highest individual energy at absolute zero coincides
with the upper limit of an allowed band. On the other hand, the Fermi levelis
equal towhen falls in the middle of an allowed band.
1486

PHYSICAL PROPERTIES OF AN ELECTRON GAS. APPLICATION TO SOLIDS1
1
2
0 E
f(E, T)
μ
Figure 3: Plot of the Fermi-Dirac distribution at absolute zero (dashed line) and at low
temperatures (solid line).
For an electron gas at absolute zero, the Fermi levelcoincides with the Fermi energy
. The curves in Figure ()by
().
()
low temperatures. At ordinary temperatures, the specic heat is essentially due to
vibrations of the ionic lattice (cf.ComplementV), since that of the electron gas
is practically negligible. However, the specic heat of the lattice approaches zero
as
3
for small. Therefore, that of the electron gas becomes preponderant at low
temperatures (around 1 K) where, for metals, a decrease that is linear with respect
tois actually observed.
. Magnetic susceptibility
Now suppose that a free electron gas is placed in a uniform magnetic eldBparallel
to. The energy of an individual stationary state then depends on the corresponding
spin state, since the Hamiltonian contains a paramagnetic spin term (cf.Chap.,
):
=2
~
(17)
where is the Bohr magneton:
=
~
2
(18)
andSis the electron spin operator. For the sake of simplicity, we shall treat (17) as
the only additional term in the Hamiltonian (the behavior of the spatial wave functions
was studied in detail in ComplementVI). Under these conditions, the stationary states
remain the same as in the absence of a magnetic eld, and the corresponding energy is
increased or decreased bydepending on the spin state. The densities of states+()
and()corresponding respectively to the spin states+and can therefore be
obtained very simply from the density()calculated in 1-a:
() =
1
2
( ) (19)
1487

COMPLEMENT C XIVE
E
F
0
2 μ
B
B
ρ
–
(E)
ρ
+
(E)
Figure 4: The densities of states+()and()corresponding respectively to the spin
states+and (is negative). At absolute zero, only the states whose energies
are less thanare occupied.
Thus, at absolute zero, we arrive at the situation shown in Figure.
Since the magnetic energyB is much smaller than, the dierence between
the number of electrons whose spins are antiparallel to the magnetic eld and the number
whose spins are parallel toBis practically, at absolute zero:
+
1
2
()2 (20)
The magnetic moment per unit volume can therefore be written:
=
1
3
( +)
=
2
1
3
() (21)
This magnetic moment is proportional to the applied eld, so that the magnetic suscep-
tibility per unit volume is equal to:
=
=
2
1
3
() (22)
or, using expression (9) for():
=
3
2
3
2
(23)
1488

PHYSICAL PROPERTIES OF AN ELECTRON GAS. APPLICATION TO SOLIDS
Comments:
()We have assumed the system to be at absolute zero, but result (23) remains
valid at low temperatures, since the modications of the number of occupied
states (Fig.b) are practically the same for both spin orientations. We there-
fore nd a temperature-independent magnetic susceptibility. This is indeed
what is observed for metals.
()As in the preceding section, we see that the system behavior in the presence
of a magnetic eld is essentially determined by the electrons whose energies
are close to. This is another manifestation of Pauli's principle. When the
magnetic eld is applied, the electrons in the+spin state tend to go into
the state, which is energetically more favorable. But most of them are
prevented from doing so by the exclusion principle, since all the neighbouring
states are already occupied.
1-c. Periodic boundary conditions
. Introduction
The functions given by formula (1a) have a completely dierent structure
from that of the plane wavese
kr
which usually describe the stationary states of free
electrons. This dierence arises solely from the boundary conditions imposed by the
walls of the box, since, inside the box, the plane waves satisfy the same equation as the
:
~
2
2
(r) =(r) (24)
The functions (1a) are less convenient to handle than plane waves; this is why the latter
are preferably used. To do so, we impose on the solutions of equation (24) new, articial,
boundary conditions which do not exclude plane waves. Of course, since these conditions
are dierent from those actually created by the walls of the box, this changes the physical
problem. However, we shall show in this section that we can nd the most important
physical properties of the initial system in this way. For this to be true, it is necessary
for the new boundary conditions to lead to a discrete set of possible values ofksuch
that:
()The system of plane waves corresponding to these values ofkconstitutes a basis
on which can be expanded any function whose domain is inside the box.
()The density of states()associated with this set of values ofkis identical to the
density of states()calculated in 1-a from the true stationary states.
Of course, the fact that the new boundary conditions are dierent from the real
conditions means that the plane waves cannot correctly describe what happens near the
walls (surface eects). However, it is clear that they can, because of condition(), lead
to a very simple explanation of the volume eects, which, according to what we have
seen in 1-b, depend only on the density of states(). Moreover, because of condition
(i), the motion of any wave packet far from the walls can be correctly described by
superposing plane waves, since, between two collisions with the walls, the wave packet
propagates freely.
1489

COMPLEMENT C XIV
. The Born-von Karman conditions
We shall no longer require the individual wave functions to go to zero at the walls
of the box, but, rather, to be periodic with a period:
(+ ) =( ) (25)
with analogous relations inand. Wave functions of the forme
kr
satisfy these
conditions if the components of the vectorksatisfy:
=
2
=
2
=
2
(26)
where, now,,andare positive or negative integers or zero. We therefore intro-
duce a new system of wave functions:
(r) =
1
32
e
2
( )
(27)
which are normalized inside the volume of the box. The corresponding energy, according
to (24), can be written:
=
~
2
2
4
2
2
(
2
+
2
+
2
) (28)
Any wave function dened inside the box can be extended into a periodic function
in of period. Since this periodic function can always be expanded in a Fourier
series (cf.Appendix, ), the (r)system constitutes a basis for wave
functions with a domain inside the box. To each vectork , whose components
are given by (26), there corresponds a well-dened value of the energy, given
by (28). Note, however, that the vectorsk can now have positive, negative or
zero components, and that their tips divide space into elementary cubes whose edges are
twice that found in .
In order to show that boundary conditions (25) lead to the same physical results
(as far as the volume eects are concerned) as those of 1-a, it suces to calculate
the number()of stationary states of energy less than, and nd the value (5) [the
Fermi energyand the density of states()can be derived directly from()]. We
evaluate()in the same way as in , taking into account the new characteristics
of the vectorsk . Since the components ofkcan now have arbitrary signs, the
volume of the sphere of radius
2 ~
2
must no longer be divided by 8. However, this
modication is compensated by the fact that the volume element (2)
3
associated with
each of the states (27) is eight times larger than the one corresponding to the boundary
conditions of 1-a. Consequently,()is the same as expression (5) for().
The periodic boundary conditions (25) therefore permit us to meet conditions
()and()of the preceding section. They are usually called the Born-Von Karman
conditions (B.V.K. conditions).
1490

PHYSICAL PROPERTIES OF AN ELECTRON GAS. APPLICATION TO SOLIDS
Comment:
Consider a truly free electron (not enclosed in a box). The eigenfunctions of the three
components of the momentumP(and, consequently, those of the Hamiltonian=
P
2
2) form a continuous basis:
1
2~
32
e
pr~
(29)
We have already indicated several times that the states for which the form (29) is valid
in all space are not physical states, but can be used as mathematical intermediaries in
studying the physical states, which are wave packets.
We sometimes prefer to use the discrete basis (27) rather than the continuous basis
(29). To do so, we consider the electron to be enclosed in a ctitious box of edge,
much larger than any dimension involved in the problem, and we impose the B.V.K.
conditions. Any wave packet, which will always be inside the box for suciently large,
can be as well expanded on the discrete basis (27) as on the continuous basis (29). The
states (27) can therefore, like the states (29), be considered to be intermediaries of the
calculation; however, they present the advantage of being normalized inside the box. We
must, of course, check, at the end of the calculations, that the various physical quantities
obtained (transition probabilities, cross sections,...) do not depend on, provided that
is suciently large.
Obviously, for a truly free electron,has no physical meaning and can be arbitrary, as
long as it is suciently large for the states (27) to form a basis on which the wave packets
involved in the problem can be expanded [condition (i) of ]. On the other hand,
in the physical problem which we are studying here,
3
is the volume inside which the
electrons are actually conned and has, consequently, a denite value.
2. Electrons in solids
2-a. Allowed bands
The model of a free electron gas enclosed in a box can be applied rather well to
the conduction electrons of a metal. These electrons can be considered to move freely
inside the metal, the electrostatic attraction of the crystalline lattice preventing them
from escaping when they approach the surface of the metal. However, this model does
not explain why some solids are good electrical conductors while others are insulators.
This is a remarkable experimental fact: the electric properties of crystals are due to the
electrons of the atoms of which they are composed; yet, the intrinsic conductivity can
vary by a factor of10
30
between a good insulator and a pure metal. We shall see, in a
very qualitative way, how this can be explained by Pauli's principle and by the existence
of energy bands arising from the periodic nature of the potential created by the ions (cf.
ComplementsIIIandXI).
We showed in ComplementXIthat if, in a rst approximation, we consider the
electrons of a solid to be independent, their possible individual energies are grouped into
allowed bands, separated byforbidden bands. Assuming that each electron is subjected to
the inuence of a linear chain of regularly spaced positive ions, we found, in the strong-
bond approximation, a series of bands, each one containing2levels, whereis the
number of ions (the factor 2 arises from the spin).
1491

COMPLEMENT C XIV
The situation, of course, is more complex in a real crystal, in which the positive
ions occupy the nodes of a three-dimensional lattice. The theoretical understanding of
the properties of a solid requires a detailed study of the energy bands, a study which
is based on the spatial characteristics of the crystalline lattice. We shall not treat in
detail these specic problems of solid state physics. We shall content ourselves with a
qualitative discussion of the phenomena.
2-b. Position of the Fermi level and electric conductivity
Knowing the band structure and the number of states per band, we obtain the
ground state of the electron system of a solid by successively lling the individual states
of the various allowed bands, beginning, of course, with the lowest energies. The electron
system is really in the ground state only at absolute zero. However, as we pointed out
in , the characteristics of this ground state permit the semi-quantitative under-
standing of the behavior of the system at non-zero temperatures often, up to ordinary
temperatures. Like the thermal and magnetic properties (cf.), the electrical prop-
erties of the system are principally determined by the electrons whose individual energies
are very close to the highest value. If we place the solid in an electric eld, an elec-
tron whose initial energy is much lower thancannot gain energy by being accelerated,
since the states it would reach in this way are already occupied. It is therefore essential
to know the position ofrelative to the allowed energy bands.
First of all, we shall assume (Fig.a) thatfalls in the middle of an allowed
band. The Fermi levelis then equal to[cf.comment (i) of ]. The electrons
whose energies are close tocan easily be accelerated, in this case, since the slightly
higher energy states are empty and accessible. Consequently,a solid for which the Fermi
level falls in the middle of an allowed band is a conductor. The electrons with the highest
energies then behave approximately like free parlicles.
Consider, on the other hand, a solid for which the ground state is composed of
entirely occupied allowed bands (Fig.b). is then equal to the upper limit of an
allowed band, and the Fermi levelfalls inside the adjacent forbidden band [cf.comment
(i) of ]. In this case, no electrons can be accelerated, since the energy states
immediately above theirs are forbidden. Therefore,a solid for which the Fermi level
falls inside a forbidden band is an insulator. The larger the intervalbetween the
last occupied band and the rst empty allowed band, the better the insulator. We shall
return to this point later.
The deep allowed bands, completely occupied by electrons and, consequently, inert
from an electrical and thermal point of view, are calledvalence bands. They are generally
narrow. In a strong-bond model (cf.ComplementXI, ), these bands arise from the
atomic levels of lowest energies, which are only slightly aected by the presence of the
other atoms in the crystal. On the other hand, the higher bands are wider; a partially
occupied band is called aconduction band.
For a solid to be a good insulator, the last occupied band must not only be entirely
full in the ground state, but also, separated from the immediately higher allowed band
by a suciently wide forbidden band. As we have indicated ( ), at non-zero
temperatures, some states of energy lower thancan empty, while some higher energy
states ll (Fig.b). For the solid to remain an insulator at the temperature, the width
of the forbidden band, which prevents this excitation of electrons, must be much
larger than. Ifis less than or of the order of, a certain number of electrons leave
1492

PHYSICAL PROPERTIES OF AN ELECTRON GAS. APPLICATION TO SOLIDSE
F
, μ
μ
E
F
∆E
a b
Figure 5: Schematic representation of the individual levels occupied by the electrons at
absolute zero (in grey).is the highest individual energy. In a conductor (g. a),
(which then coincides with the Fermi level) falls inside an allowed band, called the
conduction band. The electrons whose energies are nearcan then be accelerated
easily, since the slightly higher energy states are accessible to them. In an insulator (g.
b),falls on the upper boundary of an allowed band called the valence band (the
Fermi levelis then situated in the adjacent forbidden band). The electrons can be
excited only by crossing the forbidden band. This requires an energy at least equal to the
widthof this band.
the last valence band to occupy states of the immediately higher allowed band (which
would be completely empty at absolute zero). The crystal then possesses conduction
electrons, but in restricted numbers: it is asemiconductor(such a semiconductor is
calledintrinsic; see comment below). For example, diamond, for whichis close
to 5 eV, remains an insulator at ordinary temperatures, while silicon and germanium,
although quite similar to diamond, are semiconductors: their forbidden bands have a
widthless than 1 eV. These considerations, while very qualitative, enable us to
understand why the electrical conductivity of a semiconductor increases very rapidly
with the temperature; with more quantitative arguments, we indeed nd a dependence
of the forme
2
.
The properties of semiconductors also reveal an apparently paradoxical phenomenon.
It is as if, in addition to the electrons which have crossed the forbidden bandat a
temperature, there existed in the crystal an equal number of particles with a positive
charge. These particles also contribute to the electric current, but their contribution to
the Hall eect
1
, for example, is opposite in sign to what would be expected for electrons.
1
Recall what the Hall eect is: in a sample carrying a current and placed in a magnetic eld
perpendicular to this current, the moving charges are subjected to the Lorentz force. In the steady
state, this causes a transverse electric eld to appear (perpendicular to the current and to the magnetic
1493

COMPLEMENT C XIV
This can be explained very well by band theory, and constitutes a spectacular demon-
stration of Pauli's principle. To understand this qualitatively, we must recall that the
last valence band, when it is completely full in the vicinity of absolute zero, does not
conduct any current (Pauli's principle forbids the corresponding electrons from being
accelerated). When, by thermal excitation, certain electrons move into the conduction
band, they free the states they had occupied in the valence band. These empty states in
an almost full band are called holes. Holes behave like particles of charge opposite to
that of the electron. If an electric eld is applied to the system, the electrons remaining
in the valence band can move, without leaving this band, and occupy the empty states.
In this way, they ll holes but also leave new holes behind them. Holes therefore
move in the direction opposite to that of the electrons, that is, as if they had a positive
charge. This very rough argument can be made more precise, and it can indeed be shown
that holes are in every way equivalent to positive charge carriers.Conduction
band
Forbidden
band
Valence
band
a : type n b : type p
Acceptor level
Donor level
∆E
d
↕
↕∆E
a
Figure 6: Extrinsic semiconductors: donor atoms (g. a) bring in electrons which move
easily into the conduction band, since their ground states are separated from it only by
an energy interval which is much smaller than the width of the forbidden band.
Acceptor atoms (g. b) easily capture valence band electrons, since, for this to happen,
these electrons need only an excitation energy which is much smaller than that
needed to reach the conduction band. This process creates, in the valence band, holes
which can conduct current.
Comment:
We have been speaking only of chemically pure and geometrically perfect crystals. How-
ever, in practice, all solids have imperfections and impurities, which often play an impor-
tant role, particularly in semiconductors. Consider, for example, a quadrivalent silicon or
germanium crystal, in which certain atoms are replaced by pentavalent impurity atoms,
such as phosphorus, arsenic or antimony (this often happens, without any important
eld).
1494

PHYSICAL PROPERTIES OF AN ELECTRON GAS. APPLICATION TO SOLIDS
change in the crystal structure). An atom of such an impurity possesses one too many
outer electrons relative to the neighboring silicon or germanium atoms: it is called an
electrondonor. The binding energyof the additional electron is considerably lower
in the crystal than in the free atom (it is of the order of a few hundredths of an eV);
this is due essentially to the large dielectric constant of the crystal, which reduces the
Coulomb force (cf.ComplementVII, ). The result is that the excess electrons
brought in by the donor atoms move more easily into the conduction band than do the
normal electrons which occupy the valence band (Fig.a). The crystal thus becomes
a conductor at a temperature much lower than would pure silicon or germanium. This
conductivity due to impurities is calledextrinsic. Analogously, a trivalent impurity (like
boron, aluminium or gallium) behaves in silicon or germanium like an electronacceptor:
it can easily capture a valence band electron (Fig.b), leaving a hole which can conduct
the current. In a pure (intrinsic) semiconductor, the number of conduction electrons is
always equal to the number of holes in the valence band. An extrinsic semiconductor, on
the other hand, can, depending on the relative proportion of donor and acceptor atoms,
contain more conduction electrons than holes (it is then said to be of then-type, since the
majority of charge carriers are negative), or more holes than conduction electrons (p-type
semiconductors with a majority of positive charge carriers). These properties serve as
the foundation of numerous technological applications (transistors, rectiers, photoelec-
tric cells, etc.). This is why impurities are often intentionally added to a semiconductor
to modify its characteristics: this is called doping.
References and suggestions for further reading:
See section 8 of the bibliography, especially Kittel (8.2) and Reif (8.4).
For the solid state physics part, see Feynman III (1.2), Chap. 14 and section 13 of
the bibliography.
1495

COMPLEMENT D XIV
Complement DXIV
Exercises
1. Let0be the Hamiltonian of a particle. Assume that the operator0acts only
on the orbital variables and has three equidistant levels of energies0,~0,2~0(where
0is a real positive constant) which are non-degenerate in the orbital state space(in
the total state space, the degeneracy of each of these levels is equal to2+ 1, whereis
the spin of the particle). From the point of view of the orbital variables, we are concerned
only with the subspace ofspanned by the three corresponding eigenstates of0.
. Consider a system of three independent electrons whose Hamiltonian can be
written:
=0(1) +(2) +0(3)
Find the energy levels ofand their degrees of degeneracy.
. Same question for a system of three identical bosons of spin 0.
2. Consider a system of two identical bosons of spin= 1placed in the same
central potential(). What are the spectral terms (cf.ComplementXIV, )
corresponding to the1
2
,12,2
2
congurations?
3. Consider the state space of an electron, spanned by the two vectorsand
which represent two atomic orbitals,and, of wave functions(r)and
(r)(cf.ComplementVII, ):
(r) =() = sincos()()
(r) =() = sincos()()
. Write, in terms of and , the state that represents the
orbital pointing in the direction of theplane that makes an anglewith.
. Consider two electrons whose spins are both in the+state, the eigenstate
ofof eigenvalue+~2.
Write the normalized state vectorwhich represents the system of these two
electrons, one of which is in the stateand the other, in the state.
. Same question, with one of the electrons in the stateand the other one
in the state , whereandare two arbitrary angles. Show that the state vector
obtained is the same.
. The system is in the stateof questionCalculate the probability density
(,,;,,)of nding one electron at( )and the other one at( ).
Show that the electronic density( ) [the probability density of nding any electron
at( )] is symmetric with respect to revolution about theaxis. Determine the
probability density of having=0, where0is given. Discuss the variation of
this probability density with respect to0.
1496

EXERCISES
4. Collision between two identical particles
The notation used is that of of Chapter .
. Consider two particles (1) and (2), with the same mass, assumed for the
moment to have no spin and to be distinguishable. These two particles interact through
a potential()that depends only on the distancebetween them. At the initial time
0, the system is in the state1 :e; 2 :e. Let(0)be the evolution operator
of the system. The probability amplitude of nding it in the state1 :n; 2 :nat
time1is:
(n) =1 :n; 2 :n(10)1 :e; 2 :e
Letandbe the polar angles of the unit vectornin a system of orthonormal axes
. Show that(n)does not depend on. Calculate in terms of(n)the probability
of nding any one of the particles (without specifying which one) with the momentum
nand the other one with the momentumn. What happens to this probability if
is changed to?
. Consider the same problem [with the same spin-independent interaction po-
tential()], but now with two identical particles, one of which is initially in the state
e , and the other, in the statee (the quantum numbers and
refer to the eigenvalues~and~of the spin component along). Assume
that = . Express in terms of(n)the probability of nding, at time1, one
particle with momentumnand spinand the other one with momentumnand
spin. If the spins are not measured, what is the probability of nding one particle
with momentumnand the other one with momentumn? What happens to these
probabilities whenis changed to?
. Treat problembfor the case=. In particular, examine the=2
direction, distinguishing between two possibilites, depending on whether the particles are
bosons or fermions. Show that, again, the scattering probability is the same in theand
directions.
5. Collision between two identical unpolarized particles
Consider two identical particles, of spin, which collide. Assume that their initial
spin states are not known: each of the two particles has the same probability of being in
the2+ 1possible orthogonal spin states. Show that, with the notation of the preceding
exercise, the probability of observing scattering in thendirection is:
(n)
2
+(n)
2
+
2+ 1
[(n)(n) +]
(= +1for bosons,1for fermions).
6. Possible values of the relative angular momentum of two identical
particles
Consider a system of two identical particles interacting by means of a potential
that depends only on their relative distance, so that the Hamiltonian of the system can
be written:
=
P
2
1
2
+
P
2
1
2
+(R1R2)
1497

COMPLEMENT D XIV
As in B of Chapter, we set:
R=
1
2
(R1+R2) P=P1+P2
R=R1R2 P=
1
2
(P1+2)
then becomes:
= +
with:
=
P
2
4
=
P
2
+()
. First, we assume that the two particles are identical bosons of zero spin (
mesons, for example).
. We use therrbasis of the state spaceof the system, composed of
common eigenvectors of the observablesRandR. Show that, if21is the permutation
operator of the two particles:
21rr=rr
. We now go to the p; basis of common eigenvectors of
P L
2
and (L=RPis the relative angular momentum of the two parti-
cles). Show that these new basis vectors are given by expressions of the form:
p; =
1
(2~)
32
d
3
e
pr~
d
3
()()rr
Show that:
21p; = (1)p;
. What values ofare allowed by the symmetrization postulate?
. The two particles under consideration are now identical fermions of spin 1/2
(electrons or protons).
. In the state space of the system, we rst use therGr;SM basis of
common eigenstates ofRRS
2
and, whereS=S1+S2is the total spin of the
system (the kets of the spin state space were determined in ).
Show that:
21rr; = (1)
+1
rr;
. We now go to thep; ; basis of common eigenstates of
P,,L
2
,,S
2
and.
1498

EXERCISES
As in question-, show that:
21p; ; = (1)
+1
(1)p; ;
. Derive the values ofallowed by the symmetrization postulate for each of
the values of(triplet and singlet).
.(more dicult)
Recall that the total scattering cross section in the center of mass system of two
distinguishable particles interacting through the potential()can be written:
=
4
2
=0
(2+ 1) sin
2
where theare the phase shifts associated with()[cf.Chap. , formula (C-58)].
. What happens if the measurement device is equally sensitive to both parti-
cles (the two particles have the same mass)?
. Show that, in the case envisaged in question, the expression forbecomes:
=
16
2
even
(2+ 1) sin
2
. For two unpolarized identical fermions of spin 1/2 (the case of question),
prove that:
=
4
2
even
(2+ 1) sin
2
+ 3
odd
(2+ 1) sin
2
7. Position probability densities for a system of two identical particles
Let and be two normalized orthogonal states belonging to the orbital
state spacerof an electron, and let+and be the two eigenvectors, in the spin
state space, of thecomponent of its spin.
. Consider a system of two electrons, one in the state+and the other, in
the state . Let(rr) d
3
d
3
be the probability of nding one of them in
a volumed
3
rcentered at pointr, and the other in a volumed
3
rcentered atr(two-
particle density function). Similarly, let(r) d
3
rbe the probability of nding one of
the electrons in a volumed
3
rcentered at pointr(one-particule density function). Show
that:
(rr) =(r)
2
(r)
2
+(r)
2
(r)
2
(r) =(r)
2
+(r)
2
Show that these expressions remain valid even ifand are not orthogonal
inr.
Calculate the integrals over all space of(r)and(rr). Are they equal to 1?
Compare these results with those which would be obtained for a system of two
distinguishable particles (both spin 1/2), one in the state+and the other in the
1499

COMPLEMENT D XIV
state ; the device which measures their positions is assumed to be unable to dis-
tinguish between the two particles.
. Now assume that one electron is in the state+and the other one, in the
state+. Show that we then have:
(rr) =(r)(r)(r)(r)
2
(r) =(r)
2
+(r)
2
Calculate the integrals over all space of(r)and(rr).
What happens toand if and are no longer orthogonal in?
. Same questions for two identical bosons, either in the same spin state or in two
orthogonal spin states.
8. The aim of this exercise is to demonstrate the following point: once the state
vector of a system ofidentical bosons (or fermions) has been suitably symmetrized
(or antisymmetrized), it is not indispensable, in order to calculate the probability of any
measurement result, to perform another symmetrization (or antisymmetrization) of the
kets associated with the measurement. More precisely, provided that the state vector
belongs to(or), the physical predictions can be calculated as if we were confronted
with a system of distinguishable particles studied by imperfect measurement devices
unable to distinguish between them.
Let be the state vector of a system ofidentical bosons (all of the following
reasoning is equally valid for fermions). We have:
= (1)
I.
. Let be the normalized physical ket associated with a measurement in which
thebosons are found to be in the dierent and orthonormal individual states,
. Show that:
=
!1 :; 2 :;;: (2)
. Show that, because of the symmetry properties of:
1 :; 2 :;;:
2
= :;:;;:
2
where is an arbitrary permutation of the numbers 1, 2, . . . ,
. Show that the probability of nding the system in the statecan be written:
2
=!1 :; 2 :;;:
2
= :;:;;:
2
(3)
1500

EXERCISES
where the summation is performed over all permutations of the numbers 1, 2, . . . ,
. Now assume that the particles are distinguishable, and that their state is de-
scribed by the ket. What would be the probability of nding any one of them in
the state , another one in the state, . . . , and the last one in the state?
Conclude, by comparison with the results of, that, for identical particles, it is
sucient to apply the symmetrization postulate to the state vectorof the system.
. How would the preceding argument be modied if several of the individual
states constituting the statewere identical? (For the sake of simplicity, consider
only the case where= 3).
II.(more dicult)
Now, consider the general case, in which the measurement result being considered
is not necessarily dened by the specication of individual states, since the measurement
may no longer be complete. According to the postulates of Chapter , we must proceed
in the following way in order to calculate the corresponding probability:

state space is then. Then let be the subspace ofassociated with the
measurement result envisaged and the measurement being performed with devices
incapable of distinguishing between the particles;
denoting an arbitrary ket of, we construct the set of kets
which constitutes a vector space(is the projection ofonto); if the
dimension ofis greater than 1, the measurement is not complete;

projection ontoof the ketdescribing the state of theidentical particles.
. Ifis an arbitrary permutation operator of theparticles, show that, by
construction of:
Show that is globally invariant under the action ofand thatis simply the in-
tersection ofand.
. We construct an orthonormal basis in:
1 2 +1
the rstvectors of which constitute a basis of. Show that the kets , where
+ 166, must be linear combinations of the rstvectors of this basis. Show,
by taking their scalar products with the bras
1
,
2
, that these kets
(with>+ 1) are necessarily zero.
1501

COMPLEMENT D XIV
. Show from the preceding results that the symmetric nature ofimplies that:
=1
2
=
=1
2
that is:
=
where and denote respectively the projectors ontoand.
Conclusion:The probabilities of the measurement results can be calculated from
the projection of the ket(belonging to) onto an eigensubspacewhose kets
do not all belong to, but in which all the particles play equivalent roles.
9. One- and two-particle density functions in an electron gas at
absolute zero
I.
. Consider a system ofparticles12 with the same spin. First
of all, assume that they are not identical. In the state space()of particle(), the ket
:r0 represents a state in which particle()is localized at the pointr0in the spin
state (~: the eigenvalue of).
Consider the operator:
(r0) =
=1
:r0 :r0
=
()
where()is the identity operator in the space().
Let be the state of the-particle system. Show that (r0) d
represents the probability of nding any one of the particles in the innitesimal volume
elementdcentered atr0, the component of its spin being equal to~.
. Consider the operator:
(r0r
0) =
=1=
:r0;:r
0 :r0;:r
0
=
()
What is the physical meaning of the quantity (r0r
0) dd, where
danddare innitesimal volumes?
The average values (r0) and (r0r
0) will be written,
respectively,(r0)and (r0r
0)and will be called the one- and two-particle density
functions of the-particle system.
The preceding expressions remain valid when the particles are identical, provided
that is the suitably symmetrized or antisymmetrized state vector of the system (cf.
preceding exercise).
1502

EXERCISES
II.
Consider a system ofparticles in the normalized and orthogonal individual states
1,2 . The normalized state vector of the system is:
=
!1 :1; 2 :2;;:
whereis the symmetrizer for bosons and the antisymmetrizer for fermions. In this
part, we want to calculate the average values in the stateof symmetric one-particle
operators of the type:
=
=1
()
=
()
or of symmetric two-particle operators of the type:
=
=1=
()
=
()
. Show that:
=1 :1; 2 :2;;: 1 :1; 2 :2;;:
where = +1for bosons, and +1 or1for fermions, depending on whether the
permutationis even or odd.
Show that the same expression is valid for the operator
. Derive the relations:
=
=1
:():
=
=1=
:;:():;:
+ :;:():;:
with= +1for bosons,=1for fermions.
III.
We now want to apply the results of part II to the operators(r0)and (r0r
0)
introduced in part I. The physical system under study is a gas offree electrons en-
closed in a cubic box of edgeat absolute zero (ComplementXIV, ). By applying
periodic boundary conditions, we obtain individual states of the formk , where
the wave function associated withkis a plane wave
1
32e
kr
, and the components
ofksatisfy relations (26) of ComplementXIV. We shall call=~
22
2the Fermi
energy of the system and= 2 , the Fermi wavelength.
1503

COMPLEMENT D XIV
. Show that the two one-particle density functions
+(r0)and(r0)are both
equal to:
+(r0) =(r0) = k(r0)
2
where the summation overkis performed over all values ofkof modulus less then,
satisfying the periodic boundary conditions. By using XIV, show
that
+(r0) =(r0) =
3
6
2
=2
3
. Could this result have been predicted simply?
. Show that the two two-particle density functions
+(r0r
0)and
+(r0r
0)
are both equal to:
kk
k(r0)k(r
0)
2
=
2
4
6
where the summations overkandkare dened as above. Give a physical interpretation.
. Finally, consider the two two-particle density functions
++(r0r
0)and (r0r
0).
Prove that they are both equal to:
k =k
k(r0)k(r
0)
2
k(r
0)
k(r0)k(r0)k(r
0)
Show that the restrictionk=kcan be omitted, and show that the two two-particle
density functions are equal to:
2
4
6
k
k(r0)k(r
0)
2
=
2
4
6
[1
2
()]
with=r0r
0, where the function()is dened by:
() =
3
3
[sin cos]
(can be replaced by an integral overk)
How do the two-particle density functions
++(r0r
0)and (r0r
0)vary with
respect to the distancebetweenr0andr
0? Show that it is practically impossible to
nd two electrons with the same spin separated by a distance much smaller than.
1504

FOURIER SERIES AND FOURIER TRANSFORMS
Appendix I
Fourier series and Fourier transforms
1 Fourier series
1-a Periodic functions
1-b Expansion of a periodic function in a Fourier series
1-c The Bessel-Parseval relation
2 Fourier transforms
2-a Denitions
2-b Simple properties
2-c The Parseval-Plancherel formula
2-d Examples
2-e Fourier transforms in three-dimensional space
In this appendix, we shall review a certain number of denitions, formulas and
properties which are useful in quantum mechanics. We do not intend to enter into the
details of the derivations, nor shall we give rigorous proofs of the mathematical theorems.
1. Fourier series
1-a. Periodic functions
A function()of a variable is said to beperiodicif there exists a real non-zero
numbersuch that, for all:
(+) =() (1)
is called theperiodof the function().
If()is periodic with a period of, all numbers, whereis a positive or
negative integer, are also periods of(). Thefundamental period0of such a function
is dened as being its smallest positive period (the term period is often used in physics
to denote what is actually the fundamental period of a function).
Comment:
We can take a function()dened only on a nite interval[]of the real axis
and construct a function()which is equal to()inside[]and is periodic,
with a period( ). The function()is continuous if()is and if:
() =() (2)
We know that thetrigonometric functionsare periodic. In particular:
cos 2
and sin 2 (3)
1505
Quantum Mechanics, Volume II, Second Edition. C. Cohen-Tannoudji, B. Diu, and F. Laloë.
© 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

APPENDIX I
have fundamental periods equal to
Other particularly important examples of periodic functions are theperiodic ex-
ponentials. For an exponentialeto have a period of, it is necessary and sucient,
according to denition (1), that:
e= 1 (4)
that is:
= 2 (5)
whereis an integer. There are therefore two exponentials of fundamental period:
e
2
(6)
which are, furthermore, related to the trigonometric functions (3) which have the same
period:
e
2
= cos 2
sin 2 (7)
The exponentiale
2
also has a period of, but its fundamental period is.
1-b. Expansion of a periodic function in a Fourier series
Let()be a periodic function with a fundamental period of. If it satises
certain mathematical conditions (as is practically always the case in physics), it can be
expanded in a series of imaginary exponentials or trigonometric functions.
. Series of imaginary exponentials
We can write()in the form:
() =
+
=
e (8)
with:
=
2
(9)
The coecientsof the Fourier series (8) are given by the formula:
=
1
0+
0
de ()
(10)
where0is an arbitrary real number.
To prove (10), we multiply (8) bye and integrate between0and0+:
0+
0
de () =
+
=
0+
0
de
( )
(11)
1506

FOURIER SERIES AND FOURIER TRANSFORMS
The integral of the right-hand side is zero for=and equal tofor=. Hence formula
(10). It can easily be shown that the value obtained foris independent of the number0
chosen.
The set of valuesis called theFourier spectrumof(). Note that()is real
if and only if:
= (12)
. Cosine and sine series
If, in the series (8), we group the terms corresponding to opposite values of, we
obtain:
() =0+
=1
e+ e (13)
that is, according to (7):
() =0+
=1
(cos+sin) (14)
with:
0=0
=+
=( )
0 (15)
The formulas giving the coecientsandcan therefore be derived from (10):
0=
1
0+
0
d()
=
2
0+
0
d() cos
=
2
0+
0
d() sin (16)
If()has a denite parity, expansion (14) is particularly convenient, since:
= 0 if()is even
= 0 if()is odd (17)
Moreover, if()is real, the coecientsandare real.
1-c. The Bessel-Parseval relation
It can easily be shown from the Fourier series (8) that:
1
0+
0
d()
2
=
+
=
2
(18)
1507

APPENDIX I
This can be shown using equation (8):
1
0+
0
d()
2
=
1
0+
0
de
( )
(19)
As in (11), the integral of the right-hand side is equal to. This proves (18).
When expansion (14) is used, the Bessel-Parseval relation (18) can also be written:
1
0+
0
d()
2
=0
2
+
1
2
=1
2
+
2
(20)
If we have two functions,()and(), with the same period, whose Fourier
coecients are, respectively.and, we can generalize relation (18) to the form:
1
0+
0
d()() =
+
=
(21)
2. Fourier transforms
2-a. Denitions
. The Fourier integral as the limit of a Fourier series
Now, consider a function()which is not necessarily periodic. We dene()to
be the periodic function of periodwhich is equal to()inside the interval[22].
The function()can be expanded in a Fourier series:
() =
+
=
e (22)
whereis dened by formula (9), and:
=
1
0+
0
de () =
1
+
22
de () (23)
Whenapproaches innity,()becomes the same as(). We shall therefore let
approach innity in the expressions above.
Denition (9) ofthen yields:
+1 =
2
(24)
We shall now replace1by its expression in terms of(+1 )in (23), and substitute
this value ofinto the series (22):
() =
+
=
+1
2
e
+
22
de () (25)
1508

FOURIER SERIES AND FOURIER TRANSFORMS
When ,+1 approaches zero [cf.(24)], so that the sum overis trans-
formed into a denite integral;()approaches(). The integral appearing in (25)
becomes a function of the continuous variable. If we set:
~
() =
1
2
+
de () (26)
relation (25) can be written in the limit of innite:
() =
1
2
+
de
~
() (27)
()and
~
()are calledFourier transformsof each other.
. Fourier transforms in quantum mechanics
In quantum mechanics, we actually use a slightly dierent convention. If()is
a (one-dimensional) wave function, its Fourier transform
()is dened by:() =
1
2~
+
de ()
(28)
and the inverse formula is:
() =
1
2~
+
de
~
()
(29)
To go from (26) and (27) to (28) and (29), we set:
=~ (30)
(has the dimensions of a momentum ifis a length), and:
() =
1
~
~
() =
1
~
~
~
(31)
In this appendix, as is usual in quantum mechanics, we shall use denition (28) of
the Fourier transform instead of the traditional denition, (26). To return to the latter
denition, furthermore, all we need to do is replace~by 1 andbyin all the following
expressions.
2-b. Simple properties
We shall state (28) and (29) in the condensed notation:
() =[()] (32a)
() =
[()] (32b)
The following properties can easily be demonstrated:
()
( 0) =[e
0~
()] (33)
e
0~
() =[( 0)]
1509

APPENDIX I
This follows directly from denition (28).
()
() =[()] =[()] =
1(34)
To see this, all we need to do is change the integration variable:
= (35)
In particular:
[()] =
() (36)
Therefore, if the function()has a denite parity, its Fourier transform has the same
parity.
() ()real [
()]=() (37a)
()pure imaginary[
()]=() (37b)
The same expressions are valid if the functionsand
are inverted.
()If
()
denotes theth derivative of the function, successive dierentiations inside
the summation yield, according to (28) and (29):
[
()
()] =
~() (38a)
()
() =~
() (38b)
()Theconvolutionof two functions1()and2()is, by denition, the function()
equal to:
() =
+
d 1()2( ) (39)
Its Fourier transform is proportional to the ordinary product of the transforms of
1()and2():
() =2~1()
2() (40)
This can be shown as follows.
We take the Fourier transform of expression (39):
() =
1
2~
+
de
~
+
d 1()2( ) (41)
and perform the change of integration variables:
= = (42)
If we multiply and divide bye
~
we obtain:
() =
1
2~
+
de
~
1()
+
de
~
2() (43)
1510

FOURIER SERIES AND FOURIER TRANSFORMS
which proves ().
()When()is a peaked function of width, the widthof
()satises:
&~ (44)
(see , where this inequality is analyzed, and ComplementIII).
2-c. The Parseval-Plancherel formula
A function and its Fourier transform have the same norm:
+
d()
2
=
+
d
()
2
(45)
To prove this, all we need to do is use (28) and (29) in the following way:
+
d()
2
=
+
d ()
1
2~
+
de
~
()
=
+
d
()
1
2~
+
de
~
()
=
+
d
()() (46)
As in , the Parseval-Plancherel formula can be generalized:
+
d ()() =
+
d
()
()
(47)
2-d. Examples
We shall conne ourselves to three examples of Fourier transforms, for which the
calculations are straightforward.
()Square function
() =
1
for
22
= 0 for
2() =
1
2~
sin (2~)
2~
(48)
()Decreasing exponential
() = e
() =2~
1
(
2
~
2
) + (1
2
)
(49)
()Gaussian function
() = e
22
() =2~
e
22
4~
2
(50)
(note the remarkable fact that the Gaussian form is conserved by the Fourier transform).
1511

APPENDIX I
Comment:
In each of these three cases, the widthsandcan be dened for()and
()respectively, and they verify inequality (44).
2-e. Fourier transforms in three-dimensional space
For wave functions(r)which depend on the three spatial variables,,, (28)
and (29) are replaced by:
(p) =
1
(2~)
32
d
3
e
pr~
(r) (51a)
(r) =
1
(2~)
32
d
3
e
pr~
(p) (51b)
The properties stated above () can easily be generalized to three dimensions.
Ifdepends only on the modulusof the radius-vectorr,
depends only on the
modulusof the momentumpand can be calculated from the expression:
() =
1
2~
2
0
r dsin
~
() (52)
Proof:
First, we shall nd using (51a) the value of
for a vectorpobtained frompby an
arbitrary rotation:
p=p (53)
(p) =
1
(2~)
32
d
3
e
pr~
() (54)
In this integral, we replace the variablerbyrand set:
r=r (55)
Since the volume element is conserved under rotation, we have:
d
3
= d
3
(56)
In addition, the functionis unchanged, since the modulus ofrremains equal to;
nally:
pr=pr (57)
since the scalar product is rotation-invariant. We thus nd:
(p) =(p) (58)
that is,
depends only on the modulus ofpand not on its direction.
1512

FOURIER SERIES AND FOURIER TRANSFORMS
We can then choosepalongto evaluate
():() =
1
(2~)
32
d
3
e
~
()
=
1
(2~)
32
0
2
d()
2
0
d
0
dsine
cos~
=
1
(2~)
32
0
2
d() 2
2~
sin
~
=
1
2~
2
0
d() sin
~
(59)
This proves (52).
Assume for instance that ()is given by the following (non normalized) function:
() =
e
(60)
whereis positive. Relation (52) then becomes:
() =
1
2~
1
0
dee
~
e
~
=
1
2~
11
+~
1
~
=
2~
3
1
2
+
2
~
2
(61)
A central potential that varies withas the right-hand side of (60) is called a Yukawa
potential. When= 0, it becomes a Coulomb potential, whose gradient gives an
electric eld. If we take the gradient of (51b), we obtain the Fourier transformation
correspondence between two following vector functions (we now use variablekinstead
ofp, and therefore set~= 1):
1
er
FT2
k
2
+
2
(62)
The limit0then provides:
r
3
FT2
k
2
(63)
References and suggestions for further reading:
See, for example, Arfken (10.4), Chaps. 14 and 15, or Butkov (10.8), Chaps. 4
and 7; Bass (10.1), vol. I, Chaps. XVIII through XX: section 10 of the bibliography,
especially the subsection Fourier transforms; distributions.
1513

THE DIRAC -FUNCTION
Appendix II
The Dirac-function
1 Introduction; principal properties
1-a Introduction of the-function
1-b Functions that approach . . . . . . . . . . . . . . . . . . . .
1-c Properties of . . . . . . . . . . . . . . . . . . . . . . . . . .
2 The -function and the Fourier transform
2-a The Fourier transform of . . . . . . . . . . . . . . . . . . .
2-b Applications
3 Integral and derivatives of the -function
3-a is the derivative of the unit step-function
3-b Derivatives of. . . . . . . . . . . . . . . . . . . . . . . . . .
4 The -function in three-dimensional space
The-function is actually a distribution. However, like most physicists, we shall
treat it like an ordinary function. This approach, although not mathematically rigorous,
is sucient for quantum mechanical applications.
1. Introduction; principal properties
1-a. Introduction of the -function
Consider the function
()
()given by (cf.Fig.):
()
() =
1
for
22
= 0 for
2
(1)1
ε
δ
(ε)
(x)
2
– +
ε
2
ε x
Figure 1: The function
()
(): a square
function of widthand height 1/, centered
at= 0.
1515

APPENDIX II
whereis a positive number. We shall evaluate the integral:
+
d
()
()() (2)
where()is an arbitrary function, well-dened for= 0. Ifis suciently small, the
variation of()over the eective integration interval[22]is negligible, and()
remains practically equal to(0). Therefore:
+
d
()
()()(0)
+
d
()
() =(0) (3)
The smaller, the better the approximation. We therefore examine the limit= 0and
dene the-function by the relation:
+
d()() =(0) (4)
which is valid for any function()dened at the origin. More generally,( 0)is
dened by:
+
d( 0)() =(0) (5)
Comments:
()Actually, the integral notation in (5) is not mathematically justied.is
dened rigorously not as a function but as a distribution. Physically, this
distinction is not an essential one as it becomes impossible to distinguish
between
()
()and()as soon asbecomes negligible compared to all the
distances involved in a given physical problem
1
: any function()which we
might have to consider does not vary signicantly over an interval of length.
Whenever a mathematical diculty might arise, all we need to do is assume
that()is actually
()
()[or an analogous but more regular function, for
example, one of those given in (7), (8), (9), (10), (11)], withextremely small
but not strictly zero.
()For arbitrary integration limitsand, we have:
d()() =(0) if 0[]
= 0 if 0[] (6)
1
The accuracy of present-day physical measurements does not, in any case, allow us to investigate
phenomena on a scale of less than a fraction of a Fermi (1 Fermi =10
15
m).
1516

THE DIRAC -FUNCTION
1-b. Functions that approach
It can easily be shown that, in addition to
()
()dened by (1), the following
functions approach(), that is, satisfy (5), when the parameterapproaches zero from
the positive side:
()
1
2
e (7)
()
1
2
+
2
(8)
()
1
e
22
(9)
()
1
sin()
(10)
()
sin
2
()
2
(11)
We shall also mention an identity which is often useful in quantum mechanics
(particularly in collision theory):
Lim
0+
1
=
1
() (12)
wheredenotes the Cauchy principal part, dened by
2
[()is a regular function at
= 0]:
+
d
() = Lim
0+
+
+
+
d
(); 0 (13)
To prove (12), we separate the real and imaginary parts of1( ):
1
=
2
+
2
(14)
Since the imaginary part is proportional to the function (), we have:
Lim
0+
2
+
2
= () (15)
As for the real part, we shall multiply it by a function()that is regular at the origin, and
integrate over:
Lim
0+
+
d
2
+
2
() = Lim
0+
Lim
0+
+
+
+
+
+
d
2
+
2
() (16)
2
One often uses one of the following relations:
+
d
() =
+
d
()
+
d
() =
+
d
()(0)
+(0) Log
where() = [()()]2is the odd part of(). These formulas allow us to explicitly eliminate
the divergence at the origin.
1517

APPENDIX II
The second integral is zero:
Lim
0+
+
d
2
+
2
() =(0) Lim
0+
1
2
Log(
2
+
2
)
+
= 0 (17)
If we now reverse the order of the evaluation of the limits in (16), the 0limit presents no
diculties in the two other integrals. Thus:
Lim
0+
+
d
2
+
2
() = Lim
0+
+
+
+
d
() (18)
This establishes identity ().
1-c. Properties of
The properties we shall now state can be demonstrated using (5). Multiplying
both sides of the equations below by a function()and integrating, we see that the
results obtained are indeed equal.
() () =() (19)
() () =
1
() (20)
and, more generally:
[()] =
1
()
( ) (21)
where()is the derivative of()and theare the simple zeros of the function():
() = 0
()= 0 (22)
The summation is performed over all the simple zeros of(). If()has zeros of multiple
order [that is, for which()is zero], the expression[()]makes no sense.
() ( 0) =0( 0) (23)
and, in particular:
() = 0 (24)
The converse is also true and it can be shown that the equation:
() = 0 (25)
has the general solution:
() =() (26)
whereis an arbitrary constant.
1518

THE DIRAC -FUNCTION
More generally:
()( 0) =(0)( 0) (27)
()
+
d( )( ) =( ) (28)
Equation (28) can be understood by examining functions
()
()like the one shown in
Figure. The integral:
()
() =
+
d
()
( )
()
( ) (29)
is zero as long as , that is, as long as the two square functions do not overlap (Fig.).δ
(ε)
(x – y) δ
(ε)
(x – z)
1
ε
ε ε
xzy
Figure 2: The functions
()
( )and
()
( ): two square functions of widthand
height1, centered respectively at=and=.
The maximum value of the integral, obtained for=, is equal to1. Between this
maximum value and 0, the variation of
()
()with respect to is linear (Fig.). We see
immediately that
()
()approaches( )when 0.
Comment:
A sum of regularly spaced-functions:
+
=
( ) (30)
can be considered to be a periodic function of period. By applying formulas
(8), (9) and (10) of Appendix, we can write it in the form:
+
=
( ) =
1
+
=
e
2
(31)
1519

APPENDIX IIF
(ε)
(y, z)
y – z
1
ε
– ε + ε
Figure 3: The variation with respect toof the scalar product
()
()of the two
square functions shown in Figure. This scalar product is zero when the two functions
do not overlap( ), and maximal when they coincide.
()
()approaches
( )when 0.
2. The -function and the Fourier transform
2-a. The Fourier transform of
Denition (28) of Appendix 5) enable us to calculate directly the
Fourier transform
0
()of( 0):
0
() =
1
2~
+
de
~
( 0) =
1
2~
e
0~
(32)
In particular, that of()is a constant:
0() =
1
2~
(33)
The inverse Fourier transform [formula (29) of Appendix] then yields:
( 0) =
1
2~
+
de
( 0)~
=
1
2
+
de
( 0)
(34)
This result can also be found by using the function
()
()dened by (1) or any of the functions
given in . For example, (48) of Appendix
()
() =
1
2~
+
de
~sin(2~)
2~
(35)
If we letapproach zero, we indeed obtain ().
1520

THE DIRAC -FUNCTION
2-b. Applications
Expression (34) for the-function is often very convenient. We shall show, for ex-
ample, how it simplies nding the inverse Fourier transform and the Parseval-Plancherel
relation [formulas (29) and (45) of Appendix].
Starting with:
() =
1
2~
+
de
~
() (36)
we calculate:
1
2~
+
de
~
() =
1
2~
+
d()
+
de
( )~
(37)
In the second integral, we recognize( ), so that:
1
2~
+
de
~
() =
+
d()( ) =() (38)
which is the inversion formula of the Fourier transform.
Similarly:
()
2
=
1
2~
+
de
~
()
+
de
~
() (39)
If we integrate this expression over, we nd:
+
d
()
2
=
1
2~
+
d ()
+
d()
+
de
( )~
(40)
that is, according to (34):
+
d
()
2
=
+
d ()
+
d()( ) =
+
d()
2
(41)
which is none other than the Parseval-Plancherel formula.
We can obtain the Fourier transform of a convolution product in an analogous way
[cf.formulas (39) and (40) of Appendix].
3. Integral and derivatives of the-function
3-a. is the derivative of the unit step-function
We shall evaluate the integral:
()
() =
()
() d (42)
where the function
()
()is dened in (1). It can easily be seen that
()
()is equal to 0
for
2
, to 1 for
2
, and to
1
+
2
for
22
. The variation of
()
()
1521

APPENDIX IIθ
(ε)
(x)
1
2
– +
ε
2
ε x
Figure 4: Variation of the function
()
(), whose derivative
()
()is shown in Figure.
When 0,
()
()approaches the Heaviside step-function().
with respect tois shown in Figure. When 0,
()
()approaches theHeaviside
step-function(), which, by denition, is equal to:
() = 1 if 0
() = 0 if 0 (43)
()
()is the derivative of
()
(). By considering the limit0, we see that()is
the derivative of():
d
d
() =() (44)
Now, consider a function()which has a discontinuity0at= 0:
Lim
0+
()Lim
0
() =0 (45)
Such a function can be written in the form() =1()() +2()(), where1()
and2()are continuous functions which satisfy1(0) 2(0) =0. If we dierentiate
this expression, using (44), we obtain:
() =
1()() +
2()() +1()() 2()()
=
1()() +
2()() +0() (46)
according to properties (19) and (27) of. For a discontinuous function, there is then
added to the ordinary derivative [the rst two terms of (46)] a term proportional to the
-function, the proportionality coecient being the magnitude of the function's discon-
tinuity
3
.
3
Of course, if the function is discontinuous at=0, the additional term is of the form:[1(0)
2(0)]( 0).
1522

THE DIRAC -FUNCTION
Comment:
The Fourier transform of the step-function()can be found simply from (12).
We get:
+
() ed= Lim
0+
0
de
(+)
= Lim
0+
+
=
1
+()(47)
3-b. Derivatives of
By analogy with the expression for integration by parts, the derivative()of the
-function is dened by the relation
4
:
+
d()() =
+
d()() =(0) (48)
From this denition, we immediately get:
() =() (49)
and:
() =() (50)
Conversely it can be shown that the general solution of the equation:
() =() (51)
can be written:
() =() +() (52)
where the second term arises from the homogeneous equation [cf.formulas (25) and
(26)].
Equation (34) allows us to write()in the form:
() =
1
2~
+
d
~
e
~
=
2
+
de (53)
Theth-order derivative
()
()can be dened in the same way:
+
d
()
()() = (1)
()
(0) (54)
Relations (49) and (50) can then be generalized to the forms:
()
() = (1)
()
() (55)
and:
()
() =
(1)
() (56)
4
()can be considered to be the limit, for0, of the derivative of one of the functions given
in .
1523

APPENDIX II
4. The -function in three-dimensional space
The-fonction in three-dimensional space, which we shall write simply as(r), is
dened by an expression analogous to (4):
d
3
(r)(r) =(0) (57)
and, more generally:
d
3
(rr0)(r) =(r0) (58)
(rr0)can be broken down into a product of three one-dimensional functions:
(rr0) =( 0)( 0)( 0) (59)
or, if we use polar coordinates:
(rr0) =
1
2
sin
( 0)( 0)( 0)
=
1
2
( 0)(coscos0)( 0) (60)
The properties stated above for()can therefore easily be generalized to(r).
We shall mention, in addition, the important relation:

1
=4(r) (61)
whereis the Laplacian operator.
Equation (61) can easily be understood if it is recalled that in electrostatics, an electrical
point chargeplaced at the origin can be described by a volume density(r)equal to:
(r) =(r) (62)
We know that the expression for the electrostatic potential produced by this charge is:
(r) =
40
1
(63)
Equation (61) is thus simply the Poisson equation for this special case:
(r) =
1
0
(r) (64)
To prove (61) rigorously, it is necessary to use mathematical distribution theory. We shall
conne ourselves here to an elementary proof.
First of all, note that the Laplacian of1is everywhere zero, except, perhaps, at the
origin, which is a singular point:
d
2
d
2
+
2d
d
1
= 0 for = 0 (65)
1524

THE DIRAC -FUNCTION
Let(r)be a function equal to1whenris outside the sphere, centered atand
of a radius, and which takes on values (of the order of1) inside this sphere such that(r)
is suciently regular (continuous, dierentiable, etc.). Let(r)be an arbitrary function ofr
which is also regular at all points in space. We now nd the limit of the integral:
() =d
3
(r) (r) (66)
for 0. According to (65), this integral can receive contributions only from inside the sphere
, and:
() = d
3
(r) (r) (67)
We choosesmall enough for the variation of(r)insideto be negligible. Then:
()(0) d
3
(r) (68)
Transforming the integral so obtained into an integral over the surfaceSof, we obtain:
()(0)
S
r(r)dn (69)
Now, since(r)is continuous on the surfaceS, we get:
[r(r)]
=
=
1
2
=
e=
1
2
e (70)
(whereeis the unit vectorr). This yields:
()(0)4
2 1
2
4(0) (71)
that is:
Lim
0
d
3
(r)(r) =4(0) (72)
According to denition (57), this is simply (61).
Equation (61) can be used, for example, to derive an expression which is useful in
collision theory (cf.Chap. ):
( +
2
)
e
=4(r) (73)
To do so, it is sucient to considere as a product:

e
=
1
(e ) + e
1
+ 2r
1
r(e) (74)
Now:
r(e) = e
r
(e ) =
2
e
2
e (75)
1525

APPENDIX II
We therefore nd, nally:
( +
2
)
e
=
2
2
2
4(r)
2
2
() +
2
e
=4e (r)
=4(r) (76)
according to (27).
Equation (61) can, furthermore, be generalized: the Laplacian of the function
()
+1
involvesth-order derivatives of(r). Consider, for examplecos
2
. We
know that the expression for the electrostatic potential created at a distant point by an
electric dipole of moment D directed alongis
40
cos
2
. Ifis the absolute value of
each of the two charges which make up the dipole andis the distance between them,
the modulusof the dipole moment is the product, and the corresponding charge
density can be written:
(r) = r
2
e r+
2
e (77)
(whereedenotes the unit vector of theaxis). If we letapproach zero, while
maintaining=nite, this charge density becomes:
(r)
0
(r) (78)
Therefore, in the limit where0, the Poisson equation (64) yields:

cos
2
=4(r) (79)
Of course, this formula could be justied as (61) was above, or proven by distribution
theory. Analogous reasoning could be applied to the function()
+1
which gives
the potential created by an electric multipole momentlocated at the origin (comple-
mentX).
References and suggestions for further reading:
See Dirac (1.13) 15, and, for example, Butkov (10.8), Chap. 6, or Bass (10.1), vol.
I, 21.7 and 21.8; section 10 of the bibliography, especially the subsection Fourier
transforms; distributions.
1526

LAGRANGIAN AND HAMILTONIAN IN CLASSICAL MECHANICS
Appendix III
Lagrangian and Hamiltonian in classical mechanics
1 Review of Newton's laws
1-a Dynamics of a point particle
1-b Systems of point particles
1-c Fundamental theorems
2 The Lagrangian and Lagrange's equations
3 The classical Hamiltonian and the canonical equations
3-a The conjugate momenta of the coordinates
3-b The Hamilton-Jacobi canonical equations
4 Applications of the Hamiltonian formalism
4-a A particle in a central potential
4-b A charged particle placed in an electromagnetic eld
5 The principle of least action
5-a Geometrical representation of the motion of a system
5-b The principle of least action
5-c Lagrange's equations as a consequence of the principle of least
action
We shall review the denition and principal properties of the Lagrangian and
the Hamiltonian in classical mechanics. This appendix is not meant to be a course in
analytical mechanics. Its goal is simply to indicate the classical basis for applying the
quantization rules (cf.Chap.) to a physical system. In particular, we shall concern
ourselves essentially with systems of point particles.
1. Review of Newton's laws
1-a. Dynamics of a point particle
Non-relativistic classical mechanics is based on the hypothesis that there exists at
least one geometrical frame, called theGalileanorinertial frame, in which the following
law is valid:
The fundamental law of dynamics: a point particle has, at all times, an acceleration
which is proportional to the resultantFof the forces acting on it:
F= (1)
The constantis an intrinsic property of the particle, called itsinertial mass.
It can easily be shown that if a Galilean frame exists, all frames which are in
uniform translational motion with respect to it are also Galilean frames. This leads us
to theGalilean relativity principle: there is no absolute frame; there is no experiment
which can give one inertial frame a privileged role with respect to all others.
1527

APPENDIX III
1-b. Systems of point particles
If we are dealing with a system composed ofpoint particles, we apply the fun-
damental law to each of them
1
:
r=F;= 12 (2)
The forces that act on the particles can be classed in two categories:internal
forcesrepresent the interactions between the particles of the system, andexternal forces
originate outside the system. The internal forces are postulated to satisfy theprinciple of
action and reaction: the force exerted by particle()on particle () is equal and opposite
to the one exerted by()on(). This principle is true for gravitational forces (Newton's
law) and electrostatic forces, but not for magnetic forces (whose origin is relativistic).
If all the forces can be derived from a potential, the equations of motion (2) can
be written:
r=r (3)
whererdenotes the gradient with respect to thercoordinates, and the potential
energyis of the form:
=
=1
(r) + (rr) (4)
(the rst term in this expression corresponds to the external forces, and the second one
to the internal forces).In cartesian coordinates, the motion of the system is therefore
described by the3dierential equations:
=
==
= 12 (5)
1-c. Fundamental theorems
We shall rst review a few denitions. Thecenter of massorcenter of gravityof
a system is the pointwhose coordinates are:
r=
=1
r
=1
(6)
1
In mechanics, a simplied notation is generally used for the time-derivatives; by denition,_=
dd
,
=
d
2
d
2
, etc...
1528

LAGRANGIAN AND HAMILTONIAN IN CLASSICAL MECHANICS
The totalkinetic energyof the system is equal to:
=
=1
1
2
_r
2
(7)
where_ris the velocity of particle(). Theangular momentumwith respect to the origin
is the vector:
LLL=
=1
r _r (8)
The following theorems can then be easily proven:
()The center of mass of a system moves like a point particle with a mass equal
to the total mass of the system, subject to a force equal to the resultant of all the forces
involved in the system:
=1
r=
=1
F (9)
()The time-derivative of the angular momentum evaluated at a xed point is
equal to the moment of the forces with respect to this point:
d
d
LLL=
=1
rF (10)
()The variation of the kinetic energy between time1and2is equal to the
work performed by all the forces during the motion between these two times:
(2)(1) =
2
1=1
F_rd (11)
If the internal forces satisfy the principle of action and reaction, and if they are
directed along the straight lines joining the interacting particles, their contribution to
the resultant [equation (9)] and to the moment with respect to the origin [equation (10)]
is zero. If, in addition, the system is isolated (that is, if it is not subject to any external
forces), the total angular momentumLLLis constant, and the center of mass is in uniform
rectilinear motion. This means that the total mechanical momentum:
=1
_r (12)
is also a constant of the motion.
1529

APPENDIX III
2. The Lagrangian and Lagrange's equations
Consider a system ofparticles in which the forces are derived from a potential energy
[cf.formula (4)], which we shall write simply(r). TheLagrangian, orLagrange's
function, of this system is the function of6variables ; ___;= 12
given by:
(r_r) =
=
1
2
=1
_r
2
(r) (13)
It can immediately be shown that the equations of motion written in (5) are identical to
Lagrange's equations:
d
d_
= 0
d
d_
= 0 (14)
d
d_
= 0
A very interesting feature of Lagrange's equations is that they always have the
same form, independent of the type of coordinates used (whether they are cartesian or
not). In addition, they can be applied to systems which are more general than particle
systems. Many physical systems (including for example one or several solid bodies) can
be described at a given timeby a set ofindependent parameters(= 12 ),
calledgeneralized coordinates. Knowledge of thepermits the calculation of the position
in space of any point of the system. The motion of this system is therefore characterized
by specifying thefunctions of time(). The time-derivatives_()are called the
generalized velocities. The state of the system at a given instant0is therefore dened
by the set of(0)and_(0). If the forces acting on the system can be derived from a
potential energy(12 ), the Lagrangian(12 ; _1_2 _)is again
the dierence between the total kinetic energyand the potential energy. It can be
shown that, for any choice of the coordinates, the equations of motion can always be
written:
d
d_
= 0
(15)
where
d
d
denotes thetotal time-derivative
d
d
=+
=1
_+
=1

_
(16)
Furthermore, it is not really necessary for the forces to be derived from a potential for us
to be able to dene a Lagrangian and use Lagrange's equations (we shall see an example
1530

LAGRANGIAN AND HAMILTONIAN IN CLASSICAL MECHANICS
of this situation in ). In the general case, the Lagrangian is a function of the
coordinatesand the velocities_, and can also be explicitly time-dependent
2
. We shall
then write it:
(_;) (17)
Lagrange's equations are important in classical mechanics for several reasons. For
one thing, as we have just indicated, they always have the same form, independent of
the coordinates which are used. Furthermore, they are more convenient than Newton's
equations when the system is complex. Finally, they are of considerable theoretical
interest, since they form the foundation of the Hamiltonian formalism (cf.
and since they can be derived from a variational principle (). The rst two points are
secondary as far as quantum mechanics is concerned, since quantum mechanics treats
particle systems almost exclusively and since the quantization rules are stated in cartesian
coordinates (cf.Chap., ). However, the last point is an essential one, since the
Hamiltonian formalism constitutes the point of departure for the quantization of physical
systems.
3. The classical Hamiltonian and the canonical equations
For a physical system described bygeneralized coordinates, Lagrange's equations (15)
constitute a system ofcoupled second-order dierential equations withunknown
functions, the(). We shall see that this system can be replaced by a system of2
rst-order equations with2unknown functions.
3-a. The conjugate momenta of the coordinates
The conjugate momentum of the generalized coordinateis dened as:
=
_
(18)
is also called thegeneralized momentum. In the case of a particle system for which
the forces are derived from a potential energy, the conjugate momenta of the position
variablesr(1 )are simply [see (13)] the mechanical momenta:
p= _r (19)
However, we shall see in that this is no longer true in the presence of a magnetic
eld.
Instead of dening the state of the system at a given timeby thecoordinates
()and thevelocities_(), we shall henceforth characterize it by the2variables:
()();= 12 (20)
2
The Lagrangian is not unique: two functions(_;)and(_;)may lead, using (15), to
the same equations of motion. This is true, in particular, if the dierence betweenandis the total
derivative with respect to time of a function(;)..
=
d
d
()+ _
1531

APPENDIX III
This amounts to assuming that from the2parameters()and(), we can determine
the_()uniquely. These variables may be considered as the2coordinates of a point
dening the state of the system at every time, and moving in a2dimensional space
called thephase space.
3-b. The Hamilton-Jacobi canonical equations
Theclassical Hamiltonian, orHamilton's function, of the system is, by denition:
=
=1
_ (21)
In accordance with convention (20), we eliminate the_and consider the Hamiltonian
to be a function of the coordinates and their conjugate momenta. Like,may be
explicitly time-dependent:
(;) (22)
The total dierential of the function:
d=
d+d+d (23)
is equal to, using denitions (21) and (18):
d= [d _+ _d]
d
_
d _d
= _d
dd (24)
Setting (23) and (24) equal, we see that the change from the_variables to
the variables leads to:
= (25a)= _ (25b)= (25c)
Furthermore, using (18) and (25a), we can write Lagrange's equations (15) in the form:
d
d
= (26)
By grouping terms in (25b) and (26), we obtain the equations of motion:
d
d
=
d
d
=
(27)
1532

LAGRANGIAN AND HAMILTONIAN IN CLASSICAL MECHANICS
which are called the Hamilton-Jacobi canonical equations. As we said, (27) is a system
of 2rst-order dierential equations for 2unknown functions, the()and().
These equations determine the motion of the point in the phase space.
For an-particle system whose potential energy is(r), we have, according to
(13):
=
=1
p_r
=
=1
p_r
1
2
=1
_r
2
+(r) (28)
To express the Hamiltonian in terms of the variablesrandp, we use (19). This yields:
(rp) =
=1
p
2
2
+(r) (29)
Note that the Hamiltonian is thus equal to thetotal energyof the system. The canonical
equations:
dr
d
=
p
dp
d
=r (30)
are equivalent to Newton's equations, (3).
4. Applications of the Hamiltonian formalism
4-a. A particle in a central potential
Consider a system composed of a single particle of masswhose potential energy
()depends only on its distance from the origin. In polar coordinates( ), the
components of the particle's velocity on the local axes (Fig.) are:
= _
=
_
(31)
=sin_
so that the Lagrangian, (13), can be written:
( ; _
_
_) =
1
2
_
2
+
2_2
+
2
sin
2
_
2
() (32)
1533

APPENDIX IIIz
M
O
x
y
θ
φ
r
e
r
e
φ
e
θ
Figure 1: The unit vectorser,e,eof the
local axes associated with point, where
is dened by its spherical coordinates,,
.
The conjugate momenta of the three variablescan then be calculated:
=
_
=_ (33a)
=
_
=
2_
(33b)
=
_
=
2
sin
2
_ (33c)
To obtain the Hamiltonian of the particle, we use denition (21). This amounts to adding
()to the kinetic energy, expressed in terms ofand . We nd:
( ; ) =
2
2
+
1
2
2
2
+
2
sin
2
+() (34)
1534

LAGRANGIAN AND HAMILTONIAN IN CLASSICAL MECHANICS
The system of canonical equations [formulas (27)] can be written here:
d
d
== (35a)
d
d
==
2
(35b)
d
d
==
2
sin
2
(35c)
d
d
==
1
3
2
+
2
sin
2
(35d)
d
d
==
2
cos
2
sin
3
(35e)
d
d
== 0 (35f)
The rst three of these equations simply give (33); the last three are the real equations
of motion.
Now, consider the angular momentum of the particle with respect to the origin:
LLL=r v (36)
Its local components can easily be calculated from (31):
L= 0
L= =
2
sin_=
sin
L= =
2_
= (37)
so that:
LLL
2
=
2
+
2
sin
2
(38)
From the angular momentum theorem [formula (10)], we know thatLLLis a vector which
is constant over time, since the force derived from the potential()is central, that is,
collinear at each instant
3
with the vectorr.
By comparing (34) and (38), we see that the Hamiltoniandepends on the angular
variables and their conjugate momenta only through the intermediary ofLLL
2
:
( ; ) =
2
2
+
1
2
2
LLL
2
( ) +() (39)
Now, assume that the initial angular momentum of the particle isLLL0. Since the angular
momentum remains constant, the Hamiltonian (39) and the equation of motion (35d)
3
This conclusion can also be derived from (35e) and (35f) by calculating the time-derivatives of the
components ofLLLon the xed axes,,.
1535

APPENDIX III
are the same as they would be for a particle of mass, in a one-dimensional problem,
placed in the eective potential:
e() =() +
LLL
2
0
2
2
(40)
4-b. A charged particle placed in an electromagnetic eld
Now, consider a particle of massand chargeplaced in an electromagnetic eld
characterized by the electric eld vectorE(r)and the magnetic eld vectorB(r).
. Description of the electromagnetic eld. Gauges
E(r)andB(r)satisfy Maxwell's equations:
rE=
0
(41a)
rE=
B
(41b)
rB= 0 (41c)
rB=0j+00
E
(41d)
where(r)andj(r)are the volume charge density and the current density producing
the electromagnetic eld. The eldsEandBcan be described by a scalar potential
(r)and a vector potentialA(r), since equation (41c) implies that there exists a
vector eldA(r)such that:
B=rA(r) (42)
(41b) can thus be written:
r E+
A
=0 (43)
Consequently, there exists a scalar function(r)such that:
E+
A
=r(r) (44)
The set of the two potentialsA(r)and(r)constitutes what is called agaugefor
describing the electromagnetic eld. The electric and magnetic elds can be calculated
from theA gauge by:
B(r) =rA(r) (45a)
E(r) =r(r)
A(r) (45b)
A given electromagnetic eld, that is, a pair of eldsE(r)andB(r), can be
described by an innite number of gauges, which, for this reason, are said to be equivalent.
1536

LAGRANGIAN AND HAMILTONIAN IN CLASSICAL MECHANICS
If we know one gauge,A , which yields the eldsEandB, all the equivalent gauges,
A , can be found from thegauge transformation formulas:
A(r) =A(r) +r(r) (46a)
(r) =(r)
(r) (46b)
where(r)is any scalar function.
First of all, it is easy to show from (46) that:
rA(r) =rA(r)
r(r)
A(r) =r(r)A(r)
(47)
Any gauge,A , which satises (46) therefore yields the same electric and magnetic elds
asA .
Conversely we shall show that if two gauges,A andA , are equivalent, there
must exist a function(r)which establishes relations (46) between them. Since, by hypothesis:
B(r) =rA(r) =rA(r) (48)
we have:
r(AA) =0 (49)
This implies thatAAis the gradient of a scalar function:
AA=r(r) (50)
(r)is, for the moment, determined only to within an arbitrary function of,(). Further-
more, the fact that the two gauges are equivalent means that:
E(r) =r(r)
A(r) =r(r)A(r) (51)
that is:
r( ) +
(AA) =0 (52)
According to (50), we must have:
r( ) =r
(r) (53)
Consequently, the functions and
(r)can dier only by a function of; thus, we
can choose()so as to make them equal:
=
(r) (54)
This completes the determination of the function(r)(to within an additive constant). Two
equivalent gauges must therefore satisfy relations of the form (46).
1537

APPENDIX III
. Equations of motion and the Lagrangian
In the electromagnetic eld, the charged particle is subject to theLorentz force:
F=[E+vB] (55)
(wherevis the velocity of the particle at the time). Newton's law therefore gives the
equations of motion in the form:
r=[E(r) +_rB(r)] (56)
Projecting this equation ontoand using (45), we obtain:
=[+ _ _]
=
+ __ (57)
It can easily be shown that these equations can be derived from the Lagrangian
by using (15):
(r_r) =
1
2
_r
2
+_rA(r) (r) (58)
Therefore, although the Lorentz force is not derived from a potential energy, we can nd
a Lagragian for the problem.
Let us show that Lagrange's equations (15) do yield the equations of motion (56), using
the Lagrangian (58). To do so, we shall rst calculate:
_
=_+ (r)=_rA(r)(r) (59)
Lagrange's equation for the-coordinate can therefore be written:
d
d
[_+ (r)]_rA(r) +(r) = 0 (60)
Writing this equation explicitly and using (16), we again get (57):
+
+ _+ _+ __+ _+ _+= 0 (61)
that is:
=
+ __ (62)
1538

LAGRANGIAN AND HAMILTONIAN IN CLASSICAL MECHANICS
. Momentum. The classical Hamiltonian
The Lagrangian (58) enables us to calculate the conjugate momenta of the cartesian
coordinates,,of the particle. For example:
=
_
=_+ (r) (63)
Themomentum of the particle, which is, by denition, the vector whose components are
( ),is no longer equal, as it was in (19), to the mechanical momentum_r:
p=_r+A(r) (64)
Finally, we shall write the classical Hamiltonian:
(rp;) =p_r
=p
1
(pA)
1
2
(pA)
2
(pA)A+ (65)
that is:
(rp;) =
1
2
[pA(r)]
2
+(r) (66)
Comment:
Hamiltonian formalism therefore uses the potentialsAand, and not the elds
EandBdirectly. The result is that the description of the particle depends on
the gauge chosen. It is reasonable to expect, however, since the Lorentz force is
expressed in terms of the elds, that predictions concerning the physical behavior
of the particle must be the same for two equivalent gauges. The physical conse-
quences of the Hamiltonian formalism are said to begauge-invariant. The concept
of gauge invariance is analyzed in detail in ComplementIII.
5. The principle of least action
Classical mechanics can be based on a variational principle, the principle of least action. In
addition to its theoretical importance, the concept of action serves as the foundation of the
Lagrangian formulation of quantum mechanics(cf.ComplementIII). This is why we shall now
briey discuss the principle of least action and show how it leads to Lagrange's equations.
5-a. Geometrical representation of the motion of a system
First of all, consider a particle constrained to move along theaxis. Its motion can be
represented by tracing, in the()plane, the curve dened by the law of motion which yields
().
More generally, let us study a physical system described bygeneralized coordinates
(for an-particle system in three-dimensional space,= 3). It is convenient to interpret
theto be the coordinates of a pointin an-dimensional Euclidean space. There is
then a one-to-one correspondence between the positions of the system and the points of.
1539

APPENDIX III
With each motion of the system is associated a motion of pointin, characterized by the
-dimensional vector function()whose components are the(). As in the simple case of
a single particle moving in one dimension, the motion of point, that is, the motion of the
system, can be represented by the graph of(), which is a curve in an(+ 1)-dimensional
space-time (the time axis is added to thedimensions of). This curve characterizes the
motion being studied.
5-b. The principle of least action
The()can be xed arbitrarily; this gives pointand the system an arbitrary motion.
But their real behavior is dened by the initial conditions and the equations of motion. Suppose
that we know that, in the course of the real motion.is at1at time1and at2at a
subsequent time2(as is shown schematically by Figure):
(1) =1
(2) =2 (67)
There is an innite number ofa prioripossible motions which satisfy conditions (67). They
are represented by all the curves
4
, orpaths in space time, which connect the points(11)and
(22)(cf.Fig.).Q
Q
2
Q
1
t
1
t
2
t
Figure 2: The path in space-time which is associated with a given motion of the physical
system. The -axis represents the time and the -axis,(which symbolizes the set
of generalized coordinates).
Consider such a path in space-time, characterized by the vector function()which
satises (67
(12 ; _1_2 _;)(
_
;) (68)
4
Excluding, of course, the curves which go backward, that is, which would give two distinct
positions offor the same time.
1540

LAGRANGIAN AND HAMILTONIAN IN CLASSICAL MECHANICS
is the Lagrangian of the system, theactionwhich corresponds to the pathis, by denition:
=
2
1
d ()
_
(); (69)
[the function to be integrated depends only on; it is obtained by replacing theand_by the
time-dependent coordinates of()and
_
()in the Lagrangian (68)].
Theprinciple of least actioncan then be stated in the following way: of all the paths in
space-time connecting(11)with (22), the one which is actually followed (that is, the one
which characterizes the real motion of the system) is the one for which the action is minimal.
In other words, when we go from the path which is actually followed to one innitely close to
it, the action does not vary to rst order. Note the analogy with other variational principles,
such as Fermat's principle in optics.
5-c. Lagrange's equations as a consequence of the principle of least action
In conclusion, we shall show how Lagrange's equations can be deduced from the principle
of least action.
Suppose that the real motion of the system under study is characterized by thefunc-
tions of time(), that is by the path in space-timeconnecting the points(11)and(22).
Now consider an innitely close path,(g.), for which the generalized coordinates are equal
to:
() =() +() (70)
where the()are innitesimally small and satisfy conditions (), that is:
(1) =(2) = 0 (71)
The generalized velocities_()corresponding tocan be obtained by dierentiating relations
(70):
_() = _() +
d
d
() (72)
Thus, their increments_()are simply:
_() =
d
d
() (73)
We now calculate the variation of the action in going from the pathto the path:
=
2
1
d
=
2
1
d
+
_
_
=
2
1
d
+
_
d
d
(74)
1541

APPENDIX IIIQ
Q
2
Q
1
Γ
Γ
t
1
t
2 t
Figure 3: Two paths in space-time which pass through the points(11)and(22):
the solid-line curve is the path associated with the real motion of the system, and the
dashed-line curve is another, innitely close, path.
according to (73). If we integrate the second term by parts, we obtain:
=
_
2
1
+
2
1
d
d
d_
=
2
1
d
d
d_
(75)
since the integrated term is zero, because of conditions (71).
Ifis the path in space-time which is actually followed during the real motion of the
system, the incrementof the action is zero, according to the principle of least action. For
this to be so, it is necessary and sucient that:
d
d_
= 0 ;= 12 (76)
It is obvious that this condition is sucient. It is also necessary, since, if there existed a
time interval during which expression (76) were non-zero for a given valueof the index,
the()could be chosen so as to make the corresponding incrementdierent from zero.
(It would suce, for example, to choose them so as to make the product
d
d_
always positive or zero). Consequently, the principle of least action is equivalent to Lagrange's
equations.
1542

LAGRANGIAN AND HAMILTONIAN IN CLASSICAL MECHANICS
References and suggestions for further reading:
See section 6 of the bibliography, in particular Marion (6.4). Goldstein (6.6),
Landau and Lifshitz (6.7).
For a simple presentation of the use of variational principles in physics, see Feyn-
man II (7.2), Chap. 19.
For Lagrangian formalism applied to a classical eld, see Bogoliubov and Chirkov
(2.15), Chap. I.
1543

Bibliography
BIBLIOGRAPHY OF VOLUMES I and II
1. QUANTUM MECHANICS: GENERAL REFERENCES
A - INTRODUCTORY TEXTS
Quantum Physics
(1.1) Berkeley Physics Course, Vol. 4: Quantum Physics, McGraw-
Hill, New York (1971).
(1.2) The Feynman Lectures on
Physics, Vol. III: Quantum Mechanics, Addison-Wesley, Reading, Mass. (1965).
(1.3) Quantum Physics of Atoms, Molecules, Solids,
Nuclei and Particules, Wiley, New York (1974).
(1.4) Fundamental University Physics, Vol. III: Quantum
and Statistical Physics, Addison Wesley, Reading, Mass. (1968).
(1.5) Basic Physics of Atoms and Molecules, Wiley, New York
(1959).
(1.6) Quantum Theory of Matter, McGraw-Hill, New York (1968).
Quantum mechanics
(1.7) Fundamentals of Quantum Mechanics, Benjamin, New York (1967).
(1.8) Quantum Mechanics, Vol. I: Old Quantum Theory, North
Holland, Amsterdam (1962).
(1.9) Introduction to Quantum Mechanics, McGraw-
Hill, New York (1935).
(1.10) Cours de Mécanique Quantique, Dunod, Paris
(1969).
(1.11) Introduction to Quantum Mechanics, McGraw-Hill, New York
(1963).
(1.12) The Quantum Theory of Atoms, Molecules and Photons, McGraw-Hill,
London (1972).
1545
Quantum Mechanics, Volume II, Second Edition. C. Cohen-Tannoudji, B. Diu, and F. Laloë.
© 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

Bibliography
B - MORE ADVANCED TEXTS:
(1.13) The Principles of Quantum Mechanics, Oxford University Press
(1958).
(1.14) Introduction to Quantum Mechanics, Addison-
Wesley, Reading, Mass. (1966).
(1.15) Quantum Mechanics, D. Reidel, Dordrecht (1964).
(1.16) Quantum Mechanics,Wiley, New York (1970).
(1.17) Mécanique Quantique, Vols 1 et 2, Dunod, Paris (1964).
(1.18) Quantum Mechanics, McGraw-Hill, New York (1968).
(1.19) Quantum Mechanics, Nonrelativistic Theory,
Pergamon Press, Oxford (1965).
(1.20) Quantum Mechanics, Translated, edited and with additions by
D. Ter HAAR, Pergamon Press, Oxford (1965).
(1.21) Intermediate Quantum Mechanics, Benjamin,
New York (1968).
(1.22) Quantum Mechanics, North Holland, Amsterdam (1958).
C - PROBLEMS IN QUANTUM MECHANICS
(1.23)Selected Problems in Quantum Mechanics, Collected and edited by D. Ter HAAR,
Infosearch, London (1964).
(1.24) Practical Quantum Mechanics, I and II, Springer-Verlag, Berlin (1971).
D - ARTICLES
(1.25) Scientic American189, 52 (Sept. 1953).
(1.26) Scientic American198, 51 (Jan.
1958).
(1.27) Scientic American201, 74 (July 1959).
(1.28) Physics Today21, p.
51 (Aug. 1968).
(1.29)
Mechanics,Physics Today22, 23 (April 1969).
(1.30) Physics
Today25, 38 (March 1972).
(1.31) Physik. Z.18,121 (1917).
1546

Bibliography
(1.32)
Motion Pictures of One-Dimensional Quantum-Mechanical Transmission and Re-
ection Phenomena,Am. J. Phys.,35, 177 (1967).
(1.33)
Representation of the Schrödinger Equation for Solving Maser Problems,J. Appl.
Phys.28, 49 (1957).
(1.34) Am. J. Phys.27, 554
(195
2. QUANTUM MECHANICS: MORE SPECIALIZED REFERENCES
A - COLLISIONS
(2.1) Quantum Theory of Scattering, Prentice Hall, En-
glewood Clis (1962).
(2.2) Scattering Theory of Waves and Particles, McGraw-Hill, New
York (1966).
(2.3) Advanced Quantum Theory, Addison-Wesley, Reading, Mass. (1965).
(2.4) Collision Theory, Wiley, New York
(1964).
(2.5) The Theory of Atomic Collisions, Oxford
University Press (1965).
B - RELATIVISTIC QUANTUM MECHANICS
(2.6) Relativistic Quantum Mechanics, McGraw-
Hill, New York (1964).
(2.7) Advanced Quantum Mechanics, Addison-Wesley, Reading, Mass.
(1967).
(2.8) Relativistic
Quantum Theory, Pergamon Press, Oxford (1971).
C - FIELD THEORY. QUANTUM ELECTRODYNAMICS
(2.9) Introduction to Quantum Field Theory, Wiley Interscience, New York
(1959).
(2.10) Relativistic Quantum Fields, McGraw-Hill,
New York (1965).
(2.11) Introductory Quantum Electrodynamics,Longmans, London (1964).
(2.12) Quantum Electrodynamics, Benjamin, New York (1961).
1547

Bibliography
(2.13) The Quantum Theory of Radiation, Clarendon Press, Oxford (1954).
(2.14) Quantum Electrodynamics, Wiley
Interscience, New York (1965).
(2.15) Introduction to the Theory of Quan-
tized Fields, Interscience Publishers, New York (1959).
(2.16) An Introduction to Relativistic Quantum Field Theory, Harper
and Row, New York (1961).
(2.17)
ics,Am. J. Phys.40,1363 (1972).
D - ROTATIONS AND GROUP THEORY
(2.18) Group Theory, North Holland, Amsterdam
(1962).
(2.19) Elementary Theory of Angular Momentum, Wiley, New York (1957).
(2.20) Multipole Fields, Wiley, New York (1955).
(2.21) Angular Momentum in Quantum Mechanics, Princeton Univer-
sity Press (1957).
(2.22) Group Theory and Quantum Mechanics, McGraw-Hill, New York
(1964).
(2.23) Group Theory and its Application to the Quantum Mechanics of
Atomic Spectra, Academic Press, New York (1959).
(2.24) Am. J. Phys.36, 577
(1968).
E - MISCELLANEOUS
(2.25) Quantum Mechanics and Path Integrals,
McGraw-Hill, New York (1965).
(2.26) Elements of Advanced Quantum Theory, Cambridge University Press
(1969).
(2.27) Concepts in Quantum Mechanics, Academic Press, New York
(1965).
F - ARTICLES
(2.28) Scientic American196,45 (April
1957).
(2.29) Sci-
entic American209, 36 (Oct. 1963).
1548

Bibliography
(2.30) Scientic American213, 28
(Dec. 1965).
(2.31)
Operator Techniques,Rev. Mod. Phys.29,74 (1957).
(2.32) Rept. Progr.
Phys.24, 304 (1961).
(2.33)
auf Grund der Diracschen Lichttheorie,Z. Physik63,54 (1930).
(2.34)
between Atoms by Perturbation Theory,Proc. Roy. Soc.A 233, 70 (1955).
(2.35)
Disturbances,Proc. Roy. Soc.A 238, 269 (1957).
(2.36) Annals of
Physics(New York),6, 156 (1959).
(2.37)
velopments in Perturbation Theory, inAdvances in Quantum Chemistry, P. O.
LOWDIN ed., Vol. I, Academic Press, New York (1964).
(2.38)
Rev. Mod. Phys.,20, 367 (1948).
(2.39)
tering in Systems of Interacting Particles,Phys. Rev.95,249 (1954).
3. QUANTUM MECHANICS: FUNDAMENTAL EXPERIMENTS
Interference eects with weak light:
(3.1) Proc. Camb. Phil. Soc.
15,114 (1909).
(3.2)
Produced by Single Photons,Nuovo Cimento61 B,355 (1969).
Experimental verication of Einstein's law for the photoelectric eect; measure-
ments of
(3.3) Phil. Trans. Roy.
Soc.212, 205 (1912).
(3.4) Phys.
Rev.7355 (1916).
1549

Bibliography
The Franck-Hertz experiment:
(3.5)
den Molekullen des Quecksilberdampfes und die Ionisierungsspannung desselben,
Verhandlungen der Deutschen Physikalischen Gesellschaft,16, 457 (1914).
Über Kinetik von Elektronen und Ionen in Gasen,Physikalische Zeitschrift17,
409 (1916).
The proportionality between the magnetic moment and the angular momentum:
(3.6)
Molekularströme,Verhandlungen der Deutschen Physikalischen Gesellschaft17,
152 (1915).
(3.7)
Annalen der Physik(Leipzig)60, 109 (1919).
The Stern-Gerlach experiment:
(3.8)
telung im Magnetfeld,Zeitschrift für Physik9, 349 (1922).
The Compton eect:
(3.9)
Elements,Phys. Rev.,21, 483 (1923).
Wavelength Measurements of Scattered X-Rays,Phys. Rev.,21, 715 (1923).
Electron diration:
(3.10)
Nickel,Phys. Rev.30, 705 (1927).
The Lamb shift:
(3.11)
Atom,
I -Phys. Rev.79, 549 (1950),
II -Phys. Rev.81, 222 (1951).
Hyperne structure of the hydrogen atom:
(3.12)
of Ground State Atomic Hydrogen,Phys. Rev. Letters11, 338 (1963).
1550

Bibliography
Some fundamental experiments are described in:
(3.13) Scientic American212, 58 (May 1965).
4. QUANTUM MECHANICS: HISTORY
(4.1) Annales de Physique(Paris),
3, 22 (1925).
(4.2)
Essays 1958-1962 on Atomic Physics and Human Knowledge, Vintage, New York
(1966).
(4.3) Physics and Beyond: Encounters and Conversations, Harper
and Row, New York (1971).
La Partie et le Tout, Albin Michel, Paris (1972).
(4.4)Niels Bohr, His life and work as seen by his friends and colleagues, S. ROZENTAL,
ed., North Holland, Amsterdam (1967).
(4.5) Correspondance 1916-1955, Editions du Seuil,
Paris (1972). See alsoLa Recherche,3, 137 (fev. 1972).
(4.6)Theoretical Physics in the Twentieth Century,M. FIERZ and V. F. WEISSKOPF
eds., Wiley Interscience, New York (1960).
(4.7)Sources of Quantum Mechanics, B. L. VAN DER WAERDEN ed., North Holland,
Amsterdam (1967); Dover, New York (1968).
(4.8) The Conceptual Development of Quantum Mechanics, McGraw-
Hill, New York (1966). This book traces the historical development of quantum
mechanics. Its very numerous footnotes provide a multitude of references. See also
(5.13).
ARTICLES
(4.9) Scientic American186,47 (March
1952).
(4.10) Physics Today
19,23 (Nov. 1966).
(4.11) Physics Today27, 38 (Feb.
1974).
Reference (5.12) contains a large number of references to the original articles.
1551

Bibliography
5. QUANTUM MECHANICS: DISCUSSION OF ITS FOUNDATIONS
A - GENERAL PROBLEMS
(5.1) Quantum Theory, Constable, London (1954).
(5.2) Foundations of Quantum Mechanics, Addison-Wesley, Reading,
Mass. (1968).
(5.3) Conceptual Foundations of Quantum Mechanics, Benjamin,
New York (1971);Conceptions de la Physique Contemporaine. Les Interprétations
de la Mécanique Quantique et de la Mesure, Hermann, Paris (1965).
(5.4)
IL;Foundations of Quantum Mechanics, B. D'ESPAGNAT ed., Academic Press,
New York (1971).
(5.5) Physics Today23, 30, (Sept.
1970).
(5.6) Physics Today24, 36 (April 1971). See also (1.28).
(5.7) Do we really understand quantum mechanics?, Cambridge University
Press, (second edition 2019).
See also (1.28).
B - MISCELLANEOUS INTERPRETATIONS
(5.8)
inA. Einstein: Philosopher-Scientist, P. A. SCHILPP ed., Harper and Row, New
York (1959).
(5.9) Natural Philosophy of Cause and Chance, Oxford University Press,
London (1951); Clarendon Press, Oxford (1949).
(5.10) Une Tentative d'Interprétation Causale et Non Linéaire de la
Mécanique Ondulatoire: la Théorie de la Double Solution, Gauthier-Villars, Paris
(1956);Etude Critique des Bases de l'Interprétation Actuelle de la Mécanique On-
dulatoire, Gauthier-Villars, Paris (1963).
(5.11)The Many-Worlds Interpretation of Quantum Mechanics, B. S. DEWITT and N.
GRAHAM eds., Princeton University Press (1973).
A very complete set of references with comments may be found in:
(5.12)
of Quantum Mechanics,Am. J. Phys.39,724 (1971).
(5.13) The Philosophy of Quantum Mechanics, Wiley-interscience, New
York (1974). General presentation of the dierent interpretations and formalisms
of quantum mechanics. Contains many references.
1552

Bibliography
C - MEASUREMENT THEORY
(5.14) Quantum Mechanics, Vol. I, Benjamin, New York (1966).
(5.15) Principes Essentiels de la Mécanique Quantique, Dunod,
Paris (1968).
(5.16) Am. J. Phys.,31, 755
(1963). See also (5.13), Chap. 11.
D - HIDDEN VARIABLES AND PARADOXES
(5.17)
scription of Physical Reality Be Considered Complete?,Phys. Rev.47, 777 (1935).
N. BOHR, Can Quantum Mechanical Description of Physical Reality Be Consid-
ered Complete?,Phys. Rev.48, 696 (1935).
(5.18)Paradigms and Paradoxes, the Philosophical Challenge of the Quantum Domain,
R. G. COLODNY ed., University of Pittsburgh Press (1972).
(5.19) Rev.
Mod. Phys.38, 447 (1966).
See also Ref. (4.8), as well as (5.12) and Chap. 7 of (5.13).
6. CLASSICAL MECHANICS
A - INTRODUCTORY LEVEL
(6.1) Fundamental University Physics, Vol. I: Mechanics,
Addison-Wesley, Reading, Mass. (1967).
(6.2) Berkeley Physics Course,
Vol. 1: Mechanics, McGraw-Hill, New York (1962).
(6.3) The Feynman Lectures on
Physics, Vol. I: Mechanics, Radiation, and Heat, Addison-Wesley, Reading, Mass.
(1966).
(6.4) Classical Dynamics of Particles and Systems, Academic Press,
New York (1965).
B - MORE ADVANCED LEVEL
(6.5) Lectures on Theoretical Physics, Vol. I: Mechanics, Academic
Press, New York (1964).
(6.6) Classical Mechanics, Addison-Wesley, Reading, Mass. (1959).
1553

Bibliography
(6.7) Mechanics, Pergamon Press, Oxford (1960);
Mécanique, 3
e
éd., Ed. Mir, Moscou (1969).
7. ELECTROMAGNETISM AND OPTICS
A - INTRODUCTORY LEVEL
(7.1) Berkeley Physics Course, Vol. 2: Electricity and Magnetism,
McGraw-Hill, New York (1965).
F. S. CRAWFORD JR.,Berkeley Physics Course, Vol. 3: Waves, McGraw-Hill,
New York (1968).
(7.2) The Feynman Lectures on
Physics, Vol. II: Electromagnetism and Matter, Addison-Wesley, Reading, Mass.
(1966).
(7.3) Fundamental University Physics, Vol. II: Fields and
Waves, Addison-Wesley, Reading, Mass. (1967).
(7.4) Optics, Addison-Wesley, Reading, Mass. (1974).
B - MORE ADVANCED LEVEL
(7.5) Classical Electrodynamics, 2
e
edition Wiley, New York (1975).
(7.6) Classical Electricity and Magnetism,
Addison-Wesley, Reading, Mass. (1964).
(7.7) Electromagnetic Theory, McGraw-Hill, New York (1941).
(7.8) Principles of Optics, Pergamon Press, London (1964).
(7.9) Lectures on Theoretical Physics, Vol. IV: Optics, Academic
Press, New York (1964).
(7.10) Optique, 5
e
completed by A. KASTLER, Masson, Paris (1954).
(7.11) The Classical Theory of Fields, Addison-Wesley,
Reading, Mass. (1951); Pergamon Press, London (1951);Théorie du champ, 2
e
éd.,
Ed. Mir, Moscou (1966).
(7.12) Electrodynamics of Continuous Media, Perg-
amon Press, Oxford (1960).
(7.13) Wave Propagation and Group Velocity, Academic Press, New York
(1960).
1554

Bibliography
8. THERMODYNAMICS. STATISTICAL MECHANICS
A - INTRODUCTORY LEVEL
(8.1) Berkeley Physics Course, Vol. 5: Statistical Physics, McGraw-Hill, New
York (1967).
(8.2) Thermal Physics, Wiley, New York (1969).
(8.3) Thermodynamique, 5édition remaniée par A. KASTLER, Masson,
Paris (1962). See also (1.4), part 2, and (6.3).
B - MORE ADVANCED LEVEL
(8.4) Fundamentals of Statistical and Thermal Physics, McGraw-Hill, New
York (1965).
(8.5) Thermodynamique Statistique, Masson, Paris (1970).
(8.6) Thermal Physics, Benjamin, New York (1964).
(8.7) Statistical Mechanics, North Holland, Amsterdam and Wiley, New York
(1965).
(8.8) Course of Theoretical Physics, Vol. 5:
Statistical Physics, Pergamon Press, London (1963).
(8.9) Thermodynamics, Wiley, New York (1961).
(8.10) The Elements of Classical Thermodynamics, Cambridge Univer-
sity Press (1957).
(8.11) The Principles of Statistical Mechanics, Oxford University Press
(1950).
9. RELATIVITY
A - INTRODUCTORY LEVEL
(9.1) Introduction to Special Relativity, Benjamin, New York (1965). See
also references (6.2) and (6.3).
B - MORE ADVANCED LEVEL
(9.2) Relativity: The Special Theory, North Holland, Amsterdam (1965).
(9.3) Relativistic Mechanics, Benjamin, New York (1970).
(9.4) The Special Theory of Relativity, Oxford University Press, London
(1959).
1555

Bibliography
(9.5) The Theory of Relativity, Oxford University Press, London (1972).
(9.6) Introduction to the Theory of Relativity, Prentice Hall, Engle-
wood Clis (1960).
(9.7) Gravitation, Freeman, San
Francisco (1973).
See also references on Electromagnetism, in particular (7.5) and (7.11).
Other useful references are :
(9.8) Quatre Conférences sur la Théorie de la Relativité, Gauthier-Villars,
Paris (1971).
(9.9) La Théorie de la Relativité Restreinte et Générale. La Relativité et
le Problème de l'Espace, Gauthier-Villars, Paris (1971).
(9.10) The Meaning of Relativity, Methuen, London (1950).
(9.11) Relativity, the Special and General Theory, a Popular Exposition,
Methuen, London (1920); H. Holt, New York (1967).
A much more complete set of references can be found in:
(9.12) Am. J. Phys.
30,462 (1962).
10. MATHEMATICAL METHODS
A - ELEMENTARY GENERAL TEXTS
(10.1) Cours de Mathématiques, Vols. I, II and III, Masson, Paris (1961).
(10.2) Compléments de Mathématiques, Revue d'Optique, Paris (1961).
(10.3) Mathematics for Scientists, Benjamin, New
York (1966).
(10.4) Mathematical Methods for Physicists, Academic Press, New York
(1966).
(10.5) Mathematics for Quantum Mechanics, Benjamin, New York
(1962).
B - MORE ADVANCED GENERAL TEXTS
(10.6) Mathematical Methods of Physics, Benjamin,
New York (1970).
(10.7) Méthodes Mathématiques pour les Sciences Physiques, Hermann,
Paris (1965).Mathematics for the Physical Sciences,Hermann, Paris (1968).
1556

Bibliography
(10.8) Mathematical Physics,Addison-Wesley, Reading, Mass (1968).
(10.9) Théorie Elémentaire des Fonctions Analytiques d'une ou Plusieurs
Variables Complexes, Hermann, Paris (1961).Elementary Theory of Analytic
Functions of One or Several Complex Variables, Addison-Wesley, Reading, Mass.
(1966).
(10.10) Mathematical Foundations of Quantum Mechanics,Princeton
University Press (1955).
(10.11) Methods of Mathematical Physics, Vols. I and
II, Wiley, Interscience, New York (1966).
(10.12) A Course of Modern Analysis, Cam-
bridge University Press (1965).
(10.13) Methods of Theoretical Physics, McGraw-Hill,
New York (1953).
C - LINEAR ALGEBRA. HILBERT SPACES
(10.14) Determinants and Matrices, Oliver and Boyd, Edinburgh (1956).
(10.15) Matrix Algebra for Physicists, Plenum Press, New York
(1966).
(10.16) Methods of Matrix Algebra, Academic Press, New York (1965).
(10.17) Linear Operators in Hilbert Space, Gordon and Breach, New York
(1967).
(10.18) Linear Operators in Hilbert Space, Academic Press, New York
(1965).
(10.19) Theory of Linear Operators in Hilbert
Space, Ungar, New York (1961).
D - FOURIER TRANSFORMS. DISTRIBUTIONS
(10.20) Introduction to Fourier Analysis, Chapman and Hall, London (1969).
(10.21) Introduction to Fourier Analysis and Generalized Functions,
Cambridge University Press (1964).
(10.22) Théorie des Distributions, Hermann, Paris (1967).
(10.23) Generalized Functions, Academic Press, New
York (1964).
(10.24) Tabellen zur Fourier Transformation, Springer-Verlag,
Berlin (1957).
1557

Bibliography
E - PROBABILITY AND STATISTICS
(10.25)Elements of Probability Theory, Academic Press, New York (1966).
(10.26) Introduction to Probability Theory,
Houghton- Miin, Boston (1971).
(10.27) An Introduction to Probability and Mathematical Statistics, Aca-
demic Press, New York (1965).
(10.28) Probability, Benjamin, New York (1966).
(10.29) An Introduction to Probability Theory and its Applications, Wiley,
New York (1968).
(10.30) Probability, Addison-Wesley, Reading, Mass. (1968).
F - GROUP THEORY
Applied to physics:
(10.31) Lectures on Group Theory, Gordon and Breach, New York (1967).
(10.32) Group Theory and its Application to Physical Problems, Addison-
Wesley, Reading, Mass. (1962).
See also (2.18), (2.22), (2.23) or reference (16.13), which provides a simple intro-
duction to continuous groups in physics.
More mathematical:
(10.33) Groupes,Presses Universitaires de Bruxelles, Bruxelles (1961);Groups,
Macmillan, New York (1964).
(10.34) The Theory of Groups, Chelsea, New York (1960).
(10.35) Topological Groups, Gordon and Breach, New York (1966).
G - SPECIAL FUNCTIONS AND TABLES
(10.36) A Treatise on Bessel Functions and their Appli-
cations to Physics, Dover, New York (1966).
(10.37) Special Functions, Macmillan, New York (1965).
(10.38) Formulas and Theorems
for the Special Functions of Mathematical Physics, Springer-Verlag, Berlin (1966).
(10.39) Higher Transcendental Functions, Vols.
I, II and III, A. ERDELYI ed., McGraw-Hill, New York (1953).
(10.40) Handbook of Mathematical Functions,
Dover, New York (1965).
1558

Bibliography
(10.41) Chambers's Shorter Six-Figure Mathematical Tables, Chambers,
London (1966).
(10.42) Tables of Functions,Dover, New York (1945).
(10.43) Tables of Laguerre
Polynomials and Functions, Pergamon Press, Oxford (1966).
(10.44) Tables of Integrals and Other Mathematical Data, Macmillan, New
York (1965).
(10.45) Nouvelles Tables d'Intégrales Dénies, Hafner, New York
(1957).
(10.46) Tables of Laplace Transforms, Springer-
Verlag, Berlin (1973).
(10.47) Tables of Integral Transforms, Vols.I
and II, A. ERDELYI ed., McGraw-Hill, New York (1954).
(10.48) The
3-j and 6-j symbols, M.I.T. Technology Press (1959); Crosby Lockwood and Sons,
London.
11. ATOMIC PHYSICS
A - INTRODUCTORY LEVEL
(11.1) Atomic Spectra, Longman, London (1969).
(11.2) Modern Atomic Physics, Vol. 1 :
Fundamental Principles,and 2 :Quantum Theory and its Application, Macmillan,
London (1975).
(11.3) Resonance Radiation and Excited
Atoms, Cambridge University Press, London (1961).
(11.4) Atomic Physics, Blackie and Son, London (1951).
(11.5) Introduction to Atomic Spectra, McGraw-Hill, New York (1934).
(11.6) La Structure des Atomes et des Molécules, Masson, Paris
(1964). See also (1.3) and (12.1).
B - MORE ADVANCED LEVEL
(11.7) The Spectrum of Atomic Hydrogen, Oxford University Press, Lon-
don (1957).
(11.8) Quantum Theory of Atomic Structure, Vols. I and II, McGraw-Hill,
New York (1960).
1559

Bibliography
(11.9) Atoms, Molecules and Quanta, Vols. I and II,
Dover, New York (1964).
(11.10)Handbuch der Physik, Vols. XXXV and XXXVI, Atoms, S. FLÜGGE ed., Springer-
Verlag Berlin (1956 and 1957).
(11.11) Molecular Beams, Oxford University Press, London (1956).
(11.12) Introduction to the Theory of Atomic Spectra, Pergamon Press,
Oxford (1972).
(11.13) The Theory of Atomic Spectra, Cambridge
University Press (1953).
C - ARTICLES
Many references to articles and books, with comments, can be found in:
(11.14) Am.
J. Phys.32, 721 (1964).
See also (3.13).
(11.15) Scientic American,219,
60 (Sept. 1968).
(11.16) Scientic American218, 72 (Jan.
1968).
(11.17) Scientic American208, 94 (June
1963).
(11.18)
servation of the Phase Shift of a Neutron due to Precession in a Magnetic Field,
Phys. Rev. Letters35, 1053 (1975).
See also: H. RAUCH, A. ZEILINGER, G. BADUREK A. WILFING, W. BAUPIESS
and U. BONSE,Physics Letters54 A, 425 (1975).
D - EXOTIC ATOMS
(11.19) Scientic Amer-
ican191, 88 (Dec. 1954).
(11.20) Scientic American214, 93, (April 1966).
Muonium,Physics Today20, 29 (Dec. 1967).
(11.21) Scientic American195, 93 (Oct. 1956).
(11.22) Scientic American227, 102 (Nov. 1972).
(11.23) Atomic Physics, B.
Bederson, V. W. Cohen and F. M. Pichanick eds., Plenum Press, New York (1969).
1560

Bibliography
(11.24)
SON and V. L. TELEGDI, Measurement of the Muonium Hfs Splitting and of the
Muon Moment by Double Resonance, and New Value of,Phys. Rev. Letters
25, 1779 (1970).
(11.25)
Lyman-Radiation,Phys. Rev. Letters34, 177 (1975). Fine-Structure Measure-
ment in the First Excited State of PositroniumPhys. Rev. Letters34, 1541
(1975).
(11.26) Phys.
Rev. Letters84, 1136 (2000).
12. MOLECULAR PHYSICS
A - INTRODUCTORY LEVEL
(12.1) Atoms and Molecules, Benjamin, New York
(1970).
(12.2) The Nature of the Chemical Bond, Cornell University Press (1948).
See also (1.3), Chap. 12; (1.5) and (11.6).
B - MORE ADVANCED LEVEL
(12.3) Quantum Chemistry, Allyn and Bacon, Boston (1970).
(12.4) Molecular Spectra and Molecular Structure, Vol. I:Spectra of Di-
atomic Moleculesand Vol. II:Infrared and Raman Spectra of Polyatomic Molecules,
D. Van Nostrand Company, Princeton (1963 and 1964).
(12.5) Quantum Chemistry, Wiley, New
York (1963).
(12.6) Valence, Oxford at the Clarendon Press (1952).
(12.7) Quantum Theory of Molecules and Solids,Vol. 1 :Electronic
Structure of Molecules,McGraw-Hill, New York (1963).
(12.8)Handbuch der Physik, Vol. XXXVII, 1 and 2, Molecules, S. FLÜGGE, ed., Springer
Verlag, Berlin (1961).
(12.9) Theory of Van der Waals Attraction, Springer Tracts in Modern
Physics, Vol. 72, Springer Verlag, Berlin (1974).
(12.10) Microwave Spectroscopy, McGraw-Hill,
New York (1955).
1561

Bibliography
(12.11) Les Molécules Interstellaires, Delachaux et Niestlé, Neuchâtel
(1974).
See also (11.9), (11.11) and (11.14).
C - ARTICLES
(12.12) Scientic American203, 47
(July 1960).
(12.13) Scientic American222, 54 (April 1970).
(12.14) Scientic American228,51 (March 1973).
(12.15) Physics Today26, 32 (March 1973).
See also (16.25).
13. SOLID STATE PHYSICS
A - INTRODUCTORY LEVEL
(13.1) Elementary Solid State Physics, Wiley, New York (1962).
(13.2) Introduction to Solid State Physics, 3
rd
ed., Wiley, New York (1966).
(13.3) Principles of the Theory of Solids, Cambridge University Press,
London (1972).
(13.4) Modern Theory of Solids, McGraw-Hill, New York (1940).
B - MORE ADVANCED LEVEL
General texts:
(13.5) Quantum Theory of Solids, Wiley, New York (1963).
(13.6) Quantum Theory of Solids, Oxford University Press, London
(1964).
(13.7) The Theory of the Properties of Metals and Alloys,
Clarendon Press, Oxford (1936); Dover, New York (1958).
More specialized texts:
(13.8) Dynamical Theory of Crystal Lattices, Oxford Univer-
sity Press, London (1954).
(13.9) Electrons and Phonons, Oxford University Press, London (1960).
(13.10) The Theory of Brillouin Zones and Electronic States in Crystals, North
Holland, Amsterdam (1962).
1562

Bibliography
(13.11) Energy Band Theory, Academic Press, New York (1964).
(13.12) Wave Mechanics of Crystalline Solids, Chapman and Hall, London
(1967).
(13.13) The Theory of Quantum Liquids, Benjamin, New
York (1966).
(13.14) Semi-Conductors, Associated Book Publishers, London (1966).
(13.15) Semi-Conductors, Cambridge University Press, London (1964).
C - ARTICLES
(13.16) Scientic American207,
92 (Dec. 1962).
(13.17) Scientic American209,
110 (July 1963).
(13.18)
Eects,Scientic American214, 30 (May 1966).
(13.19) Scientic American215, 64 (Oct. 1966).
(13.20) Scientic American217, 85
(Aug. 1967).
(13.21) Scientic American217, 80 (Sept. 1967).
(13.22)
in Metals,Scientic American228, 88 (Jan. 1973).
(13.23) Physics Today22, 23 (Oct. 1969).
14. MAGNETIC RESONANCE
(14.1) The Principles of Nuclear Magnetism, Clarendon Press, Oxford
(1961);
(14.2) Principles of Magnetic Resonance, Harper and Row, New York
(1963).
(14.3) Paramagnetic Resonance, Benjamin, New York (1962).
See also Ramsey (11.11), Chaps. V, VI and VII.
ARTICLES
(14.4)
II,Am. J. Phys.18, 438 and 473 (1950).
1563

Bibliography
(14.5) Am. J. Phys.22, 1 (1954).
(14.6) Scientic American199, 58 (Aug. 1958).
(14.7) Physics
Today23, 43 (April 1970).
(14.8) Phys. Rev.70, 460 (1946).
Numerous other references, in particular to the original articles, can be found in:
(14.9)
and Electron Paramagnetic Resonance,Am. J. Phys.33, 71 (1965).
15. QUANTUM OPTICS ; MASERS AND LASERS
A - OPTICAL PUMPING. MASERS AND LASERS
(15.1) Optical Pumping: An Introduction, Benjamin, New York (1965).
Contient de nombreuses références. De plus, plusieurs articles originaux y sont re-
produits.
(15.2)Quantum Optics and Electronics, Les Houches Lectures 1964, C. DE WITT, A.
BLANDIN and C. COHEN-TANNOUDJI eds., Gordon and Breach, New York
(1965).
(15.3)Quantum Optics, Proceedings of the Scottish Universities Summer School 1969, S.
M. KAY and A. MAITLAND eds., Academic Press, London (1970).
The proceedings of these two summer schools contain several lectures related to
optical pumping and quantum electronicss.
(15.4) Quantum Mechanical Ampliers, inLectures in Theoretical
Physics, Vol. II, W. BRITTIN and D. DOWNS eds., Interscience Publishers, New
York (1960).
(15.5) Laser Physics, Addison-
Wesley, New York (1974).
(15.6) An Introduction to Lasers and Masers, McGraw-Hill, New York
(1971).
(15.7) Essentials of Lasers, Pergamon Press, Oxford (1969). This short book
contains the reprints of several original articles related to lasers.
(15.8) Optical Resonance and Two-Level Atoms, Wiley
Interscience, New York (1975).
(15.9) Quantum Electronics, Wiley, New York (1967).
1564

Bibliography
(15.10) Introduction to Quantum Optics, Gordon and Breach, Lon-
don (1973).
B - ARTICLES
Two Resource Letters give and comment a large number of references :
(15.11)
and on Optical Pumping,Am. J. Phys.32, 589 (1964).
(15.12)
of Light,Am. J. Phys.,31, 321 (1963).
Reprints of many important papers on lasers have been collected in:
(15.13)Laser Theory, F. S. BARNES ed., I.E.E.E. Press, New York (1972).
(15.14) Scientic American196,71 (Feb. 1957).
(15.15) Scientic American199,42 (Dec. 1958).
(15.16) Scientic American203, 72 (Oct. 1960).
(15.17) Scientic American204,52 (June 1961).
Advances in Optical Masers,Scientic American209,34 (July 1963). Laser
Light,Scientic. American.,219,120 (Sept. 1968).
(15.18) Scientic American
229, 69 (Dec. 1973).
C - NON-LINEAR OPTICS
(15.19) An Introduction to Non-Linear Optics, Plenum Press, New York
(1969).
(15.20) Applied Non-Linear Optics, Wiley Inter-
science, New York (1973).
(15.21) Non-Linear Optics, Benjamin, New York (1965).
See also the lectures of this author in references (15.2) and (15.3).
D - ARTICLES
(15.22) Scientic American
210, 38 (Apr. 1964).
Non-Linear Optics,Physics Today22, 39 (Jan. 1969).
1565

Bibliography
16. NUCLEAR PHYSICS AND PARTICLE PHYSICS
A - INTRODUCTION TO NUCLEAR PHYSICS
(16.1) Physique Subatomique: Noyaux et Particules, Hermann, Paris
(1975).
(16.2) Introductory Nuclear Physics, Wiley, New York (1960).
(16.3) The Atomic Nucleus, McGraw-Hill, New York (1955).
(16.4) Physics of the Nucleus, Addison-Wesley, Reading, Mass. (1962).
(16.5) Nuclei and Particles, Benjamin, New York (1965).
B - MORE ADVANCED NUCLEAR PHYSICS TEXTS
(16.6) Theoretical Nuclear Physics, Vol. 1: Nuclear
Structure, Wiley, New York (1974).
(16.7) Theoretical Nuclear Physics, Wiley, New
York (1963).
(16.8) Shell Theory of the Nucleus, Princeton University Press (1955).
(16.9) Nuclear Structure, Benjamin, New York
(1969).
C - INTRODUCTION TO PARTICLE PHYSICS
(16.10) Elementary Particles, Van Nostrand,
Princeton (1964).
(16.11) The Fundamental Particles, Addison-Wesley, Reading, Mass. (1965).
(16.12) Theory of Fundamental Processes, Benjamin, New York (1962).
(16.13) Introduction à l'Etude des Particules Elémentaires, Ediscience, Paris
(1970).
(16.14) Fundamental Particles, Benjamin, New York (1964).
D - MORE ADVANCED PARTICLE PHYSICS TEXTS
(16.15) Qu'est-ce qu'une Particule Elémentaire?Masson, Paris (1965).
(16.16) Invariance Principles and Elementary Particles, Princeton Uni-
versity Press (1964).
(16.17) Elementary Particle Physics, Addison-Wesley, Reading, Mass. (1964).
(16.18) Elementary Particle Theory, North Hol-
land, Amsterdam (1970).
1566

Bibliography
(16.19) Muons, North Holland, Amsterdam (1967).
E - ARTICLES
(16.20) Scientic American184, 22
(March 1951).
(16.21) Scientic American200, 75 (Jan. 1959).
(16.22) Physics
Today,26, 34 (June 1973).
(16.23) Scientic American195, 93 (Oct. 1956).
(16.24) Scientic American202, 72 (April
1960).
(16.25) Scientic American225, 86 (Oct.
1971).
(16.26) Scientic American205, 46 (July 1961).
(16.27) Scientic American202, 98 (March 1960).
(16.28) Scientic
American197, 72 (July 1957).
(16.29)
Particles,Scientic American210, 74 (Feb. 1964).
(16.30) Scientic American218, 15
(May 1968).
(16.31) Scientic American229, 36
(Nov. 1973).
(16.32) Scientic
American231, 50 (July 1974).
(16.33) Scientic
American232,50 (June 1975).
(16.34) Physics Today22, 47 (Jan.
1969).
(16.35) Physics Reports(Amsterdam),
9C, 1 (1973).
1567

Index[The notation (ex.) refers to an exercise]
Absorption
and emission of photons,
collision with,
of a quantum, a photon, ,
of eld,
of several photons,
rates,
Acceptor (electron acceptor),
Acetylene (molecule),
Action,, ,
Addition
of angular momenta, ,
of spherical harmonics,
of two spins 1/2,
Adiabatic
branching of the potential,
Adjoint
matrix,
operator,
Algebra (commutators),
Allowed energy band,, ,
Ammonia (molecule),,
Amplitude
scattering amplitude,,
Angle (quantum),
Angular momentum
addition of momenta, ,
and rotations,
classical,
commutation relations,,
conservation,,,
coupling,
electromagnetic eld, ,
half-integral,
of identical particles, (ex.)
of photons,
orbital,,,
quantization,
quantum,
spin,,
standard representation,,
two coupled momenta,
Anharmonic oscillator,,
Annihilation operator,,,,
Annihilation-creation (pair), ,
Anomalous
average value, ,
dispersion,
Zeeman eect,
Anti-normal correlation function, ,
1789
Anti-resonant term,
Anti-Stokes (Raman line),,
Antibunching (photon),
Anticommutation,
eld operator,
Anticrossing of levels,,
Antisymmetric ket, state, ,
Antisymmetrizer, ,
Applications of the perturbation theory,
1231
Approximation
central eld approximation,
secular approximation,
Argument (EPR),
Atom(s),seehelium, hydrogenoid
donor,
dressed, ,
many-electron atoms, ,
mirrors for atoms,
muonic atom,
single atom uorescence,
Atomic
beam (deceleration),
orbital,, (ex.)
parameters,
Attractive bosons,
Autler-Townes
doublet,
eect,
Autoionization,
Average value (anomalous),
Azimuthal
quantum number,
Band (energy),
Bardeen-Cooper-Schrieer,
Barrier (potential barrier),,,
Basis
1569
Quantum Mechanics, Volume II, Second Edition. C. Cohen-Tannoudji, B. Diu, and F. Laloë.
© 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

INDEX [The notation (ex.) refers to an exercise]
change of bases,
characteristic relations,,
continuous basis in the space of states,
99
mixed basis in the space of states,
BCHSH inequalities, ,
BCS,
broken pairs and excited pairs,
coherent length,
distribution functions,
elementary excitations,
excited states,
gap, , ,
pairs (wave function of),
phase locking, , ,
physical mechanism,
two-particle distribution,
Bell's
inequality,
theorem, ,
Benzene (molecule),,
Bessel
Bessel-Parseval relation,
spherical Bessel function,
spherical equation,
spherical function,
Biorthonormal decomposition,
Bitter,
Blackbody radiation,
Bloch
equations,, ,
theorem,
Bogolubov
excitations,
Hamiltonian,
operator method,
phonons, spectrum,
transformation,
Bogolubov-Valatin transformation, ,
1919
Bohr,
electronic magneton,
frequencies,
magneton,seefront cover pages
model,,
nuclear magneton,
radius,
Boltzmann
constant,seefront cover pages
distribution,
Born
approximation,,,
Born-Oppenheimer approximation,,
1177,
Born-von Karman conditions,
Bose-Einstein
condensation, , ,
condensation (repulsive bosons),
condensation of pairs,
distribution,,
statistics,
Bosons,
at non-zero temperature,
attractive,
attractive instability,
condensed,
in a Fock state,
paired,
Boundary conditions (periodic),
Bra,,,
Bragg reection,
Brillouin
formula,
zone,
Broadband
detector,
optical excitation,
Broadening (radiative),
Broken pairs and excited pairs (BCS),
1920
Brossel,
Bunching of bosons,
C.S.C.O.,,,,
Canonical
commutation relations,,,
ensemble,
Hamilton-Jacobi canonical equations,
214
Hamilton-Jacobi equations,
Cauchy principal part,
Center of mass,,
Center of mass frame,
Central
1570

INDEX [The notation (ex.) refers to an exercise]
eld approximation,
potential,
Central potential,,
scattering,
stationary states,
Centrifugal potential,,,
Chain (von Neumann),
Chain of coupled harmonic oscillators,
Change
of bases,,,
of representation,
Characteristic equation,
Characteristic relation of an orthonormal
basis,
Charged harmonic oscillator in an elec-
tric eld,
Charged particle
in an electromagnetic eld,
Charged particle in a magnetic eld,,
321,
Chemical bond,,, ,
Chemical potential, ,
Circular quanta,,
Classical
electrodynamics,
histories,
Clebsch-Gordan coecients, ,
Closure relation,,
Coecients
Clebsch-Gordan,
Einstein, ,
Coherences (of the density matrix),
Coherent length (BCS),
Coherent state (eld),
Coherent superposition of states,,,
307
Collision,
between identical particles, , (ex.)
between identical particles in classi-
cal mechanics,
between two identical particles,
cross section,
scattering states,
total scattering cross section,
with absorption,
Combination
of atomic orbitals,
Commutation,
canonical relations,,
eld operator,
of pair eld operators,
relations,
Commutation relations
angular momentum,,
eld, ,
Commutator algebra,
Commutator(s),,,,
of functions of operators,
Compatibility of observables,
Complementarity,
Complete set of commuting observables
(C.S.C.O.),,,
Complex variables (Lagrangian),
Compton wavelength of the electron,,
1235
Condensates
relative phase,
with spins,
Condensation
BCS condensation energy,
Bose-Einstein, , ,
Condensed bosons,
Conduction band,
Conductivity (solid),
Congurations,
Conjugate momentum, , , ,
1983, ,
Conjugation (Hermitian),
Conservation
local conservation of probability,
of angular momentum,,,
of energy,
of probability,
Conservative systems,,
Constants of the motion,,
Contact term,
Contact term (Fermi), ,
Contextuality,
Continuous
spectrum,,,,
variables (in a Lagrangian),
Continuum of nal states, , ,
1380
Contractions,
1571

INDEX [The notation (ex.) refers to an exercise]
Convolution product of two functions,
Cooling
Doppler,
down atoms,
evaporative,
Sisyphus,
sub-Doppler,
subrecoil,
Cooper model,
Cooper pairs,
Cooperative eects (BCS),
Correlation functions, ,
anti-normal, ,
dipole and eld,
for one-photon processes,
normal, ,
of the eld, spatial,
Correlations,
between two dipoles,
between two physical systems,
classical and quantum,
introduced by a collision,
Coulomb
eld,
gauge,
Coulomb potential
cross section,
Coupling
between angular momenta,
between two angular momenta,
between two states,
eect on the eigenvalues,
spin-orbit coupling, ,
Creation and annihilation operators,,
513,, ,
Creation operator (pair of particles), ,
1846
Critical velocity,
Cross section
and phase shifts,
scattering cross section,,,,
972
Current
metastable current in superuid,
of particles,
of probability,
probability current in hydrogen atom,
851
Cylindrical symmetry,(ex.)
Darwin term, ,
De Broglie
relation,
wavelength,seefront cover pages,,
35
Decay of a discrete state,
Deceleration of an atomic beam,
Decoherence,
Decomposition (Schmidt),
Decoupling (ne or hyperne structure),
1262,
Degeneracy
essential,,,
exchange degeneracy,
exchange degeneracy removal,
lifted by a perturbation,
rotation invariance,
systematic and accidental,
Degenerate eigenvalue,,,,
Degereracy
lifted by a perturbation,
parity,
Delta Dirac function,
potential well and barriers,85(ex.)
use in quantum mechanics,,,
280
Density
Lagrangian,
of probability,
of states,, , ,
operator,,
operator and matrix,
particle density operator,
Density functions
one and two-particle, (ex.)
Depletion (quantum),
Derivative of an operator,
Detection probability amplitude (photon),
2166
Detectors (photon),
Determinant
Slater determinant, ,
Deuterium,, (ex.)
Diagonalization
1572

INDEX [The notation (ex.) refers to an exercise]
of a22matrix,
of an operator,
Diagram (dressed-atom),
Diamagnetism,
Diatomic molecules
rotation,
Diusion (momentum),
Dipole
-dipole interaction, ,
-dipole magnetic interaction,
electric dipole transition,
electric moment,
Hamiltonian,
magnetic dipole moment,
magnetic term,
trap,
Dirac,seeFermi
delta function,,,,
equation,
notation,
Direct
and exchange terms, , , ,
1646,
term, ,
Discrete
bases of the state space,
spectrum,,
Dispersion (anomalous),
Dispersion and absorption (eld),
Distribution
Boltzmann,
Bose-Einstein,
Fermi-Dirac,
function (bosons),
function (fermions),
functions, ,
functions (BCS),
Distribution law
Bose-Einstein,
Divergence (energy),
Donor atom,,
Doppler
cooling,
eect,
eect (relativistic),
free spectroscopy,
temperature,
Double
condensate,
resonance method,
spin condensate,
Doublet (Autler-Townes),
Down-conversion (parametric),
Dressed
states and energies,
Dressed-atom, ,
diagram,
strong coupling,
weak coupling,
E.P.R., (ex.)
Eckart (Wigner-Eckart theorem),seeWigner
Eect
Autler-Townes,
Mössbauer,
photoelectric,
Eective Hamiltonian,
Ehrenfest theorem,,,
Eigenresult,
Eigenstate,,
Eigenvalue,,,,
degenerate,,
equation,,
of an operator,
Eigenvector,
of an operator,
Einstein,
coecients, , ,
EPR argument,,
model,,
Planck-Einstein relations,
temperature,
Einstein-Podolsky-Rosen, ,
Elastic
scattering,
scattering (photon),
scattering, form factor, (ex.)
total cross section,
Elastically bound electron model,
Electric
conductivity of a solid,
Electric dipole
Hamiltonian,
interaction,
1573

INDEX [The notation (ex.) refers to an exercise]
matrix elements,
moment,
selection rules,
transition and selection rules,
transitions,
Electric eld (quantized), ,
Electric polarisability
NH3,
Electric polarizability
of the1state in Hydrogen,
Electric quadrupole
Hamiltonian,
moment,
transitions,
Electric susceptibility
bound electron,
of an atom,
Electrical
susceptibility, (ex.)
Electrodynamics
classical,
quantum,
Electromagnetic eld
and harmonic oscillators,
and potentials,
angular momentum, ,
energy,
Lagrangian, ,
momentum, ,
polarization,
quantization,,
Electromagnetic interaction of an atom
with a wave,
Electromagnetism
elds and potentials,
Electron spin,,
Electron(s)
congurations,
gas in solids,
in solids, ,
mass and charge,seefront cover pages
Electronic
conguration,
paramagnetic resonance, (ex.)
shell,
Elements of reality,
Emergence of a relative phase, ,
Emission
of a quantum,
photon,
spontaneous, ,
stimulated (or induced),
Energy,seeConservation, Uncertainty
and momentum of the transverse elec-
tromagnetic eld,
band,
bands in solids, ,
conservation,
electromagnetic eld,
Fermi energy,
ne structure energy levels,
free energy,
levels,
levels of harmonic oscillator,
levels of hydrogen,
of a paired state,
recoil energy,
Ensemble
canonical,
grand canonical,
microcanonical,
statistical ensembles,
Entanglement
quantum, , , ,
swapping,
Entropy,
EPR, ,
elements of reality,
EPRB,
paradox/argument,
Equation of state
ideal quantum gas,
repulsive bosons,
Equation(s)
Bloch,
Hamilton-Jacobi, , ,
Lagrange, ,
Lorentz,
Maxwell,
Schrödinger,,,
von Neumann,
Essential degeneracy,,
Ethane (molecule),
Ethylene (molecule),,
1574

INDEX [The notation (ex.) refers to an exercise]
Evanescent wave,,,,,
Evaporative cooling,
Even operators,
Evolution
eld operator,
of quantum systems,
of the mean value,
operator,,
operator (expansion),
operator (integral equation),
Exchange,
degeneracy,
degeneracy removal,
energy,
hole,
integral,
term, , ,
Excitations
BCS,
Bogolubov,
vacuum,
Excited states (BCS),
Exciton,
Exclusion principle (Pauli), , ,
1463,
Extensive (or intensive) variables,
Fermi
contact term,
energy, , , ,
gas,
golden rule,
level, ,
radius,
surface (modied),
,seeFermi-Dirac
Fermi level
and electric conductivity,
Fermi-Dirac
distribution, , ,
statistics,
Fermions,
in a Fock state,
paired,
Ferromagnetism,
Feynman
path,
postulates,
Fictitious spin,,
Field
absorption,
commutation relations, ,
dispersion and absorption,
intense laser,
interaction energy,
kinetic energy,
normal variables,
operator,
operator (evolution), ,
pair eld operator,
potential energy,
quantization, ,
quasi-classical state,
spatial correlation functions,
Final states continuum, ,
Fine and hyperne structure,
Fine structure
constant,seefront cover pages,
energy levels,
Hamiltonian, , ,
Helium atom,
Hydrogen,
of spectral lines,
of the states1,2et2,
Fletcher,
Fluctuations
boson occupation number,
intensity,
vacuum,,
Fluorescence (single atom),
Fluorescence triplet,
Fock
space, ,
state, , , ,
Forbidden,seeBand
energy band,,,
transition,
Forces
van der Waals,
Form factor
elastic scattering, (ex.)
Forward scattering (direct and exchange),
1874
Fourier
1575

INDEX [The notation (ex.) refers to an exercise]
series and transforms,
Fragmentation (condensate), ,
Free
electrons in a box,
energy,
particle,
quantum eld (Fock space),
spherical wave,,,
spherical waves and plane waves,
Free particle
stationary states with well-dened an-
gular momentum,
stationary states with well-dened mo-
mentum,
wave packet,,,
Frequency
Bohr,
components of the eld (positive and
negative),
Rabi's frequency,
Friction (coecient),
Function
of operators,
periodic functions,
step functions,
Fundamental state,
Gap (BCS), , ,
Gauge, , , ,
Coulomb,
invariance,
Lorenz,
Gaussian
wave packet,,,
Generalized velocities,,
Geometric quantization,
Gerlach,seeStern
GHZ state, ,
Gibbs-Duhem relation,
Golden rule (Fermi),
Good quantum numbers,
Grand canonical, ,
Grand potential, , ,
Green's function,,, , ,
1789
evolution,
Greenberger-Horne-Zeilinger,
Groenewold's formula,
Gross-Pitaevskii equation, ,
Ground state,
harmonic oscillator,,
Hydrogen atom, (ex.)
Group velocity,,,
Gyromagnetic ratio,,
orbital,
spin,
H
+
2
molecular ion,(ex.),,
Hadronic atoms,
Hall eect,
Hamilton
function,
function and equations,
Hamilton-Jacobi canonical equations,,
1532, , ,
Hamiltonian,,, , , ,
1995
classical,
eective,
electric dipole, ,
electric quadrupole,
ne structure, ,
hyperne, ,
magnetic dipolar,
of a charged particle in a vector po-
tential,
of a particle in a central potential,
806,
of a particle in a scalar potential,
of a particle in a vector potential,
225,,
Hanbury Brown and Twiss,
Hanle eect, (ex.)
Hard sphere
scattering,,(ex.)
Harmonic oscillator,
in an electric eld,
in one dimension,,
in three dimensions,
in two dimensions,
innite chain of coupled oscillators,
611
quasiclassical states,
thermodynamic equilibrium,
1576

INDEX [The notation (ex.) refers to an exercise]
three-dimensional,,(ex.)
two coupled oscillators,
Hartree-Fock
approximation, ,
density operator (one-particle),
equations, ,
for electrons,
mean eld, ,
potential,
thermal equilibrium, ,
time-dependent, ,
Healing length,
Heaviside step function,
Heisenberg
picture,,
relations,,,,,,,
Helicity (photon),
Helium
energy levels,
ion,
isotopes,
isotopes
3
He and
4
He, ,
solidication,
Hermite polynomials,,,
Hermitian
conjugation,
matrix,
operator,,,
Histories (classical),
Hole
creation and annihilation,
exchange,
Holes,
Hybridization of atomic orbitals,
Hydrogen,
atom,
atom in a magnetic eld,,,
862
atom, relativistic energies,
Bohr model,,
energy levels,
ne and hyperne stucture,
ionisation energy,seefront cover pages
ionization energy,
maser,
molecular ion,(ex.),,
quantum theory,
radial equation,
Stark eect,
stationary states,
stationary wave functions,
Hydrogen-like systems in solid state physics,
837
Hydrogenoid systems,
Hyperne
decoupling,
Hamiltonian, ,
Hyperne structure,seeHydrogen, muo-
nium, positronium, Zeeman ef-
fect,
Muonium,
Ideal gas, , , ,
correlations,
Identical particles, ,
Induced
emission, , ,
emission of a quantum,
emission of photons,
Inequality (Bell's),
Innite one-dimensional well,
Innite potential well,
in two dimensions,
Innitesimal unitary operator,
Insulator,
Integral
exchange integral,
scattering equation,
Intense laser elds,
Intensive (or extensive) variables,
Interaction
between magnetic dipoles,
dipole-dipole interaction, ,
electromagnetic interaction of an atom
with a wave,
eld and particles,
eld and atom,
magnetic dipole-dipole interaction,
picture,, ,
tensor interaction,
Interference
photons,
two-photon, ,
Ion H
+
2
,
1577

INDEX [The notation (ex.) refers to an exercise]
Ionization
photo-ionization,
tunnel ionization,
Isotropic radiation,
Jacobi,seeHamilton
Kastler, ,
Ket,seestate,,
for identical particles,
Kuhn,seeThomas
Lagrange
equations, , ,
fonction and equations,
multipliers,
Lagrangian, ,
densities,
electromagnetic eld, ,
formulation of quantum mechanics,
339
of a charged particle in an electro-
magnetic eld,
particle in an electromagnetic eld,
323
Laguerre-Gaussian beams,
Lamb shift,, , ,
Landau levels,
Landé factor, , (ex.), ,
Laplacian,
of1,
of()
+1
,
Larmor
angular frequency,
precession,,,,,,
1071
Laser, ,
Raman laser,
saturation,
trap,
Lattices (optical),
Least action
principle of,
Legendre
associated function,
polynomial,
Length (healing),
Level
anticrossing,,
Fermi level,
Lifetime,,,
of a discrete state,
radiative,
Lifting of degeneracy by a perturbation,
1125
Light
quanta,
shifts, , , ,
Linear,seeoperator
combination of atomic orbitals,
operators,,,
response, , ,
superposition of states,
susceptibility,
Local conservation of probability,
Local realism, ,
Longitudinal
elds,
relaxation,
relaxation time,
Lorentz equations,
Lorenz (gauge),
Magnetic
dipole term,
dipole-dipole interaction,
eect of a magnetic eld on the lev-
els of the Hydrogen atom,
hyperne Hamiltonian,
interactions, ,
quantum number,
resonance,
susceptibility, ,
Magnetic dipole
Hamiltonian,
transitions and selection rules, ,
1098,
Magnetic dipoles
interactions between two dipoles,
Magnetic eld
and vector potential,
charged particle in a,,
eects on hydrogen atom,,
harmonic oscillator in a,(ex.)
Hydrogen atom in a magnetic eld,
1263,
1578

INDEX [The notation (ex.) refers to an exercise]
multiplets,
quantized, ,
Magnetism (spontaneous),
Many-electron atoms,
Maser,, ,
hydrogen,
Mass correction (relativistic),
Master equation,
Matrice(s),,
diagonalization of a22matrix,
Pauli matrices,
unitary matrix,
Maxwell's equations,
Mean eld (Hartree-Fock), , ,
1725
Mean value of an observable,
evolution,
Measurement
general postulates,,
ideal von Neumann measurement,
of a spin 1/2,
of observables,
on a part of a physical system,
state after measurement,,
Mendeleev's table,
Metastable superuid ow,
Methane (molecule),
Microcanonical ensemble,
Millikan,
Minimal wave packet,,,
Mirrors for atoms,
Mixing of states, ,
Model
Cooper model,
Einstein model,
elastically bound electron,
vector model of atom,
Modes
vibrational modes,,
Modes (radiation), ,
Molecular ion,
Molecule(s)
chemical bond,,,,,
883,
rotation,
vibration,,
vibration-rotation,
Mollow,
Moment
quadrupole electric moment, (ex.)
Momentum,
conjugate,,, , ,
diusion,
electromagnetic eld, ,
mechanical momentum,
Monogamy (quantum),
Mössbauer eect, ,
Motional narrowing,
condition, , ,
Multiphoton transition, , ,
Multiplets, , ,
Multipliers (Lagrange),
Multipolar waves,
Multipole
moments,
Multipole operators
introduction, ,
parity,
Muon,,,
Muonic atom,,
Muonium,
hyperne structure,
Zeeman eect,
Narrowing (motional), ,
condition,
Natural width,,
Need for a quantum treatment, ,
Neumann
spherical function,
Neutron mass,seefront cover pages
Non-destructive detection of a photon,
2159
Non-diagonal order (BCS),
Non-locality,
Non-resonant excitation,
Non-separability,
Nonlinear
response, ,
susceptibility,
Norm
conservation,
of a state vector,,
of a wave function,,,
1579

INDEX [The notation (ex.) refers to an exercise]
Normal
correlation function, ,
variables,,,,
variables (eld),
Nuclear
multipole moments,
Bohr magneton,
Nucleus
spin,
volume eect, ,
Number
occupation number, ,
photon number,
total number of particles in an ideal
gas,
Observable(s),
C.S.C.O.,,
commutation,
compatibility,
for identical particles, ,
mean value,
measurement of,,
quantization rules,
symmetric observables,
transformation by permutation,
whose commutator is},,
Occupation number, ,
operator,
Odd operators,
One-particle
Hartree-Fock density operator,
operators, , , ,
Operator(s)
adjoint operator,
annihilation operator,,,,
1597
creation and annihilation,
creation operator,,,,
derivative of an operator,
diagonalization,,
even and odd operators,
evolution operator,,
eld,
function of,
Hermitian operators,
linear operators,,,
occupation number,
one-particle operator, , , ,
1756
parity operator,
particle density operator,
permutation operators, ,
potential,
product of,
reduced to a single particle,
representation,
restriction,
restriction of,
rotation operator,
symmetric, ,
translation operator,
two-particle operator, , , ,
1756
unitary operators,
Weyl operator,
Oppenheimer,seeBorn, ,
Optical
excitation (broadband),
lattices,
pumping, ,
Orbital
angular momentum (of radiation),
atomic orbital, (ex.)
hybridization,
linear combination of atomic orbitals,
1172
quantum number,
state space,
Order parameter for pairs,
Orthonormal basis,,,,
characteristic relation,
Orthonormalization
and closure relations,,
relation,
Oscillation(s)
between two discrete states,
between two quantum states,
Rabi,
Oscillator
anharmonic,
harmonic,
strength,
Pair(s)
1580

INDEX [The notation (ex.) refers to an exercise]
annihilation-creation of pairs, ,
1874,
BCS, wave function,
Cooper,
of particles (creation operator), ,
1846
pair eld (commutation),
pair eld operator,
pair wave function,
Paired
bosons,
fermions,
state energy,
states,
states (building),
Pairing term,
Paramagnetism,
Parametric down-conversion,
Parity,
degeneracy,
of a permutation operator,
of multipole operators,
operator,
Parseval
Parseval-Plancherel equality,
Parseval-Plancherel formula, ,
Partial
reection,
trace of an operator,
waves in the potential,
waves method,
Particle (current),
Particles and holes,
Partition function, , ,
Path
integral,
space-time path,
Pauli
exclusion principle, , , ,
1481
Hamiltonian, (ex.)
matrices,,
spin theory,
spinor,
Penetrating orbit,
Penrose-Onsager criterion, , ,
Peres,
Periodic
boundary conditions,
classication of elements,
functions,
potential (one-dimensional),
Permutation operators, ,
Perturbation
applications of the perturbation the-
ory,
lifting of a degeneracy,
one-dimensional harmonic oscillator,
1131
random perturbation, , ,
sinusoidal,
stationary perturbation theory,
Perturbation theory
time dependent,
Phase
locking (BCS), ,
locking (bosons), ,
relative phase between condensates,
2237,
velocity,
Phase shift (collision),, (ex.)
with imaginary part,
Phase velocity,
Phonons,,
Bogolubov phonons,
Photodetection
double, ,
single, ,
Photoelectric eect, (ex.),
Photoionization, ,
rate, ,
two-photon,
Photon,,,, , ,
absorption and emission,
angular momentum,
antibunching,
detectors,
non-destructive detection,
number,
scattering (elastic),
scattering by an atom,
vacuum,
,seeAbsorption, Emission
Picture
1581

INDEX [The notation (ex.) refers to an exercise]
Heisenberg,,
interaction, ,
Pitaevskii (Gross-Pitaevskii equation), ,
1657
Plancherel,seeParseval
Planck
constant,seefront cover pages,
law ,
Planck-Einstein relations,,
Plane wave,,,,
Podolsky (EPR argument),,
Pointer states,
Polarizability
of the1state in Hydrogen,
Polarization
electromagnetic eld,
of Zeeman components,
space-dependent,
Polynomial method (harmonic oscillator),
555,
Polynomials
Hermite polynomials,,,
Position and momentum representations,
181
Positive and negative frequency compo-
nents,
Positron,
Positronium,
hyperne structure,
Zeeman eect,
Postulate (von Neumann projection),
Postulates of quantum mechanics,
Potential
adiabatic branching,
barrier,,,,
centrifugal potential,,,
Coulomb potential, cross section,
cylindrically symmetric,(ex.)
Hartree-Fock,
innite one-dimensional well,
operator,
scalar and vector potentials, ,
1960,
scattering by a,
self-consistent potential,
square potential,
square well,
step,,,,
well,,
well (arbitrary shape),
well (innite one-dimensional),
well (innite two-dimensional,
Yukawa potential,
Precession
Larmor precession,,
Thomas precession,
Preparation of a state,
Pressure (ideal quantum gas),
Principal part,
Principal quantum number,
Principle
of least action, ,
of spectral decomposition,,
of superposition,
Probability
amplitude,,,
conservation,
current,,,,,
current in hydrogen atom,
density,,
uid,
of photon absorption,
of the measurement results,,
transition probability,
Process (pair annihilation-creation), ,
1887
Product
convolution product of functions,
of matrices,
of operators,
scalar product,,,,
state (tensor product),
tensor product,
tensor product, applications,
Projection theorem,
Projector,,,,,, (ex.)
Propagator
for the Schrödinger equation,
of a particle, ,
Proper result,
Proton
mass,seefront cover pages
spin and magnetic moment, ,
Pumping,
1582

INDEX [The notation (ex.) refers to an exercise]
Pure (state or case),
Quadrupolar electric moment, , (ex.)
Quanta (circular),,
Quantization
electrodynamics,
electromagnetic eld,,,
of a eld,
of angular momentum,,
of energy,,,,
of measurement results,,,
of the measurement results,
rules,,,,
Quantum
angle,
electrodynamics, , ,
entanglement, ,
monogamy,
number
orbital,
principal quantum number,
numbers (good),
resonance,
treatment needed, ,
Quasi-classical
eld states,
states,,,
states of the harmonic oscillator,
Quasi-particles, ,
Bogolubov phonons,
Quasi-particle vacuum,
Rabi
formula,,, ,
formula),
frequency,
oscillation,
Radial
equation,
equation (Hydrogen),
equation in a central potential,
integral,
quantum number,
Radiation
isotropic,
pressure,
Radiative
broadening,
cascade of the dressed atom,
Raman
eect,,, (ex.)
laser,
scattering,
scattering (stimulated),
Random perturbation, , ,
Rank (Schmidt),
Rate (photoionization), ,
Rayleigh
line,
scattering,,
Realism (local), ,
Recoil
blocking,
eect of the nucleus,
energy, ,
free atom,
suppression,
Reduced
density operator,
mass,
Reduction of the wave packet,,
Reection on a potential step,
Refractive index,
Reiche,seeThomas
Relation (Gibbs-Duhem),
Relative
motion,
particle,
phase between condensates, ,
phase between spin condensates,
Relativistic
corrections, ,
Doppler eect,
mass correction,
Relaxation,, , , , (ex.)
general equations,
longitudinal,
longitudinal relaxation time,
transverse,
transverse relaxation time,
Relay state, , ,
Renormalization,
Representation(s)
change of,
in the state space,
1583

INDEX [The notation (ex.) refers to an exercise]
of operators,
position and momentum,,
Schrödinger equation,185
Repulsion between electrons,
Resonance
magnetic resonance,
quantum resonance,,
scattering resonance,,,(ex.)
two resonnaces with a sinusoidal ex-
citation,
width,
with sinusoidal perturbation,
Restriction of an operator,,
Rigid rotator,, (ex.)
Ritz theorem,
Root mean square deviation
general denition,
Rosen (EPR argument),,
Rotating frame,
Rotation(s)
and angular momentum,
invariance and degeneracy,
of diatomic molecules,
of molecules,,
operator(s),,
rotation invariance,
rotation invariance and degeneracy,
1072
Rotator
rigid rotator,, (ex.)
Rules
quantization rules,
selection rules,
Rutherford's formula,
Rydberg constant,seefront cover pages
Saturation
of linear response,
of the susceptibility,
Scalar
and vector potentials,,
interaction between two angular mo-
menta,
observable, operator,,
potential,
product,,,,,,
product of two coherent states,
Scattering
amplitude,,
by a central potential,
by a hard sphere,,(ex.)
by a potential,
cross section,,,
cross section and phase shifts,
inelastic,
integral equation,
of particles with spin,
of spin 1/2 particles, (ex.)
photon,
Raman,
Rayleigh,,
resonance,,(ex.)
resonant,
stationary scattering states,
stationary states,
stimulated Raman,
Schmidt
decomposition,
rank,
Schottky anomaly,
Schrödinger,
equation,,,,
equation in momentum representa-
tion,
equation in position representation,
183
equation, physical implications,
equation, resolution for conservative
systems,
picture,
Schwarz inequality,
Second
quantization,
harmonic generation,
Secular approximation, ,
Selection rules,,, ,
electric quadrupolar,
magnetic dipolar, ,
Self-consistent potential,
Semiconductor,,
Separability, ,
Separable density operator,
Shell (electronic),
Shift
1584

INDEX [The notation (ex.) refers to an exercise]
light shift,
of a discrete state,
Singlet, ,
Sinusoidal perturbation, ,
Sisyphus
cooling,
eect,
Slater determinant, ,
Slowing down atoms,
Solids
electronic bands,
energy bands of electrons,
energy bands of electrons in solids,
381
hydrogen-like systems in solid state
physics,
Space (Fock),
Space-dependent polarization,
Space-time path,,
Spatial correlations (ideal gas),
Specic heat
of an electron gas,
of metals,
of solids,
two level system,
Spectral
decomposition principle,,,
function,
terms,
Spectroscopy (Doppler free),
Spectrum
BCS elementary excitation,
continuous,,
discrete,,
of an observable,,
Spherical
Bessel equation,
Bessel function,,
free spherical waves,
free wave,
Neumann function,
wave,
waves and plane waves,
Spherical harmonics,,
addition of,
expression for= 012,
general expression,
Spin
and magnetic moment of the proton,
1237
angular momentum,
electron,,
ctitious,
gyromagnetic ratio,,,
nuclear,
of the electron,
Pauli theory,,
quantum description,,
rotation operator,
scattering of particles with spin,
spin 1 and radiation, , ,
system of two spins,
Spin 1/2
density operator,
ensemble of,
ctitious,
interaction between two spins,
preparation and measurement,
scattering of spin 1/2 particles, (ex.)
Spin-orbit coupling, , , ,
Spin-statistics theorem,
Spinor,
rotation,
Spontaneous
emission,,, , ,
emission of photons,
magnetism of fermions,
Spreading of a wave packet,,
Square
barrier of potential,,
potential,,,,
potential well,,
spherical well,(ex.)
Standard representation (angular momen-
tum),,
Stark eect in Hydrogen atom,
State(s),seeDensity operator
density of,, , ,
Fock, , , ,
ground state,
mixing of states by a perturbation,
1121
orbital state space,
paired,
1585

INDEX [The notation (ex.) refers to an exercise]
pointer states,
quasi-classical states,,,,
801
relay state, , ,
stable and unstable states,
state after measurement,
state preparation,
stationary,,,
stationary state,,
stationary states in a central poten-
tial,
unstable,
vacuum state,
vector,,
Stationary
perturbation theory,
phase condition,,
scattering states,,
states,,,,
states in a periodic potential,
states with well-dened angular mo-
mentum,,
states with well-dened momentum,
943
Statistical
entropy,
mechanics (review of),
mixture of states,,,,
Statistics
Bose-Einstein,
Fermi-Dirac,
Step
function,
potential,,,,
Stern-Gerlach experiment,
Stimulated
(or induced) emission, , ,
2081
Raman scattering,
Stokes Raman line,,
Stoner (spontaneous magnetism),
Strong coupling (dressed-atom),
Subrecoil cooling,
Sum rule (Thomas-Reiche-Kuhn),
Superuidity, ,
Superposition
of states,
principle,,
principle and physical predictions,
Surface (modied Fermi surface),
Susceptibility,seeLinear, nonlinear, ten-
sor
electric susceptibility of an atom,
electrical susceptibility,,e1223
electrical susceptibility of NH3,
magnetic susceptibility,
tensor, , (ex.)
Swapping (entanglement),
Symmetric
ket, state, ,
observables, ,
operators, , , , ,
1628, ,
Symmetrization
of observables,
postulate,
Symmetrizer, ,
System
time evolution of a quantum system,
223
two-level system,
Systematic
and accidental degeneracies,
degeneracy,
Temperature (Doppler),
Tensor
interaction,
product,,
product of operators,
product state,,
product, applications,
susceptibility tensor,
Term
direct and exchange terms, , ,
1634, ,
pairing,
spectral terms, ,
Theorem
Bell, ,
Bloch,
projection,
Ritz,
Wick, ,
1586

INDEX [The notation (ex.) refers to an exercise]
Wigner-Eckart, , ,
Thermal wavelength,
Thermodynamic equilibrium,
harmonic oscillator,
ideal quantum gas,
spin 1/2,
Thermodynamic potential (minimization),
1715
Thomas precession,
Thomas-Reiche-Kuhn sum rule,
Three-dimensional harmonic oscillator,,
841,(ex.)
Three-level system, (ex.)
Three-photon transition,
Time evolution of quantum systems,
Time-correlations (uorescent photons),
2145
Time-dependent
Gross-Pitaevskii equation,
perturbation theory,
Time-energy uncertainty relation,,,
345, ,
Torsional oscillations,
Torus (ow in a),
Total
elastic scattering cross section,
reection,,
scattering cross section (collision),
Townes
Autler-Townes eect,
Trace
of an operator,
partial trace of an operator,
Transform (Wigner),
Transformation
Bogolubov,
Bogolubov-Valatin, ,
Gauge,
of observables by permutation,
Transition,seeProbability, Forbidden, Elec-
tric dipole, Magnetic dipole,
Quadrupole
electric dipole,2056
magnetic dipole transition,
probability,, , ,
probability per unit time,
probability, spin 1/2,
three-photon transition,
two-photon,
virtual,
Translation operator,,,
Transpositions,
Transverse
elds,
relaxation,
relaxation time,
Trap
dipolar,
laser,
Triplet, ,
uorescence triplet,
Tunnel
eect,,,,,,
ionization,
Two coupled harmonic oscillators,
Two-dimensional
harmonic oscillator,
innite potential well,
wave packets,
Two-level system,,,,
Two-particle operators, , , ,
1756
Two-photon
absorption, (ex.)
interference, ,
transition, (ex.),
Uncertainty
relation,,,,,,
time-energy uncertainty relation,
Uniqueness of the measurement result,
2201
Unitary
matrix,,
operator,,
transformation of operators,
Unstable states,
Vacuum
electromagnetism,,
excitations,
uctuations,
photon vacuum,
quasi-particule vacuum,
state,
Valence band,
1587

INDEX [The notation (ex.) refers to an exercise]
Van der Waals forces,
Variables
intensive or extensive,
normal variables,,,,
Variational method, , , (ex.)
Vector
model,
model of the atom, ,
observable, operator,
operator,
potential,
potential of a magnetic dipole,
Velocity
critical,
generalized velocities,,
group velocity,,
phase velocity,,
Vibration(s)
modes,,
modes of a continuous system,
of molecules,,
of nuclei in a crystal,,,
of the nuclei in a molecule,
Violations of Bell's inequalities, ,
Virial theorem,,
Virtual transition,
Volume eect,,, ,
Von Neumann
chain,
equation,
ideal measurement,
reduction postulate,
statistical entropy,
Vortex in a superuid,
Water (molecule),,
Wave (evanescent),
Wave function,,,
BCS pairs, ,
Hydrogen,
norm,
pair wave functions,
particle,
Wave packet(s)
Gaussian,,
in a potential step,
in three dimensions,
minimal,,,
motion in a harmonic potential,
one-photon,
particle,
photon,
propagation,,,,
reduction,,,,
spreading,,,,(ex.)
two-dimension,
two-photons,
Wave(s)
de Broglie wavelength,,
evanescent,
free spherical waves,
multipolar,
partial waves,
plane,,,
wave function,,,,
Wave-particle duality,,
Wavelength
Compton wavelength,
de Broglie,
Weak coupling (dressed-atom),
Well
potential square well,
potential well,
Weyl
operator,
quantization,
Which path type of experiments,
Wick's theorem, ,
Wigner transform,
Wigner-Eckart theorem, , ,
Young (double slit experiment),
Yukawa potential,
Zeeman
components, polarizations,
eect,,,, , , ,
1261,
polarization of the components,
slower,
Zeeman eect
Hydrogen,
in muonium,
in positronium,
Muonium,
1588

INDEX [The notation (ex.) refers to an exercise]
Zone (Brillouin zone),
1589

QUANTUMMECHANICS
olumeIII
Fermions, Bosons, Photons, Correlations, and Entanglement
Translated from the French byNicole Ostrowsky and Dan Ostrowsky
V
Claude Cohen-Tannoudji, Bernard Diu,
and Franck Laloë

Authors
Prof. Dr. Claude Cohen-Tannoudji
Laboratoire Kastler Brossel (ENS)
24 rue Lhomond
75231 Paris Cedex 05
France
Prof. Dr. Bernard Diu
4 rue du Docteur Roux
91440 Boures-sur-Yvette
France
Prof. Dr. Frank Laloë
Laboratoire Kastler Brossel (ENS)
24 rue Lhomond
75231 Paris Cedex 05
France
Cover Image
© antishock/Getty Images
First Edition
All books published by Wiley-VCH are
carefully produced. Nevertheless, authors,
editors, and publisher do not warrant the
information contained in these books,
including this book, to be free of errors.
Readers are advised to keep in mind that
statements, data, illustrations, procedural
details or other items may inadvertently be
inaccurate.
Library of Congress Card No.:
applied for
British Library Cataloguing-in-Publication
Data:
A catalogue record for this book is available
from the British Library.
Bibliographic information published by the
Deutsche Nationalbibliothek
The Deutsche Nationalbibliothek lists this
publication in the Deutsche Nationalbiblio
graﬁe; detailed bibliographic data are avail
able on the Internet at http://dnb.d-nb.de.
© 2020 WILEY-VCH Verlag GmbH &
Co. KGaA, Boschstr. 12, 69469 Weinheim,
Germany
All rights reserved (including those of trans
lation into other languages). No part of
this book may be reproduced in any form –
by photoprinting, microﬁlm, or any other
means – nor transmitted or translated into
a machine language without written per
mission from the publishers. Registered
names, trademarks, etc. used in this book,
even when not speciﬁcally marked as such,
are not to be considered unprotected by
law.
Print ISBN978-3-527-34555-7
ePDF ISBN978-3-527-82274-4
ePub ISBN978-3-527-82275-1
Cover DesignTata Consulting Services
Printing and BindingCPI Ebner & Spiegel
Printed on acid-free paper.

Directions for Use
This book is composed of chapters and their complements:
The chapterscontain the fundamental concepts. Except for a few
additions and variations, they correspond to a course given in the last
year of a typical undergraduate physics program (Volume I) or of a
graduate program (Volumes II and III). The 21 chapters arecomplete in
themselvesand can be studied independently of the complements.
The complementsfollow the corresponding chapter. Each is labelled
by a letter followed by a subscript, which gives the number of the chapter
(for example, the complements of Chapter V are, in order, AV, BV, CV,
etc.). They can be recognized immediately by the symbolthat appears
at the top of each of their pages.
The complements vary in character. Some are intended to expand the
treatment of the corresponding chapter or to provide a more detailed
discussion of certain points. Others describe concrete examples or in-
troduce various physical concepts. One of the complements (usually the
last one) is a collection of exercises.
Thedicultyof the complements varies. Some are very simple examples
or extensions of the chapter. Others are more dicult and at the grad-
uate level or close to current research. In any case, the reader should
have studied the material in the chapter before using the complements.
The complements are generally independent of one another.The student
should not try to study all the complements of a chapter at once. In
accordance with his/her aims and interests, he/she should choose a small
number of them (two or three, for example), plus a few exercises. The
other complements can be left for later study. To help with the choise,
the complements are listed at the end of each chapter in a reader's
guide, which discusses the diculty and importance of each.
Some passages within the book have been set in small type, and these
can be omitted on a rst reading.

Foreword
Foreword
Quantum mechanics is a branch of physics whose importance has continually in-
creased over the last decades. It is essential for understanding the structure and dynamics
of microscopic objects such as atoms, molecules and their interactions with electromag-
netic radiation. It is also the basis for understanding the functioning of numerous new
systems with countless practical applications. This includes lasers (in communications,
medicine, milling, etc.), atomic clocks (essential in particular for the GPS), transistors
(communications, computers), magnetic resonance imaging, energy production (solar
panels, nuclear reactors), etc. Quantum mechanics also permits understanding surpris-
ing physical properties such as superuidity or supraconductivity. There is currently a
great interest in entangled quantum states whose non-intuitive properties of nonlocality
and nonseparability permit conceiving remarkable applications in the emerging eld of
quantum information. Our civilization is increasingly impacted by technological appli-
cations based on quantum concepts. This why a particular eort should be made in the
teaching of quantum mechanics, which is the object of these three volumes.
The rst contact with quantum mechanics can be disconcerting. Our work grew
out of the authors' experiences while teaching quantum mechanics for many years. It
was conceived with the objective of easing a rst approach, and then aiding the reader
to progress to a more advance level of quantum mechanics. The rst two volumes, rst
published more than forty years ago, have been used throughout the world. They remain
however at an intermediate level. They have now been completed with a third volume
treating more advanced subjects. Throughout we have used a progressive approach to
problems, where no diculty goes untreated and each aspect of the diverse questions is
discussed in detail (often starting with a classical review).
This willingness to go further without cheating or taking shortcuts is built into
the book structure, using two distinct linked texts:chaptersandcomplements. As we
just outlined in the Directions for use, the chapters present the general ideas and
basic concepts, whereas the complements illustrate both the methods and concepts just
exposed.
Volume I presents a general introduction of the subject, followed by a second
chapter describing the basic mathematical tools used in quantum mechanics. While
this chapter can appear long and dense, the teaching experience of the authors has
shown that such a presentation is the most ecient. In the third chapter the postulates
are announced and illustrated in many of the complements. We then go on to certain
important applications of quantum mechanics, such as the harmonic oscillator, which
lead to numerous applications (molecular vibrations, phonons, etc.). Many of these are
the object of specic complements.
Volume II pursues this development, while expanding its scope at a slightly higher
level. It treats collision theory, spin, addition of angular momenta, and both time-
dependent and time-independent perturbation theory. It also presents a rst approach
to the study of identical particles. In this volume as in the previous one, each theoretical
concept is immediately illustrated by diverse applications presented in the complements.
Both volumes I and II have beneted from several recent corrections, but there have also
been additions. Chapter
perturbations, and a complement concerning relaxation has been added.
ii

Foreword
Volume III extends the two volumes at a slightly higher level. It is based on the
use of the creation and annihilation operator formalism (second quantization), which is
commonly used in quantum eld theory. We start with a study of systems of identical
particles, fermions or bosons. The properties of ideal gases in thermal equilibrium are
presented. For fermions, the Hartree-Fock method is developed in detail. It is the base
of many studies in chemistry, atomic physics and solid state physics, etc. For bosons, the
Gross-Pitaevskii equation and the Bogolubov theory are discussed. An original presen-
tation that treats the pairing eect of both fermions and bosons permits obtaining the
BCS (Bardeen-Cooper-Schrieer) and Bogolubov theories in a unied framework. The
second part of volume III treats quantum electrodynamics, its general introduction, the
study of interactions between atoms and photons, and various applications (spontaneous
emission, multiphoton transitions, optical pumping, etc.). The dressed atom method is
presented and illustrated for concrete cases. A nal chapter discusses the notion of quan-
tum entanglement and certain fundamental aspects of quantum mechanics, in particular
the Bell inequalities and their violations.
Finally note that we have not treated either the philosophical implications of quan-
tum mechanics, or the diverse interpretations of this theory, despite the great interest
of these subjects. We have in fact limited ourselves to presenting what is commonly
called the orthodox point of view. It is only in Chapter
questions concerning the foundations of quantum mechanics (nonlocality, etc.). We have
made this choice because we feel that one can address such questions more eciently
after mastering the manipulation of the quantum mechanical formalism as well as its nu-
merous applications. These subjects are addressed in the bookDo we really understand
quantum mechanics?(F. Laloë, Cambridge University Press, 2019); see also section 5 of
the bibliography of volumes I and II.
iii

Foreword
Acknowledgments:
Volumes I and II:
The teaching experience out of which this text grew were group eorts, pursued
over several years. We wish to thank all the members of the various groups and partic-
ularly Jacques Dupont-Roc and Serge Haroche, for their friendly collaboration, for the
fruitful discussions we have had in our weekly meetings and for the ideas for problems
and exercises that they have suggested. Without their enthusiasm and valuable help, we
would never have been able to undertake and carry out the writing of this book.
Nor can we forget what we owe to the physicists who introduced us to research,
Alfred Kastler and Jean Brossel for two of us and Maurice Levy for the third. It was in
the context of their laboratories that we discovered the beauty and power of quantum
mechanics. Neither have we forgotten the importance to us of the modern physics taught
at the C.E.A. by Albert Messiah, Claude Bloch and Anatole Abragam, at a time when
graduate studies were not yet incorporated into French university programs.
We wish to express our gratitude to Ms. Aucher, Baudrit, Boy, Brodschi, Emo,
Heywaerts, Lemirre, Touzeau for preparation of the mansucript.
Volume III:
We are very grateful to Nicole and Daniel Ostrowsky, who, as they translated this
Volume from French into English, proposed numerous improvements and clarications.
More recently, Carsten Henkel also made many useful suggestions during his transla-
tion of the text into German; we are very grateful for the improvements of the text
that resulted from this exchange. There are actually many colleagues and friends who
greatly contributed, each in his own way, to nalizing this book. All their complementary
remarks and suggestions have been very helpful and we are in particular thankful to:
Pierre-François Cohadon
Jean Dalibard
Sébastien Gleyzes
Markus Holzmann
Thibaut Jacqmin
Philippe Jacquier
Amaury Mouchet
Jean-Michel Raimond
Félix Werner
Some delicate aspects of Latex typography have been resolved thanks to Marco
Picco, Pierre Cladé and Jean Hare. Roger Balian, Edouard Brézin and William Mullin
have oered useful advice and suggestions. Finally, our sincere thanks go to Geneviève
Tastevin, Pierre-François Cohadon and Samuel Deléglise for their help with a number of
gures.
iv

Table of contents
Table of contents
I WAVES AND PARTICLES. INTRODUCTION TO THE BASIC
IDEAS OF QUANTUM MECHANICS
READER'S GUIDE FOR COMPLEMENTS 33
AIOrder of magnitude of the wavelengths associated with material
particles
BIConstraints imposed by the uncertainty relations
CIHeisenberg relation and atomic parameters
DIAn experiment illustrating the Heisenberg relations
EIA simple treatment of a two-dimensional wave packet
FIThe relationship between one- and three-dimensional problems
GIOne-dimensional Gaussian wave packet: spreading of the wave packet
HIStationary states of a particle in one-dimensional square potentials
JIBehavior of a wave packet at a potential step
KIExercises
***********
II THE MATHEMATICAL TOOLS OF QUANTUM MECHANICS
READER'S GUIDE FOR COMPLEMENTS 159
AIIThe Schwarz inequality
BIIReview of some useful properties of linear operators
CIIUnitary operators
DIIA more detailed study of the r and p representations
EIISome general properties of two observables,and, whose commu-
tator is equal to~ 187
FIIThe parity operator
v
Volume I

Table of contents
GIIAn application of the properties of the tensor product: the two-
dimensional innite well
HIIExercises
III THE POSTULATES OF QUANTUM MECHANICS
READER'S GUIDE FOR COMPLEMENTS 267
AIIIParticle in an innite one-dimensional potential well
BIIIStudy of the probability current in some special cases
CIIIRoot mean square deviations of two conjugate observables
DIIIMeasurements bearing on only one part of a physical system
EIIIThe density operator
FIIIThe evolution operator
GIIIThe Schrödinger and Heisenberg pictures
HIIIGauge invariance
JIIIPropagator for the Schrödinger equation
KIIIUnstable states. Lifetime
LIIIExercises
MIIIBound states in a potential well of arbitrary shape
NIIIUnbound states of a particle in the presence of a potential well or
barrier
OIIIQuantum properties of a particle in a one-dimensional periodic struc-
ture 375
***********
IV APPLICATIONS OF THE POSTULATES TO SIMPLE CASES:
SPIN 1/2 AND TWO-LEVEL SYSTEMS
READER'S GUIDE FOR COMPLEMENTS 423
AIVThe Pauli matrices
BIVDiagonalization of a22Hermitian matrix
CIVFictitious spin 1/2 associated with a two-level system
vi

Table of contents
DIVSystem of two spin 1/2 particles
EIVSpin12density matrix
FIVSpin 1/2 particle in a static and a rotating magnetic elds: magnetic
resonance
GIVA simple model of the ammonia molecule
HIVEects of a coupling between a stable state and an unstable state
JIVExercises
***********
V THE ONE-DIMENSIONAL HARMONIC OSCILLATOR
READER'S GUIDE FOR COMPLEMENTS 525
AVSome examples of harmonic oscillators
BVStudy of the stationary states in the x representation. Hermite poly-
nomials
CVSolving the eigenvalue equation of the harmonic oscillator by the
polynomial method
DVStudy of the stationary states in the momentum representation
EVThe isotropic three-dimensional harmonic oscillator
FVA charged harmonic oscillator in a uniform electric eld
GVCoherent quasi-classical states of the harmonic oscillator
HVNormal vibrational modes of two coupled harmonic oscillators
JVVibrational modes of an innite linear chain of coupled harmonic
oscillators; phonons
KV Vibrational modes of a continuous physical system. Photons
LVOne-dimensional harmonic oscillator in thermodynamic equilibrium
at a temperature 647
MVExercises
***********
VI GENERAL PROPERTIES OF ANGULAR MOMENTUM IN QUAN-
TUM MECHANICS
vii

Table of contents
READER'S GUIDE FOR COMPLEMENTS 703
AVISpherical harmonics
BVIAngular momentum and rotations
CVIRotation of diatomic molecules
DVIAngular momentum of stationary states of a two-dimensional har-
monic oscillator
EVIA charged particle in a magnetic eld: Landau levels
FVIExercises
***********
VII PARTICLE IN A CENTRAL POTENTIAL, HYDROGEN ATOM
READER'S GUIDE FOR COMPLEMENTS 831
AVIIHydrogen-like systems
BVIIA soluble example of a central potential: The isotropic three-dimensional
harmonic oscillator
CVIIProbability currents associated with the stationary states of the hy-
drogen atom
DVIIThe hydrogen atom placed in a uniform magnetic eld. Paramag-
netism and diamagnetism. The Zeeman eect
EVIISome atomic orbitals. Hybrid orbitals
FVIIVibrational-rotational levels of diatomic molecules
GVIIExercises
INDEX 901
***********
viii

Table of contents
VOLUME II
Table of contents
VIII AN ELEMENTARY APPROACH TO THE QUANTUM THEORY
OF SCATTERING BY A POTENTIAL
READER'S GUIDE FOR COMPLEMENTS 957
AVIIIThe free particle: stationary states
with well-dened angular momentum
BVIIIPhenomenological description of collisions with absorption
CVIIISome simple applications of scattering theory
***********
IX ELECTRON SPIN
READER'S GUIDE FOR COMPLEMENTS 999
AIX Rotation operators for a spin 1/2 particle
BIX Exercises
***********
X ADDITION OF ANGULAR MOMENTA
READER'S GUIDE FOR COMPLEMENTS 1041
AX Examples of addition of angular momenta
BX Clebsch-Gordan coecients
CX Addition of spherical harmonics
DX Vector operators: the Wigner-Eckart theorem
EX Electric multipole moments
FX Two angular momenta J1andJ2coupled by
an interactionJ1J2 1091
GX Exercises
ix
Volume II

Table of contents
***********
XI STATIONARY PERTURBATION THEORY
READER'S GUIDE FOR COMPLEMENTS 1129
AXIA one-dimensional harmonic oscillator subjected to a perturbing
potential in,
2
,
3
1131
BXIInteraction between the magnetic dipoles of two spin 1/2
particles
CXIVan der Waals forces
DXIThe volume eect: the inuence of the spatial extension of the nu-
cleus on the atomic levels
EXIThe variational method
FXIEnergy bands of electrons in solids: a simple model
GXIA simple example of the chemical bond: the H
+
2
ion
HXIExercises
***********
XII AN APPLICATION OF PERTURBATION THEORY: THE FINE
AND HYPERFINE STRUCTURE OF THE HYDROGEN ATOM
READER'S GUIDE FOR COMPLEMENTS 1265
AXIIThe magnetic hyperne Hamiltonian
BXIICalculation of the average values of the ne-structure Hamiltonian
in the1,2and2states
CXIIThe hyperne structure and the Zeeman eect for muonium and
positronium
DXIIThe inuence of the electronic spin on the Zeeman eect of the
hydrogen resonance line
EXIIThe Stark eect for the hydrogen atom
***********
x

Table of contents
XIII APPROXIMATION METHODS FOR TIME-DEPENDENT
PROBLEMS
READER'S GUIDE FOR COMPLEMENTS 1337
AXIIIInteraction of an atom with an electromagnetic wave
BXIIILinear and non-linear responses of a two-level system subject to a
sinusoidal perturbation
CXIIIOscillations of a system between two discrete states under the
eect of a sinusoidal resonant perturbation
DXIIIDecay of a discrete state resonantly coupled to a continuum of nal
states
EXIIITime-dependent random perturbation, relaxation
FXIIIExercises
***********
XIV SYSTEMS OF IDENTICAL PARTICLES
READER'S GUIDE FOR COMPLEMENTS 1457
AXIVMany-electron atoms. Electronic congurations
BXIVEnergy levels of the helium atom. Congurations, terms, multi-
plets
CXIVPhysical properties of an electron gas. Application to solids
DXIVExercises
***********
APPENDICES
I Fourier series and Fourier transforms
II The Dirac -function
III Lagrangian and Hamiltonian in classical mechanics
BIBLIOGRAPHY OF VOLUMES I AND II
INDEX
***********
xi

Table of contents
VOLUME III
Table of contents
XV CREATION AND ANNIHILATION OPERATORS FOR IDENTI-
CAL PARTICLES
A General formalism
B One-particle symmetric operators
C Two-particle operators
READER'S GUIDE FOR COMPLEMENTS 1617
AXVParticles and holes
1 Ground state of a non-interacting fermion gas
2 New denition for the creation and annihilation operators
3 Vacuum excitations
BXVIdeal gas in thermal equilibrium; quantum distribution functions
1 Grand canonical description of a system without interactions
2 Average values of symmetric one-particle operators
3 Two-particle operators
4 Total number of particles
5 Equation of state, pressure
CXVCondensed boson system, Gross-Pitaevskii equation
1 Notation, variational ket
2 First approach
3 Generalization, Dirac notation
4 Physical discussion
DXVTime-dependent Gross-Pitaevskii equation
1 Time evolution
2 Hydrodynamic analogy
3 Metastable currents, superuidity
EXVFermion system, Hartree-Fock approximation
1 Foundation of the method
2 Generalization: operator method
xii
Volume III

Table of contents
FXVFermions, time-dependent Hartree-Fock approximation
1 Variational ket and notation
2 Variational method
3 Computing the optimizer
4 Equations of motion
GXVFermions or Bosons: Mean eld thermal equilibrium
1 Variational principle
2 Approximation for the equilibrium density operator
3 Temperature dependent mean eld equations
HXVApplications of the mean eld method for non-zero temperature
1 Hartree-Fock for non-zero temperature, a brief review
2 Homogeneous system
3 Spontaneous magnetism of repulsive fermions
4 Bosons: equation of state, attractive instability
***********
XVI FIELD OPERATOR
A Denition of the eld operator
B Symmetric operators
C Time evolution of the eld operator (Heisenberg picture)
D Relation to eld quantization
READER'S GUIDE FOR COMPLEMENTS 1767
AXVISpatial correlations in an ideal gas of bosons or fermions
1 System in a Fock state
2 Fermions in the ground state
3 Bosons in a Fock state
BXVISpatio-temporal correlation functions, Green's functions
1 Green's functions in ordinary space
2 Fourier transforms
3 Spectral function, sum rule
CXVIWick's theorem
1 Demonstration of the theorem
2 Applications: correlation functions for an ideal gas
***********
XVII PAIRED STATES OF IDENTICAL PARTICLES
A Creation and annihilation operators of a pair of particles
B Building paired states
C Properties of the kets characterizing the paired states
D Correlations between particles, pair wave function
E Paired states as a quasi-particle vacuum; Bogolubov-Valatin transformations
xiii

Table of contents
READER'S GUIDE FOR COMPLEMENTS 1843
AXVIIPair eld operator for identical particles
1 Pair creation and annihilation operators
2 Average values in a paired state
3 Commutation relations of eld operators
BXVIIAverage energy in a paired state
1 Using states that are not eigenstates of the total particle number
2 Hamiltonian
3 Spin 1/2 fermions in a singlet state
4 Spinless bosons
CXVIIFermion pairing, BCS theory
1 Optimization of the energy
2 Distribution functions, correlations
3 Physical discussion
4 Excited states
DXVIICooper pairs
1 Cooper model
2 State vector and Hamiltonian
3 Solution of the eigenvalue equation
4 Calculation of the binding energy for a simple case
EXVIICondensed repulsive bosons
1 Variational state, energy
2 Optimization
3 Properties of the ground state
4 Bogolubov operator method
***********
XVIII REVIEW OF CLASSICAL ELECTRODYNAMICS
A Classical electrodynamics
B Describing the transverse eld as an ensemble of harmonic oscillators
READER'S GUIDE FOR COMPLEMENTS 1977
AXVIIILagrangian formulation of electrodynamics
1 Lagrangian with several types of variables
2 Application to the free radiation eld
3 Lagrangian of the global system eld + interacting particles
***********
xiv

Table of contents
XIX QUANTIZATION OF ELECTROMAGNETIC RADIATION
A Quantization of the radiation in the Coulomb gauge
B Photons, elementary excitations of the free quantum eld
C Description of the interactions
READER'S GUIDE FOR COMPLEMENTS 2017
AXIXMomentum exchange between atoms and photons
1 Recoil of a free atom absorbing or emitting a photon
2 Applications of the radiation pressure force: slowing and cooling atoms
3 Blocking recoil through spatial connement
4 Recoil suppression in certain multi-photon processes
BXIXAngular momentum of radiation
1 Quantum average value of angular momentum for a spin 1 particle
2 Angular momentum of free classical radiation as a function of normal variables2047
3 Discussion
CXIXAngular momentum exchange between atoms and photons
1 Transferring spin angular momentum to internal atomic variables
2 Optical methods
3 Transferring orbital angular momentum to external atomic variables
***********
XX ABSORPTION, EMISSION AND SCATTERING OF PHOTONS
BY ATOMS
A A basic tool: the evolution operator
B Photon absorption between two discrete atomic levels
C Stimulated and spontaneous emissions
D Role of correlation functions in one-photon processes
E Photon scattering by an atom
READER'S GUIDE FOR COMPLEMENTS 2095
AXXA multiphoton process: two-photon absorption
1 Monochromatic radiation
2 Non-monochromatic radiation
3 Discussion
BXXPhotoionization
1 Brief review of the photoelectric eect
2 Computation of photoionization rates
3 Is a quantum treatment of radiation necessary to describe photoionization?
4 Two-photon photoionization
5 Tunnel ionization by intense laser elds
xv

Table of contents
CXXTwo-level atom in a monochromatic eld. Dressed-atom method
1 Brief description of the dressed-atom method
2 Weak coupling domain
3 Strong coupling domain
4 Modications of the eld. Dispersion and absorption
DXXLight shifts: a tool for manipulating atoms and elds
1 Dipole forces and laser trapping
2 Mirrors for atoms
3 Optical lattices
4 Sub-Doppler cooling. Sisyphus eect
5 Non-destructive detection of a photon
EXXDetection of one- or two-photon wave packets, interference
1 One-photon wave packet, photodetection probability
2 One- or two-photon interference signals
3 Absorption amplitude of a photon by an atom
4 Scattering of a wave packet
5 Example of wave packets with two entangled photons
***********
XXI QUANTUM ENTANGLEMENT, MEASUREMENTS, BELL'S IN-
EQUALITIES
A Introducing entanglement, goals of this chapter
B Entangled states of two spin-12systems
C Entanglement between more general systems
D Ideal measurement and entangled states
E Which path experiment: can one determine the path followed by the photon
in Young's double slit experiment?
F Entanglement, non-locality, Bell's theorem
READER'S GUIDE FOR COMPLEMENTS 2215
AXXIDensity operator and correlations; separability
1 Von Neumann statistical entropy
2 Dierences between classical and quantum correlations
3 Separability
BXXIGHZ states, entanglement swapping
1 Sign contradiction in a GHZ state
2 Entanglement swapping
CXXIMeasurement induced relative phase between two condensates
1 Probabilities of single, double, etc. position measurements
2 Measurement induced enhancement of entanglement
3 Detection of a large numberof particles
xvi

DXXIEmergence of a relative phase with spin condensates; macroscopic
non-locality and the EPR argument
1 Two condensates with spins
2 Probabilities of the dierent measurement results
3 Discussion
***********
APPENDICES
IV Feynman path integral
1 Quantum propagator of a particle
2 Interpretation in terms of classical histories
3 Discussion; a new quantization rule
4 Operators
V Lagrange multipliers
1 Function of two variables
2 Function of variables
VI Brief review of Quantum Statistical Mechanics
1 Statistical ensembles
2 Intensive or extensive physical quantities
VII Wigner transform
1 Delta function of an operator
2 Wigner distribution of the density operator (spinless particle)
3 Wigner transform of an operator
4 Generalizations
5 Discussion: Wigner distribution and quantum eects
BIBLIOGRAPHY OF VOLUME III
INDEX
xvii

Chapter XV
Creation and annihilation
operators for identical particles
A General formalism
A-1 Fock states and Fock space
A-2 Creation operators . . . . . . . . . . . . . . . . . . . . . .
A-3 Annihilation operators. . . . . . . . . . . . . . . . . . . . .
A-4 Occupation number operators (bosons and fermions)
A-5 Commutation and anticommutation relations
A-6 Change of basis
B One-particle symmetric operators
B-1 Denition
B-2 Expression in terms of the operatorsand. . . . . . . . .
B-3 Examples
B-4 Single particle density operator
C Two-particle operators
C-1 Denition
C-2 A simple case: factorization
C-3 General case
C-4 Two-particle reduced density operator
C-5 Physical discussion; consequences of the exchange
Introduction
For a system composed of identical particles, the particle numbering used in Chapter
XIV, the last chapter of Volume II [2], does not really have much physical signicance.
Furthermore, when the particle number gets larger than a few units, applying the sym-
metrization postulate to numbered particles often leads to complex calculations. For
Quantum Mechanics, Volume III, First Edition. C. Cohen-Tannoudji, B. Diu, and F. Laloë.
© 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

CHAPTER XV CREATION AND ANNIHILATION OPERATORS FOR IDENTICAL PARTICLES
example, computing the average value of a symmetric operator requires the symmetriza-
tion of the bra, the ket, and nally the operator, which introduces a large number of
terms
1
. They seem dierent, a priori, but at the end of the computation many are found
to be equal, or sometimes cancel each other. Fortunately, these lengthy calculations may
be avoided using an equivalent method based on creation and annihilation operators in
a Fock space. The simple commutation (or anticommutation) rules satised by these
operators are the expression of the symmetrization (or antisymmetrization) postulate.
The non-physical particle numbering is replaced by assigning occupation numbers to
individual states, which is more natural for treating identical particles.
The method described in this chapter and the following is sometimes called second
quantization
2
. It deals with operators that no longer conserve the particle number,
hence acting in a state space larger than those we have previously considered; this new
space is called the Fock space (). These operators which change the particle number
appear mainly in the course of calculations, and often regroup at the end, keeping the
total particle number constant. Examples will be given () for one-particle symmetric
operators, such as the total linear momentum or angular momentum of a system of
identical particles. We shall then study two-particle symmetric operators (), such as
the energy of a system of interacting identical particles, their spatial correlation function,
etc. In quantum statistical mechanics, the Fock space is well adapted to computations
performed in the grand canonical ensemble, where the total number of particles may
uctuate since the system is in contact with an external reservoir. Furthermore, as we
shall see in the following chapters, the Fock space is very useful for describing physical
processes where the particle number changes, as in photon absorption or emission.
A. General formalism
We denote the state space of a system ofdistinguishable particles, which is the
tensor product ofindividual state spaces1:
=1(1) 1(2) 1() (A-1)
Two sub-spaces ofare particularly important for identical particles, as they contain
all their accessible physical states: the space()of the completely symmetric states
for bosons, and the space()of the completely antisymmetric states for fermions.
The projectors onto these two sub-spaces are given by relations (B-49) and (B-50) of
Chapter :
=
1
!
(A-2)
and:
=
1
!
(A-3)
1
For a one-particle symmetric operator, which includes the sum ofterms, both the ket and bra
contain!terms. The matrix element will therefore involve(!)
2
terms, a very large number once
exceeds a few units.
2
A commonly accepted but a somewhat illogical expression, since no new quantication comes in
addition to that of the usual postulates of Quantum Mechanics; its essential ingredient is the sym-
metrization of identical particles.
1592

A. GENERAL FORMALISM
where theare the!permutation operators for theparticles, andthe parity
of(in this chapter we have added for clarity the indexto the projectors S and A
dened in Chapter XIV).
A-1. Fock states and Fock space
Starting from an arbitrary orthonormal basis of the state space for one
particle, we constructed in
identical particles. Its vectors are characterized by the occupation numbers, with:
1+2+++= (A-4)
where1is the occupation number of the rst basis vector1(i.e. the number of
particles in1),2that of2, ..,that of. In this series of numbers, some
(even many) may be zero: a given state has no particular reason to always be occupied.
It is therefore often easier to specify only the non-zero occupation numbers, which will
be noted . This series indicates that the rst basis state that has at least
one particle isand it containsparticles; the second occupied state iswith a
population, etc. As in (A-4), these occupation numbers add up to.
Comment:
In this chapter we constantly use subscripts of dierent types, which should not be
confused. The subscripts,,,, ..denote dierent basis vectorsof the state
space1of a single particle; they span values given by the dimension of this state space,
which often goes to innity. They should not be confused with the subscripts used to
number the particles, which can takedierent values, and are labeled,, etc.
Finally the subscriptdistinguishes the dierent permutations of theparticles, and
can therefore take!dierent values.
A-1-a. Fock states for identical bosons
For bosons, the basis vectors can be written as in (C-15) of Chapter :
= 1 :; 2 :;;:;+ 1 :; +:; (A-5)
whereis a normalization constant; on the right-hand side,particles occupy the state
,the state, etc... (because of symmetrization, their order does not matter).
Let us calculate the norm of the right-hand term. It is composed of!terms,
coming from each of the!permutations included in, but only some of them are
orthogonal to each other: all the permutationsleading to redistributions of the
rst particles among themselves, of the nextparticles among themselves, etc. yield
the same initial ket. On the other hand, if a permutation changes the individual state
of one (or more than one) particle, it yields a dierent ket, actually orthogonal to the
initial ket. This means that the dierent permutations contained incan be grouped
into families of!!!equivalent permutations, all yielding the same ket; taking
into account the factor!appearing in the denition of, the coecient in front of
this ket becomes!!!!and its contribution to the norm of the ket is equal
to the square of this number. On the other hand, the number of orthogonal kets is
1593

CHAPTER XV CREATION AND ANNIHILATION OPERATORS FOR IDENTICAL PARTICLES
!!!!. Consequently ifwas equal to1in formula (A-5), the ket thus dened
would have a norm equal to:
!
!!!
1
!
!!!
2
=
!!!
!
(A-6)
We shall therefore choose forthe inverse of the square root of that number, leading to
the normalized ket:
=
!!!!
1 :; 2 :;;:;+ 1 :;;+:;
(A-7)
These states are called the Fock states, for which the occupation numbers are well
dened.
For the Fock states, it is sometimes handy to use a slightly dierent but equivalent
notation. In (A-7), these states are dened by specifying the occupation numbers of all
the states that are actually occupied (1). Another option would be to indicate all
the occupation numbers including those which are zero
3
this is what we have explicitly
done in (A-4). We then write the same kets as:
12 (A-8)
Another possibility is to specify a list ofoccupied states, whereis repeatedtimes,
repeatedtimes, etc. :
-times-times
(A-9)
As we shall see later, this latter notation is sometimes useful in computations involving
both bosons and fermions.
A-1-b. Fock states for identical fermions
In the case of fermions, the operatoracting on a ket where two (or more)
numbered particles are in the same individual state yields a zero result: there are no
such states in the physical space(). Hence we concentrate on the case where all
the occupation numbers are either1or0. We denote ,,..,, .. all the states
having an occupation number equal to1. The equivalent for fermions of formula (A-7)
is written:
=
! 1 :; 2 :;;:; if all theare dierent
0 if twoare identical
(A-10)
3
Remember that, by convention,0! = 1.
1594

A. GENERAL FORMALISM
Taking into account the1!factor appearing in denition (A-3) of, the right-
hand side of this equation is a linear superposition, with coecients1
!, of!kets
which are all orthogonal to each other (as we have chosen an orthonormal basis for the
individual states); hence its norm is equal to1. Consequently, Fock states for
fermions are dened by (A-10). Contrary to bosons, the main concern is no longer how
many particles occupy a state, but whether a state is occupied or not. Another dierence
with the boson case is that, for fermions, the order of the states matters. If for instance
the rst two statesandare exchanged, we get the opposite ket:
= (A-11)
but it obviously does not change the physical meaning of the ket.
A-1-c. Fock space
The Fock states are the building blocks used to construct this whole chapter. We
have until now considered separately the spaces()associated with dierent values
of the particle number. We shall now regroup them into a single space, called the
Fock space, using the direct sum
4
formalism. For bosons:
Fock=(0) (1) (2) () (A-12)
and, for fermions:
Fock=(0) (1) (2) () (A-13)
(the sums go to innity). In both cases, we have included on the right-hand side a rst
term associated with a total number of particles equal to zero. The corresponding space,
(0), is dened as a one-dimensional space, containing a single state called vacuum
and denoted0orvac. For bosons as well as fermions, an orthonormal basis for the Fock
space can be built with the Fock states1 2 , relaxing the constraint (A-
4): the occupation numbers may then take on any (integer) values, including zeros for all,
which corresponds to the vacuum ket0. Linear combinations of all these basis vectors
yield all the vectors of the Fock space, including linear superpositions of kets containing
dierent particle numbers. It is not essential to attribute a physical interpretation to such
superpositions since they can be considered as intermediate states of the calculation.
Obviously, the Fock space contains many kets with well dened particle numbers: all
those belonging to a single sub-space()for bosons, or()for fermions. Two kets
having dierent particle numbersare necessarily orthogonal; for example, all the kets
having a non-zero total population are orthogonal to the vacuum state.
Comments:
(i) Contrary to the distinguishable particle case, the Fock space is not the tensor product
of the spaces of states associated with particles numbered1,2,...,, etc. First of all, for a
4
The direct sum of two spaces(with dimension) and(with dimension) is a space+
with dimension+, spanned by all the linear combinations of a vector from the rst space with a
vector from the second. A basis for+may be simply obtained by grouping together a basis for
and one for. For example, vectors of a two-dimensional plane belong to a space that is the direct
sum of the one-dimensional spaces for the vectors of two axes of that plane.
1595

CHAPTER XV CREATION AND ANNIHILATION OPERATORS FOR IDENTICAL PARTICLES
xed, it only includes the totally symmetric (or antisymmetric) subspace of this tensor
product; furthermore, the Fock space is the direct sum of such subspaces associated with
each value of the particle number.
The Fock space is, however, the tensor product of Fock spaces
Fock
associated with the
individual orthogonal states, each
Fock
being spanned by the ketswheretakes
on all integer values (from zero to innity for bosons, from zero to one for fermions):
Fock
=
1
Fock
2
Fock Fock (A-14)
This is because the Fock states, which are a basis for
Fock
, may be written as the tensor
product:
1 2 = 1 2 (A-15)
It is often said that each individual state denes a mode of the system of identical
particles. Decomposing the Fock state into a tensor product allows considering the modes
as describing dierent and distinguishable variables. This will be useful on numerous
occasions (see for example ComplementsXV,XVandXV).
(ii) One should not confuse a Fock state with an arbitrary state of the Fock space. The
occupation numbers of individual states are all well dened in a Fock state (also called
number state), whereas an arbitrary state of the Fock space is a linear superposition
of these eigenstates, with several non-zero coecients.
A-2. Creation operators
Choosing a basis of individual states, we now dene the action in the Fock
space of the creation operator
5
on a particle in the state.
A-2-a. Bosons
For bosons, we introduce the linear operatordened by:
12 =
+ 112 + 1 (A-16)
As all the states of the Fock space may be obtained by a linear superposition of1 2 ,
the action ofis dened in the entire space. It adds a particle to the system, which
goes from a state of()to a state of(+ 1), and in particular from the vacuum
to a state having one single occupied state.
Creation operators acting on the vacuum allow building occupied states. Recurrent
application of (A-16), leads to:
12 =
1
1!2!!
1
1
2
2
0 (A-17)
Comment:
Why was the factor
+ 1introduced in (A-16)? We shall see later () that, together
with the factors of (A-7), it simplies the computations.
5
A similar notation was used for the harmonic oscillator.
1596

A. GENERAL FORMALISM
A-2-b. Fermions
For fermions, we dene the operatorby:
= (A-18)
where the newly created stateappears rst in the list of states in the ket on the
right-hand side. If we start from a ket where the individual stateis already occupied
(= 1), the action ofleads to zero, as in this case (A-10) gives:
= = 0 (A-19)
Formulas (A-16) and (A-17) are also valid for fermions, with all the occupation numbers
equal to0or1(or else both members are zero).
Comment:
Denition (A-18) must not depend on the specic order of the individual states
in the ket on which the operatoracts. It can be easily veried that any permutation
of the states simply multiply by its parity both members of the equality. It therefore
remains valid independently of the order chosen for the individual states in the initial
ket.
A-3. Annihilation operators
We now study the Hermitian conjugate operator of, that we shall simply call
since taking twice in a row the Hermitian conjugate of an operator brings you back
to the initial operator.
A-3-a. Bosons
For bosons, we deduce from (A-16) that the only non-zero matrix elements of
in the Fock states orthonormal basis are:
12 + 1 12 =
+ 1 (A-20)
They link two vectors having equal occupation numbers except for, which increases
by one going from the ket to the bra.
The matrix elements of the Hermitian conjugate ofare obtained from relation
(A-20), using the general denition (B-49) of Chapter. The only non-zero matrix
elements ofare thus:
12 12 + 1=
+ 1 (A-21)
Since the basis we use is complete, we can deduce the action of the operatorson kets
having given occupation numbers:
12 =
12 1 (A-22)
(note that we have replacedby 1). As opposed to, which adds a particle
in the state, the operatortakes one away; it yields zero when applied on a ket
where the stateis empty to begin with, such as the vacuum state:
0= 0 (A-23)
We callthe annihilation operator for the state.
1597

CHAPTER XV CREATION AND ANNIHILATION OPERATORS FOR IDENTICAL PARTICLES
A-3-b. Fermions
For fermions, relation (A-18) allows writing the matrix elements:
= 1 (A-24)
The only non-zero elements are those where all the individual occupied states are left
unchanged in the bra and the ket, except for the stateonly present in the bra, but
not in the ket. As for the occupation numbers, none change, except forwhich goes
from0(in the ket) to1(in the bra).
The Hermitian conjugation operation then yields the action of the corresponding
annihilation operator:
= (A-25)
or, if initially the stateis not occupied:
= 0 (A-26)
Relations (A-22) and (A-23) are also valid for fermions, with the usual condition that all
occupation numbers should be equal to0or1; otherwise, the relations amount to0 = 0.
Comment:
To use relation (A-25) when the stateis already occupied but not listed in the rst
position, we rst have to bring it there; if it requires an odd permutation, a change of
sign will occur. For example:
212= 1 (A-27)
For fermions, the operatorsandtherefore act on the individual state that is listed in
the rst position in the-particle ket;destroys the rst state in the list, andcreates
a new state placed at the beginning of the list. Forgetting this could lead to errors in
sign.
A-4. Occupation number operators (bosons and fermions)
Consider the operatordened by:
= (A-28)
and its action on a Fock state. For bosons, if we apply successively formulas (A-22)
and (A-16), we see that this operator yields the same Fock state, but multiplied by its
occupation number. For fermions, ifis empty in the Fock state, relation (A-26)
shows that the action of the operatoryields zero. If the stateis already occupied,
we must rst permute the states to bringto the rst position, which may eventually
change the sign in front of the Fock space ket. The successive application on this ket of
(A-25) and (A-19) shows that the action of the operatorleaves this ket unchanged;
we then move the stateback to its initial position, which may introduce a second
change in sign, canceling the rst one. We nally obtain for fermions the same result
as for bosons, except that thecan only take the values1and0. In both cases the
Fock states are the eigenvectors of the operatorwith the occupation numbers as
1598

A. GENERAL FORMALISM
eigenvalues; consequently, this operator is named the occupation number operator of the
state. The operatorassociated with the total number of particles is simply the
sum:
= = (A-29)
A-5. Commutation and anticommutation relations
Creation and annihilation operators have very simple commutation (for the bosons)
and anticommutation (for the fermions) properties, which make them easy tools for taking
into account the symmetrization or antisymmetrization of the state vectors.
To simplify the notation, each time the equations refer to a single basis of individual
states, we shall writeinstead of. If, however, it can lead to ambiguity, we will
return to the full notation.
A-5-a. Bosons: commutation relations
Consider, for bosons, the two operatorsand. If both subscriptsandare
dierent, they correspond to orthogonal statesand . Using twice (A-16) then
yields:
12 =
+ 1+ 112 + 1 + 1 (A-30)
Changing the order of the operators yields the same result. As the Fock states form a
basis, we can deduce that the commutator ofandis zero if=. In the same way,
it is easy to show that both operator productsand acting on the same ket yield
the same result (a ket having two occupation numbers lowered by1);andthus
commute if=. Finally the same procedure allows showing thatandcommute
if=. Now, if=, we must evaluate the commutator ofand. Let us apply
(A-16) and (A-22) successively, rst in that order, and then in the reverse order:
12 = (+ 1)12
12 = ()12
(A-31)
The commutator of and is therefore equal to1for all the values of the
subscript. All the previous results are summarized in three equalities valid for bosons:
[ ] = 0 = 0 = (A-32)
A-5-b. Fermions: anticommutation relations
For fermions, let us rst assume that the subscriptsandare dierent. The
successive action ofandon an occupation number ket only yields a non-zero ket if
== 0; using twice (A-18) leads to:
= (A-33)
but, if we change the order:
= = (A-34)
1599

CHAPTER XV CREATION AND ANNIHILATION OPERATORS FOR IDENTICAL PARTICLES
Consequently the sign change that goes with the permutation of the two individual
states leads to:
= if= (A-35)
If we dene the anticommutator[]
+
of two operatorsandby:
[]
+
= + (A-36)
(A-35) may be written as:
+
= 0if= (A-37)
Taking the Hermitian conjugate of (A-35), we get:
= if= (A-38)
which can be written as:
[ ]
+
= 0if= (A-39)
Finally, we show by the same method that the anticommutator ofandis zero
except when it acts on a ket where= 1and= 0; those two occupation numbers
are then interchanged. The computation goes as follows:
= = (A-40)
and:
=
= = (A-41)
Adding those two equations yields zero, hence proving that the anticommutator is zero:
+
= 0si= (A-42)
In the case where=, the limitation on the occupation numbers (0or1) leads to:
2
= 0 and
2
= 0 (A-43)
Equalities (A-37) and (A-39) are still valid ifandare equal. We are now left with
the computation of the anticommutator ofand. Let us rst examine the product
; it yields zero if applied to a ket having an occupation number= 1, but leaves
unchanged any ket with= 0, since the particle created byis then annihilated by
. We get the inverse result for the productwhere the order has been inverted:
it yields zero if= 0, and leaves the ket unchanged if= 1. Finally, whatever the
occupation number ket is, one of the terms of the anticommutator yields zero, the other
1, and the net result is always1. Therefore:
+
= 1 (A-44)
All the previous results valid for fermions are summarized in the following three
relations, which are for fermions the equivalent of relations (A-32) for bosons:
[ ]
+
= 0
+
= 0
+
= (A-45)
1600

A. GENERAL FORMALISM
A-5-c. Common relations for bosons and fermions
To regroup the results valid for bosons and fermions in common relations, we
introduce the notation:
[]= (A-46)
with:
= 1for bosons
=1for fermions
(A-47)
so that (A-46) is the commutator of A and B for bosons, and their anticommutator for
fermions. We then have:
= 0for alland(equal or dierent)
= 0for alland(equal or dierent)
(A-48)
and the only non-zero combinations are:
= = (A-49)
A-6. Change of basis
What are the eects on the creation and annihilation operators of a change of
basis for the individual states? The operatorsand have been introduced by their
action on the Fock states, dened by relations (A-7) and (A-10) for which a given basis
of individual stateswas chosen. One could also choose any another orthonormal
basis and dene in the same way bases for the Fock state and creationand
annihilationoperators. What is the relation between these new operators and the
ones we dened earlier with the initial basis?
For creation operators acting on the vacuum state0, the answer is quite straight-
forward: the action ofon0yields a one-particle ket, which can be written as:
0=1 := 1 := 0 (A-50)
This result leads us to expect a simple linear relation of the type:
= (A-51)
with its Hermitian conjugate:
= (A-52)
Equation (A-51) implies that creation operators are transformed by the same unitary
relation as the individual states. Commutation or anticommutation relations are then
conserved, since:
= = (A-53)
1601

CHAPTER XV CREATION AND ANNIHILATION OPERATORS FOR IDENTICAL PARTICLES
which amounts to (as expected):
= = (A-54)
Furthermore, it is straightforward to show that the creation operators commute (or
anticommute), as do the annihilation operators.
Equivalence of the two bases
We have not yet shown the complete equivalence of the two bases, which can be done
following two dierent approaches. In the rst one, we use (A-51) and (A-52) to dene
the creation and annihilation operators in the new basis. The associated Fock states are
dened by replacing theby thein relations (A-17) for the bosons, and (A-18) for
the fermions. We then have to show that these new Fock states are still related to the
states with numbered particles as in (A-18) for bosons, and (A-10
will establish the complete equivalence of the two bases.
We shall follow a second approach where the two bases are treated completely symmet-
rically. Replacing in relations (A-7) and (A-10) theby the, we construct the new
Fock basis. We next dene the operatorsby transposing relations (A-17) and (A-18)
to the new basis. We then must verify that these operators obey relation (A-51), without
limiting ourselves, as in (A-50), to their action on the vacuum state.
(i) Bosons
Relations () and (A-17) lead to:
0
=
! 1 :; 2 :;;:;+ 1 :;;+:; (A-55)
where, on the right-hand side, therst particles occupy the same individual state,
the followingparticles, numbered from+ 1to+, the individual state, etc.
The equivalent relation in the second basis can be written:
0
=
! 1 :; 2 :;;:;+ 1 :;;+:; (A-56)
with:
++=++= (A-57)
Replacing on the right-hand side of (A-56), the rst ketby:
=
i
i i (A-58)
we obtain:
i
i
!1 :i; 2 :;;:;+ 1 :;;+:; (A-59)
1602

B. ONE-PARTICLE SYMMETRIC OPERATORS
Following the same procedure for all the basis vectors of the right-hand side, we can
replace it by:
i
i
i
i
j
j
j
j
!1 :i; 2 :i;;+ 1 :j;+ 2 :j;
(A-60)
or else
6
, taking into account (A-55):
i
i
i
i
j
j
j
j 0
(A-61)
We have thus shown that the operators act on the vacuum state in the
same way as the operators dened by (A-51), raised to the powers,, ..
When the occupation numbers,, .. can take on any values, the kets (A-56) span the
entire Fock space. Writing the previous equality forand+ 1, we see that the action
on all the basis kets ofand of yields the same result, establishing
the equality between these two operators. Relation (A-52) can be readily obtained by
Hermitian conjugation.
(ii) Fermions
The demonstration is identical, with the constraint that the occupation numbers are0or
1. As this requires no changes in the operator or state order, it involves no sign changes.
B. One-particle symmetric operators
Using creation and annihilation operators makes it much easier to deal, in the Fock space,
with physical operators that are thus symmetric ( of Chapter ). We rst
study the simplest of such operators, those which act on a single particle and are called
one-particle operators.
B-1. Denition
Consider an operatordened in the space of individual states;()acts in the
state space of particle. It could be for example the momentum of the-th particle, or
its angular momentum with respect to the origin. We now build the operator associated
with the total momentum of the-particle system, or its total angular momentum,
which is the sum overof all the()associated with the individual particles.
A one-particle symmetric operator acting in the space()for bosons - or()
for fermions - is therefore dened by:
()
=
=1
() (B-1)
6
In this relation, the rstsums are identical, as are the nextsums, etc.
1603

CHAPTER XV CREATION AND ANNIHILATION OPERATORS FOR IDENTICAL PARTICLES
(contrary to states, which are symmetric for bosons and antisymmetric for fermions,
the physical operators are always symmetric). The operatoracting in the Fock space
is dened as the operator
()
acting either in()or in(), depending on the
specic case. Since the basis for the entire Fock space is the union of the bases of these
spaces for all values of, the operatoris thus well dened in the direct sum of all
these subspaces. To summarize:
()
;= 123 = (B-2)
Using (B-1) directly to compute the matrix elements ofoften leads to tedious manip-
ulations. Starting with an operator involving numbered particles, we place it between
states with numbered particles; we then symmetrize the bra, the ket, and take into
account the symmetry of the operator (cf. footnote). This introduces several summa-
tions (on the particles and on the permutations) that have to be properly regrouped to
be simplied. We will now show that expressingin terms of creation and annihilation
operators avoids all these intermediate calculations, taking nevertheless into account all
the symmetry properties.
B-2. Expression in terms of the operatorsand
We choose a basis for the individual states. The matrix elementsof the
one-particle operatorare given by:
= (B-3)
They can be used to expand the operator itself as follows:
() = : : (): := : : (B-4)
B-2-a. Action of
()
on a ket withparticles
Using in (B-1) the expression (B-4) for()leads to:
()
=
=1
: : (B-5)
The action of
()
on a symmetrized ket written as (A-9) therefore includes a sum over
andof terms:
=1
: : (B-6)
with coecients. Let us use (A-7) or (A-10) to compute this ket for given values
ofand. As the operator contained in the bracket is symmetric with respect to the
exchange of particles, it commutes with the two operatorsand ( C-4-a-of
Chapter )), and the ket can be written as:
!!!!
=1
: :
1 :; 2 :;;:;+ 1 :;;:; (B-7)
1604

B. ONE-PARTICLE SYMMETRIC OPERATORS
In the summation over, the only non-zero terms are those for which the individual
statecoincides with the individual stateoccupied in the ket on the right by the
particle labeled q; there aredierent values ofthat obey this condition (i.e. none or
one for fermions). For theseterms, the operator: :transforms the state
into, then(or) reconstructs a symmetrized (but not normalized) ket:
!!!!
1 :;;+ 1 :;;:; (B-8)
This ket is always the same for all the numbersamong theselected ones (for fermions,
this term might be zero, if the statewas already occupied in the initial ket). We
shall then distinguish two cases:
(i) For=, and for bosons, the ket written in (B-8) equals:
+ 1
(B-9)
where the square root factor comes from the variation in the occupation numbers
and, which thus change the numerical coecients in the denition (A-7) of the Fock
states. As this ket is obtainedtimes, this factor becomes
(+ 1). This is exactly
the factor obtained by the action on the same symmetrized ket of the operator,
which also removes a particle from the stateand creates a new one in the state.
Consequently, the operatorreproduces exactly the same eect as the sum over.
For fermions, the result is zero except when, in the initial ket, the statewas
occupied by a particle, and the stateempty, in which case no numerical factor
appears; as before, this is exactly what the action of the operatorwould do.
(ii) if=, for bosons the only numerical factor involved is, coming from the
number of terms in the sum overthat yields the same symmetrized ket. For fermions,
the only condition that yields a non-zero result is for the stateto be occupied, which
also leads to the factor. In both cases, the sum overamounts to the action of the
operator.
We have shown that:
=1
: := (B-10)
The summation overandin (B-5) then yields:
()
= = (B-11)
B-2-b. Expression valid in the entire Fock space
The right-hand side of (B-11) contains an expression completely independent of
the space()or()in which we dened the action of the operator
()
. Since
we dened operatoras acting as
()
in each of these subspaces having xed, we
can simply write:
= (B-12)
1605

CHAPTER XV CREATION AND ANNIHILATION OPERATORS FOR IDENTICAL PARTICLES
This is the expression of one-particle symmetric operators we were looking for. Its form
is valid for any value ofand the particles are no longer numbered; it contains equal
numbers of creation and annihilation operators, which only act on occupation numbers.
Comment:
Choosing the proper basis , it is always possible to diagonalize the Hermitian
operatorand write:
= (B-13)
Equality (B-11) is then simply written as:
= = (B-14)
where= is the occupation number operator in the statedened in (A-28).
B-3. Examples
A rst very simple example is the operator, already described in (A-29), and
corresponding to the total number of particles:
= = (B-15)
As expected, this operator does not depend on the basischosen to count the
particles, as we now show. Using the unitary transformations of operators (A-51) and
(A-52), and with the full notation for the creation and annihilation operators to avoid
any ambiguity, we get:
= = (B-16)
which shows that:
= = (B-17)
For a spinless particle one can also dene the operator corresponding to the prob-
ability density at pointr0:
=r0r0 (B-18)
Relation (B-12) then leads to the particle local density (or single density) operator:
(r0) = (r0)(r0) (B-19)
The same procedure as above shows that this operator is independent of the basis
chosen in the individual states space.
1606

B. ONE-PARTICLE SYMMETRIC OPERATORS
Let us assume now that the chosen basis is formed by the eigenvectorskof a
particle's momentum}k, and that the corresponding annihilation operators are noted
k. The operator associated with the total momentum of the system can be written as:
P=
k
}k
kk=
k
}kk (B-20)
As for the kinetic energy of the particles, its associated operator is expressed as:
0=
k
}
2
k
2
2
kk=
k
}
2
k
2
2
k (B-21)
B-4. Single particle density operator
Consider the average valueof a one-particle operatorin an arbitrary-
particle quantum state. It can be expressed, using relation (B-12), as a function of the
average values of operator products:
= (B-22)
This expression is close to that of the average value of an operator for a physical system
composed of a single particle. Remember (ComplementIII, ) that if a system is
described by a single particle density operator1(1), the average value of any operator
(1)is written as:
(1)=Tr(1)1(1)= 1 (B-23)
The above two expressions can be made to coincide if, for the system of identical particles,
we introduce a density operator reduced to a single particle1whose matrix elements
are dened by:
1 = (B-24)
This reduced operator allows computing average values of all the single particle operators
as if the system consisted only of a single particle:
=Tr 1 (B-25)
where the trace is taken in the state space of a single particle.
The trace of the reduced density operator thus dened is not equal to unity, but
to the average particle number as can be shown using (B-24) and (B-15):
Tr1= = (B-26)
1607

CHAPTER XV CREATION AND ANNIHILATION OPERATORS FOR IDENTICAL PARTICLES
This normalization convention can be useful. For example, the diagonal matrix element
of1in the position representation is simply the average of the particle local density
dened in (B-19):
r01r0= (r0) (B-27)
It is however easy to choose a dierent normalization for the reduced density operator:
its trace can be made equal to1by dividing the right-hand side of denition (B-24) by
the factor.
C. Two-particle operators
We now extend the previous results to the case of two-particle operators.
C-1. Denition
Consider a physical quantity involving two particles, labeledand. It is as-
sociated with an operator()acting in the state space of these two particles (the
tensor product of the two individual state's spaces). Starting from this binary operator,
the easiest way to obtain a symmetric-particle operator is to sum all the()over
all the particlesand, where the two subscriptsandrange from1to. Note,
however, that in this sum all the terms where=add up to form a one-particle
operator of exactly the same type as those studied in . Consequently, to obtain a
real two-particle operator we shall exclude the terms where=and dene:
()
=
1
2
=1;=
() (C-1)
The factor12present in this expression is arbitrary but often handy. If for example the
operator describes an interaction energy that is the sum of the contributions of all the
distinct pairs of particles,()and()corresponding to the same pair are equal
and appear twice in the sum overand: the factor12avoids counting them twice.
Whenever() =(), it is equivalent to write
()
in the form:
()
= () (C-2)
As with the one-particle operators, expression (C-1) denes symmetric operators
separately in each physical state's space having a given particle number. This denition
may be extended to the entire Fock space, which is their direct sum over all. This
results in a more general operator, following the same scheme as for (B-2):
()
;= 123 = (C-3)
1608

C. TWO-PARTICLE OPERATORS
C-2. A simple case: factorization
Let us rst assume the operator()can be factored as:
() =()() (C-4)
The operator written in (C-1) then becomes:
()
=
1
2
=1;=
()() =
1
2
=1
()
=1
()
=1
()() (C-5)
The right-hand side of this expression starts with a product of one-particle operators,
each of which can be replaced, following (B-11), by its expression as a function of the
creation and annihilation operators:
=1
() = and
=1
() = (C-6)
As for the last term on the right-hand side of (C-5), it is already a single particle operator:
=1
()() = (C-7)
This leads to:
()
=
1
2
(C-8)
We can then use general relations (A-49) to transform the operator product:
= + = + (C-9)
Including this form in the rst term on the right-hand side of (C-8) yields, for the
contribution:
= (C-10)
which exactly cancels the second term of (C-8). Consequently, we are left with:
()
=
1
2
(C-11)
As the right-hand side of this expression has the same form in all spaces having a xed
, it is also valid for the operatoracting in the entire Fock space.
1609

CHAPTER XV CREATION AND ANNIHILATION OPERATORS FOR IDENTICAL PARTICLES
C-3. General case
Any two-particle operator()may be decomposed as a sum of products of
single particle operators:
() = ()() (C-12)
where the coecientsare numbers
7
. Hence expression (C-1) can be written as:
()
=
1
2
=1;=
() =
1
2
=1=1;=
()() (C-13)
In this linear combination with coecients, each term (corresponding to a given
and) is of the form (C-5) and can therefore be replaced by expression (C-11). This
leads to:
()
=
1
2
(C-14)
The right-hand side of this equation has the same form in all the spaces of xed; hence
it is valid in the entire Fock space. Furthermore, we recognize in the summation over
andthe matrix element ofas dened by (C-12):
=1 :; 2 :(12)1 :; 2 := (C-15)
The nal result is then:
=
1
2
1 :; 2 :(12)1 :; 2 :
(C-16)
which is the general expression for a two-particle symmetric operator.
As for the one-particle operators, each term of expression (C-16) for the two-
particle operators contains equal numbers of creation and annihilation operators. Con-
sequently, these symmetric operators do not change the total number of particles, as was
obvious from their initial denition.
C-4. Two-particle reduced density operator
Relation (C-16) implies that the average value of any two-particle operator may
be written as:
=
1
2
1 :; 2 :(12)1 :; 2 : (C-17)
7
The two-particle state space is the tensor product of the two spaces of individual states (see
of Chapter). In the same way, the space of operators acting on two particles is the tensor product of
the spaces of operators acting separately on these particles. For example, the operator for the interaction
potential between two particles can be decomposed as a sum of products of two operators: the rst one
is a function of the position of the rst particle, and the second one of the position of the second particle.
1610

C. TWO-PARTICLE OPERATORS
Figure 1: Physical interaction between two identical particles: initially in the states
and (schematized by the lettersand), the particles are transferred to the states
and (schematized by the lettersand)
This expression is similar to the average value of an operator(12)for a two-particle
system having a density operator2(12):
(12)= 1 :; 2 :(12)1 :; 2 :
1 :; 2 : 2(12)1 :; 2 : (C-18)
which leads us to dene a two-particle reduced density operator2:
1 :; 2 : 21 :; 2 := (C-19)
In this denition we have left out the factor12of (C-17) since this will lead
to a normalization of2often more handy: the matrix element of2in the position
representation yields directly the double density (as well as the eld correlation function
that we shall study in ). The trace of2is then written:
Tr2= =
= 1 (C-20)
It is obviously possible to divide the right-hand side of the denition of2either by the
factor2, or else by the factor 1if we wish its trace to be equal to1.
C-5. Physical discussion; consequences of the exchange
As mentioned in the introduction of this chapter, the equations no longer contain
labeled particles, permutations, symmetrizers and antisymmetrizers; the total number of
particleshas also disappeared. We may now continue the discussion begun in
of Chapter
longer specify the total particle number.
1611

CHAPTER XV CREATION AND ANNIHILATION OPERATORS FOR IDENTICAL PARTICLES
C-5-a. Two terms in the matrix elements
Consider a physical process (schematized in Figure) where, in a system of
identical particles, an interaction produces a transfer from the two statesand
towards the two states and ; we assume that the four states we are dealing
with are dierent. In the summation over of (C-16), the only terms involved in
this process are those where the bra contains either=and=, or the opposite
= and=; as for the ket, it must contain either= and=, or the
opposite=and=. We are then left with four terms:
1
2
1 :; 2 : 1 :; 2 :
1
2
1 :; 2 : 1 :; 2 :
1
2
1 :; 2 : 1 :; 2 :
1
2
1 :; 2 : 1 :; 2 :
(C-21)
However, since the numbers used to label the particles are dummy variables, the rst
two matrix elements shown in (C-21) are equal and so are the last two. In addition, the
product of creation and annihilation operators obey the following relations, for bosons
(= 1) as well as for fermions (=1):
=
= =
(C-22)
These relations are obvious for bosons since we only commute either creation operators
or annihilation operators. For fermions, as we assumed all the states were dierent, the
anticommutation of operatorsor of operators, leads to sign changes; these may
cancel out depending on whether the number of anticommutations is even or odd. If we
now double the sum of the rst and last term of (C-21), we obtain the nal contribution
to (C-16):
1 :; 2 : 1 :; 2 :
+ 1 :; 2 : 1 :; 2 : (C-23)
Hence we are left with two terms whose relative sign depends on the nature (bosons
or fermions) of the identical particles. They correspond to a dierent switching point
for the incoming and outgoing individual states (Fig.).
For bosons, the product of the4operators in (C-23) acting on an occupation
number ket introduces the square root:
+ 1(+ 1) (C-24)
For large occupation numbers, this square root may considerably increase the value of
the matrix element. For fermions, however, this amplication eect does not occur.
Furthermore, if the direct and exchange matrix elements ofare equal, they will cancel
each other in (C-23) and the corresponding transition amplitude of this process will be
zero.
1612

C. TWO-PARTICLE OPERATORS
Figure 2: Two diagrams representing schematically the two terms appearing in equation
(C-23); they dier by an exchange of the individual states of the exit particles. They
correspond, in a manner of speaking, to a dierent switching point for the incoming
and outgoing states. The solid lines represent the particles' free propagation, and the
dashed lines their binary interaction.
C-5-b. Particle interaction energy, the direct and exchange terms
Many physics problems involve computing the average particle interaction energy.
For the sake of simplicity, we shall only study here spinless particles (or, equivalently,
particles being in the same internal spin state so that the corresponding quantum number
does not come into play) and assume their interactions to be binary. These interactions
are then described by an operatorint, diagonal in ther1r2rbasis (eigenstates
of all the particles' positions), which multiplies each of these states by the function:
int(r1r2r) = 2(rr) (C-25)
In this expression, the function2(rr)yields the diagonal matrix elements of the
operator2(RR)associated with the two-particle()interaction, whereRis
the quantum operator associated with the classical positionr. The matrix elements of
this operator in the;basis is simply obtained by inserting a closure relation for
each of the two positions. This leads to:
1 :; 2 : 2(R1R2)1 :; 2 :
=d
3
1d
3
22(r1r2)(r1)(r2)(r1)(r2) (C-26)
. General expression:
Replacing in (C-16) operator(12)byint(R1R2)and taking (C-26) into ac-
count, we get:
int=
1
2
d
3
1d
3
22(r1r2)(r1)(r2)(r1)(r2) (C-27)
1613

CHAPTER XV CREATION AND ANNIHILATION OPERATORS FOR IDENTICAL PARTICLES
We can thus write the average value of the interaction energy in any normalized state
as:
int= 2=
1
2
d
3
1d
3
22(r1r2)2(r1r2) (C-28)
where2(r1r2)is the spatial correlation function dened by:
2(r1r2) = (r1)(r2)(r1)(r2) (C-29)
Consequently, knowing the correlation function2(r1r2)associated with the quantum
stateallows computing directly, by a double spatial integration, the average interac-
tion energy in that state.
Actually, as we shall see in more detail in ,2(r1r2)is
simply thedouble density, equal to the probability density of nding any particle in
r1and another one inr2. The physical interpretation of (C-28) is simple: the average
interaction energy is equal to the sum over all the particles' pairs of the interaction energy
int(r1r2)of a pair, multiplied by the probability of nding such a pair at pointsr1
andr2(the factor12avoids the double counting of each pair).
. Specic case: the Fock states
Let us assume the stateis a Fock state, with specied occupation numbers:
= 1:1;2:2;;:; (C-30)
We can compute explicitly, as a function of the, the average values:
(C-31)
contained in (C-29). We rst notice that to get a non-zero result, the two operators
must create particles in the same states from which they were removed by the two
annihilation operators. Otherwise the action of the four operators on the ketwill
yield a new Fock state orthogonal to the initial one, and hence a zero result. We must
therefore impose either=and=, or the opposite=and=, or eventually
the special case where all the subscripts are equal. The rst case leads to what we call
the direct term, and the second, the exchange term. We now compute their values.
(i) Direct term,=and=, shown on the left diagram of Figure. If
===, the four operators acting onreconstruct the same ket, multiplied by
the factor( 1); this yields a zero result for fermions. If=, we can move the
operator=just to the right of the rst operatorto form the particle number
operator^. This permutation in the operators' order does not change anything: for
bosons, we are moving commuting operators, and for fermions, two anticommutations
introduce two minus signs, which cancel each other. The same goes for the operators
with subscript, leading to the particle number^. Finally, the direct term is equal to:
dir
2(r1r2) =
=
(r1)
2
(r2)
2
+ ( 1)(r1)
2
(r2)
2
(C-32)
1614

C. TWO-PARTICLE OPERATORS
Figure 3: Schematic representation of a direct term (left diagram where each particle
remains in the same individual state) and an exchange term (right diagram where the
particles exchange their individual states). As in Figure, the solid lines represent the
particles free propagation, and the dashed lines their binary interaction.
where the second sum is zero for fermions (is equal to0or1).
(ii) Exchange term,=and=, shown on the right diagram of Figure. The
case where all four subscripts are equal is already included in the direct term. To get the
operators' product^^starting from the product , we just have to permute the
two central operators; when=this operation is of no consequence for bosons,
but introduces a change of sign for fermions (anticommutation). The exchange term can
therefore be written as
ex
2(r1r2), with:
ex
2(r1r2) =
=
(r1)(r2)(r2)(r1) (C-33)
Finally, the spatial correlation function (or double density)2(r1r2)is the sum
of the direct and exchange terms:
2(r1r2) =
dir
2(r1r2) +
ex
2(r1r2) (C-34)
where the factorin front of the exchange term is1for bosons and1for fermions.
The direct term only contains the product(r1)
2
(r2)
2
of the probability densities
associated with the individual wave functions(r1)and(r2); it corresponds to non-
correlated particles. We must add to it the exchange term, which has a more complex
mathematical form and reveals correlations between the particles, even when they do
not interact with each other. These correlations come from explicitly taking into account
the fact that the particles are identical (symmetrization or antisymmetrization of the
state vector). They are sometimes called statistical correlations and their spatial
dependence will be studied in more detail in ComplementXVI.
Conclusion
The creation and annihilation operators introduced in this chapter lead to compact and
general expressions for operators acting on any particle number. These expressions
involve the occupation numbers of the individual states but the particles are no longer
1615

CHAPTER XV CREATION AND ANNIHILATION OPERATORS FOR IDENTICAL PARTICLES
numbered. This considerably simplies the computations performed on -body sys-
tems, likeinteracting bosons or fermions. The introduction of approximations such
as the mean eld approximation used in the Hartree-Fock method (ComplementXV)
will also be facilitated.
We have shown the complete equivalence between this approach and the one where
we explicitly take into account the eect of permutations between numbered particles.
It is important to establish this link for the study of certain physical problems. In spite
of the overwhelming eciency of the creation and annihilation operator formalism, the
labeling of particles is sometimes useful or cannot be avoided. This is often the case for
numerical computations, dealing with numbers or simple functions that require numbered
particles and which, if needed, will be symmetrized (or antisymmetrized) afterwards.
In this chapter, we have only considered creation and annihilation operators with
discrete subscripts. This comes from the fact that we have only used discrete bases
or for the individual states. Other bases could be used, such as the position
eigenstatesrof a spinless particle. The creation and annihilation operators will then
be labeled by a continuous subscriptr. Fields of operators are thus introduced at each
space point: they are called eld operators and will be studied in the next chapter.
1616

COMPLEMENTS OF CHAPTER XV, READER'S GUIDE
AXV: PARTICLES AND HOLES In an ideal gas of fermions, one can dene cre-
ation and annihilation operators of holes (ab-
sence of a particle). Acting on the ground state,
these operators allow building excited states.
This is an important concept in condensed mat-
ter physics.
Easy to grasp, this complement can be consid-
ered to be a preliminary to ComplementXV.
BXV: IDEAL GAS IN THERMAL EQUILIBRIUM;
QUANTUM DISTRIBUTION FUNCTIONS
Studying the thermal equilibrium of an ideal
gas of fermions or bosons, we introduce the
distribution functions characterizing the phys-
ical properties of a particle or of a pair of
particles. These distribution functions will be
used in several other complements, in particular
GXVandXV. Bose-Einstein condensation is
introduced in the case of bosons. The equation
of state is discussed for both types of particles.
The list of complements continues on the next page
1617

Series of four complements, discussing the behavior of particles interacting through a mean
eld created by all the others. Important, since the mean eld concept is largely used throughout
many domains of physics and chemistry.
CXV: CONDENSED BOSON SYSTEM, GROSS-
PITAEVSKII EQUATION
DXV: TIME-DEPENDENT GROSS-PITAEVSKII
EQUATION
EXV: FERMION SYSTEM, HARTREE-FOCK
APPROXIMATION
FXV : FERMIONS, TIME-DEPENDENT
HARTREE-FOCK APPROXIMATION
CXV: This complement shows how to use a
variational method for studying the ground
state of a system of interacting bosons. The
system is described by a one-particle wave
function in which all the particles of the system
accumulate. This wave function obeys the
Gross-Pitaevskii equation.
DXV: This complement generalizes the previous
one to the case where the Gross-Pitaevskii wave
function is time-dependent. This allows us
to obtain the excitation spectrum (Bogolubov
spectrum), and to discuss metastable ows
(superuidity).
EXV: An ensemble of interacting fermions
can be treated by a variational method, the
Hartree-Fock approximation, which plays an
essential role in atomic, molecular and solid
state physics. In this approximation, the
interaction of each particle with all the others
is replaced by a mean eld created by the other
particles. The correlations introduced by the
interactions are thus ignored, but the fermions'
indistinguishability is accurately treated. This
allows computing the energy levels of the system
to an approximation that is satisfactory in many
situations.
FXV: We often have to study an ensemble
of fermions in a time-dependent situation, as
for example electrons in a molecule or a solid
subjected to an oscillating electric eld. The
Hartree-Fock mean eld method also applies
to time-dependent problems. It leads to a set
of coupled equations of motion involving a
Hartree-Fock mean eld potential, very similar
to the one encountered for time-independent
problems.
The list of complements continues on the next page
1618

The mean eld approximation can also be used to study the properties, at thermal equi-
librium, of systems of interacting fermions or bosons. The variational method amounts to opti-
mizing the one-particle reduced density operator. It permits generalizing to interacting particles
a number of results obtained for an ideal gas (ComplementXV).
GXV: FERMIONS OR BOSONS: MEAN FIELD
THERMAL EQUILIBRIUM
HXV: APPLICATIONS OF THE MEAN FIELD
METHOD FOR NON-ZERO TEMPERATURES
(FERMIONS AND BOSONS)
GXV: The trial density operator at non-zero
temperature can be optimized using a varia-
tional method. This leads to self-consistent
Hartree-Fock equations, of the same type as
those derived in ComplementXV. We thus
obtain an approximate value for the thermody-
namic potential.
HXV: This complement discusses various
applications of the method described in the
previous complement: spontaneous magnetism
of an ensemble of repulsive fermions, equation of
state for bosons and instability in the presence
of attractive interactions.
1619

PARTICLES AND HOLES
Complement AXV
Particles and holes
1 Ground state of a non-interacting fermion gas
2 New denition for the creation and annihilation operators
3 Vacuum excitations
Creation and annihilation operators are frequently used in solid state physics where
the notion of particle and hole plays an important role. A good example is the study of
metals or semiconductors, where we talk about an electron-hole pair created by photon
absorption. A hole means an absence of a particle, but it has properties similar to a
particle, like a mass, a momentum, an energy; the holes obey the same fermion statistics
as the electrons they replace. Using creation or annihilation operators allows a better
understanding of the hole concept. We will remain in the simple framework of a free
particle gas, but the concepts can be generalized to the case of particles placed in an
external potential or a Hartree-Fock mean potential (ComplementXV).
1. Ground state of a non-interacting fermion gas
Consider a system of non-interacting fermions in their ground state. We assume for
simplicity that they are all in the same spin state, and thus introduce no spin index
(generalization to several spin states is fairly simple). As we showed in ComplementXIV,
this system in its ground state is described by a state where all the occupation numbers
of the individual states having an energy lower than the Fermi energyare equal to
1, and all the other individual states are empty. In momentum space, the only occupied
states are all the individual states whose wave vectorkis included in a sphere (called
the Fermi sphere) of radius(the Fermi radius) given by
1
:
}
2
()
2
2
= =
}
2
2
6
2
3
23
(1)
where we have used the notation of formula (7) in ComplementXIV: is the Fermi
energy (proportional to the particle density to the power23), andthe edge length
of the cube containing theparticles. When the system is in its ground state, all the
individual states inside the Fermi sphere are occupied, whereas all the other individual
states are empty. Choosing for the individual states basisthe plane wave basis,
noted kto explicit the wave vectork, the occupation numbers are:
k= 1 ifk
k= 0 ifk
(2)
1
In ComplementXIVwe had assumed that both spin states of the electron gas were occupied,
whereas this is not the case here. This explains why the bracket in formula (1) contains the coecient
6
2
instead of3
2
.
1621

COMPLEMENT A XV
In a macroscopic system, the number of occupied states is very large, of the order of the
Avogadro number (10
23
). The ground state energy is given by:
0=
k
k (3)
with:
k=
}
2
()
2
2
(4)
The sum overkin (3) must be interpreted as a sum over all thekvalues that obey the
boundary conditions in the box of volume
3
, as well as the restriction on the length of
the vectorkwhich must be smaller or equal to.
2. New denition for the creation and annihilation operators
We now consider this ground state as a new vacuum0and introduce creation op-
erators that, acting on this vacuum, create excited states for this system. We dene:
ifk
k
=k
k=
k
ifk
k=k
k
=
k
(5)
Outside the Fermi sphere, the new operators
k
andkare therefore simple operators
of creation (or annihilation) of a particle in a momentum state that is not occupied in
the ground state. Inside the Fermi sphere, the results are just the opposite: operator
k
creates a missing particle, that we shall call a hole; the adjoint operatorkrepopulates
that level, hence destroying the hole. It is easy to show that the anticommutation
relations for the new operators are:
kk
+
=
kk
+
= 0
kk
+
=kk
(6)
as well as:
kk
+
=
kk
+
= 0
kk
+
=kk
(7)
which are the same as for ordinary fermions. Finally, the cross anticommutation relations
are:
kk
+
=
kk
+
= kk
+
=
kk
+
= 0 (8)
1622

PARTICLES AND HOLES
3. Vacuum excitations
Imagine, for example, that with this new point of view we apply an annihilation operator
k, withk , to the new vacuum0. The result must be zero since it is
impossible to annihilate a non-existent hole. From the old point of view and according to
(5), this amounts to applying the creation operator
k
to a system where the individual
statekis already occupied, and the result is indeed zero, as expected. On the other
hand, if we apply the creation operator
k
, withk , to the new vacuum, the
result is not zero: from the old point of view, it removes a particle from an occupied
state, and in the new point of view it creates a hole that did not exist before. The two
points of view are consistent.
Instead of talking about particles and holes, one can also use a general term,
excitations (or quasi-particles). The creation operator of an excitation ofk is
the creation operator
k
=kof a hole ; the creation operator of an excitation ofk
is the creation operator
k
=
k
of a particle. The vacuum state dened initially
is a common eigenvector of all the particle annihilation operators, with eigenvalues zero;
in a similar way, the new vacuum state0is a common eigenvector of all the excitation
annihilation operators. We therefore call it the quasi-particle vacuum.
As we have neglected all particle interactions, the system Hamiltonian is written
as:
=
k
kk=
k
kkk=
k
kkk
+
k
kkk (9)
Taking into account the anticommutation relations between the operatorskand
k
we
can rewrite this expression as:
0=
k
kkk+
k
kkk (10)
where0has been dened in (3) and simply shifts the origin of all the system energies.
Relation (10) shows that holes (excitations withk ) have a negative energy,
as expected since they correspond to missing particles. Starting from its ground state,
to increase the system energy keeping the particle number constant, we must apply
the operator
kk
that creates both a particle and a hole: the system energy is then
increased by the quantity ; inversely, to decrease the system energy, the adjoint
operatorkkmust be applied.
Comments:
(i) We have discussed the notion of hole in the context of free particles, but nothing in
the previous discussion requires the one-particle energy spectrum to be simply quadratic
as in (4). In semi-conductor physics for example, particles often move in a periodic
potential, and occupy states in the valence band when their energy is lower than the
Fermi level, whereas the others occupy the conduction band, separated from the
previous band by an energy gap. Sending a photon with an energy larger than this
gap allows the creation of an electron-hole pair, easily studied in the formalism we just
introduced.
1623

COMPLEMENT A XV
A somewhat similar case occurs when studying the relativistic Dirac wave equation, where
two energy continuums appear: one with energies greater than the electron rest energy
2
(whereis the electron mass, andthe speed of light), and one for negative energies
less than
2
associated with the positron (the antiparticle of the electron, having the
opposite charge). The energy spectrum is relativistic, and thus dierent from formula
(4), even inside each of those two continuums. However, the general formalism remains
valid, the operators
k
andkdescribing now, respectively, the creation and annihilation
of a positron. The Dirac equation however leads to diculties by introducing for example
an innity of negative energy states, assumed to be all occupied to avoid problems. A
proper treatment of this type of relativistic problems must be done in the framework of
quantum eld theory.
(ii) An arbitrary-particle Fock statedoes not have to be the ground state to be
formally considered as a quasi-particle vacuum. We just have to consider any annihi-
lation operator on an already occupied individual state as a creation operator of a hole
(i.e. of an excitation); we then dene the corresponding hole (or excitation) annihila-
tion operators, which all have in common the eigenvectorwith eigenvalue zero. This
comment will be useful when studying the Wick theorem (ComplementXVI). In
of Chapter , we shall see another example of a quasi-particle vacuum, but where,
this time, the new annihilation operators are no longer acting on individual states but
on states of pairs of particles.
1624

IDEAL GAS IN THERMAL EQUILIBRIUM; QUANTUM DISTRIBUTION FUNCTIONS
Complement BXV
Ideal gas in thermal equilibrium; quantum distribution functions
1 Grand canonical description of a system without interactions 1626
1-a Density operator
1-b Grand canonical partition function, grand potential
2 Average values of symmetric one-particle operators
2-a Fermion distribution function
2-b Boson distribution function
2-c Common expression
2-d Characteristics of Fermi-Dirac and Bose-Einstein distributions
3 Two-particle operators
3-a Fermions
3-b Bosons
3-c Common expression
4 Total number of particles
4-a Fermions
4-b Bosons
5 Equation of state, pressure
5-a Fermions
5-b Bosons
This complement studies the average values of one- or two-particle operators for an
ideal gas, in thermal equilibrium. It includes a discussion of several useful properties of
the Fermi-Dirac and Bose-Einstein distribution functions, already introduced in Chapter
XIV.
To describe thermal equilibrium, statistical mechanics often uses the grand canon-
ical ensemble, where the particle number may uctuate, with an average value xed by
the chemical potential(cf. Appendix, where you will nd a number of useful con-
cepts for reading this complement). This potential plays, with respect to the particle
number, a role similar to the role the inverse of the temperature term= 1 plays
with respect to the energy (is the Boltzmann constant). In quantum statistical me-
chanics, Fock space is a good choice for the grand canonical ensemble as it easily allows
changing the total number of particles. As a direct application of the results of
C , we shall compute the average values of symmetric one- or two-particle
operators for a system of identical particles in thermal equilibrium.
We begin in
show in
in terms of the Fermi-Dirac and Bose-Einstein distribution functions, increasing their
application range and hence their importance. In , we shall study the equation of
state for an ideal gas of fermions or bosons at temperatureand contained in a volume
.
1625

COMPLEMENT B XV
1. Grand canonical description of a system without interactions
We rst recall how a system of non-interacting particles is described, in quantum sta-
tistical mechanics, by the grand canonical ensemble; more details on this subject can be
found in Appendix, .
1-a. Density operator
Using relations (42) and (43) of Appendix, we can write the grand canonical
density operator(whose trace has been normalized to1) as:
=
1
(1)
whereis the grand canonical partition function:
=Tr (2)
In these relations,= 1()is the inverse of the absolute temperaturemultiplied by
the Boltzmann constant, and, the chemical potential (which may be xed by a large
reservoir of particles). Operatorsandare, respectively, the system Hamiltonian and
the particle number operator dened by (B-15) in Chapter.
Assuming the particles do not interact, equation (B-1) of Chapter
ing the system Hamiltonianas a sum of one-particle operators, in each subspace having
a total number of particles equal to:
=
=1
() (3)
Let us call the basis of the individual states that are the eigenstates of the operator
. Notingandthe creation and annihilation operators of a particle in these states,
may be written as in (B-14):
= = (4)
where theare the eigenvalues of. Operator (1) can also be written as:
=
1
( )
=
1 ( )
(5)
We shall now compute the average values of all the one- or two-particle operators for a
system described by the density operator (1).
1626

IDEAL GAS IN THERMAL EQUILIBRIUM; QUANTUM DISTRIBUTION FUNCTIONS
1-b. Grand canonical partition function, grand potential
In statistical mechanics, the grand potentialassociated with the grand canon-
ical equilibrium is dened as the (natural) logarithm of the partition function, multiplied
by (cf.Appendix, ):
= ln (6)
whereis given by (2). The trace appearing in this equation is easily computed in the
basis of the Fock states built from the individual states, as we now show. The
trace of a tensor product of operators (Chapter, ) is simply the product of the
traces of each operator. The Fock space has the structure of a tensor product of the
spaces associated with each of the(each being spanned by kets having a population
ranging from zero to innity see comment (i) of ); we must
thus compute a product of traces in each of these spaces. For a xed, we sum all the
diagonal elements over all the values of, then take the product over all's, which
leads to:
= exp [ ( )] (7)
. Fermions
For fermions, ascan only take the values0or1(two identical fermions never
occupy the same individual state), we get:
fermions= 1 +
( )
(8)
and:
fermions= ln1 +
( )
(9)
The indexmust be summed over all the individual states. In case these states are
also labeled by orbital and spin subscripts, these must also be included in the summation.
Let us consider for example particles having a spinand contained in a box of volume
with periodic boundary conditions. The individual stationary states may be written as
k, wherekobeys the periodic boundary conditions (ComplementXIV, ) and
the subscripttakes(2+ 1)values. Assuming the particles to be free in the box (no
spin Hamiltonian), eachvalue yields the same contribution tofermions; in the large
volume limit, expression (9) then becomes:
fermions=(2+ 1)
(2)
3
d
3
ln1 +
( )
(10)
. Bosons
For bosons, the summation overin (7) goes from= 0to innity, which
introduces a geometric series whose sum is readily computed. We therefore get:
bosons=
1
1
( )
(11)
1627

COMPLEMENT B XV
which leads to:
bosons= ln1
( )
(12)
For a system of free particles with spin, conned in a box with periodic boundary
conditions, we obtain, in the large volume limit:
bosons= (2+ 1)
(2)
3
d
3
ln1
( )
(13)
In a general way, for fermions as well as bosons, the grand potential directly yields
the pressure, as shown in relation (61) of AppendixVI:
= (14)
Using the proper derivatives with respect to the equilibrium parameters (temperature,
chemical potential, volume), it also yields the other thermodynamic quantities such as
the energy, the specic heats, etc.
2. Average values of symmetric one-particle operators
Symmetric quantum operators for one, and then for two particles, were introduced in
a general way in Chapter ). The general expression for a one-particle
operatoris given by equation (B-12) of that chapter. We can thus write:
= (15)
with, when the state of the system is given by the density operator (1):
=Tr =
1
Tr (16)
This trace can be computed in the Fock state basis1 associated with
the eigenstates basis of. If=, operator destroys a particle in the
individual stateand creates another one in the dierent state; it therefore
transforms the Fock state1 into a dierent, hence orthogonal, Fock state
1 1 + 1. Operatorthen acts on this ket, multiplying it by a constant.
Consequently, if=, all the diagonal elements of the operator whose trace is taken in
(16) are zero; the trace is therefore zero. If=, this average value may be computed
as for the partition function, since the Fock space has the structure of a tensor product
of individual state's spaces. The trace is the product of thevalue contribution by all
the othervalues contributions. We can thus write, in a general way:
=
1
exp [( )]
=
exp [ ( )](17)
For=, this expression yields the average particle number in the individual state.
1628

IDEAL GAS IN THERMAL EQUILIBRIUM; QUANTUM DISTRIBUTION FUNCTIONS
2-a. Fermion distribution function
As the occupation number only takes the values0and1, the rst bracket in
expression (17) is equal to
( )
; as for the other modes (=) contribution,
in the second bracket, it has already been computed when we determined the partition
function. We therefore obtain:
=
1 ( )
=
1 +
( )
(18)
Multiplying both the numerator and denominator by1 +
( )
allows reconstructing
the functionin the numerator, and, after simplication by, we get:
=
( )
1 +
( )
= ( ) (19)
We nd again the Fermi-Dirac distribution function( XIV):
( ) =
( )
1 +
( )
=
1
( )
+ 1
(20)
This distribution function gives the average population of each individual statewith
energy; its value is always less than1, as expected for fermions.
The average value at thermal equilibrium of any one-particle operator is now read-
ily computed by using (19) in relation (15).
2-b. Boson distribution function
The mode=contribution can be expressed as:
=0
exp [( )] =
1
=0
exp [( )]
=
1
1
1exp [( )]
(21)
We then get:
=
( )
1
( )
2
=
1
1
( )
(22)
which, using (11), amounts to:
= ( ) (23)
where the Bose-Einstein distribution functionis dened as:
( ) =
( )
1
( )
=
1
( )
1
(24)
1629

COMPLEMENT B XV
This distribution function gives the average population of the individual statewith
energy. The only constraint of this population, for bosons, is to be positive. The
chemical potential is always less than the lowest individual energy. In case this energy
is zero,must always be negative. This avoids any divergence of the function.
Hence for bosons, the average value of any one-particle operator is obtained by
inserting (23) into relation (15).
2-c. Common expression
We dene the functionas equal to either the functionfor fermions, or the
function for bosons. We can write for both cases:
( ) =
1
( )
(25)
where the numberis dened as:
=1for fermions
= +1for bosons (26)
2-d. Characteristics of Fermi-Dirac and Bose-Einstein distributions
We already gave in ComplementXIV(Figure) the form of the Fermi-Dirac
distribution. Figure
distribution. For the sake of comparison, it also includes the variations of the classical
Boltzmann distribution:
Boltzmann
( ) =
( )
(27)
which takes on intermediate values between the two quantum distributions. For a non-
interacting gas contained in a box with periodic boundary conditions, the lowest possible
energyis zero and all the others are positive. Exponential
( )
is therefore always
greater than. We are now going to distinguish several cases, starting with the most
negative values for the chemical potential.
(i) For a negative value of, with a modulus large compared to1(i.e. for
, which corresponds to the right-hand side of the gure), the exponential in
the denominator of (25) is always much larger than1(whatever the energy), and the
distribution reduces to the classical Boltzmann distribution (27). Bosons and fermions
have practically the same distribution; the gas is said to be non-degenerate.
(ii) For a fermion system, the chemical potential has no upper boundary, but the
population of an individual state can never exceed1. Ifis positive, with :
for low values of the energy, the factor1is much larger than the exponential
term; the population of each individual state is almost equal to1, its maximum value.
if the energyincreases to values of the order, the population decreases and
when , it becomes practically equal to the value predicted by the Boltzmann
exponential (27).
Most of the particles occupy, however, the individual states having an energy less
or comparable to, whose population is close to1. The fermion system is said to be
degenerate.
1630

IDEAL GAS IN THERMAL EQUILIBRIUM; QUANTUM DISTRIBUTION FUNCTIONS
Figure 1: Quantum distribution functions of Fermi-Dirac(for fermions, lower
curve) and of Bose-Einstein (for bosons, upper curve) as a function of the di-
mensionless variable( ); the dashed line intermediate curve represents the classical
Boltzmann distribution
( )
. In the right-hand side of the gure, corresponding to
large negative values of, the particle number is small (the low density region) and the
two distributions practically join the Boltzmann distribution. The system is said to be
non-degenerate, or classical. Asincreases, we reach the central and left hand side of
the gure, and the distributions become more and more dierent, reecting the increasing
gas degeneracy. For bosons,cannot be larger than the one-particle ground state en-
ergy, assumed to be zero in this case. The divergence observed for= 0corresponds to
Bose-Einstein condensation. For fermions, the chemical potentialcan increase without
limit, and for all the energy values, the distribution function tends towards1(but never
exceeding1due to the Pauli exclusion principle).
(iii) For a boson system, the chemical potential cannot be larger than the lowest0
individual energy value, which we assumed to be zero. Astends towards zero through
negative values and 0, the distribution function denominator becomes very
small leading to very large populations of the corresponding states. The boson gas is
then said to be degenerate. On the other hand, for energies of the order or larger than
, and as was the case for fermions, the boson distribution becomes practically equal to
the Boltzmann distribution.
(iv) Finally, for situations intermediate between the extreme cases described above,
the gas is said to be partially degenerate.
3. Two-particle operators
For a two-particle symmetric operatorwe must use formula (C-16) of Chapter,
which yields:
=
1
2
1 :; 2 :(12)1 :; 2 : (28)
1631

COMPLEMENT B XV
with:
=
1
Tr (29)
As the exponential operator in the trace is diagonal in the Fock basis states
1 , this trace will be non-zero on the double condition that the states
andassociated with the creation operator be exactly the same as the statesand
associated with the annihilation operators, whatever the order. In other words, to get a
non-zero trace, we must have either=and=, or=and=, or both.
3-a. Fermions
As two fermions cannot occupy the same quantum state, the productis zero
if=; we therefore assume=which allows, using forexpression (5) (which is
a product), to perform independent calculations for the dierent modes. The case=
and=yields, using the anticommutation relations:
= (30)
and the case=and=yields:
= + (31)
We begin with term (30). Asandare dierent, operatorsand act on dierent
modes, which belong to dierent factors in the density operator (5). The average value
of the product is thus simply the product of the average values:
= (32)
= ( )( ) (33)
As for the second term (31), it is just the opposite of the rst one. Consequently, we
nally get:
= [ ] ( )( ) (34)
The rst term on the right-hand side is called the direct term. The second one is the
exchange term, and has a minus sign, as expected for fermions.
3-b. Bosons
For bosons, the operatorscommute with each other.
. Average value calculation
If=, a calculation, similar to the one we just did, yields:
= [ + ] ( )( ) (35)
1632

IDEAL GAS IN THERMAL EQUILIBRIUM; QUANTUM DISTRIBUTION FUNCTIONS
which diers in two ways from (34): the result now involves the Bose-Einstein distribu-
tion, and the exchange term is positive.
If=, only one individual state comes into a new calculation, which we now
perform. Using forexpression (5) we get, after summing as in (11) a geometric
series:
=
1
=0
( 1) exp [ ( )]
=
1
1
( )
(36)
The sum appearing in this equation can be written as:
=0
( 1) exp [ ( )] =
1
2
2
2
1
=0
exp [( )]
=
1
2
2
2
11
1
( )
(37)
The rst order derivative term yields:
1
1
1
( )
=
( )
1
( )
2
(38)
and the second order derivative term is:
1
2
2
2
1
1
( )
=
( )
1
( )
2
+ 2
( )
2
1
( )
3
(39)
Summing these two terms yields:
2
( )
2
1
( )
3
=
2
1
( )
( )
2
(40)
Multiplying by11
( )
the product at the end of the right-hand side of (36)
yields the partition function, which cancels out the rst factor1. We are then left
with:
= 2 ( )
2
(41)
This result proves that (35) remains valid even in the case=.
. Physical discussion: occupation number uctuations
For two dierent physical statesand, the average value for an ideal
gas is simply equal to the product of the average values= ( )and
= ( ); this is a consequence of the total absence of interaction between
the particles. The same is true for the average value.
1633

COMPLEMENT B XV
Now if=, we note the factor2in relation (41). As we now show, this factor
leads to the presence of strong uctuations associated with the operator, the particle
number in the state. The calculations shows that:
()
2
= = +
= 2 ( )
2
+ ( ) (42a)
The square of the root mean square deviationis therefore given by:
()
2
=()
2 2
= ( )
2
+ ( ) =
2
+ (42b)
The uctuations of this operator are therefore larger than its average value, which implies
that the population of each stateis necessarily poorly dened
1
at thermal equilib-
rium. This is particularly true for large: in an ideal boson gas, a largely populated
individual state is associated with a very large population uctuation. This is due to
the shape of the Bose-Einstein distribution (24), a decreasing exponential which is max-
imum at the origin: the most probable occupation number is always= 0. Hence it is
impossible to get a very large averagewithout introducing a distribution spreading
over manyvalues. ComplementXV() discusses certain consequences of these
uctuations for an ideal gas. It also shows that as soon as a weak repulsive particle inter-
action is introduced, the uctuations greatly diminish and almost completely disappear,
since their presence would lead to a very large increase in the potential energy.
3-c. Common expression
To summarize, we can write in all cases:
= [ + ]( )( ) (43)
with:
for fermions =1, =
for bosons = +1, =
(44)
As shown in relation (C-19) of Chapter, this average value is simply the matrix
element1 :; 2 : 21 :; 2 :of the two-particle reduced density operator. To
get the general expression for the average of any symmetric two-particle operator, we
simply use (43) in (28). Consequently, for independent particles, the average values
of all these operators are simply expressed in terms of the quantum Fermi-Dirac and
Bose-Einstein distribution functions.
ComplementXVIwill show how the Wick theorem allows generalizing these re-
sults to operators dealing with any number of particles.
1
A physical observable is said to have a well dened value in a given quantum state if, in this state,
its root mean square deviation is small compared to its average value.
1634

IDEAL GAS IN THERMAL EQUILIBRIUM; QUANTUM DISTRIBUTION FUNCTIONS
4. Total number of particles
The operatorcorresponding to the total number of particles is given by the sum over
all the individual states:
=
=1
(45)
and its average value is given by:
=Tr =
=1
( ) (46)
Asincreases as a function of, the total number of particles is controlled (for xed
) by the chemical potential.
4-a. Fermions
For the sake of simplicity, we study the ideal gas properties without taking into
account the spin, which assumes that all particles are in the same spin state (the spin
can easily be accounted for by adding the contributions of the dierent individual spin
states). For a large physical system, the energy levels are very close and the discrete sum
in (46) can be replaced by an integral. This leads to:
= () (47)
where the function()is dened as (the subscriptstands for ideal gas):
() =
(2)
3
d
3 1
( )
+ 1
(48)
Figure ()as a function of, for xed
values ofand the volume.
To deal with dimensionless quantities, one often introduces the thermal wave-
lengthas:
=}
2
=}
2
(49)
We can then use in the integral of (48) the dimensionless variable:
=
2
(50)
and write:
() =
()
332() (51)
1635

COMPLEMENT B XV 0
Figure 2: Variations of the particle number()for an ideal fermion gas, as
a function of the chemical potential, and for dierent xed temperatures(=
1()). For= 0(lower dashed line curve), the particle number is zero for neg-
ative values of, and proportional to
32
for positive values of. For a non-zero
temperature=1(thick line curve), the curve is above the previous one, and never
goes to zero. Also shown are the curves obtained for temperatures twice (= 21) and
three times (= 31) as large. The units chosen for the axes are the thermal energy
1associated with the thick line curve, and the particle number1=(
1
)
3
, where
1
is the thermal wavelength at temperature1.
Largely negative values ofcorrespond to the classical region where the fermion gas is
not degenerate; the classical ideal gas equations are then valid to a good approximation.
In the region where , the gas is largely degenerate and a Fermi sphere shows up
clearly in the momentum space; the total number of particles has only a slight temperature
dependence and varies approximately as
32
.
This gure was kindly contributed by Geneviève Tastevin.
with
2
:
32() =
32
d
3 1
2
+ 1
=
2
0
d
+ 1
(52)
where, in the second equality, we made the change of variable:
=
2
(53)
Note that the value of
32only depends on a dimensionless variable, the product.
If the particles have a spin12, both contributions+and from the
two spin states must be added to (46); in the absence of an external magnetic eld, the
2
The subscript32refers to the subscript used for more general functions(), often called the
Fermi functions in physics. They are dened by() =
=1
(1)
+1
, whereis the fugacity
= . Expanding in terms of the function11 +
1
= 1 + and using the
properties of the Euler Gamma function, it can be shown that
32() =
32().
1636

IDEAL GAS IN THERMAL EQUILIBRIUM; QUANTUM DISTRIBUTION FUNCTIONS
individual particle energies do not depend on their spin direction, and the total particle
number is simply doubled:
= ++ = 2 () (54)
4-b. Bosons
For the sake of simplicity, we shall also start with spinless particles, but including
several spin states is fairly straightforward. For bosons, we must use the Bose-Einstein
distribution (24) and their average number is therefore:
= ( ) =
1
( )
1
(55)
We impose periodic boundary conditions in a cubic box of edge length. The lowest
individual energy
3
is= 0. Consequently, for expression (55) to be meaningful,must
be negative or zero:
0 (56)
Two cases are possible, depending on whether the boson system is condensed or not.
. Non-condensed bosons
When the parametertakes on a suciently negative value (much lower than
the opposite of the individual energy1of the rst excited level), the function in the
summation (55) is suciently regular for the discrete summation to be replaced by an
integral (in the limit of large volumes). The average particle number is then written as:
= () (57)
with:
() =
(2)
3
d
3 1
( )
1
(58)
Performing the same change of variables as above, this expression becomes:
() =
()
332() (59)
with
4
:
32() =
32
d
3 1
2
1
=
2
0
d
1
(60)
3
Dening other boundary conditions on the box walls will lead in general to a non-zero ground state
energy; choosing that value as the common origin for the energies and the chemical potential will leave
the following computations unchanged.
4
The subscript32refers to the subscript used for the functions(), often called, in physics, the
Bose functions (or the polylogarithmic functions). They are dened by the series() =
=1
.
The exact value of the numberdened in (61) is thus given by the series
=1
32
.
1637

COMPLEMENT B XV
The variations of()as a function ofare shown in Figure. Note that the
total particle number tends towards a limit
3
astends towards zero through
negative values, whereis the number:
=
32(0) = 2612 (61)
As the function increases with, we can write:
()
()
3
(62)
There exists an insurmountable upper limit for the total particle number of a non-
condensed ideal Bose gas.
Figure 3: Variations of the total particle number()in a non-condensed ideal
Bose gas, as a function ofand for xed= 1(). The chemical potential is always
negative, and the gure shows curves corresponding to several temperatures=1(thick
line),= 21and= 31. Units on the axes are the same as in Figure: the thermal
energy 1associated with curve=1, and the particle number1=(
1
)
3
, where
1is the thermal wavelength for this same temperature1. As the chemical potential
tends towards zero, the particle numbers tend towards a nite value. For=1, this
value is equal to1(shown as a dot on the vertical axis), whereis given by (61).
This gure was kindly contributed by Geneviève Tastevin
. Condensed bosons
Asgets closer to zero, the population0of the ground state becomes:
0 (0) =
1
1 0
1
(63)
1638

IDEAL GAS IN THERMAL EQUILIBRIUM; QUANTUM DISTRIBUTION FUNCTIONS
This population diverges in the limit= 0and, when gets small enough, it can
become arbitrarily large. It can, for example, become proportional
5
to the volume, in
which case it adds a nite contribution0to the particle numerical density (particle
number per unit volume) as .
This particularity is limited to the ground state, which, in this case, plays a very
dierent role than the other levels. Let us show, for example, that the rst excited state
population does not yield a similar eect. Assuming the system to be contained in a cubic
box
6
of edge length, the population of the rst excited energy level1
2
}
2
(2
2
)
can be written as:
(1) =
1
(1)
1 0
1
1
2
(64)
(we assume the box to be large enough so that , which means11); this
population can therefore be proportional only to the square of, i.e. to the volume to
the power23. It shows that this rst excited level cannot make a contribution to the
particle density in the limit; the same is true for all the other excited levels
whose contributions are even smaller. The only arbitrary contribution to the density
comes from the ground state.
This arbitrarily large value as0obviously does not appear in relation (59),
which predicts that the density()is always less than a nite value as shown by
(62). This is not surprising: as the population varies radically from the rst energy level
to the next, we can no longer compute the average particle number by replacing in (55) the
discrete summation by an integral and a more precise calculation is necessary. Actually,
only the ground state population must be treated separately, and the summation over
all the excited states (of which none contributes to the density divergence) can still be
replaced by an integral as before. Consequently, to get the total population of the physical
system we simply add the integral on the right-hand side of (57) to the contribution0
of the ground level:
= (0) +0 (65)
where0is dened in (63).
As 0, the total population of all the excited levels (others than the ground
level) remains practically constant and equal to its upper limit (62); only the ground
state has a continuously increasing population0, which becomes comparable to the
total population of all the excited states when the right-hand sides of (63) and (62) are
of the same order of magnitude:
&
3
0&
3
(66)
(being of course always negative). When this condition is satised, a signicant frac-
tion of the particles accumulates in the individual ground level, which is said to have a
5
The limit where while the density remains constant is often called the thermodynamic
limit.
6
As above, we assume periodic conditions on the box walls. Another choice would be to impose zero
values for the wave functions on the walls: the numerical coecients of the individual energies would be
changed, but not the line of reasoning.
1639

COMPLEMENT B XV
macroscopic population (proportional to the volume). We can even encounter situa-
tions where the majority of the particles all occupy the same quantum state. This phe-
nomenon is called Bose-Einstein condensation (it was predicted by Einstein in 1935,
following Bose's studies of quantum statistics applicable to photons). It occurs when the
total densityreaches the maximum predicted by formula (62), that is:
=
3
2612
3
(67)
This condition means that the average distance between particles is of the order of the
thermal wavelength.
Initially, Bose-Einstein condensation was considered to be a mathematical curios-
ity rather than an important physical phenomenon. Later on, people realized that it
played an important role in superuid liquid Helium4, although this was a system with
constantly interacting particles, hence far from an ideal gas. For a dilute gas, Bose-
Einstein condensation was observed for the rst time in 1995, and in a great number of
later experiments.
5. Equation of state, pressure
The equation of state of a uid at thermal equilibrium is the relation that links, for a
given particle number, its pressure, volume, and temperature= 1 . We
have just studied the variations of the total particle number. We shall now examine the
pressure of a fermion or boson ideal gas.
5-a. Fermions
The grand canonical potential of a fermion ideal gas is given by (9). Equation (14)
indicates that, for a system at thermal equilibrium, this grand potential is equal to the
opposite of the product of the volumeand the pressure. We thus have:
= ln1 +
( )
=
(2)
3
d
3
ln1 +
( )
(68)
(where the second equality is valid in the limit of large volumes). Simplifying by=
3
,
we get the pressure of a fermion system contained in a box of macroscopic dimension:
= (2)
3
d
3
ln1 +
( )
=
1
352() (69)
with:
52() =
32
d
3
ln1 +
2
=
2
0
d
ln1 + (70)
1640

IDEAL GAS IN THERMAL EQUILIBRIUM; QUANTUM DISTRIBUTION FUNCTIONS
wherehas been dened in (53).
To obtain the equation of state, we must nd a relation between the pressure, the
volume, and the temperatureof the physical system, assuming the particle number to
be xed. We have, however, used the grand canonical ensemble (cf. Appendix), where
the temperature is determined by the parameterand the volume is xed, but where
the particle number can vary: its average value is a function of a parameter, the chemical
potential(for xed values ofand). Mathematically, the pressureappears as a
function of,andand not as the function of,and the particle number we were
looking for. We can nevertheless vary, and obtain values of the pressure and particle
number of the system and consequently explore, point by point, the equation of state in
this parametric form. To obtain an explicit form of the equation of state would require
the elimination of the chemical potential using both (47) and (69); there is generally no
algebraic solution, and people just use the parametric form of the equation of state, which
allows computing all the possible state variables. There also exists a virial expansion in
powers of the fugacity, which allows the explicit elimination ofat all the successive
orders; its description is beyond the scope of this book.
5-b. Bosons
The pressure of an ideal boson gas is derived from the grand potential (12), taking
into account its relation (14) to the pressureand volume:
= ln1
( )
=
(2)
3
3
ln1
( )
(71)
(the second relation being valid in the limit of large volumes). This leads to:
=
(2)
3
3
ln1
( )
=
1
352() (72)
with:
52() =
32
d
3
ln1
2
=
2
0
d
ln1 (73)
As 0, the contribution0of the ground level to the pressure written in (71)
is:
0=
ln1ln[] (74)
When the chemical potential tends towards zero as in (66), it leads to:
0
ln
3
(75)
1641

COMPLEMENT B XV
which therefore goes to zero in the limit of large volumes. For a large system, the ground
level contribution to the pressure remains negligible compared to that of all the other
individual energy levels, whose number gets bigger as the system gets larger. Contrary
to what we encountered for the average total particle number, the condensed particles'
contribution to the pressure goes to zero in the limit of large volumes.
As we have seen for fermions, the equation of state must be obtained by elimi-
nating the chemical potentialbetween equations (72) yielding the pressure and (65)
yielding the total particle number. As opposed to an ideal fermion gas, whose particle
number and pressure increase without limit asand the density increase, the pressure in
a boson system is limited. As soon as the system condenses, only the particle number in
the individual ground state continues to grow, but not the pressure. In other words, the
physical system acquires an innite compressibility, and becomes a marginally patho-
logical system (a system whose pressure decreases with its volume is unstable). This
pathology comes, however, from totally neglecting the bosons' interactions. As soon as
repulsive interactions are introduced, no matter how small, the compressibility will take
on a nite value and the pathology will disappear.
This complement is a nice illustration of the simplications incurred by the sys-
tematic use, in the calculations, of the creation and annihilation operators. We shall
see in the following complements that these simplications still occur when taking into
account the interactions, provided we stay in the framework of the mean eld approxi-
mation. ComplementXVIwill even show that for an interacting system studied without
using this approximation, the ideal gas distribution functions are still somewhat useful
for expressing the average values of various physical quantities.
1642

CONDENSED BOSON SYSTEM, GROSS-PITAEVSKII EQUATION
Complement CXV
Condensed boson system, Gross-Pitaevskii equation
1 Notation, variational ket
1-a Hamiltonian
1-b Choice of the variational ket (or trial ket)
2 First approach
2-a Trial wave function for spinless bosons, average energy
2-b Variational optimization
3 Generalization, Dirac notation
3-a Average energy
3-b Energy minimization
3-c Gross-Pitaevskii equation
4 Physical discussion
4-a Energy and chemical potential
4-b Healing length
4-c Another trial ket: fragmentation of the condensate
The Bose-Einstein condensation phenomenon for an ideal gas (no interaction) of
identical bosons was introduced in of ComplementXV. We show in the present
complement how to describe this phenomenon when the bosons interact. We shall look
for the ground state of this physical system within the mean eld approximation, using a
variational method (see ComplementXI). After introducing in
variational ket, we study in
is simple and the introduction of the creation and annihilation operators does not lead
to any major computation simplications. This will lead us to a rst version of the
Gross-Pitaevskii equation. We will then come back in
creation operators, to deal with the more general case where each particle may have a
spin. Dening the Gross-Pitaevskii potential operator, we shall obtain a more general
version of that equation. Finally, some properties of the Gross-Pitaevskii equation will be
discussed in , as well as the role of the chemical potential, the existence of a relaxation
(or healing) length, and the energetic consequences of condensate fragmentation
(these terms will be dened in ).
1. Notation, variational ket
We rst dene the notation and the variational family of state vectors that will lead to
relatively simple calculations for a system of identical interacting bosons.
1643

COMPLEMENT C XV
1-a. Hamiltonian
The Hamiltonian operatorwe consider is the sum of operators for the kinetic
energy0, the one-body potential energyext, and the interaction energyint:
=0+ext+ int (1)
The rst term0is simply the sum of the individual kinetic energy operators associated
with each of the particles:
0= 0() (2)
where :
0() =
P
2
2
(3)
(Pis the momentum of particle). Similarly,extis the sum of the external potential
operators1(R), each depending on the position operatorRof particle:
ext=
=1
1(R) (4)
Finally,intis the sum of the interaction energy associated with all the pairs of particles:
int=
1
2
==1
2(RR) (5)
(this summation can also be written as a sum over, while removing the prefactor
12).
1-b. Choice of the variational ket (or trial ket)
Let us choose an arbitrary normalized quantum state:
= 1 (6)
and callthe associated creation operator. The-particle variational kets we consider
are dened by the family of all the kets that can be written as:
=
1
!
0 (7)
where can vary, only constrained by (6). Consider a basis of the individual
state space whose rst vector is1=. Relation (A-17) of Chapter
this ket is simply a Fock state whose only non-zero occupation number is the rst one:
= 1= 2= 03= 0 (8)
An assembly of bosons that occupy the same individual state is called a Bose-Einstein
condensate.
Relation (8) shows that the kets are normalized to1. We are going to vary
, and therefore , so as to minimize the average energy:
= (9)
1644

CONDENSED BOSON SYSTEM, GROSS-PITAEVSKII EQUATION
2. First approach
We start with a simple case where the bosons have no spin. We can then use the wave
function formalism and keep the computations fairly simple.
2-a. Trial wave function for spinless bosons, average energy
Assuming one single individual state to be populated, the wave function (r1r2r)
is simply the product offunctions(r):
(r1r2r) =(r1)(r2)(r) (10)
with:
(r) =r (11)
This wave function is obviously symmetric with respect to the exchange of all particles
and can be used for a system of identical bosons.
In the position representation, each operator0()dened by (3) corresponds to
}
2
2r, whereris the Laplacian with respect to the positionr; consequently,
we have:
0=
}
2
2
=1
d
3
1 d
3
d
3
(r1) (r) (r)(r1)(r)(r) (12)
In this expression, all the integral variables others thanrsimply introduce the square of
the norm of the function(r), which is equal to1. We are just left with one integral over
r, in whichrplays the role of a dummy variable, and thus yields a result independent
of. Consequently, all thevalues give the same contribution, and we can write:
0=
}
2
2
d
3
(r)(r) (13)
As for the one-body potential energy, a similar calculation yields:
ext= d
3
(r)1(r)(r) (14)
Finally, the interaction energy calculation follows the same steps, but we must keep
two integral variables instead of one. The nal result is proportional to the number
( 1)2of pairs of integral variables:
2=
( 1)
2
d
3
d
3
(r)(r)2(rr)(r)(r) (15)
The variational average energyis the sum of these three terms:
= 0+ ext+ 2 (16)
1645

COMPLEMENT C XV
2-b. Variational optimization
We now optimize the energy we just computed, so as to determine the wave func-
tions(r)corresponding to its minimum value.
. Variation of the wave function
Let us vary the function(r)by a quantity:
(r)(r) + (r) (17)
where(r)is an innitesimal function andan arbitrary number. A priori,(r)must
be chosen to take into account the normalization constraint (6), which forces the integral
of the(r)modulus squared to remain constant. We can, however, use the Lagrange
multiplier method (Appendix) to impose this constraint. We therefore introduce the
multiplier(we shall see in
potential) and minimize the function:
= d
3
(r)(r) (18)
This allows considering the innitesimal variation(r)to be free of any constraint. The
variationof the functionis now the sum of4variations, coming from the three
terms of (16) and from the integral in (18). For example, the variation of0yields:
0=
}
2
2
d
3
(r) (r) + (r) (r) (19)
which is the sum of a term proportional toand another proportional to. This
is true for all4variations and the total variationcan be expressed as the sum of two
terms:
= 1 + 2 (20)
the rst being the(r)contribution and the second, that of(r). Now if is
stationary,must be zero whatever the choice of, which is real. Choosing for example
= 0imposes1+ 2= 0, and the choice=2leads (after multiplication by) to
1 2= 0. Adding and subtracting the two relations shows that both coecients1
and2must be zero. In other words, we can imposeto be zero as just(r)varies
but not(r) or the opposite
1
.
. Stationary condition: Gross-Pitaevskii equation
We choose to impose the variationto be zero as only(r)varies and for= 0.
We must rst add contributions coming from (13) and (14), then from (15). For this last
contribution, we must add two terms, one coming from the variations due to(r), and
the other from the variation due to(r). These two terms only dier by the notation
1
This means that the stationary condition may be found by varying indierently the real or imaginary
part of(r).
1646

CONDENSED BOSON SYSTEM, GROSS-PITAEVSKII EQUATION
in the integral variable and are thus equal: we just keep one and double it. We nally
add the term due to the variation of the integral in (18), and we get:
= d
3
(r)
}
2
2
+1(r) + ( 1)d
3
2(rr)(r)(r)(r)(21)
This variation must be zero for any value of(r); this requires the function that
multiplies(r)in the integral to be zero, and consequently that(r)be the solution
of the following equation, written for(r):
}
2
2
+1(r)+ ( 1)d
3
2(rr)(r)
2
(r) =(r)
(22)
This is the time-independent Gross-Pitaevskii equation. It is similar to an eigenvalue
Schrödinger equation, but with a potential term:
1(r) + (1)d
3
2(rr)(r)
2
(23)
which actually contains the wave functionin the integral over d
3
; it is therefore a
nonlinear equation. The physical meaning of the potential term in2is simply that,
in the mean eld approximation, each particle moves in the mean potential created by
all the others, each of them being described by the same wave function(r); the factor
(1)corresponds to the fact that each particle interacts with(1)other particles.
The Gross-Pitaevskii equation is often used to describe the properties of a boson system
in its ground state (Bose-Einstein condensate).
. Zero-range potential
The Gross-Pitaevskii equation is often written in conjunction with an approxima-
tion where the particle interaction potential has a microscopic range, very small compared
to the distances over which the wave function(r)varies. We can then substitute:
2(rr) = (rr) (24)
where the constantis called the coupling constant; such a potential is sometimes
known as a contact potential or, in other contexts, a Fermi potential. We then get:
}
2
2
+1(r) + (1)(r)
2
(r) =(r) (25)
Whether in this form
2
or in its more general form (22), the equation includes a cubic
term in(r). It may render the problem dicult to solve mathematically, but it also is
the source of many interesting physical phenomena. This equation explains, for example,
the existence of quantum vortices in superuid liquid helium.
2
Strictly speaking, in what is generally called the Gross-Pitaevskii equation, the coupling constant
is replaced by4}
2
0, where0is the scattering length; this length is dened when studying the
collision phase shift()(Chapter VIII, ), as the limit of0() 0when 0. This scattering
length is a function of the interaction potential2(rr), but generally not merely proportional to it, as
opposed to the matrix elements of2(rr). It is then necessary to make a specic demonstration for
this form of the Gross-Pitaevskii equation, using for example the pseudo-potential method.
1647

COMPLEMENT C XV
. Other normalization
Rather than normalizing the wave function(r)to1in the entire space, one
sometimes chooses a normalization taking into account the particle number by setting:
d
3
(r)
2
= (26)
This amounts to multiplying by
the wave function we have used until now. At each
pointrof space, the particle (numerical) density(r)is then given by:
(r) =(r)
2
(27)
With this normalization, the factor(1)in (25) is replaced by(1), which can
generally be taken equal to1for large. The Gross-Pitaevskii equation then becomes:
}
2
2
+1(r) +(r)
2
(r) =(r)
(28)
As already mentioned, we shall see in is simply the chemical potential.
3. Generalization, Dirac notation
We now go back to the previous line of reasoning, but in a more general case where the
bosons may have spins. The variational family is the set of the-particle state vectors
written in (7). The one-body potential may depend on the positionr, and, at the same
time, act on the spin (particles in a magnetic eld gradient, for example).
3-a. Average energy
To compute the average energy value , we use a basis of the indi-
vidual state space, whose rst vector is1=.
Using relation (B-12) of Chapter, we can write the average value0as:
0= 0 (29)
Since is a Fock state whose only non-zero population is that of the state1, the
ket is non-zero only if= 1; it is then orthogonal to if= 1. Consequently,
the only term left in the summation corresponds to== 1. As the operator
11
multiplies the ket by its population, we get:
0= 101 (30)
With the same argument, we can write:
ext= 111 (31)
1648

CONDENSED BOSON SYSTEM, GROSS-PITAEVSKII EQUATION
Using relation (C-16) of Chapter, we can express the average value of the
interaction energy as
3
:
2=
1
2
1 :; 2 : 2(12)1 :; 2 : (32)
In this case, for the second matrix element to be non-zero, both subscriptsandmust
be equal to1and the same is true for both subscriptsand( otherwise the operator
will yield a Fock state orthogonal to ). When all the subscripts are equal to1, the
operator multiplies the ket by( 1). This leads to:
2=
( 1)
2
1 :1; 2 :1 2(12)1 :1; 2 :1 (33)
The average interaction energy is therefore simply the product of the number of pairs
( 1)2that can be formed withparticles and the average interaction energy of
a given pair.
We can replace1by, since they are equal. The variational energy, obtained
as the sum of (30), (31) and (33), then reads:
= [0+1]+
( 1)
2
1 :; 2 :2(12)1 :; 2 : (34)
3-b. Energy minimization
Consider a variation of:
+ (35)
where is an arbitrary innitesimal ket of the individual state space, andan arbi-
trary real number. To ensure that the normalization condition (6) is still satised, we
impose andto be orthogonal:
= 0 (36)
so that remains equal to1(to the rst order in). Inserting (35) into (34) to
obtain the variation dof the variational energy, we get the sum of two terms: the rst
one comes from the variation of the ket, and is proportional to; the second one
comes from the variation of the bra, and is proportional to. The result has the
form:
= 1+ 2 (37)
The stationarity condition formust hold for any arbitrary real value of. As before
( ), it follows that both1and 2are zero. Consequently, we can impose the
variationto be zero as just the bravaries (but not the ket), or the opposite.
Varying only the bra, we get the condition:
0 = [0+1]+
( 1)
2
[1 :; 2 :2(12)1 :; 2 :
+1 :; 2 : 2(12)1 :; 2 :]
(38)3
We use the simpler notation2(12)for2(R1R2).
1649

COMPLEMENT C XV
As the interaction operator2(12)is symmetric, the last two terms within the bracket
in this equation are equal. We get (after simplication by):
0 = [0+1]+ ( 1)1 :; 2 :2(12)1 :; 2 : (39)
3-c. Gross-Pitaevskii equation
To deal with equation (39), we introduce the Gross-Pitaevskii operator, de-
ned as a one-particle operator whose matrix elements in an arbitrary basisare
given by:
= ( 1)1 :; 2 :2(12)1 :; 2 : (40)
which leads to:
= ( 1)1 :; 2 :2(12)1 :; 2 : (41)
whereand are two arbitrary one-particle kets this can be shown by expanding
these two kets on the basisand using relation (40). Note that this potential opera-
tor does not include an exchange term; this term does not exist when the two interacting
particles are in the same individual quantum state. Equation (39) then becomes:
0 = 0+1+ (42)
This stationarity condition must be veried for any value of the bra, with only the
constraint that it must be orthogonal to(according to relation (36)). This means
that the ket resulting from the action of the operator0+1+ onmust have
zero components on all the vectors orthogonal to; its only non-zero component must
be on the ketitself, which means it is necessarily proportional to. In other words,
must be an eigenvector of that operator, with eigenvalue(real since the operator is
Hermitian):
0+1+ = (43)
We have just shown that the optimal valueofis the solution of the Gross-Pitaevskii
equation:
[0+1+ ]= (44)
which is a generalization of (28) to particles with spin, and is valid for one- or two-
body arbitrary potentials. For each particle, the operatorrepresents the mean eld
created by all the others in the same state.
Comment:
The Gross-Pitaevskii operatoris simply a partial trace over the second particle:
(1) = (1)Tr2 (2)2(12) (45)
where(2)is the projection operator(2)of the state of particle2onto:
(2) = 1 :1 : 2 :2 := 1 :; 2 :1 :; 2 : (46)
1650

CONDENSED BOSON SYSTEM, GROSS-PITAEVSKII EQUATION
To show this, let us compute the partial trace on the right-hand side of (45). To obtain
this trace (ComplementIII, ), we choose for particle2a set of basis states
whose rst vector1coincides with:
Tr2 (2)2(12) = 1 :; 2 : (2)2(12)1 :; 2 : (47)
Replacing(2)by its value (46) yields the product of(for the scalar product as-
sociated with particle1) and 1(for the one associated with particle2). This leads
to:
Tr2 (2)2(12) =1 :; 2 :2(12)1 :; 2 : (48)
which is simply the initial denition (40) of . Relation (45) is therefore another
possible denition for the Gross-Pitaevskii potential.
4. Physical discussion
We have established which conditions the variational wave function must obey to make
the energy stationary, but we have yet to study the actual value of this energy. This will
allow us to show that the parameteris in fact the chemical potential associated with
the system of interacting bosons. We shall then introduce the concept of a relaxation (or
healing) length, and discuss the eect, on the nal energy, of the fragmentation of a
single condensate into several condensates, associated with distinct individual quantum
states.
4-a. Energy and chemical potential
Since the ketis normalized, multiplying (44) by the braand by, we get:
[0+1+ ]= (49)
We recognize the rst two terms of the left-hand side as the average values of the kinetic
energy and the external potential. As for the last term, using denition (41) for, we
can write it as:
=( 1)1 :; 2 :2(12)1 :; 2 : (50)
which is simply twice the potential interaction energy given in (33) when1=. This
leads to:
= 0+ 1+ 2 2=+ 2 (51)
To nd the energy, note that2is the sum of2and of half the kinetic and
external potential energies. Adding the missing halves, we nally get for:
=
2
[+[0+1]] (52)
An advantage of this formula is to involve only one- (and not two-) particle operators,
which simplies the computations. The interaction energy is implicitly contained in the
factor.
1651

COMPLEMENT C XV
The quantitydoes not yield directly the average energy, but it is related to it,
as we now show. Taking the derivative, with respect to, of equation (34) written for
=, we get:
d
d
= [0+1]+
1
2
2(12) (53)
For large, one can safely replace in this equation( 12)by( 1); after multi-
plication by, we obtain a sum of average energies:
d
d
= 0+ 1+ 2 2 (54)
Taking relation (51) into account, this leads to:
d
d
= (55)
We know (Appendix VI, ) that in the grand canonical ensemble, and at zero tem-
perature, the derivative of the energy with respect to the particle number (for a xed
volume) is equal to the chemical potential. The quantity, introduced mathematically
as a Lagrange multiplier, can therefore be simply interpreted as this chemical potential.
4-b. Healing length
The healing length is an important concept that characterizes the way a solution
of the time-independent Gross-Pitaevskii equation reacts to a spatial constraint (for
example, the solution can be forced to be zero along a wall, or along the line of a vortex
core). We now calculate an approximate order of magnitude for this length.
Assuming the potential1(r)to be zero in the region of interest, we divide equation
(28) by(r)and get:
}
2
2
(r)
(r)
+ (r)
2
= (56)
Consequently, the left-hand side of this equation must be independent ofr. Let us assume
(r)is constant in an entire region of space where the density is0, independent ofr:
0=(r)
2
(57)
but constrained by the boundary conditions to be zero along its border. For the sake of
simplicity, we shall treat the problem in one dimension, and assume(r)only depends on
the rst coordinateofr; the wave function must then be zero along a plane (supposed
to be at= 0). We are looking for an order of magnitude of the distanceover which
the wave function goes from a practically constant value to zero, i.e. for the spatial range
of the wave function transition regime. In the region where(r)is constant, relation
(56) yields:
= 0 (58)
1652

CONDENSED BOSON SYSTEM, GROSS-PITAEVSKII EQUATION
Figure 1: Variation as a function of the positionof the wave function()in the
vicinity of a wall (at= 0) where it is forced to be zero. This variation occurs over
a distance of the order of the healing lengthdened in (61); the stronger the particle
interactions, the shorter that length. Asincreases, the wave function tends towards a
constant plateau, of coordinate
0, represented as a dashed line.
On the other hand, in the whole region where(r)has signicantly decreased, and in
particular close to the origin, we have:
}
2
2
(r)
(r)
= 0 (59)
In one dimension
4
, we then get the dierential equation:
}
2
2
d
2
d
2
() 0() (60)
whose solutions are sums of exponential functions, with:
=
}
22 0
(61)
The solution that is zero for= 0is the dierence between these two exponentials;
it is proportional to sin(), a function that starts from zero and increases over a
characteristic length. Figure
wall where it is forced to be zero.
The stronger the interactions, the shorter this healing length; it varies as the
inverse of the square root of the product of the coupling constantand the density0.
From a physical point of view, the healing length results from a compromise between the
repulsive interaction forces, which try to keep the wave function as constant as possible
in space, and the kinetic energy, which tends to minimize its spatial derivative (while the
wave function is forced to be zero at= 0);is equal (except for a2coecient) to
the de Broglie wavelength of a free particle having a kinetic energy comparable to the
repulsion energy0in the boson system.
4
A more precise derivation can be given by verifying that() =
0tanh2is a solution of
the one-dimensional equation (56).
1653

COMPLEMENT C XV
4-c. Another trial ket: fragmentation of the condensate
We now show that repulsive interactions do stabilize a boson condensate where
all the particles occupy the same individual state, as opposed to a fragmented state
where some particles occupy a dierent state, which can be very close in energy. Instead
of using a trial ket (7), where all the particles form a perfect Bose-Einstein condensate
in a single quantum state, we can fragment this condensate by distributing the
particles in two distinct individual states. Consequently, we take a trial ket where
particles are in the stateand= in the orthogonal state:
=
1
!!
0 (62)
We now compute the change in the average variational energy. In formula (29)
giving the average kinetic energy, for the operatorto yield a Fock state identical to
, we must have either==, or==. This leads to:
0= 0 + 0 (63)
The computation of the one-body potential energy is similar and leads to:
ext= 1 + 1 (64)
In both cases, the contributions of two populated states are proportional to their respec-
tive populations, as expected for energies involving a single particle.
As for the two-body interaction energy, we use again relation (32). It contains
the operator , which will reconstruct the Fock state in the following three
cases:
-====oryields the contribution:
( 1)
2
1 :; 2 : 2(12)1 :; 2 :
+
( 1)
2
1 :; 2 : 2(12)1 :; 2 : (65)
-==and==, or==and==; these two possibilities
yield the same contribution (since the2operator is symmetric), and the12factor
disappears, leading to the direct term:
1 :; 2 : 2(12)1 :; 2 : (66)
- Finally,==and==, or==and==, yield two
contributions whose sum introduces the exchange term (here again without the factor
12):
1 :; 2 : 2(12)1 :; 2 : (67)
The direct and exchange terms have been schematized in Figure
placingby, andby), with the direct term on the left, and the exchange
term on the right.
1654

CONDENSED BOSON SYSTEM, GROSS-PITAEVSKII EQUATION
The variational energy can thus be written as:
= [[0+1]] +[[0+1]]
+
( 1)
2
1 :; 2 : 2(12)1 :; 2 :
+
( 1)
2
1 :; 2 : 2(12)1 :; 2 :
+ 1 :; 2 : 2(12)1 :; 2 :
+ 1 :; 2 : 2(12)1 :; 2 : (68)
As above, the interaction between particles in the same statecontributes a term
proportional to( 1)2, the number of pairs of particles in that state; the same
is true for the interaction term between particles in the same state. The direct term
associated with the interaction between two particles in distinct states is proportional to
, the number of such pairs. But to this direct term we must add an exchange term,
also proportional to, corresponding to an additional interaction. This increased
interaction is due to the bunching eect of two bosons in dierent quantum states, that
will be discussed in more detail in XVI. As they are indistin-
guishable, two bosons occupying individual orthogonal states show correlations in their
positions; this increases the probability of nding them at the same point in space. This
increase does not occur when the two bosons occupy the same individual quantum state.
We now assume the diagonal matrix elements of[0+1]between the two states
and to be practically the same. For example, if these two states are the lowest
energy levels of spinless particles in a cubic box of edge, the corresponding energy
dierence is proportional to1
2
hence very small in the limit of large. We also
assume all the matrix elements of2(12)to be equal, which is the case if the (micro-
scopic) range of the particle interaction potential is very small compared to the distances
over which the wave functions of the two states vary. We can therefore replace in all
the matrix elements the ketsand by the same ket. Since+ =, we
obtain:
= [0+1]
+
1
2
[( 1) +( 1) + 2 ]1 :; 2 :2(12)1 :; 2 :
+ 1 :; 2 :2(12)1 :; 2 : (69)
However:
( 1) = (+) (+ 1) =( 1) +( 1) + 2 (70)
so that:
= [0+1]+
( 1)
2
1 :; 2 :2(12)1 :; 2 :+ (71)
with:
= 1 :; 2 :2(12)1 :; 2 : (72)
1655

COMPLEMENT C XV
We nd again result (34), but with an additional term, the exchange term.
Two cases are then possible, depending on whether the particle interactions are attractive
or repulsive. In the rst case, the fragmentation of the condensate lowers the energy and
leads to a more stable state. Consequently, when the particle interactions are attractive, a
condensate where only one individual state is occupied tends to split into two condensates,
which might each split again, and so on. This means that the initial single condensate
is unstable (we will come back and discuss this instability in XV
for the more general case of thermal equilibrium at non-zero temperature). On the
contrary, for repulsive interactions the fragmentation increases the energy and leads to
a less stable state: repulsive interactions therefore tend to stabilize the condensate in a
single individual quantum state
5
. This result will be interpreted in
AXVIin terms of changes of the particle position correlation function (bunching eect of
bosons). As for the ideal gas, an intermediate case between the two previous ones, it is a
marginal borderline case: adding any innitesimal attractive interaction, no matter how
small, destabilizes any condensate.
5
We are discussing here the simple case of spinless bosons, contained in a box. When the bosons
have several internal quantum states, and in other geometries, more complex situations may arise where
the ground state is fragmented [4].
1656

TIME-DEPENDENT GROSS-PITAEVSKII EQUATION
Complement DXV
Time-dependent Gross-Pitaevskii equation
1 Time evolution
1-a Functional variation
1-b Variational computation: the time-dependent Gross-Pitaevskii
equation
1-c Phonons and Bogolubov spectrum
2 Hydrodynamic analogy
2-a Probability current
2-b Velocity evolution
3 Metastable currents, superuidity
3-a Toroidal geometry, quantization of the circulation, vortex
3-b Repulsive potential barrier between states of dierent. . .
3-c Critical velocity, metastable ow
3-d Generalization; topological aspects
In this complement, we return to the calculations of ComplementXV, concerning
a system of bosons all in the same individual state. We now consider the more general
case where that state is time-dependent. Using a variational method similar to the one
we used in ComplementXV, we shall study the time variations of the-particle state
vector. This amounts to using a time-dependent mean eld approximation. We shall
establish in
some of its predictions such as the small oscillations associated with Bogolubov phonons.
In , we shall study local conservation laws derived from this equation for which we will
give a hydrodynamic analogy, introducing a characteristic relaxation length. Finally, we
will show in
ows and superuidity.
1. Time evolution
We assume that the ket describing the physical system ofbosons can be written using
relation (7) of ComplementXV:
^
()=
1
!
()0 (1)
but we now suppose that the individual ketis a function of time(). The creation
operator()in the corresponding individual state is then time-dependent:
()0=() (2)
We will let the ket()vary arbitrarily, as long as it remains normalized at all times:
()()= 1 (3)
1657

COMPLEMENT D XV
We are looking for the time variations of()that will yield for
^
()variations as
close as possible to those predicted by the exact-particle Schrödinger equation. As
the one-particle potential1may also be time-dependent, it will be written as1().
1-a. Functional variation
Let us introduce the functional of ():
[ ()] =
1
0
d ()}
() ()
+
}
2
(0) (0) (1) (1) (4)
It can be shown that this functional is stationary when ()is solution of the exact
Schrödinger equation (an explicit demonstration of this property is given in
plementXV. If ()belongs to a variational family, imposing the stationarity of this
functional allows selecting, among all the family kets, the one closest to the exact solution
of the Schrödinger equation. We shall therefore try and make this functional station-
ary, choosing as the variational family the set of kets
^
()written as in (1) where the
individual ket()is time-dependent.
As condition (3) means that the norm of
^
()remains constant, the second
bracket in expression (4) must be zero. We now have to evaluate the average value of
the Hamiltonian()that, actually, has been already computed in (34) of Complement
CXV:
^
()[()]
^
()= ()[0+1()]()
+
( 1)
2
1 :(); 2 :()2(12)1 :(); 2 :() (5)
The only term left to be computed in (4) contains the time derivative.
This term includes the diagonal matrix element:
^
()}
d
d
^
()=
}
!
0()
1
=0
()
d
d
()
1
0 (6)
For an innitesimal time, the operator is proportional to the dierence(+
) (), hence to the dierence between two creation operators associated with two
slightly dierent orthonormal bases. Now, for bosons, all the creation operators commute
with each other, regardless of their associated basis. Therefore, in each term of the
summation over, we can move the derivative of the operator to the far right, and
obtain the same result, whatever the value of. The summation is therefore equal to
times the expression:
1
!
0() ()
1d
d
0 (7)
Now, we know that:
()
1
()0=!1 :()=!()0 (8)
1658

TIME-DEPENDENT GROSS-PITAEVSKII EQUATION
Using in (6) the bra associated with that expression, multiplied by, we get:
^
()}
d
d
^
()=}0()
d
d
0= ()}
d
d
() (9)
Regrouping all these results, we nally obtain:
^
()=
1
0
d ()[0+1()]}
d
d
()
+
( 1)
2
1 :(); 2 :()2(12)1 :(); 2 :() (10)
1-b. Variational computation: the time-dependent Gross-Pitaevskii equation
We now make an innitesimal variation of():
() ()+ () (11)
in order to nd the kets()for which the previous expression will be stationary. As
in the search for a stationary state in ComplementXV, we get variations coming from
the innitesimal ket()and others from the innitesimal bra(); asis
chosen arbitrarily, the same argument as before leads us to conclude that each of these
variations must be zero. Writing only the variation associated with the innitesimal bra,
we see that the stationarity condition requires()to be a solution of the following
equation, written for():
}
d
d
()= [0+1() +()]() (12)
The mean eld operator()is dened as in relations (45) and (46) of Complement
CXVby a partial trace:
(1) = (1)Tr2
()
(2)2(12) (13)
where
()
is the projector onto the ket():
()
=()() (14)
As we take the trace over particle2whose state is time-dependent, the mean eld is also
time-dependent. Relation (12) is the general form of the time-dependent Gross-Pitaevskii
equation.
Let us return, as in XV, to the simple case of spinless bosons,
interacting through a contact potential:
2(rr) =(rr) (15)
Using denition (13) of the Gross-Pitaevskii potential, we can compute its eect in the
position representation, as in ComplementXV. The same calculations as in and
2-b-of that complement allow showing that relation (12) becomes the Gross-Pitaevskii
1659

COMPLEMENT D XV
time-dependent equation (is supposed to be large enough to permit replacing1
by):
}
(r) =
}
2
2
+1(r) + (r)
2
(r) (16)
Normalizing the wave function(r)to:
d
3
(r)
2
= (17)
equation (16) simply becomes:
}
(r) =
}
2
2
+1(r) +(r)
2
(r) (18)
Comment:
It can be shown that this time evolution does conserve the norm of(), as required by
(3). Without the nonlinear term of (16), it would be obvious since the usual Schrödinger
equation conserves the norm. With the nonlinear term present, it will be shown in
that the norm is still conserved.
1-c. Phonons and Bogolubov spectrum
Still dealing with spinless bosons, we consider a uniform system, at rest, of particles
contained in a cubic box of edge length. The external potential1(r)is therefore zero
inside the box and innite outside. This potential may be accounted for by forcing the
wave function to be zero at the walls. In many cases, it is however more convenient to
use periodic boundary conditions (Complement CXIV, ), for which the wave function
of the individual lowest energy state is simply a constant in the box. We thus consider
a system in its ground state, whose Gross-Pitaevskii wave function is independent ofr:
(r) =0() =
1
32
}
(19)
with avalue that satises equation (16):
=
3
= 0 (20)
where0=
3
is the system density. Comparing this expression with relation (58) of
ComplementXVallows us to identifywith the ground state chemical potential. We
assume in this section that the interactions between the particles are repulsive (see the
comment at the end of the section):
0 (21)
1660

TIME-DEPENDENT GROSS-PITAEVSKII EQUATION
. Excitation propagation
Let us see which excitations can propagate in this physical system, whose wave
function is no longer the function (19), uniform in space. We assume:
(r) =0() +(r) (22)
where(r)is suciently small to be treated to rst order. Inserting this expression
in the right-hand side of (16), and keeping only the rst-order terms, we nd in the
interaction term the rst-order expression:
2
(r)(r)= (20)
0+
2
0
= 02+
2 }
(23)
We therefore get, to rst-order:
}
(r) =
}
2
2
+ 20(r) +0
2 }
(r) (24)
which shows that the evolution of(r)is coupled to that of(r). The complex
conjugate equation can be written as:
}
(r) =
}
2
2
20 (r) 0
2 }
(r) (25)
We can make the time-dependent exponentials on the right-hand side disappear
by dening:
(r) =
(r)
}
(r) =
(r)
}
(26)
This leads us to a dierential equation with constant coecients, which can be simply
expressed in a matrix form:
}
(r)(r)
=
}
2
2
+ 0 0
0
}
2
2
0
(r)(r)
(27)
where we have used denition (20) forto replace20 by 0. If we now look for
solutions having a plane wave spatial dependence:
(r) =(k)
kr
(r) =(k)
kr
(28)
the dierential equation can be written as:
}
(k)
(k)
=
}
22
2
+ 0 0
0
}
22
2
0
(k)
(k)
(29)
1661

COMPLEMENT D XV
The eigenvalues}(k)of this matrix satisfy the equation:
}
22
2
+ 0}(k)
}
22
2
0}(k)+ (0)
2
= 0 (30)
that is:
[}(k)]
2}
22
2
+ 0
2
+ (0)
2
= 0 (31)
The solution of this equation is:
}(k) =
}
222
+ 0
2
(0)
2
=
}
222
}
22
2
+ 20 (32)
(the opposite value is also a solution, as expected since we calculate at the same time
the evolution of()and of its complex conjugate; we only use here the positive value).
Setting:
0=
2
}0 (33)
relation (32) can be written:
(k) =
}
22
(
2
+
2
0
) (34)
The spectrum given by (32) is plotted in Figure, where one sees the intermediate regime
between the linear region at low energy, and the quadratic region at higher energy. It is
called the Bogolubov spectrum of the boson system.
. Discussion
Let us compute the spatial and time evolution of the particle density(rt)when
(r)obeys relation (28). The particle density at each pointrof space is the sum of
the densities associated with each particle, that istimes the squared modulus of the
wave function(r). To rst-order in(r), we obtain:
(rt) = 0()
}
[(r)] +c.c. (35)
(where c.c. stands for complex conjugate). Using (26) and (28), we can nally write:
(rt) =
0()
} }
(k0)
[kr(k)]
+c.c.
=
32
(k0)
[kr(k)]
+c.c. (36)
Consequently, the excitation spectrum we have calculated corresponds to density waves
propagating in the system with a phase velocity(k).
In the absence of interactions, (=0= 0), this spectrum becomes:
}(k) =
}
22
2
(37)
1662

TIME-DEPENDENT GROSS-PITAEVSKII EQUATION
Figure 1: Bogolubov spectrum: variations of the function(k)given by equation (32)
as a function of the dimensionless variable= 0. When 1, we get a linear
spectrum (the arrow in the gure shows the tangent to the curve at the origin), whose
slope is equal to the sound velocity; when 1, the spectrum becomes quadratic, as
for a free particle.
which simply yields the usual quadratic relation for a free particle. Physically, this means
that the boson system can be excited by transferring a particle from the individual ground
state, with wave function0(r)and zero kinetic energy, to any statek(r)having an
energy}
22
2.
In the presence of interactions, it is no longer possible to limit the excitation to a
single particle, which immediately transmits it to the others. The system's excitations
become what we call elementary excitations, involving a collective motion of all the
particles, and hence oscillations in the density of the boson system. If0, we see
from (34) that:
(k) (38)
whereis dened as:
=
}
2
0=
0
(39)
For small values of, the interactions have the eect of replacing the quadratic spectrum
(37) by a linear spectrum. The phase velocity of all the excitations in thisvalue domain
is a constant. It is called the sound velocity in the interacting boson system, by
analogy with a classical uid where the sound wave dispersion relation is linear, as
predicted by the Helmholtz equation. We shall see in plays a
fundamental role in the computations related to superuidity, especially for the critical
velocity determination. If, on the other hand,0, the spectrum becomes:
}(k)
}
22
2
+ 0+ (40)
(the following corrections being in
2
0
2
,
4
0
4
, etc.). We nd again, within a small
correction, the free particle spectrum: exciting the system with enough energy allows
1663

COMPLEMENT D XV
exciting individual particles almost as if they were independent. Figure
complete variation of the spectrum (32), with the transition from the linear region at
low energies, to the quadratic region at high energies.
Comment:
As we assumed the interactions to be repulsive in (21), the square roots in () and
(39) are well dened. If the coupling constantbecomes negative, the sound velocity
will become imaginary, and, as seen from (32), so will the frequencies()(at least for
small values of). This will lead, for the evolution equation (29), to solutions that are
exponentially increasing or decreasing in time, instead of oscillating. An exponentially
increasing solution corresponds to an instability of the system. As already encountered in
XV, we see that a boson system becomes unstable in the presence
of attractive interactions, however small they might be. In XV,
we shall see that this instability persists even for non-zero temperature. In a general
way, an attractive condensate occupying a large region in space tends to collapse onto
itself, concentrating into an ever smaller region. However, when it is conned in a nite
region (as is the case for experiments where cold atoms are placed in a magneto-optical
trap), any change in the wave function that brings the system closer to the instability
also increases the gas energy; this results in an energy barrier, which allows the system
of condensed attractive bosons to remain in a metastable state.
2. Hydrodynamic analogy
Let us return to the study of the time evolution of the Gross-Pitaevskii wave function
and of the density variations(r), without assuming as in
stays very close to uniform equilibrium. We will show that the Gross-Pitaevskii equation
can take a form similar to the hydrodynamic equation describing a uid's evolution. In
this discussion, it is useful to normalize the Gross-Pitaevskii wave function to the particle
number, as in equation (17). Equation (16) can then be written as:
}
(r) =
}
2
2
+1(r) +(r)(r) (41)
where the local particle density(r)is given by:
(r) =(r)
2
(42)
2-a. Probability current
Since:
(r) =(r)(r) +(r)(r) (43)
the time variation of the density may be obtained by rst multiplying (41) by(r),
then its complex conjugate by(r), and then adding the two results; the potential
terms in1(r)and(r)cancel out, and we get:
(r) =
}
2
[(r)(r)(r)(r)] (44)
1664

TIME-DEPENDENT GROSS-PITAEVSKII EQUATION
Let us now dene a vectorJ(r)by:
J(r) =
}
2
[(r)r(r)(r)r(r)] (45)
If we compute the divergence of this vector, the terms inr rcancel out and we
are left with terms identical to the right-hand side of (44), with the opposite sign. This
leads to the conservation equation:
(r) +rJ(r) = 0 (46)
J(r)is thus the probability current associated with our boson system. Integrating over
all space, using the divergence theorem, and assuming(r)(hence the current) goes
to zero at innity, we obtain:
d
3
(r) =d
3
(r)
2
= 0 (47)
This shows, as announced earlier, that the Gross-Pitaevskii equation conserves the norm
of the wave function describing the particle system.
We now set:
(r) =
(rt)
(r)
(48)
The gradient of this function is written as:
r(r) =
(rt)
r
(r) +(r)r(r) (49)
Inserting this result in (45), we get:
J(r) =
}
(r)r(r) (50)
or, dening the particle local velocityv(r)as the ratio of the current to the density:
v(r) =
J(r)
(r)
=
}
r(r) (51)
We have dened a velocity eld, similar to the velocity eld of a uid in motion in a
certain region of space; this eld velocity is irrotational (zero curl everywhere).
2-b. Velocity evolution
We now compute the time derivative of this velocity. Taking the derivative of (48),
we get:
}
(r) =
(rt)
}
(r)}(r)
(r)
(r) (52)
so that we can isolate the time derivative of(rt)by the following combination:
}(r)
(r)(r)(r)=2}(r)(r) (53)
1665

COMPLEMENT D XV
The left-hand side of this relation can be computed with the Gross-Pitaevskii equation
(18) and its complex conjugate, as we now show. We rst take the divergence of the
gradient (49) to obtain the Laplacian:
(r) =rr(r) =
(r)

(r) + 2r(r)r(r)
+
(r)(r)(r) (r(r))
2
(54)
We then insert the time derivative of(r)given by the Gross-Pitaevskii equation (18)
in the left-hand side of relation (53), which becomes:
}
2
2
[(r)(r) +(r)(r)] + 2 [1(r) +(r)](r)
2
=
}
2
2
2
(r) (r)2(r) (r(r))
2
+ 21(r) +(r)(r) (55)
This result must be equal to the right-hand side of (53). We therefore get, after dividing
both sides by2(r):
}
(rt) =
}
2
2
1
(r)

(r)(r(r))
2
[1(r) +(r)](56)
Using (51), we nally obtain the evolution equation for the velocityv(r):
v(r) =r 1(r) +(r) +
v
2
(r)
2
+
}
2
2
1
(r)

(r)(57)
This equation looks like the classical Newton equation. Its right-hand side includes
the sum of the forces corresponding to the external potential1(r), and to the mean
interaction potential with the other particles(r); the third term in the gradient is the
classical kinetic energy gradient
1
(as in Bernoulli's equation of classical hydrodynamics).
The only purely quantum term is the last one, as shown by its explicit dependence on}
2
.
It involves spatial derivatives of(r), and is only important if the relative variations
of the density occur over small enough distances (for example, this term is zero for
a uniform density). This term is sometimes called quantum potential, or quantum
pressure term or, in other contexts, Bohm potential. A frequently used approximation
is to consider the spatial variations of(r)to be slow, which amounts to ignoring this
quantum potential term: this is the so-called Thomas-Fermi approximation.
We have found for a system ofparticles a series of properties usually associated
with the wave function of a single particle, and in particular a local velocity directly
proportional to its phase gradient
2
. The only dierence is that, for the-particle case,
1
It is a total derivative term (the derivative describing, in a uid, the motion of each particle). As
the velocity eld has a zero curl according to (51), a simple vector analysis calculation shows this term
to be equal to(vr)v; it can therefore be accounted for by replacing on the left-hand side of (57)
the partial derivativeby the total derivative dd= +vr.
2
The quantum potential is still present for a single particle, since making= 0in (57) does not change
this potential. For= 0, the Gross-Pitaevskii equation simply reduces to the standard Schrödinger
equation, valid for a single particle.
1666

TIME-DEPENDENT GROSS-PITAEVSKII EQUATION
Figure 2: A repulsive boson gas is contained in a toroidal box. All the bosons are supposed
to be initially in the same quantum state describing a rotation around theaxis. As we
explain in the text, this rotation can only slow down if the system overcomes a potential
energy barrier that comes from the repulsive interactions between the particles. This
prevents any observable damping of the rotation over any accessible time scale; the uid
rotates indenitely, and is said to be superuid.
we must add to the external potential1(r)a local interaction potential(r), which
does not signicantly change the form of the equations but introduces some nonlinearity
that can lead to completely new physical eects.
3. Metastable currents, superuidity
Consider now a system of repulsive bosons contained in a toroidal box with a rotational
axis(Figure); the shape of the torus cross-section (circular, rectangular or other) is
irrelevant for our argument and we shall use cylindrical coordinates,and. We rst
introduce solutions of the Gross-Pitaevskii equation that correspond to the system rotat-
ing inside the toroidal box, around theaxis. We will then show that these rotational
states are metastable, as they can only relax towards lower energy rotational states by
overcoming a macroscopic energy barrier: this is the physical origin of superuidity.
3-a. Toroidal geometry, quantization of the circulation, vortex
To prevent any confusion with the azimuthal angle, we now callthe Gross-
Pitaevskii wave function. The time-independent Gross-Pitaevskii equation then becomes
(in the absence of any potential except the wall potentials of the box):
}
2
2
1
+
1
2
2
2
+
2
2
+ (r)
2
(r) =(r) (58)
We look for solutions of the form:
(r) =() (59)
1667

COMPLEMENT D XV
whereis necessarily an integer (otherwise the wave function would be multi-valued).
Such a solution has an angular momentum with a well dened component along, equal
to}per atom. Inserting this expression in (58), we obtain the equation for():
}
2
2
1()
+
2
()
2
+ ()
2
+
2
}
2
2
2
()
= () (60)
which must be solved with the boundary conditions imposed by the torus shape to obtain
the ground state (associated with the lowest value of). The term in
2
}
2
2
2
is simply
the rotational kinetic energy around. If the tore radiusis very large compared to
the size of its cross-section, the term
22
may, to a good approximation, be replaced
by the constant
22
. It follows that the same solution of (60) is valid for any value of
as long as the chemical potential is increased accordingly. Each value of the angular
momentum thus yields a ground state and the larger, the higher the corresponding
chemical potential. All the coecients of the equation being real, we shall assume, from
now on, the functions()to be real.
As the wave function is of the form (59), its phase only depends on, and expres-
sion (51) for the uid velocity is written as:
v=
1
}
e (61)
whereeis the tangential unit vector (perpendicular both torand theaxis). Con-
sequently, the uid rotates along the toroidal tube, with a velocity proportional to. As
vis a gradient, its circulation along a closed loop equivalent to zero (i.e. which can be
contracted continuously to a point) is zero. If the closed loop goes around the tore, the
path is no longer equivalent to zero and its circulation may be computed along a circle
whereandremain constant, andvaries from0to2; as the path length equals2,
we get:
vds=
2}
(62)
(with a+sign if the rotation is counterclockwise and asign in the opposite case). As
is an integer, the velocity circulation around the center of the tore is quantized in units
of . This is obviously a pure quantum property (for a classical uid, this circulation
can take on a continuous set of values).
To simplify the calculations, we have assumed until now that the uid rotates as
a whole inside the toroidal ring. More complex uid motions, with dierent geometries,
are obviously possible. An important case, which we will return to later, concerns the
rotation around an axis still parallel to, but located inside the uid. The Gross-
Pitaevskii wave function must then be zero along a line inside the uid itself, which thus
contains a singular line. This means that the phase may change by2as one rotates
around this line. This situation corresponds to what is called a vortex, a little swirl of
uid rotating around the singular line, called the vortex core line. As the circulation of
the velocity only depends on the phase change along the path going around the vortex
core, the quantization relation (62) remains valid. Actually, from a historical point of
view, the Gross-Pitaevskii equation was rst introduced for the study of superuidity
and the quantization of the vortices circulation.
1668

TIME-DEPENDENT GROSS-PITAEVSKII EQUATION
3-b. Repulsive potential barrier between states of dierent
A classical rotating uid will always come to rest after a certain time, due to the
viscous dissipation at the walls. In such a process, the macroscopic rotational kinetic en-
ergy of the whole uid is progressively degraded into numerous smaller scale excitations,
which end up simply heating the uid. Will a rotating quantum uid of repulsive bosons,
described by a wave function(r), behave in the same way? Will it successively evolve
towards the state1(r), then2(r), etc., until it comes to rest in the state0(r)?
We have seen in XVthat, to avoid the energy cost of
fragmentation, the system always remains in a state where all the particles occupy the
same quantum state. This is why we can use the Gross-Pitaevskii equation (18).
. A simple geometry
Let us rst assume that the wave function(r)changes smoothly from(r)to
(r)according to:
(r) =()(r) +()(r) (63)
where the modulus of()decreases with time from1to0, whereas()does the
opposite. Normalization imposes that at all times:
()
2
+()
2
= 1 (64)
In such a state, let us show that the numerical density( ;)now depends on
(this was not the case for either statesorseparately). The transverse dependence
of the density as a function of the variablesand, is barely aected
3
. The variations
of( ;)are given by:
( ;) =()()+()()
2
=()
2
()
2
+()
2
()
2
+()()()()
()
+c.c. (65)
where c.c. stands for the complex conjugate of the preceding factor. The rst two terms
are independent of, and are just a weighted average of the densities associated with
each of the statesand. The last term oscillates as a function ofwith an amplitude
() (), which is only zero if one of the two coecients()or()is zero.
Callingthe phase of the coecient(), this last term is proportional to:
()()
()
+c.c.= 2() ()cos [()+ ] (66)
Whatever the phases of the two coecients()and(), the cosine will always oscillate
between1and1as a function of. Adjusting those phases, one can deliberately change
the value offor which the density is maximum (or minimum), but this will always occur
somewhere on the circle. Superposing two states necessarily modulates the density.
Let us evaluate the consequences of this density modulation on the internal repul-
sive interaction energy of the uid. As we did in relation (15), we use for the interaction
3
or not at all, if we suppose the functions()and()to be equal.
1669

COMPLEMENT D XV
energy the zero range potential approximation, and insert it in expression (15) of Com-
plementXV. Taking into account the normalization (17) of the wave function, we get:
2=
2
d
3
(r)
4
=
2
0
d
2
0
d
+
d[( ;)]
2
(67)
We must now include the square of (65) in this expression, which will yield several terms.
The rst one, in()
4
, leads to the contribution:
()
4
2 (68)
where 2is the interaction energy for the state(r). The second contribution is
the similar term for the state, and the third one, a cross term in2()
2
()
2
.
Assuming, to keep things simple, that the densities associated with the statesand
are practically the same, the sum of these three terms is just:
()
2
+()
2
2
2= 2 (69)
Up to now, the superposition has had no eect on the repulsive internal interaction
energy. As for the cross terms between the terms independent ofin (65) and the terms
in
()
, they will cancel out when integrated over. We are then left with the cross
terms in
() ()
, whose integral overyields:
2()
2
()
2
()
2
()
2
(70)
Assuming as before that the densities associated with the statesandare practically
the same, we obtain, after integration overand:
2()
2
()
2
2 (71)
Adding (69), we nally obtain:
2=1 + 2()
2
()
2
2 (72)
We have shown that the density modulation associated with the superposition of
states always increases the internal repulsion energy: this modulation does lower the
energy in the low density region, but the increase in the high energy region outweighs
the decrease (since the repulsive energy is a quadratic function of the density). The
internal energy therefore varies between2and the maximum(32) 2, reached
when the moduli of()and()are both equal to1
2.
. Other geometries, dierent relaxation channels
There are many other ways for the Gross-Pitaevskii wave function to go from
one rotational state to another. We have limited ourselves to the simplest geometry to
introduce the concept of energy barriers with minimal mathematics. The uid could
1670

TIME-DEPENDENT GROSS-PITAEVSKII EQUATION
transit, however, through more complex geometries, such as the frequently observed
creation of a vortex on the wall, the little swirl we briey talked about at the end of
3-a. A vortex introduces a2phase shift around a singular line along which the wave
function is zero. Once the vortex is created, and contrary to what was the case in (62),
the velocity circulation along a loop going around the torus is no longer independent of
its path: it will change by2}depending on whether the vortex is included in the
loop or not. Furthermore, as the vortex moves in the uid from one wall to another,
it can be shown that the proportion of uid conserving the initial circulation decreases
while the proportion having a circulation where the quantum numberdiers by one unit
increases. Consequently, this vortex motion changes progressively the rotational angular
momentum. Once the vortex has vanished on the other wall, the nal result is a decrease
by one unit of the quantum numberassociated with the uid rotation.
The continuous passage of vortices from one wall to another therefore yields another
mechanism that allows the angular moment of the uid to decrease. The creation of a
vortex, however, is necessarily accompanied by a non-uniform uid density, described by
the Gross-Pitaevskii equation (this density must be zero along the vortex core). As we
have seen above, this leads to an increase in the average repulsive energy between the
particles (the uid elastic energy). This process thus also encounters an energy barrier
(discussed in more detail in the conclusion). In other words, the creation and motion of
vortices provide another relaxation channel for the uid velocity, with its own energy
barrier, and associated relaxation time.
Many other geometries can be imagined for changing the uid ow. Each of them
is associated with a potential barrier, and therefore a certain lifetime. The relaxation
channel with the shortest lifetime will mainly determine the damping of the uid velocity,
which may take, in certain cases, an extraordinarily long time (dozens of years or more),
hence the name of superuid.
3-c. Critical velocity, metastable ow
For the sake of simplicity, we will use in our discussion the simple geometry of
a. The transposition to other geometries involving, for example, the creation of vortices
in the uid would be straightforward. The main change would concern the height of the
energy barrier
4
.
With this simple geometry, the potential to be used in (60) is the sum of a repulsive
potential()
2
and a kinetic energy of rotation around, equal to
2
}
2
2
2
. We
now show that, in a givenstate, these two contributions can be expressed as a function
of two velocities. First, relation (61) yields the rotation velocityassociated with state
:
=
1
}
(73)
and the rotational energy is simply written as:
=
2
}
2
2
2
=
1
2
()
2
(74)
4
When several relaxation channels are present, the one associated with the lowest barrier mainly
determines the time evolution.
1671

COMPLEMENT D XV
As for the interaction term (term inon the left-hand side), we can express it in a more
convenient way, dening as before the numerical density0:
0=()
2
(75)
and using the denition (39) for the sound velocity. It can then be written in a form
similar to (74):
0=
2
(76)
The two velocitiesandallow an easy comparison of the respective importance of the
kinetic and potential energies in a state.
We now compare the contributions of these two terms either for states with a given
, or for a superposition of states (63). To clarify the discussion and be able to draw a
gure, we will use a continuous variable dened as the averageof the component
alongof the angular momentum:
=}()
2
+}()
2
(77)
This expression varies continuously between}and}when the relative weights of
()
2
and()
2
are changed while imposing relation (64); the continuous variable:
= } (78)
allows making interpolations between the discrete integer values of.
Using the normalization relation (64) of the wave function (63), we can express
as a function of()
2
:
= ( )()
2
+ (79)
The variablecharacterizes the modulus of each of the two components of the variational
function (63). A second variable is needed to dene the relative phase between these two
components, which comes into play for example in (66). Instead of studying the time
evolution of the uid state vector inside this variational family, we shall simply give
a qualitative argument, for several reasons. First of all, it is not easy to characterize
precisely the coupling between the uid and the environment by a Hamiltonian that can
change the uid rotational angular momentum (for example, the wall's irregularities may
transfer energy and angular momentum from the uid to the container). Furthermore,
as the time-dependent Gross-Pitaevskii equation is nonlinear, its precise solutions are
generally found numerically. This is why we shall only qualitatively discuss the eects of
the potential barrier found in 3-b. The higher this barrier, the more dicult it is for
to go fromto. Let us evaluate the variation of the average energy as a function of.
For integer values of, relation (74) shows that the average rotational kinetic
energy varies as the square of; in between, its value can be found by interpolation as in
(77). As for the potential energy, we saw that a continuous variation of()and()
necessarily involves a coherent superposition, which has an energy cost and increases the
repulsive potential interaction. In particular, this interaction energy is multiplied by the
factor32when the moduli of()and()are equal (i.e. whenis an integer plus
12). As a result, to the quadratic variation of the rotational kinetic energy, we must
1672

TIME-DEPENDENT GROSS-PITAEVSKII EQUATION
add an oscillating variation of the potential energy, minimum for all the integer values
of, and maximum half-way between. The oscillation amplitude is given by:
0
2
=
1
2
2
(80)
Figure
of the average value. The lowest one, shown as a dotted line, corresponds to a
superposition of the statewith the state=1, for a very small value of the coupling
constant(weak interactions, gas almost ideal). In this case and according to (39), the
sound velocity is also very small and we are in the case. Comparing (74) and (80)
then shows that the potential energy contribution is negligible compared to the variation
of the rotational kinetic energy between the two states. As a result, the modulation on
this dotted line is barely perceptible, and this curve presents a single minimum at= 0:
whatever the initial rotational state, no potential barrier prevents the uid rotational
velocity from returning to zero (for example under the eect of the interactions with the
irregularities of the walls containing the uid).
The other two curves in Figure , hence,
according to (39), to a much higher value of. There are now several values offor
whichis small compared to. The dashed line corresponds, as for the previous curve,
to a superposition of the two states= 1and=1; the solid line (for the same value
of) to a superposition of= 3and= 0, corresponding to the case where the system
goes directly from the state= 3to the rotational ground state in the torus, with= 0.
It is obviously this last curve that presents the lowest energy barrier starting from= 3
(shown with a circle in the gure). This is normal since this is the curve that involves
the largest variation in the kinetic energy, in a sense opposite to that of the potential
energy variation. It is thus the direct transition from= 3to= 0that will determine
the possibility for the system to relax towards a state of slower rotation. Let us again
use (74) and (80) to compare the kinetic energy variation and the height of the repulsive
potential barrier. All the states, with velocitiesmuch larger than, have a kinetic
energy much bigger than the maximum value of the potential energy: no energy barrier
can be formed. On the other hand, all the stateswith velocitiesmuch smaller than
cannot lower their rotational state without going over a potential barrier.
In between these two extreme cases, there exists (for a given) a critical value
corresponding to the onset of the barrier. It is associated with a critical velocity
=}, of the order of the sound velocity, xing the maximum value of
for which this potential barrier exists. If the uid rotational velocity in the torus is
greater than, the liquid can slow down its rotation without going over an energy
barrier, and dissipation occurs as in an ordinary viscous liquid the uid is said to
be normal. If, however, the uid velocity is less than the critical velocity, the physical
system must necessarily go over a potential barrier (or more) to continuously tend towards
= 0. As this barrier results from the repulsion between all the particles and their
neighbors, it has a macroscopic value. In principle, any barrier can be overcome, be it
by thermal excitation, or by the quantum tunnel eect. However the time needed for
this passage may take a gigantic value. First of all, it is extremely unlikely for a thermal
uctuation to reach a macroscopic energy value. As for the tunnel eect, its transition
probability decreases exponentially with the barrier height and becomes extremely low
for a macroscopic object. Consequently, the relaxation times of the uid velocity may
1673

COMPLEMENT D XV
Figure 3: Plots of the energy of a rotating repulsive boson system, in a coherent superpo-
sition of the stateand the state, as a function of its average angular momentum,
expressed in units of}. The lower dotted curve corresponds to the case where= 1
and the interaction constantis small (almost ideal gas). The potential energy is then
negligible and the total energy presents a single minimum in= 0. Consequently,
whatever the initial rotational state of the uid, it will relax to a motionless state= 0
without having to go over any energy barrier, and its rotational kinetic energy will dis-
sipate: it behaves as a normal uid. The other two curves correspond to a much larger
value of therefore, according to (39) to a much higher value of. The dashed curve
still corresponds to a superposition of the rotational statesand= 1, and the solid
line to the direct superposition of the state= 3(shown with a circle in the gure) and the
ground state= 0. The solid line curve presents the smallest barrier, hence determining
the metastability of the current.
The higher the coupling constant, the morestates presenting a minimum in the po-
tential energy appear. They correspond to ow velocities in the torus that are smaller
than the critical velocity. To go from the rotational state= 1to the motionless state
= 0, the system must go over a macroscopic energy barrier, which only occurs with a
probability so small it can be considered equal to zero. The rotational current is therefore
permanent, lasting for years, and the system is said to be superuid. On the other hand,
the states with higher values of, for which the curve presents no minima, correspond to
a normal uid, whose rotation can slow down because of the viscosity (dissipation of the
kinetic energy into heat).
become extraordinarily large, and, on the human scale, the rotation can be considered
to last indenitely. This phenomenon is called superuidity.
3-d. Generalization; topological aspects
Our argument remained qualitative for several reasons. To begin with, we showed
the existence for the uid of a critical velocity, of the order of, without giving its
precise value. It would require a more detailed study of the potential curves such as the
ones plotted in Figure, to obtain the precise values of the parameters for which the
potential barrier appears or disappears. We also limited ourselves to simple geometries
that could be described by a single variable, not taking into account other possible
1674

TIME-DEPENDENT GROSS-PITAEVSKII EQUATION
deformations of the wave function. Various situations could occur, such as the creation of
vortices or more complex processes, which would require a more elaborate mathematical
treatment. In other words, we would have to take into account the existence of other
relaxation channels for the moving uid to come to rest, and look for the one leading to
the lowest potential barrier, thereby determining the lifetime of the superuid current.
There is, however, a more general way to address the problem, which shows that
our basic conclusions are not limited to the particular case we have studied. It is based
on the topological aspects of the wave function phase. When this phase varies by2as
we go around the torus, it expresses a topological property characterized by the winding
number, which is an integer and cannot vary continuously. This is why, as long as
the phase is well dened everywhere i.e. as long as the wave function does not go to
zero we cannot go continuously fromto1. We already saw this in the particular
example of the wave function (63): when the modulus of()varies in time from1to
0, while the modulus of()does the opposite, we necessarily went through a situation
where the wave function went to zero through interference, in a plane corresponding to
a certain value of; but the phase of the wave function is undetermined in this plane,
and as we cross it, the phase undergoes a discontinuous jump. Now the canceling of the
wave function of a great number of condensed bosons means the density must also be
zero at that point, hence larger in other points of space. This spatial density variation
introduces an energy increase, due to the nite compressibility of the uid (as we saw in
, the energy increase in the high density regions is larger than the energy decrease
in low density regions). This means there is an energy barrier opposing the change in the
number of turnsof the phase. The height of this barrier must now be compared with
the kinetic energy variation. As seen above, there is a drastic change in the ow regime,
depending on whether the uid velocity is smaller or larger than a certain critical velocity
. In the rst case, superuidity allows a current to ow without dissipation, lasting
practically indenitely. In the second, no energy consideration opposes dissipation, and
the rotation slows down progressively, as in an ordinary liquid.
The essential idea to remember is that superuidity comes from the repulsive
interactions, and for two reasons. First of all, they explain the presence of the energy
barrier, responsible for the metastability. The second reason, even more essential, is that
the repulsion between bosons constantly tends to put all the uid particles in the same
quantum state - see XV; thanks to this property, we were able
to characterize the intermediate rotational states by a very simple wave function (63).
This implies that the quantum uid can only occupy a very limited number of states,
compared to a situation where the particles would be distinguishable. Consequently, it
has a hard time dissipating its kinetic energy into heat, as a classical uid would do, and
it therefore maintains its rotation over such long times that a slowing down is practically
impossible to observe.
1675

FERMION SYSTEM, HARTREE-FOCK APPROXIMATION
Complement EXV
Fermion system, Hartree-Fock approximation
1 Foundation of the method
1-a Trial family and Hamiltonian
1-b Energy average value
1-c Optimization of the variational wave function
1-d Equivalent formulation for the average energy stationarity
1-e Variational energy
1-f Hartree-Fock equations
2 Generalization: operator method
2-a Average energy
2-b Optimization of the one-particle density operator
2-c Mean eld operator
2-d Hartree-Fock equations for electrons
2-e Discussion
Introduction
Computing the energy levels of a system ofelectrons, interacting with each other
through the Coulomb force, and placed in an external potential1(r)is a very important
problem in physics and chemistry. It is encountered in the determination of the energy
levels of atoms (in which case the external potential for the electrons
1
is the Coulomb
potential created by the nucleus
2
40), or of molecules as well, or of electrons in
a solid (submitted to a periodic potential), or in an aggregate or a nanocristal, etc. It
is a problem where two ingredients simultaneously play an essential role: the fermionic
character of the electrons, which forbids them to occupy the same individual state, and
the eects of their mutual interactions. Ignoring the Coulomb repulsion between electrons
would make the calculation fairly simple, and similar to that of 1 in ComplementXIV,
concerning free fermions in a box; the free plane wave individual states would have to
be replaced by the energy eigenstates of a single particle placed in the potential1(r).
This would lead to a3-dimensional Schrödinger equation, which can be solved with very
good precision, although not necessarily analytically.
However, be it in atoms or in solids, the repulsion between electrons plays an
essential role. Neglecting it would lead us to conclude, for example, that, asincreases,
the size of atoms decreases due to the attractive eect of the nucleus, whereas the opposite
occurs
2
! Forinteracting particles, even without taking the spin into account, an exact
1
We assume the nucleus mass to be innitely larger than the electron mass. The electronic system
can then be studied assuming the nucleus xed and placed at the origin.
2
The Pauli exclusion principle is not sucient to explain why an atom's size increases with its atomic
number. One can evaluate the approximate size of a hypothetical atom with non-interacting electrons
(we consider the atom's size to be given by the size of the outermost occupied orbit). The Bohr radius
1677

COMPLEMENT E XV
computation would require solving a Schrödinger equation in a3-dimensional space;
this is clearly impossible whenbecomes large, even with the most powerful computer.
Hence, approximation methods are needed, and the most common one is the Hartree-
Fock method, which reduces the problem to solving a series of3-dimensional equations.
It will be explained in this complement for fermionic particles.
The Hartree-Fock method is based on the variational approximation (Comple-
mentXI), where we choose a trial family of state vectors, and look for the one that
minimizes the average energy. The chosen family is the set of all possible Fock states de-
scribing the system offermions. We will introduce and compute the self-consistent
mean eld in which each electron moves; this mean eld takes into account the repulsion
due to the other electrons, hence justifying the central eld method discussed in Com-
plementXIV. This method applies not only to the atom's ground state but also to all
its stationary states. It can also be generalized to many other systems such as molecules,
for example, or to the study of the ground level and excited states of nuclei, which are
protons and neutrons in bound systems.
This complement presents the Hartree-Fock method in two steps, starting in
with a simple approach in terms of wave functions, which is then generalized in
using Dirac notation and projector operators. The reader may choose to go through both
steps or go directly to the second. In , we deal with spinless particles, which allows
discussing the basic physical ideas and introducing the mean eld concept keeping the
formalism simple. A more general point of view is exposed in , to clarify a number
of points and to introduce the concept of a one-particle (with or without spin) eective
Hartree-Fock Hamiltonian. This Hamiltonian reduces the interactions with all the other
particles to a mean eld operator. More details on the Hartree-Fock methods, and in
particular their relations with the Wick theorem, can be found in Chapters 7 and 8 of
reference [5].
1. Foundation of the method
Let us rst expose the foundation of the Hartree-Fock method in a simple case where
the particles have no spin (or are all in the same individual spin state) so that no spin
quantum number is needed to dene their individual states, specied by their wave
functions. We introduce the notation and dene the trial family of the-particle state
vectors.
1-a. Trial family and Hamiltonian
We choose as the trial family for the state of the-fermion system all the states
that can be written as:
=
1 2
0 (1)
where
1
,
2
,...,are the creation operators associated with a set of normalized
individual states1,2, ..., all orthogonal to each other (and hence distinct). The
state is therefore normalized to1. This set of individual states is, at the moment,
arbitrary; it will be determined by the following variational calculation.
0varies as1, whereas the highest value of the principal quantum numberof the occupied states
varies approximately as
13
. The size
2
0we are looking for varies approximately as
13
.
1678

FERMION SYSTEM, HARTREE-FOCK APPROXIMATION
For spinless particles, the corresponding wave function (r1r2r)can be writ-
ten in the form of a Slater determinant (Chapter XIV, C-3-c-):
(r1r2r) =
1
!
1(r1)2(r1) (r1)
1(r2)2(r2) (r2)
1(r)2(r) (r)
(2)
The system Hamiltonian is the sum of the kinetic energy, the one-body potential
energy and the interaction energy:
=0+ext+ int (3)
The rst term,0, is the operator associated with the fermion kinetic energy, sum of
the individual kinetic energies:
0=
=1
(P)
2
2
(4)
whereis the particle mass andP, the momentum operator of particle. The second
term,ext, is the operator associated with their energy in an applied external potential
1:
ext=
=1
1(R) (5)
whereRis the position operator of particle. For electrons with chargeplaced in
the attractive Coulomb potential of a nucleus of chargepositioned at the origin (
is the nucleus atomic number), this potential is attractive and equal to:
1(r) =
2
40
1
r
(6)
where0is the vacuum permittivity. Finally, the termintcorresponds to their mutual
interaction energy:
int=
1
2
=
2(RR) (7)
For electrons, the function2is given by the Coulomb repulsive interaction:
2(rr) =
2
40
1
rr
(8)
The expressions given above are just examples; as mentioned earlier, the Hartree-Fock
method is not limited to the computation of the electronic energy levels in an atom.
1679

COMPLEMENT E XV
1-b. Energy average value
Since state (1) is normalized, the average energy in this state is given by:
= (9)
Let us evaluate successively the contributions of the three terms of (3), to obtain an
expression which we will eventually vary.
. Kinetic energy
Let us introduce a complete orthonormal basisof the one-particle state space
by adding to the set of states(= 1,2, ...,) other orthonormal states; the subscript
now ranges from1to, dimension of this space (may be innite). We can then
expand0as in relation (B-12) of Chapter:
0=
P
2
2
(10)
where the two summations overandrange from1to. The average value in of
the kinetic energy can then be written:
0=
P
2
2
0
2 1
1 2
0 (11)
which contains the scalar product of the ket:
1 2
0= 12 (12)
by the bra:
0
2 1
= 12 (13)
Note however that in the ket, the action of the annihilation operatoryields zero
unless it acts on a ket where the individual state is already occupied; consequently, the
result will be dierent from zero only if the stateis included in the list of the
states1,2, ..... Taking the Hermitian conjugate of (13), we see that the same
must be true for the state, which must be included in the same list. Furthermore,
if=the resulting kets have dierent occupation numbers, and are thus orthogonal.
The scalar product will therefore only dier from zero if=, in which case it is simply
equal to1. This can be shown by moving to the front the stateboth in the bra and
in the ket; this will require two transpositions with two sign changes which cancel out,
or none if the statewas already in the front. Once the operators have acted, the bra
and the ket correspond to exactly the same occupied states and their scalar product is
1. We nally get:
0=
=1
P
2
2
(14)
1680

FERMION SYSTEM, HARTREE-FOCK APPROXIMATION
Consequently, the average value of the kinetic energy is simply the sum of the average
kinetic energy in each of the occupied states.
For spinless particles, the kinetic energy operator is actually a dierential operator
~
2
2acting on the individual wave functions. We therefore get:
0=
}
2
2
d
3
=1
(r) (r) (15)
. Potential energy
As the potential energy1is also a one-particle operator, its average value can be
computed in a similar way. We obtain:
ext=
=1
1(R) (16)
that is, for spinless particles:
ext=d
3
1(r)
=1
(r)
2
(17)
As before, the result is simply the sum of the average values associated with the individual
occupied states.
. Interaction energy
The average value of the interaction energy2in the state has already been
computed in . We just have to replace, in the relations (C-28) as
well as (C-32) to (C-34) of that chapter, theby1for all the occupied states, by
zero for the others, and to rename the wave functions(r)as(r). We then get:
int= int =
1
2
d
3
d
3
2(rr)
=1
(r)
2
(r)
2
(r)(r)(r)(r)
(18)
We have left out the condition=, no longer useful since the=terms are zero. The
second line of this equation contains the sum of the direct and the exchange terms.
The result can be written in a more concise way by introducing the projector
over the subspace spanned by thekets:
=
=1
(19)
Its matrix elements are:
r r=
=1
(r)(r) (20)
1681

COMPLEMENT E XV
This leads to:
int=
1
2
d
3
d
3
2(rr)r rr r r rr r
(21)
Comment:
The matrix elements of are actually equal to the spatial non-diagonal correlation
function1(rr), which will be dened in Chapter ). This correlation
function can be expressed as the average value of the product of eld operators (r):
1(rr) = (r) (r) (22)
For a system offermions in the states1,2, ..,, we can write:
1(rr) =12 (r) (r)12
= 12 (r) (r) 12 =
=1
(r)(r)
(23)
Inserting this relation in (18) we get:
int=
1
2
d
3
d
3
2(rr) 1(rr)1(rr) 1(rr)1(rr) (24)
Comparison with relation (C-28) of Chapter, which gives the same average value,
shows that the right-hand side bracket contains the two-particle correlation function
2(rr). For a Fock state, this function can therefore be simply expressed as two prod-
ucts of one-particle correlation functions at two points:
2(rr) =1(rr)1(rr) 1(rr)1(rr) (25)
1-c. Optimization of the variational wave function
We now vary to determine the conditions leading to a stationary value of the
total energy:
= 0+ ext+ int (26)
where the three terms in this summation are given by (15), (16) and (18). Let us vary
one of the kets,being arbitrarily chosen between1and:
+ (27)
or, in terms of an individual wave function:
(r) (r) +(r) (28)
This will yield the following variations:
0=
}
2
2
d
3
[(r) (r) +(r) (r)] (29)
1682

FERMION SYSTEM, HARTREE-FOCK APPROXIMATION
and:
ext=d
3
1(r) [(r)(r) +(r)(r)] (30)
As for the variation ofint, we must take from (18) two contributions: the rst one
from the terms=, and the other from the terms=. These contributions are
actually equal as they only dier by the choice of a dummy subscript. The factor12
disappears and we get:
int=d
3
d
3
2(rr)
=1
(r)(r) +(r)(r)(r)
2
(r)(r)(r)(r) (r)(r)(r)(r) (31)
The variation ofis simply the sum of (29), (30) and (31).
We now consider variations, which can be written as:
(r) = (r)with (32)
(whereis a rst order innitely small parameter). These variations are proportional
to the wave function of one of the non-occupied states, which was added to the occupied
states to form a complete orthonormal basis; the phaseis an arbitrary parameter. Such
a variation does not change, to rst order, either the norm of, or its scalar product
with all the occupied states; it therefore leaves unchanged our assumption that
the occupied states basis is orthonormal. The rst order variation of the energyis
obtained by insertingand its complex conjugateinto (29), (30) and (31); we then
get terms inin the rst case, and terms inin the second. Forto be stationary,
its variation must be zero to rst order for any value of; now the sum of a term in
and another inwill be zero for any value ofonly if both terms are zero. It follows
that we can imposeto be zero (stationary condition) considering the variations of
and to be independent. Keeping only the terms in, we obtain the stationary
condition of the variational energy:
d
3
(r)
}
2
2
(r) +1(r)(r) +
+d
3
2(rr)
=1
(r)(r)
2
(r)(r)(r)= 0
(33)
or, taking (20) into account:
d
3
(r)
}
2
2
(r) +1(r)(r)+
+d
3
2(rr) [r r(r)r r(r)]= 0
(34)
This relation can also be written as::
d
3
(r)[(r)] = 0 (35)
1683

COMPLEMENT E XV
where the integro-dierential operatoris dened by its action on an arbitrary function
(r):
[(r)] =
}
2
2
+1(r) +d
3
2(rr)r r (r)
d
3
2(rr)r r(r) (36)
This operator depends on the diagonalr rand non-diagonalr r
spatial correlation functions associated with the set of states occupied by thefermions.
Relation (35) thus shows that the action of the dierential operatoron the
function(r)yields a function orthogonal to all the functions(r)for . This
means that the function[(r)]only has components on the wave functions of the
occupied states: it is a linear combination of these functions. Consequently, for the
energyto be stationary there is a simple condition: the invariance under the action of
the integro-dierential operatorof the-dimensional vector spaceF, spanned by
all the linear combinations of the functions(r)with= 12.
Comment:
One could wonder why we limited ourselves to the variationswritten in (32), propor-
tional to non-occupied individual states. The reason will become clearer in , where
we use a more general method that shows directly which variations of each individual
states are really useful to consider (see in particular the discussion at the end of ).
For now, it can be noted that choosing a variationproportional to the same wave
function(r)would simply change its norm or phase, and therefore have no impact on
the associated quantum state (in addition, a change of norm would not be compatible
with our hypotheses, as in the computation of the average values we always assumed
the individual states to remain normalized). If the state does not change, the energy
must remain constant and writing a stationary condition is pointless. Similarly, to
give(r)a variation proportional to another occupied wave function(r)(whereis
included between1and) is just as useless, as we now show. In this operation, the
creation operatoracquires a component on(Chapter ), but the state
vector expression (1) remains unchanged. The state vector thus acquire a component
including the square of a creation operator, which is zero for fermions. Consequently, the
stationarity of the energy is automatically ensured in this case.
1-d. Equivalent formulation for the average energy stationarity
Operatorcan be diagonalized in the subspaceF, as can be shown
3
from its
denition (36) a more direct demonstration will be given in . We call(r)its
3
As any Hermitian operator can be diagonalized, we simply show that (36) leads to matrix elements
obeying the Hermitian conjugation relation. Let us verify that the two integrals
3
1
(r)[2(r)]
and
3
2
(r)[1(r)]are complex conjugates of each other. For the contributions to these matrix
elements of the kinetic and potential (in1) energy, we simply nd the usual relations insuring the
corresponding operators are Hermitian. As for the interaction term, the complex conjugation is obvious
for the direct term; for the exchange term, a simple inversion of the integral variables
3
and
3
, plus
the fact that2(rr)is equal to2(rr)allows verifying the conjugation.
1684

FERMION SYSTEM, HARTREE-FOCK APPROXIMATION
eigenfunctions. These functions(r)are linear combinations of the(r)corresponding
to the states appearing in the trial ket (1), and therefore lead to the same-particle
state, because of the antisymmetrization
4
. The basis change from the(r)to the(r)
has no eect on the projectoronto to the subspace, whose matrix elements
appearing in (36) can be expressed in a way similar to those in (20):
r r=
=1
(r)(r) (37)
Consequently, the eigenfunctions of the operatorobey the equations:
}
2
2
+1(r) +d
3
2(rr)
=1
(r)
2
(r)
d
3
2(rr)
=1
(r)(r)(r)= (r)
(38)
whereare the associated eigenvalues. These relations are called the Hartree-Fock
equations.
For the average total energy associated with a state such as (1) to be stationary, it
is therefore necessary for this state to be built fromindividual states whose orthogonal
wave functions1,2, .. ,are solutions of the Hartree-Fock equations (38) with= 1,
2, .. ,. Conversely, this condition is sucient since, replacing the(r)by solutions
(r)of the Hartree-Fock equations in the energy variation (34) yields the result:
d
3
(r)(r) (39)
which is zero for all(r)variations, since, according to (32), they must be orthogonal
to thesolutions(r). Conditions (38) are thus equivalent to energy stationarity.
1-e. Variational energy
Assume we found a series of solutions for the Hartree-Fock equations, i.e. a set
ofeigenfunctions(r)with the associated eigenvalues. We still have to compute
the minimal variational energy of the-particle system. This energy is given by the
sum (26) of the three terms of kinetic, potential and interaction energies obtained by
replacing in (15), (16) and (18) the(r)by the eigenfunctions(r):
= 0 + ext + int (40)
4
A determinant value does not change if one adds to one of its column a linear combination of the
others. Hence we can add to the rst column of the Slater determinant (2) the linear combination of
the2(r),3(r), ... that makes it proportional to1(r). One can then add to the second column the
combination that makes it proportional to2(r), etc. Step by step, we end up with a new expression for
the original wave function (r1r2r), which now involves the Slater determinant of the(r). It
is thus proportional to this determinant. A demonstration of the strict equality (within a phase factor)
will be given in .
1685

COMPLEMENT E XV
(the subscriptsindicate we are dealing with the average energies after the Hartree-
Fock optimization, which minimizes the variational energy). Intuitively, one could expect
this total energy to be simply the sum of the energies, but, as we are going to show,
this is not the case. Multiplying the left-hand side of equation (38) by(r)and after
integration over d
3
, we get:
=d
3
(r)
}
2
2
+1(r)(r)
+d
3
2(rr)
=1
(r)
2
(r) (r)(r)(r) (41)
We then take a summation over the subscript, and use (15), (16) and (18), thebeing
replaced by the:
=1
= 0 + ext + 2 int (42)
This expression does not yield the stationary value of the total energy, but rather a sum
where the particle interaction energy is counted twice. From a physical point of view, it
is clear that if each particle energy is computed taking into account its interaction with
all the others, and if we then add all these energies, we get an expression that includes
twice the interaction energy associated with each pair of particles.
The sum of thedoes contain, however, useful information that enables us to
avoid computing the interaction energy contribution to the variational energy. Eliminat-
ing intbetween (40) and (42), we get:
=
1
2
=1
+ 0 + ext (43)
where the interaction energy is no longer present. One can then compute0 and
ext using the solutions of the Hartree-Fock equations (38), without worrying about
the interaction energy. Using (15) and (17) in this relation, we can write the total energy
as:
=
1
2
=1
+
=1
d
3
(r)
2
1(r)
}
2
2
=1
d
3
(r) (r)
(44)
The total energy is thus half the sum of the, of the average kinetic energy, and nally
of the one-body average potential energy.
1-f. Hartree-Fock equations
Equation (38) may be written as:
}
2
2
+1(r) +
dir
(r)(r) d
3 ex
(rr)(r) = (r)
(45)
1686

FERMION SYSTEM, HARTREE-FOCK APPROXIMATION
where the direct
dir
(r)and exchange
ex
(rr)potentials are dened as:
dir
(r) =
=1
d
3
(r)
2
2(rr)
ex
(rr) =
=1
(r)(r)2(rr)
(46)
Note that the terms=coming from the two potentials cancel each other; hence
they can be eliminated from the two summations, without changing the nal result.
The contribution of the direct potential is sometimes called the Hartree term, and the
contribution of the exchange potential, the Fock term. The rst is easy to understand:
with the exception of the term=, it corresponds to the interaction of a particle
at pointrwith all the others at pointsr, averaged for each of them by its density
distribution(r)
2
. As for the exchange potential, and in spite of its name, this term
is not, strictly speaking, a potential; it is not diagonal in the position representation,
even though it basically comes from a particle interaction which is diagonal in that
representation. This peculiar non-diagonal form actually comes from the combination
of the fermion antisymmetrization and the variational approximation. This exchange
potential is homogeneous to a potential divided by the cube of a length. It is obviously a
Hermitian operator as it is derived from a potential2(rr)which is real and symmetric
with respect torandr.
A more intuitive and simplied version of these equations was suggested by Hartree,
in which the exchange potentials are ignored in (45). Without the integral term, these
equations become very similar to a series of Schrödinger equations for independent par-
ticles, each of them moving in the mean potential created by all the others (still with
the exception of the term=in the summation). Including the Fock term should,
however, lead to more precise calculations.
Using for the potentials their expressions (46), the Hartree-Fock equations (45)
become a set ofcoupled equations. They are nonlinear, since the direct and exchange
potentials depend on the functions(r). Even though they look like linear eigenvalue
equations with eigenfunctions(r)as solutions, a linear resolution would actually re-
quire knowing in advance the solutions, since these functions also appear in the potentials
(46). The term self-consistent is used to characterize this type of situation and the
solutions(r)it leads to.
There are no general analytical methods to solve nonlinear self-consistent equations
of this type, even in their simplied Hartree version, and numerical methods using suc-
cessive approximations are commonly used. We start from a series of plausible functions
(0)
(r), and compute with (46) the associated potentials. Considering them to be xed,
we obtain linear eigenvalue equations which can be solved quite readily with computers
(the single very complicated equation in a3-dimensional space has been replaced by
independent3-dimensional equations); we have to diagonalize a Hermitian operator
to get a new series of orthonormal functions, resulting from the rst iteration, and called
(1)
(r)and
(1)
. The second iteration starts from these
(1)
(r), to compute the new
potential values, and get new linear dierential equations. Solving these equations yields
the next order
(2)
(r),
(2)
, etc. After a few iterations, one expects the
()
(r)and
()
to vary only slightly with the iteration order(), in which case the Hartree-Fock
1687

COMPLEMENT E XV
equations have been solved to a good approximation. Using (44) we can then compute
the energy we were looking for. It is also possible that physical arguments can help us
choose directly adequate trial functions(r)without any iteration. Inserting them in
(44) then directly provides the energy.
Comments:
(i) The solutions of the Hartree-Fock equations may not be unique. Using the iteration
process described above, one can easily wind up with dierent solutions, depending on
the initial choice for the
(0)
(r)functions. This multiplicity of solutions is actually one
of the method's advantages, as it can help us nd not only the ground level but also the
excited levels.
(ii) As we shall see in , taking into account the12spin of the electrons in an atom does
not bring major complications to the Hartree-Fock equations. It is generally assumed
that the one-body potential is diagonal in a basis of the two spin states, labeled+and
, and that the interaction potential does not act on the spins. We then simply assemble
+equations, for+wave functions
+
(r)associated with the spin+particles, with
other equations, forwave functions(r)associated with spinparticles. These
two sets of equations are not independent, since they contain the same direct potential
(computed using (46), whose rst line includes a summation overof all the= ++
wave functions). As for the exchange potential, it does not lead to any coupling between
the two sets of equations: in the second line of (46), the summation overonly includes
particles in the same spin state for the following reason. If the particles have opposite
spins, they can be recognized by the direction of their spin (the interaction does not act
on the spins), and they no longer behave as indistinguishable particles. The exchange
eects only arise for particles having the same spin.
2. Generalization: operator method
We now describe the method in a more general way, using an operator method that leads
to more concise expressions, while taking into account explicitly the possible existence
of a spin which plays an essential role in the atomic structure. We will identify more
precisely the mathematical object, actually a projector, which we vary to optimize the
energy. Physically, this projector is simply the one-particle density operator dened in
. This will lead to expressions both more compact and general
for the Hartree-Fock equations. They contain a Hartree-Fock operator acting on a
single particle, as if it were alone, but which includes a potential operator dened by
a partial trace which reects the interactions with the other particles in the mean eld
approximation. Thanks to this operator we can get an approximate value of the entire
system energy, computing only individual energies; these energies are obtained with
calculations similar to the one used for a single particle placed in a mean eld. With
this approach, we have a better understanding of the way the mean eld approximately
represents the interaction with all the other particles; this approach can also suggest
ways to make the approximations more precise.
We assume as before that the-particle variational ket is written as:
=
1 2
0 (47)
1688

FERMION SYSTEM, HARTREE-FOCK APPROXIMATION
This ket is derived fromindividual orthonormal kets, but these kets can now
describe particles having an arbitrary spin. Consider the orthonormal basisof
the one-particle state space, in which the set of(= 1,2, ...) was completed by
other orthonormal states. The projectoronto the subspaceis the sum of the
projections onto the rstkets:
=
=1
(48)
This is simply the one-particle density operator dened in
(normalized by a trace equal to the particle numberand not to one), as we now show.
Relation (B-24) of that chapter can be written in thebasis:
1 = (49)
where the average value is taken in the quantum state (47). In this kind of Fock
state, the average value is dierent from zero only when the creation operator reconstructs
the population destroyed by the annihilation operator, hence if=, in which case it is
equal to the populationof the individual states. In the variational ket (47), all
the populations are zero except for the rststates(= 1,2, ...), where they are
equal to one. Consequently, the one-particle density operator is represented by a matrix,
diagonal in the basis, and whoserst elements on the diagonal are all equal to
one. It is indeed the matrix associated with the projector, and we can write:
1= (50)
As we shall see, all the average values useful in our calculation can be simply expressed
as a function of this operator.
2-a. Average energy
We now evaluate the dierent terms included in the average energy, starting with
the terms containing one-particle operators.
. Kinetic and external potential energy
Using relation (B-12) of Chapter, we obtain for the average kinetic energy
0:
0=
P
2
2
(51)
The same argument as that for the evaluation of the matrix elements (49) shows that
the average value in the state (47) is only dierent from zero if=; in that
case, it is equal to one when, and to zero otherwise. This leads to:
0=
=1
P
2
2
=Tr1
P
2
2
(52)
1689

COMPLEMENT E XV
The subscript1was added to the trace to underline the fact that this trace is taken in
the one-particle state space and not in the Fock space. The two operators included in
the trace only act on that same particle, numbered arbitrarily1; the subscript1could
obviously be replaced by the subscript of any other particle, since they all play the same
role. The average potential energy coming from the external potential is computed in a
similar way and can be written as:
ext=
=1
1 =Tr1 1 (53)
. Average interaction energy, Hartree-Fock potential operator
The average interaction energyintcan be computed using the general expression
(C-16) of Chapter
int=
1
2
1 :; 2 : 2(12)1 :; 2 : (54)
For the average value in the Fock state to be dierent from zero, the
operator must leave unchanged the populations of the individual statesand. As
in , two possibilities may occur: either=and=(the direct
term), or=and=(the exchange term). Commuting some of the operators, we
can write:
= +
= [ ] (55)
whereandare the respective populations of the statesand . Now these
populations are dierent from zero only if the subscriptsandare between1and,
in which case they are equal to1(note also that we must have=to avoid a zero
result). We nally get
5
:
int=
1
2
=
1 :; 2 : 2(12)1 :; 2 :
1 :; 2 : 2(12)1 :; 2 : (56)
(the constraint=may be ignored since the right-hand side is equal to zero in this
case). Here again, the subscripts1and2label two arbitrary, but dierent particles, that
could have been labeled arbitrarily. We can therefore write:
int=
1
2
=1
1 :; 2 : 2(12) [1ex(12)]1 :; 2 : (57)
whereex(12)is the exchange operator between particle1and2(the transposition which
permutes them). This result can be written in a way similar to (53) by introducing a
5
As in the previous complement, we have replaced2(R1R2)by2(12)to simplify the notation
1690

FERMION SYSTEM, HARTREE-FOCK APPROXIMATION
Hartree-Fock potential, similar to an external potential acting in the space of
particle1; this potential is dened as the operator having the matrix elements:
(1)=
=1
1 :; 2 : 2(12) [1ex(12)]1 :; 2 : (58)
This operator is Hermitian, since, as the two operatorsexand 2are Hermitian and
commute, we can write:
(1)=
=1
1 :; 2 : 2(12) [1ex(12)]1 :; 2 :
=
=1
1 :; 2 :[1 ex(12)]2(12)1 :; 2 :
= (1) (59)
Furthermore, we recognize in (58) the matrix element of a partial trace on particle2
(Complement III, 5-b):
(1) =Tr2 (2)2(12) [1ex(12)] (60)
where the projectorhas been introduced inside the trace to limit the sum overto
its rstterms, as in (57). The one-particle operator(1)is thus the partial trace
over a second particle (with the arbitrary label2) of a product of operators acting on
both particles. As the summation overis now taken into account, we are left in (57)
with a summation over, which introduces a trace over the remaining particle1, and we
get:
int=
1
2
Tr1 (1) (1) (61)
This average value depends on the subspace chosen with the variational ket in two
ways: explicitly as above, via the projector(1)that shows up in the average value
(61), but also implicitly via the denition of the Hartree-Fock potential in (60).
. Role of the one-particle reduced density operator
All the average values can be expressed in terms of the projectoronto the
subspaceof the space the individual states spanned by theindividual states1,
2, ...., which means, according to (50), in terms of the one-particle reduced density
operator1= . Hence it is this operator that is the pertinent variable to optimize
rather than the set of individual states: certain variations of those states do not change
, and are meaningless for our purpose.
Furthermore, the choice of the trial ket is equivalent to that of. In other
words, the variational ket built in (1) does not depend on the basis chosen in the
subspace: if we choose in this subspace any orthonormal basisother than the
basis, and if we replace in (1) theby the, the ket will remain the same
1691

COMPLEMENT E XV
(to within a non-relevant phase factor) as we now show. As seen in
XV, each operatoris a linear combination of the, so that in the product of all
the (= 1,2, ..) we will nd products ofoperators. Relation (A-43) of
Chapter
means that the only non-zero products are those including once and only once each of
thedierent operators. Each term is then proportional to the ket built from
the. Consequently, the two variational kets built from the two bases are necessarily
proportional. As denition (1) ensures they are also normalized, they can only dier by
a phase factor, which means they are equivalent from a physical point of view. It is thus
the operator=1that best embodies the trial ket .
2-b. Optimization of the one-particle density operator
We now vary =1to look for the stationary conditions for the total energy:
= 0+ 1+ int=Tr1
P
2
2
+1+
1
2
(62)
We therefore consider the variation:
+ (63)
which leads to the following variations for the average values of the one-particle operators:
0+ 1=Tr1
P
2
2
+1 (64)
As for the interaction energy, we get two terms:
int=
1
2
Tr1 +
1
2
Tr1 (65)
which are actually equal since:
Tr1 (1) (1)=Tr12 (1)(2)2(12) [1ex(12)] (66)
and we recognize in the right-hand side of this expression the trace:
Tr2 (2) (2) (67)
As we can change the label of the particle from2to1without changing the trace, the
two terms of the interaction energy are equal. As a result, we end up with the energy
variation:
=Tr1
P
2
2
+1+ (68)
To vary the projector, we choose a value0ofand make the change:
0 0+ (69)
1692

FERMION SYSTEM, HARTREE-FOCK APPROXIMATION
where is any ket from the space of individual states, andany real number; no
other individual state vector varies except for
0
. The variation ofis then written
as:
=
0
+
0
(70)
We assume has no components on any, that is no components in, since this
would change neither, nor the corresponding projector. We therefore impose:
= 0 (71)
which also implies that the norm of
0
remains constant
6
to rst order in. Inserting
(70) into (68), we obtain:
=Tr1
P
2
2
+1+
0
+ Tr1
P
2
2
+1+
0
(72)
For the energy to be stationary, this variation must remain zero whatever the choice of
the arbitrary number. Now the linear combination of two exponentialsand
will remain zero for any value ofonly if the two factors in front of the exponentials are
zero themselves. As each term can be made equal to zero separately, we obtain:
0 =Tr1
P
2
2
+1+
0
=
P
2
2
+1+
0
(73)
This relation must be satised for any ketorthogonal to the subspace.
This means that if we dene the one-particle Hartree-Fock operator as:
=
P
2
2
+1+
(74)
the stationary condition for the total energy is simply that the ket
0
must belong
to:
0
(75)
As this relation must hold for any
0chosen among the1,2, ...., it follows
that the subspaceis stable under the action of the operator (74).
2-c. Mean eld operator
We can then restrict the operatorto that subspace:

=(1)
P
2
2
+1(1) + (1)(1) (76)
6
Since (71) shows that is orthogonal to any linear combinations of the, we can write
(
0
+ ) (
0
+ ) =
0 0
+ = 1+second order terms.
1693

COMPLEMENT E XV
This operator, acting in the subspacespanned by thekets, is a Hermitian
linear operator, hence it can be diagonalized. We callits eigenvectors (= 1,2,
..), which are linear combinations of the kets. The stationary condition for the
energy (75) amounts to imposing theto be not only eigenvectors of

, but also
of the operator dened by (74) in the entire one-particle state space (without the
restriction to); consequently, themust obey:
= (77)
Operator is dened in (74), where the operatoris given by (60) and depends
on the projector. This last operator may be expressed as a function of thein
the same way as with the, and relation (48) may be replaced by:
=
=1
(78)
Relations (77), together with denition (60) where (78) has been inserted, are a
set of equations allowing the self-consistent determination of the; they are called
the Hartree-Fock equations. This operator form (77) is simpler than the one obtained in
; it emphasizes the similarity with the usual eigenvalue equation for a single particle
moving in an external potential, illustrating the concept of a self-consistent mean eld.
One must keep in mind, however, that via the projector (78) included in , this
particle moves in a potential depending on the whole set of states occupied by all the
particles. Remember also that we did not carry out an exact computation, but merely
presented an approximate theory (variational method).
The discussion in depends on the,
the Hartree-Fock equations have an intrinsic nonlinear character, which generally requires
a resolution by successive approximations. We start from a set ofindividual states
0
to build a rst value ofand the operator, which are used to compute the
Hamiltonian (74). Considering this Hamiltonian now xed, the Hartree-Fock equations
(77) become linear, and can be solved as usual eigenvalue equations. This leads to new
values
1
for the, and nishes the rst iteration. In the second iteration, we use
the
1
in (78) to compute a new value of the mean eld operator; considering
again this operator as xed, we solve the eigenvalue equation and obtain the second
iteration values
2
for the, and so on. If the initial values
0
are physically
reasonable, one can hope for a rapid convergence towards the expected solution of the
nonlinear Hartree-Fock equations.
The variational energy can be computed in the same way as in . Multiplying
on the left equation (77) by the bra, we get:
=
P
2
2
+1+ (79)
After summing over the subscript, we obtain:
=1
=
=1
P
2
2
+1+ =Tr1 (1)
P
2
2
+1+ (80)
1694

FERMION SYSTEM, HARTREE-FOCK APPROXIMATION
Taking into account (51), (53), and (61), we get:
=1
= 0 + 1 + 2 int (81)
where the particle interaction energy is counted twice. To compute the energy, we can
eliminateintbetween (26) and this relation and we nally obtain:
=
1
2
=1
+ 0 + 1 (82)
2-d. Hartree-Fock equations for electrons
Assume the fermions we are studying are particles with spin12, electrons for
example. The basisrof the individual states used in
basis formed with the ketsr, whereis the spin index, which can take2distinct
values noted12, or more simply. To the summation over d
3
we must now add a
summation over the2values of the index spin. A vectorin the individual state
space is now written:
=
=12
d
3
(r)r (83)
with:
(r) =r (84)
The variablesrandplay a similar role but the rst one is continuous whereas the
second is discrete. Writing them in the same parenthesis might hide this dierence, and
we often prefer noting the discrete index as a superscript of the function, and write:
(r) =r (85)
Let us build anparticle variational state fromorthonormal states,
with= 1,2, .. ,. Each of the describes an individual state including the spin
and position variables; the rst+values of(= 12 +) are equal to+12, the
lastare equal to12, with++ =(we assume+and are xed for the
moment but we may allow them to vary later to enlarge the variational family). In the
space of the individual states, we introduce a complete basiswhose rstkets
are the , but where the subscriptvaries from1to innity
7
.
We assume the matrix elements of the external potential1to be diagonal for;
these two diagonal matrix elements can however take dierent values
1
(r), which allows
including the eventual presence of a magnetic eld coupled with the spins. We also assume
the particle interaction2(12)to be independent of the spins, and diagonal in the
position representation of the two particles, as is the case, for example, for the Coulomb
7
The subscriptdetermines both the orbital and the spin state of the particle; the indexis not
independent since it is xed for each value of.
1695

COMPLEMENT E XV
interaction between electrons. With these assumptions, the Hamiltonian cannot couple
states having dierent particle numbers+and.
Let us see what the general Hartree-Fock equations become in ther repre-
sentation. In this representation, the eect of the kinetic and potential operators are
well known. We just have to compute the eect of the Hartree-Fock potential. To
obtain its matrix elements, we use the basis1 :r; 2 : to write the trace in (60):
r (1)r =
=1
1 :r; 2 :
=1
2 : 2 :
2(12) [1ex(12)]1 :r; 2 : (86)
As the right-hand side includes the scalar product2 :2 : which is equal to,
the sum overdisappears and we get:
r (1)r
=
=1
1 :r; 2 : 2(12) [1ex(12)]1 :r; 2 : (87)
(i) We rst deal with the direct term contribution, hence ignoring in the bracket
the term inex(12). We can replace the ket2 : by its expression:
2 : =d
3
2(r2)2 :r2 (88)
As the operator is diagonal in the position representation, we can write:
2(12)1 :r; 2 := 2(12)d
3
2(r2)1 :r; 2 :r2
=d
3
2(r2)2(rr2)1 :r; 2 :r2
(89)
The direct term of (87) is then written:
d
3
2
=1
2(rr2)(r2)1 :r; 2 :1 :r; 2 :r2 (90)
where the scalar product of the bra and the ket is equal to(rr)(r2). We
nally obtain:
(rr)d
3
22(rr2)
=1
(r2)
2
= (rr)dir(r) (91)
with:
dir(r) =
=1
d
3
2(rr) (r)
2
(92)
1696

FERMION SYSTEM, HARTREE-FOCK APPROXIMATION
This component of the mean eld (Hartree term) contains a sum over all occupied states,
whatever their spin is; it is spin independent.
(ii) We now turn to the exchange term, which contains the operatorex(12)in the
bracket of (87). To deal with it, we can for example commute in (87) the two operators
2(12)andex(12); this last operator will then permute the two particles in the bra.
Performing this operation in (90), we get, with the minus sign of the exchange term:
3
2
=1
2(rr2)(r2)1 :; 2 :r1 :r; 2 :r2 (93)
The scalar product will yield the products of(rr2), making the integral over
3
2disappear; this term is zero if=, hence the factor. Since2(rr) =
2(rr), we are left with:
=
2(rr)(r) (r)= (rr) (94)
where the sum is over the values offor which==(hence, limited to the rst
+values of, or the last, depending on the case); the exchange potential
exhas
been dened as:
ex(rr) =
=
2(rr) (r) (r) (95)
As is the case for the direct term, the exchange term does not act on the spin. There are
however two dierences. To begin with, the summation overis limited to the states
having the same spin; second, it introduces a contribution which is non-diagonal in
the positions (but without an integral), and which cannot be reduced to an ordinary
potential (the term non-local potential is sometimes used to emphasize this property).
We have shown that the scalar product of equation (77) withrintroduces three
potentials (in addition to the the one-body potential
1
), a direct potentialdir(r)and
two exchange potentials
ex(rr)with=12. Equation (77) then becomes, in the
r representation, a pair of equations:
}
2
2
+
1
(r) +dir.(r) (r) d
3
ex(rr)(r) = (r)
(96)
These are the Hartree-Fock equations with spin and in the position representation, widely
used in quantum physics and chemistry. It is not necessary to worry, in these equations,
about the term in which the subscriptin the summation appearing in (92) and (95) is
the same as the subscript(of the wave function we are looking for); the contributions
=cancel each other exactly in the direct and exchange potentials.
Both the Hartree term giving the direct potential contribution, and the Fock
term giving the exchange potential, can be interpreted in the same way as above (
1-f). The Hartree term contains the contributions of all the other electrons to the mean
potential felt by one electron. The exchange potential, on the other hand, only involves
electrons in the same spin state, and this can be simply interpreted: the exchange ef-
fect only occurs for two totally indistinguishable particles. Now if these particles are in
1697

COMPLEMENT E XV
orthogonal spin states, and as the interactions do not act on the spins, one can in prin-
ciple determine which is which and the particles become distinguishable: the quantum
exchange eects cancel out. As we already pointed out, the exchange potential is not
a potential stricto sensu. It is not diagonal in the position representation, even though
it basically comes from a particle interaction that is diagonal in position. It is the an-
tisymmetrization of the fermions, together with the chosen variational approximation,
which led to this peculiar non-diagonal form. It is however a Hermitian operator, as can
be shown using the fact that the initial potential2(rr)is real and symmetric with
respect torandr.
2-e. Discussion
The resolution of the nonlinear Hartree-Fock equations is generally done by the
successive iteration approximate method discussed in . There is no particular reason
for the solution of the Hartree-Fock equations to be unique
8
; on the contrary, they can
yield solutions that depend on the states chosen to begin the nonlinear iterations. They
can actually lead to a whole spectrum of possible energies for the system. This is how
the ground state and excited state energies of the atom are generally computed. The
atomic orbitals discussed in ComplementVII, the central eld approximation and the
electronic congurations discussed in ComplementXIVcan now be discussed in a
more precise and quantitative way. We note that the exchange energy, introduced in this
complement for a two-electron system, is a particular case of the exchange energy term
of the Hartree-Fock potential. There exist however many other physical systems where
the same ideas can be applied: nuclei (the Coulomb force is then replaced by the nuclear
interaction force between the nucleons), atomic aggregates (with an interatomic potential
having both repulsive and attractive components, see ComplementsXIandXI), and
many others.
Once a Hartree-Fock solution for a complex problem has been found, we can go
further. One can use the basis of the eigenfunctions just obtained as a starting point
for more precise perturbation calculations, including for example correlations between
particles (Chapter XI). In atomic spectra, we sometimes nd cases where two congu-
rations yield very close mean eld energies. The eects of the interaction terms beyond
the mean eld approximation will then be more important. Perturbation calculations
limited to the space of the congurations in question permits obtaining better approx-
imations for the energy levels and their wave functions; one then speaks of mixtures,
or of interactions between congurations.
Comment:
The variational method based on the Fock states is not the only one that leads to the
Hartree-Fock equations. One could also start from an approximation of the two-particle
density operatorby a function of the one-particle density operatorand write:
(12)
1
2
[1 ex(12)](1) (2) (97)
Expressing the energy of the-particle system as a function of, we minimize it by
varying this operator, and nd the same results as above. This method amounts to a
closure of the hierarchy of the-body equations ( ). We have in
8
They all yield, however, an upper limit for the ground state energy
1698

FERMION SYSTEM, HARTREE-FOCK APPROXIMATION
fact already seen with equation (21) and in
amounts to expressing the two-particle correlation functions as a function of the one-
particle correlation functions. In terms of correlation functions (ComplementXVI),
this amounts to replacing the two-particle function (four-point function) by a product
of one-particle functions (two-point function), including an exchange term. Finally, an-
other method is to use the diagram perturbation theory; the Hartree-Fock approximation
corresponds to retaining only a certain class of diagrams (class of connected diagrams).
Finally note that the Hartree-Fock method is not the only one yielding approximate
solutions of Schrödinger's equation for a system of interacting fermions; in particular,
one can use the electronic density functional theory (a functional is a function of
another function, as for instance the actionin classical lagrangian mechanics). The
method is used to obtain the electronic structure of molecules or condensed phases in
physics, chemistry, and materials science. Its study nevertheless lies outside the scope
of this book, and the reader is referred to [6], which summarizes the method and gives a
number of references.
1699

FERMIONS, TIME-DEPENDENT HARTREE-FOCK APPROXIMATION
Complement FXV
Fermions, time-dependent Hartree-Fock approximation
1 Variational ket and notation
2 Variational method
2-a Denition of a functional
2-b Stationarity
2-c Particular case of a time-independent Hamiltonian
3 Computing the optimizer
3-a Average energy
3-b Hartree-Fock potential
3-c Time derivative
3-d Functional value
4 Equations of motion
4-a Time-dependent Hartree-Fock equations
4-b Particles in a single spin state
4-c Discussion
The Hartree-Fock mean eld method was introduced in ComplementXVfor a
time-independent problem: the search for the stationary states of a system of interacting
fermions (the search for its thermal equilibrium will be discussed in ComplementXV.
In this complement, we show how this method can be used for time-dependent problems.
We start, in , by including a time dependence in the Hartree-Fock variational ket
(time-dependent Fock state). We then introduce in
that can be used for solving the time-dependent Schrödinger equation. We then compute,
in , the function to be optimized for a Fock state; the same mean eld operator as
the one introduced in ComplementXVwill here again play a very useful role. Finally,
the time-dependent Hartree-Fock equations will be obtained and discussed in . More
details on the Hartree-Fock methods in general can be found, for example, in Chapter 7
of reference [5], and especially in its Chapter 9 for time-dependent problems.
1. Variational ket and notation
We assume the-particle state vector
^
()to be of the form:
^
()=
1()2() ()
0 (1)
where the
1()
,
2()
, ...,
()
are the creation operators associated with an arbitrary
series of orthonormal individual states1(),2(), ...,()which depend on time
. This series is, for the moment, arbitrary, but the aim of the following variational
calculation is to determine its time dependence.
1701

COMPLEMENT F XV
As in the previous complements, we assume that the Hamiltonianis the sum
of three terms: a kinetic energy Hamiltonian, an external potential Hamiltonian, and a
particle interaction term:
=0+ext() +int (2)
with:
0=
=1
(P)
2
2
(3)
(is the particles' mass,Pthe momentum operator of particle), and:
ext() =
=1
1(R) (4)
and nally:
int=
1
2
=
2(RR) (5)
2. Variational method
Let us introduce a general variational principle; using the stationarity of a functional
of the state vector (), it will yield the time-dependent Schrödinger equation.
2-a. Denition of a functional
Consider an arbitrarily given Hamiltonian(). We assume the state vector ()
to have any time dependence, and we note
()the ket physically equivalent to (),
but with a constant norm:
()=
()
() ()
(6)
The functionalof
()is dened as
1
: ()=
1
0
dRe ()}
d
d
()
()
=
1
0
d
}
2 ()
d
d ()
d
d () () ()() ()
(7)
where0and1are two arbitrary times such that0 1. In the particular case where
the chosen
()is equal to a solution ()of the Schrödinger equation:
}
d
d ()=() () (8)
1
The notation where the dierential operator ddis written between a bra and a ket means that
the operator takes the derivative of the ket that follows (and not of the bra just before).
1702

FERMIONS, TIME-DEPENDENT HARTREE-FOCK APPROXIMATION
the bracket on the rst line of (7) obviously cancels out and we have:
()= 0 (9)
Integrating by parts the second term
2
of the bracket in the second line of (7), we
get the same form as the rst term in the bracket, plus an already integrated term. The
nal result is then:
()=
1
0
d ()}
d
d
()
()
+
}
2 (0) (0) (1) (1)
=
1
0
d
()}
d
d
()
() (10)
where we have used in the second line the fact that the norm of
()always remains
equal to unity. This expression foris similar to the initial form (7), but without the
real part.
2-b. Stationarity
Suppose now
()has an arbitrary time dependence between0and1, while
keeping its norm constant, as imposed by (6); the functional then takes a certain value
, a priori dierent from zero. Let us see under which conditionswill be stationary
when
()changes by an innitely small amount (): () ()+ () (11)
For what follows, it will be convenient to assume that the variation
()is free; we
therefore have to ensure that the norm of
()remains constant, equal to unity
3
. We
introduce Lagrange multipliers (Appendix)()to control the square of the norm at
every time between0and1, and we look for the stationarity of a function where the
sum of constraints has been added. This sum introduces an integral, and we the function
in question is:
()= ()
1
0
d() () ()
=
1
0
d
()}
d
d
()()
() (12)
where()is a real function of the time.
2
If we integrate by parts the rst term rather than the second, we get the complex conjugate of
equation (10), which brings no new information.
3
For the normalization of
()to be conserved to rst order, it is necessary (and sucient) for
the scalar product
() ()to be zero or purely imaginary. If this is the case, the Lagrangian
multiplier()is not needed
1703

COMPLEMENT F XV
The variation
ofto rst order is obtained by inserting (11) in (10). It yields
the sum of a rst term
1containing the ket ()and of another2containing the
bra
():1=
1
0
d ()}
d
d
()()
()2=
1
0
d ()}
d
d
()()
() (13)
We now imagine another variation for the ket:
() ()+ () (14)
which yields a variation
of; in this second variation, the term in ()becomes1=1, whereas the term in ()becomes2=2. Now, if the functional
is stationary in the vicinity of
(), the two variationsandare necessarily zero,
as are also
and+. In those combinations, only terms in1appear
for the rst one, and in
2for the second; consequently they must both be zero. As a
result, we can write the stationarity conditions with respect to variations of the bra and
the ket separately.
Let us write for example that
2= 0, which means the right-hand side of the
second line in (13) must be zero. As the time evolution between0and1of the bra
()is arbitrary, this condition imposes this bra multiplies a zero-value ket, at all
times. Consequently, the ket
()must obey the equation:
}
d
d
()()
()= 0 (15)
which is none other than the Schrödinger equation associated with the Hamiltonian
() +().
Actually,()simply introduces a change in the origin of the energies and this
only modies the total phase
4
of the state vector
(), which has no physical eect.
Without loss of generality, this Lagrange factor may therefore be ignored, and we can
set:
() = 0 (16)
A necessary condition
5
for the stationarity ofis that
()obey the Schrödinger
equation (8) or be physically equivalent (i.e. equal to within a global time-dependent
phase factor) to a solution of this equation. Conversely, assume
()is a solution of
the Schrödinger equation, and give this ket a variation as in (11). It is then obvious
from the second line of (13) that
2is zero. As for1, an integration by parts over
time shows that it is the complex conjugate of
2, and therefore also equal to zero. The4
If in (15) we set
()=
()
(), we see that()obeys the dierential equation obtained
by replacing()by()}
d
d
in (15). If we simply choose for()the integral over time of the function
(), this constant will disappear from the dierential equation.
5
The same argument as above, but starting from the variation , would lead to the complex
conjugate of (8), and hence to the same equation.
1704

FERMIONS, TIME-DEPENDENT HARTREE-FOCK APPROXIMATION
functionalis thus stationary in the vicinity of any exact solution of the Schrödinger
equation.
Suppose we choose any variational familyof normalized kets
^
(), but which
now includes a ket
^

stat
()for whichis stationary. A simple example is the case where
is a family0that contains the exact solution of the Schrödinger equation; according
to what we just saw, this exact solution will makestationary, and conversely, the ket
that makesstationary is necessarily
^

stat
0
(). In this case, imposing the variation of
to be zero allows identifying, inside the family0, the exact solution we are looking for. If
we now change the family continuously from0to, in generalwill no longer contain
the exact solution of the Schrödinger equation. We can however follow the modications
at all times of the values of the ket
^

stat
(). Starting from an exact solution of the
equation, this ket progressively changes, but, by continuity, will stay in the vicinity of
this exact solution ifstays close to0. This is why annulling the variation ofin
the familyis a way of identifying a member of that family whose evolution remains
close to that of a solution of the Schrödinger equation. This is the method we will follow,
using the Fock states as a particular variational family.
2-c. Particular case of a time-independent Hamiltonian
If the Hamiltonianis time-independent, one can look for time-independent kets
to make the functionalstationary. The function to be integrated in the denition
of the functionalalso becomes time-independent, and we can writeas:
= (1 0)
(17)
Since the two times0and1are xed, the stationarity ofis equivalent to that of the
diagonal matrix element of the Hamiltonian
. We nd again the stationarity
condition of the time-independent variational method (ComplementXI), which appears
as a particular case of the more general method of the time-dependent variations. Conse-
quently, it is not surprising that the Hartree-Fock methods, time-dependent or not, lead
to the same Hartree-Fock potential, as we now show.
3. Computing the optimizer
The family of the state vectors we consider is the set of Fock kets
^
()dened in (1).
We rst compute the function to be integrated in the functional (10) when ()takes
the value
^
().
3-a. Average energy
For the term in(), the calculation is identical to the one we already did in
XV. We rst add to the series of orthonormal states()with
= 1,2, ...,other orthonormal states()with=+ 1,+ 2, ..., to obtain
a complete orthonormal basis in the space of individual states. Using this basis, we
can express the one-particle and two-particle operators according to relations (B-12) and
(C-16) of Chapter. This presents no diculty since the average values of creation
and annihilation operator products are easily obtained in a Fock state (they only dier
1705

COMPLEMENT F XV
from zero if the product of operators leaves the populations of the individual states
unchanged). Relations (52), (53) and (57) of ComplementXVare still valid when the
become time-dependent. We thus get for the average kinetic energy:
0=
=1
()
P
2
2
() (18)
for the external potential energy:
ext()=
=1
()1(R)() (19)
and for the interaction energy:
int=
1
2
1 :(); 2 :()2(12) [1ex(12)]1 :(); 2 :() (20)
3-b. Hartree-Fock potential
We recognize in (20) the diagonal element (=) of the Hartree-Fock potential
operator (1)whose matrix elements have been dened in a general way by relation
(58) of ComplementXV:
() (1)()
= 1 :(); 2 :()2(12) [1ex(12)]1 :(); 2 :() (21)
We also noted in that complementXVthat (1)is a Hermitian operator.
It is often handy to express the Hartree-Fock potential using a partial trace:
(1) =Tr2 (2)2(12) [1ex(12)] (22)
where is the projector onto the subspace spanned by thekets():
(2) =
=1
2 :()2 :() (23)
As we have seen before, this projector is actually nothing bu the one-particle reduced
density operator1normalized by imposing its trace to be equal to the total particle
number:
(2) =1(2) (24)
The average value of the interaction energy can then be written as:
int=
1
2
Tr11(1) (1)
(25)
1706

FERMIONS, TIME-DEPENDENT HARTREE-FOCK APPROXIMATION
3-c. Time derivative
As for the time derivative term, the function it contains can be written as:
=1
0 () ()
1()
1
()
d
d
() ()0 (26)
In this summation, all terms involving the individual statesother than the state
(which is undergoing the derivation) lead to an expression of the type:
0()()0 (27)
which equals1since this expression is the square of the norm of the state()0, which
is simply the Fock state= 1. As for the state, it leads to a factor written in the
form of a scalar product in the one-particle state space:
0()
d
d
()0=()
d
d
() (28)
3-d. Functional value
Regrouping all these results, we can write the value of the functionalin the form:
^
()=
=1
1
0
d }()
d
d
() ()
P
2
2
+1()()
1
2
=1
1 :(); 2 :()2(12) [1ex(12)]1 :(); 2 :()
(29)
4. Equations of motion
We now vary the ket()according to:
() ()+ ()with (30)
As in complementXV, we will only consider variations()that lead to an actual
variation of the ket
^
(); those where()is proportional to one of the occupied
states()with yield no change for
^
()(or at the most to a phase change) and
are thus irrelevant for the value of. As we did in relations (32) or (69) of Complement
EXV, we assume that:
()=() () with (31)
where()is an innitesimal time-dependent function.
The computation is then almost identical to that of XV.
When ()varies according to (31), all the other occupied states remaining constant,
the only changes in the rst line of (29) come from the terms=. In the second line,
the changes come from either the=terms, or the=terms. As the2(12)
1707

COMPLEMENT F XV
operator is symmetric with respect to the two particles, these variations are the same
and their sum cancels the12factor. All these variations involve terms containing either
the ket (), or the bra() . Now their sum must be zero for any value
of, and this is only possible if each of the terms is zero. Inserting the variation (31) of
(), and canceling the term inleads to the following equality:
1
0
d()}()
d
d
() ()
P
2
2
+1()()
=1
1 :(); 2 :()2(12) [1ex(12)]1 :(); 2 :()= 0
(32)
As we recognize in the function to be integrated the Hartree-Fock potential operator
(1)dened in (21), we can write:
1
0
d()()}
d
d
P
2
2
1() ()()= 0 (33)
with .
4-a. Time-dependent Hartree-Fock equations
As the choice of the function()is arbitrary, for expression (33) to be zero
for any()requires the function inside the curly brackets to be zero at all times.
Stationarity therefore requires the ket:
}
d
d
P
2
2
1() ()() (34)
to have no components on any of the non-occupied states()with ( ). In other
words, stationarity will be obtained if, for all values ofbetween1and, we have:
}
d
d
()=
P
2
2
+1() + ()()+() (35)
where()is any linear combination of the occupied states()( ). As we
pointed out at the beginning of , adding to one of the()a component on the
already occupied individual states has no eect on the-particle state (aside from an
eventual change of phase), and therefore does not change the value of; consequently,
the stationarity of this functional does not depend on the value of the ket(), which
can be any ket, for example the zero ket.
Finally, if the()are equal to the solutions()of theequations:
}
d
d
()=
P
2
2
+1() + () ()
(36)
the functionalis indeed stationary for all times. Furthermore, as we saw in Comple-
mentXVthat ()is Hermitian, so is the operator on the right-hand side of (36).
1708

FERMIONS, TIME-DEPENDENT HARTREE-FOCK APPROXIMATION
Consequently, thekets()follow an evolution similar to the usual Schrödinger
evolution, described by a unitary evolution operator (ComplementIII). Such an opera-
tor does not change either the norm nor the scalar products of the kets: if the kets()
initially formed an orthonormal set, this remains true at any later time. The whole cal-
culation just presented is thus consistent; in particular, the norm of the-particle state
vector
^
()is constant over time.
Relations (36) are the time-dependent Hartree-Fock equations. Introducing the
one-particle mean eld operator allowed us not only to compute the stationary energy
levels, but also to treat time-dependent problems.
4-b. Particles in a single spin state
Let us return to the particular case of fermions all having the same spin state, as
in XV. We can then write the Hartree-Fock equations in terms of
the wave functions as:
}
(r) =
}
2
2
+1(r) +dir(r;)(r)
d
3
ex(rr;)(r) (37)
using denitions of (46) of that complement for the direct and exchange potentials, which
are now time-dependent. There is obviously a close relation between the Hartree-Fock
equations, whether they are time-dependent or not.
4-c. Discussion
As encountered in the search for a ground state with the time-independent Hartree-
Fock equations, there is a strong similarity between equations (36) and an ordinary
Schrödinger equation for a single particle. Here again, an exact solution of these equations
is generally not possible, and we must use successive approximations. Assume for example
that the external time-dependent potential1()is zero until time0and that for
0, the physical system is in a stationary state. With the time-independent Hartree-
Fock method we can compute an approximate value for this state and hence a series of
initial values for the individual states(0). This determines the initial Hartree-Fock
potential. Between time0and a slightly later time0+ , the evolution equation
(36) describes the eect of the external potential1()on the individual kets, and allows
obtaining the(0+ ). We can then compute a new value for the Hartree-Fock
potential, and use it to extend the computation of the evolution of the()until a
later time0+ 2. Proceeding step by step, we can obtain this evolution until the nal
time1. For the approach to be precise,must be small enough for the Hartree-Fock
potential to change only slightly from one time step to another.
Another possibility is to proceed as in the search for the stationary states. We
start from a rst family of orthonormal kets, now time-dependent, and which are not too
far from the expected solution over the entire time interval; we then try to improve it
by successive iterations. Inserting in (21) the rst series of orthonormal trial functions,
we get a rst approximation of the Hartree-Fock potential and its associated dynamics.
We then solve the corresponding equation of motion, with the same initial conditions at
1709

COMPLEMENT F XV
=0, which yields a new series of orthonormal functions. Using again (21), we get a
value for the Hartree-Fock potential, a priori dierent from the previous one. We start
the same procedure anew until an acceptable convergence is obtained.
Applications of this method are quite numerous, in particular in atomic, molecular,
and nuclear physics. They allow, for example, the study of the electronic cloud oscillations
in an atom, a molecule or a solid, placed in an external time-dependent electric eld
(dynamic polarisability), or the oscillations of nucleons in their nucleus. We mentioned
in the conclusion of ComplementXVthat the time-independent Hartree-Fock method is
sometimes replaced by the functional density method; this is also the case when dealing
with time-dependent problems.
In concluding this complement we underline the close analogy between the Hartree-
Fock theory and a time-independent or a time-dependent mean eld theory. In both
cases the same Hartree-Fock potential operators come into play. Even though they are
the result of an approximation, these operators have a very large range of applicability.
1710

FERMIONS OR BOSONS: MEAN FIELD THERMAL EQUILIBRIUM
Complement GXV
Fermions or Bosons: Mean eld thermal equilibrium
1 Variational principle
1-a Notation, statement of the problem
1-b A useful inequality
1-c Minimization of the thermodynamic potential
2 Approximation for the equilibrium density operator
2-a Trial density operators
2-b Partition function, distributions
2-c Variational grand potential
2-d Optimization
3 Temperature dependent mean eld equations
3-a Form of the equations
3-b Properties and limits of the equations
3-c Dierences with the zero-temperature Hartree-Fock equations
(fermions)
3-d Zero-temperature limit (fermions)
3-e Wave function equations
Understanding the thermal equilibrium of a system of interacting identical par-
ticles is important for many physical problems: conductor or semiconductor electronic
properties, liquid Helium or ultra-cold gas properties, etc. It is also essential for study-
ing phase transitions, various and multiple examples of which occur in solid and liquid
physics: spontaneous magnetism appearing below a certain temperature, changes in elec-
trical conduction, and many others. However, even if the Hamiltonian of a system of
identical particles is known, calculation of the equilibrium properties cannot, in general,
be carried to completion: these calculations present real diculties in the handling of
state vectors and interaction operators, where non-trivial combinations of creation and
annihilation operators occur. One must therefore use one or several approximations.
The most common one is probably the mean eld approximation, which, as we saw in
ComplementXV, is the base of the Hartree-Fock method. In that complement, we
showed, in terms of state vectors, how this method could be used to obtain approximate
values for the energy levels of a system of interacting particles. As we consider here the
more complex problem of thermal equilibrium, which must be treated in terms of density
operators, we show how the Hartree-Fock method can be extended to this more general
case.
We are going to see that, thanks to this approach, one can obtain compact for-
mulas for an approximate value of the density operator at thermal equilibrium, in the
framework of the grand canonical ensemble. The equations to be solved are fairly sim-
ilar
1
to those of ComplementXV. The Hartree-Fock method also gives a value of the
1
They are not simply the juxtaposition of that complement's equations: one could imagine writing
1711

COMPLEMENT G XV
thermodynamic grand potential, which leads directly to the pressure of the system. The
other thermodynamic quantities can then be obtained via partial derivatives with respect
to the equilibrium parameters (volume, temperature, chemical potential, eventually ex-
ternal applied eld, etc. see Appendix). It is clearly a powerful method even though
it still is an approximation as the particles interactions are treated via a mean eld ap-
proach where certain correlations are not taken into account. Furthermore, for bosons, it
can only be applied to physical systems far from Bose-Einstein condensation; the reasons
for this limitation will be discussed in detail in XV.
Once we have recalled the notation and a few generalities, we shall establish ()
a variational principle that applies to any density operator. It will allow us to search in
any family of operators for the one closest to the density operator at thermal equilibrium.
We will then introduce () a family of trial density operators whose form reects the
mean eld approximation; the variational principle will help us determine the optimal
operator. We shall obtain Hartree-Fock equations for a non-zero temperature, and study
some of their properties in the last section (). Several applications of these equations
will be presented in ComplementXV.
The general idea and the structure of the computations will be the same as in
ComplementXV, and we keep the same notation: we establish a variational condition,
choose a trial family, and then optimize the system description within this family. This
is why, although the present complement is self-contained, it might be useful to rst read
ComplementXV.
1. Variational principle
In order to use a certain number of general results of quantum statistical mechanics (see
Appendix refappend-6 for a more detailed review), we rst introduce the notation.
1-a. Notation, statement of the problem
We assume the Hamiltonian is of the form:
=0+ext+ int (1)
which is the sum of the particles' kinetic energy0, their coupling energyextwith an
external potential:
ext= 1() (2)
and their mutual interactionint, which can be expressed as:
int=
1
2
=
2(RR) (3)
We are going to use the grand canonical ensemble (Appendix, ), where
the particle number is not xed, but takes on an average value determined by the chemical
those equations independently for each energy level, and then performing a thermal average. We are
going to see (for example in ) that the determination of each level's position already implies
thermal averages, meaning that the levels are coupled.
1712

FERMIONS OR BOSONS: MEAN FIELD THERMAL EQUILIBRIUM
potential. In this case, the density operatoris an operator acting in the entire Fock
space(wherecan take on all the possible values), and not only in the state space
forparticles (which is is more restricted since it corresponds to a xed value of
). We set, as usual:
=
1
(4)
where is the Boltzmann constant andthe absolute temperature. At the grand
canonical equilibrium, the system density operator depends on two parameters,and
the chemical potential, and can be written as:
eq=
1
exp (5)
with the relation that comes from normalizing to1the trace ofeq:
=Trexp (6)
The functionis called the grand canonical partition function (see Appendix,
). The operatorassociated with the total particle number is dened in (B-
17) of Chapter. The temperatureand the chemical potentialare two intensive
quantities, respectively conjugate to the energy and the particle number.
Because of the particle interactions, these formulas generally lead to calculations
too complex to be carried to completion. We therefore look, in this complement, for
approximate expressions ofeqandthat are easier to use and are based on the mean
eld approximation.
1-b. A useful inequality
Consider two density operatorsand, both having a trace equal to1:
Tr=Tr = 1 (7)
As we now show, the following relation is always true:
Trln Trln (8)
We rst note that the functionln, dened for0, is always larger than the
function1, which is the equation of its tangent at= 1(Fig.). For positive values
ofandwe therefore always have:
ln1 (9)
or, after multiplying by:
ln ln (10)
the equality occurring only if=.
1713

COMPLEMENT G XV
Figure 1: Plot of the functionln. At= 1, this curve is tangent to the line= 1
(dashed line) but always remains above it; the function value is thus always larger than
1.
Let us callthe eigenvalues ofcorresponding to the normalized eigenvectors
, andthe eigenvalues ofcorresponding to the normalized eigenvectors.
Used for the positive numbersand, relation (10) yields:
ln ln (11)
We now multiply this relation by the square of the modulus of the scalar product:
2
= (12)
and sum overand. For the term inlnof (11), the summation overyields in
(12) the identity operator expanded on the basis; we then get = 1, and
are left with the sum overofln, that is the trace Trln. As for the term in
ln, the summation overintroduces:
ln =ln (13)
and we get:
ln =Trln (14)
As for the terms on the right-hand side of inequality (11), the term inyields:
= =Tr= 1 (15)
1714

FERMIONS OR BOSONS: MEAN FIELD THERMAL EQUILIBRIUM
and the one inalso yields1for the same reasons, and both terms cancel out. We
nally obtain the inequality:
Trln Trln 0 (16)
which proves (8).
Comment:
One may wonder under which conditions the above relation becomes an equality. This
requires the inequality (11) to become an equality, which means= whenever the
scalar product (12) is non-zero; consequently all the eigenvalues of the two operators
andmust be equal. In addition, the eigenvectors of each operator corresponding to
dierent eigenvalues must be orthogonal (their scalar product must be zero). In other
words, the eigenvalues and the subspaces spanned by their eigenvectors are identical,
which amounts to saying that=.
1-c. Minimization of the thermodynamic potential
The entropyassociated with any density operatorhaving a trace equal to1is
dened by relation (6) of Appendix:
= Trln (17)
The thermodynamic potential of the grand canonical ensemble is dened by the grand
potential, which can be expressed as a function ofby relation (Appendix, ):
= =Tr + ln (18)
Inserting (5) into (18), we see that the value ofat equilibrium,eq, can be directly
obtained from the partition function:
eq=Tr +
1
+ ln eq
=
1
lnTreq= ln (19)
We therefore have:
=

or= ln (20)
Consider now any density operatorand its associated functionobtained from
(18). According to (5) and (20), we can write:
=ln+ln=ln (21)
Inserting this result in (18) yields:
=Tr
1
[ln++ln]=
1
Tr[ln+ln]+ (22)
Now relation (16), used with=eq, is written as:
Tr[lnlneq] 0 (23)
1715

COMPLEMENT G XV
Relation (22) thus implies that for any density operatorhaving a trace equal to1, we
have:
eq (24)
the equality occurring if, and only if,=eq.
Relation (24) can be used to x a variational principle: choosing a family of density
operatorshaving a trace equal to1, we try to identify in this family the operator that
yields the lowest value for. This operator will then be the optimal operator within this
family. Furthermore, this operator yields an upper value for the grand potential, with
an error of second order with respect to the error made on.
2. Approximation for the equilibrium density operator
We now use this variational principle with a family of density operators that leads to
manageable calculations.
2-a. Trial density operators
The Hartree-Fock method is based on the assumption that a good approximation
is to consider that each particle is independent of the others, but moving in the mean
potential they create. We therefore compute an approximate value of the density operator
by replacing the Hamiltonianby a sum of independent particles' Hamiltonians():
=
=1
() (25)
We now introduce the basis of the creation and annihilation operators, associated with
the eigenvectors of the one-particle operator:
0= with = (26)
The symmetric one-particle operatorcan then be written, according to relation (B-14)
of Chapter:
= (27)
where the real constantsare the eigenvalues of the operator.
We choose as trial operators acting in the Fock space the set of operatorsthat can
be written in the form corresponding to an equilibrium in the grand canonical ensemble
see relation (42) of Appendix. We then set:
=
1
exp (28)
whereis any symmetric one-particle operator, the constantthe inverse of the tem-
perature dened in (4),a real constant playing the role of a chemical potential, and
the trace of:
=Trexp (29)
1716

FERMIONS OR BOSONS: MEAN FIELD THERMAL EQUILIBRIUM
Consequently, the relevant variables in our problem are the states, which form an
arbitrary orthonormal basis in the individual state space, and the energies. These
variables determine theas well as, and we have to nd which of their values
minimizes the function:
=Tr + ln (30)
Taking (27) and (28) into account, we can write:
=
1
exp ( ) (31)
The following computations are simplied since the Fock space can be considered to be
the tensor product of independent spaces associated with the individual states; con-
sequently, the trial density operator (28) can be written as a tensor product of operators
each acting on a single mode:
=
1
exp ( ) (32)
2-b. Partition function, distributions
Equality (32) has the same form as relation (5) of ComplementXV, with a simple
change: the replacement of the free particle energies=}
22
2by the energies,
which are as yet unknown. As this change does not impact the mathematical structure
of the density operator, we can directly use the results of ComplementXV.
. Variational partition function
The functiononly depends on the variational energies, since the trace of (32)
may be computed in the basis , which yields:
= exp [ ( )] (33)
We simply get an expression similar to relation (7) of ComplementXV, obtained for an
ideal gas. Since for fermionscan only take the values0and1, we get:
= 1 +
()
(34)
whereas for bosonsvaries from0to innity, so that:
=
1
1
()
(35)
In both cases we can write:
ln= ln1
()
(36)
1717

COMPLEMENT G XV
with= +1for bosons, and=1for fermions.
Computing the entropy can be done in a similar way. As the density operator
has the same form as the one describing the thermal equilibrium of an ideal gas, we can
use for a system described bythe formulas obtained for the entropy of a system without
interactions.
. One particle, reduced density operator
Let us compute the average value ofwith the density operator:
=Tr (37)
We saw in XVthat:
Tr = ( ) (38)
where the distribution functionis noted for fermions, andfor bosons:
( ) =
( ) =
1
( )
+ 1
for fermions
( ) =
1
( )
1
for bosons
(39)
When the system is described by the density operator, the average populations of the
individual statesare therefore determined by the usual Fermi-Dirac or Bose-Einstein
distributions. From now on, and to simplify the notation, we shall write simplyfor
the kets.
We can introduce a one-particle reduced density operator1(1)by
2
:
1(1) = ( )1 :1 : (40)
where the1enclosed in parentheses and the subscript1on the left-hand side emphasize
we are dealing with an operator acting in the one-particle state space (as opposed to
that acts in the Fock space); needless to say, this subscript has nothing to do with
the initial numbering of the particles, but simply refers to any single particle among all
the system particles. The diagonal elements of1(1)are the individual state populations.
With this operator, we can compute the average value overof any one-particle operator
:
=Tr (41)
as we now show. Using the expression (B-12) of Chapter
2
Contrary to what is usually the case for a density operator, the trace of this reduced operator is
not equal to1, but to the average particle number see relation (44). This dierent normalization is
often more useful when studying systems composed of a large number of particles.
1718

FERMIONS OR BOSONS: MEAN FIELD THERMAL EQUILIBRIUM
tor
3
, as well as (38), we can write:
= Tr = ( )
= 1(1) (42)
that is:
=Tr1 1(1) (43)
As we shall see, the density operator1(1)is quite useful since it allows obtaining in a
simple way all the average values that come into play in the Hartree-Fock computations.
Our variational calculations will simply amount to varying1(1). This operator presents,
in a certain sense, all the properties of the variational density operatorchosen in (28)
in the Fock space. It plays the same role
4
as the projector(which also represents the
essence of the variational-particle ket) played in ComplementXV. In a general way,
one can say that the basic principle of the Hartree-Fock method is to reduce the binary
correlation functions of the system to products of single-particle correlation functions
(more details on this point will be given in XVI).
The average value of the operatorfor the total particle number is written:
=Tr =Tr11(1)=
=1
( ) (44)
Both functionsand increase as a function ofand, for any given temperature,
the total particle number is controlled by the chemical potential. For a large physical
system whose energy levels are very close, the orbital part of the discrete sum in (44)
can be replaced by an integral. Figure XVshows the variations of the
Fermi-Dirac and Bose-Einstein distributions. We also mentioned that for a boson system,
the chemical potential could not exceed the lowest value0of the energies; when it
approaches that value, the population of the corresponding level diverges, which is the
Bose-Einstein condensation phenomenon we will come back to in the next complement.
For fermions, on the other hand, the chemical potential has no upper boundary, as,
whatever its value, the population of states having an energy lower thancannot exceed
1.
. Two particles, distribution functions
We now consider an arbitrary two-particle operatorand compute its average
value with the density operator. The general expression of a symmetric two-particle
3
We have changed the notationandof Chapter andto avoid any confusion with
the distribution functions.
4
For fermions, and when the temperature approaches zero, the distribution functionincluded
in the denition of(1)becomes a step function and(1)does indeed coincide with(1).
1719

COMPLEMENT G XV
operator is given by relation (C-16) of Chapter, and we can write:
=Tr
=
1
2
1 :; 2 :(12)1 :; 2 :Tr (45)
We follow the same steps as in XV: we use the mean eld approx-
imation to replace the computation of the average value of a two-particle operator by
that of average values for one-particle operators. We can, for example, use relation (43)
of ComplementXV, which shows that:
Tr = [ + ]( )( ) (46)
We then get:
=
1
2
( )( )
1 :; 2 :(12)1 :; 2 :+1 :; 2 :(12)1 :; 2 :
(47)
Which, according to (40), can also be written as:
=
1
2
1 : 1(1)1 : 2 : 1(2)2 :
1 :; 2 :(12) [1 +ex(12)]1 :; 2 :
(48)
whereexis the exchange operator between particles1and2. Since:
1 : 1(1)1 :2 : 1(2)2 :
=1 :; 2 :1(1) 1(2)1 :; 2 : (49)
and as the operators1(1)and1(2)are diagonal in the basis, we can write the
right-hand side of (48) as:
1
2
1 :; 2 :[1(1) 1(2)](12) [1 +ex(12)]1 :; 2 : (50)
which is simply a (double) trace on two particles1and2. This leads to:
=
1
2
Tr12[1(1) 1(2)](12) [1 +ex(12)] (51)
As announced above, the average value of the two-particle operatorcan be expressed,
within the Hartree-Fock approximation, in terms of the one-particle reduced density
operator1(1); this relation is not linear.
1720

FERMIONS OR BOSONS: MEAN FIELD THERMAL EQUILIBRIUM
Comment:
The analogy with the computations of ComplementXVbecomes obvious if we regroup
its equations (57) and (58) and write:
^
int=
1
2
Tr12[(1) (2)]2(12) [1 +ex(12)] (52)
Replacing2(12)by, we get a relation very similar to (51), except for the fact that
the projectorsmust be replaced by the one-particle operators1. In , we shall
come back to the correspondence between the zero and non-zero temperature results.
2-c. Variational grand potential
We now have to compute the grand potentialwritten in (30). As the exponential
form (28) for the trial operator makes it easy to compute ln, we see that the terms in
cancel out, and we get:
=Tr ln (53)
We now have to compute the average energy, with the density operator, of the dierence
between the Hamiltoniansandrespectively dened by (1) and (25).
We rst compute the trace:
Tr = (54)
starting with the kinetic energy contribution0in (1). We call0the individual kinetic
energy operator:
0=
P
2
2
(55)
(is the particle mass). Equality (43) applied to0yields the average kinetic energy
when the system is described by:
0=Tr1 01(1)= 0 ( ) (56)
This result is easily interpreted; each individual state contributes its average kinetic
energy, multiplied by its population.
The computation of the average valueextfollows the same steps:
ext=Tr111(1)= 1 ( ) (57)
(as in ComplementXV, operator1is the one-particle external potential operator).
To complete the calculation of the average value of, we now have to compute
the trace Trint, the average value of the interaction energy when the system is
described by. Using relation (51) we can write this average value as a double trace:
int=
1
2
Tr121(1) 1(2) [2(12)] [1 +ex(12)] (58)
1721

COMPLEMENT G XV
We now turn to the average value of. The calculation is simplied since
is, like0, a one-particle operator; furthermore, thehave been chosen to be the
eigenvectors ofwith eigenvalues see relation (26). We just replace in (56),0by
, and obtain:
= () ( ) = ( ) (59)
Regrouping all these results and using relation (36), we can write the variational
grand potential as the sum of three terms:
=1+2+3 (60)
with:
1=Tr1[0+1]1(1)
2=
1
2
Tr12[1(1) 1(2)]2(12) [1 +ex(12)]
3= ( ) + ln1
()
(61)
2-d. Optimization
We now vary the eigenenergiesand eigenstatesofto nd the value of the
density operatorthat minimizes the average valueof the potential. We start with
the variations of the eigenstates, which induce no variation of3. The computation is
actually very similar to that of ComplementXV, with the same steps: variation of the
eigenvectors, followed by the demonstration that the stationarity condition is equivalent
to a series of eigenvalue equations for a Hartree-Fock operator (a one-particle operator).
Nevertheless, we will carry out this computation in detail, as there are some dierences.
In particular, and contrary to what happened in ComplementXV, the number of states
to be varied is no longer xed by the particle number; these states form a complete
basis of the individual state space, and their number can go to innity. This means that
we can no longer give to one (or several) state(s) a variation orthogonal to all the other
; this variation will necessarily be a linear combination of these states. In a second
step, we shall vary the energies.
. Variations of the eigenstates
As the eigenstatesvary, they must still obey the orthogonality relations:
= (62)
The simplest idea would be to vary only one of them,for example, and make the
change:
+d (63)
The orthogonality conditions would then require:
d= 0for all= (64)
1722

FERMIONS OR BOSONS: MEAN FIELD THERMAL EQUILIBRIUM
preventingdfrom having a component on any ketother than: in other words,
dand would be colinear. As must remain normalized, the only possible
variation would thus be a phase change, which does not aect either the density operator
1(1)or any average values computed with. This variation does not change anything
and is therefore irrelevant.
It is actually more interesting to vary simultaneously two eigenvectors, which will
be calledand , as it is now possible to givea component on , and the
reverse. This does not change the two-dimensional subspace spanned by these two states;
hence their orthogonality with all the other basis vectors is automatically preserved. Let
us give the two vectors the following innitesimal variations (without changing their
energiesand):
d=d
d=d
with= (65)
where dis an innitesimal real number andan arbitrary but xed real number. For
any value of, we can check that the variation ofis indeed zero (it contains the
scalar products or which are zero), as is the symmetrical variation of
, and that we have:
d =d d = 0 (66)
The variations (65) are therefore acceptable, for any real value of.
We now compute how they change the operator1(1)dened in (40). In the sum
over, only the=and=terms will change. The=term yields a variation:
d ( ) + (67)
whereas the=term yields a similar variation but where( )is replaced by
( )This leads to:
d1(1) =d[( ) ( )] + (68)
We now include these variations in the three terms of (61); as the distributions
are unchanged, only the terms1and2will vary. The innitesimal variation of1is
written as:
d1=Tr1[0+1]d1(1) (69)
As for d2, it contains two contributions, one from d1(1)and one from d1(2). These
two contributions are equal since the operator2(12)is symmetric (particles1and2
play an equivalent role). The factor12in2disappears and we get:
d2=Tr12d1(1) 1(2) [2(12)] [1 +ex(12)] (70)
We can regroup these two contributions, using the fact that for any operator(12), it
can be shown that:
Tr12d1(1)(12)=Tr1d1(1)Tr2(12) (71)
1723

COMPLEMENT G XV
This equality is simply demonstrated
5
by using the denition of the partial trace Tr2(12)
of operator(12)with respect to particle2. We then get:
d =d1+2
=Tr1d1(1)0+1+Tr21(2)2(12) [1 +ex(12)] (72)
Inserting now the expression (68) for1(1), we get two terms, one proportional
to, another one to, whose value is:
d [( ) ( )]
Tr1 0+1+Tr21(2)2(12) [1 +ex(12)] (73)
Now, for any operator(1), we can write:
Tr1 (1)= (1)= (1)
= (1) (74)
so that the variation (73) can be expressed as:
d [( ) ( )]
0+1+Tr21(2)2(12) [1 +ex(12)] (75)
The term inhas a similar form, but it does not have to be computed for the
following reason. The variation dis the sum of a term inand another in:
d =d 1 +2 (76)
and the stationarity condition requires dto be zero for any choice of. Choosing= 0,
yields1+2= 0; choosing=2, and multiplying by, we get1 2= 0. Adding
and subtracting those two relations shows that both coecients1and2must be zero.
Consequently, it suces to impose the terms in, and hence expression (75), to be
zero. When=, the distribution functionsare not equal, and we get:
0+1+Tr21(2)2(12) [1 +ex(12)] = 0 (77)
(if=, however, we have not yet obtained any particular condition to be satised
6
).
5
The denition of partial traces is given in 5-b of ComplementIII. The left hand side of (71)
can be written as 1 :; 2 : (1)(12)1 :; 2 :. We then insert, after(1), a closure
relation on the kets1 :; 2 :, with=since(1)does not act on particle2. This yields:
1 : (1)1 : 1 :; 2 :(12)1 :; 2 :, where the sum overis the denition
of the matrix element between1 :and1 :of the partial trace over particle2of the operator
(12). We then get the right-hand side of (71).
6
This was expected, since this choice does not lead to any variation of the trial density operator.
1724

FERMIONS OR BOSONS: MEAN FIELD THERMAL EQUILIBRIUM
. Variation of the energies
Let us now see what happens if the energyvaries by d. The function( )
then varies by dwhich, according to relation (40), induces a variation of1:
d1= d (78)
and thus leads to variations of expressions (61) of1and2. Their sum is:
d1+d2=Tr1d1(1)0+1+Tr21(2) [2(12)] [1 +ex(12)] (79)
where the factor12in2has been canceled since the variations induced by1(1)and
1(2)double each other. Inserting (78) in this relation and using again (74), we get:
d1+d2=d 0+1+Tr21(2) [2(12)] [1 +ex(12)] (80)
As for3, its variation is the sum of a term in dcoming from the explicit presence
of the energiesin its denition (61), and a term in d. If we let only the energy
vary (not taking into account the variations of the distribution function), we get a zero
result, since:
( ) + () ()
() 1
1
d
= ( ) +( )d= 0
(81)
Consequently, we just have to vary by dthe distribution function, and we get:
d3= d (82)
Finally, after simplication by d(which, by hypothesis, is dierent from zero),
imposing the variation dto be zero leads to the condition:
d1+d2+d3
= [0+1 +Tr21(2) [2(12)] [1 +ex(12)]]= 0 (83)
This expression does look like the stationarity condition at constant energy (77), but
now the subscriptsandare the same, and a term inis present in the operator.
3. Temperature dependent mean eld equations
Introducing a Hartree-Fock operator acting in the single particle state space allows writ-
ing the stationarity relations just obtained in a more concise and manageable form, as
we now show.
1725

COMPLEMENT G XV
3-a. Form of the equations
Let us dene a temperature dependent Hartree-Fock operator as the partial trace
that appears in the previous equations:
() =Tr21(2)2(12) [1 +ex(12)] (84)
It is thus an operator acting on the single particle1. It can be dened just as well by its
matrix elements between the individual states:
()
= ( )1 :; 2 : 2(12) [1 +ex(12)]1 :; 2 : (85)
Equation (77) is valid for any two chosen valuesand, as long as=. When
is xed andvaries, it simply means that the ket:
[0+1+ ()] (86)
is orthogonal to all the eigenvectorshaving an eigenvaluedierent from; it has
a zero component on each of these vectors. As for equation (83), it yields the component
of this ket on, which is equal to. The set of(including those having the
same eigenvalue as) form a basis of the individual state space, dened by (26) as the
basis of eigenvectors of the individual operator. Two cases must be distinguished:
(i) Ifis a non-degenerate eigenvalue of, the set of equations (77) and (83)
determine all the components of the ket[0+1+ ()]. This shows that
is an eigenvector of the operator0+1+ with the eigenvalue.
(ii) If this eigenvalue ofis degenerate, relation (77) only proves that the eigen-
subspace of, with eigenvalue, is stable under the action of the operator0+1+ ;
it does not yield any information on the components of the ket (86) inside that subspace.
It is possible though to diagonalize0+1+ inside each of the eigen-subspace of
, which leads to a new eigenvectors basis, now common toand0+1+ .
We now reason in this new basis where all the[0+1+ ()] are pro-
portional to. Taking (83) into account, we get:
[0+1+ ()]= (87)
As we just saw, the basis change from theto the only occurs within the eigen-
subspaces ofcorresponding to given eigenvalues; one can then replace theby the
in the denition (40) of1(1)and write:
1(1) = ( )1 :1 : (88)
Inserting this relation in the denition (84) of ()leads to a set of equations only
involving the eigenvectors.
For all the values ofwe get a set of equations (87), which, associated with (84) and
(88) dening the potential()as a function of the, are called the temperature
dependent Hartree-Fock equations.
1726

FERMIONS OR BOSONS: MEAN FIELD THERMAL EQUILIBRIUM
3-b. Properties and limits of the equations
We now discuss how to apply the mean eld equations we have obtained, and their
limit of validity, which are more stringent for bosons than for fermions.
. Using the equations
Hartree-Fock equations concern a self-consistent and nonlinear system: the eigen-
vectors and eigenvalues of the density operator1(1)are solutions of an eigenvalue
equation (87) which itself depends on1(1). This situation is reminiscent of the one
encountered with the zero-temperature Hartree-Fock equations, and, a priori, no exact
solutions can be found.
As for the zero-temperature case, we proceed by iteration: starting from a phys-
ically reasonable density operator1(1), we use it in (84) to compute a rst value of
the Hartree-Fock potential operator. We then diagonalize this operator to get its eigen-
kets and eigenvalues. Next, we build the operator
1that has the same eigenkets,
but whose eigenvalues are the( ). Inserting this new operator
1in (84), we
get a second iteration of the Hartree-Fock operator. We again diagonalize this operator
to compute new eigenvalues and eigenvectors, on which we build the next approxima-
tion
1of1, and so on. After a few iterations, we may expect convergence towards a
self-consistent solution.
. Validity limit
For a fermion system, there is no fundamental general limit for using the Hartree-
Fock approximation. The pertinence of the nal result obviously depends on the nature
of the interactions, and whether a mean eld treatment of these interactions is a good
approximation. One can easily understand that the larger the interaction range, the
more each particle will be submitted to the action of many others. This will lead to
an averaging eect improving the mean eld approximation. If, on the other hand, each
particle only interacts with a single partner, strong binary correlations may appear, which
cannot be correctly treated by a mean eld acting on independent particles.
For bosons, the same general remarks apply, but the populations are no longer
limited to1. When, for example, Bose-Einstein condensation occurs, one population
becomes much larger than the others, and presents a singularity that is not accounted
for in the calculations presented above. The Hartree-Fock approximation has therefore
more severe limitations than for the fermions, and we now discuss this problem.
For a boson system in which many individual states have comparable populations,
taking into account the interactions by the Hartree-Fock mean eld yields as good an
approximation as for a fermion system. If the system however is close to condensation,
or already condensed, the mean eld equations we have written are no longer valid. This
is because the trial density operator in relation (31) contains a distribution function as-
sociated with each individual quantum state and varies as for an ideal gas, i.e. as an
exponentially decreasing function of the occupation numbers. Now we saw in of
ComplementXVthat, in an ideal gas, the uctuations of the particle numbers in each
of the individual states are as large as the average values of those particle numbers. If the
individual state has a large population, these uctuations can become very important,
which is physically impossible in the presence of repulsive interactions. Any population
uctuation increases the average value of the square of the occupation number (equal
1727

COMPLEMENT G XV
to the sum of the squared average value and the squared uctuation), and hence of the
interaction energy (proportional to the average value of the square). A large uctuation
in the populations would lead to an important increase of the interaction repulsive en-
ergy, in contradiction with the minimization of the thermodynamic potential. In other
words, the nite compressibility of the physical system, introduced by the interactions,
prevents any large uctuation in the density. Consequently, the uctuations in the num-
ber of condensed particles predicted by the trial Hartree-Fock density operator are not
physically acceptable, in the presence of condensation.
It is worth analyzing more precisely the origin of this Hartree-Fock approximation
limit, in terms of correlations between the particles. Relation (51) concerns any two-
particle operator. It shows that, using the trial density operator (31), the two-particle
reduced density operator can be written as:
2(12) =1(1) 1(1) [1 +ex(12)] (89)
Its diagonal matrix elements are then written:
1 :; 2 :2(12)1 :; 2 :
=1 : 11 :2 : 12 :+1 : 11 :2 : 12 : (90)
and are the sum of a direct term, and an exchange term. When=, the presence of
an exchange term is not surprising, and corresponds to the general discussion of
in Chapter. It is similar to the expression of the spatial correlation function written
in (C-34) of that chapter, which is also the sum of two contributions, a direct one (C-
32) and an exchange one (C-33). Since this last contribution is positive when1 2,
the physical consequence of the exchange is a spatial bunching of the bosons. What is
surprising though is that the exchange term still exists in (90) when=, even though
the notion of exchange is meaningless: when dealing with a single individual state, the
four expressions (C-21) of Chapter
also check that the exchange term (C-34) of Chapter
=, which means it receives no contribution from=. We shall furthermore conrm
in XVIthat bosons all placed in the same individual quantum state
are not spatially correlated, and therefore present neither bunching nor exchange eects.
The mathematical expression of the trial two-particle Hartree-Fock density operator thus
contains too many exchange terms. This does not really matter as long as the boson
system remains far from Bose-Einstein condensation: the error involved is small since
the=terms play a negligible role compared to the=terms in the summations
overandappearing in the interaction energy. However, as soon as an individual state
becomes highly populated, signicant errors can occur and the Hartree-Fock method
must be abandoned. There exist, however, more elaborate theoretical treatments better
adapted to this case.
3-c. Dierences with the zero-temperature Hartree-Fock equations (fermions)
The main dierence between the approach we just used and that of Complements
CXVandXVis that these complements were only looking for a single eigenstate of
the Hamiltonian
^
, generally its ground state. If we are now interested in several of
these states, we have to redo the computation separately for each of them. To study the
1728

FERMIONS OR BOSONS: MEAN FIELD THERMAL EQUILIBRIUM
properties of thermal equilibrium, one could imagine doing the calculations a great many
times, and then weigh the results with occupation probabilities. This method obviously
leads to heavy computations, which become impossible for a macroscopic system hav-
ing an extremely large number of levels. In the present complement, the Hartree-Fock
equations yield immediately thermal averages, as well as eigenvectors of a one-particle
density operator with their energies.
Another important dierence is that the Hartree-Fock operator now depends on
the temperature, because of the presence in (85) of a temperature dependent distribution
function or, which amounts to the same thing, of the presence in (84) of an operator
dependent on, and which replaces the projector(2)onto all the populated individual
states. The equations obtained remind us of those governing independent particles, each
nding its thermodynamic equilibrium while moving in the self-consistent mean eld
created by all the others, also including the exchange contribution (which can be ignored
in the simplied Hartree version).
We must keep in mind, however, that the Hartree-Fock potential associated with
each individual state now depends on the populations of an innity of other individual
states, and these populations are function of their energy as well as of the tempera-
ture. In other words, because of the nonlinear character of the Hartree-Fock equations,
the computation is not merely a juxtaposition of separate mean eld calculations for
stationary individual states.
3-d. Zero-temperature limit (fermions)
Let us check that the Hartree-Fock method for non-zero temperature yields the
same results as the zero temperature method explained in ComplementXVfor fermions.
In XV, we introduced for an ideal gas the concept of a
degenerate quantum gas. It can be generalized to a gas with interactions: in a fermion
system, when 1, the system is said to be strongly degenerate. As the temperature
goes to zero, a fermion system becomes more and more degenerate. Can we be certain
that the results of this complement are in agreement with those of ComplementXV,
valid at zero temperature?
We saw that the temperature comes into play in the denition (85) of the mean
Hartree-Fock potential,. In the limit of a very strong degeneracy, the Fermi-Dirac
distribution function appearing in the denition (40) of1(1)becomes practically a step
function, equal to1for energiesless than the chemical potential, and zero otherwise
(Figure XV. In other words, the only populated states (and by a single
fermion) are the states having energies less than, i.e. less than the Fermi level. Under
such conditions, the1(2)of (84) becomes practically equal to the projector(2)which,
in ComplementXV, appears in the denition (52) of the zero-temperature Hartree-Fock
potential; in other words, the partial trace appearing in this relation (85) is then strictly
limited to the individual states having the lowest energies. We thus obtain the same
Hartree Fock equations as for zero temperature, leading to the determination of a set of
individual eigenstates on which we can build a unique-particle state.
3-e. Wave function equations
Let us write the Hartree-Fock equations (87) in terms of wave functions: these
equations are strictly equivalent to (87), written in terms of operators and kets, but their
1729

COMPLEMENT G XV
form is sometimes easier to use, in particular for numerical calculations.
Assuming the particles have a spin, we shall note the wave functions(r), with:
(r) =r (91)
where the spin quantum numbercan take(2+ 1)values; according to the nature of
the particles, the possible spinsare= 0,= 12,= 1etc. As in Complement
EXV(), we introduce a complete basis for the individual state space, built
from kets that are all eigenvectors of the spin component along the quantization axis,
with eigenvalue. For each value of, the spin indextakes on a given valueand
is not, therefore, an independent index. As for the potentials, we assume here again
that1is diagonal in, but that its diagonal elements
1(r)may depend on. The
interaction potential, however, is described by a function2(rr)that only depends on
rr, but does not act on the spins.
To obtain the matrix elements of()in the representationr, we use
(85) after replacing theby the(we showed in
multiply both sides byr and r, and sum over the subscriptsand;
we recognize in both sides the closure relations:
r =r and r =r (92)
This leads to:
r ()r =
( )1 :r; 2 : 2(12) [1 +(12)]1 :r; 2 : (93)
As in , we get the sum of a direct term (the term1in the central
bracket) and an exchange term (the term inex). This expression contains the same
matrix element as relation (87) of ComplementXV, the only dierence being the pres-
ence of a coecient( )in each term of the sum (plus the fact that the summation
index goes to innity).
(i) For the direct term, as we did in that complement, we insert a closure relation
on the particle2position:
2 : =
3
2(r2)2 :r2 (94)
Since the interaction operator is diagonal in the position representation, the part of the
matrix element of (93) that does not contain the exchange operator becomes:
3
2 (r2)
2
1 :r; 2 :r2 2(12)1 :r; 2 :r2 (95)
The direct term of (93) is then written:
(rr)d
3
22(rr2) ( )(r2)
2
(96)
which is equivalent to relation (91) of ComplementXV.
1730

FERMIONS OR BOSONS: MEAN FIELD THERMAL EQUILIBRIUM
(ii) The exchange term is obtained by permutation of the two particles in the
ket appearing on the right-hand side of (93); the diagonal character of2(12)in the
position representation leads to the expression:
1 :r1 : 2 :2 :r 2(rr) (97)
For the rst scalar product to be non-zero, the subscriptmust be such that=; in
the same way, for the second product to be non-zero, we must have=. For both
conditions to be satised, we must impose=, and the exchange term (93) is equal
to:
=
( ) (r) (r) 2(rr) (98)
where the summation is on all the values ofsuch that=: this term only exists if
the two interacting particles are totally indistinguishable, which requires that they be in
the same spin state (see the discussion in ComplementXV).
We now dene the direct and exchange potentials by:
dir(r) =
3
2(rr) ( )(r)
2
ex(rr) =2(rr)
=
( ) (r) (r)
(99)
The equalities (87) then lead to the Hartree-Fock equations in the position representation:
}
2
2
+
1
(r) +dir(r) (r) +d
3
ex(rr)(r) = (r)
(100)
The general discussion of
tions are both nonlinear and self-consistent, as the direct and exchange potentials are
themselves functions of the solutions(r)of the eigenvalue equations (100). This sit-
uation is reminiscent of the zero-temperature case, and we can, once again, look for
solutions using iterative methods. The number of equations to be solved, however, is
innite and no longer equal to the nite number, as already pointed out in .
The set of solutions must span the entire individual state space. Along the same line,
in the denitions (99) of the direct and exchange potentials, the summations overare
not limited tostates, but go to innity. However, even though the number of these
wave functions is in principle innite, it is limited in practice (for numerical calculations)
to a high but nite number. As for the initial conditions to start the iteration process,
one can choose for example the states and energies of a free fermion gas, but any other
conjecture is equally possible.
Conclusion
There are many applications of the previous calculations, and more generally of the
mean eld theory. We give a few examples in the next complement, which are far from
showing the richness of the possible application range. The main physical idea is to
1731

COMPLEMENT G XV
reduce, whenever possible, the calculation of the various physical quantities to a problem
similar to that of an ideal gas, where the particles have independent dynamics. We have
indeed shown that the individual level populations, as well as the total particle number,
are given by the same distribution functionsas for an ideal gas see relations (38)
and (44). The same goes for the system entropy, as already mentioned at the end of
2-b-. If we replace the free particle energies by the modied energiesk, the analogy
with independent particles is quite strong.
If we now want to compute other thermodynamic quantities, as for example the
average energy, we can no longer use the ideal gas formulas; we must go back to the
equations of . The grand potential may be calculated by inserting in (61) the
and theobtained from the Hartree-Fock equations. Another method uses the fact that
is given by ideal gas formulas that contain the distribution, and hence do not
require any further calculations. As:
=
1
ln (101)
we can integrateover(between and the current value, for a xed value of
) to obtain ln, and hence the grand potential. From this grand potential, all the other
thermodynamic quantities can be calculated, taking the proper derivatives (for example
a derivative with respect toto get the average energy). We shall see an example of
this method in
We must however keep in mind that all these calculations derive from the mean
eld approximation, in which we replaced the exact equilibrium density operator by an
operator of the form (32). In many cases this approximation is good, even excellent, as is
the case, in particular, for a long-range interaction potential: each particle will interact
with several others, therefore enhancing the averaging eect of the interaction potential.
It remains, however, an approximation: if, for example, the particles interact via a hard
core potential (innite potential when the mutual distance becomes less than a certain
microscopic distance), the particles, in the real world, can never be found at a distance
from each other smaller than the hard core diameter; now this impossibility is not taken
into account in (32). Consequently, there is no guarantee of the quality of a mean eld
approximation in all situations, and there are cases for which it is not sucient.
1732

APPLICATIONS OF THE MEAN FIELD METHOD FOR NON-ZERO TEMPERATURE
Complement HXV
Applications of the mean eld method for non-zero temperature
(fermions and bosons)
1 Hartree-Fock for non-zero temperature, a brief review
2 Homogeneous system
2-a Calculation of the energies
2-b Quasi-particules
3 Spontaneous magnetism of repulsive fermions
3-a A simple model
3-b Resolution of the equations by graphical iteration
3-c Physical discussion
4 Bosons: equation of state, attractive instability
4-a Repulsive bosons
4-b Attractive bosons
In the previous complement, we presented the Hartree-Fock (mean eld) method
for non-zero temperatures, which has numerous applications a few of them will be
discussed in this complement. We start in
with this method in the previous complement, and which will be used in the present
complement. The general properties of a homogeneous system are then studied in , as
this particular case is often encountered, hence giving it a special importance. The last
two sections are concerned with the study of phase transitions in homogeneous systems.
Section
of a transition where a fermion system becomes spontaneously magnetic because of the
repulsion between particles (even though this repulsion is supposed to be completely
independent of the spins). Finally, the last section deals with bosons and the study of
their equation of state. This will allow us to show, in particular, the appearance of an
instability when the bosons are attractive and close to Bose-Einstein condensation.
1. Hartree-Fock for non-zero temperature, a brief review
We start with a brief review of the results obtained previously ( XV
and XV, which will be useful for what follows.
For an ideal gas, the distribution functionfor fermions, orfor bosons, is
given by:
( ) =
( ) =
1
( )
+ 1
for fermions
( ) =
1
( )
1
for bosons
(1)
where= 1 andis the chemical potential. The average total particle number
is then obtained by a sum over all the individual accessible states, labeled by the
1733

COMPLEMENT H XV
subscript:
=
=1
( ) (2)
The temperature dependent Hartree-Fock equations (mean eld equations) in the
position representation are given by relation (100) of ComplementXV:
}
2
2
+
1
(r) +dir(r) (r) +d
3
ex(rr)(r) = (r) (3)
where= +1for bosons,=1for fermions, and wheredir(r)and
ex(r)are given
by relation (99) of that same complement (we assume the interaction potential does not
act on the spin quantum numbers):
dir(r) =
3
2(rr) ( )(r)
2
ex(rr) =2(rr)
=
( ) (r) (r)
(4)
2. Homogeneous system
We assume from now on that the physical system is subjected to boundary conditions
created by a one-body potential, which connes the particles inside a cubic box of edge
length; this potential is zero (1= 0) inside the box, and takes on an innite value
outside. To take this connement into account, we shall use the periodic boundary
conditions (ComplementXIV, ), for which the normalized eigenfunctions of the
kinetic energy are written as:
1
32
kr
(5)
where the possible wave vectorskare those whose three components are integer multiples
of2. Because of the spin, the eigenvectors of the kinetic energy are labeled by the
values of bothkand, and are writtenk, with:
rk=
k(r) =
1
32
kr
(6)
The index(or) that labeled the basis vectors in the previous complements is now
replaced by two indices,kand(which are independent, as opposed to the indices
andused in XV). We shall nally assume that the particle
interaction is invariant under translation:2(r1r2)only depends onr1r2.
We are going to see that, in such a case, solutions of the Hartree-Fock equations can
be found without having to search for the eigenfunctions of the (Hartree-Fock) operator
written on the left-hand side of (3); these solutions are simply the plane waves written in
(5). Only the operator's eigenvalueskremain to be calculated, and can be interpreted
as the energies of independent objects called quasi-particles ().
1734

APPLICATIONS OF THE MEAN FIELD METHOD FOR NON-ZERO TEMPERATURE
Comment:
We shall verify that these plane waves are solutions of the Hartree-Fock equations, while
neither being necessarily the only ones, nor even those leading to the lowest energy of
the total system. A phenomenon called symmetry breaking (translation symmetry in
this case) could occur and introduce solutions whose moduli vary in space and corre-
spond to lower energies. The Wigner crystal of electrons is such an example, where the
particle density spontaneously shows a periodic spatial modulation. Another example of
spontaneous symmetry breaking will be discussed in
many other cases (in nuclear physics in particular) where the Hartree-Fock method can
be used to study symmetry breaking phenomena.
2-a. Calculation of the energies
As the plane waves are obviously eigenfunctions of the kinetic energy, and since
the potential is zero inside the box, we just have to demonstrate that they are also
eigenfunctions of the direct and exchange potentials. Inserting (5) in (4), we get:
dir(r) =
1
3
k
(k )d
3
2(rr)
=
0
3
k
(k ) (7)
where
0is dened as (with a change of variablerr=s):0=d
3
2(rr) =d
3
2(s) (8)
The direct potential is therefore a constant, independent of the positionr; multiplying
an exponential
kr
, it yields a function proportional to it. This means that
kr
is an
eigenfunction of the direct potential. As for the exchange potential, using the second
relation of (4), we get:
ex(rr) =
1
3
k
(k )
k(rr)
2(rr) (9)
The exchange potential is thus also translation-invariant (it only depends on the dierence
rr). Consequently, the last term on the left-hand side of (3) can be written as:
3
k
(k )d
3 k(rr)
2(rr)
kr
32
=
kr
32
k
(k )
3
(kk) (10)
where (with the change of variablerr=s):
(kk)=d
3 (kk)s
2(s) (11)
1735

COMPLEMENT H XV
Consequently, the exchange term simply multiplies the plane wave
kr32
by:
3
k
(k )
(kk) (12)
To sum up, we showed that, for a uniform system, the plane waves are indeed solu-
tions of the Hartree-Fock equations (3). It is no longer necessary to solve the eigenvector
equations, but we simply have to replace in (3) the
k
(r)by plane waves. This leads to:
k=+
0
3
k
(k ) +
3
k
(k )
(kk)
(13)
whereis the kinetic energy of a free particle:
=
}
2
k
2
2
(14)
We have obtained self-consistent conditions for the eigenvalues, which are a set of coupled
nonlinear equations because of the(k )dependence on the energiesk.
Comment:
The exchange term contains the Fourier transform at the spatial frequencykkof the
particle interaction potential; the direct term, however, contains the Fourier component
at zero spatial frequency. This property is easy to understand from a physical point of
view. Consider two particles, having respectively an initial momentum}kand}k. We
saw in Chapter VIII ( B-4-a) that the eect, to rst order (Born approximation), of an
interaction potential is proportional to the Fourier transform of that potential, calculated
at the value of the variation of the relative momentum between the two particles (Chapter
VII, B-2-a); this variation is none other than the momentum transfer between the
particle as they interact. Consequently, it is normal that the system energy is the sum
of two terms: a direct term where no particle changes its momentum (no momentum
transfer, hence a Fourier variable equal to zero); and another one where the two particles
exchange their momenta, so that the relative momentum changes sign and the Fourier
variable is proportional to the dierencekk.
2-b. Quasi-particules
Equations (13) yield the individual energiesk, which are the sum of the free
particle energy}
22
2and a contribution from the interactions. One can look at them
as energies of individual objects
1
, often referred to as quasi-particles. The populations
of the corresponding levels, as well as the total number of quasi-particles, are given by
the same distribution functionsas for an ideal gas see relations (39) and (44) of
ComplementXV. The same is true for the system entropy, as we already mentioned
at the end of of that complement. Provided we replace the free particle energy
by the modied energiesk, the analogy with independent particles is quite strong.
1
The concept of quasi-particle is not necessarily limited to systems whose interacting particles are
free inside a box; it remains valid for non-zero1potentials (a harmonic potential for example). The
rst term in (13) must then be replaced by the particle energy in the potential1, and the direct and
exchange terms will have a dierent expression.
1736

APPLICATIONS OF THE MEAN FIELD METHOD FOR NON-ZERO TEMPERATURE
3. Spontaneous magnetism of repulsive fermions
Consider a system of spin12fermions, contained in a box. To make the computations
easier, we shall make a few simplifying hypotheses. They lead to a simple model, giving
a good illustration of the nonlinear character of the Hartree-Fock theory. They involve
the resolution of a set of nonlinear equations, containing only two variables equations
we will write in (22).
3-a. A simple model
We assume the mutual interaction potential to be repulsive and to have a very
short range0. In relation (13) the only vectorskandkthat matter are those for which
the distributions(k )and(k )are not negligible. If, for all these vectors,
the products0and 0are very small compared to1, the product(kk)smay be
replaced by zero in (11), and we get:
(kk)0 (15)
. Energy of the quasi-particles; spin state populations
As=1for fermions, equation (13) can be simplied:
k=+
0
3
k
(k )
k
(k ) (16)
or else:
k=+
0
3
k=
(k ) (17)
Consequently, the energy of a quasi-particle with a givenis only modied by the
interaction with quasi-particles having a dierent spin component(opposite spin if
the particles have a spin= 12). This result was to be expected since if the spins
of the two quasi-particles are parallel, they cannot be distinguished; the Pauli principle
then forbids them to approach at a distance closer than0, and they cannot interact.
On the other hand, if their spins are opposite, they can be identied by the direction of
their spin (we have assumed the interaction does not act on the spins): they behave as
distinguishable particles, the exclusion principle does not apply, and they now interact.
We note +and the total particle numbers respectively in the spin state
+or:
=
k
(k ) (18)
Equation (17) shows that the energies of the+andspin states are modied according
to:
k+=+ ; k=+ + (19)
1737

COMPLEMENT H XV
where the coupling constant(having the dimension of an energy) is dened by:
=
0
3
(20)
Since the particle numbers only depend on the dierence between the energies
and the chemical potential, we can account for the terms inappearing in (19) by
keeping the energiesfor free particles, but lowering the chemical potentialsby the
quantity . Calling (), as in relation (47) of ComplementXV, the total
number of fermions in an ideal gas:
() =
2
3
d
3 1
( )
+ 1
(21)
we get, for an interacting gas:
+=
= +
(22)
These equations determine the populations of the two spin states as a function of the
parameters(or the temperature),and nally the volume. These are, however,
two coupled equations since the population+depends on and conversely.
Finding their solution is not obvious, and we shall use a change of variables and resort
to a graphic construction.
. Change of variables
It is useful to write the previous relations in terms of dimensionless variables. We
shall thus introduce the thermal wavelengthby:
=}
2
=}
2
(23)
We can now make the same change of integration variable as in
BXV:
=
2
(24)
Relation (21) then becomes:
() =
3
32() (25)
where:
32() =
32
d
3 1
2
+ 1
(26)
These relations are just the same as those written in (51) and (52) in ComplementXV.
The value of
32only depends on a dimensionless variable, the product. As opposed
1738

APPLICATIONS OF THE MEAN FIELD METHOD FOR NON-ZERO TEMPERATURE
to which is an extensive quantity (proportional
2
to the volume=
3
of the
system),
32()is an intensive quantity (independent of the volume).
We can also replace the two unknowns, which are the populationsof the
two spin states, by two dimensionless and intensive variables:
=
3
(27)
To characterize the interactions appearing in relations (22) via the constant, we intro-
duce the dimensionless parameter
:=
3
=
02}
2
(28)
Replacing in equations (22) the by the, andby
, we get the simpler form:
+=
(1)
()
=
(1)
(+)
(29)
where the function
(1)
is dened by:
(1)
() =
32(
) (30)
(this function depends not only on, but also on the parametersand
). As (22),
the system (29) contains two coupled equations:allows computing directly+, and
vice versa.
3-b. Resolution of the equations by graphical iteration
We now show how to solve equations (29) by a graphical method. The two variables
+and can be uncoupled by noting that:
+=
(1) (1)
(+) (31)
with the same equation for. We now introduce a second iterated function
(2)
of the
function
(1)
(function of the same function) as:
(2)
() =[()] (32)
This leads to:
+=
(2)
(+) (33)
Applying the function
(2)
to the variableyields the same value, which is said to
be the abscissa of a xed point of this function. Graphically, the xed points of any
functionare at the intersections of the curve representing the functionwith the rst
bisector.
2
In a more general way, Appendix
of large volumes (forandconstant), its ratio to the volumetends toward a constant; this does
not prevent the quantity from containing terms in
2
for example. On the other hand, it is said to be
intensive if, in this same limit (and without dividing by the volume), it tends toward a constant.
1739

COMPLEMENT H XV
. Iterations of a function
Consider, from a general point of view, the equation:
=() (34)
whose solutions correspond to the xed points of the function. These solutions can be
found by iteration: starting from an approximate value1of the solution, we compute
(1), then use2=(1)as a new value of the variable to compute3=(2), etc.
It can be shown that this iteration process converges toward the solution of equation
(34), hence toward the xed point on the rst bisector, if the slope of the functionat
that point is included between1and+1, that is if:
1 ()+1 (35)
whereis the derivative of the function. The xed point of the applicationis then
said to be stable. On the other hand, if that slope is outside the interval[1+1], the
xed point on the rst bisector becomes unstable; the iteration method forno longer
converges.
We can also introduce the second iterated function
(2)
() =[()]. Any
xed point ofis necessarily a xed point of
(2)
. The inverse is not true, as it is
possible to get a two-order cycle where two dierent values ofare swapped under the
eect of:
2=(1)
1=(2)
(36)
In such a case,1and2are both xed points of
(2)
, but not of(we shall see below an
example of such a situation, illustrated by Figure). These xed points can be stable for
(2)
, in which case they constitute a stable cycle of order two for the initial function.
After a certain number of iterations of, the solution converges toward a series taking
alternatively two distinct values,1and2.
The process may repeat itself: it is possible for the xed point of
(2)
to next
become unstable, and yield xed points for an iterated function of a higher order, and
hence to a stable cycle of that order.
. Form of the function
(1)
Relation (25) shows that the variations of
32()as a function ofare very similar
to the variations of()as a function of, already studied in ComplementXV
(Figure). The equality (30) shows that the plot of
(1)
()can be deduced from that
of
32():
reversing the variable(symmetry with respect to the vertical axis)
multiplying this variable by
(scale change of the abscissa )
and nally shifting to the right the abscissa origin by the value.
This leads to the solid line curve in Figure, that plots a constantly decreasing function
(for xed values of,and
).
1740

APPLICATIONS OF THE MEAN FIELD METHOD FOR NON-ZERO TEMPERATURE
As the parameter
changes (forandconstant), we get a set of curves repre-
senting
(1)
()for each value of
. For= 0, all these curves go through the same point
of ordinate
32(). If
= 0, the curve is a simple horizontal line going through this
point. When
becomes slightly positive, the curve starts decreasing but still extends
along the abscissa axis; it has a small negative slope at the origin. As
increases more
and more, the curve contracts more and more toward the ordinate axis and its slope at
the origin is more and more negative. In the limit where
goes to innity, the curve
becomes a straight vertical line.
. Form of the function
(2)
Figure
(1)
to
(2)
. For
a given value of, we start from point1of ordinate
(1)
(), and draw a horizontal
line until it crosses the rst bisector in2, which transfers the ordinate of1onto
the abscissa. From the intersection point2we draw a vertical line that intersects the
function
(1)
at point3of ordinate
(2)
(), which we simply transfer to the initial
abscissato get the nal point (surrounded by a triangle).
This construction shows that
(2)
()is an increasing function ofconned be-
tween two horizontal asymptotes: the abscissa axis, and a horizontal line of ordinate
32(). The larger the value of
, the faster the increase of
(2)
().
. Inuence of the coupling parameter on the xed points
We now discuss the inuence of the parameter
0on the stability of single or
multiple xed points.
(i) The trivial case where
0=
= 0(no interaction between the fermions) is
particularly simple: the two curves are now identical horizontal lines, whose zero slope
makes their intersection with the rst bisector obviously stable. We then get the ideal
gas results, with equal+and densities.
(ii) As long as
(hence
0) is weak enough, the slope of
(1)
at the intersection
point is small and the corresponding xed point remains stable, as shown in Figure.
This same point is obviously a xed point for
(2)
as well; as the derivative of a function
[()]with respect tois[()](), that is[()]
2
at a point where=(),
the slope of
(2)
is less than1, and that xed point is also stable.
In such a case, both functions have only one common xed point, which determines
the only solution of the equations: the numbers+and are necessarily equal since
they correspond to a xed point of
(1)
. The only eect of the fermion repulsion is to
lower in an equal way the densities associated with each of the spin states.
(iii) If now
(or
0) gets larger, we come to a situation, for a certain critical value
of, where the slope of
(1)
at the intersection with the rst bisector becomes equal
to1, and that of
(2)
now takes the value+1. The corresponding critical situation is
plotted in Figure, where the curve representing the function
(2)
is now tangent to the
rst bisector at their intersection (even osculating
3
to it, as their contact is of order two).
For both functions, the xed point is now right at the border of its stability domain.
3
The rst derivative of the function[()]is equal to() [()], and its second derivative, to
()[()]+[()]
2
[()]. At a xed point, that second derivative becomes()() [1 +()],
which cancels out when() =1.
1741

COMPLEMENT H XV
Figure 1: Plots of the functions
(1)
(solid line) and
(2)
(dashed line) as a function of.
Starting from any initial value of the variable, we place a point1on the rst iterated
(1)
curve, whose ordinate we transfer on theaxis by using the rst bisector (point
2); a new vertical intersection with the solid line (point3) yields the second iterated
value
(2)
, that must be simply transferred to the initial value of the variable(nal
point surrounded by a triangle). The whole dashed line curve can thus be constructed
point by point. This method shows that when , that curve is asymptotic to
the abscissa axis; when+, the curve now has another horizontal asymptote, of
ordinate
(1)
(0) =
32(), represented by a line with smaller dashes. The general form
of the second iterated function is plotted in the gure: a uniformly increasing function
between those two asymptotic values. In the case represented here, the coupling constant
is supposed to be weak enough for the two curves to intersect the rst bisector with
slopes of moduli less than1; we then get a unique stable solution whereand+are
equal, which corresponds to a non-polarized spin system.(iv) Beyond that situation, as shown in Figure, the curve representing
(2)
intersects the rst bisector in three points; the middle one is unstable as it corresponds to
a slope larger than1, but the two points on each sides are stable since they are associated
with slopes between1and+1. As far as the central point is concerned, the slightest
perturbation moves the iteration away from that point. On the other hand, the other
two points are xed stable points for
(2)
; they correspond to a physically acceptable
solution of equations (29). As those two points are not xed for
(1)
, two dierent values,
+and, are swapped under the action of the function
(1)
(two-cycle xed points,
represented by the arrows in the gure). We get a solution of the equations where the
spin state populations are dierent: the gas develops a spontaneous polarization when
the repulsion goes beyond a certain critical value and a phase transition occurs.
Comment:
For convenience, we discussed the emergence of the spontaneous polarization as a function
of the parameter
0, for xed values ofand: the plot of the curves is then simply
1742

APPLICATIONS OF THE MEAN FIELD METHOD FOR NON-ZERO TEMPERATURE
Figure 2: Plots of the functions
(1)
(solid line) and
(2)
(dashed line) in the critical
case=, where the function
(1)
intersects the rst bisector with a slope equal to
1; the slope of
(2)
is then equal to+1, and this function is not only tangent but also
osculating to the rst bisector (in other words it intersects this bisector at three points
grouped together).
obtained by a scale change along theaxis. In general, however, the phase transition is
observed by changing either the physical system density, or its temperature, while keeping
the interactions constant. Our line of reasoning can be applied to this case, keeping the
interactions constant while changing either the chemical potentialthat controls the
particle density, or, which is the inverse of the temperature. When either of these
two parameters gets larger, the ordinate at the origin of32()gets higher, which
increases the absolute values of the slopes of
(1)
and
(2)
; the same phenomenon as
above (instability and phase transition) will thus occur when the temperature is lowered
or the density increased. If32()1, relation (25) shows that the particle number
contained in a volume()
3
is large compared to one, which means that the average
distance between the particles is smaller than the thermal wavelength; the fermion gas
is then degenerate.
3-c. Physical discussion
As the spins carry a magnetic moment, a spontaneous polarization implies a tran-
sition towards a ferromagnetic phase. The origin of this phenomenon comes from an
equilibrium between two opposite tendencies. On one hand, the motor of the tran-
sition comes from the fact that, to minimize the repulsion energy, the system tends to
put all its particles in the same spin state (polarized system), which prevents them from
interacting. This is because the Pauli principle forbids them to be at the same point in
space, and as we assumed their interaction potential to be of zero-range, they can no
longer interact. On the other hand, the system polarization (for a xed total density) in-
creases its kinetic energy: the same number of particles must be placed in a single Fermi
sphere, instead of two, which results in a sphere with a larger radius, i.e. a higher Fermi
level. It also changes the system entropy. The compromise between gain and loss (for
1743

COMPLEMENT H XV
Figure 3: Beyond the critical point, the function
(1)
intersects the rst bisector with a
slope less than1, and there are now three distinct intersection points of the function
(2)
and the bisector. The middle point corresponding to an
(2)
slope larger than1is
unstable, but the other two points (surrounded by circles) are stable. These two points
are swapped under the action of
(1)
(two-cycle xed points, represented by the arrows).
They yield dierent values for the spin densities, which leads to the appearance of a
spontaneous spin polarization.
the grand potential) varies as a function of the parameters; when those parameters take
a value where gain and loss balance each other perfectly, a spontaneous ferromagnetic
transition occurs.
A more detailed study is possible; examining the shapes of the curves we plotted,
we deduce that the conditions that favor the transition are: strong repulsion, high density,
low temperature. It is worth noting that no Hamiltonian acting on the spins comes into
play in this phase transition. Even though the interactions are totally independent of the
spin, the Fermi-Dirac statistics has an eect on the spins, and can induce a transition
polarizing those spins.
At the critical point (Figure), the two new stable points appear at the same
place, and move away from each other in a continuous way. The phase transition is
therefore continuous, which puts it into the category of second order phase transitions.
The study of critical transitions is a very large domain of physics that we cannot discuss
here in a general way. We can, however, take the analysis a little further, without too
much diculty: we note, from the equations written above, that the distance between
the two stable points increases, beyond the critical point where= and=,
as the square root of the dierence (or ). In other words, the system
spontaneous magnetization varies as the square root of the distance to the critical point,
which is typical of the so-called Hopf bifurcation. In addition, at the critical point, the
magnetic susceptibility of the spin system diverges.
Comments:
(i) A very general concept plays a role here: spontaneous symmetry breaking. The
rst symmetry breaking concerns the two opposed directions along the quantization axis
1744

APPLICATIONS OF THE MEAN FIELD METHOD FOR NON-ZERO TEMPERATURE
. Equations (29) are invariant upon a permutation of+and; for any solution of
these equations, there exists another one where these two variables are interchanged, and
where the spin magnetization points in the opposite direction. This was to be expected
since nothing physically distinguishes those two directions. The symmetry is said to be
broken if the stable solutions of the set of equations are asymmetric, corresponding to
dierent values of+and; there are then necessarily two (or more) distinct solutions,
symmetric to one another.
Furthermore, the quantization axiswe used is arbitrary; had we chosen a dierent
direction, we would have found that the spontaneous magnetization could point in any
spatial direction. This again was to be expected since our problem is rotation invariant.
The ferromagnetic transition phenomenon we have just studied corresponds to a spon-
taneous breaking of the rotational symmetry of the usual space, often called, in terms
of group symmetry, (3)symmetry breaking. There are many other second order
transitions that break various symmetry groups, as for example the symmetry(1)for
the superuid transition, etc.
(ii) A mean eld theory like the one we used i.e. an approximate theory may identify
the existence of a critical transition (second order transition) as explained above, but
does not allow an exhaustive study of all its aspects, in particular in the vicinity of the
critical point. Several critical phenomena (large wavelength critical uctuations for ex-
ample) cannot be accounted for with such an approximation, and require more elaborate
theoretical methods.
4. Bosons: equation of state, attractive instability
For bosons, the equations (3) are very similar to those we used for fermions, except
for a change of sign of, and hence of the exchange potential. For a barely degener-
ate system, this modies the interaction eects, but does not drastically change their
consequences. On the other hand, for a system of degenerate bosons, the situation is rad-
ically dierent since expression (1) presents a singularity when( )is zero whereas
none occurs for fermions. As pointed out in ComplementXV, this is the origin of the
Bose-Einstein condensation phenomenon: as the chemical potentialincreases, the
singularity becomes signicant whengets close (through lower values) to the lowest in-
dividual energy among the, that is close to the ground level energy0. The population
of this level then increases more and more and can become extensive (proportional to
the system volumein the limit of large volumes).
Actually, using the Hartree-Fock equations for condensed boson systems leads to
some diculties, which will be briey discussed below see Comment (ii) of . We
shall limit ourselves to the study of non-condensed systems, not excluding the possibility
that they approach condensation. We assume the bosons are without spin, and, as we
did for fermions, that the range of the interaction potential2(s)appearing in relation
(11) is short enough so that:
(kk)=0 (37)
In that case, the direct and exchange contributions in (13) are equal. For a homogeneous
system, this equation then becomes:
k=+
2
0
3
k
(k ) =+
2
0
3
(38)
1745

COMPLEMENT H XV
where is the average total number of particles:
=
k
(k ) =
k
(k ) (39)
with:
=
2
0
3
(40)
We therefore nd that the average total number of particles is the same as for a boson
gas without interactions, provided the chemical potentialis replaced by an eective
chemical potential=+ . The same holds true for the average population of each
individual statek.
As in ComplementXV, we note ()the function yielding the particle
number for an ideal gas of bosons:
() =
k
(k ) =
(2)
3
d
3 1
( )
1
(41)
(the second equality is valid for large volumes). Equation (39) then becomes:
= () (+ ) (42)
4-a. Repulsive bosons
For repulsive interactions, Figure
metric construction, the system density predicted by equation (42). For a given chemical
potential, the particle number decreases because of the repulsion, which takes the sys-
tem further away from condensation; consequently, its description by the Hartree-Fock
equations is a good approximation.
To rst order in
0, relation (42) may be approximated by:
() +
()
= ()2
0
()
3
() (43)
Noting()the grand potential, relation (62) of Appendix
=
1
ln=() (44)
Integrating overrelation (43) from to the value, we get the grand potential:
() = () +
0
3
()
2
(45)
where()is the grand potential for the ideal gas, at the same temperature and
chemical potential. In addition, relation (62) of Appendix
potential is equal to minus the product of the volume and the pressure:
() = (46)
1746

APPLICATIONS OF THE MEAN FIELD METHOD FOR NON-ZERO TEMPERATURE
Figure 4: Geometric solutions of equations (40) and (42) for a gas of repulsive bosons.
On the graph plotting the total number of particles as a function of the chemical potential,
we draw a line starting from the point on the abscissa axis with chemical potential, and
having the slope
3
2
0. It intersects the curve at a point whose abscissa isand
ordinate . As
0is positive for a repulsive gas, we see that the interactions lower
the density (at constant temperature and chemical potential). The text explains how this
geometric construction yields the equation of state for the interacting gas.
This means that if, at constant, we vary the parameterin (43) and (45), we obtain in
the plane (,) a curve representing the pressure as a function of the particle number
in the volume, which is an isothermal line of the equation of state. Repeating this plot
for several values of, we get a set of curves covering the whole equation of state, taking
into account the changes introduced by the interactions.
Comments:
(i) To keep the computations as simple as possible, we limited ourselves to the rst order
in
0. It is however possible to make the graphical construction of Fig.
by including the higher order terms.
(ii) We discussed in of ComplementXVthe limits of the Hartree-Fock approxi-
mation for bosons, which can no longer be used when the physical system gets too close
to Bose-Einstein condensation. The graphical construction shown in Figure
physical meaning if the intersection point on the curve is too close to the contact point
of the curve with the vertical axis.
4-b. Attractive bosons
Attractive interactions (
00) result in an increase of the eective chemical
potential, and consequently raises the value of. This in turn increases the eective
chemical potential, and this positive feedback may even induce an avalanche eect leading
to an instability ifis too close to zero.
1747

COMPLEMENT H XV
Figure 5: Graphical construction similar to that of Figure, but for an attractive boson
gas (where
0is negative). When the attractive potential0is not too large, the line
noted1in the gure yields two possible solutions, only one of which is close to the
solution in the absence of interactions, and hence suitable for our approximation. As
0
increases, for a certain critical value we only get one solution (tangent line2), then none
(line3). In this last case, no solution signals an instability of the gas, which collapses
onto itself because of the attractive interactions. Starting from a nearly condensed ideal
gas, the closer it is to condensation, the weaker the attractive interactions necessary to
trigger the instability.
The geometrical construction that yieldsand from the intersection of a
straight line with a curve is shown on Fig.. If
0is weak enough, and for a xed
value of, we get two intersection points, corresponding to possible solutions. We only
keep the rst one, yielding the lowest value of. The other point yields a high value
of, which changes radically and increases considerably the system density; in that
case, chances are the approximate mean eld treatment of the interactions is no longer
valid. Beyond the value of
0for which the straight line becomes tangent to the curve,
the couple of equations (39) and (40) do not have a solution: there no longer exists any
stable solution.
Figure
of the attractive interactions between bosons becomes more and more important; weak
interactions are enough to render the system unstable. The reason we did not nd any
solution to the equations is that we assumed, in the computations, that the system was
perfectly homogeneous; now this homogeneity cannot be maintained beyond a certain
attraction intensity. We must therefore enlarge the theoretical framework, and include
the possibility for the system to become spontaneously inhomogeneous. A more pre-
cise study would show that the system may develop local instabilities, hence breaking
spontaneously the translation invariance symmetry. In the limit of large systems (ther-
modynamic limit), condensed bosons tend to collapse onto themselves under the eect
1748

APPLICATIONS OF THE MEAN FIELD METHOD FOR NON-ZERO TEMPERATURE
of an attractive interaction, however weak it may be
4
.
As a general conclusion, the Hartree-Fock method applied to fermions yields results
valid in a very large parameter range. As an example, it allowed computing eects of
the interactions on the particle number and the pressure of the system. In addition,
this method was able to predict the existence of phase transitions. This is also true for
non-degenerate bosons, and the mean eld method actually has a very large number of
applications that we cannot detail here. We must, however, keep in mind that when
Bose-Einstein condensation occurs, certain predictions pertaining to the condensate may
not be realistic from a physical point of view, as they depend too closely on the mean eld
approximation which does not properly account for the correlations between particles.
4
If the interaction potential is attractive at large distance, but strongly repulsive at short range (hard
core for example), the system spontaneously forms a high density liquid or solid.
1749

Chapter XVI
Field operator
A Denition of the eld operator
A-1 Denition
A-2 Commutation and anticommutation relations
B Symmetric operators
B-1 General expression
B-2 Simple examples
B-3 Field spatial correlation functions
B-4 Hamiltonian operator
C Time evolution of the eld operator (Heisenberg picture)
C-1 Contribution of the kinetic energy
C-2 Contribution of the potential energy
C-3 Contribution of the interaction energy
C-4 Global evolution
D Relation to eld quantization
Introduction
This chapter is a continuation of the previous chapter and uses the same mathematical
tools. The main dierence is that, until now, we have mainly used discrete bases in the
individual state space,or . In this chapter, we shall use a continuous basis,
which is the basis, for spinless particles, of the position eigenvectors (see Chapter II, E).
As they now depend on the positionr, the creation and annihilation operators become
eld operators depending on a continuous subscriptr. They are the operator analog of
the classical elds (which are numbers and not operators), and are often called eld
operators. They are useful for concisely describing numerous properties of identical
particle systems. They have commutation relations for bosons, and anticommutation
relations for fermions. This chapter is a preparation for Chapters , where
we will introduce the quantization of the electromagnetic eld.
Quantum Mechanics, Volume III, First Edition. C. Cohen-Tannoudji, B. Diu, and F. Laloë.
© 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

CHAPTER XVI FIELD OPERATOR
After dening these operators in , we discuss some of their properties. Their
commutation and anticommutation relations are examined in . As in Chapter
XV, we then study () symmetric operators and their expression as a function of the
eld operators; special attention will be given to the operators associated with the eld
correlation functions. In , we shall use the Heisenberg picture to study the time
dependence of these operators. As a conclusion, we shall briey come back (in ) to
the eld quantization procedure and its link to the concept of identical particles.
A. Denition of the eld operator
The eld operator is dened as the annihilation operator, but associated with a basis
of individual states labeled by the continuous position indexrinstead of a discrete index
. Our starting point will be the basis change relation (A-52) of Chapter:
= (A-1)
where the subscriptsandlabel the kets of the two orthonormal basesand
in the individual state space. In what follows, and as already done in Chapter, we
will simplify the notation and often replace the subscriptby, and the subscript
by.
A-1. Denition
We will rst dene the eld operator for spinless particles, and later generalize it
to the case with spin.
A-1-a. Spinless particles
We replace, in relation (A-1), the basis by the basis of vectorsr, where
rsymbolizes three continuous indices (the three vector components). The operator
now becomes an operator depending on the continuous indexr, that we shall call eld
operator for the system of identical particles we consider. We could write it simply as
r, but we shall prefer the commonly used classical notation (r). Like any annihilation
operator, (r)acts in the Fock space where it lowers by one unit the particle number.
In (A-1), the coecient appearing in the sum is now the wave function(r)associated
with the ket:
(r) =r (A-2)
and this relation becomes:
(r) = (r) (A-3)
Formally, denition (A-3) looks like the expansion of a wave function on the basis func-
tions(r); here, however, the componentsare operators and no longer simple
complex numbers. In the same way asannihilates a particle in the state, the
operator (r)now annihilates a particle at pointr.
1752

A. DEFINITION OF THE FIELD OPERATOR
It does not depend on the basis, i.e. on the wave functions chosen to dene it
in (A-3), as we now show. We insert in this equality the closure relation in any arbitrarily
chosen basis , and use again (A-1):
(r) = r = r (A-4)
(we temporarily came back to the explicit notation for the annihilation operators). We
can thus write:
(r) = (r) (A-5)
which means (r)satises, in the new basis, a relation similar to (A-3).
We now take the Hermitian conjugate of (A-3):
(r) = (r) (A-6)
The operator (r)creates a particle at pointr, as can be shown for example by com-
puting the ket resulting from its action on the vacuum:
(r)0= (r)0= (r) (A-7)
that is:
(r)0= r=r (A-8)
which represents, as announced, a particle localized at pointr.
One can easily invert formulas (A-3) and (A-6) by writing, for example:
d
3
(r) (r) =d
3
(r) (r)
= = (A-9)
or else, by Hermitian conjugation:
d
3
(r) (r) = (A-10)
A-1-b. Particles with spin
When the particles have a spin, the basis vectorrused above must be
replaced by the basis vectorr, whereis the spin index, which can take2+ 1
discrete values (= ,+ 1+). To all the summations over d
3
, we must now
add a summation on the2+ 1values of the spin index. As an example, a basis vector
in the individual state space must now be written:
=
+
=
d
3
(r)r (A-11)
1753

CHAPTER XVI FIELD OPERATOR
with:
(r) =r (A-12)
The variablesrandplay a similar role. The rst one is, however, continuous, whereas
the second is discrete. Writing them in the same parenthesis might hide this dierence
and one often prefers noting the discrete index as a superscript of the function, writing
for example:
(r) =r (A-13)
Let us again use relation (A-1). On the left-hand side, the indexnow symbolizes
both the positionrand the spin quantum number, which leads us to dene a eld
operator (r)having2+ 1spin components. Inserting (A-13) in the right-hand side
of (A-1), we get:
(r) = (r) (A-14)
The Hermitian conjugate operator (r)now creates a particle at pointrwith a spin:
(r)0= r =r (A-15)
As we did above, we can invert those relations. In relation (A.51) of Chapter
(basis change), we replaceby, andbyr(which means that the summation over
is replaced by an integral over d
3
and a summation over), and use equality (A-11);
we therefore get:
=
+
=
d
3
(r) (r) (A-16)
which is the analog, in the presence of spin, of relation (A-10).
A-2. Commutation and anticommutation relations
Commutation relations for eld operators are analogous to those obtained in
of Chapter, but the discrete indexis now replaced by a continuous index.
A-2-a. Spinless particles
The commutator (or anticommutator) of two eld operators:
[ (r) (r)]= (r)(r) []= 0 (A-17)
is indeed zero, as expected from expression (A-48) of Chapter. In the same way, by
Hermitian conjugation:
(r) (r)= (r)(r) = 0 (A-18)
1754

B. SYMMETRIC OPERATORS
However, when we (anti)commute the eld operator and the adjoint operator, we get:
(r) (r)= (r)(r) (A-19)
which yields, taking into account the commutation relations (A-49) of Chapter:
(r) (r)= (r)(r) = r r=rr (A-20)
Finally, we get:
(r) (r)=(rr) (A-21)
which is the equivalent of the relations (A-49) of Chapter
basis.
A-2-b. Particles with spin
Relation (A-17) now becomes:
[ (r) (r)]= (r)(r) []= 0 (A-22)
Relation (A-18) remains valid even if the eld operators have spin indices. Finally,
relation (A-19) becomes:
(r) (r)= (r)(r)
=rr (A-23)
that is:
(r) (r)= (rr) (A-24)
B. Symmetric operators
In the previous chapter, we wrote one- or two-particle symmetric operators in terms
of creation and annihilation operators in the discrete states. We are now going to
express those operators in terms of the eld operator (and its Hermitian conjugate).
B-1. General expression
We start with spinless particles. We can either directly transpose expressions
(B-12) and (C-16) of Chapter r(replacing the sums by
integrals), or insert in those expressions the form (A-9) for the operators. In both
cases, we get:
=d
3
d
3
rr (r) (r) (B-1)
1755

CHAPTER XVI FIELD OPERATOR
and:
=
1
2
d
3
d
3
d
3
d
3
1 :r; 2 :r1 :r; 2 :r (r) (r) (r) (r) (B-2)
where, as in relation (C-16) of Chapter, the order of the annihilation operators is the
inverse of the order that appears in the ket of the matrix element.
Expression (B-1) reminds us of the average value(1)of the operator(1)for
a single particle (without spin), described by the wave function(r1):
(1)=d
3
1d
3
1r1r
1 (r1)(r
1) (B-3)
Both expressions are not equivalent since (B-1) concerns any number of identical particles,
rather than a single one; furthermore, the are now operators, and their respective order
matters as opposed to the order of thein (B-3). As for formula (B-2), it can be
compared to the average value of an operator(12)acting on two particles,1and2,
both described by the same wave function. Here again, the order of the eld operators
is important, as opposed to the order in a product of wave functions.
For particles with spins, we simply complement each integral overrwith a sum
over the spin index, we include this index in the matrix elements and add a spin index
to the eld operator. As an example, relation (B-1) is generalized to:
=d
3
d
3
= =
r r (r) (r) (B-4)
As for relation (B-2), we get four summations over the spin indices, and the operator
matrix elements are taken between bras and kets where an indexis added to the variable
r.
B-2. Simple examples
We start with a few examples concerning operators for one spinless particle. For
a single particle, the operator associated with the local density at pointr0is:
r0r0 (B-5)
and its matrix elements are:
rr0r0r=(rr0)(rr0) (B-6)
The corresponding-particle operator is written as:
()
(r0) =
=1
:r0:r0 (B-7)
Replacingbyr0r0in (B-1) yields the operator acting in the Fock space:
(r0) = (r0) (r0) (B-8)
1756

B. SYMMETRIC OPERATORS
This operator annihilates a particle at pointr0, and immediately recreates it at the same
point. The average value:
(r0) =(r0) (B-9)
in a normalized stateof the-particle system yields the particle density associated
with this state at pointr0.
The operator, total number of particles, has been written in (B-15) of Chap-
ter. As the discrete summation indexis changed to a continuous indexr, the
summation becomes an integral over all space:
=d
3
(r) (r) (B-10)
As expected, it is the integral over d
3
of the operator(r).
The operator1(r)describing the one-particle potential energy is also diagonal in
the position representation; in the Fock space, it becomes the operator1:
1=d
3
1(r) (r) (r) (B-11)
As for the particle current, it can be deduced from the expression of the currentj(r0)as-
sociated with a single particle of mass; it is the product of the operators giving the local
densityr0r0and the velocityp(product that must obviously be symmetrized):
j(r0) =
1
2
r0r0
p
+
p
r0r0 (B-12)
If the particle is described by a wave function(r), a simple calculation
1
shows that the
average value of this operator yields the usual expression for the probability current
see equation (D-17) of Chapter III:
J(r0) =
}
2
[(r0)r(r0)(r0)r(r0)] (B-14)
The currentJ(r0)of a system of identical particles is obtained by replacing in (B-1) the
operatorbyj(r0):
J(r0) =
}
2
(r0)r (r0) (r0)r (r0) (B-15)
Another way of arriving at this equality is to use the substitution process mentioned in
. To obtain the operator we are looking for in terms of (r), we start from the
1
Let us calculate the average value j(r0) . The rst term on the right-hand side of (B-12)
yields:
1
2
r0r0p =
}
2
(r0)r(r0) (B-13)
since the action of the operatorpin the position representation is given by(})r. The second term is
its complex conjugate, and we therefore get (B-14).
1757

CHAPTER XVI FIELD OPERATOR
expression of the average value for a single particle, described by the wave function(r),
which is then replaced by the eld operator (r).
For particles with spin, the local density at pointr0, with spin, is written in the
same way:
()
(r0) =
=1
:r0 :r0 (B-16)
and yields, in the Fock space, the operator:
(r0) = (r0) (r0) (B-17)
The total density is obtained by a summation over:
(r0) =
=
(r0) (r0) (B-18)
For particles with spin, the operatorassociated with the total particle number, or the
operator probability currentJr0, can be obtained in a similar way.
B-3. Field spatial correlation functions
Operators associated with spatial correlation functions can also be dened with
eld operators; their average values are very useful for characterizing the eld properties
at dierent points of space. When we reason in terms of elds, we generally characterize
each correlation function by the number of points concerned, which is dierent from the
number of particles involved: the two-point functions characterize the properties of a
single particle, the four-point ones concern two particles, etc. The reason is simple: a
one-particle density operatoris characterized by non-diagonal elementsr
0r0de-
pending on two positions, a two-particle density operatorinvolves elements depending
on four positions, etc.
B-3-a. Two-point correlation functions
Dening a non-diagonal operator depending on two parametersr0andr
0, we can
generalize relation (B-5):
r0r
0 (B-19)
A calculation very similar to the one leading to equation (B-7) we simply add a prime
to the secondr0 gives the-particle symmetric operator:
()
(r0r
0) =
=1
:r0:r
0 (B-20)
which yields, in the Fock space, the operator:
(r0r
0) = (r0) (r
0) (B-21)
1758

B. SYMMETRIC OPERATORS
We thus obtain an operator that annihilates a particle at pointr
0and recreates it at a
dierent pointr0.
When the-particle system is described by a quantum state, we calltwo-point
eld correlation function,1(r0r
0), the average value:
1(r0r
0) = (r0) (r
0)= (r0) (r
0) (B-22)
which also yields the matrix element, in therrepresentation, of the one-particle
operator
2
:
(r0) (r
0)=r
0r0 (B-23)
Demonstration:
The matrix elements of the density operator
()
of particleare:
r
0
()
r0=Tr
()
:r0 :r
0= :r0 :r
0 (B-24)
For an-particle system, we dene the one-particle density operatorby a sum over
all the particles:
=
=1
()
(B-25)
(be careful: the trace of this operator is, not1). Its matrix elements are the sum of
the average values written in (B-24), i.e. the average value of the one-particle symmetric
operator obtained by summing overthe:r0:r
0. This result is simply the op-
erator
()
(r0r
0)of (B-20), which, as seen above, yields in the Fock space expression
(B-21). We then simply take the average value of each side of this expression to get
equality (B-23).
The average value (B-22) at two dierent points plays an important role in the
study of Bose-Einstein condensation. For a system at thermal equilibrium, this average
value generally tends to zero rapidly as the distance betweenrandrincreases; it only
remains non-zero in a domain of microscopic size. However, for a Bose-Einstein condensed
gas, this average value behaves quite dierently as it tends toward a non-zero value
at large distance. This dierence is actually the Penrose and Onsager condensation
criterion ; they have dened the existence of such a condensation as the appearance of
a non-zero value of the matrix element ofat large distance; this denition is quite
general as it applies not only to the ideal gas but also to systems of interacting particles.
Particles with spin:
If the particles have a non-zero spin, we use, as a basis, the ketsrwheretakes
the(2+ 1)values,+ 1, ..,+, and we add aindex to the eld operators. We
then dene(2+ 1)
2
two-point eld correlation functions as the average values:
(r0) (r
0)= 1(r0;r
0) (B-26)
2
Note the inversion of the variable order between the function1(or the variables of and )
and the matrix element of.
1759

CHAPTER XVI FIELD OPERATOR
The same computation that led to (B-23) for spinless particles, can be repeated with no
changes other than the simple replacement of the kets (or bras)rbyr; it shows
that these average values yield the matrix elements of the one-particle density operator:
r
0 r0= (r0) (r
0) (B-27)
(here again we have chosen to normalize tothe trace of the one-particle density oper-
ator).
B-3-b. Higher order correlation functions
One can also start with the two-particle operator, which now depends on four
positions:
(1 :r0r
0; 2 :r
0r
0) =1 :r01 :r
02 :r
02 :r
0 (B-28)
In this case, the expression ofis not symmetric with respect to the exchange of particles
1and2, as opposed to what happens for an interaction energy. The operatoris then
dened without the12factor of relation (C-1) of Chapter:
()
(r0r
0r
0r
0) =
=1;=
(:r0r
0;:r
0r
0) (B-29)
and yields in the Fock space the operator(r0r
0r
0r
0). Relation (B-2), without this
factor12, then leads to:
(r0r
0r
0r
0) = (r0) (r
0) (r
0) (r
0) (B-30)
In this case, the operator annihilates two particles at two points and recreates them at
two others.
A computation very similar to the one leading to (B-22) and (B-23) enables us to
show, using (B-2), that the matrix elements of the two-particle density operatorcan
be written
3
:
1 :r
0; 2 :r
0 1 :r0; 2 :r
0= (r0) (r
0) (r
0) (r
0) (B-31)
This density operator, whose trace is equal to( 1), plays an essential role in the
study of correlations between particles.
A particularly important example of a higher order correlation function corre-
sponds to the case wherer0=r
0andr
0=r
0. We then get:
()
(r0r0r
0r
0) =
=1;=
:r0:r0 :r
0 :r
0
=
=1;=
:r0;:r
0:r0;:r
0 (B-32)
3
One can also use relation (C-19) of Chapter
1760

B. SYMMETRIC OPERATORS
which yields in the Fock space the operator:
(r0r0r
0r
0) = (r0) (r
0) (r
0) (r0) (B-33)
The expression on the right-hand side of (B-32) characterizes the probability of
nding any particle atr0and any other one atr
0. In the same way as the average value
(B-9) gives the one-particle density, the average value:
(r0r0r
0r
0)=(r0r0r
0r
0)=2(r0r
0) (B-34)
gives the two-particle double density, which contains information on all the binary
correlations between the particle positions.
We are now in a position to again obtain expression (C-28) of Chapter, and
more precisely justify the interpretation we gave of the average value of the interaction
energy written in (C-27) of that chapter. We replace in (B-33) the eld operators (or
their adjoints) by their expansion (A-3) on the operators(or the); we then get
(C-28) of Chapter,r0being replaced byr1andr
0byr2.
Ifr0=r
0, we can check
4
that this operator is equal to the product of the simple
densities dened in (B-8):
(r0r0r
0r
0) =(r0)(r
0) (B-35)
Obviously this relation between operators does not mean that the double density2(r0r
0)
is merely the product(r0) (r
0)of the simple densities: the average value of a
product of operators is not, in general, equal to the product of the average values. When
we studied the function2( ), we did nd the presence of an
exchange term that introduces statistical correlations between particles, even in the
absence of interactions.
Particles with spin:
For particles with non-zero spin, we just have to add an indexto each of the kets or
bras, as well as to the eld operators; this brings up to(2+ 1)
4
the number of4-point
correlation functions. The matrix elements of the two-body density operator are then
given by the average values:
1 :r; 2 :r 1 :r; 2 :r = (r) (r) (r) (r)
(B-36)
B-4. Hamiltonian operator
We now establish the expression, in terms of the eld operator, of the Hamiltonian
operator for a system of identical (spinless) particles. Two formulas will be useful for this
computation. The rst one transposes to three dimensions the formula (34) of Appendix
II:
(rr) =
1
(2)
3
d
3 k(rr)
(B-37)
4
If the particles are bosons, we just permute the commuting operators to bring (r0)to the second
position and obtain the result. If we are dealing with fermions, two successive anticommutations are
necessary to get that result, and the corresponding two minus signs cancel each other.
1761

CHAPTER XVI FIELD OPERATOR
The second one is obtained from (B-37) by a double derivation with respect tor:
(rr) =
1
(2)
3
d
3 2k(rr)
(B-38)
The matrix elements of a single particle's kinetic energy are written as:
r
2
2
r=d
3
rk
}
22
2
kr=
}
2
2
(rr) (B-39)
In the Fock space, it corresponds to the following operators (an integration by parts
5
was used to go from the rst to the second relation):
0=
}
2
2
d
3
(r) (r) =
}
2
2
d
3
r (r)r (r) (B-40)
As in , we obtain here an expression similar to the average value of an operator
(here the kinetic energy) for one particle; but the gradient of the wave function must be
replaced by that of a eld operator, and the order of the operators can matter.
The system Hamiltonian includes in general an interaction term, which makes it
a two-particle operator and requires using formula (B-2). For a two-particle system, we
know that the interaction yields an operator that is diagonal in therrrepresenta-
tion; furthermore, it only depends on the relative positionrr(and not onrandr
separately). Consequently, the matrix element in (B-2) takes the form:
rrrr=(rr)(rr) 2(rr) (B-41)
where2(rr)is the interaction potential energy between two particles located at a
relative positionrr(this interaction is often isotropic, in which case2only depends
on the relative distancerr). Finally, starting from (B-2), we get the following
expression for the Hamiltonian operator:
=d
3}
2
2
r (r)r (r) +1(r) (r) (r)
+
1
2
d
3
d
3
2(rr) (r) (r) (r) (r)
(B-42)
The rst term corresponds to the particles' kinetic energy, the second to the external
potential1(r)acting separately on each particle, and the third one to the mutual inter-
action between particles; note that this last term involves four eld operators, whereas
the rst two involve only two. The same comment as above still applies: this expression
is reminiscent of the average energy of a system of two particles, both described by the
same wave function; but now we are dealing with operators that do not commute.
This Hamiltonian can also be expressed directly via the one-particle simple density
(r0r
0)and the two-particle double density(r0r0r
0r
0)operators, as we now show.
Inserting relations (B-21) and (B-30) in expression (B-42), we obtain:
=d
3}
2
2
rr0
rr
0
(r0r
0)
r0=r
0
=r
+1(r)(rr)
+
1
2
d
3 3
2(rr)2(rrrr)
(B-43)
5
The value of the already integrated terms must be taken at innity, and we assume that all the
states of the physical system are limited to a nite volume; as a result, those terms do not play any role
and can be ignored.
1762

C. TIME EVOLUTION OF THE FIELD OPERATOR (HEISENBERG PICTURE)
where the notationrr0
andrr
0
represents the gradient taken with respect to the vari-
ablesr0(for the rst one), andr
0(for the second); once these gradients have been
computed in the kinetic energy term of (B-43), both variables take on the same valuer.
The fact that the Hamiltonian operator can be directly expressed in terms of the simple
and double density operators can be useful. For example, to determine the ground state
energy of an-particle system, we do not have to compute the state wave function,
which involves all the correlations of order1tobetween the particles; it is sucient
to know the average values of these two densities. There exist, in certain cases, approxi-
mation methods that yield directly good estimates of these simple and double densities,
hence allowing an access to the-body energy. ComplementsXVandXVdiscuss
the Hartree-Fock method, which is based on an approximation where the two-particle
density operator is simply expressed as a function of the one-particle density operator,
i.e. the double density as a function of the simple density (ComplementXV, );
this allows convenient mean eld calculations.
C. Time evolution of the eld operator (Heisenberg picture)
The operators we have considered until now correspond to the Schrödinger picture,
where the time evolution of the system is determined by the time evolution of its state
vector. It may, however, be more convenient to adopt the Heisenberg picture (Com-
plementIII), where this time evolution is transferred to the operators associated with
the system's physical quantities. For spinless particles, let us call (r;)the operator
corresponding, in the Heisenberg picture, to (r):
(r;) =
}
(r)
}
(C-1)
(is the Hamiltonian operator), and whose time dependence follows the equation:
}
(r;) = (r;) (C-2)
We are going to compute successively the commutator of (r;)with each of the three
terms on the right-hand side of (B-42). The evolution equation for the eld operator
involves all the terms of the Hamiltonian (B-42): the kinetic, potential and interaction
energies.
C-1. Contribution of the kinetic energy
In order to determine the commutator of the eld operator with the kinetic energy,
we rst transpose the equations (A-17), (A-18) and (A-21) to the Heisenberg picture.
Actually, they can be used without any changes: the unitary transform of a product
by (C-1) is the product of the unitary transforms, that of the commutator (or of the
anticommutator) is the commutator (or the anticommutator) of the transforms, and
numbers like zero or the function(rr)are invariant. Those three relations are
therefore still valid in the Heisenberg picture, if we simply add an indexto the eld
operators. We now take their derivative with respect to the positions; only (A-21) yields
a non-zero result:
(r;)rr (r;)=rr(rr) =rr(rr) (C-3)
1763

CHAPTER XVI FIELD OPERATOR
Taking (B-40) into account, we can write the commutator to be evaluated as:
(r;)0()=
}
2
2
d
3
(r;)r (r;)r (r;) (C-4)
In the term to be integrated on the right-hand side, a signis introduced each time we
permute two eld operators, or two adjoints; when we permute a eld operator and an
adjoint, we must add to the result the right-hand side of (C-3). Adding and subtracting
two equal terms, we then obtain for the function to be integrated:
(r;)r (r;)r (r;)r (r;)( (r;)r (r;))
+r (r;)( (r;)r (r;))r (r;)r (r;) (r;)
(C-5)
that is:
(r;)r (r;) r (r;)
+r (r;)[ (r;)r (r;)]
=rr(rr)r (r) (C-6)
The integration over d
3
then yields the Laplacian atrof the eld operator, and we
nally get:
(r;) 0()=
}
2
2
(r;) (C-7)
C-2. Contribution of the potential energy
Instead of (C-4), it is now the commutator:
(r;)1()=d
3
1(r) (r;) (r;) (r;) (C-8)
which comes into play. The calculation is similar to the previous one, but without the
gradients which were applied to the eld operators depending onr. The right-hand side
of (C-6) now becomes simply:
(rr) (r) (C-9)
and the integration over d
3
is straightforward, so that:
(r;)1()=1(r) (r;) (C-10)
C-3. Contribution of the interaction energy
It is now the commutator of (r;)with a product of four eld operators that
will have to be integrated:
(r;) (r;) (r;) (r;) (r;) (C-11)
1764

D. RELATION TO FIELD QUANTIZATION
We will not go through the details of the calculation, a bit long but without any real
diculties; as in the previous two cases, it involves the repeated application of the com-
mutation relations. The result is that the commutator of the eld with the interaction
energy can be written as:
d
3
2(rr) (r;) (r;) (r;) (C-12)
C-4. Global evolution
Regrouping the three previous terms, we get the evolution equation for the eld
operator:
}
+
}
2
2
1(r) (r;) =d
3
2(rr) (r;) (r;) (r;)
(C-13)
The left hand-side includes the dierential operator of the usual Schrödinger equation
for a single particle in a potential1(r); however, as already pointed out, is not a
simple function here, but an operator. The right-hand side includes the binary inter-
action eects; its presence implies that the evolution equation of the eld operator is
not closed. Its evolution depends not only on the operator itself, but also on a term
containing the product of three elds (or their conjugates).
Analyzing in a similar way the evolution of such a product of three factors, we see
that it depends on that product, and also on the product of 5 elds (or their conjugates);
in turn, the evolution of a product of 5 elds will involve 7 others, etc. We thus get
a series of more and more complex equations, often called a hierarchyof equations.
They are in general very hard to solve exactly. This is why it is frequent to use an
approximation by truncating this hierarchy at a certain stage, or else by eliminating the
coupling term at a certain order, or replacing it by a more convenient expression. Many
dierent methods have been proposed to accomplish this, the most well-known being the
mean eld approximation (ComplementsXVandXV).
D. Relation to eld quantization
In conclusion, we make some remarks concerning the eld quantization procedures, and
their relation to the identical particle concept. Consider a single spinless particle in an
external potential well, and call(r)the wave functions associated with its stationary
states in that potential (the subscriptruns from1to innity; we assume, for the sake
of simplicity, that the spectrum is entirely discrete). The(r)form a basis on which we
can expand any particle wave function(r):
(r) = (r) (D-1)
We already noted the similarity between this formula and equality (A-3), where the only
dierence is that the numbersare replaced by the operators. Along the same line,
relations (B-8) and (B-15) are reminiscent of a particle's probability density of presence,
and of its probability current. Finally, expression (B-42) is very similar to the average
1765

CHAPTER XVI FIELD OPERATOR
value of the energy of a system of two particles, both placed in the same state described
by the wave function(r), once we replace that usual wave function by an operator
depending on the parameterr. Consequently, the creation and annihilation operator
method has an air of second quantization: we start with the quantum wave function
for one (or two) particle(s) (rst quantization), and in a second stage, we replace the
wave functions coecients by operators (second quantization). However, we must keep
in mind that we do not, in reality, quantize the same physical system twice; the main
dierence comes from the fact that we go from a very small number of particles, one or
two, to a very large number of identical particles.
Field operators can also appear when quantizing a classical eld, such as the elec-
tromagnetic eld. This is the object of Chapters , where we will show
how the concept of a photon emerges, as the elementary excitation of the electromag-
netic eld. We shall also see how the electric and magnetic elds, which were classical
functions, become operators dened at each point in space, creating and annihilating
photons.
Generally speaking, a system consisting of an ensemble of identical bosons and a
system obtained by quantizing a classical eld obey exactly the same equations. The par-
ticles of the rst system play the role of eld quanta for the second system, and the eld
operators then satisfy commutation relations. The two physical systems are therefore
perfectly equivalent. In the case of the electromagnetic eld, the particles in questions
are the photons and they have a zero mass. However, this is not necessarily the case for
all elds. Moreover, quantum elds associated with a system of identical fermions also
exist. These elds do not have a direct classical correspondence and their operators obey
anticommutation relations. In particle physics, one simultaneously takes into account
fermonic and bosonic elds, associated in general with non-zero mass particles.
1766

COMPLEMENTS OF CHAPTER XVI, READER'S GUIDE
AXVI: SPATIAL CORRELATIONS IN A BOSON
OR FERMION IDEAL GAS
In this complement we study the properties of the
spatial correlation functions in systems of fermions
or bosons. For fermions, we establish the existence
of an exchange hole which corresponds to the
impossibility for fermions with parallel spins to be
found at the same point in space. For bosons, we
discuss their tendency tobunch(group together).
Recommended in a rst reading
BXVI : SPATIO-TEMPORAL COORELATION
FUNCTIONS, GREEN'S FUNCTIONS
Green's functions are a very general tool for the
theoretical study of -body systems. In this
complement, they are rst introduced in ordinary
space, then in reciprocal space (the Fourier mo-
mentum space). Knowledge of these functions al-
lows calculation of numerous physical properties
of the system.
Slightly more dicult than the previous comple-
ment
CXVI: WICK'S THEOREM Wick's theorem permits calculating average
values of any product of creation and annihilation
operators, for an ideal gas system in thermal
equilibrium. The calculation involves a very
useful concept, the operator contraction.
1767

SPATIAL CORRELATIONS IN AN IDEAL GAS OF BOSONS OR FERMIONS
Complement AXVI
Spatial correlations in an ideal gas of bosons or fermions
1 System in a Fock state
1-a Two-point correlations
1-b Four-point correlations
2 Fermions in the ground state
2-a Two-point correlations
2-b Correlations between two particles
3 Bosons in a Fock state
3-a Ground state
3-b Fragmented state
3-c Other states
In this complement, we establish a certain number of properties of the correlation
functions, arising solely from the particle statistics (i.e. from the fact they are either
bosons or fermions), and independent of their possible interactions. To keep the cal-
culations simple, we assume that the-particle system is described by a Fock state,
characterized by the occupation numbersof each individual state. We shall see
that fermions and bosons behave very dierently: whereas the latter tend to bunch, the
former tend to avoid each other, as indicated by the existence of an exchange hole.
We give in
any hypothesis concerning the nature of the individual states; the physical system is
not necessarily homogeneous in space. In , we study successively bosons
and fermions, assuming the physical system to be contained in a box of volumein
which the particles are free. The periodic boundary conditions (ComplementXIV)
allow taking into account the connement while maintaining translation invariance (the
system is perfectly homogeneous in space), which makes the calculations easier. For
spinless particles, the individual statescorrespond to plane waves normalized in the
volume:
(r) =
1
kr
(1)
where thekare chosen to satisfy the periodic boundary conditions.
1. System in a Fock state
We assume the stateof the-particle system to be a Fock state built from the basis
of individual states, with occupation numbers(not greater than1for fermions):
= 1:1;2:2;;:; (2)
This will be the case, for example, if the particles do not interact (ideal gas) and if the
system is in a stationary state, such as its ground state.
1769

COMPLEMENT A XVI
1-a. Two-point correlations
For spinless particles, relations (A-3) and (B-21) of Chapter
(rr) = (r)(r) (3)
Now the average value in a Fock state of the operator productis zero if=: the
successive action of the two operators leads to another Fock state with the same particle
number, but with two dierent occupation numbers therefore to an orthogonal state.
If, on the other hand,=, that product becomes the particle number operatorthat
now acts on its eigenket, with eigenvalue. We thus get:
= (4)
(whereis the Kronecker delta); this yields:
1(rr) =(rr)= (r)(r) (5)
The physical interpretation of this result is the following: for a single particle in the indi-
vidual state, the function1would simply be(r)(r); for an-particle system
in a Fock state, each individual state gives a contribution, multiplied by a coecient
equal to its population.
Particles with non-zero spin:
For particles with spin, we dene the(r)by:
(r) =r (6)
When we add a discrete spin indextor, formula (3) becomes:
(r;r) = [(r)](r) (7)
We obtain, with the same reasoning:
1(r;r) =(r;r)= [(r)](r) (8)
whose physical interpretation is similar to the previous one.
One often chooses a basis of individual states, such that each ket corresponds to a
well dened value of the spin: each indexindicates both an individual orbital state and
a value of(which then becomes a functionof). In that case, for a given, the wave
function(r)is only dened for a single value of the index; conversely, for a given,
the wave functions are dierent from zero only if the index(or) belongs to a certain
domain(). In expression (8),andare xed, and the indexmust necessarily
belong to both()and(), or else the result is zero. This leads to:
1(r;r) =
()
[(r)](r) (9)
This correlation function is therefore zero if=.
1770

SPATIAL CORRELATIONS IN AN IDEAL GAS OF BOSONS OR FERMIONS
1-b. Four-point correlations
We limit ourselves to the calculation of(rrrr)in the diagonal case
r=randr=r; we callr1the common value ofrandr, andr2the common
value ofrandr. Such a diagonal correlation function was already written in (B-20)
of Chapter ; this is the only one that plays a role in
the particle interaction energy, as the associated operator is also diagonal in the position
representation.
In the absence of spin, and for a Fock state, the calculation of the correlation
function2(r1r2)was carried out in , where relations (C-32) to
(C-34) yield the value of this function:
2(r1r2) = ( 1)(r1)
2
(r2)
2
+
+
=
(r1)
2
(r2)
2
+ (r1)(r2)(r2)(r1) (10)
where= +1for bosons,=1for fermions. The second line of this equation contains
a direct term only involving the moduli squared of the wave functions; it also contains
an exchange term where the phase of the wave functions come into play, and which
changes sign depending on whether we are dealing with a system of bosons or fermions.
Particles with non-zero spin:
In the presence of spins, we must add, as previously, the corresponding spin indices,
and the correlation function becomes:
2(r11;r22) = ( 1)
1
(r1)
2 2
(r2)
2
+
+
=
1
(r1)
2
2
(r2)
2
+
1
(r1)
2
(r2)
2
(r2
1
(r1)
(11)
If we choose, as in (9), a basis of individual statesin which each ket has a well dened
value of the spin, the summation is simpler and we get:
2(r11;r22) =
12 (1)
( 1)
1
(r1)
2 2
(r2)
2
+
+
(1) (2);=
1
(r1)
2
2
(r2
2
+
+
12
1
(r1)
2
(r2)
2
(r2)
1
(r1)
(12)
As above,()corresponds to the domain of the indexfor the wave function(r)to
exist (otherwise, it is not dened). If the spin states are dierent (1=2), the only
contribution to the correlation function comes from the second line (the direct term).
The exchange term which follows only concerns particles being in the same spin state;
it changes sign depending on whether they are bosons or fermions. No exchange term
exists for particles having orthogonal spins. This comes from the fact that to behave as
strictly identical objects, two particles must occupy the same spin state; otherwise, their
spin direction could, at least in principle, be used to distinguish them.
2. Fermions in the ground state
Consider an ideal gas of fermions, contained in a volume, and having a spinequal
to12(as for electrons); the indexcan only take on two values12. If we assume
1771

COMPLEMENT A XVI
the gas to be in its ground state, this corresponds, for an ideal gas, to a Fock state: for
each of the two spin states, all the individual states with an energy lower than a certain
value (called the Fermi energy) have an occupation number equal to1, all the others
being empty. We shall proceed as in ComplementXIV(in particular for the study of
the magnetic susceptibility) and attribute to each of the two spin states a dierent Fermi
energy this is useful to account for an average spin orientation (under the inuence of a
magnetic eld for example). For all the values of the indexcorresponding to= +12,
we thus assume that the occupation numbers are equal to1if they correspond to plane
waves (1) having a wave vector smaller than the Fermi vector
+
, and zero otherwise.
This Fermi vector is linked to the Fermi energy by the relation:
+
=
}
2+
2
2
(13)
whereis the mass of each particle. In a similar way, for all the spin values=12,
we assume that only the states with wave vectors smaller than the Fermi vectorare
occupied, with a relation similar to (13) in which the index+is replaced by. The
total particle numberin each of the two spin states are then:
=
k
1 (14)
(the summation runs over all the states having a population equal to1). In the limit of
large volumes, this expression becomes an integral:
=
(2)
3
k
d
3
=
2
2
0
2
d=
3
6
2
(15)
Depending on whether
+
is larger or smaller than, the spins+orwill make up
the majority, the populations being equal if
+
=.
2-a. Two-point correlations
Let us compute the average value of the operator
(r;r)dened in (3), dis-
tinguishing the two cases whereandare equal or dierent.
(i) Same spin states
Taking (1) into account, relation (9) then yields:
1(r;r) =(r;r)=
1
k
k(rr)
(16)
where the notation for the spins is simplied from12to. In the limit of large
volumes, the summation overkbecomes an integral, and we get:
1(r;r) =
1
(2)
3
k
d
3 k(rr)
(17)
Forr=r, this function simply yields the particle density, already computed. For
r=r, the function to be computed is the Fourier transform of a function ofkthat only
1772

SPATIAL CORRELATIONS IN AN IDEAL GAS OF BOSONS OR FERMIONS
Figure 1: Plot of the function()as a function of the dimensionless variable= .
depends on its modulus. Using relation (59) of Appendix
1(r;r) =
(rr)
(0)
(18)
with
1
:
() =
3
3
0
dsin=
3
3
cos+
1
2
sin
0
=
3
3
sin cos (19)
Finally, we get:
1(r;r) =
(rr) (20)
Figure ()as a function of the mutual distancebetween the two points.
It shows that, for each spin state, the non-diagonal one-particle correlation function
presents a maximum atr=r, then rapidly decreases to zero over a distance of the order
of a few Fermi wavelengths,= 2 . A system of free fermions, in its ground state,
does not show any long-range non-diagonal order.
(ii) Opposed spin states.
Relation (9) shows that the two-point correlation function is zero between two
states of dierent spins; there is no non-diagonal order.
1
An arbitrary coecient3
3
has been introduced in the functionto make it tend towards1
when its variable tends towards zero, which allows dropping the factor(0).
1773

COMPLEMENT A XVI
2-b. Correlations between two particles
We start from relation (12). In the second line, the condition=may be ignored
as, for fermions, the=terms exactly cancel those on the third line; we can therefore
consider the indicesandas independent. Two cases must be distinguished:
(i) If1=2, the three terms in (12) remain, but their behavior as a function of
the volumeare dierent. This is because, in the limit of large volumes, each of the
summations overoris proportional to the volume, whereas the moduli squared of the
wave functions are each proportional to1. For a large system, as the rst of the three
terms only contains a single sum over, it varies as1and is thus negligible. We are
left with the two other terms:
2(r;r) =
(=)
(r)
2
2
(=)
(r) (r)
2
=
2
1
k
k(rr)
2
(21)
The same sum as (17) appears again in this relation. This leads to, in the limit of large
volumes:
2(r;r) =
2
1[(rr)]
2
(22)
The Pauli principle forbids particles in the same spin states to be at the same point in
space; as expected, expression (22) goes to zero whenr=r. As the distance between
particles increases, the function(rr)goes to zero, and the two-body correlation
function tends towards the square of the one-body density, indicating that the
long-range correlations disappear. This change of behavior occurs over a characteristic
distance of the order of, comparable to the distance over which the non-diagonal
order disappears. A plot of the spatial variations of the correlation function is given
in Figure; it shows clearly the existence of an exchange hole corresponding to the
mutual particle exclusion over this characteristic distance.
(ii) If1=2, of the three terms of (12) only the second one (the direct term) is
non-zero and yields a constant:
2(r;r) =
+
2
(23)
It is simply the product of the densities of the two kinds of spins; in the absence of
interactions, particles with dierent spins do not show any correlation. This is because
physically two particles at positionsrandrand in dierent spin states, can in principle
be identied by the direction of their spin; consequently, they no longer behave as really
indistinguishable quantum particles, and no Fermi statistical eects may be observed.
As we assumed the particles did not interact with each other, no spatial correlations can
develop.
1774

SPATIAL CORRELATIONS IN AN IDEAL GAS OF BOSONS OR FERMIONS
Figure 2: Plot, as a function of the dimensionless variablerr, of the correlation
function2(r;r)between the positionsrandrof two particles in the same spin
state, in a free fermion gas. As the Pauli principle forbids two particles to be at the same
point in space, this function goes to zero at the origin, which creates an exchange hole.
As the distance increases, the function approaches1over a distance of the order of the
inverse of the Fermi vector associated with this spin state.
Comment:
To keep the computation simple, we considered a system of non-interacting fermions,
contained in a cubic box and in its ground state. The properties we discussed are,
however, more general. In particular, it can be shown that a fermion system always
exhibits an exchange hole for particles with identical spins, whether they interact or
not; for a system at thermal equilibrium, the hole width gets smaller as the temperature
increases, and goes from the Fermi wavelength at low temperature (degenerate system)
to the thermal wavelength at high temperature (non-degenerate system).
3. Bosons in a Fock state
The situation is radically dierent for bosons, as there is no upper boundary for the
occupation numbers.
3-a. Ground state
For non-interacting spinless bosons in their ground state, the occupation number
0of the individual state0having the lowest energy is equal to the total particle
number, all the other occupation numbers being equal to zero. Relation (5) then
yields:
1(r;r) =
(rr)=
0(r)0(r) (24)
1775

COMPLEMENT A XVI
As the wave function
0(r)extends over the entire volume, the modulus of this wave
function does not decrease as the distance betweenrandrincreases and becomes compa-
rable to the size of the system, as opposed to what occurs for fermions. This asymptotic
behavior of1(r;r)has been used by Penrose and Onsager to dene a general criterion
for Bose-Einstein condensation, valid also for interacting bosons .
As for the two-particle correlation function, formula (10) yields:
2(rr) =
(rrrr)=( 1) 0(r)
2
0(r)
2
(25)
If the ground state wave function is of the form
k0r
, this function is simply equal
to the constant( 1)
2
, independent ofrandr. A system of bosons that are all
in the same quantum state does not show any spatial correlations.
3-b. Fragmented state
We now assume the-boson system to be in a fragmented state: instead of all
the particles being in the same individual state,1particles are in the state1and2
in the state2, with=1+2. Relation (5) then yields:
1(r;r) =11(r)1(r) +22(r)2(r) (26)
Whenr=r, expressions (24) and (26) contain the moduli squared of the wave functions
(r), which are all equal to1(for a system contained in a box of volumewith
periodic boundary conditions); both expressions (24) and (26) are therefore equal. On
the other hand, whenrandrare dierent, the phases of the two terms in (26) do
not coincide any longer, and (destructive) interference eects can lower the modulus of
1(r;r). Consequently, the fragmentation of a physical system into two states decreases
the modulus of the non-diagonal terms of1(r;r). Obviously, the more fragmented
states there are, the more noticeable the decrease.
Relation (10) now becomes:
2(rr) =1(11) 1(r)
2
1(r)
2
+2(21) 2(r)
2
2(r)
2
+1221(r)
2
2(r)
2
+
1(r)1(r)
2(r)2(r) +c.c. (27)
where the factor2in the second line comes from the fact that either= 1and= 2,
or the opposite; the last two terms of this expression correspond to the exchange term,
and the notation c.c. indicates the complex conjugate of the previous term. Replacing
the wave functions by
k12r
, and assuming1and2to be very large compared
to1, we obtain the square of the sum(1+2)
2
=
2
, and we can write:
2(rr)
2
2
+ 2
12
2
cos [(k2k1)(rr)] (28)
The rst term is simply the square of the one-particle density; it does not have
any spatial dependence and is what we expect in the absence of any particle correlation.
On the other hand, the exchange term is position dependent; it presents a maximum
whenr=r, and oscillates at the spatial frequencyk2k1. This exchange term
enhances the probability of nding two bosons close to one another (bunching eect
1776

SPATIAL CORRELATIONS IN AN IDEAL GAS OF BOSONS OR FERMIONS
coming from the Bose-Einstein statistics); the probability of nding them at a greater
distance is then lower, and then increases again, etc. For a short interaction range, only
the rst maximum plays a role, and increases the average value of the interaction. The
consequences of that eect, in terms of the internal interaction energy, has been discussed
in XV.
3-c. Other states
We now consider situations described by Fock states where0is still very large,
but where other statesare also occupied, with populationsmuch smaller than0.
(i) One could for instance place a nite fraction0of particles in the ground
state, and distribute the remaining fraction1 0among a large number of states,
whose individual populations remain small and vary regularly with the index. This
leads to:
1(r;r) =00(r)0(r) +(rr) (29)
where(rr)is given by:
(rr) =
1
=0
k(rr)
=
1
(2)
3
d
3
(k)
k(rr)
(30)
This function is the Fourier transform of the distribution=(k)fork= 0. As all
theare positive or zero, the function(rr)presents a maximum whenr=r,
since this is where all the exponentials
k(rr)
are in phase; it then decreases when
the dierencerrincreases, as all the phases spread out. If thedistribution is
a regular function of width(a Gaussian for example), the function(rr)tends
towards zero over a distance 1, in general much smaller than the size of the
system.
Figure 1(r;r)when the particles are contained in a box,
so that we can use (1). As we assumed0to be the ground state, the corresponding
wave function is the inverse of the square root of the volume, and (29) becomes:
1(r;r) =0+(rr) (31)
where0is the density of atoms in the ground state:
0=
0
(32)
After the decrease linked to that of(rr), and occurring over the interval1,
the function does not tend towards zero (as it would for fermions), but towards a constant
proportional to the population0. As already mentioned, this particular behavior is the
base for the Penrose and Onsager criterion that denes, in a general way, the appearance
of Bose-Einstein condensation.
1777

COMPLEMENT A XVI
Figure 3: Plot of the function1(r;r) =0+(rr)for bosons, as a function of the
distancerr. The function starts by decreasing over an interval of the order of the
inverse of; at a larger distance, it tends towards a constant0proportional to the
ground state population. The fact that it does not go to zero indicates the presence of a
long-range non-diagonal order, and the existence of a highly populated individual level.
As for the two-body correlation function, relation (10) shows the existence of three
kinds of terms for a system having only one single highly populated state0:
the terms corresponding to two particles in the highly populated state (con-
densate), which come from the rst line
2
of (10), and yield again (25), replacingby
0.
the crossed terms in0, which yield:
0
=0
2(r)
2
0(r)
2
+
0(r)0(r)(r)(r) +(r)(r)
0(r)0(r)
(33)
Inserting the value (1) for the wave functions (assumingk0= 0), we get a contribution
to2(rr)equal to:
0
2
=0
2 +
k(rr)
+
k(rr)
) (34)
Using relation
=0
= 0, we can write this result in the form:
20
0
+Re[(rr)] (35)
2
In the limit of large volumes, we have assumed that0is the only population proportional to the
volume. In the rst line of (10), the term= 0then contains the product of
2
0
and the two wave
functions squared, each proportional to the inverse of the volume; this term is therefore independent
of the volume. On the other hand, the terms= 0of the second line contains one summation over
, introducing a factor proportional to the volume (in the limit of large volumes), but also two wave
functions squared, each inversely proportional to the volume. The net result is a contribution inversely
proportional to the volume, hence negligible compared to the previous one.
1778

SPATIAL CORRELATIONS IN AN IDEAL GAS OF BOSONS OR FERMIONS
where Re means real part andis the function dened in (30).
the terms corresponding to two particles in states other than the ground state,
and which yield:
=
2
1 +
(kk)(rr)
(36)
In (34), as well as in (36), we notice that the contributions from all the states= 0have
various phases in general; they are, however, all in phase whenr=rand the correlation
function then presents a maximum. The bosons have thus a tendency for bunching, and
this eect is felt over a distance 1, as if they were attracted to one another.
It is, however, a purely statistical eect linked to the bosonic character of the particles,
since we assumed there were no interactions between the particles.
(ii) One could also imagine the population distribution to be regular, without
favoring any individual state, in which case the0contribution vanishes; we are then left
with the contribution from theterms of (36), which is maximum whenr=rfor
the same reasons as above. The general behavior of the two-body correlation function
is shown in Figure: it presents a maximum at the origin, and then tends towards zero
at large distance (in this case,0= 0). Once again, identical bosons exhibit a bunching
tendency.
Comment:
Suppose that, instead of assuming the bosonic system to be in a Fock state (a pure state),
it is at thermal equilibrium, described by the thermodynamic equilibrium. This would
lead to results similar to those we just derived, but with , the thermal wavelength
of the particles [24]. The boson bunching tendency is a quite general property.
1779

SPATIO-TEMPORAL CORRELATION FUNCTIONS, GREEN'S FUNCTIONS
Complement BXVI
Spatio-temporal correlation functions, Green's functions
1 Green's functions in ordinary space
1-a Spatio-temporal correlation functions
1-b Two- and four-point Green's functions
1-c An example, the ideal gas
2 Fourier transforms
2-a General denition
2-b Ideal gas example
2-c General expression in the presence of interactions
2-d Discussion
3 Spectral function, sum rule
3-a Expression of the one-particle correlation functions
3-b Sum rule
3-c Expression of various physical quantities
This complement discusses the properties of the spatio-temporal correlation func-
tions of an ensemble of identical particles, generalizing the spatial correlation functions
dened in ; the corresponding Green's
functions shall also be introduced. We rst study () the normal and anti-normal
spatio-temporal correlation functions, then the Green's function, and discuss some of
their properties, illustrated with the example of an ideal gas. We then study in
Fourier transforms of these functions for physical systems that are translation invariant
both in space and time; we shall write their general expression in the presence of inter-
actions. In , we nally introduce the spectral function, which leads, for interacting
particles, to very simple expressions for various physical quantities, in a form similar to
the one used for an ideal gas.
1. Green's functions in ordinary space
In the previous complement, we studied the spatial dependence of the correlation func-
tions, taken at a given time. We now take into account the temporal dependence, using
the Heisenberg picture (ComplementIII) where the operators are time-dependent. To
keep the notation simple, we assume, from now on, that either the spin is zero for bosons
(= 0), or else, in the general case (fermions and bosons), that all the particles are in
the same spin state. As mentioned before, the generalization to the case whereis non-
zero would only require adding an indexto all the eld operators. In the Heisenberg
picture, the eld operator (r)becomes a time-dependent operator (rt):
(r) =
}
(r)
}
(1)
1781

COMPLEMENT B XVI
where
}
is the evolution operator, expressed as a function of the system Hamilto-
nian(including the particle interactions when present), that we assume to be time-
independent.
Consider a system ofidentical particles, fermions or bosons, described by a
density operator. The spatio-temporal correlation functions and the Green's functions
are dened as the average values, computed with, of the products of a number of eld
operators (r)and their Hermitian conjugates (r)taken at dierent space-time
points(r),(r),... etc.
1-a. Spatio-temporal correlation functions
The density operatormay contain very complex correlations between particles,
and its time evolution can be very complicated in the presence of interactions. We are
going to dene a certain number of functions that characterize its most simple and useful
properties, as they only pertain to a small number of particles.
. Two-point normal and anti-normal functions
The one-particle spatio-temporal normal
1and anti-normal
1correlation
functions are dened by
1
:
1(r;r) =Tr (r) (r)= (r) (r)
1(r;r) =Tr (r) (r)= (r) (r)
(2)
The normal ordering is obtained when the creation operator is on the left and the anni-
hilation on the right; it is the opposite for the anti-normal order. Note that it is only the
order of the two operators that changes between
1and
1; the position and time
variables attributed to each of the two operators and remain the same.
In the particular case where=, the normal correlation function simply yields
the matrix elements of the one-particle density operator see formulas (B-22) and (B-
23) of Chapter . The normal correlation function is therefore a generalization, at
dierent times, of this matrix element, which will prove to be useful.
To understand the physical meaning of these two denitions in an intuitive way,
we start with the anti-normal function and consider the simple case where an-particle
system is in a pure state0(its ground state, for example). We then get:
1(r;r) =0 (r) (r)0
=0
}
(r)
()}
(r)
}
0 (3)
The right-hand side reads as follows (from right to left): starting from the system initial
state0, we let it evolve according to its own Hamiltonian until the time, when we
create a particle at pointr; we then let the (+ 1)-particle system freely evolve until
time, and we nally annihilate a particle at pointr. Consequently, the function
1
is the scalar product of the ket thus obtained with the state
}
0, result of the
1
To be consistent with the notation of Chapter ), we choose a notation where, for the trace,
the rst group of variables(r)of function1is associated with the operator , the second group
(r)to the operator . Note however that the opposite convention can also be found in the literature.
1782

SPATIO-TEMPORAL CORRELATION FUNCTIONS, GREEN'S FUNCTIONS
free evolution of the initial ket over the same time interval (without the creation or
annihilation of a particle). In other words, the perturbation created by the creation of a
particle, followed by its later destruction, changes the state of the physical system; the
value of
1is given by the probability amplitude of nding the system in the same
state as the one it would have reached in the absence of this perturbation.
The previous interpretation is natural when; if this is not the case, the
mathematical denition of
1is the same, but the intermediate evolution stage goes
backward in time. In this process, and as expected, the dynamics of the system remains
unchanged, including the particle interactions. Furthermore, the additional particle is
not simply a particle juxtaposed to the pre-existing system, it is indistinguishable from
the others and hence undergoes indistinguishability eects (more details on this point
will be given in ).
For the normal function, (r)acts before (r), which means this function
has a natural interpretation if: the system evolves freely until, at which time a
particle is annihilated (which amounts to the creation of a hole, see ComplementXV
AXV); the system, with1particles, then evolves freely until time, when a particle
is created (which annihilates the hole). The normal function is therefore the analog of
the anti-normal function, provided we replace the additional particle by a hole, and we
invert the times.
. Physical discussion
The denition of the correlation functions contains particle creation and annihila-
tion operators, but this does not imply that such physical processes really occur in our
system. Going from an-particle system to another one with1particles can be
mathematically useful but only plays an intermediate role since we nally return, via a
second operator, to the same number of particles. Furthermore, the action of an operator
(r)is not merely a local destruction of a particle, neither is the action of (r)the
simple creation of a particle at pointr, juxtaposed with the already present particles:
the quantum indistinguishability that concerns all the particles (including the new one)
plays an essential role.
A few simple examples
(i) For the normal correlation function, the perturbation starts with the annihilation of
a particle (creation of a hole), followed by a later creation of a particle (destruction of a
hole). At time= 0, the annihilation operator (r)can be expanded according to
formula (A-3) of Chapter :
(r) = (r) (4)
where the eect of each operatordepends on the populationof the stateas
it introduces the factor
. Consider a very simple case: a gas of bosons, all in the
same quantum state0(ideal gas of totally condensed bosons). Acting on a state0
where only the individual state0is occupied, all the terms of the sum (4) yield zero,
except for the= 0term. The eect of the operator (r)on0is to actually destroy a
particle in the state0; asrvaries, the result is still the same, simply multiplied by the
coecient0(r). If, for example, the ideal gas is placed in a trap where the individual
ground state is0, the operator (r)yields a ket with an appreciable norm only if
1783

COMPLEMENT B XVI
rfalls in a domain where0(r)is not negligible; if it falls outside this domain, the
resulting ket is practically zero. In a general way, for fermions as for bosons, (r)can
obviously destroy particles only while acting on an already occupied individual state. For
bosons, in addition, the factor
means that the operator (r)gives more weight to
the highly populated individual states rather than those with low occupation numbers.
Consequently, the creation of a hole is not a local process at pointr.
(ii) For the anti-normal correlation function, we start with the creation of a particle by
the operator (r). In the case of bosons, and because of the factor
+ 1introduced
by, (r)tends to preferentially create particles in stateshaving a high population
. Let us go back to the previous example of a large number of bosons in a trap, all in
the same individual ground state0. When 0(r)is not negligible, the supplementary
boson is created in the same ground state. If, on the other hand,ris far away from the
trap center and falls in a domain where0(r)is practically zero, the boson is actually
created at pointrbut without perturbing very much the bosons already present.
For fermions, on the contrary, it is impossible to create a particle in an already occupied
state; in a Fock state, an additional fermion can only be created in a state orthogonal to
all the initially occupied states. Let us assume the ideal gas of fermions is in its ground
state, and contained in a harmonic trap; the energy levels, up to the Fermi level, are all
occupied. The eect of the creation operator (r)at a point close to the trap center
is to create an additional particle in a state that can be expanded on all the individual
stationary states of the trap; as this state must be orthogonal to all the already occupied
states, it only has components on the non-occupied states, which have an energy higher
than the Fermi level. Now the corresponding wave functions take on small values at the
center of the trap, and are maximum in the classical turning point regions
2
, which means,
in this case, at the edge of the existing fermion cloud (or even further). If positionris
close to the center of the trap, the additional fermion will be added on the periphery or
outside the cloud of fermions. On the other hand, ifrfalls outside, in a region of space
where all the wave functions of the occupied states are practically zero, the additional
particle will be created practically right at pointr. Examples of these various situations
will be given in .
. Properties
The complex conjugate of
1is obtained by changing the eld operators' order
in (2) and replacing them by their Hermitian conjugates:
1(r;r)= (r) (r)=
1(r;r) (5)
The complex conjugation of the function
1is therefore equivalent to exchanging the
two points(r)and(r), which amounts to a parity operation on the variablesrr
and . A similar property is easily demonstrated for
1. As a result, the Fourier
transforms of these functions with respect to the variablesrrand must be real.
2
The modulus squared of a stationary state wave function yields, at each point, the probability
density of presence for this state. This probability is maximal in the regions of space where, classically,
the particle spends the most time, i.e. where its velocity is small, as is the case for the classical turning
point regions. Figure 6 of Chapter V gives an example of such a situation.
1784

SPATIO-TEMPORAL CORRELATION FUNCTIONS, GREEN'S FUNCTIONS
In addition, when the system is in a state that is translation invariant in space and time
3
,
the correlation functions only depend on the dierencesrrand .
We note that the linear combination
1(r;r)
1(r;r)involves the
average value of the operator:
(r) (r) (6)
where= +1for bosons,=1for fermions. If=, and taking into account relation
(A-19) of Chapter , this equality becomes:
1(r;r)
1(r;r) =Tr (rr)=(rr) (7)
. Temporal evolution, four-point functions
We now use, in denition (2) of
1, the Hermitian conjugate of the evolution
equation for (r), written in (C-13) of Chapter ; this leads to:
}
+
}
2
2
r 1(r)
1(r;r)
=d
3
2(rr) (r;) (r;) (r;) (r;)
=d
3
2(rr)
2(r;r;r;r) (8)
which involves the normal two-particle (or four-point) correlation function, whose general
expression is:
2(r;r;r;r ) =Tr (r) (r) (r ) (r) (9)
Taking into account a time dependence generalizes formula (B-30) of Chapter
more precisely its average value). We obtain, in a similar way, the equation giving the
variation of
1with respect to the variablesrand, or that giving the evolution of
the anti-normal function
1. The four groups of space-time variables the function
2
depends on, are, in general, independent. Most of the time, however, we only need the
diagonal part
2(r;r)of the correlation function, obtained forr=r,=
andr=r,=; this diagonal part is analogous to the two-particle correlation
function (two positions, two times) in classical statistical mechanics. When the system is
translation-invariant both in space and time, this function only depends on the dierences
rrand .
Whereas, for the function
1, a hole is created and then destroyed, for the function
2there are now two holes rst created and then destroyed; the natural order of the
increasing times is given by the time variables of the operators in (9), taken from right
to left. One could dene, in a similar way, a function
2where two particles would
be created at the beginning and then destroyed at later times (compared to the normal
correlation function, the role of particles and holes are interchanged). Consequently,
when the particles interact, the evolution equation of
1involves another higher order
3
This is the case if the system Hamiltonian is translation invariant, and if the system is in an
eigenstate of, or described by a density operatorthat is a function of.
1785

COMPLEMENT B XVI
correlation function,
2. In turn, the evolution of
2involves correlation functions
of an even higher order,
3etc. This means that, because of the interactions, the set
of equations is not closed, but includes a complete hierarchy of a large number of
equations, involving correlations of higher and higher order.
1-b. Two- and four-point Green's functions
Equation (8) is a linear partial dierential equation, with a right-hand side some-
times called a source term. In our case, this right-hand side does not contain any
singular function. But when this right-hand side is modied to include a delta function,
the new solutions of the equation are called the Green's functions. We now show
how to introduce Green's functions in the problem we are concerned with.
. Two-point Green's function
The two-point Green's function1is obtained by a combination of the two corre-
lation functions of . We saw in , the anti-normal correlation
function is the most natural as it includes the propagation of a particle fromto. On
the other hand, for, the most natural one is the normal function that involves the
propagation of a hole fromto. We can combine those two possibilities into one by
setting:
1(r;r) =( )
1(r;r) +( )
1(r;r) (10)
where( )is the Heaviside function of the variable(equal to1if 0,
zero otherwise), and whereequals+1for bosons,1for fermions; we shall see later
on, for example in , that the introduction of thefactor simplies the following
computations.
When we take the derivative of the two-point Green's function1with respect to
time, the discontinuities introduced by the Heaviside functions yield delta functions; the
precise calculation will be done in for an ideal gas, allowing us to verify that1
is indeed a Green's function. Using this type of function is quite useful in a number of
problems, for example those involving Fourier transforms, or in perturbation calculations.
. Four-point Green's function
By analogy with (9) we dene a two-particle (or four-point) Green's function:
2(r;r;r;r ) =Tr (r) (r ) (r) (r)
(11)
whereis the time ordering operator, which orders the4times by decreasing values
from left to right (by denition, this operator also includes a factor, whereis the
parity of the permutation necessary for this time ordering; this may result in a change
of sign for fermions).
1786

SPATIO-TEMPORAL CORRELATION FUNCTIONS, GREEN'S FUNCTIONS
1-c. An example, the ideal gas
When the system considered is an ideal gas, it is possible to get explicit values of
the previous functions. As before, we assume the gas to be contained in a box of volume
, with periodic boundary conditions. Using relation (A-3) of Chapter
where the(r)are plane waves, we can write the eld operator (r)as:
(r) =
kr
k (12)
where the sum covers all the wave vectorsksatisfying the periodic boundary conditions.
This expansion is convenient as, for an ideal gas, the time dependence of the operators
in the Heisenberg picture
k
is particularly simple, as we now show. The Hamiltonian
can be written as:
= }k kk (13)
with:
k=
}(k)
2
2
(14)
As the operator
kkcommutes with any annihilation operatorkpertaining to a
dierent momentum
4
, and sincek kk=k, we get:
[k] =}kk (15)
This corresponds, in the Heisenberg picture, to an evolution:
k() =
k
k (16)
The time evolution of (r)is then written as:
(r) =
(kr k)
k (17)
. Normal correlation function
Inserting this result in (2), we get:
1(r;r) =
(kr k)(kr k)
kk (18)
We now show that, because of the translation invariance, the average value
kk
must be zero wheneverk=k. Assume, for example, that the system density operator
4
For fermions, two minus signs are introduced because of the anticommutations, but they cancel
each other. If the momentum is the same, we have k
k
k=
k
kk+k= 0 +kand
k
kk= 0, so thatk
k
k=k.
1787

COMPLEMENT B XVI
is the canonical thermal equilibrium operator= . This operator is diagonal
in the basis of the Fock states, and the trace of the product
kkwill be zero unless
both annihilation and creation operators act on the same individual state: the non-zero
condition is thereforek=k. In the same way, if we are now using the grand canonical
equilibrium=
( )
, the operatoris still diagonal in the same basis, and
the same rule applies. In a general way, one can see
5
that the translation invariance of
requires:
kk=Tr
kk= (19)
whereis the average value of the population operator:
= (20)
We can then write:
1(r;r) =
1
[k(rr) k()]
(21)
The normal correlation function is simply the sum of all the contributions from the
individual states occupied by free particles, labeled by the indexk. Each contributes
proportionally to its average population, and has a spatio-temporal dependence given
by the progressive wave
[k(rr) k()]
it is associated with.
As expected for a translation invariant (both in space and time) system, this normal
function only depends on the dierencesrrand . Taking (14) into account, it
obeys the partial dierential equation:
+
}
2
r 1(r;r) = 0 (22)
which corresponds to the free propagation of particles in an ideal gas (a similar equation
exists for the variablesandr, with a change of sign for the time derivative term).
Expression (21) allows shedding new light on the physical interpretation of the
normal correlation function given in , in terms of the creation, in the-particle
system, of a hole, which then propagates until it is annihilated at a later time. In an
ideal gas, each term of the sum in (21) is a free wave plane: in the absence of interactions,
the particles can freely propagate along straight lines. Theappearing in the formula
shows that the hole can only propagate along already populated individual states in the
-particle state: a hole can only be created in an already occupied quantum state, as
pointed out already in . As a result, the created hole is not a point-like object
actually localized at pointr: it is only built from superpositions of free occupied states,
whereas for a truly point-like excitation, one would have to combine values ofkextending
to innity.
5
To prove this in a general way, we simply have to note that the operator
k
kis translation
invariant only ifk=k. To show this, one can use the expression of the translation operator as an
exponential of the operator associated with the total momentum (ComplementII, ).
1788

SPATIO-TEMPORAL CORRELATION FUNCTIONS, GREEN'S FUNCTIONS
. Anti-normal correlation function
For the anti-normal function
1(r;r), the calculation is practically the
same, the only dierence being the inversion of the order of the operatorskand
k
,
which leads to:
1(r;r) =
1
[1 +]
[k(rr) k()]
(23)
It obeys the same partial dierential equation as the normal function. It follows from
(21) and (23) that the linear combination:
1(r;r)
1(r;r) =
1
[k(rr) k()]
(24)
is independent of the state of the system; for=, it is simply equal to the function
(rr). We shall see later the relation between this expression and the spectral function.
Let us come back, here again, to the interpretation of the eect of the operator
(r), discussed in . For a fermion system (=1), relation (23) clearly shows
that the particle creation does not involve any of the already occupied individual states
k, with= 1, since the corresponding term is zero. The object created by the ex-
citation cannot have any component on the already occupied individual states, which
simply means that its wave function must remain orthogonal to those of all the fermions
already present. For a boson system, the eect is just the opposite: if, for instance, an
individual state of bosons is highly populated compared to all the others, the term in (23)
corresponding to this largewill be dominant: the additional particle will be mainly in
the same individual state as all the other particles already present.
. Two-point Green's function
Using denition (10), we obtain the Green's function1(r;r):
1(r;r)
=
1
[k(rr) k()]
( ) [1 +] +( ) (25)
If we take the derivative with respect to the time, the two Heaviside step functions
yield delta functions with opposite signs. We then get the partial dierential equation:
+
}
2
r 1(r;r) =
( )
k(rr)
[1 + ] (26)
that is:
+
}
2
r 1(r;r) =( )(rr) (27)
As expected, the right-hand side is a product of delta functions of the set of variables
characterizing a Green's function.
In the presence of interactions, this partial dierential equation is no longer valid;
we must add to its right-hand side the interaction contributions, which involve Green's
functions of a higher order.
1789

COMPLEMENT B XVI
. Four-point functions
The computation of the four-point correlation functions is very similar to the one
we just explained; we simply use again relation (17) to get their explicit expressions for an
ideal gas contained in a box. The results are the same as those obtained in Complement
AXVI, as for example in relation (21), except for the fact that we must now multiply each
spatial plane wave
kr
by the associated time evolution factor
k
.
2. Fourier transforms
From now on, we shall only study systems that are translation-invariant in space and
time. Consequently, the correlation functions only depend on the dierence of the vari-
ablesrrand , so that we can choose to cancel bothrand.
2-a. General denition
Let us introduce the two (double) Fourier transforms with respect to time and
space:
1(k) =d
3
d
( kr)
1
(00;r)
(28)
(we have setr= 0and= 0in the function
1
). These functions are real, as shown
by the parity relation (5).
The Fourier transform
1of the Green's function1introduced in (10) is called
the one-particle propagator; it is dened by:
1(k) =d
3
d
( kr)
1(00;r)
(29)
For a system contained in a volumewith periodic boundary conditions (Comple-
mentXI), the integrals over d
3
in the above formulas must be taken over the volume
. They yield the coecients of a Fourier series where the vectorsktake the discrete
valueskcorresponding to the boundary conditions. This series characterizes the spatial
dependence. As for the time dependence, the Fourier transform is a continuous function
6
.
The inverse transformation relations are:
1
(00;r) =
1
k
d
2
(kr )
1(k) (30)
where, in the limit of very large volumes, the discrete summation becomes an integral
with the coecient(2)
3
:
1
(00;r) =
d
3
(2)
3
d
2
(kr )
1(k) (31)
6
As opposed to the space variables, the time variable is not conned to a nite variation domain,
and time Fourier transforms may be singular; such an example, concerning an ideal gas, will be given in
, where we shall introduce, as a convergence factor, a decreasing exponential(wheretends
toward zero through positive values).
1790

SPATIO-TEMPORAL CORRELATION FUNCTIONS, GREEN'S FUNCTIONS
Comment:
One can also express the functions
1(), not as Fourier transforms as in (28),
but directly from the average values of products of creation
k
and annihilationk
operators in the individual statesk. Formulas (A-3) and (A-6) of Chapter
that:
(r) =
1
kr
k and (r) =
1kr
k
(32)
Inserting these relations in the denitions (2), then in (28), we get:
1(k) =
1d
3
d
( kr) kr
k k() (33)
where
k()is the operatorkin the Heisenberg picture. The integral over the volume
selects a single term,=, and cancels the volume. Furthermore, the translation
invariance means that neither the density operator, nor the system Hamiltonian, have
matrix elements between state vectors with dierent total momentum. Now the operator
k
increases that total momentum by}and
k()decreases it by}. The average
value
k k()is therefore zero unlessis equal to, and that eliminates the
summation over. Finally:
1(k) =d
kk() (34)
In a similar way, we can also show that:
1(k) =d
k()
k
(35)
2-b. Ideal gas example
For an ideal gas contained in a box (with periodic boundary conditions), thekare
discrete. Replacingkbykin (28) and using expression (21), after replacing the dummy
indexkbyk, we get a product of exponentials to be integrated. The integral over
d
3
, combined with the factor1, yields a Kronecker deltakkthat eliminates the
summation overk; the integral over dyields2( k), and we get:
1(k) = 2 ( k) (36)
In a similar way:
1(k) = 2[1 +]( k) (37)
and we can nally write:
1(k)
1(k) = 2( k) (38)
These expressions are particularly simple, but no longer valid when the particles interact.
They are, however, useful as a reference point to understand the interaction eects.
1791

COMPLEMENT B XVI
The one-particle propagator
1(k)dened in (29) can be obtained in a similar
way; the integral over d
3
is unchanged, but the integral over time now yields:
d
( k)
() [1 +] +() (39)
which does not converge. For the term in(), a classical method is to introduce a
convergence factor by changinginto+, with0through positive values; for the
()term, the change is to. We get:
1(k) =
1 +
[ k+]
+
[ k ]
=
[ k+]
+
2
[ k]
2
+
2
(40)
In the limit where0, the two fractions on the right-hand side yield principal parts
and delta functions see relation (12) of Appendix II. The rst term on the right-hand
side yields both a principal part[1( k)]and a delta function( k), whereas
the second one (in) yields only a delta function( k). If the system is diluted,
it is in the classical regime (i.e. non-degenerate), where all the occupation numbers are
small compared to1and where the exchange eects are weak; the second term, associated
with the indistinguishability between the particles, is then negligible.
2-c. General expression in the presence of interactions
A translation invariant system has a Hamiltonian that commutes with its total
momentum. We can then build a basis with state vectors that are, for each particle
numberand for each value of the total momentum}K, eigenvectors ofwith energies
; we shall noteKthese energies as it is often useful to explicitly keep track of the
values ofandKthat dene the subspace corresponding to that eigenvaluein the
spectrum of. We call the corresponding eigenvectors, where the index
accounts for possible degeneracies of these eigenvalues.
We assume the system to be in a stationary state, and translation invariant. This
means that its density operatorcannot have non-diagonal matrix elements between
eigenvectors corresponding to dierent momenta or energies; in each of the subspaces
common to both of these quantities, we can choose the basis that diagonalizes
the density operator and set:
K=
K
K (41)
We now use this basis to compute the trace appearing in (2). We rst insert expression
(12) for the eld operators (and their conjugates) as a function of the operatorsk; taking
into account the exponentials introduced by the operators in the Heisenberg picture, we
get:
1(00;r) =
1
kr
K K K

K k

K

K k
K
e
K
K } (42)
Several simplications can be made on the right-hand side of this equation. First of all,
as the operatorkdestroys a particle, we necessarily have= 1, or else the
1792

SPATIO-TEMPORAL CORRELATION FUNCTIONS, GREEN'S FUNCTIONS
matrix element ofkwould be zero; the sum overdisappears. Along the same line,
as this operator decreases the total momentum by~k, we must also haveK=Kk,
and the sum overKis also eliminated. Now, for the matrix element of
k
to be non-
zero, and since this operator adds~kto the momentum~(Kk), to recover the initial
momentum~K, we necessarily havek=k. Once these simplications have been made,
we insert the result in denition (28) of
1(k), and get:
1(k) =
1d
3
d
( kr) kr
K K

K k

1
Kk
2
e
[
1Kk K]}
(43)
On the rst line, the integral over
3
yields a Kronecker delta that forcesk=k; the
integral then yields, which cancels the volume appearing in the denominator. The
time integral in relation (43) yields a2coecient multiplied by a delta function of the
variable+
1Kk K}. We nally get:
1(k) =
2
K
K
K k

1
Kk
2
K 1Kk
}
(44)
The same type of calculation also yields:
1(k) =
2
K
K
K k
+1
K+k
2
+1K+k K
}
(45)
Let us assume, in addition, that our system is at thermal equilibrium, and described
by the grand canonical density operator:
=
1
(46)
with the classical notation:is the total particle number,= 1 (is the Boltz-
mann constant) and=Tr is the grand canonical partition function.
The two functions
1and
1then obey a simple relation, often called boundary
condition:
1(k) =
(} )
1(k)
(47)
This relation turns out to be crucial in many calculations involving Green's functions;
we shall use it in .
1793

COMPLEMENT B XVI
Demonstration:
To establish this relation, we rewrite equality (45) using the fact thatkand
k
are
Hermitian conjugates. We then get:
1(k) = 2
K K

+1
K+k k

K
2
+1K+k K
}
(48)
We now permute the indicesand, as well as the indicesand, we change the
dummy summation index to=+ 1, and nally replace the dummy variable
by=+. This yields:
1(k) = 2
K
1
Kk

K k

1
Kk
2
K 1K k
}
(49)
In this summation, just as in (44), the lowest value of the indexthat gives a con-
tribution is= 1; we therefore get the same expression as (44), with (aside from the
irrelevant change of the dummy indexinto) just one modication, the replacement
of
K by
1
Kk
. However, since:
K =
1
[ K ]
(50)
the ratio of these two diagonal elements
1
Kk K introduces in the integral a
factor:
1K k
+( 1)+
K
=
K
1K k (51)
Now, the delta function in (49) allows replacing, in the exponent of this factor, the energy
dierence by}:
[ K 1K k]
=
[} ]
(52)
This factor comes out of the summation and relation (47) is established.
2-d. Discussion
Expressions (44) and (45) give an idea of the behavior of the functions
1and
1. For an ideal gas, relations (36) and (37) show that they are singular functions,
actually delta functions forcing the energy}to take on exactly the one-particle kinetic
energy}
22
2. This is because, in an ideal gas, one can choose Fock states as stationary
states
K
, where each individual state, with a given momentum}k, has a well
dened population. In such a case, the operatorkin expression (45) of
1can only
couple, via its matrix element
K k
+1
K+k
, one state
+1
K+k
to the
initial state
K
: the Fock state, where the occupation numberkis larger by one
unit. Consequently, the energy dierence
+1K+k Kalways takes the same value,
the energy}
22
2of an individual particle with momentum}k. We nd again the
results of .
Let us assume now that the system includes mutual interactions. The matrix
element
K k
+1
K+k
then yields the scalar product of an-particle system
1794

SPATIO-TEMPORAL CORRELATION FUNCTIONS, GREEN'S FUNCTIONS
stationary state with another state where a particle with a well dened momentum}k
has been removed from a stationary state of a system with+1particles. This new state
has no reason to also be stationary, because of the particle interactions. It is therefore
expected to have a non-zero scalar product with a whole series of kets
K
with
dierent energies, which can become more numerous as the system enlarges. The sum
overno longer reduces to a single value of the energy; in the limit of large systems,
it becomes a continuous sum, which absorbs the delta function, so that the function
1(k)takes, for a givenk, non-negligible values in a wholedomain. A priori,
its variations can take any value in this domain; however, if the interactions are not
too strong and by comparison with the ideal gas, we expect the scalar product to have
non-negligible values mostly in anenergy domain close toK+}
22
2. For each
k, the interaction eect is to enlarge the energy peak, innitely narrow for an ideal gas,
by giving it a width that increases as the interaction becomes stronger. This width is
interpreted in terms of a nite lifetime of the excitation created when a free particle is
removed from a stationary state of the (+ 1)-particle system; as the moduli of the two
matrix elements
K k
+1
K+k
and
+1
K+k k

K
are equal, this
lifetime can also be interpreted as that of the excitation created when a free particle is
added to a stationary state of the-particle system.
3. Spectral function, sum rule
In view of formula (7), it is natural to introduce the real function(k)as:
(k) =
1(k)
1(k)
=d
k()
k
(53a)
(53b)
where we have used (34) and (35). We call(k)the spectral function; it is real
since both functions
1and
1are real.
For an ideal gas, formula (38) shows that we simply have:
(k) = 2( ) (54)
but this equality is no longer valid as soon as the particles interact. We are going to
show, however, that the spectral function allows expressing the correlation functions with
a general formula, reminiscent of that established for an ideal gas, even in the presence
of interactions.
3-a. Expression of the one-particle correlation functions
Inserting (47) in the denition of(k), we get:
(k) =
1
(} )
(55)
that is:
1(k) =(k)(} ) (56)
1795

COMPLEMENT B XVI
where(} )is the usual Bose-Einstein distribution for bosons, and the Fermi-Dirac
distribution for fermions:
(} ) =
1
(} )
(57)
Using again (47), we get:
1(k) =
(} )
(k)(} ) =(k)1 +(} ) (58)
Knowing the spectral function(k), we can determine the one-particle corre-
lation functions with formulas similar to those of the ideal gas, containing the same
quantum distribution functions. Note, however, that in the presence of interactions,
the energy}and momentum}kvariables are independent, whereas for an ideal gas,
they are constrained by relation (54).
3-b. Sum rule
We now insert relations (28) and (2) in (53a); since the operators and
coincide at= 0, we get:
(k) =d
3
d
( kr)
Tr (r) (r=0) (59)
Taking a summation over, a time delta function is introduced:
d(k) = 2d
3 kr
Tr (r) (r=0) (60)
that is:
d
2
(k) =d
3 kr
(r) = 1 (61)
In the presence of interactions, we do not know, a priori, how the spectral function
depends onkand. However, for each value ofk, the interactions can only modify the
frequency distribution, but not its integral over.
We have seen that for a gas of interacting particles, the eect of the interactions
is to spread the spectral function over a certain domain of frequencies, all the while
obeying the sum rule (61). There is no reasons for the spectral function to present any
particular shape, or still contain delta functions of frequency. It often happens, however,
that it presents marked peaks, whose narrowness signals the existence of excitations in
the system, behaving almost like free particles (with long lifetimes). These excitations are
called quasi-particles as they are, in a way, an extension once the interactions have
been introduced of the independent particles of the ideal gas. The peak associated with
a particle of momentum~kis not, in general, centered on the energy=}
22
2of a
free particle: in addition to the spreading, the energies of the quasi-particles are shifted
by the interactions. This spreading and shifting is reminiscent of the results obtained in
ComplementXIII, where we studied, to lowest order, the coupling of a discrete state
with a continuum. In other words, one can say that the interactions couple the state of
a free particle with momentum~k, to a continuum of states having dierent momenta,
1796

SPATIO-TEMPORAL CORRELATION FUNCTIONS, GREEN'S FUNCTIONS
which explains the analogy. Note, however, that the present results concern not a single
but an ensemble of identical particles, and its properties at thermal equilibrium; the
physical situation is therefore dierent.
To sum up, to go from the ideal gas (whereis necessarily equal to) to an
interacting gas, we simply introduce in the Green's function, and for each value ofk,
a weighting by a spectral function(k); this function distributes thedependence
over a certain frequency domain. We are going to show that from the spectral function
we can infer many properties of an interacting physical system. But this obviously
does not mean that the spectral function is easy to compute! On the contrary, in most
interacting physical systems, we do not know its exact value. Its very existence, however,
independently of its precise mathematical form, is a very useful conceptual tool.
3-c. Expression of various physical quantities
The spectral function contains information on a great number of physical proper-
ties of interacting systems, in a form much more concise than the density operator of
the-body system: that operator contains everything but is mathematically far more
complicated than a simple function.
Consider rst the particle density, given by:
(r) = (r) (r)=
1(r0;r0) (62)
according to the denition (2) of the normal function
1. For a translation invariant
system, this density is independent ofr, and we can setr= 0. The density is then the
value, at the origin, of the normal function (31), that is, taking (56) into account:
=
d
3
(2)
3
d
21(k) =
d
3
(2)
3
d
2
(k)(} ) (63)
Let us now study a quantity furnishing more precise information, the particle
momentum distribution, and compute the average numberkof particles having a
momentum}k. We assume the system is contained in a cubic box of volume. Relations
(A-9) and (A-10) of Chapter , applied to the case where the basis wave functions are
given by (12), yield:
k=
kk=
1
d
3
d
3 k(rr)
(r) (r) (64)
Replacing the integral variablerbys=rr, the average value (r) (r+s)
appears, which is independent ofrbecause of the translation invariance; we can then
replace, in this average value,rby zero, and the integral over
3
simply yields the
volume. We are left with:
k=d
3 ks
(0) (s) (65)
Now, performing the integral overof the denition (28) of
1(k), and taking (2)
1797

COMPLEMENT B XVI
into account, we get:
d
1(k) = 2d
3 kr
1(00;r0)
= 2d
3 kr
(0) (r) (66)
which is identical to (65) within a factor of2. It then follows that
7
:
k=
1
2
d
1(k) =
d
2
(k)(} ) (67)
In the same way, one can show that the system average energy
8
per unit volume
is given by:
=
d
3
(2)
3
d
2
}
+
2
(k)(} ) (68)
whereis dened in (14). Once this function is known, one can get, by integration
over, the logarithm of the partition function, which in turn yields, by derivation, all
the thermodynamic quantities (particle density, pressure, etc.). It is remarkable that
the spectral function, whose denition comes from the one-particle Green's functions
and could therefore be expected to only contain information on the one-particle density
operator, actually allows computing all these physical quantities that depend on the
correlations between the particles, and hence on their interactions. With this method, the
study of-body properties is reduced to the computation of functions mathematically
dened for a single particle. It generalizes, in a way, the ideal gas equations, while
taking rigorously into account the presence of an ensemble of indistinguishable particles
at thermal equilibrium. It is therefore quite powerful.
Nevertheless, this obviously does not solve the problem of calculating the equi-
librium properties of an interacting system; in practice, getting precise values for the
spectral function can pose a very dicult mathematical problem. Numerous approx-
imation methods have been developed to try and resolve this, using in particular the
concept of self energy and of perturbation diagrams, but this is beyond the scope of
this complement.
7
For an ideal gas of bosons, the chemical potential is always below the lowest individual energy, so
that the distribution functionnever diverges. As in relation (67)is integrated fromto+, this
divergence now seems unavoidable. Relation (55), however, shows that the spectral function(k)for
bosons goes to zero for~=, as long as the function
1remains regular at this point. Consequently,
integrals (63), (67) and (68) do not present any divergences associated with thatvalue
8
Relation (68) can be demonstrated by rst computing the time evolution of (r)and (r)
to get the expression for}
(r) (r); one then takes its average value and performs
a Fourier transformation to get the average value of the energy see for example 2.2 of reference [7].
1798

WICK'S THEOREM
Complement CXVI
Wick's theorem
1 Demonstration of the theorem
1-a Statement of the problem
1-b Recurrence relation
1-c Contractions
1-d Statement of the theorem
2 Applications: correlation functions for an ideal gas
2-a First order correlation function
2-b Second order correlation functions
2-c Higher order correlation functions
For an ideal gas at thermal equilibrium, we computed in ComplementXVthe
average values of one- and two-particle operators, and showed they could all be expressed
in terms of the one-body quantum distributions(the Fermi-Dirac distribution for
fermions, and Bose-Einstein for bosons). In this complement we establish a theorem
that allows generalizing those results to operators involving any number of particles.
The demonstration of Wick's theorem is explained in , and will be applied in
the calculation of correlation functions in an ideal gas.
1. Demonstration of the theorem
Let us consider an ideal gas at thermal equilibrium, described by the grand canonical
ensemble (Appendix, ), with the density operator:
=
1
(1)
where= 1 is the inverse of the temperature multiplied by the Boltzmann constant
,the chemical potential, andthe Hamiltonian:
=
k
with: =
~
22
2
(2)
The grand canonical partition functionis dened by:
=Tr (3)
1-a. Statement of the problem
We wish to calculate the average valueof a product of operators:
=12 (4)
1799

COMPLEMENT C XVI
where each of the operatorsis, either an annihilation operator, or a creation operator
:
= (5)
Taking into account relations (A-48) and (A-49) of Chapter, we have:
[]=
=
0ifandare both creation or annihilation operators
ifis an annihilation, anda creation operator
ifis a creation, andan annihilation operator
(6)
with= +1for bosons and=1for fermions.
Assuming the quantum state is described by the density operatorgiven in (1),
the average value ofis:
=Tr 12 (7)
As is diagonal in the basis of the Fock states associated with the, this average
value will be dierent from zero only if the series of operatorscontains, for each creation
operator, an annihilation operatorin that same individual state; they must exactly
balance one another and must therefore appear the same number of times. In particular,
the average value will always be zero ifis odd; from now on, we shall assume
that= 2,being an integer.
1-b. Recurrence relation
We have to compute:
2=Tr 12 2 (8)
We rst start by changing the order of1and2, using one of the relations (6); we will
then continue to progressively shift1towards the right, by permuting it rst with3,
then with4, etc. until the permutation with2brings it to the very last position. As
a trace is invariant under a circular permutation, the operator1can then be moved
back all the way to the rst position, ahead of; a last commutation with, that
we compute just below, returns it to its initial position, and allows computing the value
of2as a function of the average values of a product of2(1)operators. The
computation goes as follows:
2=Tr [12] 342+Tr 21342
= [12]Tr 342+[13]Tr 242
+Tr 23142
= [12]Tr 342+[13]Tr 242+
+
22
[12]Tr 23421
+
21
Tr 23421 (9)
1800

WICK'S THEOREM
Most of the terms on the right-hand side are in general zero: the rst is non-zero only if1
and2are two conjugated operators (an annihilation and a creation operator associated
with the same individual quantum state); the second is non-zero if this is also the case
for operators1and3, etc.
After a circular permutation under the trace, the last term of the sum can be
written as:
21
Tr1 23 2 (10)
We shall now relate both operators1and 1, showing that they are proportional
to each other. Assume, for example, that1is a creation operator; in the operator
dened in (1), all the terms=commute withand the change of order for the
operators leads to two expressions, to be compared:
( )
and
( )
(11)
By action on the Fock vectors, it is easy to check (as we assumed the system was in
thermal equilibrium) that:
( )
=
( ) ( )
(12)
which leads to:
=
( )
(13)
If1is an annihilation operator, the same reasoning shows that the change of order
introduces the inverse factor:
( )
. To sum up:
1=
(1)
1 (14)
with a+sign in the exponential if1is a creation operator, and asign if1is an
annihilation operator. Consequently, the last term in (9) is equal to
(1)
2,
with a factor
21
=since=1.
Moving this last term to the left-hand side, we get:
21
(1)
= [12]Tr 32
+[13]Tr 242+
+
22
[12]Tr 2321 (15)
On the right-hand side of this equality, all the (anti)commutators[12]are actually
numbers, and many of them are zero: as before, the only non-zero ones are those for
which the two concerned operatorsare conjugates of each other (a creation and an
annihilation operator for the same individual quantum state). The average value of
the product of2operators we are looking for can therefore be expressed as a linear
combination of average values of products of22operators.
1801

COMPLEMENT C XVI
1-c. Contractions
We now dene the contraction of two operatorsandas the number, written
, dened by:=
1
1
( )
[]= (( ))[] (16)
where, as above, in the denominator a+sign is chosen in the exponential if1is a
creation operator, and asign if1is an annihilation operator. The functionis the
Fermi-Dirac distributionfor fermions, and that of Bose-Einsteinfor bosons:
( ) =
1
( )
(17)
The contraction is zero if it concerns two operatorsandacting on dierent individual
quantum states; it is also zero if the operators are both creation or both annihilation
operators in the same individual quantum state. Ifis the creation operator, and
the annihilation operator in the same individual state, the contraction is simply equal to
the distribution function( )since:
=
=
1
+( )
=( ) (18)
In the opposite case (antinormal order), the contraction is given by:
=
=
1
1
( )
= 1 +( ) (19)
Relation (15) can thus be rewritten as:
2=
1232+13242++
22
12 2321 (20)
where the traces have been replaced with quantum averages.
We shall then reason by recurrence: each of the average values on the right-hand
side of (20) is of the same type as2written in (8), except for the fact thathas been
lowered by one unit. Dealing with each of the average values22as we did for2,
that last average value now appears as a double sum of terms containing two contractions
and average values24. Continuing as many times as necessary, we end up with an
average value2expressed as the sum of diverse products ofcontractions.
As an illustration, let us consider a few simple examples. If= 1, we get directly:
12=
12
(21)
This simple relation can actually be used as a denition of contractions, instead of (16).
If1is a creation operator and2the corresponding annihilation operator, we get the
result (18), equivalent to relations (19) and (23) of ComplementXV; if the operator's
order is reversed, we get (19) which comes directly from the previous result and from the
1802

WICK'S THEOREM
commutation or anticommutation relation (6). For all the other cases, we nd zero on
each side of the equality.
If= 2, we use a rst time relation (20), and obtain:
1234=
1234+1324+1423 (22)
Using again this same relation, we compute each of the average values of the product of
two operators, which yields:
1234=
1234+1324+1423
=
1234+1
234+
1
234 (23)
In the second line, we have used a generalization of the notation of products of contrac-
tions. When two operators inside a contraction are separated, a permutation is needed
to group them. For fermions, this introduces a sign given by the parity of the required
permutation, but no sign change for bosons. When two contractions are embedded, we
group together all pairs of operators belonging to the same contraction and, for fermions,
we multiply by the parity of the corresponding permutation
1
; for bosons, no sign change
is introduced. In the present case, we therefore have :
1
234=
1324 and1
234=
1423 (24)
The nal result (23) only contains products of contractions, i.e. of distribution
functions. One can easily check that, among those three products, a maximum of two
are non-zero.
Comments:
(i) Another notation is frequently used, where operators and contractions are embedded
in the same average value, for instance:
12
2= ()12 2 (25)
where, for fermions,is the parity of the permutation needed to bring operatornext
to; for bosons,= 1. This can be generalized to cases where several contraction
appear, embedded or not.
(ii) In the limit of zero temperature where, relation (16) simplies into:
= 0if the two operators are of the same nature (both creation,
or both annihilation operators) (26)
as well as:
=
1si
0si
(27)1
We multiply by1if, when writing the permutation, the number of crossings between brackets is
odd. This is for instance the case in the permutation in the left of (24), but not that on the right.
1803

COMPLEMENT C XVI
and:
=
0si
1si
(28)
(the second lines () of these relations are useful only for fermions, since for bosons
cannot be larger than).
1-d. Statement of the theorem
The recurrence overwe have been using leads to Wick's theorem:
The average value12 2is the sum of all the complete systems of
contractions that can be made on the string of operators12 2. Each
system is the product of binary contractions (16); for fermions, this product is
multiplied by parity factorsassociated with each of them.
The word complete means in this case that in every considered system of con-
tractions, each operator listed in the string ofoperators is taken in one and only
one contraction. The parity factor rst includes the parity
1of the permutation that
brings right after1the operator it is contracted with; these two operators are then
taken out of the list of the. In the remaining list, we again compute the parity
2
of the permutation needed to bring together the next two operators to be contracted,
and it is multiplied by
1
. We continue this until all the contractions have been taken
into account, and obtain the product
1 2 of all the parities involved. Among all
the system of contractions, a very large number yield zero. The only non-zero ones are
those for which every contraction contains a creation and an annihilation operator in the
same individual quantum state. This rule signicantly limits the number of contractions
involved in the nal result.
As seen above, the theorem yields again the results of ComplementXV. For
example, if (as is the case in the formula for the two-particle symmetric operators) the
rst two operatorsare creation operators, and the last two annihilation operators,
the rst system of contraction in (23) yields zero, and we are left with the last two,
corresponding to the two terms of equation (43) in ComplementXV. The main interest
of the theorem is, however, that it allows getting, almost without calculations, the average
value of the product of any number of operators.
Comment:
Until now, we assumed that the operatorswere creation or annihilation operators
associated with the basis of individual states formed by the one-particle Hamil-
tonian eigenvectors. If this is not the case, and we wish to compute the average
value of the product of creationand annihilationoperators associated with
any other basis, we rst use formulas (A-51) and (A-52) of Chapter
those operators in terms of the operators associated with the eigenbasis of the
one-particle Hamiltonian, and then use Wick's theorem.
2. Applications: correlation functions for an ideal gas
As an illustration of the use of Wick's theorem, we now compute the-order correlation
functions in an ideal gas at thermal equilibrium. Thanks to Wick's theorem, they can
1804

WICK'S THEOREM
each be expressed as simple products of rst order correlation functions. As a rst
step, we will derive, in a simpler way, a number of results already obtained in
ComplementXV; these will then be generalized to correlation functions of a higher
order.
Consider a gas of spinless particles, conned by a one-body potential inside a cubic
box of edge length; this potential is zero inside the box, and becomes innite outside.
We use periodic boundary conditions to account for this connement (ComplémentXIV,
); the normalized eigenfunctionsk(r)of the kinetic energy are then written:
k(r) =
1
32
kr
(29)
where the possible wave vectorskare those whose three components are integer multiples
of2.
2-a. First order correlation function
Relation (B-21) of Chapter 1,
which depends on the two positionsr1andr
1:
1(r1r
1) = (r1) (r
1) (30)
Using relations (A-3) and (A-6) of Chapter , the eld operator can be expressed as a
function of the annihilation operatorskin the state (29), and its adjoint, as a function
of the creation operators
k
in that same state. Taking into account (29), this leads to:
1(r1r
1) =
1
3
kk
(kr
1
kr1)
kk (31)
At thermal equilibrium, all the average valuesof operatorsare taken in the
state described by the density operatorwritten in (1):
=Tr (32)
We can then use Wick's theorem in a particularly simple case, since in (31) the only
contraction that comes into play is the one containing
kk . Relation (18) thus
applies and we get:
1(r1r
1) =
1
3
k
k(r
1
r1)
(k ) (33)
The correlation function1(rr)is therefore directly (to within a constant factor) the
Fourier transform of the distribution function(k )itself.
The denition of1can be generalized, using the expressions of the eld operators
in the Heisenberg picture; this leads to a correlation function depending on space and
time:
1(r1;r
1) = (r1) (r
1) (34)
1805

COMPLEMENT C XVI
For free particles (ideal gas), we have ( XV):
(r) =
1
32
k
(kr )
k (35)
whereis the (angular) Bohr frequency associated with the energy of a particle of mass
, with wave vectork:
=
}
2
2
(36)
For an ideal gas, we simply multiply each exponential
kr
by e to go from the
Schrödinger to the Heisenberg representation. Expression (33) is then generalized as:
1(r1;r
1) =
1
3
k
[k(r
1
r1)()]
(k ) (37)
Note that this correlation function only depends on the dierences in positions (space
homogeneity) and times (time translation invariance).
2-b. Second order correlation functions
. Application of Wick's theorem
The second order correlation function is dened as:
2(r1r
1;r2r
2) = (r1) (r2) (r
2) (r
1) (38)
Here again, the average value is computed with the density operator at thermal equilib-
rium. The same calculation as in
2(r1r
1;r2r
2) =
1
6
kkkk
(kr
1
+kr
2
kr1kr2)
kkk k (39)
As we already saw in (23), using Wick's theorem yields two contraction systems, one
where
k
is contracted withk(and hence
k
withk), and another one where
k
is
contracted withk(and hence
k
withk):
kkk k
et
kkk k
The second contraction involves an odd permutation, and introduces a factor. We
therefore obtain (after changing the dummy indexkintok):
2(r1r
1;r2r
2)
=
1
6
kk
[k(r
1
r1)+k(r
2
r2)]
+
[k(r
2
r1)+k(r
1
r2)]
(k )(k ) (40)
that is, taking (33) into account:
2(r1r
1;r2r
2) =1(r1r
1) 1(r2r
2) + 1(r1r
2) 1(r2r
1) (41)
1806

WICK'S THEOREM
This means that the second order correlation function is simply expressed as the sum of
two products of rst order correlation functions. The rst is the direct term, and corre-
sponds to totally uncorrelated particles. The second is the exchange term, a consequence
of the quantum indistinguishability of the particles; it has an opposite sign for fermions
and bosons. As in ComplementXVI, we will show that this term introduces correlations
between the particles.
. Double density
Of particular interest is the diagonal case wherer1=r
1andr2=r
2, as the
function2(r1r
1;r2r
2)becomes very simple to interpret: it is the double density
2(r1r2)characterizing the probability of nding a particle at pointr1and another one
at pointr2. The above relation takes on the simplied form:
2(r1r2) =1(r1r1) 1(r2r2) + 1(r1r2) 1(r2r1) (42)
If, in addition,r1=r2, this function indicates the probability of nding two particles at
the same point. We then get:
for fermions2(r;r) = 0
for bosons 2(r;r) = 2 [1(rr)]
2 (43)
For fermions, we see as expected that one can never nd two of them at the same
point in space, a consequence of Pauli exclusion principle. For bosons, we nd that
the double density is twice the square of the one-body density. Now, if both particles
were uncorrelated, this double density should simply be equal to the square, without
the factor two. This factor two thus indicates an increase in the probability of nding
two bosons at the same point in space; it expresses the bunching tendency of identical
bosons, a tendency that comes from a pure quantum statistical eect since we assumed
the particle's interactions to be zero. These results were already discussed in Complement
AXVI see in particular Figure.
The Hartree-Fock method (mean eld approximation), presented in Complements
EXVandXV, uses a variational ket (or a density operator) such that the binary cor-
relation function2(r1r2)is given by the sum of products of functions1, written in
(42), even in the presence of interactions. Moreover, another way of introducing the
Hartree-Fock approximation is to assume directly that the binary correlation function
keeps this form even in the presence of interactions, which then allows a simple calcu-
lation of the interaction energy. Even though this method has numerous applications,
and may be quite precise in certain cases, it does rely on an approximation: when the
particles interact with each other, there is no general reason for2to remain linked to
1by this relation, obtained with the assumption that the gas was ideal.
. Time-dependent correlation function
As we did for the rst order correlation function, we can include time dependence
in the eld operators, and dene:
2(r11;r
11;r22;r
22) = (r11) (r22) (r
22) (r
11) (44)
1807

COMPLEMENT C XVI
To include the time dependence, we simply add, as above, in each spatial exponential
with wave vectork, a time exponential with the corresponding angular frequency,
which leads to:
2(r11;r
11;r22;r
22) =
1
6
kk
e
i[k(r
1
r1) (
1 1)+k(r
2
r2) (
2 2)]
+e
i[k(r
2
r1) (
2 1)+k(r
1
r2) (
1 2)]
(k )(k )
= 1(r11;r
11)1(r22;r
22) + 1(r11;r
22)1(r22;r
11) (45)
Hence, when time dependence is included, we get a factored relation similar to
(41). As before, because of the space homogeneity and the time translation invariance,
only the dierences in the space and time variables appear in the correlation function
expression.
2-c. Higher order correlation functions
In a more general way
2
, the-order correlation functionis dened by:
(r1r
1;r2r
2;;rr) = (r1) (r2) (r) (r) (r
2) (r
1) (46)
These functions give information on the correlated behavior of groups ofparticles in
an ideal gas at equilibrium. Using Wick's theorem, each of them can be expressed in
terms of the rst order correlation function1(r1r
1). As an example, let us study the
correlation function of order three:
3(r1r
1;r2r
2;r3r
3) = (r1) (r2) (r3) (r
3) (r
2) (r
1)
=
1
9
kkkkkk
(kr
1
+kr
2
+k r
3
kr1kr2kr3)
kkk k k k (47)
Six contraction systems must be considered to compute the average values. In the rst
system,kandkare associated,kwithk, and nallykwithk. One can
then permute the three vectorsk,kandkin5dierent ways, with odd or even
permutations. In each of the terms thus obtained, the sixfold summation on the wave
vectors is reduced to a triple sum, which yields a product of functions1. This leads to:
3(r1r
1;r2r
2;r3r
3) =1(r1r
1)1(r2r
2)1(r3r
3)
+1(r1r
2)1(r2r
3)1(r3r
1) +1(r1r
3)1(r2r
1)1(r3r
2)
+ 1(r1r
1)1(r2r
3)1(r3r
2) + 1(r1r
3)1(r2r
2)1(r3r
1)
+ 1(r1r
2)1(r2r
1)1(r3r
3)
(48)
The computation can be generalized in the same way to correlation functions of
any order; in an ideal gas, they are not independent since they are all simple products
2
We only consider here the so-called normalcorrelation functions, those where the come before
the . In ComplementXVIwe introduce more general correlation functions.
1808

WICK'S THEOREM
of rst order correlation functions. In other words, the function1contains all the
information necessary for computing correlations of any order.
Finally, we can compute the triple density3(r1r2r3)by settingr1=r
1,r2=r
2
andr3=r
3in (48). The particular caser1=r2=r3where all the positions are identical
is interesting. For fermions, the triple density is zero, for the same reason as with the
double density: Pauli principle does not allow several fermions to occupy the same point
in space. For bosons, we nd:
3(rr;r) = 6 [1(rr)]
3
(49)
For three identical bosons, the bunching tendency under the eect of their quantum
statistics is even higher than for two bosons, introducing a factor6instead of2.
Comment:
The results we have obtained are valid when the system's density operator is that of an
ideal gas at thermal equilibrium as in relation (1), but they could be quite dierent if the
system is in another state. If, for example, we assume (as in ComplementXVI) that the
system is described by a Fock state, the relations between correlation functions can be
totally dierent. The simplest case is that of an ideal gas of bosons in its ground state,
where all the bosons occupy the same individual state; relations (24) and (25) of that
complement indicate that:
2(r1r2) =
1
1(r1r1) 1(r2r2) 1(r1r1) 1(r2r2) (50)
2is thus simply the product of two functions1, without the exchange term of (42);
consequently, the factor2in the second line of (43) no longer exists. In a similar way,
one can show that the factor6of relation (49) is no longer present. In a general way,
for an ensemble of bosons all in the same individual state, the bunching eects related
to the indistinguishability of the particles are not present.
Following this line of thought, note that it is not possible to get the projector onto a
Fock state (other than the vacuum) such as the one discussed above, by using the density
operator (1) at thermal equilibrium, and taking its limit as the temperaturegoes to
zero, i.e. as . This is because this density operator associates with each individual
state an occupation number distribution that is always a decreasing exponential, and
never a narrow curve centered around a high value of the particle number. Consequently,
there exist large uctuations of the particle number in each mode, and hence the presence
of the factors2in (43) and6in (49), whatever the value of.
To conclude, let us mention that Wick's theorem can take on diverse forms, in
particular at zero or non-zero temperadure (see for instance Chapter 4 of Reference [5]).
As we saw, thanks to this theorem, and when dealing with independent particles, the
computation of correlation functions of any order, time-dependent or not, can be reduced
to computing the product of rst order correlation functions. It is obviously a great
simplication. This property is reminiscent of random Gaussian variables in classical
statistics: for such variables, all moments of any order can be expressed in terms of
products of the lowest order moment. These properties are characteristic of an ideal gas:
in a system where particles interact, the correlation functions of successive orders remain
independent in general. Nevertheless, the use of Wick's theorem is not limited to ideal
gases; its range of application is much more general, and it is very useful in perturbation
calculations where power series of the interaction potential are derived [5].
1809

Chapter XVII
Paired states of identical
particles
A Creation and annihilation operators of a pair of particles
A-1 Spinless particles, or particles in the same spin state
A-2 Particles in dierent spin states
B Building paired states
B-1 Well determined particle number
B-2 Undetermined particle number
B-3 Pairs of particles and pairs of individual states
C Properties of the kets characterizing the paired states
C-1 Normalization
C-2 Average value and root mean square deviation of particle number1825
C-3 Anomalous average values
D Correlations between particles, pair wave function
D-1 Particles in the same spin state
D-2 Fermions in a singlet state
E Paired states as a quasi-particle vacuum; Bogolubov-Valatin
transformations
E-1 Transformation of the creation and annihilation operators
E-2 Eect on the kets
k
. . . . . . . . . . . . . . . . . . . . . .
E-3 Basis of excited states, quasi-particles
Introduction
Fock states were introduced in Chapter
individual state creation operators. A certain number of their properties were studied in
of Chapter XVI(exchange hole for fermions, bunching
Quantum Mechanics, Volume III, First Edition. C. Cohen-Tannoudji, B. Diu, and F. Laloë.
© 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

CHAPTER XVII PAIRED STATES OF IDENTICAL PARTICLES
eect for bosons). We also used Fock states in ComplementsXV,XVandXVas
variational kets to account, approximately, for the interactions between the particles.
This led us, both for fermions and bosons, to a mean eld theory where each particle
can be seen as propagating in the mean eld created by all the others.
We now introduce a larger class of variational states to improve the accuracy of
these results, allowing us to study many more properties of physical systems of identical
particles. It concerns the paired states obtained by the action on the vacuum of
a product of creation operators, no longer of individual particles but rather of pairs
of particles (if these particles form a molecule, we are dealing with molecule creation
operators). As we shall see in the course of this chapter, these paired states are more
general than the Fock states, since they can be reduced to Fock states for certain values
of the parameters characteristic of the pair
1
. What is sought is an improvement of the
variational method allowing us to ameliorate our treatment of the interactions, compared
to that based on Fock states.
The additional exibility introduced by the paired states plays an essential role for
the following simple reason: changing the properties of the pair wave function(r1r2)
used to build them, we modify the binary correlation function of the-particle system.
We therefore take advantage of the power of the mean eld method, while retaining the
possibility of taking into account any binary correlations. Whereas using variational Fock
states allows taking into account only statistical correlations (due to particle indistin-
guishability), the paired states enable us to add dynamic correlations (due to interac-
tions). These latter correlations are essential: when dealing with binary interactions (as
is the case with a standard Hamiltonian), these correlations actually determine the av-
erage value of the potential energy (Chapter, ). Three-body, four-body, etc..
correlations are indeed present in the system, playing their role; they are not, however,
directly involved in the energy. This explains why the optimization with paired states
of only the binary correlations can lead to fairly good results in the study of-body
systems. These possibilities have a wide range of applications for both fermions and
bosons, which will be discussed in the complements.
This chapter is centered on the study of the general properties of paired states,
and introduces the tools for handling such states. We study, in parallel, fermions and
bosons to highlight the numerous analogies between results obtained for both cases. We
rst introduce () the creation and annihilation operators for pairs of particles. We
then build () the paired states and discuss some of their properties; this permits
introducing () the concept of normal average values (average values of operators
conserving the particle number) or anomalous average values (average values of oper-
ators changing the particle number). We then show in
us to actually vary the spatial correlation functions of a system of identical particles.
This will lead us to introduce a function playing an important role in what follows (in
particular in the complements of this chapter), the pair wave functionpair, which is
related to the anomalous average values. We then study in
erty of the paired states: they can be related to the concept of quasi-particle thanks
to the introduction of new creation and annihilation operators resulting from a linear
transformation of the initial operators (Bogolubov transformation). As the paired states
are eigenkets of the new annihilation operators with a zero eigenvalue, they behave as
1
For example, we shall clarify at the end of
a particular case of the pairing method.
1812

A. CREATION AND ANNIHILATION OPERATORS OF A PAIR OF PARTICLES
a quasi-particle vacuum. Furthermore, the creation operators can associate with each
paired state an entire basis of other orthogonal states, which are interpreted as states
occupied by quasi-particles.
This study of the necessary tools for handling the paired states will be continued in
the rst two complements,XVIIandXVII. ComplementAXVIIdiscusses a complemen-
tary aspect of pairing, the introduction of the pair eld operators. These operators have
a non-zero average value in paired states, and highlight the cooperative eects existing
in those states. This can lead to the spontaneous appearance in the system of an order
parameter, described by the same pair wave functionpairas the one appearing in the
computations of correlation functions in a paired state. In addition, ComplementXVII
will show that the commutation properties of these operators are reminiscent of those
of a boson eld: in a certain sense, a composite object built from two identical particles
(whether they are bosons or fermions) behaves as a boson. It is, however, only an ap-
proximation, as can be inferred from the corrective terms appearing in the computation
of the commutators, which can sometimes play an important role. ComplementXVII
discusses the computation of the energy average value in a paired state, whose expression
is the basis of the following complements; it gives an example of how to deal with normal
and anomalous average values in these computations.
The last three complements apply these results to the variational study of inter-
acting boson or fermion systems. For fermions, the paired states play an essential role in
the BCS (Bardeen-Cooper-Schrieer, theory of supraconductivity) theory of supracon-
ductivity (ComplementXVII), and explain the appearance of a pair eld as a collective
eect; paired states also come into play noticeably in nuclear physics, and in the study
of ultra-cold fermionic atomic gases. For repulsive bosons (ComplementXVII), paired
states can be quite useful for studying the ground state properties, and to obtain, for
example, the Bogolubov linear spectrum. In that case, the paired state is associated with
another state (a coherent state, for example), whose role is to describe the condensate as
an accumulation of a notable fraction of particles in a single individual quantum state.
A. Creation and annihilation operators of a pair of particles
Let us introduce the creation or annihilation operators, no longer of a single particle, but
of two identical particles in a bound state. We rst assume the particles have no spin
(or are both in the same spin state, so that no spin variable is needed).
A-1. Spinless particles, or particles in the same spin state
Consider two identical particles (bosons or fermions in the same spin state), with
positionsr1andr2; the system is contained in a cubic box of edge lengthand volume
=
3
. These two particles occupy a bound state, characterized by a normalized wave
function(r1r2), forming a kind of binary molecule. The state of the system is
dened by this wave function (as far as its internal orbital variables are concerned), by
spin variables identical for both particles (since those spin variables are of no importance
here, they need not be written explicitly in what follows), and nally by its external
orbital variables (center of mass). The normalized wave function of a molecule having
1813

CHAPTER XVII PAIRED STATES OF IDENTICAL PARTICLES
a total momentum}Kis then:
K(r1r2) = ()
32K(r1+r2)2
(r1r2)
= ()
3
k
k
(
K
2
+k)r1(
K
2
k)r2
(A-1)
wherekis the Fourier transform of:
k=
1
32
3
3 kr
(r)
(r) =
1
32
k
k
kr
(A-2)
We assume that the individual wave functions of the particles obey the periodic boundary
conditions (ComplementXIV, 1-c); in (A-1), each component of the wave vector of
particle1or2can therefore take only the values2 ,2 and2 , where
,andare any integer number (positive, negative, or zero). The normalization
of the functionsandis written:
3
d
3
(r)
2
=
k
k
2
= 1 (A-3)
Moreover, for identical particles, the symmetrization (or antisymmetrization) requires
the function(r)and its Fourier transform(k)to have the parity:
k= k (A-4)
(= +1for bosons,=1for fermions).
In terms of kets, relation (A-1) becomes:
K(12)= ()
3
k
d
3
1d
3
2k
(
K
2
+k)r1(
K
2
k)r2
1 :r1; 2 :r2
=
k
k1 :
K
2
+k; 2 :
K
2
k (A-5)
which, taking (A-4) into account, and changing the sign of the sum variablek, can also
be written as:
K(12)=
1
2
k
k1 :
K
2
+k; 2 :
K
2
k
+1 :
K
2
k; 2 :
K
2
+k (A-6)
The expression between brackets in the summation is simply the (anti)symmetrized ket
of two particles, the rst one of momentum}(k+K2), and the other one of momentum
}(k+K2). Two cases must be distinguished:
(i) Ifk= 0, to normalize the ket between brackets, we divide it by
2; we then
get a Fock state where two individual states with dierent momenta are occupied (see
1814

A. CREATION AND ANNIHILATION OPERATORS OF A PAIR OF PARTICLES
the general denition of the Fock states in Chapter). The ket between brackets is
thus equal to:
2
K
2
+k
K
2
k
0 (A-7)
(ii) Ifk= 0and in the case of bosons, the ket between brackets is equal to twice
the Fock state where a single individual level is occupied by two particles; this ket is
equal to:
2
K
2
2
0 (A-8)
For fermions, the ket between brackets must be zero, which is indeed the case of the ket
in (A-8). To sum up, whether we are dealing with fermions or bosons, and whetherkis
zero or not, the ket between brackets can always be expressed as (A-7). This leads to:
K=
1
2
k
k K
2
+k
K
2
k
0 (A-9)
If the particles are all in the same spin state, remember that in this expression the spin
index is implicit: each creation operator is associated with an individual state whose
momentum is specied by the operator index, and whose spin state is the common spin
state of all the particles.
The creation operator
K
of a molecule having a total momentum}Kcan
therefore be written as:
K
=
1
2
k
k K
2
+k
K
2
k
(A-10)
Its action is to create two particles of momenta}[(K2)k], with amplitudes given by
the functionk. As this function has the parity, we note that:
k K
2
k
K
2
+k
= k K
2
k
K
2
+k
=k K
2
+k
K
2
k
(A-11)
Accordingly, the contributions of opposite values ofkdouble each other in (A-10). Such
a redundancy will cause a problem in , when we write a tensor product. It is thus
preferable to eliminate it right now and this is why we restrict the summation overkto
half the wave vector space. Callingthis half space, we shall write
K
in the form:
K
=
2
k
k K
2
+k
K
2
k
(A-12)
For a molecule having a zero total momentum, this relation becomes:
K=0
=
2
k
kkk
(A-13)
As for the annihilation operator of a molecule with total momentum}K, it is simply the
adjoint of (A-12):
K=
2
k
k
K
2
k
K
2
+k
(A-14)
1815

CHAPTER XVII PAIRED STATES OF IDENTICAL PARTICLES
We have reasoned in terms of molecules being created or annihilated, but the
wave function(r)and its Fourier transformkare not related to any particular bound
state, and do not imply the existence of any attraction potential between the two com-
ponents of this molecule. Actually, in what follows, thekwill play the role of freely
adjustable parameters, for example when using a variational method. To illustrate this
generality, we shall, from now on, talk about pairs.
Comment:
If we choose forkas in a Kronecker delta, with the symmetrization required by
(A-4):
k=
1
2
[kk0
+ kk0
] (A-15)
we get, according to (A-12):
K
=
1
2
K2
+k0
K
2
k0
+
K
2
k0
K
2
+k0
=
K
2
+k0
K
2
k0
(A-16)
In the right-hand side of this relation, the momenta appearing as indices of the
creation operators can take any given values, obtained by varyingKandk0.
It is therefore possible, by a suitable choice of the pair's parameters, to create
two particles in individual states having any given momenta, and thereby obtain a
Fock state. Successive applications ofoperators
K
(having, in general, dierent
values ofKandk0) can thus yield a Fock state with2particles whose momenta
can take on any values.
A-2. Particles in dierent spin states
We assume the internal state of the pair is a tensor product of an orbital state
depending onr1r2and a spin state. Equation (A-1) must then be replaced by:
12
K
(r1r2) =1 :r11; 2 :r22K
= ()
32K(r1+r2)2
(r1r2)12
= ()
3
k
k
(
K
2
+k)r1(
K
2
k)r2
12 (A-17)
This means that relation (A-1) is to be multiplied by12 ; relation (A-5) is now
written:
K(12)= ()
3
d
3
1d
3
2
k
k
(
K
2
+k)r1(
K
2
k)r2
12
12 1 :r11; 2 :r22
=
k
k
12
12 1 :
K
2
+k 1; 2 :
K
2
k 2 (A-18)
1816

A. CREATION AND ANNIHILATION OPERATORS OF A PAIR OF PARTICLES
The function(r1r2)is supposed to have an orbital parity equal to, and the spin
ket , a parity with respect to the exchange of spins equal to, with, obviously:
= (A-19)
Hence:
K(12)=
1
2
k
k
12
12 1 :
K
2
+k1; 2 :
K
2
k2
+1 :
K
2
k2; 2 :
K
2
+k1 (A-20)
which shows that the creation operator of a pair is:
K
=
1
2
k
k
12
12 K
2
+k1
K
2
k2
(A-21)
As an example, for two fermions of spin12in a singlet state:
K
=
1
2
k
k K2
+k=+
K
2
k=
K
2
+k=
K
2
k=+
(A-22)
Since=1, the functions(r)andkare even. Using this parity, we can exchange
the dummy indiceskandkin the second term on the right-hand side, and change the
order of the two creation operators, with a sign change (anticommutation of fermionic
operators). This second term then doubles the rst one, and we get:
K
=
k
k K
2
+k
K
2
k
(A-23)
with the simplied notation we shall use from now on:
k=+ noted: k
k= noted: k (A-24)
(and, of course, a similar notation for the creation operators). Note in passing that,
because of the presence of spins, no redundancy is present in the summation appearing
in (A-23), and there is no need to restrict it to a half-space.
Comments:
(i) Taking forka (symmetrized) delta function, as in (A-15), it is possible, as we pointed
out before, to construct any Fock state with arbitrary momenta by successive application
of operators
K
on the vacuum; note, however, that the total occupation numbers of the
two spin states must remain equal.
(ii) Choosing in (A-22) a functionkthat is even instead of odd for fermions, the operator
written in (A-23) creates a fermion pair with a total spin state= 1, and a= 0
component. This is because replacing the minus sign by a plus sign in the middle of the
bracket of relation (A-22) yields a triplet spin state; using the fact thatkis now an odd
function, the same reasoning leads to (A-23).
1817

CHAPTER XVII PAIRED STATES OF IDENTICAL PARTICLES
B. Building paired states
To avoid complex formulas, we will build the simplest possible paired states . We
shall be guided by the Gross-Pitaevskii variational method (ComplementXV), where
we assumed that the state of the-particle system could be obtained from the vacuum
by creatingparticles in the same individual state. However, instead of applying many
times the creation operator
k
of a single particle to the particle vacuum, we shall now
use the pair creation operator
K
. This dierence is essential, in particular for fermions.
As we know, it is impossible to create several fermions in the same individual quantum
state, since the square, cube, etc. of any creation operator acting on a given individual
state yields zero. We shall see, however, that the creation ofpairs of fermions, all in
the same quantum state, does not lead to a zero state vector.
B-1. Well determined particle number
We dene the paired state (K)as the (non-normalized) state vector where
= 2particles formpairs, each having a total momentum}K:
(K)=
K
0 (B-1)
where
K
has been dened in (A-12) or (A-21), depending on the case. To keep things
simple, we assume in what follows that all the created pairs have zero total momentum;
if this is not the case, we can change the reference frame and choose the one where the
common value of the total momentum of all the pairs of particles is zero. The paired
state is then written:
=
K=0
0 (B-2)
We shall rst study (as in ) the case of bosons or fermions in the same spin
state. As for the case of particles in several spin states (as in ), we shall, from now
on, limit our study to fermions in a singlet state; this will allows exposing the general
principle while avoiding more complex calculations. In both cases, the2-particle state
only depends on the values of the parametersk. As soon as1, we will see that the
normalization of the ket (K)does not reduce to the simple condition (A-3), which
required the sum of thek
2
to be equal to unity. This is why we shall consider from now
on that thekare totally free variational parameters. For example, multiplying them
all by the same constant, one can choose to vary at will the norm of (K). This will
oer a exibility simplifying the computations.
B-1-a. Particles in the same spin state
For particles in the same spin state, we can use (A-13), which leads to:
=
k
2kkk
0 (B-3)
whereis the summation domain dened previously (half of thekspace); remember
that the physical system is assumed to be contained in a cubic box of side lengthand
1818

B. BUILDING PAIRED STATES
volume=
3
; the periodic boundary conditions then xes all the possible values for the
summation overk. Note also that the spin index is implicit:
k
is the creation operator
in the individual state dened by the momentum}kand the unique spin state we are
concerned with.
Initially, the parameterskwere introduced as the Fourier components of the
normalized pair wave function(r); the sum of their moduli squared was xed to unity.
This condition, however, does not ensure the normalization of , as we now show.
Because of the powerof the operator appearing in (B-3), factors containing square
roots of occupation numbers will be introduced for each occurrence of the indexk; the
ket is therefore not a simple tensor product, and its norm is not simply the sum of
the squared moduli of thekraised to the power. It will be simpler for the following
computations to consider thekas entirely free parameters, and hence not impose a
normalization of the state .
On can choose to take a nite or innite number of non-zerok. The simplest
case is the one already discussed above, wherek kk0
; the ket then becomes
proportional to a simple Fock state where only two states of opposite momenta are
occupied. For other functionsk, the structure of the paired state will be more complex;
adjusting those parameters allows a ne tuning of the particle correlation properties,
which is not possible with a simple Fock state.
B-1-b. Fermions in a singlet state
Another frequently encountered case concerns fermionic particles in a singlet state;
we must then use operator (A-23). The paired state is then:
=
K=0
0=
k
kk k
0with= 2 (B-4)
The summation overkruns over all the non-zero wave vectors, without the restriction
(B-3) to the half-space(because of the spins, the pairs of statesk,kandk,
kare dierent).
Here again we see that the normalization of does not simply reduce to condi-
tion (A-3). When 1, the same indexkmay appear twice (or more) in the expansion
of the powerof the operator on the right-hand side of (B-4); the corresponding com-
ponent cancels out since the square of any fermionic creation operator is zero. The norm
of the ket is therefore a complex expression. Rather than imposing this norm to be
equal to one, it is easier to let it vary, and consider thekto be totally free variational
parameters.
B-1-c. Consequences of the symmetrization
The state vectors (B-3) for bosons, and (B-4) for fermions, are not simple jux-
tapositions ofpairs of particles, each being described by the relative wave function
(r), withkas its Fourier transform, according to (A-2). As we already saw, the sym-
metrization or antisymmetrization of the2-particle paired states strongly aect their
norm; it also aects the very structure of these states, which are not merely the tensor
product ofpair states. This is particularly obvious for fermions: expanding the sum
of operators to the powerin the curly brackets of (B-4), we get the product ofsums
1819

CHAPTER XVII PAIRED STATES OF IDENTICAL PARTICLES
over indicesk1,k2,..,k, and many terms will cancel out: all those for which two (or
more) summation indiceskare equal (in which case we get squares of creation operators,
which are zero for fermions).
There exists, however, a limiting case where the paired state practically describes
the juxtaposition ofbinary molecules. It occurs when the wave function(r1r2)
varies over a very short range, and hence has a large number of signicant Fourier com-
ponentsk. When this number is much larger than the number of pairs, most of
the terms do not contain several occurrences of the same summation indexk, and conse-
quently the paired state vector is very close to the tensor product of the pairs of particles.
This state vector describes a gas formed by very strongly bound binary molecules, each
moving in the mean eld created by all the others (it is, in a certain sense, a molecule
Fock state). This is actually a very special case; in general, when the pair wave function
does not obey that criterion, we can study many other physical situations, hence the
interest for introducing paired states.
Even though the values ofkor the wave function(r)of a molecule are math-
ematically the starting ingredient that allows building , the resulting state after
symmetrization has a complex structure, hard to describe in terms of molecules. On the
other hand, this state has a simple property: it contains exactly= 2particles, since
this is the case for each of its non-zero components; as all the particles are paired, it
contains exactlypairs.
B-2. Undetermined particle number
Computations with the ket written above (and in particular its normalization)
are not easy: a great number of individual stateskappear inside the curly brackets, which
must be raised to a very large power. This practical diculty leads us to introduce
another variational state pairedwhere the total number of particles is no longer xed.
This new state, which leads to simpler calculations
2
, is dened, starting with (B-2), by:
paired=
=0
1
!
=
=0
1
!
K=0
0 (B-5)
The are not normalized; multiplying all thek, and henceK=0, by the same
constant, changes their norm by the factor. This results in varying the relative
weights of the terms in the serie (B-5). The larger, the more weight is placed on the
high values of, which is a way, for example, of modifying the average particle number.
In (B-5) we recognize the series expansion of an exponential, so that:
paired= exp
K=0
0 (B-6)
This property will greatly simplify the following calculations and is the major reason for
letting the total particle number uctuate.
Writing (B-5), we chose a state vector that is the superposition of states corre-
sponding to dierent total particle numbers; there are actually no physical processes
taken into account in our approach that could create such a coherent superposition. This
2
This does not mean that computations with a variational state having a xed number of particles
are always impossible, as shown for example in the treatment of the BCS theory in 5.4 and Appendix
5C of the book [8].
1820

B. BUILDING PAIRED STATES
operation reminds us of the passage from the canonical to the grand canonical ensemble
where one introduces, for mathematical convenience, an (incoherent) statistical mixture
of dierentvalues. In our present case, however, we are dealing with a coherent su-
perposition, introduced arbitrarily as we just did, and we may wonder whether it might
radically change the physics of our problem. This is actually not the case for two reasons.
The rst is that, for very large values of, we are going to show that the components of
pairedare only important in a domain ofwhose width is very small compared to the
average value of the particle number; the distribution of the possible values foris thus
very narrow, in relative value, and the particle number remains quite well dened. The
second reason is that we shall compute average values of operators that, such as, con-
serve the total particle number, and for which the coherence of the state vector between
kets of dierentvalues is irrelevant. The average value in the coherent state paired
is therefore the weighted average of the average values obtained for eachwhich, when
the average value of the particle number is very large, are approximately the same (since
thedistribution is very narrow). In other words, the average values we are going
to compute are good approximations of those we would obtain by projecting paired
onto one of its main components with xed; using the coherent superposition (B-5) is
thus very convenient from a mathematical point of view, without greatly perturbing the
results from a physical point of view. A more detailed discussion of this question will be
presented in XVII.
B-2-a. Particles in the same spin state
When all the particles are in the same spin state, inserting (A-13) in (B-6) leads
to:
paired= exp
k
2kkk
0 (B-7)
The operators
kk
and
k k
commute with each other (for fermions in the same
spin state, two minus signs cancel each other as we commute products of two operators).
It then follows that the exponential of the sum is a product of exponentials, and we can
write:
paired=
k
exp
2kkk
0
=
k
k (B-8)
The state vector pairedis then simply a tensor product
3
of state vectorsk:
k= exp
2kkk
0 (B-9)
3
The Fock space is the tensor product of the states associated with all the individual quantum states
k, each having any positive occupation number. One can regroup those spaces in pairs corresponding
to opposite values ofk, and introduce spaces(k)whose tensor product is also the Fock space. To
build a basis in those spaces, one must vary two occupation numbers.
The restriction of the summation overkto a half-space, introduced above, prevents each component
of the tensor product from appearing twice in (B-8).
1821

CHAPTER XVII PAIRED STATES OF IDENTICAL PARTICLES
For fermions in a single spin state, the square of any creation operator is zero; the
exponential reduces to the sum of the rst two terms of its expansion:
k=1 +
2kkk
0 (fermions only) (B-10)
B-2-b. Fermions in a singlet state
For paired fermions in a singlet state, the state pairedwill be called the BCS
state (ComplementXVII) and noted BCS; relation (A-23) must be used withK= 0.
As the exponential of a sum of commuting operators
4
is a product of operators, we get:
BCS= exp kk k
0=
k
k (B-11)
with:
k= expkk k
0 (B-12)
As the square of any fermion creation operator is zero, the series expansion of the expo-
nential is limited to its rst two terms:
k=1 +kk k
0 (fermions only) (B-13)
B-3. Pairs of particles and pairs of individual states
Pairs of statesis an important concept not to be confused with pairs of particles.
In (B-7) as well as in (B-11), the individual states intervene as pairs of states(kk).
The number of those pairs (which can be innite ifis innite) is not related to the
particle number. For fermions in a singlet state, it is convenient to label the pair of
states by the momentumkassociated with the spin state, while remembering that the
momentum associated with the spin stateis the opposite,k. We shall systematically
use this simplication in what follows.
C. Properties of the kets characterizing the paired states
Let us examine a few properties of the stateskthat will be useful in what follows.
To keep things simple, we continue limiting in this
study, and assume the particles in the same spin states are bosons; as for the particles in
dierent spin states, we shall continue using the example of fermions in a singlet state.
The generalization to other paired cases does not introduce any particular diculties.
C-1. Normalization
The normalization of the stateskis actually simpler for fermions than for
bosons; this is because, as we shall see below, the series expansion of the exponen-
tial (B-12) contains only two terms for fermions, instead of an innity for bosons. This
is why we do not keep in this
spin 1/2 fermions.
4
The operators
k k
and
k k
associated with dierent pairs commute, since they are
products of two fermionic operators.
1822

C. PROPERTIES OF THE KETS CHARACTERIZING THE PAIRED STATES
C-1-a. Fermions in a singlet state
We choose to normalize separately each of thekby multiplying them by a
numberk. This operation amounts to replacingkby:
k= k+kk k
0 (C-1)
with:
k=kk (C-2)
The normalization condition becomes:
k
2
+k
2
= 1 (C-3)
It then becomes natural to set:
k= cosk
k
k= sink
k
(C-4)
wherekandkare the two variables
5
the ket
kdepends on. One can choosek
between0and2:
0 k
2
(C-5)
so thatcoskandsinkare positive and represent the moduli ofkandk. We saw in
k= k; the functionskandkare therefore even with respect tok.
The variational ket BCSnow becomes the normalized ket
BCS: BCS=
k
k+kk k
0
=
k
cosk
k
+ sink
k
k k
0 (C-6)
Comment:
A particular case occurs when all thekare either zero or equal to2. The ket
BCS
then reduces to a simple Fock state, whose populations of individual states are either
zero, or equal to one (for populations corresponding to states belonging to a pair for
whichk=2). In that case, the phaseskno longer play any role: instead of xing a
relative phase, they only determine the global phase of the state vector.
If, furthermore, we choosek=2for all values ofkwhose modulus is less than a given
value, and zero otherwise, the paired state now describes an ensemble of fermions
lling two Fermi spheres (one for each spin state), which is simply the ground state of
an ideal gas of fermions. The ket
BCSthen reduces to the trial ket of the Hartree-
Fock method of ComplementXV; that method appears as a particular case of the more
general pairing method used in this chapter.
5
The variablekdetermines the dierence2kbetween the phases ofkandk. We could also
introduce a variable to determine their sum, but that would be pointless: such a variable would only
change the total phase of the ket
k
, without any physical consequences.
1823

CHAPTER XVII PAIRED STATES OF IDENTICAL PARTICLES
C-1-b. Bosons in the same spin state
For bosons, the results are slightly dierent. To maintain a certain analogy, we
shall use the same parameterskandkas for fermions, but it is now the hyperbolic
sine and cosine ofkthat will come into play. Relation (B-9) leads to:
k=
=0
1
!2kkk
0=
=0
1
!
[k]
kk
0 (C-7)
with
6
:
k=
2k (C-8)
As mentioned before, the spin index that comes in addition to the indexkis not written
explicitly as its value does not change.
Consequently:
kk=
=0
1
!
2
k
2
!
4
=
=0
k
2
=
1
1 k
2
(C-9)
We assumed, to sum the series, that:
k
2
1 (C-10)
It is useful in what follows to characterize the complex variablekby two real
variables:kto dene its modulus, and an anglekthat characterizes its phase. We
therefore set:
k=tanhk
2k
with: k0 (C-11)
Inequality (C-10) is automatically satised since the modulus of a hyperbolic tangent is
always less than1; as the functionkis even see relation (A-4) so are the variable
kand the functions
7
kandk. We then get:
kk=
1
1tanh
2
k
=cosh
2
k (C-12)
The normalized kets
kcan be written as:
k=
1
coshk
k=
1
coshk
exp kkk
0 (C-13)
Replacing thekby the
k, the ket pairedbecomes normalized to1.
Initially, the ketsk, as well as their normalized version
k, have been dened
in the tensor product (B-8) only whenkbelongs to the half-space. They can, however,
be dened by relations (C-7) and (C-13) for anyk; we then have simply
k=
k,
which was to be expected since
kinvolves the two individual stateskandkin the
same way.
6
The minus sign in this denition is arbitrary a change of sign of the wave function(r)or of its
Fourier transformkhas no physical consequences but it is convenient to introduce this sign to ensure
coherence with the calculations in .
7
Furthermore, rotational invariance generally requires those functions to depend only on the modulus
ofk.
1824

C. PROPERTIES OF THE KETS CHARACTERIZING THE PAIRED STATES
C-2. Average value and root mean square deviation of particle number
The particle number in the individual statekcorresponds to the operator:
k=
kk (C-14)
We are now going to compute the average value and the root mean square deviation of
the particle number, rst in a given pair of states, then for the system as a whole.
C-2-a. Fermions in a singlet state
Let us compute the average value of the particle number in the state
BCS, which
is the tensor product of the states
k, each being associated with the pair of states
(}k= +;}k=); as dened above, each pair is labeled by the wave vectorkof
the spin+particle. The particle number in each of these pairs of states corresponds to
the operator:
(pairk)=k+ k=
kk+
k k (C-15)
with eigenvalues0,1and2. Now
k
is given by (C-1), the sum of two components,
one with zero particles, and the other with two particles. This leads to:
k(pairk)k= 2k
2
= 2 sin
2
k (C-16)
and:
k (pairk)
2
k= 4k
2
= 4 sin
2
k (C-17)
The root mean square deviation
(pairk)of the particle number in a pair is thus:

(pairk)=
4k
2
1 k
2
= 2 sinkcosk (C-18)
Consequently, the uctuations of the particle number in each pair of states can be large.
On the other hand, the uctuations of the total particle number, obtained by
summing over all the pairs, remain small. The average value of this total number is:
= 2
k
k
2
= 2
k
sin
2
k (C-19)
As we will show just below, the square of the uctuationof is given by:
[]
2
= 4
k
k
2
1 k
2
(C-20)
Since1 k
2
1, we get:
[]
2
4
k
k
2
= 2 (C-21)
1825

CHAPTER XVII PAIRED STATES OF IDENTICAL PARTICLES
so that:

2
(C-22)
Hence, for large values of, the uctuations of the particle number, in relative value,
are very small, decreasing at least as fast as the inverse of the square root of the average
value.
Demonstration:
The operator corresponding to the square of the particle number is:
2
=
k
(pairk)
2
+
k=k
(pairk)(pairk) (C-23)
As the state
BCS is a product of states of pairs, the latter are not correlated and
the average value of this operator is written:
BCS
2
BCS
=
k
k (pairk)
2
k
+
k=k
k(pairk)kk (pairk)k
(C-24)
Expression (C-1) for
k
leads to:
(pairk)
k= 2kk;k (C-25)
so that:
2
= 4
k
k
2
+ 4
k=k
k
2
k
2
(C-26)
Now the square of the average valueis equal to the last terms of this equality, but
without the constraintk=kin the summation. It follows that the root mean square
is written as:
[]
2
=
2
[]
2
= 4
k
k
2
k
4
(C-27)
which leads to (C-20).
C-2-b. Bosons in the same spin state
For bosons, each pair contains two individual states of oppositek. We show below
that for each of them, we have:
k=sinh
2
k (C-28)
and that:
[k]
2
= 2k
2
+ k (C-29)
1826

C. PROPERTIES OF THE KETS CHARACTERIZING THE PAIRED STATES
The root mean square deviation of the distribution associated with the values ofkis
thus:
k=
[k]
2
k
2
=k
2
+ k
=sinhkcoshk (C-30)
(the average value of the particle number in a pair of states is2k, and the root mean
square deviation of that number is2k).
Demonstration:
As kis symmetric with respect to the two individual stateskandk, we have:
k= k (C-31)
with:
kk k=
=0
k
2
= k
2
k
2
k k
=
k
2
1 k
2
2
(C-32)
so that:
k=
kk k
k k
=
k
2
1 k
2
(C-33)
which leads to (C-28).
The average value of the particle number squared is computed in a similar way. Using
the identity
2
=(1) +to bring up the second derivative with respect tok
2
, we
can write:
k[k]
2
k=
=0
k
2 2
= k
4
2
k
2
2
k k+ k
2
k
2
k k
=
2k
4
1 k
23
+
k
2
1 k
22
(C-34)
and hence:
[k]
2
=
k[k]
2
k
k k
= 2k
2
+ k (C-35)
The total number of particles is written:
=
k
k
2
=
k
sinh
2
k (C-36)
1827

CHAPTER XVII PAIRED STATES OF IDENTICAL PARTICLES
(as expected, each pair of states appears twice in this summation; one can also restrict
the summation to the half-spaceprovided we add a factor2). We also have:
2
=
k
k
k
k=
k
kk+ k+
k=k
k
=
k
[k]
2
+ k k+
kk=k
k k (C-37a)
where, in the last term, we have used the fact that the paired state is the product of
uncorrelated pairs. But this state is symmetric inkandk, and operatorskand k
act on it in the same way. Therefore:
2
=
k
2[k]
2
k
2
+
kk
k k (C-37b)
(the constraint in the second summmation has been eliminated by subtracting a term in
the rst summation). We then get:
2
= 2
k
[k]
2
+
2
(C-37c)
The root mean square deviationskhave been obtained in (C-30). Hence, the square
of the root mean square deviation of the total number of particles is written as:
[]
2
= 2
k
[k]
2
= 2
k
sinh
2
kcosh
2
k (C-38)
As for the fermion case, this square contains only a single summation onk, whereas the
square of the total particle number contains two. Now the number of non-zero terms in
those summations is the number of Fourier components necessary to describe the pair of
particles used, in , to build the paired state in a cube of edge length(size of the
momentum quantization box see ). This number is of the order of the cube of
the ratio betweenand the size of the pair, hence a very large number, as it is the ratio
between a macroscopic and a microscopic volume. A double summation overktherefore
contains many more terms than a simple summation, and since all the terms are positive
and of comparable magnitude, we have:
2
[]
2
(C-39)
We again nd, as for fermions, that .
C-3. Anomalous average values
For computing average values of the energy (in particular, in ComplementXVII),
we will need the average values of products of two creation or annihilation operators. For
example, for bosons we will need to calculate:
kkkk and
kkkk (C-40)
1828

C. PROPERTIES OF THE KETS CHARACTERIZING THE PAIRED STATES
We note, right away, that they concern operators that do not conserve the particle num-
ber, and this is the reason they are often called anomalous average values. One could
be surprised that such average values come into play while studying physical processes
that do not physically imply creation or destruction of particles. We will show that they
actually occur in a very natural way in the calculation of the average value of a Hamilto-
nian that conserves the particle number. The reason is that
kis only a component of
the total state vector (B-8), in which it is associated with many other
k; in the total
state vector, the particle number in the state
kmay, for example, decrease by2while
the particle number in the state
ksimultaneously increases by the same quantity. We
are, therefore, performing computations on the components of a state vector that has
the same total particle number; the anomalous character is only apparent, and is due
to the fact that we only consider part of the total state vector.
C-3-a. Fermions in a singlet state
Consider the action of the operatorkkon the ket
kwritten in (C-1).
Only one of its component, in(k), remains and, after two anticommutations, we can
write:
kk
k=kkkk k
0=k k k kk
0
=k0 (C-41)
Taking the scalar product of this ket with the bra
k, only its component(k)0
remains; the average value is thus:
k kkk=
kk= sinkcosk
2k
(C-42)
The anticommutation of these two operators then yields:
kk kk=
kk=sinkcosk
2k
(C-43)
Taking the Hermitian conjugate of (C-42), we get:
kk kk=kk= sinkcosk
2k
(C-44)
whereas the average value of
kk
is the opposite (anticommutation):
k kkk=sinkcosk
2k
(C-45)
We saw, in , that the functionskandkare even; we can therefore change the
sign ofkon the left-hand side of the previous relations without changing the right-hand
side.
C-3-b. Bosons in the same spin state
For bosons, it is easier to rst compute the average value of a product of creation
operators:
kkkk=
1
cosh
2
k
kkkk (C-46)
1829

CHAPTER XVII PAIRED STATES OF IDENTICAL PARTICLES
This expression contains the product of the ket:
kkk=
kk
=0
[k]k=;k=
=
=0
(+ 1) [k]k=+ 1;k=+ 1 (C-47)
by the bra:
=0
[
k] k=;k= (C-48)
To get a non-zero term, we must have=+ 1, which leads to:
(+ 1) [k][
k]
+1
(C-49)
whose sum overyields, taking (C-32) into account:
=0
(+ 1) [
k]k
2
= [
k] [k+ 1]kk (C-50)
We nally divide bykkas in (C-46), to obtain, inserting the value (C-11) ofk:
kkkk=tanhk
2k
1 +sinh
2
k
=
2k
sinhkcoshk (C-51)
As for the other anomalous average value
kkkk, a simple Hermitian
conjugation operation shows that it is the complex conjugate of the previous one:
kkkk=
2k
sinhkcoshk (C-52)
As was the case for fermions, the functionskandkare even, which allows
changing the sign ofkon the left-hand side of the previous equations without changing
the result.
D. Correlations between particles, pair wave function
As already mentioned in this chapter's introduction, one of the major interest of the
paired states is to allow varying the spatial correlation functions of a system of identical
particles. In addition to the purely statistical correlations, coming from the indistin-
guishability of the particles and already present in an ideal gas, we now have a way to
include dynamic correlations due to the interactions. Using paired states instead of sim-
ple Fock states allows, for example, a better optimization of the energy. We shall limit
our study to the two-particle diagonal correlation function, as it is the one that xes the
average value of the interaction Hamiltonian. This will lead us to introduce a new wave
function, that we shall name the pair wave function. In the complements following
1830

D. CORRELATIONS BETWEEN PARTICLES, PAIR WAVE FUNCTION
this chapter we shall also study non-diagonal correlation functions; it will concern the
one-particle correlation function, whose long range behavior may signal the existence of
Bose-Einstein condensation, as well as the two-particle correlation function.
In a general way, one may wonder about the physical signicance of correlation
functions computed in states pairedor BCS, since these states are coherent super-
positions of kets containing dierent particle numbers. However, correlation functions
are average values of operators keeping the particle number constant, and hence inde-
pendent of the coherence between kets of dierentvalues. Furthermore, we saw in
, the relative uctuations
of that number were negligible. In the limit of large, one can thus expect the results
obtained with pairedor BCSto be very close to those obtained with the , for
which these uctuations are strictly zero. This question will be discussed in more detail
in XVII.
When studying correlation functions in the case where the paired particles are in
the same spin state, the only relevant indices concern the orbital variables. We shall
start with this simpler case, and study later the case of paired particles in a singlet state.
D-1. Particles in the same spin state
Relation (B-34) of Chapter
function2(r1r2)can be written:
2(r1r2) = (r1) (r2) (r2) (r1) (D-1)
Replacing the eld operators and their adjoints by expressions (A-3) and (A-6) of Chapter
XVI, using as a basis the normalized plane waves, we get:
2(r1r2) =
1
6
k1k2k3k4
[(k4k1)r1+(k3k2)r2]
k1k2
k3k4
(D-2)
where the average value of the product of the4creation and annihilation operators
must be taken in a paired state. Figure
correlation function.
D-1-a. Simplications due to pairing
The computation is greatly simplied by noting that in a paired state, the popula-
tions of the two individual states having opposite wave numberskandkmust always
be equal. Consequently, only those combinations of the4operators that do not change
this equality will lead to non-zero average values. Three cases are then possible:
Case I: the two annihilation operators concern two individual states that do not belong
to the same pair (k3=k4); the two creation operators must then restore to their initial
values the populations of these two same states, or else their average value will be zero;
these are the so called forward scattering terms. We then have eitherk4=k1and
k3=k2(direct term), ork4=k2andk3=k1(exchange term).
Case II: the two annihilation operators act on the two states of a rst pair (k4=k3),
and the creation operators on the two states of another pair (k2=k1). We then talk
about a pair annihilation-creation process .
1831

CHAPTER XVII PAIRED STATES OF IDENTICAL PARTICLES
Figure 1: This diagram symbolizes the terms that come into play in the computation of
the correlation function of two particles at pointsr1andr1. The two incoming arrows
at the bottom left-hand side represent the two particles eliminated by the annihilation
operators; they are associated with a positive imaginary exponent of the position. The
two outgoing arrows on the top right-hand side represent the two particles resulting from
the action of the creation operators, associated with a negative imaginary exponent. The
correlation function is the sum of these terms over all the values of the4kvecteurs.
Case III: the two annihilation operators act on the two states of the same pair, and
the creation operators replenish these same two states (this is a special case of the one
we just discussed); another possibility is that the two annihilation operators act on the
same individual state (all the wave numberskmust then be equal).
Using these conditions on the values of the wave numbers in (D-2), we note that
the terms corresponding to cases I and II include two summations over the wave vectors,
whereas there is only one summation in the terms corresponding to case III. Consequently,
for a large (macroscopic) volume
3
, there are far fewer terms coming from case III than
from cases I and II. We shall therefore only take into account terms arising from case
I and II. For the same reason, we shall ignore in our computation of these terms the
constraintsk3=k4ork3=k1, as this amounts to adding a negligible number of
terms.
D-1-b. Expression of the correlation function
The direct term is obtained fork4=k1andk3=k2; it no longer has any spatial
dependence. Sincek1andk2are dierent, the average value of the product of operators
can also be written
k1
k1k2
k2
for fermions, the two minus signs coming from the
anticommutations cancel each other. Now relation (B-8) shows that the paired state is a
tensor product of pairs of states. This means that the average value we wish to determine
is simply the product of the average values of the rst two operators and of the last two
operators, i.e. the product of the average values of two occupation numbers. We thus
1832

D. CORRELATIONS BETWEEN PARTICLES, PAIR WAVE FUNCTION
get a rst contribution:
dir
2(r1r2) =
1
6
k1k2
k1 k2
=
2
6
(D-3)
where the summation overk1andk2are considered as independent, since as we men-
tioned above, we can neglect the constraint linking these two indices.
The exchange term is obtained fork4=k2andk3=k1; it exhibits a spatial
dependence. As we did for the direct term, we regroup the creation and annihilation
operators acting on the same individual states, but this operation now involves only one
commutation between operators. We then introduce a factor, equal to1for fermions,
and we get:
ex
2(r1r2) =
6
k1k2
(k2k1)(r1r2)
k1 k2
(D-4)
The pair annihilation-creation termk4=k3andk2=k1also exhibits a spatial
dependence, but no longer involves average values of occupation numbers. Its expression
is:
pair-pair
2
(r1r2) =
1
6
k1k4
(k4k1)(r1r2)
k1k1
k4k4
(D-5)
and its structure is schematized in Figure. Expression (D-5) contains average values
of products of operators that do not conserve the particle number, but rather annihilate
(or create) two of them. They are called anomalous average values. As we explained in
, these anomalous average values come into play quite naturally in the computation
of the average value of an operator that does conserve the particle number. Dening the
pair wave functionpairas:
pair(r) =
1
3
k
kr
kk (D-6)
this correlation function can be written as:
pair-pair
2
(r1r2) =pair(r1r2)
2
(D-7)
The complete correlation function2(r1r2)is the sum of the three previous
contributions:
2(r1r2) =
dir
2(r1r2) +
ex
2(r1r2) +
pair-pair
2
(r1r2) (D-8)
For bosons in the same spin state, we can insert in this correlation function the average
values given in (C-28), (C-51) and (C-52). We then get a binary correlation function
that explicitly depends on the parametersk, as well as on the phasesk, which both
dene the paired state. This clearly veries that these parameters introduce exibility
in the two-body correlation function. For example, we nd:
pair(r) =
1
3
k
sinhkcoshk
(kr2k)
(D-9)
1833

CHAPTER XVII PAIRED STATES OF IDENTICAL PARTICLES
Figure 2: Diagram symbolizing the pair-pair term of the binary correlation function, with
the same convention as for Figure.
The pair wave function thus directly depends on the phasesk; as we shall see in the
complements, these phases actually play a major role in the optimization of the energy.
For bosons, this wave function is always even since, as we saw in , this is the case
for the functionskandkintroduced in (C-11).
Comment:
To describe systems of interacting bosons undergoing Bose-Einstein condensation
(see XVIIand ComplementXVII, we shall add to the paired
state pairedanother highly populated state with zero momentum (k= 0). This
will introduce new terms in the correlation functions, in addition to those com-
puted in this chapter. When the population of that individual state with zero
momentum is very high, these additional terms may become dominant.
D-2. Fermions in a singlet state
For fermions with spin12, since each spin can point in two directions, there exists
a larger number of correlation functions. Several among them will be studied in 2
ComplementXVII. We shall only compute one of them here, involving opposite spins,
as it plays the most signicant role:
2(r1;r2) = (r1) (r2) (r2) (r1) (D-10)
Relation (D-2) now becomes:
2(r1;r2) =
1
6
k1k2k3k4
[(k4k1)r1+(k3k2)r2]
k1k2
k3k4
(D-11)
The diagram schematizing each term of this sum is obtained by adding spin indices to
the positions in Figure XVII.
1834

D. CORRELATIONS BETWEEN PARTICLES, PAIR WAVE FUNCTION
The computation is then similar to that of . The direct term is written:
dir
2(r1;r2) =
1
6
k1k2
k1 k2=
6
(D-12)
There is no exchange term wherek4=k2andk3=k1, as it would correspond to the
average value of an operator changing the direction of one of the spins in two dierent
pairs, hence destroying the equality between populations of opposite spins in each pair;
this term does exist, however, in the special case wherek1=k2, but its contribution
is negligible. Finally, the pair annihilation-creation term corresponds tok4=k3and
k2=k1; it is written:
pair-pair
2
(r1;r2) =
1
6
k1k4
(k4k1)(r1r2)
k1 k1
k4k4 (D-13)
Here again, the pair-pair term involves anomalous average values. As before, we can
dene a pair wave functionpairas:
pair(r) =
1
3
k
kr
k k=
1
3
k
kr
kk (D-14)
whose modulus squared appears in the correlation function:
pair-pair
2
(r1;r2) =pair(r1r2)
2
(D-15)
Inserting relations (C-42) into (D-14) yields:
pair(r) =
1
3
k
kk
kr
=
1
3
k
sinkcosk
(kr+2k)
(D-16)
The important role of this pair wave function in the BCS condensation phenomenon will
be discussed in detail in ComplementXVII. We will show in particular that this function
not only plays a role in the diagonal binary correlation function; it also determines the
long-range non-diagonal properties of the density operator, hence playing the role of
an order parameter. We noted that the parameterskandkare even functions ofk;
consequently, the functionpair(r)is also an even function ofr.
The total correlation function is then:
2(r1;r2) =
6
+pair(r1r2)
2
(D-17)
Inserting in this result expression (D-16) forpair(r), we obtain the dependence of the
correlation function on the parameterskandk. This illustrates how these parameters,
which dene the paired state, allow changing the correlation function.
Comment:
In the particular case where all thekare either zero or equal to2, we already
mentioned (see end of ) that the paired state becomes a Fock state in which
the phaseskno longer play any role. It is easy to check that the anomalous
average values are then all equal to zero, as is, obviously, the functionpair(r).
On the other hand, for a dierent choice of the parametersk, the phaseskplay
an especially important role, as will be shown for example in ComplementXVII.
1835

CHAPTER XVII PAIRED STATES OF IDENTICAL PARTICLES
E. Paired states as a quasi-particle vacuum; Bogolubov-Valatin transformations
The Hamiltonian of a noninteracting particle system can be written as:
0= (}) (E-1)
where}is the energy of an individual state labeled by the index. The ground state
0of0is an eigenvector of all the annihilation operators, with a zero eigenvalue:
0= 0 (E-2)
The paired ket pairedis not an eigenvector of the usual annihilation operators.
We shall, however, introduce in andinto new
annihilation and creation operators, and show in pairedis an eigenvector,
with a zero eigenvalue, of all the new annihilation operators. The paired state will then
appear as a particle vacuum. Furthermore, in , we shall associate with paired
a family of operators having the same form as the Hamiltonian (E-1), but where the
andare replaced by the new annihilation and creation operators. The interest of that
association is the possibility, in certain cases (illustrated in the complements), to identify
with certain approximations if needed an operator in this family with the Hamiltonian
of a given physical situation. The problem of nding the ground state and the excited
states is then solved, as if dealing with a system of independent particles. The state
pairedcan then be considered as the ground state of the Hamiltonian of independent
quasi-particles, while the new creation operators permit building a complete orthogonal
basis of excited states.
E-1. Transformation of the creation and annihilation operators
For bosons in the same spin state, the statekbelongs to the spacekassociated
with the pair(kk); this space is generated by the action of two creation operators
k
and
k
on the vacuum. This is also the case for fermions in opposite spin states, if
we simplify the notationktok, as well asktok(we have labeled each pair of
individual states by the value ofkassociated with the spin). For both cases, we now
dene two new couples of creation and annihilation operators that act ink.
We introduce the two annihilation operators
k
and k, dened fork= 0, as well
as the Hermitian conjugate operators
k
and
k
, as:
k=kk+ kk k
=
kk
+
kk
k=kk+kk k
=
kk
+
kk (E-3)
or:
k
k
k
k
=
k k
kk
kk
k k
k
k
k
k
(E-4)
As for now,kandkare any two complex numbers.
1836

E. PAIRED STATES AS A QUASI-PARTICLE VACUUM; BOGOLUBOV-VALATIN TRANSFORMATIONS
As in Chapters ,[]denotes the commutator ofandif= 1
(bosons), and their anticommutator if=1(fermions). We now compute[k k];
as
k
(anti)commutes with
k
and as
k
(anti)commutes with
k
, only the cross terms
inkkremain:
[
k k]=kk kk
+
k k (E-5)
For bosons, the commutator of
k
and
k
equals1, and hence the commutator of
k
and
k
equals1; for fermions, the two anticommutators of those operators are equal
to1, so that we obtain, in both cases:
[
k k]=kk11
= 0 (E-6)
By Hermitian conjugation, we get:
k k
= 0 (E-7)
We now compute kk
. This time, we get the two squared terms ink
2
andk
2
.
The one ink
2
contains the (anti)commutatorkk
equal to1; the one ink
2
contains, for bosons, the commutator of
k
and kwhich equals1, and for fermions,
the anticommutator of those two operators which is equal to+1. We therefore get:
kk
= k
2
k
2
(E-8)
In a similar way:
k k
= k
2
k
2
(E-9)
Finally, we are left with the computation ofk k
and kk
. The rst is
zero since
k
(anti)commutes both with
k
and itself
8
, and that
k
(anti)commutes
with itself and with
k
; the reasoning is the same for the second, so that:
k k
= 0
kk
= 0 (E-10)
To sum up, it suces to impose, for all values ofk, the condition:
k
2
k
2
= 1 (E-11)
8
For fermions, its square is identically zero.
1837

CHAPTER XVII PAIRED STATES OF IDENTICAL PARTICLES
to get the relations:
kk
= 1
k k
= 1 (E-12)
so that the operatorskand k, as well as their adjoints, obey the same relations of
(anti)commutation as the usual annihilation and creation operators of identical particles.
For fermions, we nd again condition (C-3), which allows us to simply set, as in
(C-4):
k= cosk
k
k= sink
k
(E-13)
We then see that the matrix in the right hand side of (E-4) is unitary. This unitary trans-
formation of the creation and annihilation operators is called the Bogolubov-Valatin
transformation.
For bosons, we will set:
k=coshk
k
k=sinhk
k
(E-14)
Comparison with relation (C-11) shows that:
k=
k
k
(E-15)
The transformation of the creation and annihilation operators for bosons is called the
Bogolubov transformation.
E-2. Eect on the kets
k
We now show that the vectors
kare eigenkets of two annihilation operators
k
and
k
with eigenvalues zero. This property makes them similar to a usual vacuum
state, which yields zero under the action of all the annihilation operatorsk.
E-2-a. Fermions in a singlet state
Let us compute the eect of those operators on the ket
kdened by relation
(C-1), that we write with the simplied notation already used above (kis associated with
the spin index+andkwith the spin index):
k= k+kkk
0 (E-16)
We start with the operator
k
dened in (E-3). Itskk
term yields zero when acting on
the term inkof
k; only the term inkremains, for which the operator lowers from
one to zero the occupation number of the statek, since:
kkk
0=
k
0 (E-17)
1838

E. PAIRED STATES AS A QUASI-PARTICLE VACUUM; BOGOLUBOV-VALATIN TRANSFORMATIONS
As for the operatorkk
, it yields zero when acting on thekterm of
k(for fermions,
the square of a creation operator is zero), leaving only the term ink. This leads to:
k
k= kkk kkk
0= 0 (E-18)
The computation is the same for the operator
k
, except for the fact that the
operator
k
must rst anticommute with
k
before it can be regrouped with
k
and
lower from one to zero the occupation number of the statek. The anticommutation
therefore introduces a sign change, but as the denition of
k
does not contain any
sign, we again nd:
k
k=kk k
+
k
0= 0 (E-19)
We have shown that the two operators
k
and
k
have the ket
kas an eigenvector,
with eigenvalue zero.
E-2-b. Bosons in the same spin state
Taking into account (E-15), relation (C-7) is written:
k=
=0
1
!
k
k
kk
0 (E-20)
Since:
kkk
0=
kk k
1
k
0= ()
k
1
k
0 (E-21)
we have:
kkk=k
=0
!
k
k
k
1
k
0 (E-22)
or else, since
k
commutes with all the operators in this expression:
kkk= kk
=0
1
!
k
k
k k
0
= kkk (E-23)
where we have set= 1. This leads to:
kk+kk k= 0 (E-24)
which clearly shows thatkis an eigenvector of the operatorkdened in (E-3):
kk= 0 (E-25)
The same computation leads to:
kkk= kkk (E-26)
1839

CHAPTER XVII PAIRED STATES OF IDENTICAL PARTICLES
and hence to:
kk= 0 (E-27)
As for fermions, the two operators
k
and
k
have the ket
kas an eigenvector with
an eigenvalue zero.
E-3. Basis of excited states, quasi-particles
For bosons as for fermions, we just saw that the new creation and annihilation
operators introduced in (E-3) and (E-4) have the same properties as the usual creation
and annihilation operators. In particular, the two operators:
(k) =
kk k=
kk (E-28)
have as eigenvalues all the positive or zero integers, in perfect analogy with the opera-
tors corresponding to the population of individual states. By analogy with (E-1), it is
therefore natural to introduce the operator:
=
k
}
kk+
kk (E-29)
where, for the moment, theare free parameters, as are the parameters which dene
the paired state (they will be xed later on, depending on the physical problem we study).
In relation (E-29), the summation is limited, as above, to a momentum half-space, which
avoids taking opposite momenta into account twice. The eigenvalues ofare all of the
form:
=
k
(
k) +
k} (E-30)
where(k)and(k)are any positive or zero integers for bosons, and restricted to0
or1for fermions.
The ground state0()ofis an eigenvector of all the annihilation operators
k
and
k
with eigenvalues zero. Now we saw in (B-8) for bosons, and in (B-11)
for fermions, that the paired state vector is a tensor product of statesk, which are
precisely the eigenvectors, with zero eigenvalues, of these two operators. The paired
state, pairedfor bosons
9
, or BCSfor fermions, is thus an eigenvector ofwith a
zero eigenvalue (ground state).
One can then obtain the other eigenstates of(excited states) by the action of
the creation operators
k
and
k
on0(). For bosons, each of these two operators
will be able to act any number of times. For fermions, on the other hand, we shall
only get3excited states, by the action of either
k
, or
k
, or their product; as these
operators anticommute, any higher power of those operators' product will yield zero. We
nally note that operator (E-29) shares many of the properties of the Hamiltonian of an
ensemble of particles without mutual interactions. Just as the usual creation operators
9
For bosons, in ComplementXVII, we will associate to that paired state a coherent state0to
obtain the state. But, as none of the operatorskor kact in the Fock space associated with
the individual statek= 0, the conclusions will be unchanged for.
1840

E. PAIRED STATES AS A QUASI-PARTICLE VACUUM; BOGOLUBOV-VALATIN TRANSFORMATIONS
can add particles in a system of free identical particles, the creation operators
k
and
k
can be considered as the operators adding a supplementary quasi-particle into the
physical system. These quasi-particles are not the same as particles in a system really
without interactions, as illustrated by the expression of these creation operators. They
yield, however, a basis of states in which we can reason as if there were no interactions,
which is a very powerful framework for reasoning in many domains of physics.
For the previous considerations to be relevant from a physical point of view, we
have yet to show that the Hamiltonian of the problem we study can be approximated by
an operator, provided we make a judicious choice of all the parametersk,kand.
This is not a priori easy: the Hamiltonian of an ensemble of particles includes, in general,
two-body interaction terms, and those are expressed in terms of sums of products of two
creation operators
k
and two annihilation operators
k
, hence of4operators. Now,
if we insert denitions (E-3) and (E-4) into (E-29) to expressas a function of the
old creation and annihilation operators
k
and
k
, it is clear that we shall only obtain
combinations of products of2operators. We shall need to make certain approximations
to be able to consideras a physically pertinent approximate Hamiltonian. Examples
of such situations will be given in the complements.
Conclusion
In conclusion, the paired states are a powerful tool for studying both fermions and bosons.
They provide a systematic method allowing a certain exibility in variational calculations
in the presence of interactions. Furthermore, starting from a paired ground state, we were
able to build a whole basis of excited states using creation and annihilation operators
matching that ground state. In the complements of this chapter, we shall use the paired
states to study dierent problems and compute the optimal parameters most relevant
for each situation. The physical results will be quite dierent, depending on the cases,
especially for fermions or for bosons; but the main point remains that the paired states
oer a unied framework for obtaining all these dierent results.
1841

COMPLEMENTS OF CHAPTER XVII, READER'S GUIDE
The rst two complements provide more details about a number of results given in the chapter,
concerning various properties of the pair operators and the paired states. The following three comple-
ments apply these concepts to physical phenomena involving fermions, and then bosons.
AXVII: PAIR FIELD OPERATOR FOR
IDENTICAL PARTICLES
The pair eld operator is the analog, for a pair of
particles, of the usual eld operator for a single
particle. It is a useful tool for computing average
values in a paired state. The commutation
relations of fermion pair operators are similar to
those of bosons, except for an additional term due
to the fermionic character of the pair constituents.
BXVII: AVERAGE ENERGY IN A PAIRED STATE This complement explains the computation
of the average energy in a paired state. For
bosons, we add to this paired state a condensate,
described by a coherent state. The results of this
complement are used in Complements XVIIand
EXVII.
CXVII: FERMION PAIRING, BCS THEORY Even weak attractive interactions can greatly
modify the ground state of a fermion system,
via the BCS mechanism for pair formation.
This complement discusses the theory of this
phenomena, and its eect on the particle distribu-
tion and correlation functions, as well as its link
to Bose-Einstein condensation of pairs of particles.
DXVII: COOPER PAIRS The simple Cooper model studies the bound
states of two weakly attracted particles, in the
presence of a Fermi sphere that prevents the
particles from occupying states inside that sphere.
Whereas, in general, a minimum depth of an
attractive potential is required for two particles
to form a bound state in 3-D, the presence of the
Fermi sphere ensures the existence of a bound
state, no matter how weak the attraction is. The
Cooper model accounts in a somewhat intuitive
way for a number of results of the BCS theory.
EXVII: CONDENSED REPULSIVE BOSONS For an ensemble of bosons, using paired states
as variational states leads to the same results
as the Bogolubov method based on operator
transformations. We thus obtain the Bogolubov
spectrum, compute the quantum depletion
introduced by the interactions, etc.
1843

PAIR FIELD OPERATOR FOR IDENTICAL PARTICLES
Complement AXVII
Pair eld operator for identical particles
1 Pair creation and annihilation operators
1-a Particles in the same spin state
1-b Pairs in a singlet spin state
2 Average values in a paired state
2-a Average value of a eld operator; pair wave function, and order
parameter
2-b Average value of a product of two eld operators; factorization
of the order parameter
2-c Application to the computation of the correlation function
(singlet pairs)
3 Commutation relations of eld operators
3-a Particles in the same spin state
3-b Singlet pairs
In Chapter (r)acting in the state space of a
system of identical particles. This operator was dened as a linear combination of anni-
hilation operators associated with individual states having a given momentum. It proved
to be a useful tool for various computations, and in particular for the determination of
correlation functions. We then showed, in Chapter , the relevance of paired states
where, essentially, identical particles were grouped into pairs. We introduced creation
and annihilation operators of pairs of particles in well dened momentum states,
K
and
K. Consequently, it is natural to envisage the introduction of a eld operator for pairs
of particles, which will be the operator(R)destroying a pair of particles whose center
of mass is at pointRand whose internal state is described by the wave function. Its
adjoint,(R), creates a pair of particles in that same state. In this complement, we
will dene these operators and study some of their properties.
We start in (R)and
(R)for pairs described by any orbital state. We consider the case where the
particles are either in the same spin state, or in a spin singlet state. We then study,
in , the average values, in paired states, of pair eld operators and of products of
such operators. These average values have some very interesting properties leading us,
in particular, to introduce a new wave functionpair(r), called the pair wave function,
which is not simply the two-particle wave functionpair(r)used to build the paired
state. As we shall see in , this new wave function explicitly appears in the binary
correlation function of the particles' positions. Moreover, its norm is linked to the number
of quanta present in the eld of condensed pairs. The origin of this pair function is the
fact that pairs can collectively contribute to the creation of a eld whose average value
is what we shall call an order parameter. This non-zero order parameter indicates the
existence of a macroscopic eld associated with the pairs. We will show how it relates
the anomalous average values (of operators that do not conserve particle number) to
1845

COMPLEMENT A XVII
the normal average values of a product of two eld operators(R)and(R),
that does conserve particle number. In particular, we shall use, in , the properties
of the pair eld operator to get the correlation functions in a paired BCS state, and
to study the consequences of the existence of the macroscopic eld associated with the
pairs. We shall nally study, in , the commutation properties of these operators; they
will be found to be similar to those of bosons (whether the particles building the pair are
bosons or fermions), but not completely identical as corrective terms must be added to
the boson commutator. We shall see that, since the pairs are strongly bound and have
a spatial extension much smaller than all the characteristic dimensions of the problem,
the pairs can be assimilated to bosons; if, however, the pairs are weakly bound (as is
the case, in particular, for the BCS mechanism we will discuss in ComplementXVII, it
is not possible to consider them as indivisible entities: the fermionic structure of their
components plays an important role that cannot be ignored.
1. Pair creation and annihilation operators
By analogy with the eld operator for the particles composing the pairs, we now introduce
a eld operator concerning the pairs themselves. The adjoint of this eld operator allows
the direct creation of a pair of particles at a given point, and with a given internal state;
as for the operator itself, it annihilates that same pair.
1-a. Particles in the same spin state
We dened, in Chapter , the operators
K
and
K
for pairs of particles
without spin (or in the same spin state), as:
K
=
1
2
k
k K
2
+k
K
2
k
K=
1
2
k
k
K
2
k
K
2
+k
(1)
In this expression,kis the Fourier transform of the wave function(r)characterizing
the pair:
k=
1
32
3
3 kr
(r) (2)
andis the edge length of a cube, of volume
3
, which contains the physical system.
We now generalize these denitions to the case where the pair is not necessarily in a given
orbital state, but in any state belonging to an orthonormal basis of states, with the
indexgoing from1to innity; these states each have a wave function(r)whose
Fourier transform is
k
. We therefore simply add an indexto the previous denitions,
as for example:
K
=
1
2
k
k K
2
+k
K
2
k
(3)
We saw in Chapter
building the pair requires the functions
k
to have the paritywith respect to the
1846

PAIR FIELD OPERATOR FOR IDENTICAL PARTICLES
variablek, which means
k
=
k
with:
=
+1 for bosons
1for fermions
(4)
Should
k
be of parity, it is easy to check by (anti)commutation of the operators that
expression (3) yields zero. We can then only consider the case where the
k
, and hence
the corresponding wave functions(r), have the parity. If, however, we need the basis
of statesassociated with these wave functions to be complete, we can include states
of any parity,1. We must then remember that the operators
K
are zero whenever
the indexcorresponds to a wave function of parity.
. Expression of
K
in terms of the particle eld operator
Relation (A-10) in Chapter
k
by:
k
=
1
3
d
3 kr
(r) (5)
where (r)is the adjoint of the eld operator associated with the elementary compo-
nents of the pair (the atoms of each molecule). Using twice this relation in (3), we
get:
K
=
1
2
3
k
kd
3
d
3(
K
2
+k)r(
K
2
k)r
(r) (r) (6)
or else, choosing as the integration variablesR= (r+r)2andx=rr:
K
=
1
2
3
d
3 KR
d
3
k
k
kx
(R+
x
2
) (R
x
2
) (7)
The summation overkon the right-hand side leads to expression (A-2) of Chapter
for the wave function, and we can write:
K
=
1
2
3
d
3 KR
d
3
(x) (R+
x
2
) (R
x
2
) (8)
This other form for the operator already introduced in (3) demonstrates the fact that it
creates a pair of particles in a molecular state characterized, for its external variables, by
a plane wave of wave vectorK, and for its internal variables, by the wave function.
. Pair eld
For each internal stateof the pair, we can introduce, using relation (A-3) of
Chapter , an operator(R)that creates a pair at pointRand in the internal
state:
(R) =
1
3
K
KR
K
(9)
1847

COMPLEMENT A XVII
Replacing in (8) the integral variableRbyR, and using the result in equality (9), we
get:
(R) =
1
2
3
K
d
3 K(RR)
d
3
(x) (R+
x
2
) (R
x
2
)(10)
The sum overKof
K(RR)
then yields
3
(RR), which allows integrating over
3
, and we obtain:
(R) =
1
2
d
3
(x) (R+
x
2
) (R
x
2
) (11)
This operator is therefore a product of eld operators creating successively each of the
two elements of the pair, which is easy to understand from a physical point of view. Note,
however, that the two elements are not created at the same point, but symmetrically with
respect to pointR, and with a spatial distribution whose amplitude is given by the wave
function(x)of the molecule. The spatial zone involved in the process thus extends
over a distance of the order of the range of this wave function.
As for the pair eld operator itself, which annihilates a pair, it is dened by
Hermitian conjugation of the previous relation:
(R) =
1
2
d
3
(x) (R
x
2
) (R+
x
2
) (12)
We now use relation (A-3) of Chapter
in a basis of individual states with xed momenta. Using (twice) this relation in (12),
we get:
(R) =
1
32
d
3
(x)
k1k2
k1(R
x
2)k2(R+
x
2)
k1k2 (13)
This relation will be useful in what follows.
. Inversion; expression for the interaction energy
We call the individual states corresponding to the wave functions(r)and
assume they form a complete basis. The closure relation on these states is written:
k k= (
k) (
k)=kk (14)
We now multiply relation (3) by(
k
), and sum overto get:
(
k)
K
=
1
2
K
2
+k
K
2
k
(15)
It is thus possible to invert relations (3) and express any two creation operators as a sum
of pair creation operators, according to:
k1k2
=
2
(k1k2)2 k1+k2
(16)
1848

PAIR FIELD OPERATOR FOR IDENTICAL PARTICLES
where we have replacedKpark1+k2andkby(k1k2)2. By Hermitian conjugation,
we get a similar relation for any productk3k4
of annihilation operators.
Interaction Hamiltonian:
Any operatorintfor the binary interactions between particles can therefore be written
as:
int=
k1k2k3k4
1 :k1; 2 :k2 2(12)1 :k3; 2 :k4
(k1k2)2 (k3k4)2 K K (17)
where 2(12)is the binary interaction between particles (as, for example, in Comple-
mentXV; using momentum conservation, we have set:
K=k1+k2=k3+k4 (18)
Written in terms of pair creation and annihilation operators,intis the sum of quadratic
terms, and no longer of fourth degree terms as was the case with operators for individual
particles. Note, however, that one must be careful when using relation (17) since, as
we shall see in , the pair creation and annihilation operators do not obey the usual
commutation relations. The action of an operatorKon a paired state obtained by
the action of
K
on the vacuum, does not necessarily yield zero whenK=K.
Pair creation operators are not as simple to handle as particle creation operators.
1-b. Pairs in a singlet spin state
For a pair of spin12particles in a singlet spin state, we use relation (A-23) of
Chapter to represent the internal orbital state of the pair; this
reads:
K
=
k
k K
2
+k
K
2
k
(19)
The following computations apply directly to fermions in a singlet state, for which the
functions
k
must be even with respect to the variablek. We noted however in Chapter
XVII, in comment (ii) just before , that they can also apply to fermions in a triplet
spin state, when the function
k
is odd; even though this case can be included in the
following discussion, for the sake of simplicity we will continue to talk about singlet pairs.
. Expression of
K
in terms of the particle eld operator
Relation (A-9) of Chapter
k
=
1
3
d
3 kr
(r) (20)
Inserting this equality in (19) yields:
K
=
1
3
k
kd
3
d
3(
K
2
+k)r(
K
2
k)r
(r) (r) (21)
1849

COMPLEMENT A XVII
As previously, the wave function(x)appears when we use as integral variablesR= (r+r)2
andx=rr, and we get:
K
=
1
3
d
3 KR
d
3
(x) (R+
x
2
) (R
x
2
) (22)
This yields the form of the operator creating a pair of particles in a singlet molecular
state, characterized by a plane wave of wave vectorKfor its external variables, and by
the wave functionfor its internal variables.
. Pair eld
We now insert relation (22) in (9); we get:
(R) =
1
3
K
d
3 K(RR)
d
3
(x) (R+
x
2
) (R
x
2
) (23)
As before, the sum overKof
K(RR)
yields
3
(RR), and we get:
(R) =d
3
(x) (R+
x
2
) (R
x
2
) (24)
The same comments as in can be made: this operator successively creates the
two elements of the pair at dierent points, with a probability amplitude given by the
internal wave function(x)of the distance between these points. The eld operator is
obtained by Hermitian conjugation:
(R) =d
3
(x) (R
x
2
) (R+
x
2
) (25)
It will often be convenient to come back and use the annihilation operators in a basis of
individual states of xed momenta. Using (twice) relation (A-14) of Chapter , we
get:
(R) =
1
3
d
3
(x)
k1k2
k1(R
x
2)k2(R+
x
2)
k1k2
(26)
Comment:
For singlet pairs, we could invert those relations, as we did before, and express the
interaction energy in terms of the pair creation and annihilation operators. It is, however,
a bit more complicated in this case than when the pairs were in the same spin state: as
we shall see in , it would be necessary to involve another pair creation operator
(in a triplet state). This would lead to cumbersome notation, and the computation will
not be presented here.
1850

PAIR FIELD OPERATOR FOR IDENTICAL PARTICLES
2. Average values in a paired state
We now compute the average values of pair eld operators, or of products of such oper-
ators, in the paired state we dened Chapter . We shall use relations (13) or (26),
depending on whether the particles are in the same spin state, or in a singlet spin state.
In both cases, the computation of the average value of those operators in a paired state
involves the computation of average values of products of annihilation operators i.e. of
anomalous average values as dened in Chapter .
2-a. Average value of a eld operator; pair wave function, and order parameter
Expressions for the paired kets were obtained in
products of states of pairs that are not eigenstates of the occupation number operators.
These pairs all have a zero total momentum; we therefore assume, from now on, that
K= 0. The average value computation of a pair eld operator in these states will lead
to a new wave function, that we will call the pair wave function.
. Particles in the same spin state
Relations (B-8) and (B-9) of Chapter
vector pairedfor an ensemble of a large number of particles:
paired=
kD
exp
2kkk
0 (27)
The functionkused to build this paired state is a priori totally independent of the
functions
k
dening the pair eld operators. In such paired states, the populations of
the states of the same pair are always equal; consequently, the only non-zero average
valuesk1k2are those in which the two annihilation operators act on the two states
of the same pair, which have opposite momenta. As the total momenta of each pair is
zero, we can setk1=k2in (13) and obtain:
(R)=
1
32
d
3
(x)
k1
k1x
k1k1
=d
3
(x)pair(x) (28)
where the (non normalized) pair wave function has already been dened in (D-6) of
Chapter :
pair(x) =xpair=
1
32
k
kx
kk
(29)
Changing the sign of the summation variablek, allows writing the pair wave function

pair(k)in the momentum representation as:

pair(k) =kpair=
1
322
kk (30)
1851

COMPLEMENT A XVII
Note that because of the conditionk1=k2(the total momentum of each pair is zero),
the average value(R)no longer depends onR.
The average value of the pair eld operator is thus:
(R)= = pair (31)
As expected from the translation invariance of the system, it is independent ofR. On
the other hand, it depends on the internal state, and reaches a maximum when
is equal to the normalized state
norm
pairproportional topair:
norm
pair=
pair
pairpair
(32)
This computation therefore leads to a new state
norm
pair, dierent from the state
that was used in Chapter paired. Choosing for the rst
vector of the basis1=
norm
pair, the average value of the eld is given by:

1
(R)=
1
=
pairpair
(33)
This average value
1
(R)is often called the order parameter of the pairs; its non-
zero value is important as it indicates the existence of a eld constructed collectively by
the pairs. In the present case, the average value of this eld is independent ofR, as the
paired state was built from pairs having a total momentumK= 0and whose center of
mass has a constant wave function.
The average valueskk, which according to (29) determine the pair wave
function, have been called, in , anomalous average values, as
they involve operators that do not conserve particle number. For bosons in the same
spin state, relation (C-52) of that chapter indicates that:
kk=
2k
sinhkcoshk (34)
One may wonder, of course, what the purpose of computing an anomalous average value
is, as it can only be zero in a state with a xed total particle number. We shall see,
however, in
average values of operators that do conserve the total number of particles and hence have
a direct physical interpretation.
For bosons, operatorskand kcommute, and hence the denition (29) shows
that the wave functionpair(x)is even:
pair(x) =pair(x) (35)
The eld mean value (31) is thus zero for any stateof the basis whose wave function
is odd: the postulate of symmetrization with respect to the pair components requires
that pair to be in an even orbital state
1
.
1
If the particles composing the pair were fermions in the same spin state, the conclusions would be
opposite. The wave function would be odd (because of the anticommutation of the operatorskand
k); the average values for even internal stateswould be zero.
1852

PAIR FIELD OPERATOR FOR IDENTICAL PARTICLES
. Singlet pairs
For fermions in a singlet state, the paired state vector BCSis given by relations
(B-11) and (B-12) of Chapter :
BCS=
k
expkk k
0 (36)
Following (26), we must add the spin indexto the statek, and the spin indexto
the statek. Here again, the average value of the product of annihilation operators is
dierent from zero only if their wave vectors are opposite, and relation (26) then leads
to:
(R)=
1
3
d
3
(x)
k
kx
k k
= pair (37)
with the denition (D-14) of Chapter pair, associated with the (non-
normalized) wave function:
pair(x) =xpair=
1
3
k
kx
k k (38)
In a similar way, the pair wave function in the momentum representation

pair(k)can
be dened as:

pair(k) =
1
32
kk (39)
This wave function can be interpreted in the same way as the wave function dening
the orbital variables of a pair in the singlet state. As in (32), we dene the normalized
ket
norm
pair. The eld average value (37) is zero ifis orthogonal to
norm
pairand
reaches a maximum for = 1=
norm
pair; this maximum is equal to:

1
(R)=
1
=
pairpair
(40)
and denes the order parameter of the physical system. It indicates the presence of a
eld created collectively by the pairs. As noted before, since the total momentum of each
pair is zero, this average value does not depend onR.
The average values that come into play in that denition are given by relation
(C-42) of Chapter :
k k=
kk= sinkcosk
2k
(41)
(in the BCS state, thekandkare even functions ofk). We noted, at the end of
of that chapter that, in the specic case where thekare either zero or equal to2,
the paired ket is simply a Fock state of individual particles, hence a ket without pairing.
Since (41) is then equal to zero, we see that the pair wave function is zero in the absence
of pairing.
1853

COMPLEMENT A XVII
2-b. Average value of a product of two eld operators; factorization of the order
parameter
The eld operators do not conserve particle number, as opposed to the usual
operators such as the Hamiltonian, the total momentum, the double density, etc. On the
other hand, the product of operators(R) (R)does conserve that number, and
may help characterizing the properties of the pairs while being easier to interpret from
a physical point of view.
. Particles in the same spin state
Using relation (13) we get:
(R)(R)=
1
2
6
d
3
(x)d
3
(x)
k1k2k3k4
k1(R+
x
2
)k2(R
x
2
)k3(R
x
2
)k4(R+
x
2
)
k1k2
k3k4
(42)
The integrals over d
3
and d
3
yield Fourier transforms
k
of the wave functions(x):
k=
1
32
d
3 kx
(x) (43)
and we get:
(R) (R)=
1
2
3
k1k2k3k4
k1k2
2
k4k3
2
[(k3+k4)R(k1+k2)R]
k1k2
k3k4
(44)
where, to simplify the notation, we have written(k)the Fourier transform of
k
.
Computation of the average value
k1k2
k3k4
This computation follows the same steps as the one in
the correlation between particles, as well as the one in of ComplementXVIIfor
the interaction energy. Three cases must be distinguished:
(I) The forward scattering terms are obtained either fork4=k1andk3=k2
(direct terms), ork3=k1andk4=k2(exchange terms). We assume these forward
scattering terms concern two dierent pairs, meaningk1=k2. Since:
k1k2
k2k1=
k1k2
k1k2= k1 k2 (45)
their sum yields the contribution:
(R)(R)
forward
=
1
2
3
k1k2
(k) [(k) + (k)]
K(RR)
k1 k2
(46)
1854

PAIR FIELD OPERATOR FOR IDENTICAL PARTICLES
(as in , we may consider the summations overk1andk2as
independent, since, for a large volume
3
, ignoring the constraintk1=k2leads to a
negligible error); we have used the notation:
K=k1+k2
k=
k1k2
2
(47)
When the parity of the function(k)is, the two terms in the bracket of (46)
are equal and we get the simpler relation:
(R)(R)
forward
=
1
3
k1k2
(k)(k)
K(RR)
k1 k2
(48)
This result only depends on the dierenceRR(translation invariance); it goes to zero
whenRRbecomes larger than the inverse of the momentumKdistribution width
of the function appearing on the right-hand side of (46), once it is summed overk, the
dierence in momenta.
(II) The terms corresponding to the annihilation-creation of dierent pairs are
obtained fork2=k1andk4=k3, withk4=k2. Their contribution is written:
(R)(R)
paire-paire
=
1
2
3
k1
k1k1
(k1)
k4
k4k4
(k4) (49)
Now, using (30) and the denition (2) of the Fourier components of each pair state,
we have:
1
322
k4
k4k4 (k4) =
k4
k4pair k4= pair (50)
The summation overk1is computed in a similar way, via a simple complex conjugation.
We then get on the right-hand side of (49) two scalar products, which nally yields:
(R)(R)
pair-pair
= pair pair (51)
Unlike the previous contribution, this one is independent ofRR.
(III) The terms corresponding to the annihilation-creation of the same pair
are obtained fork1=k2=k3=k4, and yield the average valuesk1 k1and
k1 k1respectively. Those terms are just a particular case of the terms appearing
in the summation (46) whenk1=k2, and do not require a specic calculation. Finally,
the termsk1=k2that we ignored in (I), and for which all thek's must be equal, contain
only one summation over the wave vectors; consequently, they are negligible compared
to (46), and will be omitted in this computation.
We are then left with the total (I) + (II), which yields:
(R) (R)=(R)(R)
forward
+(R)(R)
pair-pair
(52)
1855

COMPLEMENT A XVII
where only the second term on the right-hand side does not go to zero whenRR
becomes large, which indicates a long-range non-diagonal order. According to (51), this
second term reaches a maximum when the two internal statesand are equal
to the state
norm
pairdened in (32). It indicates the existence of a cooperative eld of
pairs that have a total momentumK= 0as their external state, and
norm
pairas their
internal state.
Comparing (31) and (51) shows that:
(R)(R)
pair-pair
=(R) (R) (53)
The pair-pair term of the two-point correlation function can thus be factored into a
product of two one-point correlation functions; for= 1, we get the same function
we previously called the order parameter. As already pointed out in , it is
because the pairs have a zero total momentum that anyRandRdependence has
disappeared from both sides of (53), but this point is not essential. It is more important
to note that introducing such an order parameter, a priori dicult to understand from a
physical point of view as it is an average value that does not conserve particle number,
is actually quite useful for computing other more physical parameters. We will make the
connection between the factorization relation (53) and the Penrose-Onsager criterion for
Bose-Einstein condensation in .
. Singlet pairs
Using (26) instead of (13) now leads to a relation very similar to (42); the factor
12is, however, missing, and we must make the substitution:
k1k2
k3k4 k1k2
k3k4
(54)
Relation (44) then becomes:
(R) (R)=
1
3
k1k2k3k4
k1k2
2
k4k3
2
[(k3+k4)R(k1+k2)R]
k1k2
k3k4
(55)
The rest of the calculation is very similar to the one we just did, and involves the sum
of several terms:
(I) The forward scattering terms are obtained fork4=k1andk3=k2. In two
dierent pairs, a particle is destroyed and then created again in the same individual state
(as we now have spin indices, there is no exchange term in this case). The computation
is the same as the one that yielded (48) for spinless particles; with the notation (47) for
the wave vectors, we get here:
(R)(R)
forward
=
1
3
k1k2
(k)(k)
K(RR)
k1 k2
(56)
(II) The terms corresponding to the annihilation-creation of dierent pairs are
obtained fork2=k1andk4=k3, withk4=k1. The computation is now the same
1856

PAIR FIELD OPERATOR FOR IDENTICAL PARTICLES
as the one that yielded (51). The right-hand side of (55) becomes:
1
3
k1k4
(k1)(k4)
k1 k1
k4k4
(57)
Using the denition (38) for the pair wave function in the singlet case, we again obtain:
(R)(R)
pair-pair
= pair pair (58)
As mentioned above, anyRdependence has disappeared from this average value
since the paired state was built from pairs having a zero total momentum.
(III) The terms corresponding to the annihilation-creation of the same pair are
obtained fork2=k3=k1=k4; they are proportional tok1 k1
and already
included in the terms (I). The terms where all thek's are equal are neglected for the
same reason as above.
To sum up, we nd as before:
(R) (R)=(R)(R)
forward
+(R)(R)
pair-pair
=(R)(R)
forward
+(R)(R) (59)
We arrive, nally, at the same results as for spinless bosons, with the same long-range
non-diagonal order of the pairs, as well as the factorization (53) of the order parameters.
We shall see in ComplementXVIIthat this long-range order parameter is intimately
linked to the nature of the BCS transition. Here again, the anomalous average values
turn out to be useful tools for computing normal average values that conserve the particle
number.
. Link with Bose-Einstein condensation of pairs
There is a close link between the order parameter of the pairs and the existence
of Bose-Einstein condensation of those pairs. To show this, it is convenient to introduce
the density operator of pairs, limiting ourselves, for the sake of simplicity, to the case
of spinless particles. In Chapter , the one-particle density operatorfor identical
particles was given, in terms of the eld operator, by its matrix elements (B-26):
r r= (r) (r) (60)
whereris the particle's position andits spin. For pairs, the corresponding relation is
written
2
:
R
pair
R =(R)(R) (61)
whereRis the position of the center of mass, andand dene the internal state
of the pair; the indexplays a role similar to that of a spin index for a single particle
(even though it corresponds to an internal orbital state).
2
As we shall see in , the pair eld operators do not exactly satisfy the boson commutation
relations. Consequently, operator (61) is not, strictly speaking, a density operator; to underline this
dierence,
pair
is sometimes called a density quasi-operator.
1857

COMPLEMENT A XVII
In the momentum representation, the diagonal matrix elements of this density
operator are:
K
pair
K =
1
3
d
3
d
3 K(RR)
(R)(R) (62)
Since(R)(R)only depends onRR, we perform the change of variables
X=RR; the integral over d
3
is then trivial and cancels the factor1
3
; we therefore
get:
K
pair
K =d
3 KX
(R)(RX) (63)
Inserting relation (52) in this result, we get the sum of a contribution from the forward
scattering term and from the pair-pair term.
(i) The rst contribution comes from inserting (48) in (63). The integral over
d
3
yields a delta functionKKand a factor
3
that cancels the same factor in the
denominator. As the sumk1+k2must now be equal toK, the double summation over
k1andk2reduces to a summation overk. We then get:
k
(k)
2
K
2
+k
K
2
k
(64)
This result is a regular function ofK, related to the wave number dependence of the
occupation numbers.
(ii) The pair-pair term contains the integral of the function (53), which is a product
of two constant order parameters; it therefore leads to:
K0
3

2
(65)
The presence of the delta functionK0shows that theK= 0level has an additional
population (number of quanta of the pair eld) that does not exist for any other value of
the momentumK; this population is simply the square of the order parameter, multiplied
by the system's volume; it is thus an extensive quantity. It indicates that the pairs
of the system undergo Bose-Einstein condensation. As the corresponding population
is proportional to the square of the order parameter, this clearly shows the close link
between the long-range non-diagonal order, the order parameter and the existence of
condensation. The factorization appearing in (53) is often called the Penrose-Onsager
condensation criterion.
2-c. Application to the computation of the correlation function (singlet pairs)
The average values of products of pair eld operators can also be used to get the
correlation functions between particles. We are going to show, in particular, that the
correlation function2is the sum of an incoherent term, independent of the positions,
and of a coherent term that involves the pair wave function dened previously. In order
to keep the demonstration short, we shall limit the discussion to the case of fermions
described by a paired state built from singlet pairs
3
, but the transposition to spinless
particles is fairly straightforward.
3
Condensed bosons will be studied in ComplementXVII. We will then show that the properties of
the paired state, built from thek= 0states, are not determined by the interactions within this paired
state, but rather by the interactions with a condensatek= 0, external to the paired state. It is therefore
a completely dierent case.
1858

PAIR FIELD OPERATOR FOR IDENTICAL PARTICLES
Relation (24) expresses the conjugate of the pair eld operator as a function of
products of creation operators for its constituent particles; we shall start by inverting
this relation.
. Inversion of the relation between elds
The closure relation for the orthonormal basis of the wave functions(r)with
= 12is written:
(x)(x) =(xx) (66)
This summation overmust include even orbital functions(x)(associated with a pair
elddescribing paired fermions in a singlet state) as well as odd functions (associated
with a pair eld describing fermions paired in a triplet state). We then multiply (24)
by(x)and perform the summation over. We recognize in the integral on the
right-hand side the closure relation (66), which yields:
(x) (R) = (R+
x
2
) (R
x
2
) (67)
This leads to:
(r1) (r2) = (r1r2)
r1+r2
2
(68)
Creating two particles of opposite spins at pointsr1andr2thus amounts to cre-
ating a coherent superposition of pairs with a center of mass at(r1+r2)2, in a singlet
or triplet spin state, and with coecients equal to the wave functionstaken at the
positionr1r2.
The average value of this expression can be computed in a paired state, using
relation (37). This leads to:
(r1) (r2)= (r1r2) pair (69)
Aspairis an even function, it easily follows that only the evenwill contribute to this
average value; the triplet pair elds have a zero average value in a singlet pair state.
As in , we can choose for thea basis whose rst ket1coincides with
the normalized pair ket
norm
pair. We then get:
(r1) (r2)=
pair(r1r2)
pairpair
pairpair
pairpair
=
pair(r1r2) (70)
.4-point correlation function
According to (68), the4-point correlation function (for opposite spins) is written
as:
(r1) (r2) (r
2) (r
1)
= (r1r2)(r
1r
2)
r1+r2
2

r
1+r
2
2
(71)
1859

COMPLEMENT A XVII
It is expressed in terms of the average values of products of pair creation and annihilation
operators, hence in terms of the average values of products of elds for which the index
plays the role of an internal state of the molecule. We will show below that it can be
expressed as:
(r1) (r2) (r
2) (r
1)=1(r1;r
1) 1(r2;r
2)
+
pair(r1r2) pair(r
1r
2) (72)
where1(r;r)is the non-diagonal one-particle correlation function, the Fourier
transform ofk:
1(r;r) =
1
3
k
k(rr)
k (73)
and with a similar denition for1(r;r), the occupation numberkbeing simply
replaced byk; the pair wave functionpairhas already been dened in (38).
The function1(r;r), being the Fourier transform of a regular function
k, tends toward zero when the dierencerris larger than a certain (micro-
scopic) limit; the only terms left are those on the second line of (72). Imagine then that
positionsr1andr2are close to each other, forming a rst group, and that the same
is true for positionsr
1andr
2, forming a second group, while these two groups are far
from each other. The non-diagonal correlation function can then be factored into a prod-
uct of functionspair. This situation is reminiscent of the Penrose-Onsager criterion for
Bose-Einstein condensation of bosons (ComplementXVI, ), but it now concerns
the 4-point (instead of 2-point) non-diagonal correlation function. As the norm ofpair
is the order parameter, it again underlines the important role of this parameter.
An important particular case of the4-point correlation function is the two-body
(diagonal) correlation function for opposite spins:
2(r1;r2) = (r1) (r2) (r2) (r1) (74)
The intensity of the pair eld is therefore written:
2(r1;r2) =
1
6
k1k2
k1 k2
+pair(r1r2)
2
=
6
+pair(r1r2)
2
(75)
We nd again relation (D-17) of Chapter , but via another method. The two-body
correlation function is the sum of a contribution independent of the positions (hence,
with no correlations) and of the modulus squared of the pair wave function. This latter
contribution comes from the term that, for pairs, indicates the existence of a long-range
non-diagonal order (Bose-Einstein condensation). This is an important property, which
is at the heart of the BCS mechanism, and which will be discussed in more detail in
ComplementXVII.
1860

PAIR FIELD OPERATOR FOR IDENTICAL PARTICLES
Demonstration
We insert in (71) relations (56) and (). In the forward scattering term, we get the
following expression:
(r1r2) r
1r
2 (k)(k)
K(r
1
+r
2
r1r2)2
= r1r2r
1r
2 k k
K(r
1
+r
2
r1r2)2
=kr1r2r
1r
2k
K(r
1
+r
2
r1r2)2
=
1
3
(k2k1)(r
1
r
2
r1+r2)2(k1+k2)(r
1
+r
2
r1r2)2
=
1
3
k2(r
1
r1)k1(r
2
r2)
(76)
wherekandKwere dened in (47). Inserting this result in (71), we get the rst term
of the right-hand side of (72)
As for the pair annihilation-creation term (58), it yields:
(r1) (r2) (r
2) (r
1)
pair-pair
= (r1r2) r
1r
2 pair pair
= r
1r
2 pair pair r1r2
=
pair(r1r2) pairr
1r
2 (77)
and we obtain the second term of the right-hand side of (72). Note that only the singlet
pair elds (associated with the even function) contribute to this term.
3. Commutation relations of eld operators
We now study the commutation relations between the pair eld operators just dened.
The spin-statistics theorem (Chapter XIV, C-1) states that particles with integer
spin are bosons, and particles with half-integer spin are fermions. If we consider two
paired fermions, the rules for adding angular momenta (Chapter X) indicate that this
composite system necessarily has an integer spin. Intuitively, one could thus expect two
bound fermions to behave like a boson; this is the question we now discuss by examining
the commutation relations between the operatorsKand
K
, and establishing the
correction factors introduced by the underlying fermionic structure.
3-a. Particles in the same spin state
Starting with spinless particles, we shall explain in this simple case the main com-
mutation properties of the pair operators. If the pairs created and annihilated by the
operatorsKandKand their Hermitian conjugates were really bosons, the commu-
tator of these two operators should be equal toKK. We are going to show that the
commutator does contain such a term, but with several additional corrections.
1861

COMPLEMENT A XVII
. Commutation relations of the
K
Any product of two creation operators commute with any product of two creation
operators (for fermions, two minus signs cancel each other when products of two opera-
tors cross each other); the same is true for two products of annihilation operators. We
therefore have:
K K
= 0
K K = 0 (78)
The commutator of
K
and
K
has yet to be computed:
K K
=
1
2
k
(
k)
k
k
K
2
k
K
2
+k K
2
+k
K
2
k
(79)
We will show below ( ) that:
K K
=KK + 2
k
(
k)
KK
2
+kK(K2)k(K2)k
=KK + 2

K
2
K
2
KK (80)
(in the second line, we have set=k+K2and used the parityof the function);
if needed, we can get rid of thecoecient on the right-hand side provided we change
the sign of the subscript of(or of).
The rst termKK , on the right-hand side of (80) , is exactly the commutator
of two bosons with internal statesand(spin states for example): this term is
dierent from zero only if both the external and internal variables are the same (in the
present case, these internal states are actually orbital states). This rst term is, however,
followed by an additional term that shows that the fermionic structure of the pairs still
plays a role. This latter term is a one-particle operator in the sense dened in
Chapter; relation (B-12) of that chapter permits computing the matrix elements of
the corresponding operator. This additional term contains creation and annihilation
operators in normal order, which means that it will go to zero when the populations of
the individual states tend to zero; in this limit, the pairs can be assimilated to bosons.
WhenK=Kand=(pairs in the same internal and external states), we get
the simpler relation:
K K
= 1 + 2

K
2
2
K (81)
with the usual denition of the population operatork:
k=
kk (82)
The corrections to a purely bosonic commutation are then proportional to the populations
of the individual states of the particles forming the pair, hence conrming the fact that
they become negligible when the sum of all these populations is small enough.
1862

PAIR FIELD OPERATOR FOR IDENTICAL PARTICLES
. Demonstration
We start by computing the commutator:
12 34
=1234 3412 (83)
where1234are indices labeling any individual states. We can write:
1234
=2314
+ 1324
=2314
+ 1324
+
3124
=2314
+ 1324
+2431+
3142
=2314
+ 1324
+2431+ 1432+
3412 (84)
so that:
12 34
=2314
+2431+ 1324
+1432 (85)
Putting all the operators in normal order, we get
4
:
12 34
=2314+ 1324+2431+1342+ 2341+1432(86)
The commutator appearing on the right-hand side of (79) is therefore equal to:
KK kk+ KK kk+ (KK)2kk
(K2)+k (K2)k
+(KK)2k+k
(K2)k (K2)+k
+(KK)2kk
(K2)k
(K2)k+ (KK)2kk
(K2)+k (K2)+k(87)
Inserting the rst two terms back into (79), we get the following contribution to the
commutator
K K
:
1
2
KK
k
(
k)
k+
k=KK
k
(
k)
k=KK (88)
where we have taken into account the parity with respect tokof the the functions
k
see relation (A-4) of Chapter
orthonormal. As mentioned above, thisKK is precisely what is expected for a
boson commutation relation.
It is, however, followed in (87) by four other terms, which are written:
1
2
k
(
k)
KK2
k
K(K2)k(K2)k
1
2
k
(
k)
KK2
k
K(K2)+k(K2)+k
2
k
(
k)
KK2
+k
K(K2)k(K2)k
2
k
(
k)
KK2
+k
K(K2)+k(K2)+k (89)
4
Since =, we have = +.
1863

COMPLEMENT A XVII
In each of them, and without modifying the result, we can change the sign of the summa-
tion dummy indexk, or change the sign of the subscript of the functionsor(provided
we introduce a factor). For example, in the second term, we can change the sign of
the subscripts of the two functionsand(two factorsthen cancel each other), then
change the summation indexkintok: this second term then doubles the rst one. As
for the third term, we simply change the sign of the subscript(KK)2 +kof the
function(which introduces a factorcanceling that same factor already present) and
reproduce the rst term. Finally, for the fourth term, a parity operation on the function
followed by a change of the summation index fromktokmakes it equal to the rst.
The four terms are therefore equal; choosing for example the expression of the third one,
and replacing the summation indexkby=k+K2, we get relation ().
. Commutation relations of pair eld operators
For the same reasons as explained above (commutation of any products of two
annihilation operators), the operators(R)all commute with each others; the same
is true for the adjoint operators(R). We have yet to examine the relations between
the(R)and the(R). Relations (11) and (12) show that:
(R)(R)=
1
2
d
3
(x)d
3
(x)
(R
x
2
) (R+
x
2
) (R+
x
2
) (R
x
2
) (90)
The computation will not be carried out explicitly (though it does not present any par-
ticular diculty); it leads to:
(R)(R)
=(RR)
+ 16d
3
[x][2 (RR)x] (2RR
x
2
) (R
x
2
)
=(RR)
+ 16d
3
(RRz)(RR+z)
(
R+R+z
2
+RR) (
R+R+z
2
+RR) (91)
In the second more symmetrical form of this commutator, we have used the notation
z=RRx. These relations are the equivalent, in the position representation, of the
commutation relations (80) in the momentum representation (as already mentioned, it
is possible to make the factorappear or disappear in front of the integral by changing
the sign of the variable of one of the two functionsor).
The commutator thus includes several terms. The rst term in(RR)
corresponds to the commutation relation of a usual bosonic eld (whether the pair con-
stituents are bosons or fermions); thereects the commutation of eld components
1864

PAIR FIELD OPERATOR FOR IDENTICAL PARTICLES
corresponding to dierent orbital internal states of the pairs. To this term must be added
a correction that depends on the structure of the pair, characterized by the functions
(r)and (r). We again nd a result similar to that obtained before: a rst simple
bosonic term, which only takes into account the simultaneous exchanges of the two con-
stituents of a molecule with the two constituents of another one. This term is followed
by a correction that comes from the possibility of exchanges other than those involving
complete pairs. Note that this correction is expressed in terms of eld operators of the
elementary constituents themselves and not of the pairs, as was to be expected since it
is the constituents themselves that are involved. This correction term is a one-particle
operator, non-diagonal in the position representation, since it destroys a particle at point
rand recreates another one at pointr+2(RR), always at the same distance.
To keep things simple, let us assume the dimensions of the molecules that dene
the pair eld for the two internal statesandwe are concerned with, are both of the
order of the same dimension0; this means that the wave functions(r)and (r)
go to zero when 0. In relation (91), the values ofxthat contribute to the integral
are those for which none of the two functions[x]and [2 (RR)x]takes a
negligible value; this requires that neitherx, nor2 (RR)x, be large compared
to0. This double condition imposesRR 0, in which case there are values of
xfor which both functions take simultaneously large values and the correction to the
commutator cannot be neglected. On the other hand, ifRR 0, there is no
common domain where both functions and take on signicant values and the
integral over
3
is practically negligible. In other words, the molecular wave function
range0also plays the role of the commutator correction range.
The limit0 0can be obtained by choosing functionsproportional to a
function(Appendix II, 1-b), whose width goes to zero as0and whose integral
equals one (it takes values of the order of1
3
in a domain of volume of the order of
3
). For the sake of simplicity, we assume that=; as it is the square of the function
that is normalized to1(and not the function itself), we must choose:
(r)
32
(r) (92)
Using this form for the functions, the integral over d
3
in (91) leads to the convolution
of two delta functions, which yields a function(RR)multiplied by the operator
(R) (R); nevertheless, the coecient
3
of this term yields zero in the limit where
0. Consequently, if the molecules' size is very small compared to all the characteristic
lengths of the system (such as the distance between molecules), the commutation relations
of the eld operator are exactly the same as for elds associated with bosons.
In conclusion, when the molecules have no spatial overlap
5
, the only relevant ex-
changes concern exchanges between both of their constituents. On the other hand, when
the two molecules do overlap, individual exchanges between their constituents become
possible. If the molecules are loosely bound, as in the example of the BCS fermion pairing
mechanism (ComplementXVII), they cannot be treated as bosons without structure,
and one must use the complete formula (91) for the commutator.
5
This does not exclude the case where the distance between molecules is small or comparable to the
de Broglie wavelength of their centers of mass: the gas of molecules may be degenerate.
1865

COMPLEMENT A XVII
3-b. Singlet pairs
We now study the case of particles in a singlet pair, as in .
. Commutation relations of the
K
As before, any products of two creation operators commute with any products of
two creation operators; the same is true for any products of two annihilation operators.
Relations (78) are thus still valid. We now have to compute the commutator:
K K
=
k
(
k)
k
k
K
2
k
K
2
+k K
2
+k
K
2
k
(93)
We are going to show that:
K K
=KK
+
k
(
k
)
KK
2
+k
K(K2)k
K
2
k
+
K(K2)k
K
2
k
=KK +

K
2
K
2
K K;#
+
K K
(94)
(with, in the last line, the notation=k+K2). Here again we nd that the commutator
of the two operatorsKand
K
includes, to begin, a purely bosonic term, followed
by corrections containing operators in normal order ( which go to zero in the limit of low
occupation numbers); a correction must be added for each of the two spin states.
Demonstration
To prove (94), we again use relation (86). As the indices1,2,3and4represent all the
quantum numbers associated with an individual state, they must now contain the spin
indices; these are added to the momentum indices, which play the same role as in the
previous calculation. It then follows that the states1and3are always orthogonal, as
are the states2and4; the only terms remaining on the right-hand side of (86) are the
terms in23and14, so that:
K K
=
k
(
k)
kk KK kk
+ (KK)2kk
K
2
k
K
2
k
+(KK)2kk
K
2
+k
K
2
+k
(95)
or else (since the basis of the functions
kis orthonormal):
KK
+
k
(
k)
KK
2
+k
K(K2)k
K
2
k
+
KK
2
+k
K(K2)+k
K
2
+k
(96)
We now modify the second term in the bracket of the summation, to make it similar to
the rst one: as the functions
khave a denite parity () with respect tok, we can
change the index signs ofand, and the sign of the dummy indexkof the summation.
The only dierence between the two terms is then the spin directions, and we therefore
obtain ().
1866

PAIR FIELD OPERATOR FOR IDENTICAL PARTICLES
. Commutation relations of pair eld operators
A similar calculation to the one that led to (91) permits obtaining:
(R)(R)=(RR)
+ 8d
3
[x][2 (RR)x]
(2RR
x
2
) (R
x
2
) + (2RR
x
2
) (R
x
2
) (97)
(a more symmetrical form of the right-hand side can be obtained by again using the
notationz=RRx). The commutator is thus equal to that of elementary bosons
plus a correction term. This latter term plays an important role over a distanceRR, of
the order of the range of the wave functions, and is the sum of contributions independent
of the two spin states.
Conclusion
In conclusion, note that the pair eld operator provides interesting insights concerning
the physical properties of paired states in a many-body system. For a-particle state,
built from a two-particle wave function, it leads to a new wave functionpairwhen
the particle indistinguishability is taken into account. In the framework of the BCS
theory, we will see how this pair wave function allows characterizing the cooperative
eects of pair interactions. Introducing an order parameter is also useful for showing
the link between anomalous average values (which do not conserve particle number)
and the normal average values. The results take on dierent forms for paired states of
bosons or fermions. There is, however, a strong analogy between the two cases, which
provides a unied framework for the study of dierent phenomena, such as Bose-Einstein
condensation of particles or pairs.
1867

AVERAGE ENERGY IN A PAIRED STATE
Complement BXVII
Average energy in a paired state
1 Using states that are not eigenstates of the total particle
number
1-a Computation of the average values
1-b A good approximation
2 Hamiltonian
2-a Operator expression
2-b Simplications due to pairing
3 Spin 1/2 fermions in a singlet state
3-a Dierent contributions to the energy
3-b Total energy
4 Spinless bosons
4-a Choice of the variational state
4-b Dierent contributions to the energy
4-c Total energy
In Chapter , the paired states were introduced in a general way, without
specifying any particular form of the Hamiltonian. In order to use the paired states
pairedin the framework of a variational method, i.e. to be able to minimize the
average value of the energy of an-particle system, we must compute the average value
of the energy in these paired states; this is the purpose of this complement. We start
() by examining the consequences of the fact that these states are not eigenvectors
of the total particle number operator. In , we clarify the notation and give the
expression of the Hamiltonian. We then deal successively with the fermion case ()
and the boson case (). This second case is slightly more complicated since it requires
the adjunction of a specic state to describe the condensate.
1. Using states that are not eigenstates of the total particle number
The paired states pairedare coherent superpositions of states containing dierent num-
bers of particles. One may wonder how the average values computed in such states can
be relevant for a physical system wherehas a xed value. As we already mentioned
in , this approach is correct for large values of the average particle
number, provided the operators, whose average values we are computing, conserve the
particle number (i.e. commute with the total particle number, as is the case for the
Hamiltonian operator). We are going to show in more detail that when these condi-
tions are met, the average values do not depend on the state vector's coherences between
dierentvalues; they can thus be obtained using the paired states.
1869

COMPLEMENT B XVII
1-a. Computation of the average values
The state paireddened in (B-5) of Chapter
where the particle number is exactly= 2:
paired=
=0
1
!
(1)
As the matrix elements of the operatorbetween eigenkets ofcorresponding to
dierent eigenvalues are zero, we have:
paired paired=
=0
1
!
2

=
=0
1
!
2
(2)
where is the energy average value in the state :
=

(3)
Consequently, if we dene the weight distribution()as:
() =
1
!
2
(4)
the diagonal element ofin pairedis given by:
paired paired=
=0
() (5)
The average value is then obtained by dividing this expression by the square of the
norm paired paired.
In a general way, the diagonal element in pairedof any operatorthat commutes
withis given by a linear combination of the average values of this operator in the states
with the weight distribution(). As an example, for any functionof the
operator, we can write:
app app=
=0
()(2) (6)
1-b. A good approximation
For a system with a xed= 2particle number, we are trying to determine the
eigenvaluesand the kets ; the most direct method would be to vary separately
1870

AVERAGE ENERGY IN A PAIRED STATE
each ket to optimize. This would lead, however, to complicated calculations.
It turns out to be much more practical to vary pairedand optimize the corresponding
energy; this leads to nearly the same results for large particle numbers, as we now explain.
We saw in
pairedare very small in relative value whenis large. This means that the distribution
()has a sharp peak around a certain value0of, which determines half the average
value of the particle number. Now if the energiesare practically constant over the
width of that distribution, the Hamiltonian diagonal matrix element (2) can be written:
paired paired 0
=0
1
!
2

=
0
paired paired (7)
Making this diagonal matrix element stationary (keeping constant the norm of paired)
is equivalent to making
0stationary. The optimal value obtained for this matrix
element, divided by the squared norm of paired, yields a good approximation of the
energy
0
we are looking for. Once pairedhas been optimized in this way, it can be
projected onto the various subspaces with xed particle numbers, and therefore obtain
the , corresponding to stationary states with xed particle numbers. In the following
complements, we shall use the paired states rather than the states with xed particle
number.
Comment:
In the following complements, rather than optimizing the average energy, it is the dif-
ference between this average energy and the average particle number multiplied by the
chemical potentialthat we shall optimize. As the two operatorsandcommute
with the total particle number, the line of reasoning we just followed also applies to that
case.
2. Hamiltonian
Consider a physical system composed of fermions or bosons, placed in a cubic box of
edge length.
2-a. Operator expression
The Hamiltonianis the same as the one used on several occasions, for example
in ComplementXV(but we assume here that there is no external potential):
=0+ int (8)
The operator0is the sum of the kinetic energy operators0()associated with each
particle:
0= 0() =
P
2
()
2
(9)
1871

COMPLEMENT B XVII
andintis the sum of the interaction energies between particles:
int=
1
2
==1
2(RR) (10)
where2(RR)only depends on the dierenceRR(translation invariance) and
does not act on the spins.
We now expressin terms of creation and annihilation operators, according to
formulas established in Chapter. We use the basis of individual statesk, where
klabels the momentum}kof a plane wave that satises the periodic conditions in the
box; the indexlabels the spin state of the particles, but if they are all in the same spin
state, it can be omitted in what follows. We get:
=
k
kk+
+
1
2
k;k;k;k
1 :k; 2 :k 2(R1R2)1 :k; 2 :k
k k k k (11)
with:
=
}
22
2
(12)
(since the interaction potential does not act on the spins, we were able to replace the
spin indexassociated withkby the index, as well as the indexassociated with
kby the index). The matrix elements of2appearing in (11) can be written:
d
3
1d
3
22(r1r2)
1
6
(kk)r1(kk)r2
(13)
We make the following change of variables:R= (r1+r2)2andr=r1r2. The integral
over d
3
of the exponential yields the Kronecker delta function:
1
3
d
3(k+kkk)R
=k+kk+k (14)
which enforces the conservation of the total momentum:
k+k=k+k (15)
The integral over d
3
introduces the Fourier transformqof the potential
1
:
q=
1
3
d
3 qr
2(r) (16)
with:
q=
(kk)(k k)
2
(17)
1
The factor1
3
in (16) comes from the normalization of the plane waves
kr32
in a cube of
edge length; it ensures the potentialhas the dimension of an energy.
1872

AVERAGE ENERGY IN A PAIRED STATE
Figure 1: Symbolic plot of a general interaction process where two particles of momenta
~kand~kare replaced, as the result of their mutual interaction, by particles of momenta
~kand~k. The indicesandlabel the spins, which are not modied by the
interaction. The horizontal line represents the momentum transfer~qwhose value is
given by (17) and (18).
or else, taking (15) into account:
q=kk=kk (18)
The momentum transferqgives the momentum variation of particle1, as well as the
opposite of the momentum variation of particle2. Since2(r1r2)is symmetric with
respect to the exchange of the variablesr1andr2, the functions2(r)andqare both
even and real.
The matrix element of the interaction potential is then:
1 :k; 2 :k 2(R1R2)1 :k; 2 :k =k+kk+k q (19)
and is schematized in Figure, where the horizontal line represents the momentum
transfer~qresulting from the interaction between the ingoing and outgoing particles.
The interaction potential operator can thus be written:
int=
1
2
kkkk
q k k k k (20)
where the summation over thekactually concerns only three wave vectors, sincek=
k+kk.
In a frequently used approximation, one assumes the interaction potential range
to be very small compared with the de Broglie wavelengths of all the particles involved
1873

COMPLEMENT B XVII
(contact potential). The variations withkof
2(k)can then be neglected, and all the
matrix elements of the interaction potential are equal to a given constant0(provided
they conserve the total momentum; otherwise, they are obviously zero):
0=
1
3
d
3
2(r) (21)
2-b. Simplications due to pairing
In general, the computation of the average value of the operator (11) is very com-
plex, due to the large number of possible interaction terms. However, as we already saw
in , some simplications occur for a paired state. The main
reason is that in the various components of a paired state on Fock states, all the paired
individual states have the same population. If the population of an individual statek
changes, the population of the individual statekmust change by the same quantity,
otherwise the average value of the operator is zero. To get a non-zero average value in
a paired state, the combination of creation and annihilation operators in the considered
interaction term must respect this parity condition.
Now the interaction operator (20) is a sum of terms containing two annihilation
operators on the right, and two creation operators on the left. Only two possibilities exist
for the population balance of all the pairs to be conserved upon the action of these four
operators: either the two creation operators re-establish the initial populations of the two
states that were depopulated by the annihilation operators (in which case none of the
populations are changed); or else, the two annihilation operators destroy particles in the
same pair of states, and the creation operators produce another pair (in which case the
population of the rst pair
2
is lowered by2, and the population of the second increased by
2). The two possibilities are combined in the particular case where the creation operators
restore precisely the pair of particles destroyed by the annihilation operators. We are then
led to the dierent cases examined in detail in : CaseI(direct
and exchange forward scattering terms), CaseII(pair annihilation-creation terms) and
CaseIII(combination of the two previous terms, yielding a negligible contribution).
3. Spin 1/2 fermions in a singlet state
We now compute the average valueof the operator, written in (11), in the state
BCSdened in . As far as the interaction energy is concerned,
we will show that the terms associated with CaseIonly yield the usual mean eld
contributions, already discussed in the previous chapters. On the other hand, the terms
associated with CaseIIare a direct consequence of the pairing, and are therefore totally
new; they play a leading role in the BCS theory. The terms associated with CaseIII,
being a particular case of the other two cases, generally play a negligible role.
3-a. Dierent contributions to the energy
The dierent contributions to the energy will be computed successively, starting
with the kinetic energy.
2
We dened in , the pair population operator^pairas the sum of the population
operators of each of the two individual states forming the pair.
1874

AVERAGE ENERGY IN A PAIRED STATE
. Kinetic energy
The rst term (kinetic energy) is, as for the particle number, the sum of the
contributions of the pairs of states, labeled byk(each of the two states having the same
kinetic energy):
0=
k
k(pairk)k= 2
k
(k)
2
= 2
k
sin
2
k (22)
. Interaction energy
The average of the interaction potential energy is the sum of the averages of the
terms on the right-hand side of (20), i.e. of terms that belong to one of the three
possibilitiesI,IIandIIIcited above; we study them successively.
Case I(the creation operators repopulate the states depopulated by the
annihilation operators)
For such terms, the occupation numbers of each individual state remain unchanged
in the course of the interaction process. They are diagonal terms (sometimes called
mean eld terms). Two cases may arise, depending on whether the spin indexis the
same as, or dierent from; we examine each of these possibilities in turn.
(i) If= , as the interaction potential does not act on the spin, we can trace each
particle using its spin direction; it is as if the particles were distinguishable. If the creation
operators repopulate exactly the same individual states depopulated by the annihilation
operators, the only possible interaction is schematized in Figure, and corresponds to a
forward scattering. As the momentum transferqis zero, the potential term includes the
constant0, and we get the following contribution to the average energy:
0
2
k=k
BCS
k k
k k BCS (23)
(the conditionk=kcomes from the fact that the pairs are dierent, each pair being
labeled by the value ofkassociated with the spin+). Two anticommutations permit
bringing the last operatorkright after the rst one
k
(with two sign changes that
cancel each other). If we now sum all the contributions from= +and from=, we
get:
0
2
k=k
kk
kkk k
kk
+
kk
kkk k
kk
(24)
We can show that the two terms inside the brackets yield the same contribution by
interchanging the two dummy indiceskandkin the summation. We thus double the
rst term, and after changing the sign ofk, we get:
0
k=k
kk
kkk k
kk
=0
k=k
(k)
2
k
2
=0
k=k
sin
2
ksin
2
k (25)
1875

COMPLEMENT B XVII
Figure 2: Schematic plot of the interaction between particles of opposite spins, which
do not belong to the same pair (forward scattering). This diagram contributes to the
particles' mean eld.
When the particles are distributed in a large number of individual states, the value of
the summation in the above expression is barely changed if we ignore the constraint
k=k. If we now use forthe expression (C-19) of Chapter , we can write this
contribution as:
0
4
2
(26)
According to relation (21), the constant0is proportional to the inverse of the volume
3
. This term can be interpreted as a mean eld term, where2particles with a spin
+interact with 2particles having a spin; a particle with a given spin direction
feels the mean eld exerted by all the particles with opposite spin, whose numerical
density is2
3
.
(ii) if=, it is no longer possible to distinguish the particles by the direction of
their spin, and the indistinguishability eects play their full role. Two cases must be
distinguished for these diagonal terms: eitherk=kandk=k, which yields a
direct term; ork=kandk=k, which yields anexchange term. In both cases, the
individual states populated in the bra and the ket are the same, and we are dealing with
diagonal processes that can be called mean eld terms.
For the direct term, no particle changes its momentum, which again corresponds to a
forward scattering (left-hand side of Figure), and the potential term again includes
the constant0written in (21). The average value of this direct term is:
0
2
k=k
BCS
k k
k k BCS (27)
Here again, sincek=k(otherwise we would have the square of an annihilation operator,
which is zero), two anticommutations let us bring the operatorkto the second position,
1876

AVERAGE ENERGY IN A PAIRED STATE
Figure 3: Interaction between particles having the same spin; the direct term (forward
scattering) is schematized on the left, and the exchange term on the right. These two
diagrams add their contributions to the diagram of Figure
eld.
and we get:
0
2
k=k
kk
kkk k
kk=0
k=k
k
2
k
2
=0
k=k
sin
2
ksin
2
k (28)
(the two values ofyield the same contribution, hence the disappearance of the factor
12on the right-hand side). As for the exchange term, we havek=kandk=k
(right-hand side of Figure); for such a momentum exchange, the transferqis no longer
zero, but equal to:
q=kk (29)
and the potential term now includeskkobtained by insertingq=kkin (16).
Furthermore, whenk=k:
k k
k k=
k k
k k (30)
Apart from this sign change, the computation is the same as for the direct term. The
sum of the two direct plus the exchange contributions nally yields:
k=k
[0 kk]k
2
k
2
=
k=k
[0 kk] sin
2
ksin
2
k (31)
In the short-range potential approximation wherekk=0, this sum is zero: the Pauli
exclusion principle prevents particles having the same spin components from interacting
via a contact potential.
1877

COMPLEMENT B XVII
Case II (particles annihilated in a pair of states and restored in another
pair)
Considering the nature of the creation and annihilation operators it contains, this
process may be called pair annihilation-creation. It plays an essential role in the BCS
pairing, as we shall see in ComplementXVII; the corresponding term in the Hamiltonian
is thus often called the pairing term.
We then have, on one sidek=kand=, and on the other,k=k, so
that, according to its denition (17), the momentum transfer isq=kk; the corre-
sponding diagram is shown in Figure. We are going to show below that its contribution
to the energy can be written as:
k=k
kksinkcosksinkcosk
2(
k k
)
(32)
This term is new, in the sense that it is not a mean eld term, like the previous ones,
but that its existence is due to the pairing process. We will show in ComplementXVII
that its contribution to the average energy plays an essential role in the BCS theory.
Figure 4: Interaction process between two particles in the same pair, which, in their nal
states, end up in another pair. In terms of creation and annihilation operators, this
process is a pair annihilation-creation (two particles of the same pair are annihilated,
while two particles are created in another pair). As opposed to the terms introduced by the
other interaction processes, this term's contribution to the energy depends on the pairing;
it is sometimes called the pairing term, and is responsible for the energy gain in the
BCS theory (ComplementXVII).
1878

AVERAGE ENERGY IN A PAIRED STATE
Demonstration:
If= +, the contribution contains a product of anomalous average values:
1
2
k=k
k k
BCS
k k k k BCS
=
1
2
k=k
k k
k k kkk k kk
(33)
that is, using (C-42) and (C-44) of Chapter :
1
2
k=k
k kkk kk
=
1
2
k=k
k ksinkcosksinkcosk
2(
k k)
(34)
If=, it is now the kets
kand
kthat come into play, and we obtain another
product of anomalous average values for which we must use (C-43) and (C-45) of Chapter
XVII kare even, as indicated in that chapter):
1
2
k=k
k k
BCS
k k k k BCS
=
1
2
k=k
k k k k k k (35)
This expression is the same as the previous one, since it only diers by the sign of the
summation dummy indiceskandk(remember thatqis even). We therefore remove
the factor12in (34) and get (32).
Case III (particles annihilated in a pair of states, then restored in the same
pair)
We then have againk=kand=, but in additionk=k(and hence
necessarilyk=k), as shown in Figure; this is another case of forward scattering.
We now check that this term can be neglected. Its contribution to the energy is:
0
2
k
BCS
k k
k k BCS (36)
If= +, we get (after two operator anticommutations):
0
2
k
k k
k
k
kk
=
0
2
k
k
2
(37)
and if=:
0
2
k
k k
k
k
kk=
0
2
k
k
2
(38)
1879

COMPLEMENT B XVII
Figure 5: Interaction process where two particles of the same pair are scattered in the
forward direction.
This term is the same as the previous one, as it only diers by the sign of the summation
dummy indexk. Taking into account expression (C-19) of Chapter
^
, we
can write the total contribution as:
0
k
k
2
=
0
2
^
(39)
This contribution is interpreted as the average attraction energy in an ensemble of
^
2
pairs. When the average particle number is large, we can neglect (39) compared to (26).
Consequently, the pairing eects we are going to discuss cannot be simply interpreted as
an attraction among an ensemble of2pairs.
3-b. Total energy
Finally, adding the terms (22), (31), (26) and the double of (34), we get the average
energy
3
:
= 2
k
sin
2
k+
0
4
2
+
k=k
[0 kk] sin
2
ksin
2
k
+
kk
kksinksinkcoskcosk
2(
k k
)
(40)
3
The summations overkhave no restrictions, contrary to the tensor product appearing in relation
(B-8) of Chapter , where the summation is limited to a half-space to avoid redundancy.
1880

AVERAGE ENERGY IN A PAIRED STATE
The rst term on the right-hand side corresponds to the kinetic energy, the second to
the mean eld for particles of opposite spins, the third one is the analogous term for
particles having the same spin (it goes to zero for a short-range potential); these three
terms were already present in the Hartree-Fock theory. The fourth term, however, is
new: it corresponds to the pair annihilation-creation (pairing term) whose average value
is non-zero only in a paired state. It is the only one that depends on the phasesk, which
will prove to be essential in the BCS theory (ComplementXVII).
4. Spinless bosons
For bosons, we must take into account the Bose-Einstein condensation phenomenon
(ComplementsXV,XVandXV): in the ground state, a large fraction of the particles
can occupy a single quantum state, the statek= 0. This is not the case for a paired
state; we must therefore choose a variational state permitting such a condensation.
We assume the interactions to be repulsive, in order to avoid the instabilities
occurring for a system of attractive bosons (ComplementXV, ).
4-a. Choice of the variational state
In ComplementXV, we used the Gross-Pitaevskii approximation to treat, in the
simplest way, Bose-Einstein condensation: the system ofbosons is supposed to be,
at a given instant, in a state that is the product of identical individual states, generally
chosen as the zero momentum state,k= 0; the system state is thus written as:
=
0
0 (41)
(
0
is the creation operator in the individual statek= 0). However, whereas such a
state is suitable for an ideal gas ground state, it can only be an approximation for a gas
of interacting particles: it is an eigenvector of the kinetic energy, but not of the oper-
ator associated with the interaction energy. The interaction potential actually couples
this state to all the states where two particles are transferred from the individual state
k= 0toward any two individual states of opposite momenta~kand~k(because of
momentum conservation), such as, for example, the state:
=
k k 0
2
0 (42)
where two states of a pair are occupied. This suggests using a state pairedas a
variational ket
4
for describing the components of the system state vector associated with
all the individual statesk= 0. We must also include the components corresponding to the
individual statek= 0; those will be described
5
by a coherent state (ComplementV).
4
The interaction potential also couples a state such as (42) to numerous states of the form
k+q q k 0
3
0, whereqcan take on any value. An exact theory would require taking
those unpaired states into account, but leads to complex calculations. This is why we limit ourselves
to a variational method in the framework of an approximation where the statesk= 0are only accessible
to pairs (we assume 0 ).
5
This individual state must be treated separately, as applying the general formula (B-9) of Chapter
XVII, used whenk=0, to obtaink=0would involve the exponential of the square of the operator
1881

COMPLEMENT B XVII
We therefore choose a variational state vector of the form:
= 0 paired= 0
k
k (43)
(the notationrefers to the name Bogolubov).
In this expression, pairedis the paired state for spinless particles (B-8) of Chapter
XVII, a tensor product of the normalized states
kdened in (B-9) and (C-13). The
domainof the tensor product in (43) is half thek-space to avoid (as seen previously) a
double appearance of each state
k; the origink= 0is excluded from. This domain
could eventually have an upper bound fork.
As for0, it is the coherent state whose expression can be found, for example,
in ComplementV, whose relations (65) and (66) provide
6
:
0=
02 0
0
0= 0 (44)
This state depends on a complex parameter0, characterized by its modulus
0and
its phase0:
0=
0
0
(45)
It is a normalized eigenvector of the operator0with the eigenvalue0:
00=00 (46)
The average particle number in the statek= 0is thus:
0000=
0000=0 (47)
The width of the corresponding distribution is
0(ComplementV), hence negligible
compared to0(supposed to be a large number).
The variational variables contained in the trial ket (43) are thus the set ofkand
k, as well as0and0.
4-b. Dierent contributions to the energy
We now compute the average energy, in the variational statewritten in (43),
of the Hamiltonian operator given by (11).
k=0
, leading to large uctuations of the particle number in the statek= 0(condensed particles). This
would necessarily yield large uctuations of the total particle number, as well as of the average repulsive
energy, whereas, as we saw in of ComplementXV, those uctuations are not possible precisely
because of this repulsion.
6
One shold be careful about the change of notation: in Chapter 0denotes
the ground state of the one particle harmonic oscillator, which here corresponds to the vacuum0= 0.
In the present complement,0is the ket associated with a large number of particles occupying the
same individual state, as is also the case of the wave function(r)of the Gross-Pitaevskii equation
(ComplementXV); with the notation of ComplementV, this ket would rather correspond to a0
state.
1882

AVERAGE ENERGY IN A PAIRED STATE
. Kinetic energy
The kinetic energy term is the sum of the contributions from the dierent individual
statesk, with no contribution from thek= 0state (since=0= 0). Each term of the
summation contains the operatork, whose average value in the factored (over thek)
state (43) is given by the average value
kkkin the state
k. This average value
is given by relation (C-33) of Chapter
2
k, which yields, for the average
valueof the kinetic energy in the statethe expression:
=
k=0
kkk=
k=0
sinh
2
k (48)
with:
=
}
22
2
(49)
whereis the particle mass.
. Interaction energy
The average value of the interaction potential energy is a sum over four indices
kkkkof the potential matrix elements described in (19). As opposed to what
happened for the kinetic energy, these elements have no particular reason to cancel out
if one (or several) of their indices is zero. We shall therefore compute the dierent
contributions, arranging them in decreasing order of the number of their zero indices.
A noticeable simplication of the computation occurs with the choice of the trial
vector, as the coherent state0is one of its factors. Any time one of the four indices
in the potential energy term is zero, the corresponding annihilation operator may be
replaced by the complex number0. This is because the trial ketis an eigenvector
of the operator0with eigenvalue0- see relation (46). In the same way, each time one
of the two indiceskorkis zero, the creation operators on the left of the product, and
hence acting on the bra, can be replaced by
0, since the Hermitian conjugate of
relation (46) is:
00
=
00 (50)
These two operators are therefore simply replaced by numbers. Let us examine in turn
all the possible cases.
(i) If the four indiceskkkkare zero, the interaction potential contributes
via the constant0(forward scattering term), dened in (16) as the integral of the
interaction potential2(r); this contribution is written as:
forward
00
=
0
2
000000=0
(0)
2
2
(51)
The corresponding term is represented in Figure.
There is no contribution from terms where three (and only three) indiceskkkk
are zero: total momentum conservation would require the fourth index to also be zero.
(ii) If among the four indiceskkkktwo are zero, one concerning an annihi-
lation operator, the other a creation operator, the momentum conservation requires the
1883

COMPLEMENT B XVII
Figure 6: Diagram symbolizing the interactions between particles in the individual state
k= 0, and which remain in that state after the interaction (forward scattering term that
yields the internal mean eld of the condensate).
other two operators to be
k
andk, with the same indexk. This can yield either a
direct term, or an exchange term.
- the direct terms contain the average value of either the product
k00k, or of the
product
0kk0; it is again a forward scattering process and the potential appears via
the constant0. The two average values can be factored into two terms0000=
0
2
and
kkk=sinh
2
k; they are thus equal and the corresponding contribution
is written as:
direct
0
=00
k=0
sinh
2
k (52)
where the subscriptsymbolizes the ensemble of the excited states, i.e. those with
momentum~k= 0that have a non-zero kinetic energy. Introducing the average total
number of particlesin these excited states:
=
k=0
kk=
k=0
sinh
2
k (53)
we can write:
direct
0
=00 (54)
This term is simply interpreted as coming from the interaction between0particles in
the condensed statek= 0and particles in the other individual states.
- the exchange terms contain
k0k0and
0k0k. We are now dealing with a
momentum transfer process, and the potential now appears via the constantkobtained
by insertingq=kin (16). Otherwise, the computation is the same as for the direct
1884

AVERAGE ENERGY IN A PAIRED STATE
terms: the two average values can be factored, and the corresponding contribution is
written:
ex
0
=0
k=0
ksinh
2
k (55)
The two terms (54) and (55) correspond to mean eld contributions associated with
the interaction betweenk= 0particles andk= 0particles, taking into account the
indistinguishability of the particles that led to the exchange term.
(iii) if the product of operators contains the annihilation operator in the statek= 0
twice, the momentum conservation requires the product to be of the form
kk00. We
are now dealing with a process where two particles in the statek= 0are replaced by a
pair(kk), which amounts to creating a pair from particles initially in the condensate,
as shown on the left-hand side of Figure; here again, the potential appears via the
constantk. The two annihilation operators introduce the factor[0]
2
= 0
20
and
the other two operators, an anomalous average value in a state
k, which we already
computed in (C-51) of Chapter . We therefore get:
0
2
k=0
ksinhkcoshk
2(0 k)
(56)
If the product of operators contains the creation operator in the statek= 0twice,
it must necessarily be of the form
00kk, which corresponds to the annihilation of
a pair whose particles are transferred to the statek= 0(right-hand side of Figure).
This product is the Hermitian conjugate of the previous one, and its average value is the
complex conjugate of the previous result. The sum of these two terms is the contribution
of the processes of creation and annihilation of pairs from the condensate:
0
k=0
ksinhkcoshkcos 2 (0 k) (57)
These terms come from the pairing of particles, as opposed to other terms that are related
to the mean eld. We shall see in ComplementXVIIthe essential role they play in the
Bose-Einstein condensation of an ensemble of bosons.
(iv) There are matrix elements of the interaction potential involving a single par-
ticle in thek= 0individual state, and three other particles in thek= 0states. The
corresponding terms have a zero average value in the statebecause of the struc-
ture of its component paired, where the occupation numbers of two paired states must
always vary together.
(v) We have yet to compute the contribution of terms where none of the wave
vectors are zero. The computation is very similar to that of , and we shall again
distinguish three cases:
Case I
Terms containing interactions where particles are created in the states from which
they were destroyed: a direct term in
kkkk, and an exchange term in
kkkk.
The computation is the same as in , except for the fact that no minus sign occurs in
1885

COMPLEMENT B XVII
Figure 7: The diagram on the left represents a process where two particles, initially in the
k= 0individual state, interact and end up in states of opposite momenta. The diagram
on the right represents the inverse process, where two particles of opposite momenta
collide and end up in thek= 0individual state (i.e. in the condensate). As opposed
to the previous terms, corresponding to the mean eld of interacting particles, the terms
corresponding to this diagram are introduced by the pairing process: they play a central
role in the Bogolubov theory (ComplementXVII).
the exchange term. Result (31) therefore becomes, for bosons
7
:
1
2
k=k
[0+kk]k
2
k
2
=
1
2
k=k
[0+kk]sinh
2
ksinh
2
k
0
2
()
2
+
1
2
kk
kksinh
2
ksinh
2
k (58)
The rst term is the direct term that, when 1, is interpreted as the eect of
the interaction mean eld between the( 1)2dierent pairs of particles (when
1). It is corrected by a second exchange term, which expresses the increased
interaction between particles due to the boson bunching eect.
Case II
The pair annihilation-creation term in
kkk k, which yields here, taking
7
As opposed to the fermion case, the contribution of the termsk=kis not zero but involves the
average value of the operatork k1in the statek, see . However,
for a large system, the number of individual stateskis very important, and this contribution is totally
negligible compared to (58). This is why we neglected this term.
As for fermions, the summations overkdo not have any restriction (no limitation to half the reciprocal
space).
1886

AVERAGE ENERGY IN A PAIRED STATE
(C-51) and (C-52) of Chapter
1
2
k=k
kk
k k kkk kkk
=
1
2
k=k
kksinhkcoshksinhkcoshk
2(k k
)
(59)
Case III
Finally, the case where only one pair is involved leads, as in the fermion case, to
a term proportional to0 , negligible compared to the term in0
2
of (58) when
the average particle number is large. We shall therefore neglect it.
4-c. Total energy
Regrouping the terms in0of (51), (54) and of (58), we get a total mean eld
term:
mean eld=
0
2
(0+)
2
(60)
From a physical point of view, it is natural that this term be proportional to the square
of the total particle number divided by2, that is to the number of waysparticles can
be associated by pairs (when1). If we now include (55), (57) and (59), we get for
the total energy:
=
k=0
sinh
2
k+
0
2
(0+)
2
+0
k=0
ksinh
2
ksinhkcoshkcos 2 (0 k)
+
1
2
kk=0
kksinh
2
ksinh
2
k+sinhksinhkcoshkcoshkcos 2 (k k)
(61)
The second summation on the right-hand side describes the eect of momentum transfers
betweenk= 0particles and the condensate, as well as the processes of annihilation and
creation of pairs from the condensate (this last process depends on the relative phases
0 kand, as already mentioned, arises from the pairing of particles). The last terms
on the right-hand side, included in the double summation overkandk, correspond
to interactions between particles in thek= 0states. Since the number of individual
states is very large, we have ignored the constraintk=kof relation (59), which has a
negligible eect; furthermore, as we noted in Chapter , it is justied to replace the
imaginary exponential by a cosine.
We have shown, in this complement, that the paired states are a useful tool for
computing the average energy of an ensemble of interacting particles. In the following
complements, we shall use these results successively for fermions and bosons.
1887

FERMION PAIRING, BCS THEORY
Complement CXVII
Fermion pairing, BCS theory
1 Optimization of the energy
1-a Function to be optimized
1-b Cancelling the total variation
1-c Short-range potential, study of the gap
2 Distribution functions, correlations
2-a One-particle distribution
2-b Two-particle distribution, internal pair wave function
2-c Properties of the pair wave function, coherence length
3 Physical discussion
3-a Modication of the Fermi surface and phase locking
3-b Gain in energy
3-c Non-perturbative character of the BCS theory
4 Excited states
4-a Bogolubov-Valatin transformation
4-b Broken pairs and excited pairs
4-c Stationarity of the energies
4-d Excitation energies
We present in this complement the BCS mechanism for the pairing of fermions
through attractive interactions. The three letters BCS refer to the names of J. Bardeen,
L.N. Cooper and J.R. Schrieer who proposed in 1957 [9] a theory for a physical phe-
nomenon already observed in 1911 by H. Kamerlingh Onnes in Leiden, but as yet unex-
plained. This latter scientist observed that, below a certain temperature, the electrical
resistivity of certain metals (mercury in his case) abruptly goes to zero as a phase transi-
tion occurs toward a so-called superconducting state. Along with this transition, many
other spectacular eects occur, such as the expulsion of magnetic elds from the material.
In this complement, we shall be concerned, with the general pairing mechanism of at-
tractive fermions in the framework of BCS theory. We shall not, however, give any detail
about the theory of metals, simply accepting the existence of an attraction between the
fermions, without justifying its precise origin. In metals, this eective attraction comes
from a coupling between electrons and phonons, and is therefore indirect, introducing an
additional complexity to the problem. Furthermore, we shall not present any calculation
of electrical resistivity, and hence not show that it can go to zero.
The BCS theory is a mean eld theory, of the same type as the Hartree-Fock theory
(ComplementsXVandXV). In this latter theory, particles are assumed to indepen-
dently propagate in the mean eld created by all the others; the system is described by
an-particle Fock state. Here, we shall assume that the particles form pairs, and this
hypothesis will lead us to use, as a variation trial ket, the ket BCSintroduced in Chap-
ter ; this complement is a direct application of the results of that chapter. The state
1889

COMPLEMENT C XVII
we will choose does indeed mathematically resemble a Fock state of molecules, each
composed of two particles. It should not be concluded, however, that this approximation
reduces to a theory where each molecule is considered as an identiable object moving in
the mean eld created by all the others. This naive picture is correct in the limit where
the molecules are very strongly bound, but we shall see that it is totally inappropriate
for loosely bound pairs such as those in the BCS theory. As already underlined in the
introduction to Chapter , the use of paired states brings a lot of exibility to the
mean eld approach, as it allows modulating the binary correlation function between
particles, and then to adapt it to interactions.
We start () by minimizing the energy to determine the optimal quantum state in
the family considered. In , we discuss some physical properties of the optimized BCS
wave function, mainly in terms of one- or two-particle correlation functions, but also in
terms of what is called non-diagonal order (ComplementsXVIandXVII). Finally,
in , we shall study in more detail the physical content of the BCS pairing mechanism
allowing the optimization of the energy of a fermion system, and in particular the role of
phase locking (spontaneous symmetry breaking). For the sake of simplicity, we assume
throughout this complement that the temperature is zero, but the BCS method can also
be extended to the study of non-zero temperatures. This will lead to the study of excited
states (), as will be briey mentioned in
Shortly before the BCS theory was established, Cooper proposed a model including
two attractive fermions. He showed that the exclusion of their wave functions from the
interior of a Fermi sphere led to a bound state having certain properties similar to those
described later by the complete BCS theory. This theory can be considered to be a
generalization toparticles of the Cooper model, highlighting the collective eects
leading to the properties of the BCS ground state. The Cooper model will be studied in
ComplementXVII, and its analogies with the-particle theory will be underlined. In
the present complement, we present the BCS theory, starting directly from the general
results of Chapitre ; we shall also use the average energy values calculated in
ComplementXVII.
We obviously cannot give here a detailed account of superconductivity theory and
its various resulting eects, which would require an entire book. Limiting a large part
of the computations to zero temperature situations already implies that numerous phe-
nomena are outside the scope of this complement. To learn more about the subject, the
reader can consult reference [8].
1. Optimization of the energy
Relation (B-11) of Chapter
1
BCS:
BCS= exp
k
kk k
0=
k
k (1)
1
This state is a superposition of components containing dierent numbers of particles. As already
mentioned in Chapter , one could also choose a variational state where the particle number is
perfectly determined ([
more complex.
1890

FERMION PAIRING, BCS THEORY
The ket BCSwas then normalized by separately normalizing each ketk, which
became the kets
k:
k= k+kk k
0 (2)
where the two functionskandkare related by:
k=kk (3)
and satisfy:
k
2
+k
2
= 1 (4)
In that chapter,kwas introduced as the Fourier transform of the wave function(r)
of the diatomic molecule used to build the paired state; until now, this state was not
specied. Here, we shall consider thekas variational parameters. Choosingk= 0
leads tok= 0andk= 1: in that case, the two individual stateskandkare
neither occupied nor paired. They will be, however, ifkis not zero. In general, the
number of non-zerokis, a priori, arbitrary (nite or innite). We can, for example,
limit their number by setting a maximum valuefor the modulus ofk, and consider
this maximum value as a supplementary variational parameter dening the trial ket.
We were led, in that same Chapter , to setk= cosk
k
and k=
sink
k
, relations that imply thatkandkhave opposite phases (a situation always
possible to obtain by changing the global phase of the ket
k, which has no physical
consequences). In the present complement, it will be more convenient to assume that
the phase of
kis chosen in order to makekreal and positive, and we will set:
k= cosk
k= sink
2k
(5)
Relation (C-19) of Chapter
BCS:
= 2
k
k
2
= 2
k
sin
2
k (6)
1-a. Function to be optimized
The average particle number in the state BCSmay be changed by varying thek
dependence of thekandk: as an example, choosingk= 1andk= 0for any value
ofk, the average numberwill be zero; on the other hand, ifkis very small and
kequals1for a great number ofkvalues, the average total particle numbercan
attain arbitrarily large values. As the energy minimization operation makes sense only
for a xed value of, we shall determine that value with the Lagrangian multiplier
(chemical potential; see Appendix, ). We will optimize thekandkchoices
by introducing the variations dkand dkand cancelling the variation of the average
value= . The volume
3
of the physical system and its chemical potential
are supposed to be xed; we can choose one of two equivalent sets of variables to be
determined, either thekand thek, or thekand thek.
1891

COMPLEMENT C XVII
Relation (40) of ComplementXVIIyields, whereas is given by (6). We
then have:
=
= 2
k
( )k
2
+
0
4
2
+
k=k
[0 kk]k
2
k
2
+
kk
kkkkkk (7)
with, according to (5):
k
2
= sin
2
k ;
kk= sinkcosk
2k
(8)
In the above expression for,is the kinetic energy of a free particle with momentum
}k:
=
}
22
2
(9)
and thekare the Fourier transforms of the interaction potential2(r):
k=
1
3
d
3 kr
2(r) (10)
As this potential is rotationally invariant, the functionkonly depends on the modulus
ofk, and it must be real (Appendix I, 2-e); as the potential is attractive, we can
assume all thekto be negative.
We saw in Chapter 7) corresponds to the kinetic
energy, the second to the mean eld (diagonal term) for particles with opposite spins,
the third one to the similar term for particles with identical spins (in which the direct
and exchange terms cancel each other for a short-range potential). Finally, the fourth
term, on the second line (which containskk) plays a particularly important role in
what follows; it comes from the pair annihilation-creation diagram schematized in Figure
4 XVII. It is often called the pairing term.
We use relation (6) to write::
d[]
2
= 2 d = 4
k
dk
2
(11)
We then get:
d= 2
k
dk
2
+
kk
kk kkd[
kk] +
kkd[
kk] (12)
where the variableis the kinetic energy with respect to the chemical potential, and
corrected by the interaction eects
2
:
= +
0
2
+
k
(0 kk)k
2
(13)
2
A factor2appears in front of the summation overkas the variations ofk
2
and
k
2
in (7)
must be added, but is included in the factor2in front of the summation containing.
1892

FERMION PAIRING, BCS THEORY
Comment:
For the applications of the BCS theory, the choice of the interaction potential to be used
in the equations is not necessarily self-evident.
This is especially true in the superconductivity theory of metals, where the fermions
involved are electrons which, isolated, interact via a repulsive Coulomb potential. In a
metal, however, the direct repulsive interaction between electrons is mostly screened and
they interact indirectly via the crystalline network deformations (phonons, see Comple-
mentV). This phenomenon leads to a long-range attractive component in their eective
interaction, and explains why pairing between electrons is possible. This eective inter-
action depends on the phonon characteristics, and in particular on the Debye frequency
of the solid under study.
This is also true in the theory of an ultra-cold diluted fermionic gas, where we do not use
directly the interatomic potential in the equations. This interatomic potential contains,
at short-range, a strongly repulsive part (often assimilated to a hard core) and, at
an intermediate distance, a strongly attractive well, permitting the formation of a large
number of molecular states. Now when the gas under study is very dilute, the three-body
collisions leading to these molecules are very rare, meaning these molecular states play
practically no role; only the long distance eects of the potential have a real importance.
In other words, the essential role is played by the asymptotic properties of the stationary
collision states, as described by the scattering amplitude (Chapter VIII, relation B-9)
and the associated phase shifts (Chapitre VIII, C). The potential used in the BCS
computations will therefore be an eective potential. Furthermore, as the collisions
occur at very low energy, this eective potential only depends on the phase shift0
associated with= 0. This phase shift is generally characterized by a scattering
length0dened as0 0when 0; the eective potential will be attractive
if this scattering length is negative.
As this complement deals mainly with the quantum mechanism for BCS pairing, and not
with the determination of a valid potential, we shall not examine this point further and
assume a pertinent choice of the interaction potential has been made.
1-b. Cancelling the total variation
It is obvious in (7) that the rst three terms on the right-hand side depend only on
the moduli of thek; only the last term (annihilation-creation) depends on the phases
k. Now the functionmust be minimal if we vary thekwithout changing thek, i.e.
when we vary only this last term:
kk
kksinkcosksinkcosk
2(
k k)
(14)
We assumed that all the potential matrix elements were negative, whereas relation (C-5)
of Chapter sinkcoskare positive. The lowest value
of this sum will be obtained when all the terms in the summation overkandkhave
the same phase in order to add coherently. This condition is called the phase locking
condition, and will be discussed in more detail in . The minimum obtained does
not depend on the absolute phase of thek, but only on their relative phases. One can
then simply choose all thekto be equal to zero, which means that all thekare real
and positive. This is the choice we shall adopt from now on.
1893

COMPLEMENT C XVII
In relation (12), the terms in d[
kk
]and d[
kk
]now become equal (thekand
kare dummy indices), and we have:
d= 2
k
d(k)
2
+ 2
kk
kkkkd(
kk) (15)
We must now vary thek. For that purpose, we introduce the quantitiesk, having the
dimension of an energy, as:
k=
k
(kk)kk (16)
Thekare real since thekandkare real, and positive since we assumed the in-
teraction potential matrix elements to be negative. They are called gaps and play an
important role in the BCS theory. It will be easier to discuss this role in the case of a
very short-range potential; this will be done in . The choice of the word gap will
also be explained later (in , see in particular Figure). The variation ofcan now
be written as:
d= 4
k
kdk2
k
k[
kdk+kd
k] (17)
The variations of dkand dkare, however, not independent since relation (4) requires
(forkandkreal) that:
2kd
k+ 2kdk= 0 (18)
This means we can replace dkby kdkk. The right-hand side of (17) then becomes:
4
k
kdk2
k
k k
(k)
2
k
dk (19)
Cancelling the variation ofwith respect to all thekleads to:
2 kk k
(k)
2
k
= 0 (20)
Multiplying by
k
, we get:
2 kk= k(k)
2
(k)
2
(21)
or else:
sin 2k= kcos 2k (22)
One can then compute the sine and the cosine, since:
[cos 2k]
2
=
[cos 2k]
2
[cos 2k]
2
+ [sin 2k]
2
=
()
2
()
2
+ (k)
2
(23)
1894

FERMION PAIRING, BCS THEORY
and we obtain:
cos 2k=
()
2
+ (k)
2
=
sin 2k=
k
()
2
+ (k)
2
=
k
(24)
where we have set:
=
()
2
+ (k)
2
(25)
We nally obtain:
[k]
2
=
1
2
[1 + cos 2k] =
1
2
1
[k]
2
=
1
2
[1cos 2k] =
1
2
1 (26)
They are many possibilities for rendering stationary the dierence of average values,
= , depending on the signchosen in each equation, and for each value
ofk. This multiplicity of solutions is not surprising since the stationarity is obtained
not only for the ground state, but also for all the possible excited states of the physical
system; those will be discussed in , and we will see that the stationarity conditions (26)
include the possibility of excited pairs ( ). For the moment, we shall concentrate
on the search for the ground state and look for the absolute minimum of the average
value (7).
Let us examine which signs must be chosen in (26) to get the ground state, i.e. the
lowest possible value ofin expression (7). The chemical potential of an ideal fermion
gas in its ground state is positive, equal to the Fermi level, and proportional to the
particle density to the power23(ComplementXIV, ). In the presence of a weak
attractive potential, the factor( )in the rst term on the right-hand side of (7) is
negative when the modulus ofkis small, and positive when . In the rst case,
to minimize, it is better to choose values of[k]
2
as large as possible, and hence the
sign in the second equation (26) sincekis negative in this case. On the other hand,
when , it is better to choose values of[k]
2
as small as possible, and it is again
thesign in the second equality that must be chosen. As a result, it is thesign that
must be taken in the second equality (26), and hence the+sign in the rst one.
As we know thatkandkare positive, we nally obtain for the ground state:
k=
12
1 +
k=
12
1 (27)
Inserting these results in (21), we verify that the stationarity relations are fullled, inde-
pendently of the sign of. They apply as long as the self-consistent condition derived
1895

COMPLEMENT C XVII
from thekdenition (16) is satised:
k=
1
2
k
kk
1
2
=
1
2
k
kk
k
()
2
+ (k)
2
(28)
As we now show, this condition takes on a simpler form for a very short-range interaction
potential.
The above computation shows that starting from any function(r), or (which
amounts to the same thing) from functionskconsidered to be entirely free variables,
the optimization procedure yields values forkandk; this in turn xes the optimalk
and determines the function(r)for building the paired state described in (1).
1-c. Short-range potential, study of the gap
The matrix elements of the interaction potentialwere dened in Complement
BXVIIas the Fourier transforms of the potential see relations (16) and (18) of that
complement. For a regular potential of range, the matrix elementnecessarily varies
whenchanges by a quantity of the order of1; in particular,0when 1.
However, in many physical applications of the BCS theory, thewave vectors involved
remain very small compared to1, and a useful approximation is to ignore the variations
of the. We therefore consider them all equal to the same constant:
k= (29)
The minus sign was introduced to makea positive number for an attractive poten-
tial; this number is inversely proportional to the volume, as shown by relation (10).
Denition (13) ofknow takes on a simplied form:
=
2
(30)
that can be inserted in the relations (27) to get the functionskandk.
Relation (16) also has a simpler version; all thektake on the same value:
=
k
kk (31)
In this case, there exists only one value of the gap, and since thekandkare real, this
value is also real. We shall see in what follows thatplays a particularly important role,
especially in the dispersion curve characterizing the system excitations (see for example
Figure ).
All the previous formulas apply, provided we replace thekby. Relations (24)
then become, with the sign choice leading to the ground state:
cos 2k=
()
2
+
2
=
sin 2k=

()
2
+
2
=

(32)
1896

FERMION PAIRING, BCS THEORY
Equalities (27) are unchanged. When the second relation (32) shows that, taking
(25) and (30) into account, the value ofkgoes to zero as:
k2 2 (33)
It will be useful in what follows to know the asymptotic behaviors of the functionsk
andk, whose values as a function ofkare given by (5). When , relation (33)
shows thatkgoes to1whereaskgoes to zero as:
k

2
+ 0(

2
2
)
1
2
+ 0(
1
4
) (34)
This ensures the convergence of the summations (C-19) and (C-26) of Chapter
giving the average values ofand of its square.
. Self-consistency condition and divergences
The self-consistent condition (28) now takes on the simpler form:
=
2
k
1
2
=
2
k

()
2
+
2
(35)
that is:
1 =
2
k
1
()
2
+
2
(36)
which is an implicit equation expressing the gapin terms of(as this latter parameter
appears in the denition of).
Choosing a large volume
3
, we can replace the discrete summation overkby an
integral. We shall assume, as mentioned in , that thekin the variational ket (1) are
zero when the modulus ofkis larger than a given cuto value. Under these conditions,
the implicit equation for the gap becomes:
1 =
2
3
(2)
3
0
d
3 1
()
2
+
2
(37)
whereis the upper limit of the wave vectors introduced in ; remember thatis
inversely proportional to the volume, and hence the right-hand side of this equation does
not depend on that volume. The integral will diverge ifis innite, since when,
the function to be integrated behaves as1 1 1
2
. The value obtained for
therefore depends on the value chosen for; this upper limit then plays an important
role.
. Calculation of the gap
We notethe equivalent, in terms of energy, of the cuto frequency:
=
}
22
2
(38)
1897

COMPLEMENT C XVII
We now choose the energyas the integration variable. We then have to consider the
density of states(), obtained
3
by taking the dierential of the denition of:
() =
3
4
2
2
}
2
32
(39)
Relation (37) then becomes:
1 =
2
0
()d
1
(
)
2
+
2
(40)
where, to simplify relation (30), we introduced a chemical potential
relative to the
mean eld energy
4
:
=+
2
(41)
In relation (40), the function to be integrated overcontains a fraction that is
maximum for=
; this fraction takes on signicant values in an energy band of width
centered, inkspace, on the surface of the Fermi sphere (see ComplementXIV)
whose radiusobeys:
}
22
2
= (42)
As for the density of states(), it takes on low values in the vicinity of the center of that
sphere, but increasingly larger ones outside. The inside of the sphere barely contributes
to the summation, the main contribution coming from the outside, in between the Fermi
surface and the cuto energy. We can then nd in this region an intermediate value
0for the density of states that can replace()without changing the integral, with:
() 0 () (43)
The density of states can be removed from the integral, and we get:
1 =
0
2
0
d
1
(
)
2
+
2
(44)
Asis inversely proportional to the volume, whereas, according to (39), the density of
states is proportional to it, this relation is independent of the volume.
In the physics of superconducting metals, the attractive interaction between the
electrons is mediated by the motions of the crystal's ions, i.e. by the phonons of the
network. A cuto energy naturally appears in the matrix elements of the interaction
potential, the Debye energy}of the phonons. One often uses a simple model where,
3
This density of states is dened in Complement CXIV, and given by formula (8) of that complement.
In our case, as we do not have to take into account the two spin states, the density of states is half the
one computed in that complement.
4
With the sign convention we chose forin (29), this mean eld energy is equal to 2per
particle.
1898

FERMION PAIRING, BCS THEORY
in (28), the potential matrix elementskkare zero as soon as the dierence in energy
is larger than}, supposed to be much smaller than
; otherwise, they are
all equal to a constant. The same computations as those that led to (44), yield in this
case the gap equation
5
for levels close to the Fermi surface:
1
2
+}}
d
1
(
)
2
+
2
= arsinh
}

(45)
where is the density of states on the Fermi surface. If, furthermore, we make the
approximation 1, we get:
=
}
sinh(1 )
2}exp
1
(46)
This important equation is called the BCS gap equation.
It is worth noting that this expression cannot be expanded in a power series of
when the interaction goes to zero: all the derivatives ofwith respect toare zero
for= 0. Consequently, this expression cannot be obtained in the framework of a
perturbation theory in powers of the interaction (this point will be discussed in ).
2. Distribution functions, correlations
Inserting expressions (27) in the trial ket, we obtain the optimal state vector BCS
that best describes the ground state. We now examine the physical implications of
that optimized quantum state, concerning the properties of the one- or two-particle
distribution functions. These properties will be used later on in this complement to
understand the origin of the energy lowering due to condensation into pairs of particles.
2-a. One-particle distribution
As we are going to show, the properties of the one-particle distribution are fairly
close to those of an ideal gas.
. Momentum space
Once the gapis obtained, we can use relations (30) and (32) to determine
the values ofkfor each value ofk; relation (C-16) of Chapter
average number of particles in each pair of states. As the two states composing the pair
play the same role, the average number of particles in each of the states is simply half
that number, that issin
2
k. Figure , the variation of the
distribution functionkobtained, which is the momentum distribution function of a
particle, once the variableskandkhave been optimized. For an ideal gas, we saw in
ComplementXVthat it is a Fermi-Dirac distribution; at zero temperature (as is the
case here, since we are studying the ground state), this distribution is a step function
equal to1for and to zero for (dotted curve). In our case, the transition
5
There are two equal contributions to the integral on the right-hand side, one from the values of
above
, the others from those below; this is why the factor12has disappeared from the second
equality.
1899

COMPLEMENT C XVII
Figure 1: Plots of the one-particle distribution functionk=k
2
in the BCS state,
as a function of the energy. In the absence of interactions, this function is equal to1for
, and zero for (dotted line step function). In the presence of interactions,
due to the pairing of fermions the curve is rounded o over an energy domain of the order
of the gap(double arrow on the horizontal axis), and the variations occur around the
value
(value ofshifted by the mean eld eect). The dashed curve plots the product
kkas a function of the same variable. This function is largest around=
, in a
domain spreading over a few.
between0and1occurs around
, hence for a value of chemical potential shifted by the
mean eld eect as indicated by relation (41); this energy shift due to the mean eld is
natural. What is more striking is that the curve no longer presents a discontinuous step,
but varies progressively over an energy domain whose width is of the order of the gap.
The interaction eect depopulates certain pairs of states in favor of other pairs having
higher kinetic energies. Certain fermions are promoted from the inside of the Fermi
surface towards the outside, this eect occurring over a depth of the order of. Ink
space, the perturbation introduced by the attractions is localized in the neighborhood
of that surface; the fermions situated close to the center of the Fermi sphere are not
concerned, whereas those close to that surface gain an energy of the order of the gap.
. Position space
Relations (B-22) and (B-23) of Chapter
function in position space:
1(r;r) = (r) (r)=r r (47)
1900

FERMION PAIRING, BCS THEORY
whereis the one-particle density operator. Taking into account formula (A-14) of
Chapter , applied to normalized plane waves, we can write:
1(r;r) =
1
3
kk
(krkr)
kk
=
1
3
kk
(krkr)
BCS kk BCS (48)
where
k
is the creation operator in the individual state of momentum}kand spin,
andkthe annihilation operator in the state of momentum}kand spin. Now, in
the state BCS, the occupation number of each momentum pair are either0or2, which
means that the average values of the product of these operators is zero whenever each of
them concerns a dierent pair, or if the two individual states are in the same pair, but
are dierent from each other. We now have:
1(r;r) =
1
3
k
k(rr)
BCS kk BCS
=
1
3
k
k(rr)
^k (49)
with:
k= BCS kk BCS=k
2
(50)
The function1is proportional to the Fourier transform of the average populationk
of the individual statek. As this average population is a function whose width is of the
order of the Fermi wave vector, the function1goes to zero whenrr 1,
i.e. as soon as the dierence in positions is no longer microscopic: in this system, there
exists no long-range non-diagonal order of the one-particle correlation function. For
r=r, we get:
1(r;r) =
2
3
^
=
BCS
2
(51)
whereBCSis the numerical particle density:
BCS=
^
3
(52)
The function1(r;r)has no spatial dependence; the factor12reects the fact that
the total densityBCSis shared equally among the two spin states. The results are the
same as for an ideal gas.
2-b. Two-particle distribution, internal pair wave function
As opposed to the one-particle distribution, the two-particle distribution is strongly
aected by the BCS mechanism, which is to be expected since it is a pairing process.
1901

COMPLEMENT C XVII
. Momentum space, peak in the distribution
Relation (C-19) of Chapter
the two-particle density operatorin an unspecied basis. In the momentum repre-
sentation, they are written:
1 :k 3; 2 :k 4 1 :k1; 2 :k2=
k1k 2k 4k 3
(53)
We shall mainly consider the diagonal elements, characterizing the correlations between
the momenta of two particles:
1 :k; 2 :k 1 :k; 2 :k =
kk k k (54)
In this expression, the creation operators repopulate precisely the same states as those
that have been depopulated by the annihilation operators.
For=, in order to obtain a non-zero result, we must havek=k(otherwise
we get the square of a fermionic operator, which is zero). We get:
1 :k; 2 :k 1 :k; 2 :k =k
2
k
2
(ifk=k) (55)
which means there exists no correlations between the momenta.
For=, whenkis dierent fromk, two dierent pairs are involved, and
we again get a product
6
:
1 :k; 2 :k 1 :k; 2 :k =k
2
k
2
(ifk=k) (56)
On the other hand, ifk=k, only one pair is concerned, destroyed and then recon-
structed by the operators (as before, this is a contribution from the diagram in Figure
of ComplementXVII; the computation then involves a single state
k, and we get:
1 :k; 2 :k 1 :k; 2 :k =k
2
(57)
This result is not the limit of the previous one whenk k, which would bek
4
; the
value we obtain is larger, sincek
2
1.
This shows that for all thevalues that do not correspond to a pair of opposite
momenta, the density operator is simply a product, involving no correlations between
the particles' momenta. This conrms what we found in the study of the one-particle
density operator: all thekstates having a momentum smaller than that of the Fermi
level are populated, with a rounding o of the functions due to the pairing phenomenon.
On the other hand, for opposite values of both the momenta and the spin values, as is
the case for a pair, we observe a discontinuity of the diagonal correlation function: it
jumps fromk
4
to the larger valuek
2
. The corresponding discontinuity
(2)
k
can be
written as:
(2)
k
=k
2
k
4
=k
2
1 k
2
= kk
2
(58)
6
If=, to get this result we use the fact that the function(k)is even. Now ifk=k, two
pairs are still involved, labeled by opposite values of momentum (remember that we chose the convention
where each pair is labeled by the momentum of the spin+particle); but, here again, the parity of(k)
leads to(k)
4
, in agreement with (56).
1902

FERMION PAIRING, BCS THEORY
Figure 2: The left part of the gure shows the distribution function of the momenta of
two particles, assuming that the two momenta}k1et}k2are parallel (or antiparallel);
a value = 110has been chosen. In the descending part of the surface, and in the
two corners wherek1+k2vanishes, one can distinguish a small crest indicating a partial
Bose-Einstein condensation. In order to see this eect more clearly, we cut the surface
along vertical planes whose trace is indicated by the dashed lines in the right part of the
gure. This leads to the curves of Figure.
We shall see below ( ) that
(2)
k
is none other than the square of thekcomponent
of the pair wave function. This discontinuity is signicant for the values ofkfor which
the productkktakes on its largest values; Figure
region around the Fermi surface, with an energy bandwidth of the order of the gap.
The momentum distribution function depends on the6components of the two
momenta, which does not allow a simple graphic representation. To simplify, we are
going to assume the two particles' momenta}k1and}k2are parallel (or antiparallel),
so that the distribution we wish to represent becomes a surface in three-dimensional
space: we plot1along one axis,2along the second, and the probability along the third
perpendicular axis. We then obtain the surface shown in the left part of Figure, where
it has been assumed that = 110.
To explore this surfact in more detail, we plot in Figure
by cutting this surface by vertical planes parallel to the rst bisector of the1and2
axes; we assume the dierence1 2to be constant (dashed lines in the right part of
Figure) and use, as the variable, the sum of these two momenta. The horizontal axis
in Figure :
=
1+2
2
(59)
Asvaries, the corresponding point in the plane1,2moves along a straight line.
For a xed value of=1 2, we must set= 0for the wave vector components
to have opposite values:1=2and2= 2. On the left-hand side of Figure,
the dierenceis chosen equal to14; we obtain an almost square curve, rounded
o by the fact thatis not zero (as was the case in Figure), and which presents
1903

COMPLEMENT C XVII
Figure 3: Plots of the distribution function for two particles of parallel (or antiparallel)
momentak1andk2, as a function of the dimensionless variable= (1+2)2; the
gure was plotted with the choice = 110.
For the curve on the left-hand side, the dierence(1 2)is chosen equal to14.
The curve looks like a bell shaped function, practically constant for small values of, and
decreasing for larger values following a rounded slope similar to the one in Figure
all the more steep as thevalue is chosen smaller). No singularity of the distribution
is clearly visible (except for a minuscule peak at the origin).
For the curve on the right-hand side, the dierence in momenta is chosen equal to2};
whenis close to zero, the two momenta take on values that both fall into the rounded
part of the one-particle distribution. A singularity at= 0is now clearly visible, signaling
an accumulation of molecules in a state where their center of mass does not move. The
height of the central peak corresponds to the population of the discrete level having a zero
total momentum, and its width reduces to zero as it is a discrete level.
a barely visible peak. But, as we saw before, the eects of the pairing are important
when1= 2 , i.e. when 2. On the right-hand side of Figure, the
momentum dierence is chosen equal to2}, so that the momenta can both fall in the
rounded part of the distributions. We observe, superposed on a pedestal, a narrow
peak indicating an additional population in the level having a zero total momentum.
The value of that population is given by the height of the peak, whose width is strictly
zero for discrete levels. The singularity of this momentum distribution is then clearly
visible.
Therefore, a singularity appears in the momentum distribution of particle pairs,
whose centers of mass present a condensation in momentum space. This is, however, a
partial condensation: as opposed to the boson case, the condensation peak appears on a
pedestal due the presence of a majority of non-condensed pairs. Actually, the only pairs
involved are those whose two components have energies falling around the Fermi level
, in a bandwidth of the order of the gap. Despite these restrictions, it is nevertheless
true that the condensation phenomenon into attractive BCS pairs has properties related
to Bose-Einstein condensation for repulsive bosons. The link between that condensation
and the appearance of an order parameter for the pair eld is discussed in of
ComplementXVII.
1904

FERMION PAIRING, BCS THEORY
. Position space, correlations described by the pair wave functions
We did not nd any eects of the interactions on1. But, as pointed out before,
since the BCS theory relies on pairing, one expects to nd more interesting properties
concerning the two-particle correlation functions. They will be studied now, limiting
ourselves to the diagonal correlation function, as dened by relation (B-33) of Chapter
XVI, including the spin variables as in (B-36). This function is written as:
2(r;r) = (r) (r) (r) (r)
=1 :r; 2 :r 1 :r; 2 :r (60)
(is the two-particle density operator), or else, as before:
2(r;r) =
1
6
kkkk
[(kk)r+(kk)r]
k k k k (61)
This expression includes the average values of the product of four operators, whose com-
putation is similar to the one explained in XVIIfor the average
interaction energy, except that, in our case, the spin indices are xed rather than ap-
pearing as summation indices. Figure 61):
the incoming arrows represent particles that disappear (annihilation operator action),
the outgoing arrows represent those that will appear (creation operator action); each
value ofkis associated with a position valuer, via an exponential
kr
for the incoming
arrows, or
kr
for the outgoing ones, as well as with a value of the spin.
Parallel spins: if=, the two destruction operators necessarily concern pairs
with dierentk. To restore the populations of these two couples of states to an even
value, the only possibility is to again give each one its initial value; otherwise the result
will be zero. We must have, eitherk=kandk=k(direct term), ork=kand
k=k(exchange term). In the rst case, we obtain (after two anticommutations whose
sign changes cancel each other) a result
7
independent of the position variables:
1
6
kk
kkkkk k kk=
1
6
kk
k
2
k
2
(62)
and in the second case (after only one anticommutation):
1
6
kk
(kk)(rr)
kkkkk k kk
=
1
6
kk
(kk)(rr)
k
2
k
2
(63)
Regrouping these two contributions, and using (6), we get:
2(r;r) =
2
3
2
1[(rr)]
2
(64)
7
If=, we must change the sign ofkandkinkand
kbut, as before, it does not change the
result since we can change the sign of summation variables.
1905

COMPLEMENT C XVII
Figure 4: This diagram symbolizes each term involved in the binary correlation function.
The two incoming arrows on the bottom represent particles that will disappear during the
interaction under the action of the two annihilation operators; the two outgoing arrows
on the top represent particles that appeared during the interaction under the action of the
two creation operators. All the arrows are associated with an imaginary exponential of
the position, with a positive argument for the incoming arrows, and a negative one for
the outgoing arrows. The indiceslabel the spins.
with an exchange term containing the (real) function:
(r) =
2
k
kr
k
2
(65)
This result has the same form as relation (22) of ComplementXVI, taking into account
the fact that the population of each spin state is half of. It shows that the correla-
tion function for two parallel spins exhibits an exchange hole very similar to the one
plotted in Figure
the functionsk
2
are no longer exactly discontinuous step functions. The width of this
exchange hole is of the order of the inverse of, the Fermi wave number related to the
Fermi energyby =}
22
2.
Opposite spins: If=, it is possible for the two annihilation or creation
operators to act on the same pair of states; we are then dealing with a pair annihilation-
creation term (term of typeIIaccording to the classication presented in
Chapter ). Figure
computation of the correlation function for opposite spins:I(forward scattering),II
(pair-pair) andIII(special cases). The computation of their sum has been performed in

2(r;r )
2
3
2
+pair(rr)
2
(66)
1906

FERMION PAIRING, BCS THEORY
We have used the following denition of the (non-normalized) pair wave function
8
pair(rr):
pair(r) =
1
3
k
kk
kr
=

2
3
k
1
()
2
+
2
kr
(67)
which is simply the pair wave function already introduced in relation (D-14) of Chapter
XVII. We nd again relations (38) and (39) of ComplementXVII, where this wave
function was obtained by a dierent method involving the two-particle eld operator.
The presence of the second term on the right-hand side of (66) is thus due to the non-zero
average value of the pair eld, introduced in that complement (non-zero order parameter).
We have just shown that, contrary to what happens in an ideal gas (Complement
AXVI, ), two particles with opposite spins may be spatially correlated. This cor-
relation is described by the modulus squared of the wave functionpair(rr), dened
by its spatial Fourier transformkk. This new wave function, dierent from the one
we used at the beginning to build the-particle trial wave function, was introduced in
, as well as in ComplementXVII, starting from the pair eld
operator. The spatial correlation it characterizes is purely dynamic, as it does not exist
in the absence of interactions. Its physical consequences, in terms of potential energy,
will be discussed in .
Physical discussion: on the right-hand side of (66), the rst term does not
contain any spatial dependence; it simply corresponds to the correlation function of an
ensemble of totally independent particles. The second term, on the other hand, depends
on the position dierencesrr; we now discuss its physical origin in terms of quantum
interference.
This second term comes from the contribution of the pair annihilation-creation
terms for which we have, in relation (61),= and= . Let us show that
cutting them in half, they look like interference terms. They include average values of
operator products that, when=, can be written:
k k k k= BCS k k k k BCS
= (k) (k) (68)
where (k)is dened as:
(k)= k k BCS (69)
Relation (66) then becomes:
2(r;r )
2
3
2
+
1
6
kk
(kk)(rr)
(k) (k) (70)
8
The factor1
3
appearing in (67) permits dening a pair wave function independent of the dimen-
sionof the physical system, in the limit of largewhere the sum over thekbecomes an integral
over d
3
multiplied by(2)
3
. As a result, the square of that wave function is not homogeneous to
the inverse of a volume, as is generally the case for a particle wave function, but to1
6
. Actually,
it should be considered as a two-particle wave function, product of a constant wave function1
32
of
the center of mass of the pair (assumed to have a zero momentum) and a wave function describing its
relative position variable.
1907

COMPLEMENT C XVII
Figure 5: Diagrams symbolizing various contributions to the binary correlation function
for opposite spins (=). The diagram of typeIcorresponds to a process where
two particles with opposite spins are destroyed and then re-created in exactly the same
individual states (forward scattering). The typeIIdiagram corresponds to the case where
two particles of the same pair are destroyed, and then two particles are created in the
states of another pair (pair annihilation-creation process). Finally, typeIIIdiagram is a
special case of the previous diagrams, and yields a negligible contribution. It is the type
IIdiagram that introduces the spatial dependance of the correlation function.
The position dependent term in the correlation function can be interpreted as resulting
from the interference between a process where two particles of the same pair(kk)are
annihilated, and a second process where the two annihilated particles are from another
pair(k k); these two processes are schematized in Figure.
According to (1) and (69), we have:
(k)=kk= 0;k= 0
k=k
k (71)
Ifk=k, the two states (k)and (k)are neither identical, nor orthogonal;
they actually have identical components on all the pairs of states dierent from(kk)
and(k k), but these two pairs have the same component only for states where the
4populations are zero. We can then write:
(k) (k)=
k k kk (72)
Inserting this result in (70) yields relation (66), whose spatial dependence does come
from the interference between the two processes schematized in Figure.
One could also use relation (68) of ComplementXVIIto express the product
(r) (r)as a function of a sum of pair annihilation operators. This is another way
of understanding the role of pairs in the determination of the binary correlation function
expression (66).
1908

FERMION PAIRING, BCS THEORY
Figure 6: Diagram symbolizing two pair annihilation processes from the initial state
BCS, leading respectively to the states (k)and (k). As these two states are
not orthogonal, an interference eect occurs that is at the origin of the position dependent
part of the binary correlation function.
2-c. Properties of the pair wave function, coherence length
The pair wave function plays an important role in the BCS theory, and not only
for the binary correlation functions, as we already mentioned. Its range determines the
coherence length of the system, and its norm is also related to the number of quanta
present in the eld of condensed pairs (ComplementXVII).
. Form of the pair wave function
As the functionskandkonly depend on the modulus ofk, we can apply the
Fourier transform formulas for this case see Appendix, relation (52). Replacing the
discrete summation by an integral, the pair wave function (67) becomes:
pair(r) =
1
2
3
d
3 kr
kk=
1
(2)
2

0
d
1
(
)
2
+
2
sin
(73)
Therefore,the pair wave function is real. Figure kkas a function of
the energy; it presents a maximum in the vicinity of the surface of the Fermi sphere,
with a peak whose width is of the order of. More details on the role this pair wave
function plays in the correlation functions are given in ComplementXVII.
The Fourier transform ofpair(r)is thus concentrated around values of the modulus
ofkof the order of the Fermi wave vector. Its spreadis such that the corresponding
variation of energyis of the order of the gap, which leads to the condition:
}
2
2
2 that is

(74)
1909

COMPLEMENT C XVII
This wave function oscillates
9
as a function of the positionr, at a spatial frequency ap-
proximately equal to the wave vector at the Fermi surface. These oscillations are damped
over a length of the order ofpairdened by (the arbitrary factor2is introduced to
match the usual denition found in the literature):
pair=
2
=
2}
2

=
14

(75)
which is of the order of the distance between fermions, multiplied by the ratio,
very large compared to1. Each fermion pair extends over a relatively large volume,
leading to a strong overlap between pairs. In a superconductor, the lengthpairis called
the coherence length
10
; it characterizes the capacity of the physical system to adapt
to spatial constraints, and plays a role analogous to the healing length in systems of
condensed bosons (ComplementXV, ).
We have shown that the pairing signicantly modies the correlation functions for
opposite spins: the particles now become correlated, whereas this was absolutely not the
case for an ideal gas. It is a positive correlation, leading to a bunching tendency (the
opposite of a Pauli exclusion); this explains the decrease in the interaction energy of
the particles ( ). On the other hand, the pairing has no signicant eect on the
correlation function of particles having the same spin direction; it remains similar to the
correlation function of an ideal gas, with an exchange hole whose width is of the order of
1. Relation (75) indicates that the width of this exchange hole is much smaller than
the distancepairover which the modications of the correlation function for opposite
spins occur.
As mentioned before, the pair wave function has little in common with the initial
wave function(r1r2)used in Chapter -particle variational ket,
since (3) shows that the Fourier transform ofisk=kk. This is not surprising:
when building a trial ket by the repetitive action of the same pair creation operator, we
do not simply juxtapose those pairs. The antisymmetrization eects are dominant, and
in each term of the expansion to the powerof operator kk k
in relation (1),
the result is zero each time the same value ofkis repeated (the square of a fermionic
creation operator is zero; two fermions cannot occupy the same individual state). This
is why the antisymmetrization eects completely remodel the pairs formed in the system
ofidentical particles.
. Norm of the pair wave function
According to (67), the component onkof the ketpairassociated with the wave
functionpair(r)is written as:
kpair=
1
32
kk (76)
9
The existence of this oscillation is conrmed by the fact that its integral over the entire space is
practically zero; this integral is proportional to00, i.e.0
1
2
0
, which is indeed practically zero
since01.
10
The coherence lengthpairshould not be confused with the (London) penetration depth that
characterizes the magnetic eld exclusion from a superconductor, and that depends on the charges of
the particles.
1910

FERMION PAIRING, BCS THEORY
(the functionskandkare even). The square of this ket's norm is therefore:
pairpair=
1
3
k
kk
2
(77)
Replacing the discrete summation by an integral, we get:
pairpair=
1
3
2
3
d
3
kk
2
=

2
4
3
0
()
d
(
)
2
+
2
(78)
where, in the second equality, we choseas the integral variable, introducing the density
of states()dened in (39). The function to be integrated converges since()
varies as
; this function is concentrated around=, spreading over a width

. We can then, to a good approximation, replace()by its valueat the
Fermi energy, and extend the lower bound of the integral to. As the integral of a
Lorentz function is known (Appendix II, 1-b):
d
(
)
2
+
2
=

(79)
we get:
pairpair=
4
3
(80)
We showed in of ComplementXVIIthat the norm
pairpairyields the
average value of the eldpair(R), hence the value of the order parameter. In
of that same complement we showed that the square of the norm, equal topairpair,
yields the large distance behavior of the average value
pair
(R) pair(R); the quan-
titypairpairis thus related
11
to the eld pair intensity (or, in other words, to the
total number of quanta in that eld).
In addition, we just saw in the above that a peak in the momentum
distribution signaled the presence of a Bose-Einstein condensation. Inserting (76) into
(58) shows that this peak height is:
(2)
k
= kk
2
=
3
kpair
2
(81)
The total particle number associated with this peak is:
k
(2)
k
=
3
pairpair=
4
(82)
Consequently, the square of the norm,pairpair, multiplied by the volume, is also
the total particle number in the condensation peak found in , which conrms the
previous interpretation.
11
The pair eld operatorspairand
pair
do not exactly satisfy the boson commutation relations
(ComplementXVII); the operator
pair
pairis thus not, stricto sensu, an operator giving the number
of quanta in the pair eld.
1911

COMPLEMENT C XVII
. Link to the interaction energy
The energy term on the third line of (7) can be written, in the zero range potential
approximation (29):
kk
kkkkkk=
k
kk
k
kk
=
6
pair(0)
2
(83)
This result yields an energy proportional toand to the probability that the two com-
ponents of a pair, described by the wave functionpair(r), are found at the same point;
this makes sense since the pair size is very large compared to the interaction potential
range.
. Non-diagonal order
When studying Bose-Einstein condensation for bosons, we showed in
plementXVIthat the one-particle non-diagonal correlation function did not go to zero
at large distance, when a signicant fraction of the particles occupy the same individual
state. Nevertheless, in
case for a system of paired fermions, where the non-diagonal order goes to zero over a
microscopic distance. This can be understood from a physical point of view, since, in
the present case, there is no accumulation of particles in the same individual quantum
state. On the other hand, we saw in that the center of mass of the pairs of parti-
cles is subject to a phenomenon of partial accumulation, reminiscent of a Bose-Einstein
condensation. It is thus natural to examine the properties of the non-diagonal functions
relative to pairs, and compute the non-diagonal position average value:
(r) (r) (r) (r) (84)
With two positionsron the right, and two positionsron the left, this expression is the
exact transposition to two particles of the one-particle non-diagonal function: a couple of
particles with opposite spins are annihilated at pointr, and then recreated at a dierent
pointr. In a more general way, we are going to evaluate the4-point average value:
(r1) (r2) (r3) (r4) (85)
which, following the same computation steps as for the one-particle functions, can be
written as:
(r1) (r2) (r
2) (r
1)
=
1
6
kkkk
[(kr
1
+kr
2)(kr2+kr1)]
k k k k (86)
1912

FERMION PAIRING, BCS THEORY
In this equality, the matrix elements have already been evaluated in . We are
going to show that this expression can be written as:
(r1) (r2) (r
2) (r
1)
=1(r1;r
1)1(r2;r
2) +
pair(r1r2)pair(r
1r
2) (87)
where the one-particle non-diagonal distribution1has been dened in relation (49) for
the case=, and the pair wave functionpairin relation (67). This equality is the
same as relation (72) of ComplementXVII, but is now obtained by another method.
Demonstration:
To compute expression (86), we distinguish several cases, as already explained on several
occasions:
(I) Forward scattering terms; if the annihilation operators do not act on two states of
the same pair ((k=k), this term will be non-zero only ifk=kandk=k, in
which case it is written:
1
6
kk
[k(r
1
r1)+k(r
2
r2)]
k
2
k
2
= 1(r1r
1)1(r2r
2) (88)
As we already mentioned, for example in Chapter ), the constraints on the
summation indices can be ignored if the size of the system,, is macroscopic; the two
summations then become independent.
(II) Terms corresponding to the annihilation-creation of dierent pairs; ifk=kand
k=kbutk=k(annihilation-creation of dierent pairs), we get the contribution:
1
6
kk
[k(r
1
r
2)k(r2r1)]
kkkk=pair(r
1r
2)
pair(r2r1) (89)
As, in addition, the functionpair(r)is even, we indeed obtain the second term of the
right-hand side of (87).
(III) Ifk=kandk=k, and furthermorek=k(annihilation-creation of the
same pair), we get:
1
6
k
k(r
1
+r2r
2
r1)
k
2
(90)
This term is negligible as it is proportional to
6
when all the positions are the
same, whereas the term (I) is proportional to
2 6
.
Let us now assume the positions can be grouped two by two:r1andr2are close
to each other, as arer
1andr
2, but that the two groups' positions are further away from
each other. Under these conditions, the rst term in1on the right-hand side of (87),
which has a microscopic range in(r1r
1)and(r2r
2), becomes very small. We are
then left with the product of the pair wave functions. It follows that the non-diagonal
1913

COMPLEMENT C XVII
correlation function is simply the product of pair wave functions calculated at the relative
positions
12
.
In the particular case wherer1=r2=randr
2=r
1=r, we get the pair
correlation function (84), which obeys:
(r) (r) (r) (r)
rr
pair(0)
2
(91)
This non-zero long distance limit signals the existence of a non-diagonal order for the two-
particle density operator. It comes, as was already the case for the pair wave function,
from contribution (II), meaning from terms corresponding to the annihilation-creation
of dierent pairs. This situation is reminiscent of what we encountered in the case of a
condensed boson gas; but in the present case, the non-diagonal order concerns the pairs
and not the individual particles.
3. Physical discussion
In an ideal gas of fermions, and as we saw in ComplementXVI, there already exist strong
correlations between the particles, simply due to their indistinguishability (a purely sta-
tistical eect). In the presence of attractive interactions, the BCS mechanism introduces
additional correlations (dynamic correlations) that lower the total system energy. We
are going to show that this decrease in energy comes from a slight imbalance between an
increase in kinetic energy and a decrease of the potential energy, the latter one slightly
surpassing the rst one.
For clarity, we shall discuss this using the short-range potential approximation (
1-c) where all the matrix elements of the interaction potential are replaced by a constant
(being positive); all thekare then equal to the same gap.
3-a. Modication of the Fermi surface and phase locking
The energy written in (7) rst includes a kinetic energy term, then a mean eld
term expressed in terms of the average particle number. If that average number is
constant, this term is independent of the quantum state of the system, and hence not
related to the BCS pairing mechanism. By contrast, the last term in (7), which is the one
we optimized in the variational calculation, is far more interesting; we call it the pairing
term, and use the words pairing potential energy or else condensation energy for
its optimized valuepaired. As thekand thekare real,pairedcan be written as:
paired=
k
kk
2
(92)
where thekand thektake the optimized values given in (27). We see that to get a
large condensation energy, the sum
kkkmust take the largest possible value.
12
As mentioned in note, the center of mass variables are not included in (87
the pairs to be at rest. If this were not the case, the long-range factorization of the non-diagonal order
would be expressed as the product of a function of the two variables(r
1
r
2
)and(r
1
+r
2
)2by the
complex conjugate of that same function of the two variables(r1r2)and(r1+r2)2(in other words
by the product of a function ofr
1
andr
2
by the complex conjugate of that same function of the two
variablesr1andr2).
1914

FERMION PAIRING, BCS THEORY
. Compromise between the kinetic and potential energies
In an ideal gas, the ground state is the one for which all the individual states
of energy lower than the Fermi level (chemical potential= ) are occupied by one
particle, and all the states above are totally empty. In thek-space, the particles each
occupy a state inside one of the two Fermi spheres of radius(with=}
22
2),
one associated with the+spin state, and one associated with thespin state. Using
the ket (1), such a state simply corresponds to the case where:
k= 0andk= 1for
k= 1andk= 0for
(ideal gas) (93)
Whatever the value ofk, one or the other of the functionskandkwill be zero, and
so is the productkk; the condensation energy of an interacting system remains equal
to zero as long as the state of the system does not dier from the ideal gas state. A
condensation energy can only be obtained via a deformation of the Fermi distribution.
It is the attractive interactions that actually distort this distribution to create an
overlap between regions where both functionskandkare dierent from zero, as can be
seen in Figure. This allows minimizing the pairing energy (92), but involves a transfer of
particles from the inside of the Fermi sphere to the outside, hence toward states of higher
kinetic energy; this has a cost in terms of kinetic energy. The optimization we performed
amounts to looking for the most favorable balance between the gain in potential energy
and the cost in kinetic energy. The condensation energy is proportional to the square
of the integral of the dashed line curve in Figure, which presents a maximum in the
vicinity of the Fermi energy. The largest contributions come from energies close to
, over a width of the order of a few- but the gure also shows that the contributions
to the condensation energy spread relatively far from the Fermi surface (the curve only
decreases as the inverse of the energy's distance from its maximum). The Fermi surface,
which was perfectly dened for an ideal gas, becomes blurred over a certain energy
domain.
The two-particle correlation function expresses in more detail this optimization of
the attractive potential energy. Relation (64) shows that, for parallel spins, no signicant
change of the correlation function occurs, when compared to that of an ideal gas (for
which an exchange hole is already present in the binary correlation function) hence no
signicant change of the corresponding interaction energy. This lack of eect comes from
the fact that the BCS wave function only pairs particles with opposite spins. On the other
hand, when the spin directions are opposite, relation (66) shows that the probability of
presence of two particles at a short distance from each other is increased; the larger the
pair wave function modulus at the origin (r=r), the higher this increase will be. It
directly yields the gain in the attractive potential energy.
In other words, the BCS gain in energy comes from the fact that, because of
the interactions, the system changes its wave function to optimize its pairing potential
energy. It develops correlations that go beyond that of an ideal gas; they are referred to as
dynamic correlations, as opposed to the statistical correlations (coming solely from the
indistinguishability of the particles, such as those studied in ComplementXVI). This
produces a deformation of the ideal gas Fermi distribution that, instead of presenting an
abrupt transition between the occupied and empty states (perfectly well dened Fermi
sphere), presents at its edge a more progressive transition region (blurred Fermi sphere).
The system's state vector then becomes a superposition of states where the particle
1915

COMPLEMENT C XVII
number in each pair of states(kk)uctuates. The potential energy term that drives
the BCS mechanism is the pair annihilation-creation term computed in of
ComplementXVII, and that is schematized on its Figure. It includes a sum of terms
containing non-diagonal matrix elements of the form:
(kk)= 2;
(k k)= 0 2(kk)= 0;
(k k)= 2 (94)
(the occupation numbers of all the other pairs remaining the same); between the ket and
the bra, a pair(kk)is replaced by another one(kk). The BCS energy gain is due
to the summation over all these non-diagonal terms; they are sensitive to the coherence
of the state vector between these two components (where the numbers of pairs uctuate
in a correlated way) and hence to their relative phase.
. Phase locking and cooperative eects
In the computation of the , the minimization of the energy led us to choose
the phases of all thekto be equal, and they simply disappeared from the following
calculations. To discuss the physical process at work, it is useful to reintroduce them
with their non-specied values before the optimization, as they appeared in (14); when
all the matrix elements of the interaction potential are equal, the average value of the
pairing energy is written:
paired=
kk
sinkcosksinkcosk
2(
k k)
(95)
We mentioned above that adding to the phasekof eachkany given common phase,
left all the results unchanged. The energy is invariant with respect to a symmetry of the
wave function, the one that concerns the global phase of thek. It is often called the
(1)symmetry, referring to the(1)unitary symmetry group of rotations around a
circle, isomorphic to the group of phase changes for a complex number.
On the other hand, changing one by one the phases of thek, leads to an obvious
reduction (in absolute value) of the pairing energy (95): in the complex plane, vectors
that were perfectly aligned, now take dierent directions and the modulus of their sum
is reduced. We saw that this term is at the origin of the gain in energy provided by the
BCS mechanism; it is clearly linked to the acquisition of a common phase by all the pairs
of individual states. This is an example of a phenomenon called in physics spontaneous
symmetry breaking: whatever the phase of eachk, which can take on any value, it is
essential that it be the same for all, otherwise most of the gain in energy will be lost. In
a similar way, in the ferromagnetic transition in a solid, the space direction along which
the spins will align is not a priori xed, but it is essential that it be the same
13
for all
the spins.
Note nally the cooperative character of the energy gain obtained, which, math-
ematically, corresponds to the presence of a double summation overkandk. Starting
13
For an ensemble of spins parallel to any given direction, each spin is in a state where the relative
phase between the components on+and is the same. For the BCS mechanism, it is the phase
between the components where the occupation number of the couple of stateskkis0or2that takes
on a value independent ofk. The corresponding energy lowering results from an interference eect
between states where two pairskandkhave respective occupation numbersk= 2
k= 0and
k= 0
k= 2; therefore it cannot be directly expressed in terms of pair populations.
1916

FERMION PAIRING, BCS THEORY
from a perfectly phase locked situation, destroying the phase locking of a single pair leads
to an energy loss proportional to the number of pairs that remained phase locked; the
individual energy of a single pair is not what is at stake. On the other hand, starting
from a situation where the phases of all the pairs are random, changing a single phasek
barely modies the average energy. We are in the presence of a cooperative eect: the
greater the number of pairs that are already phase locked, the higher the tendency for a
new pair to become phase locked; this tendency can be seen, in a way, as resulting from
a mean eld created in a cumulative way by all the other pairs. Here again we see the
analogy with a ferromagnetic material where, the greater the number of spins already
aligned, the higher the gain in energy with the alignment of a new spin.
3-b. Gain in energy
We now compute the gain in energy resulting from the pairs' formation. We rst
insert in (7) relations (27), which yield the optimal values of thekand thek, and use
the denition (25) for; we also take all the potential matrix elements to be equal to
the same constant. Since we then have:
k
2
=
1
2
1=
1
2
(96)
and:
kk=
1
2
1
2
=

2
(97)
we get:
BCS
=
4
2
+
k

2
4
kk
1
()
2
+
2
1
()
2
+
2
(98)
The rst term on the right-hand side is the mean eld term (as before, we have neglected
1compared to the total number of particles). The second one corresponds to the kinetic
energy, and the third one, to the interaction between pairs:

2
4
kk
1
()
2
+
2
1
()
2
+
2
=

2
2
k
1
()
2
+
2
(99)
where to get the second equality we have used relation (36) to eliminate the summation
overk. Using again the denition (25) for, we get:
BCS
=
4
2
+
k
1
()
2
+
2
()
2
+
2

2
2
(100)
1917

COMPLEMENT C XVII
On the right-hand side of this expression, the rst term, corresponding to the mean eld,
is of no particular interest. The second one accounts for the change in energy due to the
dynamic correlations introduced by the interactions; it characterizes the BCS mechanism.
We rst check the convergence of its summation overk, for a xed value of.
This is not the case for each term in the parenthesis, which tends toward a constant
when , leading to a summation in1 1
2
that diverges. We are now going
to show that the divergent terms cancel each other. We can write:
()
2
+
2

2
2
3
8

4
()
3
+ (101)
and thus:
1
()
2
+
2
()
2
+
2

2
2
3
8

4
()
3
+ (102)
We have just shown that the divergent terms in the innite summation ofkin relation
(100) cancel each other between the kinetic and interaction terms; the function in the
summation goes to zero, for largevalues, as1()
3
1
6
, which ensures the conver-
gence and does not require the introduction of a cuto frequency(apart from the one
we had to introduce before to ensure a nite value for). This was also the case for the
total number of particles. We thus see that once we have introduced an upper bound-
ary (cuto)in the integral determining the gap, all the other important physical
quantities remain nite, without having to reimpose this cuto frequency.
The precise determination of the energy requires, in general, the computation of
somewhat complex integrals. It will not be detailed here, but yields the result:
BCS
=
0
4
[]
21
2

2
(103)
(remember thatis the density of individual states at the surface of the Fermi sphere,
and is proportional to the volume=
3
). Finally, the energy gain resulting from the
BCS pairing is given by:
=
1
2

2
(104)
It can be shown that the values ofthat contribute most to the energy are those that
are lower than or comparable to the gap; the energy change linked to the pairing
phenomenon is mainly located in the vicinity of the Fermi surface. This result is often
interpreted by saying that, in an ensemble of pairs, each pair gains an energy of
the order of, which explains the
2
dependence of (104); while this image is simple,
it has its shortcomings (see note).
3-c. Non-perturbative character of the BCS theory
Generally, the most basic way to take the interactions into account is to use a
rst order perturbation theory (Chapter XI), where the energy correction is the average
value, in the initial non-perturbed state, of the perturbation Hamiltonian. Applied here,
the rst order correction to the energy is obtained by inserting the values (93) into (7).
1918

FERMION PAIRING, BCS THEORY
The rst term (kinetic energy) on the right-hand side of (7) is unchanged, and the third
one remains zero since, according to (93), the productkkis always zero for any value
ofk. We are left with the second term, which produces a mean eld correction. To
the next perturbation order, the eect of the potential is to change the ground state by
transferring pairs of particles, initially both inside the Fermi sphere, toward individual
states whose momenta fall outside the sphere (all the while keeping the total momentum
constant); this changes at the same time the average kinetic energy (which is increased)
and the interaction potential energy. The computations become more and more complex
as the perturbation order increases. And above all, it is clear that this approach to higher
and higher perturbation orders cannot account for the existence of the gap obtained in
(46): as the function ()has all its derivatives with respect toequal to zero at
= 0, it can not be expanded as a series in.
The BCS theory is a non-perturbative method that solves this diculty. However,
it is not an exact method since it is a variational method, but the chosen wave function
is suciently well adapted to allow the inclusion of important physical eects, without
using any perturbation theory.
4. Excited states
Up to now, we only studied the ground state of the system of attractive fermions. As soon
as the temperature is no longer zero, excited states of the system begin to be populated.
In this last section, we shall give a survey of the BCS theory predictions concerning the
excited states. A study of the BCS theory at non-zero temperature can be found in more
specialized books.
4-a. Bogolubov-Valatin transformation
Relations (E-3) and (E-4) of Chapter
formations of the creation and annihilation operators of spin12fermions. With the
notation of the present complement where the spin directions are explicit, they become:
k=kk k k
k=k k+kk
(105)
which, by conjugation, yield the denitions of the Hermitian conjugate operators
k
and
k
:
k
=
kk k k
k
=
k k
+
kk (106)
For each value ofk, we get a general transformation of the four initial creation and
annihilation operatorsorinto four new operatorsand. We showed in Chapter
XVII
We also saw in that chapter that the ket
kdened in (2):
k= k+kk k
0 (107)
1919

COMPLEMENT C XVII
is an eigenvector of the two operatorskand kwith a zero eigenvalue:
k
k=kk k k
0= 0
k
k=kk k
+
k
0= 0 (108)
It follows that the variational ket BCSof the ground state, written in (1), is an
eigenvector, for any value ofk, of all the operatorskand k, with a zero eigenvalue. It
is therefore also an eigenket of all the operators
kkand
kkwith a zero eigenvalue,
which is a minimum eigenvalue for operators dened as positive or zero. Furthermore, we
showed that the repeated action of the creation operators
k
and
k
permits obtaining
other states, which are also eigenvectors of the operators
kkand
kk. We are going
to show that these operators
kkand
kkcan be interpreted as corresponding to
the occupation numbers of the excitations present in the physical system.
4-b. Broken pairs and excited pairs
Letting
k
act on (107), we get:
k
k= k
2
k k
2
kk k
0= k
2
k
+k
2
k
0
=
k
0 (109)
which is a ket normalized to unity, and obviously non-zero (as opposed to the one resulting
from the action ofk). Similarly, if we consider the action of
k
, we get another non-zero
ket:
k
k=
k
0 (110)
These two new normalized kets are orthogonal to the initial ket
k, since they corre-
spond to an occupation number equal to1, whereas the occupation numbers of
kare
0and2. In these states, a pair has been replaced by a single particle, not belonging to
any pair; they are called broken pair states.
As the squares of the operators
k
and
k
are zero, the repeated application of
any of the two does not allow constructing new orthogonal states; however this can be
achieved with their cross product. Letting
k
act on the ket (110), we get the ket
k:
k=
kkk=
kk k k0 (111)
which is another normalized ket, and orthonormal, as can be easily checked, to
k;
letting now the two operators
k
and
k
, in the inverse order, act on
k, we get the
same ket,
k, within a change of sign. The components of the two states
kand
kcontain occupation numbers equal to0or2; the ket
kis called excited pair
state. To go from
kto
k, we simply switchkandk, change the sign ofk, and
nally take their complex conjugate (this is true for the general case, but in the BCS
pairing case, askandktake on real values, this last step becomes unnecessary).
4-c. Stationarity of the energies
Let us now show that the energies of these new states are stationary with respect
to the variational parameters.
1920

FERMION PAIRING, BCS THEORY
. Broken pair
According to (109), the action of
k
on the ground state BCSleads to the ket:
k
BCS=
k
0
BCS (112)
where
BCS
is just the ket BCSwhosekpair component has been removed from the
product:

BCS=
k=k
k (113)
The energy average value in the state (112
(i) the kinetic energy=}
22
2associated with the state
k
0
(ii) the energy associated with the state
BCS; the computation of that energy is the
same as for BCS, including the pair interaction energy, except for the fact that one less
pair is now involved in the calculation. This slightly modies the value of, and hence
the optimal value of the parametersk; however, since the relative variation ofis
inversely proportional to the number of particles, we shall ignore this slight modication.
(iii) nally, the interaction energy between the particle in the individual state
k
0
and the particles described by
BCS; the pair structure of that state means that the
only contributions are a direct term in:
1
2
k=k
0
k k
k k+
k k
k k = 0
k
k
= 0 (114)
(whereis the average particle number in the state
BCS) and an exchange term in:
1
2
k=k
kk k k
k k+
k k k k=
k=k
kk k (115)
Note again that, for an interaction that does not act on the spins, the exchange is only
possible with particles having the same spin. With the short-range potential approxima-
tion ():
0=k= (116)
the term (114) becomes equal to , the term (115) to 2, and their sum simply
yields a constant 2.
The parameters dening the variational state (113) are the set of thekand thekfor
k=k(the dependence on theparameter, which characterized the broken pair, is no
longer present). These parameters are the ones that make the energy stationary, since
we neglected the slight variation of the gap linked to the disappearance of a pair; they
play no role either in (i) or in (iii).
We have just conrmed that
k
BCSrenders the energy stationary (in the frame-
work of the variational approximation we are using). A symmetry argument shows that
this is obviously also the case for the state
k
BCS.
1921

COMPLEMENT C XVII
. Excited pair
In the stationary relations (26), the change of
kinto
ksimply amounts to
exchanging the signs(thekandkare real); the components of the ket
kare
part of the stationary solutions we have discarded in writing (27). We thus conrm that
the excited pair corresponds to a stationary energy; it is actually the highest possible
energy for the pair of states(kk).
4-d. Excitation energies
In the4-dimensional state space associated with each pair of states(kk), the
creation operatorsacting on the
kpermit building a new basis of4orthonormal
states, whose average energies are stationary. They can be considered to be the ap-
proximate eigenvectors of the system Hamiltonian. We now compute the corresponding
eigenvalues.
In the case of the broken pair, the excited state
BCS
does not contain the same
number of particles as BCS; it does not make sense to directly compare their energies.
To make a valid comparison, we must take into account the presence of a particle reservoir
whose energy increases by(chemical potential) each time it absorbs a particle. In other
words, we must evaluate the variations of the average value.
. Broken pair
We now show that the variation of the average value associated with
the breaking of the pair is simply the energydened in (25).
To do so, we compute the variation of this average value when the state BCSis replaced
by expression (112). Several terms come into play:
(i) The variation of the average value of the kinetic energy is the dierence between the
energyof a particle and that of the populationk
2
of the pair(kk), with a kinetic
energy of2; this dierence is therefore12k
2
.
(ii) As for the potential energy, the passage from BCSto the ket
BCSchanges the
average particle numberfrom2k
2
(initial population of the pair) to0, so that the
variation ofis =2k
2
. The mean eld term
2
4in (7) varies by
2, that is k
2
. The following term in (7) is zero for a short-range
potential. Finally, the breaking of a pair has an impact on the binding energy in the
last term of (7); if we change the dummy summation indexinto, andinto,
the terms that will change correspond to the terms=and the terms=, which
double each other. The breaking of the pair leads to an increase of energy equal to:
2 kk
k
kk= 2 kk
=
1
2()
2
=

2
(117)
where we have used (31) and (27).
(iii) We saw that the unpaired particle has a potential energy2 2.
This energy must be added to the variation of the mean eld term calculated above, to
give a contribution equal to 12k
2
2.
1922

FERMION PAIRING, BCS THEORY
We now sum all the previous contributions and add the variation ofwhich yields
a term 12k
2
. The total variation is then:
=
2
12k
2
+

2
(118)
or else, taking (13) into account:
= 11
+

2
=
2
+
2
= (119)
We nd the expected result
14
.
. Excited pair
We now assume that in the product that yields BCS, the ket
kis replaced by
the orthogonal ket
kwritten in (111), and which describes an excited pair. We are
going to show that the variation of the average value associated with that
excitation is2, twice the excitation associated with the breaking of the pair.
To show this, we must, here again, add several variations. The rst one comes from the
kinetic energy and yields2 k
2
k
2
, that is212k)
2
see relation (4).
The second one is the mean eld term introduced by the fact that the average value of
the total particle number varies by2 k
2
k
2
, which leads to a potential energy
variation 12k
2
. We must also account for a variation of the pair binding
energy, which comes from the sign change of the productkk, which doubles the term
(117). Because of the change in the average particle number, the term ingives a
contribution of2 k
2
k
2
, that is212k
2
. Finally, all the terms found
for the broken pair are just doubled here and we indeed nd2.
. Spectrum of the elementary excitations
We now have the energy of the three excited states associated with each pair of
states: the energy(doubly degenerate since it corresponds to both kets
k
kand
k
k) and the energy2(non-degenerate). The value of these energies is given in
(25). Figure
these excitations as a function of their momentum). The solid line corresponds to the
spectrum associated with the breaking of a pairk,k, during which a particle disappears,
as in relations (109) and (110); as the spectrum associated with the excitation of a pair
can be simply obtained by the multiplication by a factor2, it is not plotted in the gure.
The dashed lines plot those same energies for an ideal gas (no interaction) for which
= 0. The interaction eect creates a gap, which yields a minimum value for the
excitation energy that otherwise can go to zero for an ideal gas.
Upon the breaking of a pair, we saw that, on average, the system's total population
changes by12k
2
. The curve in Figure
14
The calculation would be the same for an ideal gas; in the particular case where = 0and according
to (25), we would get:= .
1923

COMPLEMENT C XVII
Figure 7: The solid line plots the variation as a function ofof the excitation energy
=
()
2
+
2
, with= and=~
22
2(for the sake of simplicity,
we assumed
, so as to neglect the dierence betweenandin the expression of
). This energy presents a minimum equal to the gapwhenis equal the the Fermi
wave vector= (wave vector for which=). The dashed line curve is the same
function, but for = 0(zero gap), hence for an ideal gas.
The BCS theory predicts that the energy associated with the breaking of a pair is, and
the energy associated with the excitation of a pair is twice that amount, i.e.2. The
minimum of the energyis therefore the minimum of energy that must be supplied to
the system in its BCS ground state to produce one of the previously computed excitations.
As explained in the text, the region on the left-hand side of the curve corresponds to hole
type excitations (the excitation leads to a particle loss for the physical system) and the
region on the right-hand side to particle type excitations (the physical system gains a
particle).
For clarity, the gure does not take into account the mean eld eects; these would lower
all the energies by the same negative quantity, and change the chemical potentialinto
dened by (41), so thatwould become=. These eects slightly shift the
curve plottingto the left.
which part of the curve we analyze. On the left-hand side (decreasing function), the solid
line curve nearly perfectly matches the dashed curve, meaningk
2
is practically equal
to1(see Figure). In that case,12k
2
1: a particle disappears in the course of
the excitation, which is said to be of the hole type. Its energyis the energy needed
to push one particle towards the reservoir that xes the chemical potential, diminished
by its initial kinetic energy, and to which we must add the mean eld correction
15
;
the excitation energy is therefore
, a value that corresponds to the left-hand side
of the curve. As for the right-hand side part (increasing function), the constantk
2
is
practically zero: the excitation adds a particle to the system, and is said to be of the
particle type. Its energy is equal to, energy necessary to promote a particle
from its energyto a state of kinetic energy(with, as before, a mean eld correction
15
This correction changes the initial energyinto 2, and hence into
.
1924

FERMION PAIRING, BCS THEORY
that changesinto
). Finally, for the central part of the curve, we have a mixed
excitation, of both hole and particle type; it is the region of the spectrum where the solid
line parts the most from the dashed line, and where the BCS mechanism, which creates
the gapplays an essential role.
From these four energy levels associated with each pair of states, quantum statis-
tical mechanics allows obtaining a density operator describing the thermal equilibrium
of the system at temperature, as well as all the various thermodynamic functions. The
corresponding development will not be exposed in this complement. We shall simply
mention that it allows extending the validity of a certain number of results obtained at
= 0, by simply introducing a gap()that depends on the temperature. At zero
temperature,(0)is still given by (46), but the gap decreases asincreases, and goes
to zero for a certain critical temperature. This cancelling of the gap corresponds to
a phase transition: as a system of attractive fermions is cooled down, when it reaches a
certain temperature the pair condensation phenomenon occurs, which leads to a number
of physical consequences. For example, the system's specic heat rst takes on values
higher than in the absence of transition, then abruptly (exponentially) goes to zero as
0.
Conclusion
In conclusion, the choice of a variational basis of paired states sheds new light on the be-
havior of an ensemble of attractive fermions. We focused on the case of weak attractions,
corresponding to electrons in superconducting metals; in such a situation,( )1
and relation (75) shows that the pair rangepairis very large compared to the distance
between fermions. The one-particle distribution shown in Figure
to the step function obtained for an ideal gas at zero temperature, the step being never-
theless rounded o over an energy band of width equal to. In other words, the BCS
pairing only slightly modies the Fermi sphere of a perfect gas. Studying the properties
of the optimal state, we were able to expose a number of important phenomena: exis-
tence of spatial dynamic correlations, which explain the increase of the average attraction
(negative) energy between fermions of opposite spins; phase locking accounting for the co-
operative aspect of the pairing (increase of the attractive energy overcoming the increase
of kinetic energy, hence leading to a decrease of the system total energy); existence of a
pair wave function describing fermions of opposite spins, and reminiscent of the Cooper
pairs (ComplementXVII); appearance of a gap in the elementary excitation spectrum
explaining the robustness of the system's ground state.
Another interesting limiting case concerns strong attractive interactions where the
pair range becomes very small compared to the distance between particles. Molecules
are then really formed with a binding energylarge (in absolute value) compared
to the Fermi energy. Saying that the size of the bound state is small compared to
the distance between particles amounts to saying that its momentum distribution width
is large compared to the Fermi wave vector. This means that due to the attractive
potential, the occupation of the individual states are spread over a large number of
dierent momenta, which dilutes the eects of Pauli exclusion principle; these eects thus
become negligible whereas they are essential in the BCS case. Instead of being positive,
the chemical potential is now negative, close to. Relations (24) then show that the
(and hence the populations of the individual statesk) always remain small, nevertheless
1925

COMPLEMENT C XVII
extending up to energiesof the order of. In that special case, the pair wave function
pair, with its Fourier components= sincos
2
, practically coincides with
the wave function having as Fourier components( ) =tg
2
, and which was
initially used for building the paired state. As the molecules contain two strongly bound
fermions, they behave as composite bosons (ComplementXVII, ), which may undergo
Bose-Einstein condensation. Paired states enable us to see the continuous passage from
one limiting case (BCS situation with a weakly perturbed Fermi sphere) to the other
(condensation of strongly bound molecules). A detailed discussion of this continuous
passage and its physical consequences is given in 4-6 of reference [10] and in [11].
In this complement, we emphasized the physical interpretation of the results ob-
tained in the detailed calculations we presented; this will give the reader the necessary
base for studying the experimental aspects of superconductivity, which are not presented
here. Among the many aspects of superconductivity that have not been studied in this
complement, we can list: transport phenomena and the disappearance of the electri-
cal resistance; behavior in the presence of a magnetic eld (Meissner-Ochsenfeld eect);
experimental study of the elementary excitation spectrum and gap measurements via
dierent methods (tunnel eect, magnetic resonance); Josephson eect.
The interested reader can refer, for example, to the books of M. Tinkham [12],
of R.D. Parks [13], or of A.J. Leggett already quoted [8]. The book of Combescot and
Shiau [14] presents a good overview of the four main theoretical methods for studying
superconductivity, the BCS variational method discussed in this complement being the
rst of this list.
1926

COOPER PAIRS
Complement DXVII
Cooper pairs
1 Cooper model
2 State vector and Hamiltonian
3 Solution of the eigenvalue equation
4 Calculation of the binding energy for a simple case
In this complement, we present the Cooper model which was a rst step towards
the complete BCS theory. It yields some results of that theory without having to deal
with the diculties inherent to an-body problem. With this simplied model, we
study the properties of two attractive fermions whose wave function, in the momentum
representation, is excluded from the Fermi sphere. The model will be presented in ,
where we show the existence of a bound state, occurring only because of the existence
of that sphere. Furthermore, we shall see that the mathematical expression for the cor-
responding binding energy is reminiscent of the expression for the gap valueobtained
in the BCS theory.
1. Cooper model
Among a large ensemble of identical fermions, we focus our attention on two of them,
supposed to attract each other, in order to study their two-body wave function and energy
levels. The presence of all the other fermions is simply accounted for by a Fermi sphere
that, because of the Pauli exclusion principle, requires the components of that wave
function to be zero inside that sphere. Such an approach is obviously not very rigorous:
isolating two fermions among a large number of other indistinguishable fermions does not
make much sense. Furthermore, it is hard to imagine why two of them would interact via
an attractive potential, whereas all the others determining the Fermi sphere would be
without interaction. However, the mathematical form of the results obtained with this
model presents interesting similarities with the variational method where all the fermions
are treated equally; it is thus useful to study this model.
2. State vector and Hamiltonian
Consider two attractive fermions in a singlet spin state= 0:
= 0=
1
2
[1 :2 :1 :2 :] (1)
The relative motion of their position variables is described by the orbital ket orb, and
their center of mass is described by a zero momentum ketK=0. Their state vector is:
=K=0 orb = 0 (2)
1927

COMPLEMENT D XVII
The state orbis characterized by a wave function orb(r):
orb(r) =r orb (3)
where:
r=r1r2 (4)
is the dierence between the positions of the two particles (relative position). As the
singlet state is even with respect to the exchange of the two particles, their fermionic
character requires the wave function orb(r)to be even with respect to particle exchange,
i.e. with respect to a sign change ofr:
orb(r) = orb(r) (5)
We assume the operator describing the attractive interaction between the two particles
to be independent of the spin. As in , we separate in the the two-
particle Hamiltonianthe motion of the center of mass from the relative motion, and
assume the center of mass is at rest. We are then left with a Hamiltonianrelthat only
acts on the space of the relative motion variables, and can be written:
rel=
^p
2
+(^r) (6)
where^r=^r1^r2is the operator associated with the relative position of the two particles;
^pis the operator associated with the momentum of the relative motion, dened as a
function of the momenta^p1and^p2of the two particles:
^p=
^p1^p2
2
(7)
As mentioned above, we assume the presence of an ensemble of non-interacting
fermions, whose Fermi level is. We must then solve the eigenvalue equation:
rel orb= (+ 2) orb (8)
when orbdoes not have any component inside the Fermi sphere of radius, related
to the Fermi levelby:
=
}
2
()
2
2
(9)
In relation (8),is the eigen-energy with respect to twice the Fermi level. It is indeed
natural to take2as an energy reference; this is the minimal energy to be given to the
two fermions under study, for their wave function orbto have zero components inside
the Fermi sphere, in the absence of interaction. With this convention for the energy
origin,simply reects the eect of the attractive interaction.
1928

COOPER PAIRS
3. Solution of the eigenvalue equation
We now expand orbon the normalized eigenvectorsk(plane waves) of the momentum
^p:
orb=
k
kk (10)
Projected ontok, the eigenvalue equation (8) reads:
2 k+
k
kk k= (+ 2)k (11)
where we have set, as usual:
=
}
22
2
(12)
The absence of components of orbinside the Fermi sphere leads to the relation:
k= 0ifk (13)
while (11) becomes:
[+ 2 ( )]k=
k
kkk (14)
The matrix elements of the interaction operatorare noted
kk:
kk=kk (15)
4. Calculation of the binding energy for a simple case
Let us further simplify the model and assume that the potential matrix elements
kk
are such that:
kk= ifkk +
kk= 0 otherwise (16)
wheredenes a wave vector domain ; these matrix elements can therefore
be factored. Note that the minus sign in front of the constantwas introduced to ensure
that, for our present attractive potential, the constantis positive. Whenk +,
the summation on the right-hand side of (14) becomes a constant independent of
k, with:
=
k +
k (17)
whereas, ifk + , this summation is zero. The solution of this equation is
simply:
k=
+ 2 ( )
if k +
k= 0 otherwise (18)
1929

COMPLEMENT D XVII
We must add the self-consistent condition we get from inserting this solution in the
denition (17) of:
=
k +
+ 2 ( )
(19)
that is, changing the sign of the denominator:
1
=
k +
1
2 ( )
(20)
This condition is also an implicit equation for obtaining the energy. Assuming
the system is enclosed in a cubic box with a very large edge length, the discrete
summation can be replaced by an integral, and we get:
1
=
3
2
2
+ 2
2 ( )
(21)
We now choose, as the integral variable, the variable:
= (22)
As d=}
2
d, this integral now includes a density of states():
() =
3
2
2
2
d
=
3
2
2
}
2
(23)
or:
() =
3
2
2
}
32 (24)
The implicit equation forthen becomes:
1
=

0
d
(+)
2
(25)
where the upper boundis dened by:
+ =
}
2
2
(+ )
2
(26)
As we assumed , we can replace
1
in (25) the density of states(+)
by(), no longer dependent on the variable, and that we simply note:
=() (27)
We can then perform the integration, which yields:
1
=
2
ln(2 )

0
=
2
ln
2
=
2
ln
2
(28)
1
Replacingby in (23), one can easily compute an order of magnitude for the density of states at
the Fermi level. We nd() , hence a value proportional to the average particle number.
1930

COOPER PAIRS
We then have:
2
=
2
(29)
The solution of this equation foris:
=2
2
1
2
(30)
which, when ()1, can be simplied to:
=2exp [2 ] (31)
We obtain a negative energy (with respect to2), as expected for a bound state
(the wave function can be normalized). As we cannot make a series expansion of the
functionexp (1)in the vicinity of= 0, this energy cannot be expressed as a power
series of the interaction potential, since all its derivative are zero at= 0; consequently,
this energy cannot be obtained by an ordinary perturbation calculation.
Note also that the energygoes to zero (through negative values) if the density
of states()goes to zero, i.e. ifgoes to zero: the existence of the bound state is
therefore linked to the presence of the Fermi sphere, whose role is to introduce a non-zero
density of states. If the Fermi sphere disappears, so does the bound state.
We nd results similar to those found in ComplementXVIIusing the BCS theory,
and in particular to the expression (46) of that complement, which yields the gap.
To obtain that expression, we had to introduce an upper bound}for the variation
(in absolute value) of the energies around the Fermi energy; this upper bound plays a
role comparable to that played by the energywe just introduced in (25). We simply
have to assume that}= for the two results to become quite similar as they only
dier by a factor2in the exponential (the sign dierence simply comes from the fact
that the gapwas dened as a positive quantity, whereas a binding energy is negative).
The interest of the Cooper model is to clearly highlight the essential role played by the
density of states(in the vicinity of the Fermi level) in the creation of the gapin
the BCS theory.
1931

CONDENSED REPULSIVE BOSONS
Complement EXVII
Condensed repulsive bosons
1 Variational state, energy
1-a Variational ket
1-b Total energy
1-c 0 approximation
2 Optimization
2-a Stationarity conditions
2-b Solution of the equations
3 Properties of the ground state
3-a Particle number, quantum depletion
3-b Energy
3-c Phase locking; comparison with the BCS mechanism
3-d Correlation functions
4 Bogolubov operator method
4-a Variational space, restriction on the Hamiltonian
4-b Bogolubov Hamiltonian
4-c Constructing a basis of excited states, quasi-particles
In this complement we study the properties of an ensemble of repulsively interacting
bosons
1
, undergoing Bose-Einstein condensation. We know that, for an ideal gas, a
system of bosons in its ground state is totally condensed: a single individual quantum
state, corresponding to the lowest energy, is occupied by all the particles. In the presence
of short-range interactions, and for a suciently diluted system, one expects its properties
to remain close to that of an ideal gas, and in particular that a large fraction of the
particles still occupy the same individual quantum state. We consider this to be the case
for the system under study, and that the population of one individual quantum state
is much larger than all the others. We shall assume that each of the states obeys the
periodic boundary conditions in a box of side length(Complement CXIV), and that
the state with the large population
2
is the statek=0(whose momentump=}kand
kinetic energy are zero). Consequently, if0is the average value of the population of
this zero momentum level, we assume that:
0 k (for anyk=0) (1)
1
We will not consider the case of attractive interactions, as they lead to an unstable physical state
see XV.
2
This hypothesis simplies the writing of the equations, but is not essential; in the case where it is a
state of non-zero momentumk0that is highly populated, one can go to the reference frame where this
momentum is zero. This amounts to adding, in the initial frame,k0to all the wave vectors appearing
in the equations.
1933

COMPLEMENT E XVII
wherek= kis the average number of particles occupying an individual statek=0.
The total particle numberis:
=0+
k=0
k (2)
We have already used, in ComplementXV, a rst approximation to study the
ground state of a condensed boson system: we described the state of the-particle
system as the product ofidentical individual state vectors. This led to the Gross-
Pitaevskii equation. This approach implies that only one single individual statek=0
is occupied (0=andk= 0for anyk), as for an ideal gas. This obviously cannot
be exact: it is clear that the interactions introduce dynamic correlations between the
particles, which cannot be accounted for by a state vector that is a simple product
of individual kets (hence without correlations). Actually, the eect of the interaction
potential on the ground state is to transfer at least a fraction of the particles
3
from the
statek=0to the statesk=0; a model involving only one individual state is necessarily
limited to the case where the potential eect is very weak, and hence0 .
In ComplementXV, we introduced another approximation, based on the Hartree-
Fock method; it is more general than the previous one as it allows taking into account a
non-zero temperature. However, it still implies that each particle moves in the mean eld
created by all the others, ignoring the dynamic correlations; its description of the ground
state is no better than the one derived from the Gross-Pitaevskii equation. Furthermore,
this latter method proved to be problematic for a boson system undergoing Bose-Einstein
condensation: we noted in of ComplementXVthat, for a system of condensed
bosons, the Hartree-Fock approximation predicts, at the grand canonical equilibrium,
very large uctuations of the number of condensed particles. In the real world, these
uctuations are strongly limited by the repulsion between the particles, which clearly
indicates that the predictions of the Hartree-Fock approximation concerning uctuations
are non-physical.
In the present complement, we shall try to address these two problems: on one
hand, we shall take into account the dynamic correlations introduced by the interactions
in the physical system; on the other, we shall not let the number of condensed particles
uctuate arbitrarily. We will use a variational method, choosing a variational state that
takes into account the binary correlations between the particles, but does not introduce
unrealistic uctuations of the particle number. This variational state will be built with
the help of a paired state, enabling us to directly use the results of Chapter . We will
add an extra component, to account for the Bose-Einstein condensation in the individual
k=0state. Obviously, this is still not an exact calculation, as it involves a variational
approximation, but it allows describing a physical situation more complex than the simple
Gross-Pitaevskii approximation. This approach also highlights the many analogies, but
also the dierences, between the pairing of condensed bosons and the pairing of fermions.
In a general way, this complement illustrates how variational methods allow chang-
ing the correlations between particle pairs. When dealing with binary interactions, as in
a standard Hamiltonian, these correlations determine the average value of the potential
energy (Chapter, ). The higher order correlations (ternary, etc.) may be
present and play a role in the system; but they are not directly involved in the energy.
3
This phenomenon is traditionally called quantum depletion and will be discussed in more detail
in
1934

CONDENSED REPULSIVE BOSONS
This is why using the paired states to optimize only the binary correlations can lead to
fairly good results.
We introduce in
parameters, and compute the corresponding average energy. In , we shall search for the
optimal values of these parameters that minimize this energy, using an approximation
where0 so that we can neglect the interactions between the particles in the
k=0individual states. In
obtained, such as the number of particles that are not in thek=0state, the energy,
and the correlation functions. We shall then develop, in , a dierent point of view,
the Bogolubov operator method. We shall choose a larger variational space, and use
the results of
directly diagonalized. This will conrm a certain number of previously obtained results.
The reader only interested in the Bogolubov operator method can go directly to this
paragraph, which is fairly self-contained. The conclusion of this complement will sum up
the results obtained and the limits of this approximation method.
1. Variational state, energy
We are now going to directly apply the results of ComplementXVII, for the choice of
the variational ket as well as for the computation of its average energy.
1-a. Variational ket
The (normalized) variational ket is of the form:
= 0 paired= 0
k
k (3)
where the subscriptrefers to the name Bogolubov. In this expression, pairedis the
paired state for spinless particles written in (B-8) of Chapter , which is a tensor
product of the normalized states (C-13):
k=
1
coshk
exp kkk
0 (4)
with:
k=tanhk
2k
(k0) (5)
The domainof the tensor product in (3) is half thek-space, which prevents (as we
saw in Chapter , ) the double appearance of each state
k=
k; the
origink=0is excluded from.
As for0, it is the coherent state already used in ComplementXVII, relation
(44):
0=
02 0
0
0= 0 (6)
This state depends on a complex parameter0, characterized by its modulus
0and
its phase0:
0=
0
0
(7)
1935

COMPLEMENT E XVII
It is the normalized eigenvector of the operator0with eigenvalue0:
00=00 (8)
The average particle number in the statek=0is then:
0000=
0000=0 (9)
The width of the corresponding distribution is
0(ComplementV), hence negligible
compared to0(this number is supposed to be large).
The variational variables contained in the trial ket (3) are thus the set ofkand
k, as well as0and0.
1-b. Total energy
Expression (61) of ComplementXVIIyields the total energy in the form:
=
k=0
sinh
2
k+
0
2
(0+)
2
+0
k=0
ksinh
2
ksinhkcoshkcos 2 (0 k)
+
1
2
kk=0
kksinh
2
ksinh
2
k+sinhksinhkcoshkcoshkcos 2 (k k)
(10)
where the matrix elementskof the particle interaction potential are dened, as in
Chapter , by:
k=
1
3
d
3 kr
2(r) (11)
The term on the second line of (10) corresponds to the momentum exchanges between
thek=0particles and thek=0condensate, as well as the pair annihilation-creation
processes originating from the condensate. The terms in the last line, with a double
summation overkandk, correspond to interaction eects betweenk=0particles.
1-c. 0 approximation
As already pointed out in the introduction, for an ideal gas in its ground state,
only one individual state is occupied, corresponding to the lowest energy; in that case,
the average total particle number is equal to0, and all the populationskof the other
kstates are zero. We are going to assume that the system we study is a dilute gas where
the interaction eects are limited so that0remains very large compared to the sum of
all the populationsk:
0 =
k=0
(12)
1936

CONDENSED REPULSIVE BOSONS
This hypothesis is more constraining than the one initially proposed in (1), since now the
population0must largely exceed the sum of all the other populations. Nevertheless it
allows a simplication of the following computations while highlighting a certain number
of general physical ideas.
Under these conditions, the interactions between particles in thek=0states and
particles in thek=0condensate are dominant compared to the interactions between
particles both in thek=0states. The interaction term on the second line of (10),
proportional to0, is therefore much larger than the term on the last line, which does
not contain0. This is why we use the approximate average value:
k=0
sinh
2
k+
0
2
(0+)
2
+0
k=0
ksinh
2
ksinhkcoshkcos 2 (0 k) (13)
We have yet to determine the optimal values of the variables appearing in (13) by
minimizing this energy average value with respect to each of them.
2. Optimization
The variational statedepends on the variables0and0associated with the indi-
vidual statek=0(condensate), as well as on the angleskand the phaseskassociated
with all the otherk=0states. On the other hand,is not a variational variable, but
a function of the previous variables determined by relation (53) of ComplementXVII:
=
k=0
kk=
k=0
sinh
2
k (14)
As in ComplementXVII, we introduce a Lagrange multiplier(chemical poten-
tial, see Appendix VI) to x the average total particle number; we thus impose the
stationarity of the dierence of two average values:
= (15)
where is the average total particle number in the variational state:
=
00+
k=0
kk=0+ (16)
The function to be minimized is therefore:
=
0
2
(0+)
2
0+
k=0
( )sinh
2
k+0
k=0
k (17)
with:
k=ksinh
2
ksinhkcoshkcos 2 (0 k) (18)
1937

COMPLEMENT E XVII
2-a. Stationarity conditions
The functionmust be made stationary with respect to all the variables. We
shall start with the phases, then the parametersk, and nally0.
. Stationarity with respect to the phases: phase locking
The phases only intervene in thek, as phase dierences0 k. Since we have
a repulsive potential, we assumekis positive; furthermore, as the variablekis always
positive according to its denition in Chapter , the product sinhkcoshkis also
always positive. Expression (18) shows that, whatever the value ofk, the minimization
of the functionwith respect to the phases0andkrequires the cosine to be equal to
1, that is:
k=0 for anyk (19)
Consequently, the phases used to build the paired states must all be equal to the phase
dening the coherent state associated withk=0. We call this equality the phase
locking condition.
. Stationarity with respect to thek
The stationarity ofwith respect to each parameterkimplies that, for anyk:
0 =0(0+)
k
+ 2 ( )sinhkcoshk+0
k
k
(20)
where the derivative ofkis taken at the phase values that satisfy relation (19). Grouping
on a rst line the terms in sinhkcoshk(including those coming from the derivative of
sinh
2
k), and on a second, those coming from the derivative of sinhkcoshk, we get:
0 = [0(0+) + +0k] 2sinhkcoshk
0kcosh
2
k+sinh
2
k (21)
Relation (21) then becomes:
0 = [ +0(0+) +0k]sinh2k 0kcosh2k (22)
that is:
tanh2k=
0k
+0(0+) +0k
(23)
. Stationarity with respect to0
We now write the stationarity ofwith respect to0. Taking into account rela-
tions (17) and (18), as well as the phase locking condition (19), we get:
0 =0(0+)+
k=0
ksinh
2
ksinhkcoshk (24)
1938

CONDENSED REPULSIVE BOSONS
This result shows that the chemical potential is equal to:
=0(0+) + (25)
which is the sum of a mean eld term0(0+)created by all the particles, and
another term:
=
k=0
ksinh
2
ksinhkcoshk (26)
This last term is also the sum of two terms of dierent signs: a positive contribution
coming from momentum transfer processes, leading to an increase of the repulsion en-
ergy due to the boson bunching eect; a negative contribution due to the creation or
annihilation of pairs from the condensatek=0(Figure XVII), and
expressing the reduction of that energy, due to the dynamic correlations induced by the
interactions.
Relation (23) then becomes:
tanh2k=
0k
+0k
(27)
with:
= (28)
2-b. Solution of the equations
The ground state we are looking for depends on two parameters that are externally
xed, the volume
3
of the physical system, and the chemical potentialthat controls
the total number of particles. We must determine the variableskfrom (27), as well as
0from (24). This last relation includes, which is not an independent variable since
it is determined by (14). We have a set of non-linear equations whose solution is not
obvious, a priori: relation (27) determines thek(0), and hence(0), as a function
of0. But0itself is determined as a function ofand the variablesk(directly, and
indirectly through) by the stationarity condition (24). Inserting thek(0)in this
relation, we get an implicit equation for0, reminiscent of the implicit equation for the
gapin the BCS theory (ComplementVII, ).
A rst approach for solving this implicit equation is to proceed by successive iter-
ations, as in the Hartree-Fock method. We start from an approximate, reasonable value
of0, such as the value0obtained by assuming thatandare both zero. Using
(27), we then get a rst approximation for thekand for, that can be inserted in (24)
to get a new value for0. Iterating the process, one can expect, as for the Hartree-Fock
non-linear equations, a convergence after a certain number of cycles.
Another approach is to not arbitrarily x the chemical potential, but rather deduce
it from the computation. We then start from an arbitrary0value, yielding the values
of the anglesk, then the value ofusing (14); this xes the total particle number
=0+, and the relations (25)-(26) yield the chemical potential. We shall use this
simpler approach in what follows.
1939

COMPLEMENT E XVII
3. Properties of the ground state
We start by computing the total particle number. To highlight the general ideas while
dealing with equations as simple as possible, we shall use a model where the potential
matrix elementskare all equal to the same constant0, or else equal to zero:
k=0ifk
k= 0ifk (29)
where the cuto valuecharacterizes the potential range( 1). To further
simplify, we shall consider, in each calculation, the case where= 0, i.e. whereis
the same as.
3-a. Particle number, quantum depletion
We use relation (14) to compute, the average number of particles in the indi-
vidual statesk=0. To get sinh
2
k, let us rst compute cosh2kusing:
cosh2k=
cosh
2
2k
cosh
2
2ksinh
2
2k
=
11tanh
2
2k
=
+0k
(+ 20k)
(30)
Since we have:
2sinh
2
k=cosh
2
k+sinh
2
k1 =cosh2k1 (31)
we can write:
sinh
2
k=
1
2
+0k
(+ 20k)
1 (32)
Inserting this relation in (14) we get:
=
1
2
k=0
+0k
(+ 20k)
1 (33)
Let us see what becomes of this expression in the simple model where theare
equal to the(is supposed to be negligible). Replacing the summation in (33) by
an integral, we get:
=
3
16
3
d
3
}
22
2
+0k
}
222
}
22
2
+ 20k
1 (34)
Using the matrix elements of the simplied potential, relation (29), the function to be
integrated only depends on the modulusofk, and goes to zero if; this means
1940

CONDENSED REPULSIVE BOSONS
that, using spherical coordinates, the integral overonly goes from0to. We dene
the integral variables:
s=
}
22 00
k=k (35)
whereis the healing length introduced
4
in XV:
=
}
22 00
(36)
Notingthe upper bound ofscoming from the upper boundof:
= (37)
we can write:
=
3
4
2
2 00
}
2
32
0
2
d
2
+ 1
2
(
2
+ 2)
1 (38)
The integral in (38) is still convergent ifgoes to innity since, when ,
one can make a limited expansion in powers of the innitely small= 1
2
and write:
2
+ 1
2
(
2
+ 2)
=
1 + 1
2
1 + 2
2
= 1 +
1
2
4
+ (39)
The integral also converges at the origin (the function to be integrated diverges as
1
,
but the dierential element is
2
d, which eliminates the divergence). This integral can
be readily calculated, and for an innite value of(very short-range potential) is equal
to
23. We then get:
=
3
3
2
00
}
2
32
(40)
We nd that is proportional to the volume and to the product00to the power
32. When the interaction potential0is zero, we conrm that all the particles are
in the individual statek=0. As0starts increasing, the non-condensed fraction
(0+)varies, at the beginning, proportionally to the power32of0.
We have found that the eect of the interaction potential is to transfer a certain
number of particles from the individual statek=0towards thek=0states. This
eect is often called quantum depletion. It has nothing to do with a thermal excitation
eect that would bring some particles from their ground state towards excited states,
as a result of the coupling with a thermal reservoir at a non-zero temperature. The
calculations we are performing in this complement concern the ground state, and we
assume the temperature is rigorously zero.
4
In ComplementXV, we dened in (61) a constantas a function of the parameterassociated
with an interaction potential in(r). Such a potential corresponds to0= (whereis the
volume), and relation (36) is indeed equivalent to=
}
2
2 0.
1941

COMPLEMENT E XVII
Comment:
Note however that the result (40) was established with the hypothesis 0. It is
therefore only valid if:
0
}
2
2
(0)
13
(41)
Ifis the range of the potential, andits order of magnitude, relation (11) shows that
the order of magnitude of the matrix elements0is
3 3
; the previous condition is
then written:
}
2
2
3
3
0
13
(42)
The result is thus valid ifremains small compared to the kinetic energy of a particle
localized within the potential range, multiplied by the ratio between the average particle
distance in the statek=0and. This requires the potential rangeto be suciently
small.
3-b. Energy
We now compute the energy, taking successively all the dierent terms of (13) into
account.
. Kinetic energy
The rst contribution comes from the kinetic energy, which according to (32) is
written:
=
k=0
sinh
2
k=
1
2
k=0
+0k
(+ 20k)
1 (43)
This term reminds us of the one we encountered in the computation ofin (33), but
the presence of the factorin the summation changes its properties. In the simplied
potential model where thekare given by (29), and if we furthermore assume that
= 0, the change of variable (35) leads to:
=
3
4
2
2
}
2
32
[00]
52
0
d
4
2
+ 1
2
(
2
+ 2)
1 (44)
According to (39), when , the function to be integrated behaves as:
4
2
+ 1
2
(
2
+ 2)
1=
4
1 +
1
2
4
+ 1=
1
2
+ 0
1
2
(45)
and tends towards a constant. Consequently its integral over dis divergent ifis
innite; ifis large but nite, the integral value depends linearly on the choice of.
1942

CONDENSED REPULSIVE BOSONS
. Interaction with the condensate
In relation (13), the mean eld term in the energy0(0+)
2
2is known
sincehas been obtained previously see relation (38). We do not need to compute
specically its contribution to the average energy.
The second term in the potential energy is proportional to0, and corresponds
to the interactions between atoms outside the condensate (k=0individual states) and
inside the condensate (population of thek=0state); this term contains the sum of
thekdened in (18). In order to evaluate that term, we need to compute the product
sinhkcoshk:
sinhkcoshk=
1
2
sinh2k=
1
2
sinh
2
2k
cosh
2
2ksinh
2
2k
=
1
2
tanh
2
2k
1tanh
2
2k
(46)
or else, according to (27):
sinhkcoshk=
0k
2(+ 20k)
(47)
Taking into account the phase locking condition (19) for the optimal variational state,
thekdened in (18) are equal to:
k=ksinh
2
ksinhkcoshk
=
k
2(+ 20k)
1 (48)
We obtain the contribution
0
to the energy:
0=0
k=0
k=
0
2
k=0
k
(+ 20k)
1 (49)
For the simple model (29) already used before, and if= 0, this results becomes:
0
=
3
4
2
00
2 00
}
2
32
0
2
d
22
+ 2
1 (50)
The function to be integrated behaves, at innity, as:
2
2
+ 2
1
2
1
1
2
+ 1=1 + 0
1
2
(51)
which means the integral is not convergent when, but depends linearly on.
1943

COMPLEMENT E XVII
. Ground state total energy
With the same approximation= 0, the sum of (43) and (49) yields:
+
0=
1
2
k=0
(+ 20k) 0k (52)
or else, using again the simplied model (29) for the interaction potential and adding
(44) and (50):
+
0
=
1
4
2
2
2
}
2
32
(00)
52
0
3
(
2
+ 2)
4 2
(53)
Relations (45) and (51) show that the function to be integrated tends towards12when
; we have again a divergent integral when (or ). Consequently,
its value is a linear function of the chosen cuto frequency. However, the fact that
the limit of the function to be integrated is negative indicates that, for large values of
, the decrease in potential energy overcomes the increase in kinetic energy.
The ground state total energygroundis the sum of all the energies we just com-
puted, including the mean eld term:
ground=
0
2
(0+)
2
++
0
(54)
where+
0
is given by (53).
Remarque:
The divergences appearing when in our calculation of the energy are not fun-
damental. They occur when one assumes that the matrix elementskof the potential
written in (29) remain constant when , while they tend to zero with a realistic
potential. It is indeed possible to perform a more careful treatment of the potential, such
as that mentioned in 4.2 of Référence [15], and to obtain a nite result.
3-c. Phase locking; comparison with the BCS mechanism
We just saw in how all the phaseskhad to become equal to the phase
0associated with the statek=0in order to minimize the repulsive energy between
any particle in thek=0states and any particle in thek=0state. Fixing the phase
dierences to be zero is what we called the phase locking condition. This situation
reminds us of the symmetry breaking of the BCS mechanism, where a common locking
of all the relative phases of all the pairs(kk)enabled the building of a gap, through
a collective eect. For bosons, the equivalent of the collective mean eld created by the
pairs is the one created by the condensate of particles in thek=0state. This is why it is
now the relative phase of the pair states with respect to that of the condensate that plays
a role; in other words, we have now an external instead of an internal mechanism.
The gain in energy will be exactly the same, whatever value is chosen for the phase0;
only the relative phasek 0is relevant. Accordingly, and just as for fermions, the
arbitrary choice of the phase leads to a symmetry breaking phenomenon. The analogy is
reinforced by the fact that, for a xed value of, we found in
1944

CONDENSED REPULSIVE BOSONS
particle number0in thek=0state is given by an implicit equation in0. Similarly,
in the BCS theory, it is also an implicit equation that xes the value of the gap.
Relations (40) and (53) include non-integer powers of the interaction potential0,
and are thus non-analytic functions of that potential. They cannot be obtained by a
perturbation theory as a power series expansion of0, and this is another analogy with
the results of ComplementXVII.
We also saw in that complement that, in the BCS mechanism, it is the energies in
the vicinity of the Fermi level that play the most important role. This is not the case for
a system of repulsive bosons. For example, in (44), the function whose integration over
yields the kinetic energy is:
() =
4
2
+ 1
2
(
2
+ 2)
1 (55)
whereas the one yielding the interaction potential energy between particles in thek=0
states and particles in thek=0state is:
0() =
2
2
2
(
2
+ 2)
1 (56)
Figure
line. It illustrates how, in the case of repulsive bosons, the eects of minimization of the
potential energy overcome those of the kinetic energy. This minimization of the repulsion
necessarily comes with a modication of the particles' position correlation function, which
must decrease at short distances; this interpretation will be substantiated in
where we study the binary correlation functions. We also note in the gure that the
accumulated gain of energy is not due to a particular energy band: all thevalues
contribute up to the limit imposed by the upper bound of the integral.
We can further rene the analysis of the energy balance by looking at the gain of
energy per individual quantum state. We must then remove from the previous relations
the factor
2
coming from the density of states, and hence remove a factor
2
from (55)
and (56). Figure is small
or of the order of1, the decrease in potential energy largely overcomes the increase in
kinetic energy; on the other hand, the two contributions balance each other when1.
According to (35), the condition.1corresponds to:
.
1
that is:.00 (57)
This means that individual states of low energy provide most of the decrease of the re-
pulsive potential energy; the corresponding energy domain is proportional both to0
and to the interaction matrix element0. Relations (27) also show that it is those en-
ergies in the system ground state that the energy minimization aects the most. From
the physical point of view, it is understandable that particles having a low kinetic en-
ergy compared to the interaction energy00are the most aected by the interactions,
whereas those with a kinetic energy large compared to00have their correlations only
slightly modied by the interaction potential. However, as we noted before, even though
the individual contribution of the highest energy state to the energy reduction is reduced,
their large number (corresponding to a density of states proportional to
2
) means that
their contribution to the total energy remains signicant.
1945

COMPLEMENT E XVII
Figure 1: Plots as a function ofof the functions whose integral overyield the ki-
netic energy(upper dashed curve), the potential energy for the interaction with the
condensate
0
(lower dashed curve), as well as their sum (solid line). The increase
in the kinetic energy is overcome by the decrease in the potential energy, which ends up
lowering the total energy.
3-d. Correlation functions
As the system is contained in a box and obeys periodic boundary conditions, we
expect the properties of the one-body correlation functions to be translation invariant.
This does not rule out a possible spatial dependence of the correlation functions, as far
as the dierences in positions are concerned. This is what we want to elucidate now.
. One particle
Expanding the eld operator (r)on the annihilation operators, according to
relation (A-3) of Chapter , we get for the one-particle correlation function1
1(rr) = (r) (r)=
1
3
kk
(krkr)

kk (58)
where expression (3) determines. Since in this state the particle number in an
individual statekis always the same as that number for the individual statek, the
average value of
kkinwill be dierent from zero only ifk=k. In the summation
overk, thek=0contribution introduces a term in0; adding to it all the other
contributions, we get:
1(rr) =
1
3
0+
k=0
k
k(rr)
(59)
Whenr=r, the function1is simply equal to the total particle densitytot:
tot=
0+
3
(60)
1946

CONDENSED REPULSIVE BOSONS
Figure 2: Plots as a function ofof the kinetic energy (upper dashed curve), the potential
energy (lower dashed curve), and the total energy (solid line) per individual state. It
shows that it is the lowest kinetic energy states that make the largest contribution to the
lowering of the energy.
Whenrandrare dierent, the function1is the sum of two terms:
one term corresponding to particles in thek=0state (condensed particles),
independent of the positions; this term does not decrease at large distance, but has an
innite range.
a second term corresponding to particles in thek=0state, which is the transform
of the particle distributionk, and therefore goes to zero whenrandrmove away
from each other (it has a microscopic range).
We nd again the Penrose-Onsager criterion according to which it is the condensed
fraction of a boson system that leads to an innite range of the non-diagonal one-body
correlation function (in the case of paired fermions, we found in ComplementXVII,
and , that this long range does not occur for the one-body correlation
function, but only for the two-body correlation function).
. Two particles
The diagonal two-particle correlation function is written:
2(rr;rr) = (r) (r) (r) (r)
=
1
6
kkkk
(k k)r(kk)r

kkkk (61)
We get the same simplications as in of ComplementXVIIin the computation
of the average values of products of creation and annihilation operators: operators0
placed on the right each yield a factor0and operators
0
placed on the left, each a
factor
0; the average values of the other operator products are given by the results
of . We must distinguish between several cases, depending on the
number of values, among the4summation indices, which are equal tok=0; we shall
proceed by decreasing values of that number.
1947

COMPLEMENT E XVII
(i) If the four operators concern thek=0state (case represented in Figure
ComplementXVII), we get the contribution:
0(01)
6
0
3
2
(62)
which is position independent.
The case where only three of the summation indices are zero is not possible, as
the corresponding term would contain the average value of an operatork(or of its
Hermitian conjugate) in the state, which is zero.
(ii) If one creation and one annihilation operator concern the individualk=0
state, two cases may occur and yield dierent types of terms:
direct terms in
0kk0or
k00k
; both contributions are equal and their
sum leads to:
20
6
(63)
which is also position independent.
exchange terms in
0k0k
or
k0k0; the corresponding terms are also
equal and their sum is written:
0
6
k=0
k(rr)
k+
k=0
k(rr)
k
=
20
6
k=0
kcos [k(rr)] (64)
which now depends on the dierence in the positionsrandr. These terms reect the
existence of a bunching eect between bosons; relation (C-28) of Chapter
the value ofk:
k=sinh
2
k (65)
(iii) If the operators corresponding tok=0are of the same nature (both creation
or both annihilation operators), we get terms such as the ones represented on Fig
of ComplementXVII, corresponding to the creation or annihilation of a pair from the
condensatek=0:
for a product of the type
kk00wherekandkare not zero but opposite
(for the same reason as explained before), we get an anomalous average value in the
state
k, multiplied by the average value of the product of two operators0. The rst
average value is given by relation (C-51) of Chapter , and the second yields(0)
2
,
that is0
20
according to (7).
for a product of the type
00k k
wherekandkare not zero but
opposite, we get the complex conjugate of the previous result.
1948

CONDENSED REPULSIVE BOSONS
The sum of the two previous results is then written:
0
6
k=0
k(rr)
sinhkcoshk
2(0 k)
+
k=0
k(rr)
sinhkcoshk
2(0 k
)
=
20
6
k=0
sinhkcoshkcos [k(rr) + 2 (0 k)] (66)
(iv) Finally we have terms where none of the wave vectors is zero, corresponding
to cases where the particles are ink=0states before and after the interaction. They
include a direct term:
()
2
6
(67)
which is constant, an exchange term, and nally a pair annihilation-creation term. Com-
pared to the previous terms (which are proportional0), their relative value is of the
order of 0. Taking into account the exchange term and the pair creation-annihilation
term leads to simple calculations of a type already performed; however, to be consistent
with (12) and the corresponding energy approximations, we shall ignore those terms.
The sum of (62), (63) and (67) yield the constant
26
, to which we add (64)
and (66) and get:
2(rr;rr)
2
6
+
20
6
k=0
sinh
2
kcos [k(rr)]
sinhkcoshkcos [k(rr) + 2 (0 k)] (68)
The position dependent term on the second line shows how the relative phases introduced
incontrol the relative particle position in the physical system; it conrms that the
choicek=0does indeed decrease the probability of nding two particles close to each
other.
When the phase locking condition (19) is satised, the position dependent contri-
bution becomes:
20
6
k=0
sinh
2
ksinhkcoshkcos [k(rr)] (69)
Since sinhkcoshk, the cosine in each term has a negative coecient; it does decrease
the probability2(rr;rr)of nding two particles at the same pointr: the dynamic
correlations appearing in the system tend to antibunch the particles, and hence re-
duce their repulsive interactions. The nal result is a compromise between a sinh
2
k
term that leads to bunching (as for a non-interacting boson gas) and a antibunching
term in sinhkcoshkthat is larger, and involves anomalous average values (creation or
annihilation of particle pairs in the condensate).
1949

COMPLEMENT E XVII
Comments:
(i) The correlation function (68) is invariant with respect to the exchange ofrandr,
as seen from the way the termskandkare accounted for in the summation. Its
Fourier transform only contains terms incos [k(rr)], which can take on any value
by an appropriate choice of thekand thek. As mentioned in the introduction, the
variational state can lead to any correlation function; the results discussed above concern
the optimal value of this correlation function.
(ii) On several occasions, we assumed the chemical potential correction, dened in
(26), to be zero, which enabled us to replace theby the. Let us now check that a
non-zero value of this correction does not radically change the results we obtained.
Using the model (29) where the non-zero matrix elementskare all equal to the same
constant0, expression (26) is simply written:
=0
k=0
sinh
2
ksinhkcoshk= 0
k=0
sinhk
k
0 (70)
where the summation is limited to vectorskhaving a modulus less than. Setting:
= 00 (71)
with 0, relation (27) that xes thekbecomes:
tanh2k=
00
+ 00(1 +)
(72)
The corrective eect ofand hence, is to lower thek; this correction is however
negligible if 00. Consequently, the populations of the individual statesk(which
are equal to sinh
2
k) decrease when. 00, but remain practically unchanged in
the opposite case. The quantities resulting from a summation overk, such as, are
then barely aected: the change in the function to be integrated only occurs for small
values of, whose contributions, in any case, are weak because of the factor
2
in the
integral (38). As for the energy, this is accentuated since the integral in (53) diverges
ifis innite, which means it mainly depends on the large values of(if 1).
Turning now to the correlation functions computed in , they contain summations
overkthat lead to the same integrals; they are thus fairly insensitive to the value of.
This explains why, aside from the predictions concerning the populations of small wave
vectors, the approximation= 0used in
To push the analysis a step further we need to compute the value of thecoecient.
This requires improving the precision of the calculations, and in particular taking into
account the interactions of the particles in thek=0individual states. This is beyond
the scope of this complement, and we shall simply accept thatis small and note that
only the smallpopulations are changed whenis not equal to zero.
4. Bogolubov operator method
We now present a dierent point of view and introduce the Bogolubov method; it is
based on the search for a readily diagonalizable operator form of the Hamiltonian (or of
an approximate expression of this Hamiltonian). This method not only applies to the
ground state, but it also enables the study of the excited states. We shall use the results
of
the Hamiltonian.
1950

CONDENSED REPULSIVE BOSONS
4-a. Variational space, restriction on the Hamiltonian
The variational set we consider has been dened in (3); we assume:
= 0 paired (73)
where0is the coherent state (6), and paired, any paired state in the Fock space
spanned by all the individual states others thank=0. We call(0)the ensemble of
kets expressed as (73).
We now take the general Hamiltonian operatorwritten in (8) of Complement
BXVII, and consider its action restricted to such states; the corresponding matrix elements
are of the type:
(74)
whereandare any two kets of(0). In the computation of this matrix
element, the same simplications as in 0on
the right can be replaced by the number0, any creation operator
0
on the left by the
complex conjugate
0. We will further simplify the problem by assuming, as in (12), that
the total populationof thek=0individual levels is much smaller than0= 0
2
,
and keeping only certain terms among the Hamiltonian interaction terms.
First, we study the forward scattering terms, which are the terms (k=kand
k=k) in relation (20) of ComplementXIII. Their expression is:
0
2
kk
kkkk=
0
2
kk
kkkk kkkk=
0
2
1 (75)
In this equality,is the total number of particles operator:
=0+ (76)
where0=
00is the operator associated with the population of the individual state
k=0, andthat associated with the total number of particles in the statesk=0:
=
k=0
kk (77)
As for all the other interactions terms, we shall proceed as in
keep the terms that contain0 the others correspond to interactions between particles
in thek=0individual states, assumed to be negligible when inequality (12) is satised.
In all the terms we keep, there are either four or two creation or annihilation operators
concerning thek=0state.
Those containing the product
0000, or one of the two products
k00kand
0kk0, are already taken into account in the mean eld term (75). We simply have
to add:
the terms containing the products
k0k0or
0k0k, i.e. the exchange terms
of in ComplementXVII; they yield a contribution:
0kkk (78)
1951

COMPLEMENT E XVII
the terms in
kk00, corresponding to the pair creation from the condensate,
or the terms in
00kkcorresponding to the annihilation of pairs into the condensate;
their contribution is:
0k
2
e
20
kk
+ e
20
kk (79)
With the above conditions, we get a simplied version of the Hamiltonian, which
becomes a reduced Hamiltonian:
=
0( 1)
2
+
k=0
kk+0k kk+
1
2
e
20
kk
+ e
20
kk (80)
If the number of particles is xed, the rst term in the right hand side (mean eld)
introduces only the same displacement of all energies, without physical consequence.
In the Bogolobov approximation, where the condition0 is assumed, one often
merely replacesby0in this term, which amounts to restricting the sum of (75) to
the termsk=k=0. Since the kets (73) are eigenvectors of0with eigenvalue
0,
we may then replace the mean eld operator (75) by the number0
2
02. If, moreover,
one assumes that0= 0, one obtains the simpler expression:
=
0
2
0
2
+
k=0
kk+0k kk+
kk
+
kk
2
(81)
Either (80) or (81) can be used as the Hamiltonian within the Bogolubov approximation.
Neither of these operators conserves the total particle number, because of its terms
proportional to the product of two creation or two annihilation operators. Complement
BXVIIexplained how such anomalous terms can, nevertheless, account for the interac-
tion eects within the framework of certain approximations. We are now going to show
that this expression can be put in the form of a Hamiltonian of independent particles,
provided the operators undergo the transformation introduced in .
4-b. Bogolubov Hamiltonian
We obtained in Chapter E-29) of the Hamiltonian operator
:
=
k
}
kk+
kk (82)
which includes the Bogolubov operators for bosons:
k=kk+kk
k=kk+kk
(83)
1952

CONDENSED REPULSIVE BOSONS
Remember that is half the momentum space, avoiding double counting of the same
pairs of states in (82). Relation (E-15) of Chapter kandkin terms
of the two parameterskandk:
k=coshk
k
k=sinhk
k
(84)
As for the value of the parameter, it will be xed later.
We then have:
kk= k
2
kk+k
2
kk
+
kkkk
+kkkk (85)
and:
kk= k
2
kk+k
2
kk
+
kkkk
+kkkk (86)
The operators in these equalities can be rearranged in normal order, using the proper
commutation; adding them both, we get:
kk+
kk= k
2
+k
2
kk+
kk
+ 2k
2
+ 2
kkkk
+ 2kkkk (87)
that is, taking (84) into account:
kk+
kk=cosh2k kk+
kk
+ 2sinh
2
k+sinh2k kk
2k
+
kk
2k
(88)
Operator can therefore be written:
=
k
}cosh2k kk+
kk
+2sinh
2
k+sinh2k kk
2k
+
kk
2k
(89)
Now this Hamiltonian may be identied with the approximate Hamiltonian (80).
To see this, we replace cosh2kby expression (30), sinh2kby the double of (47), and
sinh
2
kby (32); we are still in the simplied model where= 0, and hence theare
replaced by the. Finally we choose forthe value:
}=
(+ 20k) (90)
and we assume that all thekare zero. We then get:
=
k
(+0k)
kk+
kk
+
0k
2
kk
+
kk fond (91)
1953

COMPLEMENT E XVII
with, again using the value (32) for sinh
2
k:
ground=2
k
}sinh
2
k=
k=0
}sinh
2
k
=
1
2
k=0
(+ 20k) 0k (92)
Comparison with relations (52) and (54) shows thatgroundis none other than the
energygroundalready obtained, shifted by the mean eld value:
ground=ground
0
2
2
(93)
Finally, taking (80) into account, if thekare all chosen equal to zero (phase locking
condition), we simply have:
= +ground (94)
4-c. Constructing a basis of excited states, quasi-particles
Asgroundis a number, it introduces a simple energy shift in the eigenvalues of
compared to those of, with no eect on the eigenvectors. Now we saw in
of Chapter are known, and can be written as:
=
k
[(k) +(k)]} (95)
where(k)and(k)are any positive or zero integers. As for the eigenstates associated
with these energies, they can be simply obtained by the action on the ground state of
the following product of creation operators:
k
k
(k)
k
(k)
(96)
All things considered, the operatorshares a lot of properties with the Hamil-
tonian of an ensemble of non-interacting particles. Just as the usual creation operators
permit adding particles in a system of free identical particles, the
k
and
k
creation
operators can be considered as adding an extra quasi-particle to the physical system.
When acting on the ground state, the operator
k
yields a ket where both the energy
and the momentum are well-dened: the energy is increased by the amount~specied
in (90), the momentum is increased by}kwith respect to the zero momentum of the
ground state. This exact change of momentum occurs because the action of
k
on any
ket creates two components: one component where one particle with momentum~kis
added, the other component where one particle with momentum~kis suppressed. In
both cases, the total momentum has increased
5
by the same amount~k. The operator
k
5
This result can also be veried by calculating the commutatorP
k
of the total momen-
tumP=
k
~k
kk
with
k
. One obtainsP
k
=
k
~k
kkk
+
k
~k
k k
k, or:
P
k
=~k
kk
+
k
k=~k
k
. As a consequence, the eect of
k
on any eigenstate ofPis to
increase its eigenvalue by~k.
1954

CONDENSED REPULSIVE BOSONS
therefore creates a quasi-particle of well-dened energy and momentum, andkannihi-
lates it; of course,
k
and khave the same properties for the quasi-particle of opposite
momentum. These quasi-particles do not coincide with particles of a system without in-
teractions, as can be seen from the expression of those creation operators. They yield,
however, a basis of states that permits reasoning as if there were no interactions; this
provides a very powerful point of view in many elds of physics.
We can assume, as in ComplementXV, that the interaction potential has a zero
range:
2(rr) =(rr) (97)
Relation (11) then becomes:
k=
1
3
d
3 qr
2(r) =
3
(98)
and equality (90) is written as:
}(k) =
(+ 20) =
}
2
22
(
2
+
2
0
) (99)
with:
0=
2
} 0=2
(100)
In this last equation,is the healing length dened in (36). This equation is the same as
relation (34) of ComplementXV, whose Figure
When the modulusof the wave vectorkis smaller than the wave vector0, we get a
linear spectrum whose slope corresponds to the sound velocity in the boson system; for
values larger than0, the spectrum becomes quadratic, as for free particles.
Conclusion
The calculations presented in this complement illustrate the analogy between the pairing
phenomena for attractive fermions and for repulsive bosons. In both cases, binary posi-
tion correlations are introduced by the dynamic interactions, resulting in a decrease of
the interaction potential energy of the physical system; the paired states are a valuable
tool for understanding this eect. In both cases, a relative phase locking phenomenon
occurs, but the precise nature of that locking is, however, dierent.
For fermions, the energy gain is due to a collective eect, involving the pair-pair
interactions and the relative phase of every pair of states(kk); each contributes to
the value of the gapwhich, in turn, has an eect on all the others this is translated
mathematically by the presence of a double summation overkandkin the energy. This
is reminiscent of a ferromagnetic system, where each spin contributes to the collective
exchange eld that act on all its neighbors. As the interactions are supposed to be
attractive, the phase locking to zero maximizes the pair-pair interactions, and hence
minimizes the energy.
For bosons, the major role is played by the relative phase of the pairs with respect
to that of the reservoir composed of all the particles in thek=0state (condensate).
1955

COMPLEMENT E XVII
The physical process involved is illustrated in Figure XVII, where two
particles emerge from the condensate to form a pair, or vice-versa mathematically, the
energy term contains only one summation overk. The relative phase locking it introduces
will minimize the repulsion between these pairs and the condensate, and hence the total
energy. Compared to the fermion case, the presence of a condensate independent of the
pairs radically changes the nature of the phase locking.
1956

Chapter XVIII
Review of classical
electrodynamics
A Classical electrodynamics
A-1 Basic equations and relations
A-2 Description in the reciprocal space
A-3 Elimination of the longitudinal elds from the expression of
the physical quantities
B Describing the transverse eld as an ensemble of harmonic
oscillators
B-1 Brief review of the one-dimensional harmonic oscillator
B-2 Normal variables for the transverse eld
B-3 Discrete modes in a box
B-4 Generalization of the mode concept
Introduction
In the three previous chapters, we studied ensembles of identical particles, which allowed
us to introduce the concept of quantum eld operators. We now begin a new series of
three chapters where this quantum eld concept is applied to an important particular
case: the electromagnetic eld, made of identical bosons called photons. We start by
noting that, in classical electromagnetism, the dynamics of the dierent eld modes is
exactly similar to that of a series of harmonic oscillators. Each of these modes may be
quantized by the same method as that used for an elementary harmonic oscillator, for a
single particle; this method has the great advantage of simplicity. It requires, however,
establishing beforehand the equivalence between modes of the classical electrodynamic
eld and harmonic oscillators; this is the main purpose of the present chapter.
For the presentation to be self-contained, we rst review a certain number of prop-
erties of classical electromagnetism. One complement is also devoted to a synthetic
Quantum Mechanics, Volume III, First Edition. C. Cohen-Tannoudji, B. Diu, and F. Laloë.
© 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

CHAPTER XVIII REVIEW OF CLASSICAL ELECTRODYNAMICS
presentation of the Lagrangian formalism applied to this case. The reader already fa-
miliar with those aspects of classical electrodynamics may wish to go directly to the
quantum treatment presented in Chapter .
We start in
evolution of the electric eldE(r), the magnetic eldB(r)and the coordinates
and speeds of the particles acting as source for this electromagnetic eld
1
. We shall
give the expressions for a certain number of constants of motion, such as the energy,
or the linear and angular momenta of the global system eld + particles. The vector
potentialA(r)and scalar potential(r)will also be introduced, as well as the gauge
transformations that can be performed on these potentials.
We shall then show that it is useful to take the spatial Fourier transforms of these
elds, since in the reciprocal space, Maxwell's equations have a simpler form. For a
free electromagnetic eld (in the absence of charged particles), they are no longer par-
tial dierential equations, as in ordinary space, but ordinary time-dependent dierential
equations. Furthermore, the concept of longitudinal or transverse eld vectors has a clear
geometrical signicance in the reciprocal space
2
. A eld vector
~
V(k)is longitudinal
if
~
V(k)is parallel tokat every pointkof the reciprocal space, transverse if
~
V(k)
is perpendicular tokat every pointk. We will show that two of the four Maxwell's
equations yield the value of the longitudinal electrical and magnetic elds, whereas the
other two describe the evolution of the transverse elds. It will become clear that the
longitudinal electric eld is simply the Coulomb electrostatic eld created by the charged
particles. Consequently, it is not an independent eld variable since it only depends on
the coordinates of the particles
3
. Furthermore, choosing the Coulomb gauge amounts
to choosing the longitudinal potential vector equal to zero; this permits eliminating the
longitudinal elds from the expressions for all the physical quantities.
In , we establish the equivalence between the radiation eld and an ensemble of
one-dimensional harmonic oscillators. Maxwell's equations for transverse elds enable in-
troducing linear combinations of the vector potentials and transverse electric elds, whose
time evolution, in the absence of particles, is of the formwhere=. These vari-
ables, callednormal variables, thus describe the eigenmodes of the free eld vibrations.
The dynamics of each of these eigenmodes is similar to that of a one-dimensional har-
monic oscillator. The normal mode variable is the equivalent of the linear combination
of the position and velocity of the associated operator, and becomes, in the quantization
process, the annihilation operator, fundamental in the quantum theory of the harmonic
oscillator. Replacing the normal variables and their complex conjugates by annihila-
tion and creation operators will yield, in Chapter , the expressions for the various
operators of the quantum theory.
1
We assume that the speeds of the particles are small compared to the speed of light, so as to use a
non-relativistic description.
2
We shall note
~
(k)the spatial Fourier transform of(r), the symbol tilde allowing a clear
distinction between the functions in ordinary and reciprocal space.
3
As for the longitudinal magnetic eld, it is simply zero.
1958

A. CLASSICAL ELECTRODYNAMICS
A. Classical electrodynamics
A-1. Basic equations and relations
A-1-a. Maxwell's equations
There are four Maxwell's equations in vacuum, and in the presence of sources:
rE(r) =
1
0
(r) (A-1a)
rB(r) = 0 (A-1b)
rE(r) =
B(r) (A-1c)
rB(r) =
1
2
E(r) +
1
0
2
j(r) (A-1d)
whereis the velocity of light in vacuum and0the vacuum permittivity. These
equations yield the divergence and the curl of the electric eldE(r)and the magnetic
eldB(r). The charge density(r)and current densityj(r)appearing in those
equations can be expressed, in the non-relativistic limit, in terms of the positionsr()
and the speedsv() = dr()dof the various particlesof the system, each having a
mass and a charge:
(r) = [rr()] (A-2a)
j(r) = v()[rr()] (A-2b)
A-1-b. Lorentz Equations
Lorentz equations describe the dynamics of each particlesubmitted to the electric
and magnetic forces exerted by the eldsE(r)andB(r):
d
2
d
2
r() =[E(r()) +v()B(r())] (A-3)
The particle and eld evolutions are coupled: the particles move under the eect of the
forces the elds exert on them, but they also act as sources for the evolution of those
elds.
A-1-c. Constants of motion
Denitions (A-2a) of(r)and (A-2b) ofj(r)lead to the continuity equation:
(r) +rj(r) = 0 (A-4)
which implies the time invariance of the total charge of the particle system:
=d
3
(r) = (A-5)
1959

CHAPTER XVIII REVIEW OF CLASSICAL ELECTRODYNAMICS
Other constants of motion exist: the total energy, the total momentumPand the
total angular momentumJof the system eld + particles. They are respectively given
by:
=
1
2
v
2
() +
0
2
d
3
E
2
(r) +
2
B
2
(r) (A-6a)
P= v() +0d
3
E(r)B(r) (A-6b)
J= r() v() +0d
3
r[E(r)B(r)] (A-6c)
Using (A-1) and (A-3), we can verify that the derivatives with respect to time of,P
andJare indeed zero (forandP, see for example exercise 1 in Complement CIof
[16] and its correction).
A-1-d. Scalar and vector potentials: gauge transformations
As we already saw in ComplementIII, the eldsE(r)andB(r)can always
be written in the form:
E(r) =r(r)
A(r) (A-7a)
B(r) =rA(r) (A-7b)
whereA(r)and(r)are the vector and scalar potentials dening agauge. For any
function(r)ofrand of, the transformation of these potentials obeying the relations:
A(r)A(r) =A(r) +r(r) (A-8a)
(r) (r) =(r)
(r) (A-8b)
leads to the same expression forE(r)andB(r); the same physical elds can therefore
be represented by several dierent potentialsA(r)and(r). The transformation (A-
8) associated with the function(r)is called agauge transformation.
Relations (A-8) allow a exibility on the choice of the gaugeA , which allows
introducing an additional condition. The Coulomb gauge, which we will use in this
chapter and the following, is dened by the condition:
rA(r) = 0 (A-9)
A geometrical interpretation of condition (A-9) in the reciprocal space will be given later.
A-2. Description in the reciprocal space
Using Fourier transforms, the equations of electrodynamics can be put in a form
that simplies calculations.
1960

A. CLASSICAL ELECTRODYNAMICS
A-2-a. Spatial Fourier transforms
Let us introduce the Fourier transform of the electric eldE(r):
~
E(k) =
1
(2)
32
d
3
E(r)
kr
(A-10)
which enables us to writeE(r)as:
E(r) =
1
(2)
32
d
3~
E(k)
kr
(A-11)
Analogous expressions can be written for all the physical quantities we just introduced:
magnetic eld, charge and current densities, scalar and vector potentials.
It will be useful in what follows to recall the Parseval-Plancherel relation (Appendix
I, 2-c) showing the identity of the scalar products of two functions expressed in position
space or in reciprocal space
4
:
d
3
(r)(r) =d
3~
(k)
~
(k) (A-12)
and the fact that the product of two functions in reciprocal space, is the Fourier transform
of their convolution in position space:
~
(k)
~
(k)
FT
1
(2)
32
d
3
(r)(rr) (A-13)
A-2-b. Maxwell's equations in reciprocal space
Maxwell's equations take on a simpler form in the reciprocal space, clearly showing
the dierences between the longitudinal and transverse components of the various elds.
Any vector eld
~
V(k)can be decomposed into a longitudinal eld
~
V(k), parallel at
any pointkto the vectork, and a transverse eld
~
V(k)perpendicular tok:
~
V(k) =
~
V(k) +
~
V(k) (A-14)
with:
~
V(k) =
~
V(k)=kk
~
V(k)
2
(A-15a)
~
V(k) =
~
V(k)
~
V(k) (A-15b)
where
=k (A-16)
is the unit vector alongk.
4
The space of the vectorsr( ordinary space) is called position space whereas reciprocal space
is the space of the wave vectorsk.
1961

CHAPTER XVIII REVIEW OF CLASSICAL ELECTRODYNAMICS
As the operatorrin position space corresponds to the operatorkin reciprocal
space, Maxwell's equations (A-1) become in reciprocal space:
k
~
E(k) =
1
0
~(k) (A-17a)
k
~
B(k) = 0 (A-17b)
k
~
E(k) =
~
B(k) (A-17c)
k
~
B(k) =
1
2
~
E(k) +
1
0
2
~
j(k) (A-17d)
Taking into account denitions (A-15) for the longitudinal and transverse com-
ponents of a vector eld, the rst two equations (A-17a) and (A-17b) determine the
longitudinal parts, projections of the elds
~
E(k)and
~
B(k)ontok:
~
E(k) =
0
~(k)
k
2
(A-18a)
~
B(k) =0 (A-18b)
The last two equations (A-17c) and (A-17d) yield the rate of change
~
E(k)and
~
B(k)of the elds
~
E(k)and
~
B(k), and are the equations of motion of these
elds. In the absence of sources (
~
j(k) =0), i.e. for what we will call a free eld, they
are time-dependent dierential equations, and no longer partial derivative equations as
is the case in position space.
A-2-c. Longitudinal electric and magnetic elds
Equation (A-18b) shows the longitudinal magnetic eld
~
B(k)is zero. Equation
(A-18a) expresses
~
E(k)as a product of two functions ofk,~(k)andk0
2
whose
Fourier transforms are written (relation (63) of Appendix):
~(k)
FT
(r) (A-19a)
0
k
2
FT
(2)
32
40
r
3
(A-19b)
Using relation (A-13) then leads to:
E(r) =
1
40
d
3
(r)
rr
rr
3
=
1
40
rr()
rr()
3
(A-20)
This means that at time, the longitudinal electric eld coincides with the Coulomb eld
produced by the charge distribution(r), computed as if this distribution were static
and xed at that instant.
1962

A. CLASSICAL ELECTRODYNAMICS
Comment
The fact that the longitudinal electric eld instantaneously follows the evolution of the
charge distribution(r)should not lead us to believe in anaction at a distancepropa-
gating at an innite speed. The contribution of the transverse eld must also be taken into
account, as only the total electric eldE=E+Ehas a real physical meaning. It can
be shown that the transverse electric eld also has an instantaneous component, which
balances exactly the longitudinal component so that the total eld is always retarded (to
), as the electromagnetic interactions propagate at the speed of light(see
exercise 3 and its correction in Complement CIof reference [16]).
The previous results show that the longitudinal elds are not independent quan-
tities: they are either zero (in the case of the longitudinal magnetic eld), or simply
related to the particle coordinatesr()(in the case of the longitudinal electric eld,
whose expression is given by (A-20)).
A-2-d. Time evolution of the transverse elds
Now that we showed that the rst two Maxwell's equations determine the longi-
tudinal part of the elds, let us consider the last two equations (A-17c) and (A-17d) and
focus on their transverse components. SincekE=kE, they can be rewritten as:
~
B(k) =k
~
E(k) (A-21a)
~
E(k) =
2
k
~
B(k)
1
0
~
j(k) (A-21b)
which yield the time evolution of the transverse elds
~
E(k)and
~
B(k).
Comment
One can also study the longitudinal projections of the two Maxwell's equations ( )
and (A-17d). The result is trivial for the rst one: as both sides of the equation are
transverse, their longitudinal projections are zero. As for the second equation, (A-17d),
it leads to:
~
E(k) +
1
0
~
j(k) = 0 (A-22)
Taking the scalar product ofkwith each side of this equation, using (A-18a) and the fact
thatk
~
j=k
~
j, we nd:
~(k) +k
~
j(k) = 0 (A-23)
which is simply the continuity equation (A-4) in the reciprocal space, and does not provide
any new information.
A-2-e. Potentials
In the reciprocal space, relations (A-7a) and (A-7b) between elds and potentials
become:
~
E(k) =k
~
(k)
~
A(k) (A-24a)
~
B(k) =k
~
A(k) (A-24b)
1963

CHAPTER XVIII REVIEW OF CLASSICAL ELECTRODYNAMICS
and the gauge transformations relations (A-8a) and (A-8b
~
A(k)
~
A(k) =
~
A(k) +k~(k) (A-25a)
~
(k)
~
(k) =
~
(k)
~(k) (A-25b)
where~(k)is the Fourier transform of(r).
Since the last term in (A-25a) is a longitudinal vector, it is clear that a gauge
transformation does not change the transverse part
~
A(k), which thus denes a gauge
invariant physical eld:
~
A(k) =
~
A(k) (A-26)
Sincek
~
A=0, the transverse projections of relations (A-24a) and (A-24b) yield the
equations:
~
E(k) =
~
A(k) (A-27a)
~
B(k) =k
~
A(k) (A-27b)
Note that equation (A-27b) allows expressing
~
A(k)as a function of
~
B(k), as we
now show. Taking the vector product ofkwith each side of this equation, and using the
identity:
a(bc) = (ac)b(ab)c (A-28)
and the fact thatk
~
A(k) = 0, we get:
~
A(k) =
2
k
~
B(k) (A-29)
This equation, together with equation (A-27a), allow rewriting the two time evo-
lution equations (A-21a) and (A-21b) for the transverse elds in a form only involving
~
E(k)and
~
A(k):
~
A(k) =
~
E(k) (A-30a)
~
E(k) =
22~
A(k)
1
0
~
j(k) (A-30b)
In the absence of sources (
~
j(k) =0), we get two coupled time evolution equations for
the transverse elds
~
E(k)and
~
A(k). They will be useful later on for introducing
the eld normal variables, and for the demonstration of the equivalence between the
transverse eld and an ensemble of harmonic oscillators.
Time evolution equation for the transverse potential vector
The time evolution equation for
~
Acan be obtained by replacing
~
Ein (A-30b) by
~
A . We obtain:
2
2
+
22~
A(k) =
1
0
~
j(k) (A-31)
which is written, in the position space:
1
2
2
2
A(r) =
1
0
2
j(r) (A-32)
1964

A. CLASSICAL ELECTRODYNAMICS
A-2-f. Coulomb gauge
ConditionrA(r) = 0, which dened in (A-9) the Coulomb gauge, becomes in
the reciprocal space:
k
~
A(k) = 0
~
A(k) = 0 (A-33)
In the Coulomb gauge, the longitudinal vector potential is therefore equal to zero; there
only remains the transverse vector potential, which, as mentioned above, is a physical
eld.
What can be said about the scalar potentialin the Coulomb gauge? Let us
consider the longitudinal part of each side of equation (A-24a). As the last term on the
right-hand side is transverse in the Coulomb gauge, we get
~
E(k) =k
~
(k), which
reads, in position space,E(r) =r(r). The scalar potential is the potential
whose gradient yields the longitudinal electric eld. Equation (A-20) then shows that,
to within a constant,(r)is equal to:
(r) =
1
40
1
rr()
(A-34)
which is the Coulomb potential created by the charge distribution.
Lorenz gauge
In the present chapter and the next one, we shall mainly use the Coulomb gauge. Another
gauge often used, in particular in the clearly covariant formulations of electrodynamics,
is theLorentz gauge
5
dened by the condition:
rA(r) +
1
2
(r) = 0 (A-35)
which can be written, using covariant notation:
= 0 (A-36)
The condition dening the Lorenz gauge thus keeps the same form in every Lorentz refer-
ence frame, which is not the case for the Coulomb gauge (since in relativity, a transverse
eld of zero divergence in one reference frame is no longer necessarily transverse in another
frame). Nevertheless, an advantage of the Coulomb gauge is that it allows the immediate
identication, in a given reference frame, of the eld variables that are really independent.
A-3. Elimination of the longitudinal elds from the expression of the physical quantities
It will be useful for the following discussion to eliminate the longitudinal elds
from the expressions of the total energyand the total momentum given by equations
(A-6a) and (A-6b). We shall express these physical quantities only in terms of the truly
independent variables, such as particle coordinates and speeds, and transverse elds.
5
The danish physicist Ludwig Lorenz is often confused with the dutch physicist Hendrik Lorentz.
1965

CHAPTER XVIII REVIEW OF CLASSICAL ELECTRODYNAMICS
A-3-a. Total energy
We start by eliminating the longitudinal electric eld from the last term in expres-
sion (A-6a). Using the Parseval-Plancherel equality (A-12) and the fact that
~
E(k)
~
E(k) = 0, we can rewrite this term as:
0
2
d
3
E
2
(r) +
2
B
2
(r)=long+trans (A-37)
where:
long=
0
2
d
3~
E(k)
~
E(k) (A-38a)
trans=
0
2
d
3~
E(k)
~
E(k) +
2~
B(k)
~
B(k) (A-38b)
In (A-38a), we replace
~
E(k)by expression (A-18a). We get, taking (A-12) and (A-13)
into account:
long=
1
20
d
3
~(k)~(k)
2
=
1
80
d
3
d
3
~(r)~(r)
rr
=
Coul+
1
80
=
rr
=Coul (A-39)
The longitudinal eld energy is thus equal to the Coulomb electrostatic energyCoulof
the charge distribution(r). In addition to the Coulomb interaction energy between
dierent particlesand,Coulalso contains the energy
Coul
of the Coulomb eld of
each particle, which diverges for point particles.
Expression (A-38b transcan be rewritten as a function of the variables
~
E(k) =
_~
A(k)and
~
A(k)introduced above for the transverse eld:
trans=
0
2
d
3_~
A(k)
_~
A(k) +
2~
A(k)
~
A(k) (A-40)
Finally, the energy of the global system eld + particles can be expressed in the
form:
=
1
2
_r
2
() +Coul+trans (A-41)
where we used the simplied notation_r() = dr()d=v(). It is the sum of
the kinetic energy of the particles, of their Coulomb energy, and of the energy of the
transverse eld.
1966

A. CLASSICAL ELECTRODYNAMICS
A-3-b. Total momentum
Similar computations can be carried out for the total momentumP. The eld
contribution contained in the last term of (A-6b) can be written as:
0d
3~
E(k)
~
B(k) =0d
3~
E(k)
~
B(k)
Plong
+0d
3~
E(k)
~
B(k)
Ptrans
(A-42)
where we have separated the contributions toPcoming from the longitudinal and trans-
verse components of the electric eld
6
. Using (A-18a A-27b), taking into account
identity (A-28) and the fact thatk
~
A(k) = 0, we get:
Plong=0d
3
~(k)
0
k
2
k
~
A(k)
=d
3
~(k)
~
A(k) (A-43)
We then have:
Plong=d
3
(r)A(r)
= A(r) (A-44)
As we did above for (A-40), we can rewrite the expression ofPtransas a function
of the variables
~
E(k) =
_~
A(k)and
~
A(k)of the transverse eld:
Ptrans= 0d
3_~
A(k)k
~
A(k)
= 0d
3
k
_~
A(k)
~
A(k) (A-45)
The momentum of the global system eld + particles can be written in the form:
P= [_r() +A(r)] +Ptrans (A-46)
Let us nally introduce the quantity:
p() =_r() +A(r) (A-47)
We shall see later that, in the Coulomb gauge electrodynamics,p()is the conjugate
momentum ofr(), hence dierent from the mechanical momentum_r(). Expressed
6
The notationPlongshould not lead us to believe thatPlongis a longitudinal eld vector itself: it
is actually the vector yielding the longitudinal electric eld contribution to the momentum vector; the
same comment applies toPtrans.
1967

CHAPTER XVIII REVIEW OF CLASSICAL ELECTRODYNAMICS
as a function ofp(), the total energy (A-41) and total momentum (A-46) are written
as:
=
1
2
[p A(r)]
2
+Coul+trans (A-48)
P= p() +Ptrans (A-49)
wheretransandPtranswere introduced in equations (A-38b) and (A-42). We shall
see thatactually coincides with the Hamiltonian in the Coulomb gauge of the global
system eld + particles.
A-3-c. Total angular momentum
Calculations similar to ones just presented, but that will not be detailed here
7
, show that
the contribution of the longitudinal electric eld to the total angular momentum is equal
to:
Jlong=0d
3
r(E B) = rA(r) (A-50)
AddingJlongto the particles' angular momenta, we get, taking (A-47) into account:
r _r+Jlong= rp (A-51)
so that we can nally write:
J= rp+Jtrans (A-52)
where:
Jtrans=0d
3
[r(E B)] (A-53)
B. Describing the transverse eld as an ensemble of harmonic oscillators
B-1. Brief review of the one-dimensional harmonic oscillator
The energy of a harmonic oscillator is given by:
=
1
2
_
2
+
1
2
22
(B-1)
where2is the oscillation frequency, and_the oscillator velocity:
d
d
= _ (B-2)
7
These calculations can be found in 1 of Complement BIin [16].
1968

B. DESCRIBING THE TRANSVERSE FIELD AS AN ENSEMBLE OF HARMONIC OSCILLATORS
This velocity obeys:
d
d
_=
2
(B-3)
so that the equation of motion ofis:
+
2
= 0 (B-4)
Consequently, the time evolution of()is given by a (real) linear combination ofcos()
andsin().
The dynamic state of the classical harmonic oscillator is dened at each instant by
two real variables()and_(). It is often useful to combine them into a single complex
variable()by setting:
() = () +
_()
(B-5)
whereis an arbitrary (time-independent) constant. Relations (B-2) and (B-3) show
that()obeys the rst order dierential equation:
_=( _ ) = +
_
= (B-6)
The time dependence of the new variable()is therefore simply.
One can invert the system formed by equation (B-5) and its complex conjugate
yielding, and computeand_as a function ofand. Inserting the expressions
thus obtained in equation (B-1) for the energy, we obtain by a simple calculation
8
:
=
2
4
2
(+ ) (B-7)
The constantcan be chosen so that:
2
4
2
=
~
2
(B-8)
This leads, after quantization, to the Hamiltonian operator:
^
=
~
2
(^^+ ^^) (B-9)
which is the Hamiltonian of a harmonic oscillator
9
.
B-2. Normal variables for the transverse eld
B-2-a. Vibration eigenmodes of the free transverse eld
In the reciprocal space, expression (A-40) for the free transverse eld energytrans
is a sum of quadratic functions of
_~
A(k)and
~
A(k). For each value ofk, we get
a harmonic oscillator Hamiltonian. The evolution introduces no coupling between the
various spatial Fourier components of the transverse eld. We see the advantage of
1969

CHAPTER XVIII REVIEW OF CLASSICAL ELECTRODYNAMICS
Figure 1: For each vectork, the transverse elds can have two polarizations characterized
by unit vectors"1(k)and"2(k)perpendicular both to each other and tok.
working in the reciprocal space: it enables us to identify the eigenmodes of the eld
vibrations, in the absence of sources.
Actually, for eachk, the transverse eld can have two dierent polarizations
10
characterized by unit vectors"1(k)and"2(k), both perpendicular tokand to each
other, so that we can write for
~
A(k), as an example:
~
A(k) =
~
"1(k)(k)"1(k) +
~
"2(k)(k)"2(k) =
"(k)
~
"(k)(k)"(k)(B-10)
with:
~
"(k)(k) ="(k)
~
A(k) (B-11)
The setk"(k)denes what we shall call in this chapter a free eldmode; they are
the eigenmodes of the free eld vibration, with a frequency:
= (B-12)
To simplify the notation, we shall write the last summation in (B-10) in a more
compact form:
"(k)
~
"(k)(k)"(k)
"
~
"(k)" (B-13)
Let us rewrite expression (A-40) for
~
transexpliciting the components of the elds
A(k)and
_
A(k)on the polarization vectors. We get:
trans=
0
2
d
3
"
_~
"(k)
_~
"(k) +
2~
"(k)
~
"(k) (B-14)
8
In view of the quantization whereand will be replaced by non-commuting operators^and^,
we keep the sequence ofand as they appear in the computations.
9
If^and^obey the canonical commutation relation[^^] =~, relation (B-8
also leads to the commutation relation[^^] = 1.
10
We choose real vectors"1(k)and"2(k)corresponding to linear polarizations, but the choice of
these two polarizations is arbitrary, since they can always be rotated by any angle aroundk. It is also
possible to perform a more general change of basis with complex vectors dening elliptical polarizations,
for instance the right and left circular polarizations"= ("1"2)
2. Circular polarizations are
useful when discussing electromagnetic spin see XIX. If complex (orthonormal)
polarizations are used,"(k)should be replaced by"(k)in the right side hand of relation (B-11).
1970

B. DESCRIBING THE TRANSVERSE FIELD AS AN ENSEMBLE OF HARMONIC OSCILLATORS
Note that the components on the two polarizations"are truly independent dynamic
variables (generalized coordinates and velocities). This is not the case for the Carte-
sian components
~
(k)and
_~
(k)(with= ), because of the transversality
condition. For example, the components
~
(k)must obey
~
= 0.
Constraints on the dynamic variables in the reciprocal space
Since the elds are real in real space, we have the condition
~
A(k) =
~
A(k).
In half the reciprocal space, the variables
~
"(k)and
~
"(k)can be considered as
independent .
B-2-b. Denition of the normal variables, free eld case
Let us rst assume that we are in the free eld case (
~
j=0), and we can replace
the eld
~
E(k)by
_~
A(k)in equations (A-30a) and (A-30b). As=, we get two
equations exactly similar to those of a harmonic oscillator (B-2) and (B-3), withA(k)
instead of(). This analogy suggests introducing, as in (B-5), a new transverse variable:
(k) =()
~
A(k) +
_~
A(k)
=()
k
2
~
B(k)
1
~
E(k) (B-15)
where()is a real constant, not yet specied, which can depend on(its value will
be chosen at the beginning of the next chapter). This denition, together with (A-30b),
yields the equation of motion for(k):
_(k) +(k) = 0 (B-16)
As opposed toA(k)that, according to (A-31), obeys a second order equation, this
new variable(k)obeys a rst order equation. It is a complex variable whose time
evolution is proportional to, and not, as is the case for the variableA(k), to a
linear superposition ofand
+
. It will be useful in what follows to consider the
complex conjugate of equation (B-15):
(k) =()
~
A(k)
_~
A(k)
=()
~
A(k)
_~
A(k) (B-17)
To go from the rst to the second line of (B-17), we used the fact thatAis real in the
real space, which leads to:
~
A(k) =
~
A(k) (B-18)
A similar relation exists for
_~
A. The transverse variables(k)and(k)are called
the transverse eldnormal variables. We will see in the next chapter that the quantization
process will transform these variables into annihilation and creation operators of photons.
1971

CHAPTER XVIII REVIEW OF CLASSICAL ELECTRODYNAMICS
B-2-c. Equation of motion for the normal variables in the presence of sources
In the presence of sources,
~
jis no longer zero. We can still dene the normal
variables(k)by relations (B-15), but we must now keep the term in
~
j(k)on the
right-hand side of equation (A-30b). The same transformation that led us from equations
(A-30a) and (A-30b) to (B-16) now yields a new equation of motion in the presence of
sources:
_(k) +(k) =
()
0
~
j(k) (B-19)
This equation is strictly equivalent to Maxwell's equations for the transverse elds. One
can see this by taking the time derivative of equations (B-22a) and (B-22b) given below,
and using (B-19) to get the time-dependent evolution equations (A-30a) and (A-30b) of
these elds.
Independence of the normal variables
Another interest of the normal variables is that they are independent: there is no re-
lation between(k)and(k)such as the one that exists between
~
A(k)and
~
A(k). This is because the real and imaginary parts of(k)depend on two in-
dependent degrees of freedom,
~
A(k)and its time derivative. It is easy to check, by
changing the sign ofkin (B-15) and by using (B-18) that:
(k) =()
~
A(k) +
_~
A(k)=(k) (B-20)
The knowledge of the(k)in the entire reciprocal space does not entail the knowledge
of the(k). Consequently, the integrals overkof the normal variables must be taken
over the entire space, and not be limited to half the reciprocal space.
B-2-d. Expression of the physical quantities in terms of the normal variables
We are going to show that all the physical quantities can be expressed in terms of
the normal variables.
. Transverse elds in the reciprocal space
Replacingkbyk, we can rewrite equation (B-17) as:
(k) =()
~
A(k)
_~
A(k) (B-21)
Using (B-15) and (B-21), we can now express
~
A(k)and
_~
A(k)as a function of
(k)and(k). We get:
~
A(k) =
1
2()
[(k) +(k)] (B-22a)
_~
A(k) =
2()
[(k)(k)] (B-22b)
1972

B. DESCRIBING THE TRANSVERSE FIELD AS AN ENSEMBLE OF HARMONIC OSCILLATORS
. Energy and momentum of the transverse eld
We insert relations (B-22a) and (B-22b) for
~
A(k)and
_~
(k)in the expression
(A-40) fortrans, using the more compact notation:
=(k) ;=(k) (B-23)
We get:
trans=
0
2
d
3
2
4
2
()
()() + (+)(+)
=
0
2
d
3
2
4
2
()
2+ 2 (B-24)
(in these equations, we keep the ordered sequence ofandas they appear in the
computations, even thoughandare commuting numbers; the reason is that similar
computations can be carried out in the quantum theory whereandwill be replaced
by non-commuting operators). A change of variablek kin the integral of the terms
inyields an integral of. We then get:
trans=0d
3
2
4
2
()
[+] (B-25)
Expliciting the components ofandon the two polarization vectors"perpendicular
tok, and using the simplied notation (B-13), we nally get:
trans=0d
3
"
2
4
2
()
[
"(k)"(k) +"(k)
"(k)] (B-26)
This expression looks a lot like a sum of harmonic oscillator Hamiltonians; a suitable
choice for the constantwill be made in the next chapter.
Similar calculations can be carried out for the transverse eld momentumPtrans
11
.
Using equations (A-45), (B-22a) and (B-22b), we get:
Ptrans=0d
3
"
4
2
()
k[
"(k)"(k) +"(k)
"(k)] (B-27)
. Transverse elds in real space
Let us consider rst the transverse potential vector
~
A(k), whose expression in
terms of the normal variables is given by (B-22a). To get its expression in real space,
one must, taking (A-11) into account, multiply (B-22a) by(2)
32kr
and integrate
overk. Making the change of variablek kin the integral containing(k), we
nally get:
A(r) =
1
(2)
32
d
3
"
1
2()
"(k)"
kr
+
"(k)"
kr
(B-28)
11
The expression of the angular momentumJtransof the transverse eld, in terms of the normal
variables, will be computed in ComplementXIX.
1973

CHAPTER XVIII REVIEW OF CLASSICAL ELECTRODYNAMICS
This relation (as well as the next two equations) is written in the general case where the
polarizations may be complex, elliptical or circular (cf. note). This is why the term
in
"(k)contains a complex conjugate polarization".
Similar calculations can be carried out for the transverse electric eld as well as
for the magnetic eld. They yield:
E(r) =
1
(2)
32
d
3
"
2()
"(k)"
kr
"(k)"
kr
(B-29)
B(r) =
1
(2)
32
d
3
"
2()
"(k)"
kr
"(k)"
kr
(B-30)
wherehas been dened in (A-16) as the unit vector parallel tok.
B-3. Discrete modes in a box
So far, we have considered radiation propagating in an innite space and used
continuous Fourier transforms; in relation (A-11), the electric eld is expanded on a
continuous basis of normalized plane waves
kr
(2)
32
. It is often useful, however, to
use a discrete basis, assuming the radiation to be contained in a box of nite volume,
generally dened as a cube of edge length; this will frequently occur in the next two
chapters when dealing with the quantized radiation. The components of each wave vector
must obey the boundary conditions in the box
12
, and hence take on discrete values:
= 2 (B-31)
At the end of the computation, nothing prevents us from choosing a very large value of
in order to check that the nal result does not depend on.
Instead of continuous spatial Fourier transforms, one must now introduce discrete
Fourier series where each physical quantity is expanded in terms of normalized plane
waves
kr32
. The expansion (A-11) of the electric eld then becomes:
E(r) =
1
32
k
~
Ek()
kr
(B-32)
with
13
:
~
Ek() =
1
32
d
3
E(r)
kr
(B-33)
The summation in (B-32) is discrete, and the integral in (B-33) is now limited to the
volumeof the box.
12
One can choose to impose the eld being zero on the walls, but it is generally easier to enforce
periodic boundary conditions (B-31), which lead to the samedensity of states.
13
In Appendix I, we used a slightly dierent denition for the Fourier series, with which the factor
1
32
would be missing from (B-32), but where (B-33) would contain a factor1
3
. The denition we
use here is chosen to directly yield an expansion ofE(r)on plane waves normalized in the cube.
1974

B. DESCRIBING THE TRANSVERSE FIELD AS AN ENSEMBLE OF HARMONIC OSCILLATORS
Note that if the eld is zero outside the box, it is obviously possible to use the con-
tinuous Fourier transform (A-10) to get the eld component
~
E(k); however, this latter
component is dierent from the discrete component
~
Ek(), because of the coecients
introduced in the denitions. The two components are related by:
~
Ek() =
2
32
~
E(k) (B-34)
The same changes can be made on the Fourier transforms of all the other physical quan-
tities such as the magnetic eld, the vector potential, as well as the charge and current
densities. The equations in the reciprocal space such as equations (A-17), (A-21), (A-25)
and the following, remain valid if we replace the continuous variableskby discrete vari-
ables, since each side of those equations are multiplied by the same factor(2)
32 32
.
In the case of a zero eld outside the box, the"(k)are also replaced by :
k"() =
2
32
"(k) (B-35)
Coming back to ordinary space(r)via the inverse Fourier transform, we must
use relations of the type (B-32) instead of (A-11). Consequently, once we replace in
the integral over d
3
the
~
E
(k)by the
~
Ek(), we must also introduce a multiplicative
factor
14
:
d
3
=
2
32
k
(B-36)
B-4. Generalization of the mode concept
In the absence of sources, the solution of the equation of motion (B-16) for the nor-
mal variable"(k)is very simple, since it is an exponential with an angular frequency
=:
"(k) ="(k0) (B-37)
Inserting (B-37) in the expressions we just obtained for the transverse elds and the other
physical quantities, we see that the elds are linear superpositions of progressive plane
waves, propagating independently of each other. The free eld energy and momentum
are the sum of the squared moduli of the various normal variables, each being time-
independent and proportional to"(k0)
2
.
The modesk"introduced in this chapter permit expanding the free transverse
elds on progressive plane waves. Nevertheless, other expansions on monochromatic
waves that are not necessarily plane waves are also possible; they involve other families
of modes, as we are now show, coming back to equation (A-31). In the absence of sources,
any monochromatic solution of this equation, of the formA
(+)
(r) , necessarily obeys
equation:
( +
2
)A
(+)
(r) = 0 (B-38)
14
The product of the multiplicative factor of (B-34) and that of (B-35) yields the usual factor(2)
3
,
obtained directly from (B-31).
1975

CHAPTER XVIII REVIEW OF CLASSICAL ELECTRODYNAMICS
(which is simply the Helmholtz equation) with= . The plane waves
kr
are
a possible basis of eigenfunctions for this eigenvalue equation, but not the only one.
There exists other bases, such as the basis of stationary wavescoskrandsinkr, the
basis of multipolar waves (radiation modes with a specic angular momentum, whereas
plane waves have a specic linear momentum), or the basis corresponding to Gaussian
modes. More generally, any linear combination of plane waves with the same modulus
kcan become a mode. Whatever basis is chosen, the transverse eld energy will be
a sum of the squared moduli of normal variables introduced in the expansions of the
transverse elds on the eigenfunctions of that basis. The expression of the other physical
quantities, however, will only have a simple form in a particular basis. As an example,
the momentum of the transverse eld is a sum of squared moduli only in the basis of
progressive plane waves, whereas the eld angular momentum has a simple form only in
the basis of multipolar waves.
Note nally that the eld can be contained in a cavity with well dened bound-
ary conditions. Finding the eigenfunctions of equation (B-38) obeying these boundary
conditions is a way to determine the eigenmodes of this cavity.
To conclude this chapter, we can say that the free radiation eld is equivalent to
an ensemble of one-dimensional harmonic oscillators associated with the modesk"
labeled by their wave vector and their transverse polarization. Each mode is associated
with a eld normal variable, similar to the classical variable of the corresponding classical
oscillator, and which will become, in the quantization process, the oscillator annihilation
operator. The results established in this chapter will be the simple starting point for the
radiation quantization explained in the next chapter.
1976

COMPLEMENT OF CHAPTER XVIII, READER'S GUIDE
AXVIII : LAGRANGIAN FORMULATION OF
ELECTRODYNAMICS
The dynamic equations for the electrodynamic
eld (Maxwell's equations) can be obtained from
the Lagrangian formalism based on a principle of
least action. This enables introducing expressions
for the conjugate momenta of the various eld
variables, as well as for the eld Hamiltonian
when coupled to charged particles. The results of
this complement are not indispensable for reading
the other chapters and complements. They oer,
however, an overview of a more general approach
to quantum electrodynamics, which is essential
for a relativistic treatment of these problems and
for the use of path integrals (Appendix IV).
1977

LAGRANGIAN FORMULATION OF ELECTRODYNAMICS
Complement AXVIII
Lagrangian formulation of electrodynamics
1 Lagrangian with several types of variables
1-a Lagrangian formalism with discrete and real variables
1-b Extension to complex variables
1-c Lagrangian with continuous variables
2 Application to the free radiation eld
2-a Lagrangian densities in real and reciprocal spaces
2-b Lagrange's equations
2-c Conjugate momentum of the transverse potential vector
2-d Hamiltonian; Hamilton-Jacobi equations
2-e Field commutation relations
2-f Creation and annihilation operators
2-g Discrete momentum variables
3 Lagrangian of the global system eld + interacting particles 1992
3-a Choice for the Lagrangian
3-b Lagrange's equations
3-c Conjugate momenta
3-d Hamiltonian
3-e Commutation relations
Introduction
As shown in Appendix III, the dynamics of a system of point particles in an external
potential can be described either by Newton's equations, or by a Lagrangian with the
principle of least action leading to Lagrange's equations, equivalent to Newton's equa-
tions. An advantage of the Lagrangian formalism is that it facilitates the quantization of
the theory: it directly leads to the denition of the conjugate momenta of the particles'
coordinates, and of the system's Hamiltonian, which is a function of the coordinates
and the conjugate momenta. It then naturally introduces the canonical commutation
relations, fundamental for the quantum description of the system. This complement will
show, in a succinct way, how the Maxwell-Lorentz equations, studied in this chapter and
the next, can be deduced from a Lagrangian and a principle of least action. This will give
a more general justication for the expression of the Hamiltonian of the system eld +
particles postulated in Chapter
in that chapter
1
. Another advantage of the Lagrangian formalism, that we shall not
exploit here, is that it is well suited to a relativistic description of the system eld +
particles which is why it is used in the quantum theory of relativistic elds.
1
The relations postulated in Chapter
correct Heisenberg equations for the quantum operators associated with the particles and the elds.
1979

COMPLEMENT A XVIII
We start in
system coordinates are complex and not real, even though the Lagrangian remains a real
quantity. We will also show that the principle of least action and Lagrange's equations
can be generalized to the eld case, that is to a case where the system coordinates no
longer depend on a discrete but on a continuous index, such as the pointrin real space.
The discussion will be illustrated in 2, which studies the Lagrangian of the free
radiation eld in the absence of sources. The eld will be described by its components
in reciprocal space, which are complex quantities. The Lagrangian then depends only on
the eld components and their time derivatives, hence making the computations easier
than if the elds were described by the real components in real space (this is because
the Lagrangian in real space depends not only on the eld components and their time
derivatives, but also on their spatial derivatives). In this study, we shall establish the
expression for the eld Hamiltonian, and the canonical commutation relations of the
components of that eld.
Finally, we give in 3 the expression for the electrodynamic Lagrangian in the
Coulomb gauge in the presence of sources; we show how Lagrange's equations deduced
from this Lagrangian coincide with the Maxwell-Lorentz equations studied in Chapter
XVIII. Several important relations for the quantization of the theory will be established:
expression for the conjugate momenta of the particles and elds; expression for the
Hamiltonian of the global system eld + particles; canonical commutation relations.
The results obtained in this complement give a base for the quantization process more
general than the simplied approach of Chapitre XIX. The interested reader can nd a
more detailed description of the electrodynamic Lagrangian and Hamiltonian formalism
in Chapter II of reference [16] and its complements.
1. Lagrangian with several types of variables
1-a. Lagrangian formalism with discrete and real variables
The Lagrangianis a real function of dynamical variables composed of general-
ized coordinates()labeled by a discrete indexand of the corresponding generalized
velocities_() = d()d.is written:
[1()2() (); _1()_2()_()] (1)
Consider a possible motion of the system where the coordinates()follow a
certain pathbetween an initial timeinand a nal time2. The integral ofalong
the pathis, by denition,the actionassociated with this path:
=
2
1
d[1()2() (); _1()_2()_()] (2)
Theprinciple of least actionpostulates that, among all the possible paths start-
ing from the same initial conditions described by(in)and arriving at the same nal
conditions described by(2), the system will follow the one for whichpresents an
extremum (if the path varies,is stationary). Consider an innitesimal variation()
and_()of the dynamical variables around this path of extremum action, which does
not change the initial and nal values of the coordinates, i.e. such that:
(in) =(2) = 0 (3)
1980

LAGRANGIAN FORMULATION OF ELECTRODYNAMICS
The corresponding variation of the action
=
2
1
()
+_()
_
d (4)
must be zero to rst order in()and_(). We replace, in the last term of (4),_()
by:
_() =
d
d
() (5)
and integrate by parts the corresponding term. The integrated part is zero because of
(3). We then get:
=
2
1
()
d
d_
d (6)
As must be zero for any variation(), the path actually followed by the system
must obey theequations:
d
d_
= 0 (7)
These relations are calledLagrange's equations; they can be shown to be equivalent to
Newton's equations (Appendix III).
The next step of the Lagrangian formalism is to introduce theconjugate momenta
of the coordinates, dened by the equations:
=
_
(8)
as well as theHamiltonianequal to:
= _ (9)
Let us take the dierential of
d= d _1+ _d
d
_1
d _
= [ _d _d] (10)
To go from the rst to the second line of (10), we used (7) and (8) to replace
by_and _by. We assume the_can be expressed as a function of theand
the. The Hamiltonianis then a function of the coordinatesand the conjugate
momenta, whose evolution obeys, taking (10) into account, the2equations:
d
d
=
d
d
= (11)
1981

COMPLEMENT A XVIII
called theHamilton-Jacobi equations.
Let us nally recall the canonical quantization process. One associates with the
coordinatesand the conjugate momentathe operators^and^obeying the com-
mutation relations:
[^^] =~ (12)
all the other commutators being equal to zero. These results are valid only if the coor-
dinatesare Cartesian components (see comment in B-5 of Chapter III).
1-b. Extension to complex variables
Let us assume, to keep things simple, that the indextakes only= 2values.
With the two real coordinates1()and2()we build the complex variables:
() =
1
2
[1() +2()] () =
1
2
[1() 2()] (13)
whose real and imaginary parts are, within a factor1
2, equal to1()and2()for
(),1()and 2()for(). Equations (13) can be inverted and yield:
1() =
1
2
[() +()] 2() =
2
[() ()] (14)
Analogous equations can be written, relating
_
()and
_
()to_1()and_2()and vice
versa:
_
() =
1
2
[ _1() +_2()]
_
() =
1
2
[ _1()_2()] (15)
_1() =
1
2
_
() +
_
() _2() =
2
_
()
_
() (16)
Inserting in the Lagrangian (1) expressions (14) and (16) for the variables, we get a
Lagrangian of the form()()
_
()
_
()that depends on complex variables.
Note however that just as in (1), this Lagrangian, even though it depends on complex
variables, is still a real quantity since its time integral along a pathis an action (which
is a real quantity). We now study what becomes of all the results established earlier
using (1) when they are expressed as a function of
_
and
_
.
. Lagrange's equations
It is important, for what follows, to relateand
_
to 1, 2,
_1and _2. Using (14) and (16), we can write:
=
1
1
+
2
2
=
1
2
12
(17)
_
=
_1
_1
_
+
_2
_2
_
=
1
2_1_2
(18)
1982

LAGRANGIAN FORMULATION OF ELECTRODYNAMICS
Subtracting from equation (17) the time derivative of equation (18), we get:
d
d_
=
1
2
1
d
d_12
2
d
d_2
(19)
The two parentheses on the right-hand side of this equation are zero since1and2
obey Lagrange's equation (7). It then follows that the left-hand side is also zero, as is
its complex conjugate
2
:
d
d_
= 0
d
d_
= 0 (20)
which proves thatand also obey Lagrange's equations
3
.
. Conjugate momenta
In (18) we replace _1and _2by1and2(see Eq. (8)). We get:
_
=
1
2
(1 2) (21)
The complex conjugate of equation (21) is written:
_
=
1
2
(1+2) (22)
To choose the denition of the conjugate momentumof the complex variable
, it is useful to compare the way the conjugate momentumand the velocity
_
are
transformed upon the change of dynamical variables12 . We compare
the rst equation (15) and the two equations (21) and (22). The velocity
_
becomes
_1+_2; a wise choice would be to dene the associate momentumin such a way
that its transformation yields1+2. Equations (21) and (22) then clearly show that
must not be dened as
_
, but rather as
_
; the complex conjugateis
then equal to
_
:
=
_
=
_
(23)
This is the denition we shall use in the rest of this complement.
. Hamiltonian
The quantity_11+ _22appears in the denition (9) of the Hamiltonian. This
quantity can be rewritten by replacing_1and_2by their expressions (16) as a function
of
_
and
_
, as well as1and2by analogous expressions as a function ofand:
_11+ _22=
1
2
(
_
+
_
)
1
2
(+) +
2
(
__
)
2
( )
=
_
+
_
(24)
2
Sinceis real, we have( )= and an analogous equation for
_
.
3
These results could have been obtained directly by the variational calculation leading to (6), con-
sideringand to be independent variables.
1983

COMPLEMENT A XVIII
We can then write:
= _11+ _22 =
_
+
_
(25)
We now take the dierential of:
d=
_
d+d
_
+
_
d+d
_
dd
_
d
_
_
d
_
(26)
Using (20) and (23), we get:
d=
_
d
_
d+
_
d
_
d (27)
If
_
and
_
can be expressed in terms of the variables,,,, the Hamiltonian
only depends on those variables and we deduce from (27) the Hamilton-Jacobi equations:
d
d
=
d
d
= (28)
and the complex conjugate equations forand. Note that it is the partial derivative
with respect to(and not with respect to) that is equal to the total derivative of.
. Canonical commutation relations
Upon quantization, the dierent variables become operators^1,^2,^1,^2,
^
,
^
,
^
,
^
. The commutation relations between the operators
^
,
^
,
^
,
^
are obtained
by expressing those operators in terms of^1,^2,^1,^2and using the commutation
relations (12). We can easily check that the only non-zero commutators are[
^^
]and
[
^^
] =[
^^
]. Using (21), (22) and (23), we obtain:
[
^^
] =
1
2
[^1+^2^1^2]
=
1
2
([^1^1] + [^2^2]) =~ (29)
1-c. Lagrangian with continuous variables
We now assume that the dynamical variables of the system depend on a continuous
index, such as the pointrin real space (or the pointkin the reciprocal space when the real
space is innite). In other words, they constitute a eld(r), where the discrete index
labels the component of the eld if we are dealing with a vector eld; in the reciprocal
space, this eld becomes
~
(k). We shall only establish here Lagrange's equations for
a real eld. In the upcoming 2, we shall study the radiation eld described by its
complex components in the reciprocal space. We will then generalize to a complex eld
all the results established earlier for discrete and complex variables. This will yield the
expression for the free eld Hamiltonian and the commutation relations for the free eld,
which are essential for the quantum description of the eld.
The Lagrangianof a real eld is now the integral in real space of aLagrangian
densityL:
=d
3
L(r) (30)
1984

LAGRANGIAN FORMULATION OF ELECTRODYNAMICS
The Lagrangian density is a function of the eld(r)and of its partial derivatives with
respect toand to the components ofr:
L(r) =L (r)
_
(r) (r) (31)
with the notation (= ):
_
(r) =(r) (32)
(r) =(r) (33)
Consider a possible pathfor the eld, going from the value(rin)at an initial
timeinto the nal value(r2)at a nal time2. The actionassociated with this
path is, by denition:
=
2
1
dd
3
L (r)
_
(r) (r) (34)
The principle of least action postulates that among all the possible paths starting from
the same initial state and ending at the same nal state
4
, the path(s) actually followed by
the system is the one (or are those) for whichpresents an extremum. Let us compute
the variationof the action for an innitesimal variation of the path, characterized by
the innitesimal variations(r),((r))and((r)).
=
2
1
dd
3
(r)
L
+
_
(r)
L
_
+ ((r))
L
()
(35)
Using:
_
(r) =((r))
((r)) =((r)) (36)
and performing an integration by parts of the terms proportional to((r))
and((r)), we nd that the integrated terms are zero because of the boundary
conditions for(r)at the initial and nal times, and forr . The remaining
terms are therefore all proportional to(r). Grouping them all, we nd
5
:
=
2
1
dd
3
(r)
L
d
d
L
_
L
()
(37)
Asmust be zero for any time or spatial variations of(r), we can deduce that:
L
d
d
L
_
L
()
= 0 (38)
which are the Lagrange equations for the eld.
4
We also assume that the Lagrangian density is zero or tends to zero fast enough whenrgoes to
innity.
5
The function
L
_
does not directly depend on time. It nevertheless depends indirectly onif
we replace, as in (31), the elds and their derivatives by their values for a given history of the eld.
By convention, we then denote by
d
d
L
_
the time derivative of this function at each point of space. It
contains the sum of the contributions of the partial derivatives of the function with respect to all the
initial variables (the elds and their derivatives).
1985

COMPLEMENT A XVIII
2. Application to the free radiation eld
We now study the free radiation eld (in the absence of sources) starting from its La-
grangian density in reciprocal space. We choose the Coulomb gauge so that the longitudi-
nal vector potentialAis zero andAis reduced toA; furthermore, the scalar potential
is also zero since, in the Coulomb gauge, it would be the potential corresponding to the
Coulomb eld created by the charges (see relation (A-34) of Chapter ), and we are
assuming that there are no charges. The only elds we have to consider are thus the
transverse electric eld and the magnetic eld related to the transverse potential vector
by the following equations in real space:
E(r) =
_
A(r) B(r) =rA(r) (39)
and in the reciprocal space:
~
E(k) =
_~
A(k)
~
B(k) =k
~
A(k) (40)
2-a. Lagrangian densities in real and reciprocal spaces
The Lagrangian density most commonly used in real space is
6
:
L(r) =
0
2
E
2
(r)
2
B
2
(r) (41)
whereis the speed of light. Using (39), we see that this Lagrangian density depends
both onA(r)and
_
A(r), as well as on the spatial derivatives ofA(r).
We now go to the reciprocal space. The Lagrangianis then written as:
=d
3~
L(k) (42)
where the Lagrangian density
~
L(k)in the reciprocal space is obtained from (41),
rewriting the elds in the reciprocal space
7
. Let us evaluate the contribution to the La-
grangian of the two terms in the bracket of (41). Using (40) and the Parseval-Plancherel
equality see relation (A-12) of Chapter
d
3
E(r)E(r) =d
3_~
A(k)
_~
A(k)
d
3
B(r)B(r) =d
3
(k
~
A(k))(k
~
A(k)) (43)
As the two vectorsk
~
A(k)andk
~
A(k)are in the plane perpendicular tok,
we have:
d
3
(k
~
A(k)(k
~
A(k)=d
3 2~
A(k)
~
A(k) (44)
Finally, the Lagrangian density in the reciprocal space is written as:
~
L(k) =
0
2
_~
A(k)
_~
A(k)
22~
A(k)
~
A(k) (45)
6
This density has the advantage of being a relativistic invariant (Lorentz scalar).
7
We use the notation
~
L(k)for this Lagrangian density even though this function is not the Fourier
transform ofL(r).
1986

LAGRANGIAN FORMULATION OF ELECTRODYNAMICS
The generalized coordinates of the eld can be seen as components of the transverse po-
tential vector, and the generalized velocities as the time derivatives of these coordinates.
As opposed toL(r),
~
L(k)depends only on
~
A(k)and
_~
A(k), and not on the
partial derivatives of
~
A(k)with respect to thekcomponents. The computations will
thus be simpler in the reciprocal space.
It will be useful for what follows to introduce the Cartesian components of
~
A(k)
in the reference frame formed by=kand the two polarization unit vectors"1(k)
and"2(k)in the plane perpendicular tok. Since
~
A(k)is transverse, it does not have
any component along, and we can write:
~
L(k) =
0
2
"
_~
"(k)
_~
"(k)
22~
"(k)
~
"(k) (46)
where
"
is a simplied notation representing the summation over the two transverse
polarizations"1(k)and"2(k) see relation (B-13) of Chapter .
2-b. Lagrange's equations
Equation (38) becomes here:
~
L(k)
~
"
(k)
d
d
~
L(k)
_~
"
(k)
= 0 (47)
We then get, taking (46) into account:
~
"(k) +
22~
"(k) = 0 (48)
This equation coincides with equations (A-30a) and (A-30b) of Chapter , which
give the time evolution of the transverse vector potential of a free eld in the absence of
sources (we set
~
j=0). We recover, as expected, the predictions of Maxwell's equations
in the usual formulation of classical electrodynamics.
2-c. Conjugate momentum of the transverse potential vector
To dene the conjugate momentum
~
"(k)of the complex variable
~
"(k),
we use expression (23). Note however that the velocity
_~
"(k)appears several times
in the integral overkof
~
L(k). Consequently, we must add all the corresponding
contributions of the partial derivatives of
~
L(k)with respect to these various velocities
in the denition of the conjugate momentum
~
"(k). This situation results from the
fact that the elds are real in real space. The Fourier transform properties then lead to
(cf.relation (B-18) of Chapter ):
A(r) =A(r)
~
A(k) =
~
A(k) (49)
and to an equivalent relation for the time derivatives of the components of the trans-
verse potential vector. In the integral overkof
~
(k)we get, in addition to the term
_~
"
(k)
_~
"(k), the term
_~
"
(k)
_~
"(k)which, according to (49), is equal to
1987

COMPLEMENT A XVIII
_~
"(k)
_~
"(k)and therefore doubles the rst term. If we ignore the terms ink,
we must then double the contribution of the terms ink, which yields:
~
"(k) = 2
~
L(k)
_~
"
(k)
=0
_~
"(k) = 0
~
"(k) (50)
The conjugate momentum of the transverse potential is seen to be equal, within a factor
0, to the transverse electric eld.
Another equivalent way to obtain (50) is to use, in the Lagrangian expression, only
independent variables. The reality condition (49) ensures that, if one knows the variables
in half the reciprocal space, one knows them in the entire space. One can deneas the
integral over only half the reciprocal space (where all the variables are independent) of
a Lagrangian density, noted

L(k), equal to twice the initial density. Writing the
integral over half a space (the bar indicating that thekspace is divided into two parts),
we get:
=d
3
L(k) (51)
with:

L(k) =0
"
_~
"(k)
_~
"(k)
2~
"(k)
~
"(k)= 2
~
L(k) (52)
so that one can also dene the conjugate momentum of the transverse potential vector
as:
~
"(k) =

L(k)
_~
"
(k)
=0
_~
"(k) (53)
2-d. Hamiltonian; Hamilton-Jacobi equations
The Hamiltonianof the free radiation eld is obtained by generalizing, to con-
tinuous variables, expression (25) established for discrete variables. To only include
independent variables in the integral overkof the Hamiltonian density

(k), this
integral is taken over only half the reciprocal space:
=d
3
(k) (54)
where the Hamiltonian density

(k)is equal to:

(k) =

L(k) +
"
_~
"(k)
~

"(k) +
_~
"(k)
~
"(k) (55)
which yields for, taking (53) and (52) into account:
=0d
3
"
_~
"(k)
_~
"(k) +
22~
"(k)
~
"(k) (56)
1988

LAGRANGIAN FORMULATION OF ELECTRODYNAMICS
If the integral over the half-space in (56) is extended to the entire space, one must replace
0by02; we then get the same expression as (B-14) of Chapter
of the free transverse eld. The Hamiltonian obtained with the Lagrangian formalism
coincides with the eld energy.
We can write expression (55) for

(k)as a function of only the variables
~
"(k)
and
~
"(k). Using (53), we get:

(k) =
"
1
0
~

"(k)
~
"(k) +0
22~
"(k)
~
"(k) (57)
Equations (28) can then be generalized and yield the Hamilton-Jacobi equations for
~
"(k)and
~
"(k):
_~
"(k) =

(k)
~

"
(k)
=
1
0
~
"(k) (58a)
_~
"(k) =

(k)
~
"
(k)
= 0
22~
"(k) (58b)
It is easy to check that the two equations (58a) and (58b) are the same as Maxwell's
equations (A-30a) and (A-30b) of Chapter , which describes the evolution of the
transverse elds in the absence of sources. Equation (58a) of the present complement and
equations (A-30a) of Chapter
as a function of the time derivative of the transverse potential vector. As for equation
(58b) of this complement and equation (A-30b) of Chapter , they are the same
when
~
j=0. They describe the evolution of the transverse electric eld.
2-e. Field commutation relations
Generalizing the canonical commutation relation (29) to the case of continuous
variables, we get:
^~
"(k)
^~

"(k)=~""(kk) (59)
all the other commutators being equal to zero. The Kronecker delta""of the vectors
"and"is equal to 1 if both these vectors are the same, and to 0 if they are dierent.
Comment
The canonical commutation relations only apply to independent conjugate variables, which
is the case for the components of the various elds along the transverse polarization di-
rections. Now the eld components on an arbitrary xed reference framee,e,e,
~
(k)with= , are not independent because of the transversality condition
~
(k) = 0. Therefore:
^~
(k)
^~
(k)=~ (kk) (60)
To get the correct commutation relation between
^~
(k)and
^~
(k), we must express
both quantities as functions of their components along the two polarization vectors"and
1989

COMPLEMENT A XVIII
"
0
, which are perpendicular both to each other and tok(here we choose a basis of linear,
real, polarizations), and then use (59). As an example:
^~
(k) =
^~
"(k) +
^~
"(k) (61)
where:
=e" =e" (62)
We thus get:
^~
(k)
^~
(k)=~(+ )(kk) (63)
This equation can be further transformed by noting that","
0
andkform an orthonor-
mal basis, so that:
+ +
2
= (64)
Finally, the correct commutation relation between
^~
(k)and
^~
(k)is written:
^~
(k)
^~
(k)=~
2
(kk) (65)
We multiply both sides of (65) by
kr kr
(2)
3
and integrate overkandk. We then
get on the left-hand side the commutator of the elds in real space
8
,[
^
(r)
^
(r)],
and on the right-hand side the Fourier transform of the function(
2
)which is
thetransverse delta function
9
(rr):
^
(r)
^
(r)
=
~
(2)
3
d
3
d
3 krkr
2
(kk)
=
~
(2)
3
d
3 k(rr)
2
~(rr) (66)
2-f. Creation and annihilation operators
The normal variable(k)introduced in equation (B-15) of Chapter
time evolution, where=for the free eld case. According to
XVIII, this variable becomes, upon quantization, the annihilation operator^"(k)of
the harmonic oscillator associated with the modek". As for the complex conjugate
of this normal variable, it becomes the creation operator^
"(k). These two operators
are therefore written as:
^"(k) =()
^
"(k) +
0
^
"(k)
^
"(k) =()
^
"
(k)
0
^

"
(k) (67)
8
For the term in
^~
(k), we use the reality condition
^~
(k) =
^~
(k)and changekintok
in the integral overk.
9
The interested reader can nd details on the properties of that function in Complement AIof
reference [16].
1990

LAGRANGIAN FORMULATION OF ELECTRODYNAMICS
where we have used (53) to write
_
^
"(k) = (10)
^
"(k). The quantity()is a
normalization constant, arbitrary for now. It can, however, be determined by imposing
for the commutator of the two operators (67) a generalization of the well-known relation
[^^] = 1for the harmonic oscillator. We thus use the two equations (67) to compute
the commutator^"(k)^
"
(k)as a function of the commutators of the elds
^
and
^
and of their adjoints. Since the only non-zero commutators are between
^
and
^

as well as between
^
and
^
, we get:
^"(k)^
"
(k)=
2
()
0
^
"(k)
^

"
(k)
^
"(k)
^
"
(k)
=
2
()
0
2~""(kk)
=
2
()
2~
0
""(kk) (68)
To go from the rst to the second line of (68), we used (59) and its complex conjugate.
The constant()is nally determined by imposing the commutators between the
annihilation and creation operators to be equal to""(kk), which yields:
() =
02~
=
02~
(69)
Inserting this relation in equality (B-25) of Chapter , we nd that the contribution
to the classical energy of the modek"is:
}
2
[
"(k)"(k) +"(k)
"(k)] (70)
We shall see in Chapter
quantized radiation Hamiltonian.
2-g. Discrete momentum variables
We examined, in , the case where the radiation is contained
in a box of nite volume
3
, which leads to a discrete summation over the momenta.
Relation (59) then become:
^~
k
^~

k =~""kk (71)
Applying the substitution (B-34) or (B-35) of that chapter to both sides of (67), the two
coecients(2)
3
cancel each other, and these relations remain unchanged (aside from
the fact thatkis now a discrete index rather than a continuous variable).
As for the relations (68), they become:
^k^
k
=
2
()
2}
0
""kk (72)
With the choice (69) for(), we check that the commutator is equal to unity ifk=k
and"=".
1991

COMPLEMENT A XVIII
3. Lagrangian of the global system eld + interacting particles
We now study the Lagrangian of the total system, including the interactions between the
particles and the electromagnetic eld.
3-a. Choice for the Lagrangian
We choose a Lagrangian expressed as:
= + + (73)
where depends only on the radiation variables,only on the particle variables,
andon both types of variables as it describes the interactions between particles and
radiation.
For, we shall take the Lagrangian introduced above for the free eld see
relations (51) and (52):
=d
3
L(k) =0d
3_~
"(k)
_~
"(k)
2~
"(k)
~
"(k) (74)
For the Lagrangianof the particles, labeled by the index, we shall use the usual
Lagrangian for a system of particles, i.e. the dierence between their kinetic energy and
their potential interaction energy which comes from the Coulomb forces they exert on
each others:
=
1
2
_r()
2
Coul (75)
Finally, the interaction Lagrangian will be chosen as:
=d
3
j(r)A(r) (76)
wherej(r)is the particle current density given by the expression:
j(r) = _r()(rr()) (77)
andis the charge of particle. This expression does not contain terms includingA,
the charge density(r)or the scalar potential(r). This is because of our choice
of the Coulomb gauge in whichAis zero, so that the energy of the longitudinal electric
eld only depends on the particle variables; the same is true for the scalar potential,
which is at the origin of the termCoulincluded in the particle Lagrangian (75).
For the following computations, it will be useful to give other equivalent expressions
for; depending on the problem we focus on, we shall use the most suitable expression.
Inserting (77) into (76), we get:
=d
3
j(r)A(r) = _r()A(r) (78)
Now, using the Parseval-Plancherel identity, we get:
=d
3
j(r)A(r) =d
3~
j(k)
~
A(k) (79)
1992

LAGRANGIAN FORMULATION OF ELECTRODYNAMICS
In the integral overin (79) we get the sum:
~
j(k)
~
A(k) +
~
j(k)
~
A(k) =
~
j(k)
~
A(k) +
~
j(k)
~
A(k)(80)
Limiting the integral to half the reciprocal space, we can write:
=d
3~
j(k)
~
A(k)
=d
3~
j(k)
~
A(k) +
~
j(k)
~
A(k) (81)
3-b. Lagrange's equations
We now show that the Lagrange's equations associated with (73) coincide with the
Maxwell-Lorentz equations, which will be a justication for the choice of.
. Field Lagrange's equations
The time derivative of the transverse potential vector
_~
"only appears in the
Lagrangian density

L(k)written in (74). Writing

Lthe Lagrangian density of the
total system, we can write:

L
_~
"
(k)
=

L
_~
"
(k)
=0
_~
"(k) (82)
Now
~
"(k)appears both in

Land in

Lwhich is the function to be integrated in
the last integral of (81). We get:

L
~
"
(k))
=

L
~
"
(k))
+

L
~
"
(k)
= 0
22~
"(k) +~
"(k) (83)
The eld Lagrange's equation is obtained by setting equal the time derivative of () and
relation (83), which yields:
~
"(k) +
22~
"(k) =
1
0
~
"(k) (84)
We thus get the time evolution equation of the transverse potential vector in the presence
of sources, which we already obtained in (A-31) of Chapter .
. Lagrange's equations for the particles
The velocity_rof particleappears in bothand. Noting_the component
of_ron theaxis, we get:
_
=
_
+
_
= _+ (r()) (85)
We now compute the time derivative of this expression. The time dependence of the
second term of (85) is explicit via the time dependence of the vector potential, implicit
1993

COMPLEMENT A XVIII
via the time dependence of the pointr()where this potential is evaluated. We therefore
have:
d
d_
=+
(r)
+_rr (r) (86)
The partial derivative of the transverse vector potential with respect toleads to the
transverse electric eld:
(r)
= (r) (87)
As for the last term of (86), it can be written:
_rr (r) = _
+ _+ _ (88)
We now compute the partial derivative ofwith respect to the componentof
r, which appears in the termCoulofas well as in(via the position dependence
ofA). We obtain:
=+=
Coul
+_r
A
(89)
The term Coul is the Coulomb force exerted on the particle, i.e. the force due
to the electrostatic eld created by the charge distribution and acting on the charge
at pointrwhere the particleis located. It can also be written as:
Coul
= (r) (90)
whereEis the longitudinal electric eld, since, as we saw in Chapter , the longi-
tudinal electric eld is equal to the electrostatic eld created by the charge distribution.
Finally, let us explicitly write the last term of (89):
_r
A
= _+ _+ _ (91)
Lagrange's equation for particle:
d
d_
= (92)
can thus be written, using the previous results:
= (r) _rr +_r
A
(93)
where the total electric eld at pointris :
E(r) =E(r) +E(r) (94)
1994

LAGRANGIAN FORMULATION OF ELECTRODYNAMICS
We can write explicitly the last term of (93) by regrouping equations (88) and (91). We
then get:
_rr +_r
A
= _
_
=[_r(rA)]=(_rB) (95)
This yields the Lorentz magnetic force exerted by the magnetic eld on the particle with
velocity_r. To sum up, Lagrange's equation for particleis written:
r=E(r) +_rB(r) (96)
and coincides with the Lorentz equation (A-3) of ChapterXVIII. The Lagrangian we
chose above therefore leads to the right equations for the eld and the particles.
3-c. Conjugate momenta
The equation (82) established above can be used to compute the conjugate mo-
menta
~
"(k)of the eld variables
~
"(k):
~
"(k) =

L
_~
"
(k)
=

L
_~
"
(k)
=0
_~
"(k)) =0
~
"(k) (97)
In a similar way, the computation of the conjugate momentum of the coordinate
rof particleis the same as the one leading to equation (85):
p=
_r
=
_r
+
_r
= _r+A(r) (98)
3-d. Hamiltonian
Since
_~
"and
~
=0
_~
"appear only in the radiation LagrangianL, which
only depends on the radiation variables, the Hamiltonian of the global system must
contain a termidentical to the expression (56) found for the free eld.
To obtain the other terms coming from the particle conjugate momenta and from
the subtraction ofand, we rst compute_rp. Equation (98) yields:
_rp= _r
2
+_rA(r) (99)
We must now subtract from (99) the values ofandgiven by equations (75) and
(78). The term coming fromcancels the last term of the right-hand side of (99) and
we are left with:
Coul+
1
2
_r
2
=Coul+
[p A(r)]
2
2
(100)
1995

COMPLEMENT A XVIII
Finally, the global system Hamiltonian is given by:
= +Coul+
[p A(r)]
2
2
(101)
where has the same form as (56) for the free radiation. This result is a justication
for the expression ofgiven in equation (A-41) of Chapter .
3-e. Commutation relations
Since the radiation variables and their conjugate momenta are the same as for the
free eld, the commutation relations (59) established for the free eld remain valid:
^~
"(k)
^~

"(k)=~""(kk) (102)
all the other commutators being equal to zero. As for the commutation relations for the
positions and conjugate momenta of the particles, they are the usual relations:
[^^] =~ (103)
where the indiceslabel the particles and the indices= the Cartesian
components ofrandp.
As in , one can extend these commutation relations to the case where the
momenta are discrete.
1996

Chapter XIX
Quantization of
electromagnetic radiation
A Quantization of the radiation in the Coulomb gauge
A-1 Quantization rules
A-2 Radiation contained in a box
A-3 Heisenberg equations
B Photons, elementary excitations of the free quantum eld
B-1 Fock space of the free quantum eld
B-2 Corpuscular interpretation of states with xed total energy
and momentum
B-3 Several examples of quantum radiation states
C Description of the interactions
C-1 Interaction Hamiltonian
C-2 Interaction with an atom. External and internal variables
C-3 Long wavelength approximation
C-4 Electric dipole Hamiltonian
C-5 Matrix elements of the interaction Hamiltonian; selection rules
Introduction
This chapter presents a quantum description of the electromagnetic eld and its in-
teractions with an ensemble of charged particles. Such a description is necessary for
interpreting certain physical phenomena such as the spontaneous emission of a photon
by an excited atom, which cannot be carried out with the semiclassical treatments we
have used previously
1
(classical description for the eld, and quantum description for the
1
See for example in Complement XIIIthe study of the interaction between an atom and an
electromagnetic wave.
Quantum Mechanics, Volume III, First Edition. C. Cohen-Tannoudji, B. Diu, and F. Laloë.
© 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

CHAPTER XIX QUANTIZATION OF ELECTROMAGNETIC RADIATION
particles). Imagine, for example, that a monochromatic eld with angular frequency
is described by a classical eldE0cos; its interaction with an atom is then described
by the Hamiltonian=DE0cos, whereDis an operator (the electric dipole
moment) whereasE0remains a classical quantity
2
. Such a treatment is adequate for
understanding how the eld can excite the atom from its ground statewith energy
towards an excited stateof energy; the processus is resonant ifis close to
the atomic Bohr frequency0= ( )}. Imagine now that the atom is initially
in the excited state, in the absence of any incident radiation. The classical eldE0
is then identically zero and, consequently, so is the interaction Hamiltonian. The
Hamiltonian of the total system is then reduced to the atomic Hamiltonian. Since
this operator is time-independent, its eigenstates are stationary, including, in particular,
the excited state. The semiclassical theory predicts that an atom, initially excited in a
statein the absence of incident radiation, will remain indenitely in that state. But this
is not what is experimentally observed: after a certain time, the atom spontaneously falls
into a lower level, emitting a photon whose frequency is close to0= ( )}.
This process is called spontaneous emission and happens after an average time called
theradiative lifetimeof the excited state. This is a rst example of a situation where
a radiation quantum treatment is indispensable. It is far from being the only exam-
ple: numerous experiments, more and more elaborate, have created situations where the
quantum description of the electromagnetic eld is necessary.
This chapter presents the base of this quantum description, while following an
approach that is as simple as possible a more general presentation of the quantization of
the electromagnetic eld is possible with the Lagrangian formulation of electrodynamics
(ComplementXVIII. In the previous chapter, we underlined the analogy between the
eigenmodes of the radiation eld vibrations and an ensemble of harmonic oscillators. We
shall use this analogy in
this ensemble of oscillators. With each eigenmodeof the classical eld, described by
normal variablesand, we shall associate annihilationand creationoperators,
obeying the well-known commutation relations = 1. We shall also propose a
plausible form for the quantum Hamiltonian of the system eld + particles, starting
from the classical energy of that system established in the previous chapter. We will see
that the equations of evolution
3
for these various quantities in the Heisenberg picture
(ComplementIII) are the transposition of the Maxwell-Lorentz equations to operators
describing elds and particles, properly symmetrized. This will yield an a posteriori
justication for the simple quantization procedure we used.
Several important properties of the free eld (in the absence of sources) are de-
scribed in . The state space of this eld has the structure of a tensor product of Fock
spaces, analogous to those studied in Chapter; the elementary excitations of the eld
are called photons. A few important states of the eld will be described: the photon
vacuum, where no photons are present (but where there exists, nonetheless, a uctuating
eld throughout the entire space, with a zero average value), the one-photon states, and
the quasi-classical states, which reproduce the properties of a given classical eld.
Finally,
and particles, in particular when those are neutral atoms (such as the Hydrogen atom
2
For the sake of clarity, we use in the entire chapter and its complements the symbol hat to
distinguish an operatorfrom its corresponding classical quantity.
3
More concisely, we shall call them Heinsenberg equations.
1998

A. QUANTIZATION OF THE RADIATION IN THE COULOMB GAUGE
where the positive and negative charges of the atom's constituents balance each other).
It is then possible to distinguish between two types of atomic variables: the center of
mass variables (external variables) and the relative motion variables in the center of
mass frame (internal variables). We shall also study the electric dipole approximation,
valid when the radiation wavelength is large compared to the atomic sizes, as well as the
selection rules associated with the interaction Hamiltonian.
A. Quantization of the radiation in the Coulomb gauge
A-1. Quantization rules
In the previous chapter, we established in relation (B-26) the following expression
for the energy of the classical transverse eld:
trans=0d
3
"
2
4
2
()
[
"(k)"(k) +"(k)
"(k)] (A-1)
where"(k)and
"(k)are the normal variables describing the transverse eld,=
, and()a real normalization constant that appeared in the equations dening the
normal variables in terms of the transverse potential vector and its time derivative:
(k) =()
~
A(k) +
_~
A(k)
(k) =()
~
A(k)
_~
A(k) (A-2)
The analogy between the free transverse eld and an ensemble of classical harmonic
oscillators of frequencyassociated with the modesk"is clearly seen in expression
(A-1).
To quantize the eld, this analogy suggests replacing the normal variables"(k)
and
"(k)by annihilation and creations operators. We shall use in this
Schrödinger picture where these operators are time-independent and where the time
dependence only appears in the evolution of the state vector. The quantization proce-
dure will consist in replacing the"(k= 0)by time-independent annihilation operators
^"(k), and of course the
"(k= 0)by the adjoint creation operators^
"(k). Once this
operation is performed on (A-1), we obtain a quantum Hamiltonian identical to a sum of
standard harmonic oscillator Hamiltonians, provided the factor
2
4
2
()multiplying
the bracket on the right-hand side of (A-1) is equal to~20. We therefore choose for
()the value:
() =
02~
=
02~
(A-3)
This relation is the same as relation (69) of ComplementXVIII, obtained from the
commutation relations. We now replace in (A-1) the classical normal variables"(k)
and
"(k)by the operators^"(k)and^
"(k)obeying the commutation relations:
^"(k)^
"
(k)=""(kk) (A-4a)
[^"(k)^"(k)] =^
"(k)^
"
(k)= 0 (A-4b)
1999

CHAPTER XIX QUANTIZATION OF ELECTROMAGNETIC RADIATION
This yields the Hamiltonian operator (as this operator will be frequently used, we
simplify the notation and replace
^
transby
^
):
^ ^
trans=d
3
"
~
2
^
"(k)^"(k) + ^"(k)^
"(k) (A-5)
which has the expected form for the quantum Hamiltonian of the transverse eld.
Extending this procedure, we now replace the classical normal variables by anni-
hilation and creation operators in all the classical expressions established in the previous
chapter for the various physical quantities. The transverse momentum see equation
(B-27) of Chapter
^
Ptrans=d
3
"
~k
2
^
"(k)^"(k) + ^"(k)^
"(k) (A-6)
As for the transverse elds, written in (B-29), (B-30) and (B-28) of Chapter , they
become linear combinations of creation and annihilation operators:
^
E(r) =
d
3
(2)
32
"
~
2"0
12
^"(k)"
kr
^
"(k)"
kr
(A-7)
^
B(r) =
d
3
(2)
32
"
~
2"0
12
^"(k)"
kr
^
"(k)"
kr
(A-8)
^
A(r) =
d
3
(2)
32
"
~
2"0
12
^"(k)"
kr
+ ^
"(k)"
kr
(A-9)
Comment:
As in Chapter , these relations are written in the general case where the polarizations
may be complex (elliptical or circular). Complex conjugate"of the polarization vectors are
therefore associated with the creation operators. It is of course necessary to check that the
quantication procedure is independent of the arbitrary choice of the polarization basis. If a
quantization is performed with a given basis of polarizations, by substitution one can calculate
the operators multiplying"and"in the new basis, and check that the commutation relations
of these operators are indeed those of standard creation and annihilation operators. This ensures
the polarization basis independence.
Finally, relation (A-48) of Chapter
elds becomes:
^
=
1
2
^p
^
A(^r)
2
+
^
Coul+d
3
"
~
2
^
"(k)^"(k) + ^"(k)^
"(k)
(A-10)
which is a plausible form for the quantum Hamiltonian of the system particles + elds.
The position^rand momentum^poperators dened using equation (A-47) of Chapter
XVIII
[(^r)(^p)] =~ (A-11a)
[(^r)(^r)] = [(^p)(^p)] = 0 (A-11b)
2000

A. QUANTIZATION OF THE RADIATION IN THE COULOMB GAUGE
The quantization rules we just heuristically introduced have the advantage of simplic-
ity. We are going to show in addition that the Heisenberg equations for the various
operators describing the particles and the elds, deduced from the Hamiltonian (A-10)
as well as from the commutation relations (A-4), (A-11a) and (A-11b), are indeed the
Maxwell-Lorentz equations for operators. This result justies a posteriori the quantiza-
tion procedure exposed in this chapter.
A-2. Radiation contained in a box
If the real space is innite,kis a continuous variable, and there exists a continuous
innity of modes. However, as we mentioned in , it is often more
convenient to consider the eld to be contained in a cube of edge lengthwith periodic
boundary conditions; the variablekis now discrete:
= 2 (A-12)
where are positive, negative or zero integers. All the physical predictions must be
independent ofwhen it is large enough. In such an approach, we replace the Fourier
integrals by Fourier series and the integrals overkby discrete summations. For a classical
eld, the continuous variables"(k)then become discrete variablesk"(). If the eld
is zero outside the box, relation (B-35) of chapter
factor that must be used to go from one type of variable to the other.
The system is then quantized as we just explained. In the Schrödinger picture,
each classical coecientk"(= 0)in a Fourier series becomes an annihilation operator
^k"; each coecient
k"
(= 0)becomes a creation operator^
k"
. This latter operator
creates a quantum in a eld mode conned inside the box (instead of spreading over the
entire space). The commutation relations (A-4) are then written:
^k"^
k"
=""kk (A-13a)
[^k"^k"] =^
k"
^
k"
= 0 (A-13b)
Relation (B-36) of Chapter
been inserted in the expressions for the elds, the following rule must be applied to go
from a continuous to a discrete summation:
d
3
=
2
32
k
(A-14)
Expressions (A-7) to (A-9) must be modied. As an example, relation (A-7) becomes:
^
E(r) =
k"
~
2"0
3
12
^k""
kr
^
k"
"
kr
(A-15)
This means that in addition to replacing the integral by a discrete summation, and
multiplying by a factor(2)
32
, one must divide the eld expansion by the square root
of the volume
3
. Both relations (A-8) and (A-9) undergo the same changes.
2001

CHAPTER XIX QUANTIZATION OF ELECTROMAGNETIC RADIATION
A-3. Heisenberg equations
A-3-a. Heisenberg equations for massive particles
We start with the equation for the evolution of^r():
_
^r() =
1
~
^r()
^
(A-16)
The only term in Hamiltonian (A-10) that does not commute with^ris the rst one.
Using the commutation relation deduced from (A-11a) and (A-11b) :
[(^r)((^p))] =}
(^p)
(A-17)
we get:
_
^r() =
1
~
^r()
1
2
^p()
^
A(^r)
2
=
1
^p()
^
A(^r) (A-18)
This equality is simply the operator form:
^p() =
_
^r() +
^
A(^r) (A-19)
of the classical equation relating the generalized (or canonical) momentumpand the
mechanical momentum _r. We then dene the velocity operator^vof particleby:
^v() =
1
^p()
^
A(^r) (A-20)
Consider now the Heisenberg equation for the evolution of this operator. It yields
the equation of motion of that particle:
_
^v() =

^r() =
~
^v()
^
(A-21)
We shall compute below the commutator^v()
^
; it leads to the quantum equation
of motion for particle:

^r=
^
E(^r) +
2
^v
^
B(^r)
^
B(^r)^v (A-22)
which is simply the quantum Lorentz equation describing the motion of particles in-
teracting with the magnetic eld
^
Band the total electric eld
^
E=
^
E+
^
E. The
special form of the magnetic force^v
^
B(^r)
^
B(^r)^v2comes, as shown in
the computation below, from using the Heisenberg equations, and from the fact that the
operator^v
^
B(^r)is not Hermitian. To make that operator Hermitian, we must add
its adjoint^v
^
B(^r)), which is simply
^
B(^r)^v, and divide the result by 2.
2002

A. QUANTIZATION OF THE RADIATION IN THE COULOMB GAUGE
Demonstration of equation (A-22)
To compute the commutator of ^v~with the rst term of
^
, it is useful to rst
calculate the following commutators:
2
[(^v)(^v)] = (^p)(
^
A(^r)) (
^
A(^r))(^p)
=~ (
^
A(^r)) (
^
A(^r))
=~ (
^
B(^r)) (A-23)
where is the completely antisymmetric tensor that allows writing the cross product
components of two vectorsaandbin the form(ab)= . We then get:
~
(^v) (^v)
2
2=
2
2~
(^v)[(^v)(^v)] +(^v)(^v)(^v)
=
2
(^v)(
^
B(^r))+ (
^
B(^r))(^v)
(A-24)
The last line in (A-24) can be rewritten in the form:
2
^v
^
B(^r)
^
B(^r)^v (A-25)
and is thus the component along theaxis of the symmmetrized magnetic force.
The commutator of ^v~with the second term of
^
is written:
~
[(^v) Coul] =
1
~
[(^p) Coul] =
(^r)
Coul=(
^
E(^r)) (A-26)
It describes the interaction between particleand the longitudinal electric eld.
We nally have to compute the commutator of^v~with the last term of
^
. Using
the commutation relations (A-4) and expressions (A-9) and (A-7) for
^
Aand
^
E, we get:
~
(^v) d
3
"
~^
"(k)^"(k) + 12
= d
3
"
(
^
A(^r))^
"(k)^"(k)
=(
^
E(^r)) (A-27)
This term describes the interaction of particlewith the transverse electric eld. Finally
grouping (A-25), (A-26) and (A-27) leads to (A-22).
A-3-b. Heisenberg equations for elds
As all the elds are linear combinations of the operators^"(k)and^
"(k), we
simply have to consider the Heisenberg equation for^"(k):
_
^"(k) =
1
~
^"(k)
^
(A-28)
2003

CHAPTER XIX QUANTIZATION OF ELECTROMAGNETIC RADIATION
We assume that the polarizations"are real (linear polarizations). The commutator with
the rst term of
^
yields, with the use of(A-4a) and (A-20):
1
~
^"(k)
^v
2
2
=
2~
^v
A
"(k)
+
A
"(k)
^v
=
2~}20(2)
3
"^v
k^r
+
k^r
^v (A-29)
whereA
"(k)denotes the coecient of^
"(k)in the integral (A-9) of
^
A(r),
which is nothing but the coecient of
"(k= 0)in the classical expression ofA(r).
We introduce the current operator (symmetrized to make it Hermitian):
^
j(r) =
1
2
[^v(r^r) +(r^r)^v] (A-30)
The right hand side term of equation (A-29) can then be rewritten in the form:
2
20~(2)
3
"^v
k^r
+
k^r
^v=
20~(2)
3
d
3 kr
"
^
j(r)
=
20~
"
^~
j(k) (A-31)
The commutator with the second term of
^
is zero, whereas the commutator with the
third term yields, using (A-4):
1
~
^"(k)d
3
"
~^
"
(k)^"(k) + 12= ^"(k) (A-32)
Finally, regrouping (A-31) and (A-32) yields:
_
^"(k) +^"(k) =
20~
"
^~
j(k) (A-33)
This equation is, for the operator^(k), an equation of motion of the same form as the
equation of motion of the classical normal variables(k), which is given by equation
(B-19) of Chapter . As this latter equation is equivalent to Maxwell's equations for
the transverse elds, we may conclude that the Heisenberg equations for the quantum
transverse elds are simply the usual Maxwell's equations applied to the eld operators.
B. Photons, elementary excitations of the free quantum eld
We now study a certain number of properties of the electromagnetic eld we just quan-
tized, starting with the simplest case: the eld in the absence of charged particles.
B-1. Fock space of the free quantum eld
The state space of the total system eld + particles is the tensor product of the
particle state spaceand the radiation eld state space. This latter space is itself
2004

B. PHOTONS, ELEMENTARY EXCITATIONS OF THE FREE QUANTUM FIELD
the tensor product of the state spaces of the harmonic oscillators associated with the
dierent modesk":
=k11 k22 k (B-1)
wherekis the state space of the harmonic oscillator associated with the modek",
with frequency.
As in , we assume the radiation to be contained in a box of edge length.
The operators^(k)depending on the variableskare then transformed into operators
^k"depending only on discrete variables. We can even use a more compact notation
^, where the indexlabels
4
the whole set of indicesk"; the operators^"(k)are now
simply written^. In this section, it is convenient to use the Heisenberg picture; the time
dependence of the^and^is then particularly simple, since we have:
^() = exp(
^
~) ^exp(
^
~) = ^ (B-2)
as well as the Hermitian conjugate relation.
Once the discrete variables have been inserted in the continuous expressions of
the elds, we must use rule (A-14) to transform the continuous integrals into discrete
summations. The expansions of these elds in term of normal variables are then:
^
E(r) =
~
20
3
12
^"
(kr )
^"
(kr )
(B-3)
^
B(r) =
~
20
3
12
^"
(kr )
^"
(kr )
(B-4)
^
A(r) =
~
20
3
12
^"
(kr )
+ ^"
(kr )
(B-5)
^
=
~
2
^^+ ^^= ~^^+
1
2
(B-6)
^
Ptrans=
~k
2
^^+ ^^= ~k^^ (B-7)
Note that the last term in (B-7) does not contain the factor12, sincek=0.
B-2. Corpuscular interpretation of states with xed total energy and momentum
Consider rst the mode. The eigenvalues of the operator^^appearing in
expressions (B-6) and (B-7) for
^
and
^
Ptransare all the positive or zero integers:
^^ = = 012 (B-8)
4
For eachk, there exists two polarization vectors"1and"2perpendicular tokand perpendicular
to each other. The compact notation must be interpreted as a summation overkand, for each
value ofk, as a sum over"1and"2.
2005

CHAPTER XIX QUANTIZATION OF ELECTROMAGNETIC RADIATION
Remember the well-known actions of operators^and^on the states:
^ =
+ 1
^ =
1
^0= 0 (B-9)
As^^commutes with^^, the eigenstates of
^
and
^
Ptransare the tensor
products of the eigenstates1 = 1 of the creation and annihilation
operators^
1
^1,....^^...:
^
1 = +
1
2
~ 1 (B-10a)
^
Ptrans1 = ~k1 (B-10b)
The eld's ground state corresponds to all theequal to zero, and will be noted0:
0=010 (B-11)
the states1 being obtained by the action of a certain number of creation oper-
ators on this0state:
1 =
(^
1
)
1
1!
(^)
!
0 (B-12)
With respect to the eld ground state, the state1 has an energy ~and a
momentum ~k. It can be interpreted as describing an ensemble of1particles of
energy~1and momentum~k1,.....,particles of energy~and momentum~k
. These particles characterize the elementary excitations of the quantum eld and are
calledphotons. The quantum number is therefore the number of photons occupying
the mode, so that the ground state0, corresponding to all theequal to zero, can
be called the photonvacuum.
Whereas there exists for photons eigenstates of momentum and energy, there are no
quantum states of the electromagnetic eld where the position can be perfectly known; no
position operator is associated with this eld. This is a dierent situation from what we
encounter with massive particles, which have both a position and a momentum operator;
the wave functions in the two representations are related by a simple Fourier transform.
This non-existence of a position operator is linked to the impossibility of building, by a
linear superposition of transverse electromagnetic waves, a vector wave perfectly localized
at a point in space. The relativistic and transverse character of the electromagnetic eld
yields commutation relations between its components that involve the transverse delta
function (ComplementXVIII, ) instead of the usual delta function.
B-3. Several examples of quantum radiation states
We now study several examples of states of quantum radiation.
2006

B. PHOTONS, ELEMENTARY EXCITATIONS OF THE FREE QUANTUM FIELD
B-3-a. Photon vacuum
The presence of the12term in the parenthesis on the right-hand side of equation
(B-10a) shows that the vacuum state energy is not zero, but equal to~2; this
sum is an innite quantity. We encounter here a rst example of the diculties linked
to the divergences appearing in quantum electrodynamics. They can be resolved by
renormalizationtechniques, whose presentation is outside the scope of this book. We shall
avoid this diculty by only considering energy dierences with respect to the vacuum.
If we consider a single modeof the eld, the energy~2of the vacuum state
for this mode is nite, and reminiscent of the zero-point energy of a harmonic oscillator
of frequency. As you may recall, this zero-point energy is due to the impossibility of
having simultaneously zero values for the positionand momentumof that oscillator,
because of the Heisenberg relations. The lowest energy state of the oscillator results
from a compromise between the kinetic energy, proportional to
2
, and the potential
energy, proportional to
2
(this problem is discussed in ). The same
arguments can be presented for the contribution, at a given pointr, of modeto the
electric
^
E(r)and magnetic
^
B(r)elds; according to (B-3) and (B-4), those elds
are represented by two dierent linear superpositions of operators^and^, which thus
do not commute. Consequently, one cannot have simultaneously a zero value for the
electric energy proportional to
^
E
2
, and for the magnetic energy proportional to
^
B
2
.
One can further calculate the average value and variance of the contribution of
modeto the electric eld
^
E(r)at pointr. Since^and^changeby1, a simple
calculation yields:
0
^
E(r)0mode i= 0 (B-13a)
0
^
E
2
(r)0mode i=
~
20
3
(B-13b)
Similar calculations can be done for the magnetic eld. They show that in the photon
vacuum state, the average value of both the electric and magnetic elds is zero, but not
their variance. Since result (B-13b) is proportional to~, the non-zero variance of the
elds in the vacuum is a quantum eect.
Comments
(i) The summation over all the modes of expressions (B-13) yields, once we have replaced
the discrete sum by an integral:
0
^
E(r)0= 0 (B-14a)
0
^
E
2
(r)0=
~
20
3
=
~
20
2
0
3
d (B-14b)
This means that the variance of the electric eld diverges as the fourth power of the upper
boundary of the integral overappearing in the summation of the modes of frequency
=. This divergence is the same as that mentioned above.
(ii) To characterize the dynamics of these eld uctuations, it is possible to compute
the eld correlation functions in vacuum
5
. This calculation shows that the electric and
magnetic elds uctuate very rapidly around their zero average value. These uctuations
are called thevacuum uctuations. Certain radiative corrections, such as the Lamb shift
5
See for example III-C-3-c and Complement CIIIof reference [16].
2007

CHAPTER XIX QUANTIZATION OF ELECTROMAGNETIC RADIATION
in atoms, can be interpreted from a physical point of view, as resulting from the vibration
of the atom's electron caused by its interaction with this uctuating electric eld. This
vibration leads the electron to explore the nucleus Coulomb potential over the range of its
vibrational motion. The corresponding correction to its binding energy depends on the
energy level it occupies; this explains why the degeneracy between the212and212
states of the hydrogen atom, predicted by the Schrödinger and Dirac equations, can be
lifted by the interaction with the vacuum uctuations
6
.
B-3-b. Field quasi-classical states
The state and observables of a classical eld are characterized by the normal
variables introduced in . The coherent states of a one-
dimensional harmonic oscillator studied in ComplementV, can be used to build the
eld quantum states whose properties are closest to those of the classical eld.
The coherent state, supposed to be normalized, of a one-dimensional harmonic
oscillator is the eigenstate of the annihilation operator^, with eigenvalue:
^= (B-15)
The eigenvaluemay be a complex number since operator^is not Hermitian. Equation
(B-15) leads to:
^= ^ = (B-16)
More generally, the average value of any function of^and^, once put in thenormal
order, i.e. where all the annihilation operators are positioned to the right of the creation
operators (ComplementXVI, ), is equal to the expression obtained by replacing
operator^byand operator^by. As an example:
^^= (B-17)
Consider then the eld quantum state:
1 2 = 12 (B-18)
where each modeis in the coherent statecorresponding to the classical normal
variable. Using equations (B-16) and (B-17), we can obtain the average values of the
various eld operators (B-3), (B-4) and (B-5) in the state (B-18); they coincide with the
values of these various physical quantities for a classical eld described by the normal
variables. The same is true for the observables (B-6) and (B-7) corresponding to
the energy and momentum of the transverse eld. This is why the quantum state (B-18),
which yields average values identical to all the properties of a classical eld, is called a
quasi-classical state
7
. We shall see later that the correlation functions of the quantum
and classical elds involved in various photodetection signals also coincide when the eld
state is a quasi-classical state.
6
See for example [17].
7
For more details on the properties of the radiation quasi-classical states, see III-C-4 of reference
[16].
2008

C. DESCRIPTION OF THE INTERACTIONS
B-3-c. Single photon state
Consider the state vector:
= 1
=
0 (B-19)
which is a linear superposition of kets where a modecontains one photon, whereas all
the other modes=are empty. Such a ket is an eigenket of the operator total number
of photons
^
= ^^with an eigenvalue equal to1. It is therefore a single photon
state. However, except in special cases, it is not a stationary state since it is not an
eigenstate of the eld energy
^
. It describes a single photon propagating in space with
velocity. We shall see later (ComplementXX) that, when the eld is in the state
(B-19), a photodetector placed in a small region of space yields a signal corresponding
to the passage, in that region, of a wave packet.
C. Description of the interactions
C-1. Interaction Hamiltonian
The Hamiltonian
^
of the system particles + eld has been given above. In its
expression (A-10), we now separate the terms that depend only on the particle variables
or only on the eld variables, and those that depend on both. We can then write
^
=
^
+
^
+
^
, where the particle Hamiltonian is:
^
=
^p
2
2
+
^
Coul (C-1)
whereas the radiation one is:
^
= ~^^+
1
2
(C-2)
Finally, the interaction Hamiltonian is the sum:
^
=
^
1+
^
2 (C-3)
with:
^
1=
2
^p
^
A(^r) +
^
A(^r)^p (C-4)
^
2=
2
2
^
A(^r)
2
(C-5)
(we have separated the linear and quadratic terms with respect to the elds).
To that interaction Hamiltonian, we must further add the term:
^
1=
^
M
^
B(^r) (C-6)
2009

CHAPTER XIX QUANTIZATION OF ELECTROMAGNETIC RADIATION
describing the interaction of the spin magnetic moments of the various particles with the
magnetic eld of the radiation (ComplementXIII, ):
^
M=
2
^
S (C-7)
whereis the Landé g-factor of particlewhose spin is noted
^
S.
Comment
Even with this additional term, all the possible interactions are not contained in that
Hamiltonian: missing for example are the electron spin-orbit coupling, the hyperne in-
teraction between the electron and the nucleus, etc. see comment (iii) of . The
Hamiltonian we wrote is however sucient in a great number of cases.
C-2. Interaction with an atom. External and internal variables
Consider the case where the particle system is a single atom, assumed to be neutral,
formed by an electronand a nucleuswhich have opposite charges (= =) and
whose masses are notedand . This is the case for example of the hydrogen atom.
It is standard practice (see for example ) to separate the variables
^
R
and
^
Pof the system's center of mass and the variables^rand^pof the relative motion.
These two types of variables commute with each other and are given by equations:
R=
r+ r
r=rr
P=p+p
p
=
pp (C-8)
where we have notedthe total mass of the system, andits reduced mass:
= + ; =
(C-9)
Expressed as a function of these new variables, the particle Hamiltonian is written:
^
=
^
P
2
2
+
^p
2
2
+
^
Coul(^) (C-10)
The center of mass variables, also calledexternal variables, describe the global
motion of the atom, whereas the variables of the relative motion, also calledinternal
variables, describe the motion in the center of mass reference frame.
C-3. Long wavelength approximation
The interaction Hamiltonians (C-4), (C-5) and (C-6) contain elds evaluated at
the electronrand nucleusrpositions. These positions can be described with respect
to the position of the center of mass and we can write for example:
^
A(^r) =
^
A(
^
R+^r
^
R) (C-11)
In an atom, the distance between the position of the electron or the nucleus and the atom's
center of mass is of the order of the atom's size, i.e. just a fraction of a nanometer. Now
2010

C. DESCRIPTION OF THE INTERACTIONS
the radiation wavelengths that can have a resonant interaction with the atom are of the
order of a fraction of a micron, much larger than the atomic dimension. One can thus
neglect the variation of the elds over distances of the order ofrR(orr R)
and write:
^
A(^r)
^
A(
^
R)
^
A(^r)
^
A(
^
R) (C-12)
Such an approximation is called the long wavelength approximation (or dipole approxi-
mation).
Using this approximation in the interaction Hamiltonian
^
1, yields:
^
1=
^p
^
A(^r)^p
^
A(^r)
^p
^p
^
A(
^
R)
=
^p
^
A(
^
R) (C-13)
We used the relation= =as well as denition (C-8) for the relative momentum.
As for Hamiltonian
^
2, it becomes with this approximation:
^
2=
2
2
^
A
2
(^r) +
2
2
^
A
2
(^r)
2
2
^
A
2
(
^
R) (C-14)
Comment
When we include the Hamiltonian describing the spin magnetic coupling
^
1written in
(C-6), we also replace all the^rby
^
R. This is however insucient: we must add other
terms of the same order, obtained by including rst order terms ink(^r
^
R)in
^
1and
^
2, and representing corrections to the long wavelength approximation. This is because
a computation analogous to the one in XIIIshows that these
corrections yield new interaction terms of the same order as
^
1: interaction between
the atomic orbital momentumLand the radiation magnetic eld; electric quadrupole
interaction.
C-4. Electric dipole Hamiltonian
Using the long wavelength approximation, the global Hamiltonian for the system
atom + eld is written:
^
=
^
P
2
2
+
1
2
^p
^
A(
^
R)
2
+
^
Coul+
~
2
^^+ ^^ (C-15)
We are going to perform a unitary transformation on this Hamiltonian, leading to a new
interaction Hamiltonian, composed of a single term of the form
^
D
^
E(
^
R), where
^
D
is the electric dipole moment of the atom:
^
D=^r (C-16)
2011

CHAPTER XIX QUANTIZATION OF ELECTROMAGNETIC RADIATION
and
^
E(
^
R), the quantum eld given by expression (B-3). This new interaction Hamil-
tonian is called theelectric dipole Hamiltonian.
To nd this unitary transformation, it is useful to start with the simpler case where
the radiation eld is treated classically.
C-4-a. Electric dipole Hamiltonian for a classical eld
When the radiation eld is treated classically, as an external eld whose dynamic
is externally imposed and hence has a xed time dependence, the last term of relation
(C-15) does not exist; operator
^
A(
^
R), which appears in the second term, must be
replaced by the external eldA(
^
R). The system Hamiltonian is then written:
^
=
^
P
2
2
+
1
2
^pA(
^
R)
2
+
^
Coul (C-17)
We are looking for a unitary transformation that performs a translation of^pby
a quantityA(
^
R), so that the second term in (C-17) is reduced to^p
2
2. Such a
transformation reads:
^
() = exp
~
^rA(
^
R) (C-18)
We can check this since, using[^p(^r)] =~ ^rand the fact that the internal
variable^rcommutes with the external variable
^
R, we have:
^
()^p
^
() =^p+A(
^
R) (C-19)
As they do not depend on^p, the other terms of (C-17) are unchanged by the transfor-
mation. On the other hand, since this transformation has an explicit time dependence
via the termA(
^
R), the new Hamiltonianthat governs the evolution of the new
state vector:
()=
^
() () (C-20)
is given by:
^
() =
^
()
^
()
^
() +~
d
^
()
d
^
() (C-21)
As we have in addition:
~
d
^
()
d
^
() =^r
A(
^
R)
=
^
DE(
^
R) (C-22)
where
^
D=^ris the electric dipole moment of the atom, we nally obtain:
^
() =
^
P
2
2
+
^p
2
2
+Coul
^
DE(
^
R) (C-23)
where the last term has the expected form for an electric dipole Hamiltonian.
2012

C. DESCRIPTION OF THE INTERACTIONS
C-4-b. Electric dipole Hamiltonian for a quantum eld
The results we just obtained suggest using the unitary transformation:
^
= exp
~
^r
^
A(
^
R) (C-24)
where it is now the operator
^
A(
^
R)that appears in the exponential. One can check
that this operator is still a translation operator for^p, so that the second term in (C-15)
is now simply of the form^p
2
2.
As the transformation (C-24) no longer has an explicit time dependence, the term
analogous to (C-22) does not exist anymore. On the other hand, we must study the
transformation of the last term of (C-15), which represents the energy
^
of the trans-
verse quantum eld. We therefore rewrite expression (C-24) using the expansion (B-5)
of
^
A(
^
R)as a function of^and^:
^
= exp ^ ^ (C-25)
with:
=
20~
3
"
^
D
k
^
R
(C-26)
In this form, operator
^
does appear as a translation operator (ComplementV,
it obeys the equations:
^
^
^
= ^+
^
^
^
= ^+ (C-27)
To prove relations (C-27), one can use (ComplementIIBII, ) the identity:
(+)
=
[]2
(C-28)
valid ifandcommute with their commutator[], as well as the commutation
relation^(^)= ^. The transformation of the last term in (C-15) then yields:
^^^
=
~
2
(^+)(^+) + (^+)(^+) (C-29)
The terms on the right-hand side of (C-29) that are independent ofandyield again
^
. The terms linear inandyield:
~ ^+^=
~20
3
^"
k
^
R
^"
k
^
R^
D
=
^
E(
^
R)
^
D (C-30)
where we have used (B-3). We thus get the expected electric dipole form for the inter-
action Hamiltonian:
^
=
^
E(
^
R)
^
D (C-31)
2013

CHAPTER XIX QUANTIZATION OF ELECTROMAGNETIC RADIATION
Finally, the terms quadratic inandintroduce a term we shall note
^
dip:
^
dip= ~ =
1
20
3
("
^
D)("
^
D) (C-32)
It represents a dipolar energy intrinsic to the atom.
To sum up, regrouping all the previous terms, we get for the transformed Hamil-
tonian:
^
=
^
P
2
2
+
^p
2
2
+Coul+
^ ^
D
^
E(
^
R) +
^
dip (C-33)
This is a form similar to (C-23), with an additional term
^
dip.
Comments
(i)
dierent representations, deduced from one another by a unitary transformation. As an
example, the operator
^
E(
^
R)appearing in (C-31), does not represent the transverse
electric eld in the new point of view, which should be
^
E(
^
R)transformed by
^
, written
as
^^
E(
^
R)
^
, and hence dierent from
^
E(
^
R). Actually, one can show that the operator
^
E(
^
R)represents in the new point of view the physical quantity
^
D(
^
R)0where
^
D(
^
R)
is called theelectric displacement eld(see Complement AIVof [16]).
(ii)
^
dipis given by an integral over, which diverges at inn-
ity. This integral must however be limited to values offor which the long wavelength
approximation is still valid.
C-5. Matrix elements of the interaction Hamiltonian; selection rules
Consider an initial state where the atom is described by
int
infor its internal state,
ext
infor its external state, and where the radiation is in the state
in. The interaction
Hamiltonian (C-31) couples this initial state to a nal state where the atomic internal and
external variables, as well as the radiation variables are respectively in the states
int
n
,
ext
n
, and
n
. As the operator
^
E(
^
R)appearing in (C-31) is a linear superposition of
annihilation^and creation^operators, the matrix element of
^
describes two types
of processes: the absorption processes associated with operator^where one photon
disappears, and the emission processes associated with operator^where a new photon
appears. This matrix element can be factored into a product of three matrix elements
concerning the three types of variables; they are written, for the absorption processes:
~20
3
int
n"
^
D
int
in
ext
nexp(k
^
R)
ext
in n^
in (C-34)
and for the emission processes:
~20
3
int
n"
^
D
int
in
ext
nexp(k
^
R)
ext
in n^
in (C-35)
The central term in these expressions is a matrix element concerning the exter-
nal atomic variables; it expresses the conservation of the global momentum as we now
2014

C. DESCRIPTION OF THE INTERACTIONS
show. Operatorexp(k
^
R)translates the momentum by a quantity~k. If the
atom's center of mass has an initial momentum~Kin, once it absorbs a photon, its nal
momentum will be~Kn=~Kin+~k; the momentum~kof the absorbed photon is
therefore transferred to the atom during the absorption process. In a similar way, one
can show that the atom's momentum decreases by the quantity~kwhen a photon is
emitted.
In the rst matrix element of (C-34), which concerns the internal atomic variables,
operator
^
Dis an odd operator. The matrix element will be dierent from zero only if
the initial and nal internal atomic states have opposite parity, as for instance the1
and2states of the hydrogen atom. We rediscover here a second conservation law, the
conservation of parity. In addition, as the operator
^
Dis a vector operator, it leads to
selection rules on the internal angular momentum which will be studied in Complement
CXIX.
Comments
(i)
expressions (C-34) and (C-35). One may wonder whether this result is only valid for
the approximate form (C-31) of the interaction Hamiltonian used to establish those
equations. Actually it can be shown, using the commutation relations[p(r)] =
~ rand[^^^] =^, that the interaction Hamiltonian
^
1written is (C-4)
(without the long wavelength approximation) commutes with the system total momen-
tum ^p+ ~k^^. The same result is true for all the terms of the interaction
Hamiltonian. Consequently, the exact (without approximation) interaction Hamiltonian
has non-zero matrix elements only between states having the same total momentum. The
fact that the total momentum commutes with all the terms in the Hamiltonian is related
to the system invariance with respect to spatial translation. The properties of the sys-
tem are unchanged upon the translation by the same quantity of the particles and the
elds. Similar considerations apply to the rotational invariance and cause the interaction
Hamiltonian to only connect states with the same total angular momentum. These results
are important for understanding in a simple fashion the exchanges of linear and angular
momenta between atoms and photons, which will be discussed in ComplementsXIXand
CXIX.
(ii)
energy conservation, shows that the energy of the absorbed photon is dierent from the
energy separating the two internal levels involved in the transition. Two eects account
for this dierence: the Doppler eect, and the recoil eect (ComplementXIX); they play
an important role in laser cooling methods.
(iii)
tional terms for the interaction Hamiltonian, describing the interaction between the radia-
tion magnetic eld and the atomic orbital or spin magnetic moments (ComplementXIII,
). Some of these terms have already been written in (C-6). Transitions, called magnetic
dipole transitions, may occur between levels having the same parity, as opposed to the
electric dipole transitions studied above. Other types of transitions may also be observed
at higher orders, such as the quadrupole transitions.
Note nally that, if the initial radiation state already containsphotons, the
last two matrix elements of (C-34) and (C-35) are equal to 1^ =
and
+ 1^ =
+ 1. In the presence ofincident photons, the probability of
the absorption process is thus proportional to, whereas the emission probability is
2015

CHAPTER XIX QUANTIZATION OF ELECTROMAGNETIC RADIATION
proportional to+ 1. We shall see in Chapter
existence of two types of emission, the stimulated emission and the spontaneous emission.
With the knowledge of the various Hamiltonians
^
,
^
and
^
, as well as their
matrix elements, we can now solve Schrödinger's equation to compute the transition
amplitude between an initial state and a nal state of the system atom + eld. This
will be done in the next chapter, where we study various processes, such as the absorp-
tion or emission of photons for an incident radiation either monochromatic or having a
large spectral band, the photoionization phenomenon, multiphoton processes and photon
scattering.
2016

COMPLEMENTS OF CHAPTER XIX, READER'S GUIDE
The processes of photon absorption and emission by atoms must obey conservation laws for the
total linear or angular momentum. This has implications that are highlighted in the three complements
of this chapter. When an atom absorbs (or emits) a photon, it gains (or loses) an energy, a momentum
and an angular momentum equal to the energy, momentum and angular momentum of the photon. This
allows manipulating several properties of the atoms. It is, for example, at the base of optical pumping
and laser cooling methods.
AXIX: MOMENTUM EXCHANGE BETWEEN
ATOMS AND PHOTONS
Momentum exchange between atoms and photons
plays an important role in determining, for
example, the Doppler eect and the shape of the
spectral lines emitted or absorbed by gases. As an
atom continually absorbs and re-emits photons,
its momentum can be greatly aected. The atom
can be decelerated, which allows slowing down
and even bringing to rest an atomic beam over
short distances. Other uses of the Doppler eect
include the introduction of a friction force that
slows down atoms to form ultra-cold gases. When
the atoms are conned in a trap, the nature of the
Doppler shift may be greatly changed and even
completely disappear (Mössbauer eect). Muli-
tiphoton processes, in which the total momentum
of the absorbed photons is zero, are also discussed.
BXIX: ANGULAR MOMENTUM OF RADIATION This complement is more technical than the
previous one. It shows how the photon can
be seen as a spin1particle; this particle also
has an orbital angular momentum. The photon
thus possesses two types of angular momentum.
These concepts are useful for reading the next
complement.
CXIX: ANGULAR MOMENTUM EXCHANGE
BETWEEN ATOMS AND PHOTONS
This complement studies the exchanges between
the photon spin angular momentum (related to
the light beam polarization) and the internal
degrees of freedom of the atoms. These exchanges
obey selection rules for the atomic transitions.
Such rules are essential for many experimental
methods in atomic physics, such as optical
pumping where a polarized light beam allows
accumulating the atoms of a gas in specic
Zeeman sublevels. A number of applications
of these methods are briey reviewed in this
complement.
2017

MOMENTUM EXCHANGE BETWEEN ATOMS AND PHOTONS
Complement AXIX
Momentum exchange between atoms and photons
1 Recoil of a free atom absorbing or emitting a photon
1-a Conservation laws
1-b Doppler eect, Doppler width
1-c Recoil energy
1-d Radiation pressure force in a plane wave
2 Applications of the radiation pressure force: slowing and
cooling atoms
2-a Deceleration of an atomic beam
2-b Doppler laser cooling of free atoms
2-c Magneto-optical trap
3 Blocking recoil through spatial connement
3-a Assumptions concerning the external trapping potential
3-b Intensities of the vibrational lines
3-c Eect of the connement on the absorption and emission spectra2038
3-d Case of a one-dimensional harmonic potential
3-e Mössbauer eect
4 Recoil suppression in certain multi-photon processes
We studied, in , the matrix elements of the interaction Hamil-
tonian between an atom and a eld. This led us to establish selection rules based on the
conservation of the total momentum of the system atom + eld during the absorption
or emission of photons by the atom. The study in Chapter
emission amplitudes will show that the global energy of the system is also conserved dur-
ing these processes. The goal of this complement is to show how these conservation laws
1
provide an interesting view on many aspects of momentum exchange between atoms and
photons.
We start in
to any external potential. We shall establish the expressions for the Doppler shift and
the recoil energy that appear in the equation yielding the frequency of the absorbed or
emitted photons. In a gas containing a large number of atoms with dierent velocities,
the dispersion of these velocities yields a Doppler broadening of the emission and absorp-
tion lines. This broadening, as well as the shift related to the recoil energy, introduce
perturbations in the lines observed using high resolution spectroscopy, hence limiting its
precision. In addition, when the atom constantly absorbs and re-emits lots of photons, its
momentum change per unit time can become very large. This results in a force generated
1
The rst study along this line concerned the Compton eect (1922), where the scattering of a photon
by an electron is considered as a collision between two particles. Assigning the photon an energy
and a momentum}k, withk= 2 , one writes the conservation equations for the total momentum
and energy during the collision. This yields the change of frequency of the photon as it is scattered in a
given direction, in complete agreement with experimental observations.
2019

COMPLEMENT A XIX
by the radiation pressure. We shall calculate the order of magnitude of that force and
show that it can produce an acceleration or deceleration of the atom a hundred thousand
times larger than the one due to gravity.
In , we will show that this force is able to slow down and immobilize a beam
of atoms propagating in the direction opposed to that of the light beam. The velocity
dependence of the radiation pressure force, due to the Doppler eect, is also very interest-
ing. It allows, using two light beams in opposite directions, but with the same intensity
and frequency, to generate a friction force on the atom, provided the light frequency is
lower than the atomic frequency: the radiation pressure force is zero when the atomic
velocityis zero, but opposite towhen it is dierent from zero, therefore producing a
damping of that velocity. This is the principle of one of the rst laser cooling mechanisms
observed experimentally. The very low temperatures obtained, millions of times lower
than room temperature, explain the increasing number of application of the ultra-cold
atoms thus obtained. We also explain in
which involves a position dependence of the radiation pressure force.
We describe in
suppress or circumvent the eect related to the recoil. Conning the atom in a trap,
as studied in , may prevent the atom's recoil if the trapping is strong enough. If
the transition is multi-photonic, for example if two photons having the same energy but
opposite momenta are absorbed in the transition, no recoil is experienced by the atom,
and there is no Doppler shift. An important example of this method is the Doppler-free
two-photon spectroscopy ().
1. Recoil of a free atom absorbing or emitting a photon
Consider rst an atom that is not subjected to any external potential (free atom). The
Hamiltonian
^
extfor the external variables is reduced to the kinetic energy term
^
P
2
2,
where
^
Pis the momentum of the center of mass andthe total mass of the atom. The
eigenstates of
^
extcan be chosen as states having well dened momentumPand energy
2
2.
1-a. Conservation laws
Let us express the radiation eld as a function of plane waves of wave vectorkand
frequency=. The eigenstates of the radiation Hamiltonian
^
can be described in
terms of photons having an energy~and a momentum~k. The interaction Hamilto-
nian
^
studied in
2
under spatial translation of the total
system atom + eld. Consequently, it commutes with the system's total momentum,
and can induce transitions only between states having the same total momentum. Fur-
thermore, the transition amplitude associated with an interaction lasting a timecan
only connect states of the total system having the same total energy, within~(this
point will be further discussed in the next chapter). These two conservation laws can be
used to study the inuence of the motion of the center of mass on the frequencies of the
photons it can absorb or emit.
2
The interaction Hamiltonian involves eld values at points where particles are located. It is therefore
invariant when both elds and particles are shifted (by the same quantity).
2020

MOMENTUM EXCHANGE BETWEEN ATOMS AND PHOTONS
Consider rst an absorption process where the atom goes from an internal state
to another internal stateby absorbing a photon of energy~and momentum~k. We
shall note:
~0= (1)
the energy dierence between the two internal states. The initial (nal) momentum of the
center of mass before (after) the photon absorption is notedPin(Pn). The conservation
of the total momentum leads to:
Pn=Pin+~k (2)
The total energy in the initial state is the sum of the photon energy, the internal energy
of the atom and its translation energy
2
in2:
in=~++
2
in
2
(3)
The total energy in the nal state reduces to the atom's energy since the photon has
been absorbed:
n=+
2
n
2
(4)
The conservation of the total energy means thatin=n. Using (1) and (2), this last
relation reads:
~=~0+
~kPin
+
~
22
2
(5)
The last two terms in (5) represent the variation of the external energy of the atom
during the transition i.e. the variation of the atom's center of mass kinetic energy
between the nal state where it is equal to(Pin+~k)
2
2, and the initial state, where
it is equal toP
2
in2. This equation can be rewritten as:
=0+kvin+
rec
~
(6)
wherevin=Pinis the initial velocity of the center of mass, and where:
rec=
~
22
2
=~R (7)
is therecoil energy; it is the energy an atom, initially at rest, will acquire upon the
absorption of a photon having a momentum~.
The same type of calculation holds for the emission process during which an atom,
whose center of mass has an initial momentumPin, goes from the internal stateto the
internal stateby emitting a photon of energy~and momentum~k. Equation (6)
must now be replaced by:
=0+kvin
rec
~
(8)
where only one sign is changed with respect to (6).
2021

COMPLEMENT A XIX
1-b. Doppler eect, Doppler width
The termkvinin equations (6) and (8) is simply the Doppler shift, due to the
motion of the atom, of the frequencies it absorbs or emits. Setting:
0= 2( 0) = 2 and = 2
(9)
we get for the frequency variationdue to the Doppler eect:

=
vin
(10)
where=kis the unit vector dening the propagation direction of the photon. Note
that a radiation quantum theory is not needed to account for this frequency shift, which
can be predicted by a classical theory. This was to be expected since, among the last
two terms of (6) and (8),kvinis the only one that does not go to zero when~tends
toward zero, as opposed torec~=~
2
2(which is proportional to~).
For an ensemble of atoms in a dilute gas at thermal equilibrium at temperature,
the velocities are distributed according to the Maxwell-Boltzmann law, and the velocity
dispersionis of the order of
, whereis the Boltzmann constant. The
shiftsof the frequencies emitted or absorbed by the atoms are distributed following
a Gaussian curve
3
whose width(standard deviation of the frequency distribution,
equal to the square root of the variance), called theDoppler width, is given by:

0
=
2
(11)
In general, in the optical domain and for temperatures around 300 K, the Doppler
widthis of the order of 1GHz=10
9
Hz, much smaller than the frequency0(of the
order of 10
15
Hz), but much larger than the natural width, of the order of 10
7
Hz. In
this domain, the resolution of spectroscopic measurements of line frequencies emitted by
a dilute gas is generally limited by the Doppler broadening of the lines.
Relativistic Doppler eect
The previous calculations are only valid in the non-relativistic limit (). The Doppler
shift expression can be generalized to any value ofby noting that the four quantities
are the four components of a four-vector. Let us assume the atom is
at rest in a reference frameand emits a photon of frequencyalong theaxis (we
ignore here the recoil energy). An observer, in a reference framemoving with velocity
along theaxis, sees the atom moving away at velocityand measures a frequency
for the photon emitted by the atom. According to the relativistic expressions for the
transformations of four-vector components, we have:
=
1
22
(12)
To rst order in, replacingkvinby , we again nd the Doppler shift of equation
(8) the relativistic correction being (in relative value) of the order of
22
for .
3
Actually, this distribution is the convolution of a Gaussian and a curve of width, whereis the
natural width (due to the spontaneous emission) of the line emitted or absorbed by the atoms. For a
gas at room temperature,is much smaller than the Doppler width.
2022

MOMENTUM EXCHANGE BETWEEN ATOMS AND PHOTONS
1-c. Recoil energy
Imagine the atom is initially at rest, so that the terms inkvinin (6) and (8)
are zero. When the atom absorbs a photon, its momentum increases by a quantity~k
equal to the momentum of the absorbed photon. Consequently, the atom recoils with
a velocityrec=~ in the direction of the incident photon, and its kinetic energy
becomesn=
2
rec2 =rec. The energy~of the incident photon is used both to
increase the internal energy of the atom by~0(since the atom goes fromto), and
to increase its kinetic energy from0torec. We then have~=~0+rec, which is a
particular case of (6) for a zero initial velocity. However, for the emission process where
the atom goes fromtoby emitting a photon, this relation is modied. As the photon
leaves the atom with a momentum~k, the atom recoils with the same momentum but in
the opposite direction, and acquires a kinetic energyrec. The loss of internal energy of
the atom, equal to~0, must now be used both to increase the radiation energy by~
(the energy of the emitted photon), and to increase the kinetic energy of the atom from
zero torec. We then have~0=~+rec, that is~=~0 rec, which coincides
with (8) with the minus sign on the right-hand side.!D
Emission Absorption
!!0
!0!R !0+!R
Figure 1: Because of the recoil eect of the atom, the absorption and emission lines do
not coincide, but form a doublet called therecoil doublet; they are centered at+ for
the absorption line and for the emission line. In a gas, their width is the Doppler
width.
The recoil of the atom with a kinetic energyreccauses the centers of the ab-
sorption and emission lines to be dierent (Figure); their position is0+ for the
absorption line and0 for the emission line, with= rec~. For a gas con-
taining a large number of moving atoms, the velocity dispersion around a zero average
gives each of these two lines a Doppler width. In the optical domain, the recoil
frequencyis of the order of a few kHz, much smaller than the Doppler width at room
temperature, and also smaller than the natural width. Consequently the recoil doublet
shown in Figure
less than their width. However, when one studies the lines emitted by nuclei in the X-ray
or-ray domains, the recoil frequency (which increases as
2
) becomes comparable to or
2023

COMPLEMENT A XIX
even larger than the Doppler width (which only increases as). The two lines in Figure
1
in an excited statehas very little chance of being absorbed by another identical nucleus
in the lower state. We shall see later how the recoil of a nucleus can be blocked when
the atom having that nucleus is suciently strongly bound to other atoms in a crystal
(Mössbauer eect).
1-d. Radiation pressure force in a plane wave
Each time an atom absorbs a photon, it gains a momentum~k. If
_
absis the
number of absorptions per unit time, the atom gains a momentum
_
abs~k, per unit
time. In a steady state, the number
_
absof absorptions per unit time is equal to the
number of emissions per unit time
_
em. This latter number is equal to, where
is the natural width of the atom's excited state, and where the diagonal elementof
the atom's density operator inis the occupation probability of that state. This gain
in momentum per unit time of the atom can be considered as coming from the action of
a force, associated with the radiation pressure exerted by the light beam on the atom.
One often calls this force the radiation pressure force. According to what we just saw,
it is equal to
4
:
F=
_
abs~k= ~k (13)
To get an idea of the order of magnitude of this force, let us assume the light
intensity is very high; the atomic transition is then saturated, meaning the occupation
probabilitiesand of the higher leveland the lower levelare both equal to12.
We then have:
F=~k

2
(14)
Such a force can communicate an accelerationAequal toF to the atom of mass.
Taking (14) into account, this acceleration is equal to:
A=
~k

2
=
vrec
2
(15)
wherevrec=~kis the recoil velocity of an atom absorbing or emitting a photon, and
= 1is the radiative lifetime of the excited state.
Let us calculate the value of this acceleration for a sodium atom. The recoil velocity
is of the order of310
2
msand the radiative lifetime of the order of16210
9
s,
meaning the accelerationis of the order of10
6
ms
2
, which is10
5
times larger (!)
than the acceleration due to gravity (of the order of10ms
2
). This high value for the
accelerationarises because the velocity change of the atomrecfor each absorption-
emission cycle, though very small by itself, accumulates during the very large number of
cycles,12, occurring in one second.
4
To compute the momentum change, we only considered here the photon absorption processes.
The spontaneous emission processes also change the atom's momentum, as the atom recoils when it
emits a photon. However, the spontaneously emitted photons can have any direction in space, and the
momentum change of the atom has a zero average. This phenomenon, however, gives rise to a diusion
of momentum, hence increasing the velocity dispersion of the atoms. We shall see in that this
momentum diusion must be taken into account when evaluating the limits of laser cooling.
2024

MOMENTUM EXCHANGE BETWEEN ATOMS AND PHOTONS
2. Applications of the radiation pressure force: slowing and cooling atoms
We now study three applications of the radiation pressure force using either one laser
beam, or two laser beams with the same intensity and same frequency, propagating
in opposite directions along theaxis. We shall see in
laser beam, the radiation pressure force exerted by the beam on the atoms in an atomic
beam propagating in the opposite direction can be used to slow down, and even bring to
rest, the atoms of the beam. With two laser beams propagating in opposite directions,
interesting eects occur if one can introduce an imbalance between the two radiation
pressure forces, depending either on the atom's velocityalong theaxis, or on its
position. We study in
of the two forces exerted by the two waves, zero for= 0, becomes, for= 0but
suciently small, a linear function of; it can then be expressed as. For a proper
detuning of the lasers' frequency, the coecientcan be negative, so that the resulting
forceacts as a friction force that damps the velocities of all the atoms in the beam,
and hence cools them down. This is the principle of laser cooling. In , we shall see
how a position dependence can be obtained, and how the resulting force, zero for= 0,
becomes dierent from zero if= 0and equal toifis suciently small. Ifis
negative, this force becomes a restoring force that can trap the atoms around= 0.
This is the principle of the magneto-optical trap.
2-a. Deceleration of an atomic beam
Imagine that an atomic beam is irradiated by a resonant laser beam propagating
in the direction opposite to that of the beam. Due to the radiation pressure force, the
atoms of the beam will slow down. Is it possible to bring them to rest? It is important
to notice that even if the laser beam is initially resonant, it will not be when the atoms'
velocities change, since the Doppler eect takes the atoms out of resonance; this will
signicantly lower the radiation pressure force, and hence the slowing down eect. For
the sake of simplicity, we shall rst ignore the Doppler eect following the change in the
atoms' velocities; we shall see later how it can actually be circumvented.
We rst assume the radiation pressure force does not change in the course of the
deceleration process, and that the laser intensity is high enough for the atomic transition
to be saturated; we can then use the orders of magnitude calculated in the previous
paragraph. We found that for sodium atoms, the decelerationis of the order of
10
6
ms
2
. If the atoms of the beam have an initial velocity of the order of10
3
ms, their
velocity will be zero after a timeof the order of10
3
s, after they traveled a distance
2
2of the order of05m, which shows that the size of such an experiment is not, a
priori, excessive.
To solve the problem of the atoms going out of resonance because of the Doppler
eect which changes in the course of the slowing down, an ingenious method was pro-
posed and demonstrated [18]. It is based on the propagation of the atoms in a spatially
inhomogeneous magnetic eld. More precisely, the atomic beam travels along the axis of
a spatially varying solenoid coil (Figure). The magnetic eld produced by the solenoid
is parallel to the beam direction, and its intensity varies along the beam axis. As an
atom propagates along this eld, it undergoes a variable Zeeman shift of its resonance
frequency. One can then adjust the prole of the eld so that as the atom is slowed
down, the Zeeman shift of the atomic frequency balances the Doppler shift of the laser
2025

COMPLEMENT A XIX Atomic
beam
Solenoidwithvaryingdiameter
Laser
Figure 2: Schematic diagram of a Zeeman slower. The atomic beam is cooled by a laser
beam propagating in the opposite direction. It travels along the axis of a solenoid composed
of a set of magnetic coils with decreasing diameters, whose cross sections are shown in the
gure (the current in the coils ows perpendicularly to the gure). While they propagate,
the atoms are submitted to a larger and larger magnetic eld. The Zeeman shift of their
resonance frequency can thus follow the Doppler shift of the apparent laser frequency in
their own reference frame. Consequently, instead of going o resonance, they can be
slowed down during their entire propagation through the solenoid, and even come to rest.
frequency: at each point, the eld is calculated so that both Doppler and Zeeman shifts
exactly balance each other. This type of apparatus is called a Zeeman slower.
2-b. Doppler laser cooling of free atoms
. Doppler laser cooling principle
The slower described above concerns the mean velocity of atoms, which can be
brought down to zero. However, the root mean square of the atomic velocities remains
non-zero, as does the temperature which is characterized not by the mean velocity but
by its root mean square. We now describe a method using the velocity dependence of
the radiation pressure force, due to the Doppler eect, and which permits reducing the
dispersion of the atomic velocities around their mean value, and hence really cooling
down the atoms. As this method uses the Doppler eect, it is called Doppler laser
cooling. It was proposed in 1975 for free atoms [19] and for trapped ions [20]. Here, we
shall only study the case of free atoms.
The idea is to use two laser waves 1 and 2, having the same angular frequency
and the same intensity, counterpropagating along theaxis, wave 1 toward negative
, and wave 2 toward positive(Figure). Imagine an atom also propagating along the
axis with, for example, a positive velocity. We call0the angular frequency of the
atomic transition excited by the laser. We assume the lasers are red-detuned,
meaning that:
0 (16)
In the reference frame where the atom is at rest, the apparent frequency of wave(with
= 12) is Doppler shifted, and equal tokiv. Asis positive, the atom and wave
1 propagate in opposite directions, so thatk1vis negative. The apparent frequency
2026

MOMENTUM EXCHANGE BETWEEN ATOMS AND PHOTONS
Figure 3: Principle of Doppler laser cooling. An atom moving with velocityalong the
axis interacts with two laser waves 1 and 2 having the same angular frequency;
this frequency is red-detuned, meaning0. The two waves have the same intensity
but propagate in opposite directions along theaxis. The direction of the velocityis
opposite to that of wave 1. In the atom's frame of reference and because of the Doppler
eect, the frequency of wave 1 is shifted closer to resonance, whereas the frequency of
wave 2 is shifted away from it. Consequently, the modulus of force1exerted by wave 1
increases, whereas that of the force2exerted by wave 2 decreases. The resulting force,
zero for= 0, is opposed towhen= 0, and proportional tofor small enough values
of. It is therefore equivalent to a friction force.
kvof wave 1 is therefore increased. The Doppler eect brings the apparent frequency
of wave 1 and the atomic resonance frequency closer together; consequently, the modulus
of the radiation pressure force1exerted by wave 1 on the atom and directed, like wave
1, towards the negative, increases with respect to its value for= 0. The conclusions
are just the opposite for wave 2, whose frequency is shifted away from resonance by the
Doppler eect; the force2it creates along the positiveis weaker than its value for
= 0.
The sum of the two forces
5
is zero for= 0(both forces have the same modulus
and opposite directions), but no longer zero when= 0. For positive values of, it has
the same direction as1since the modulus of1is larger than that of2(Figure);
for negative values of, it is just the opposite since the two waves have switched their
roles. The sum=1+2of the two radiation pressure forces exerted by the two
waves is thus in the direction opposite to that of the velocity. For small values of, it
is proportional toand can be written:
= (17)
whereis a positive friction coecient.
Under the eect of this friction force, the atomic velocities are constantly reduced.
Their dispersion, however, does not tend towards zero for long interaction times, because
of the unavoidable uctuations in the emission and absorption processes. There is actu-
ally a competition between the friction eect we just described, which tends to cool down
the atoms, and the momentum diusion that tends to heat them up. We evaluate in the
next paragraphs the eect of these two processes to estimate the order of magnitude of
the temperatures that can be obtained by Doppler laser cooling.
5
We shall see below that the eects of interference between the two waves can be neglected.
2027

COMPLEMENT A XIX
. Estimation of the friction coecient
We will now seek an estimation of the friction coecient, in order to calculate
the evolution of the momentum of the atom, as well as of its energy. In a rst step
and for the sake of simplicity, we will limit ourselves to a calculation of the average
eect of the photon absorption and emission cycles by the atom. This will allow us to
determine the evolution of its average momentum
. Nevertheless, the absorption and
emission processes of photons by an atom are actually uctuating processes, as we will
discuss below. Ssuch a calculation is theferore not sucient to obtain the evolution of
the average
2
of the square of the momentum, that is of the average kinetic energy,
since the average value of a square diers in general from the square of the average value.
In a second step, we will use a calculation that takes the uctuations of the momentum
transfers betwee photons and atoms into account.
We set:
= 0+ (18)
where~is the recoil energy dened in (7);is the detuning between the laser frequency
and the atomic frequency0 . We assume from now on that the intensity of the
two lasers is weak; the atomic transition is not saturated and consequently the population
of the excited levelremains low. Its variation as a function of the detuningfollows
a Lorentzian curve with a total width at half maximum equal to:
() =(0)
(2)
2
[(2)
2
+
2
]
(19)
We shall not need the expression for(0), since it cancels out in the expressions for the
friction and diusion coecients we shall obtain; it can however be found in Chapter V
of reference [21] (optical Bloch equations).
In a perturbative approach to the problem, two types of terms must be considered:
the square terms, coming from the interaction between the dipole induced by wave
with that same wave; the cross terms coming from the interaction of the dipole
induced by wavewith the other wave=. The cross terms correspond to interference
between the eects of the two waves. However, as these two waves do not have the same
spatial dependence (they propagate in opposite directions), these interference eects vary
rapidly in space asexp(2). They consequently vanish when the forces are averaged
over distances of the order of the laser wavelengths, as we shall assume. It is then possible
to consider that the radiation pressure force acting on the atom is simply the sum of the
radiation pressure forces exerted by each wave, in the absence of the other.
When the atom is at rest, the two waves have the same frequency in the atom's
reference frame, and hence the same detuning; remember thatis supposed to be
negative in a laser cooling experiment see (16). If the atom moves with a velocity
0, we just saw that the apparent frequency of wave 1 is increased by a quantity
(whereandare positive), so that the detuning for the interaction of the atom with
wave 1 is equal to:
1=+ (20)
while it is2= for wave 2. For this atom, the population of the excited state
is the sum of two contributions: that of wave 1, obtained by replacingwith+in
2028

MOMENTUM EXCHANGE BETWEEN ATOMS AND PHOTONS
Figure 4: Populationsbb(1)andbb(2)excited in the upper stateby wave 1, with
detuning1=+, and wave2, with detuning2= . As the detuningfor= 0
is assumed to be negative, the ordinatebb(1)of pointis higher than the ordinate
bb(2)of point.
expression (19), and which is proportional to the ordinate of pointwhose abscissa is
1=+in Figure; that of wave 2, which requires to replacewith in (19),
and which is therefore proportional to the the ordinate of pointhaving the abscissa
in that same gure. Computations similar to those of
total forceacting on the atom, as the sum of the two forces exerted by each wave,
added independently of each other:
=~(+) + ~( ) (21)
where()is the function dened on the right-hand side of (19).
When is small compared to the widthof the curve in Figure, one can expand
( )to rst order inand obtain:
=2~
d
d
() (22)
The last factor on the right-hand side of (22), which is the slope at point(with abscissa
) of the curve representing(), can be computed from (19). For the point=2
where the slope is maximum, we nd:
d
d
() =
2()

(23)
Inserting (23) into (22), we get:
= (24)
where the friction coecientis given by:
= 4~
2
() (25)
2029

COMPLEMENT A XIX
Using equation (24), we can also compute the damping of the momentum
and of
its square. As= d
dand=, we have:
d
d
= (26)
We can also write:
d
d
2
= 2
d
d
=22 (27)
Remember, however, that the average value of the kinetic energy is proportional
to
2
, the average value of the square of the momentum, and not to the square of its
average value
2
. We should therefore not conclude from relation (27) that the ultimate
temperature that can be reached by Doppler laser cooling whenis zero. Equa-
tion (27) was obtained by considering only the average eect on the atom's velocity of
the light beams and of the successive spontaneous emission processes, which introduces
a continuous average evolution. But, in reality, the absorption and emission processes
uctuate and yield photons emitted in random directions. Even though their total mo-
mentum has a zero average (meaning
is not aected), these uctuations will change2
. This eect can be considered as a source of noise (also called momentum diusion)
that increases
2
, and acts in the direction opposite to the friction. It is therefore the
competition between these two opposed mechanisms that leads to an equilibrium state,
whose energy
2
2determines the ultimate temperature that can be obtained.
. Momentum diusion
We now study the diusion of the momentum of one atom, due to the spontaneously
emitted photons; the ensemble of particles discussed above then reduces to one single
atom (=). Let us consider a time interval dwhose value will be specied later. We
call d1and d2the photon numbers from waves 1 and 2 that are absorbed during that
time interval. As we assume the friction has acted long enough to cancel the average
velocity, the detuninghas become the same for the two beams. We then have, on the
average:
d
1=d2=d (28)
Each absorbed photon then yields a spontaneously emitted photon. We use here a simple
one-dimension model: each photon is emitted spontaneously in a random way, either in
the positivedirection (the atom's recoil is then negative), or in the negativedirection
(the recoil is then positive). The variation of the atom's momentum is then}, with
=1in the rst case and= +1in the second. There are no correlations in the
directions of two consecutive photons. The total momentum dgained by the atom
during the absorption and re-emission of d1+d2= 2dphotons is equal to the sum
of the momentum gained by the absorption of photons from beam 1, of the momentum
gained by the absorption from beam 2, and nally of the momentum coming from the
spontaneous emissions:
d
=~d2d1+
2d
=1
(29)
2030

MOMENTUM EXCHANGE BETWEEN ATOMS AND PHOTONS
(i) To begin with, we neglect the uctuations of the number of photons absorbed
from each wave (we shall discuss this point later). The numbers d1and d2are then
equal to their average values, and we have:
d
=~
2d
=1
(30)
The summation overin (29) yields a zero contribution to d
, since the sum of the
is zero on the average, but this is not the case for the contribution to d
2
that we now
compute. Taking the square of the summation overin (30), the cross terms
with
=are zero on the average, because there are no correlations between the signs of the
and the. We are left with the square terms
2
which are all equal to1. We obtain:
d
2
= 2d~
22
(31)
which is obviously non-zero.
Let us specify the time intervald, which should be long enough for the numberdN
of absorptions and spontaneous emissions to be large during this time, but suciently
small for the variations ofto stay negligible. We can then write:
d=
_
d= ()d (32)
where we used in the last equality the fact that the average number of photons absorbed
per unit time in each wave is equal to(), as we have seen above. Inserting this
result in (31), we get the increase of
2
during the time intervald:
d
2
= 2~
22
d= 2~
22
()d (33)
We nally get:
d
2
d
sp
= 2~
22
() =sp (34)
The subscript sp of the parenthesis reminds us that this increase of
2
per unit time
is due to spontaneous emission processes. This expression is often called the diusion
coecientsp.
(ii) We now study the eect of uctuations on the numbers d1and d2of ab-
sorbed photons in each wave during the time interval d. If these uctuations are no
longer neglected, we must write:
d=d
+ =d+ (35)
with= 12. In this equation,is the uctuation of the number of absorbed photons
in wave(by denition, the average value ofis zero). The total momentum dthe
atom receives during the absorption of these photons is equal to:
d=~(d2d1) =~(2 1) (36)
Since the average values of1and 2are zero, the uctuations in the absorption
process do not change the average value of
, but this is not true for the average value
2031

COMPLEMENT A XIX
2
. Taking the square of (36) and using the fact that there are no correlations between
the uctuations of1and 2(and hence the average value of their product is zero),
we get:
d
2
=
2
1
+
2
2
~
22
(37)
To compute
2
, we take the square of equation (35). This leads to:
d
2
=d
2
+ 2d +
2
(38)
We now take the average value of each side of this equation. Using the fact that the
average value ofis zero, we get:
2
=d
2
d
2
(39)
The quantity d
2
d
2
is simply the variance of the number of photons absorbed from
the wave. In general, for Poisson statistics, we have
6
:
d
2
d
2
=d=d (40)
We deduce that
2
is simply equal to d, so that equation (37) is simply written:
d
2
= 2d~
22
(41)
We therefore obtain for the increase of
2
due to uctuations in the absorption processes
a result identical to (31). The computations leading from (31) to (34) can be repeated
and we get the following result for the increase, per unit time, of
2
due to the absorption
processes:
d
2
d
abs
= 2~
22
() =abs (42)
where the diusion coecientabsdue to the absorption processes has the same value
asspfor the spontaneous emission processes:
abs=sp= 2~
22
() (43)
To evaluate the global rate of variation of
2
, we must add to the rates of variation
(34) and (42) the one due to the cooling process(d
2
d)cool. This variation is simply
the variation of
2
in the absence of uctuations during the momentum exchanges, so
that it can be obtained by assuming that
2
and
2
are simply equal. Using (27), we can
write this variation as:
d
2
d
cool
=2
2
(44)
6
One could evaluate the eects of the deviations from Poisson statistics, but it will not be done here
to keep things simple; this is legitimate for the low laser intensities (unsaturated transition) we assumed
here.
2032

MOMENTUM EXCHANGE BETWEEN ATOMS AND PHOTONS
Adding (34), (42) and (44), we nally obtain a total rate of change equal to:
d
2
d
tot
=2
2
+sp+abs (45)
This rate of change goes to zero when :
2
= (sp+abs)2
(46)
Dividing both sides of this equation by2and using expressions (25) and (43) for the
friction and diusion coecients, we nally get
7
the expression for the average kinetic
energy in the stationary regime:
2
2
=
~
4
(47)
This result indicates that the ultimate temperature is directly related to the natural
width of the excited level.
. Doppler temperature
In a Doppler laser cooling experiment, and for a one-dimensional treatment of the
problem, the average kinetic energy of the velocity uctuations around its average value
is related to the Doppler temperature:
2
2
=
2
(48)
whereis the Boltzmann constant. Using (47) and (48), we nd that the equilibrium
temperature that can be reached by Doppler laser cooling is given by:
=
~
2
(49)
For sodium atoms, this temperature is equal to23510
6
K, that is10
6
times lower
than room temperature (of the order of300K)!
Our treatment of Doppler laser cooling is based on several approximations, as for
example the one-dimensional treatment and the simplied description of spontaneous
emission occurring only in two opposite directions. Nevertheless, more precise calcula-
tions lead to the conclusion that the average kinetic energy reached in a stationary regime
is indeed of the order of~, within numerical coecients of a few units.
Finally, equation (47) yields an estimation of the velocityof the cold atoms
thus obtained:

~ (50)
This means that the Doppler shiftsof the apparent frequencies of waves1and2
are such that the separations of the dotted lines in Figure
~.
7
As the diusion and friction coecients are all proportional to(), this factor disappears from
(46) and is no longer present in (47).
2033

COMPLEMENT A XIX
We can compare these Doppler shifts with the widthof the curve plotted in Figure
and obtain:
~~
2
rec~
(51)
whererecis the recoil energy given in (7). The ratiorec~is in general small for the
atomic resonance lines used in laser cooling; it is of the order of1100for sodium, which
means thatremains small compared toand shows the validity of the limited series
expansion used in equation (22).
Other laser cooling methods
Until now we have described laser cooling methods using only the Doppler eect. Other
methods have been proposed and demonstrated, such as Sisyphus cooling (
plementXXand [22]), the subrecoil cooling and evaporative cooling [23]. The reader
interested in the two latter methods may read for instance 13.3 of [24].
2-c. Magneto-optical trap
We wish to introduce an imbalance, depending on the atom's position, between
two laser beams1and2, with the same frequency but propagating in opposite directions
along theaxis. This requires achieving detunings (between the lasers' frequency
and the atomic frequency) that depend on the positionof the atom along the
axis; these detunings must be the same for both lasers when= 0, and dierent when
= 0. We must then necessarily use an atom with several Zeeman sublevels and dierent
polarizations for the two beams1and2. The principle of the method, suggested for the
rst time by Jean Dalibard in 1986, is schematized in Figure. We assume the atomic
transition is between a ground state with a zero angular momentum (= 0) and an
excited state with an angular momentum equal to1(= 1). The solid lines in Figure
5 +1,0and 1of the excited state,
and of the sublevel0of ground state, in an inhomegeneous magnetic eldapplied
along theaxis. This eld is zero at= 0and varies linearly witharound= 0;
it can be created, for example, by two circular coils having the same axis, placed
symmetrically with respect to= 0, and carrying currents of opposite directions. The
quantization axis is chosen alongand allows dening the magnetic quantum numbers
and of the sublevels in the excited and ground states. The two laser waves1and
2propagate in opposite directions and have opposite circular polarizations with respect
to the quantization axis
8
. Wave1, with polarization+, excites the transition0 +1
whereas wave2, with polarization, excites the transition0 1. The energy~0
of the zero-eld atomic transition is equal to the dierence between the energies of states
0and0(solid horizontal lines in the gure). The detuning~=~~0between the
energy~of the laser photons and that of the zero-eld atomic transition is shown as
the dierence in height between the dashed and solid horizontal lines in the gure. We
8
One generally denes the right-hand and left-hand circular polarization with respect to the prop-
agation direction of the photon. In that case, the polarization of the photons of wave1and wave2
in Figure +(in both cases the electric eld of the wave turns around the direction of
propagation following the right-hand rule). When studying the selection rules of the various transitions
resulting from the conservation of spin angular momentum (see ComplementsXIXandXIX), it is
best to dene both the quantum numbers and the polarizations of all the beams with respect to the
same axis, chosen here to be the quantization axis.
2034

MOMENTUM EXCHANGE BETWEEN ATOMS AND PHOTONS
assume here that the detuning is negative (the laser's frequencyis shifted towards the
red of the atomic transition0).
At point= 0, the energies of states+1and 1are equal, as are the detunings
of waves1and2which excite the transitions0 +1and0 1. The radiation
pressure forces exerted by the two waves are equal in intensity and opposite in direction,
so that the resulting force is zero. This balance does not hold as soon as we move away
from= 0. For example, at=+, wave1which excites transition0 +1, is at
resonance with this transition and the force it exerts is at its maximum. On the other
hand, at that same point, wave2, which excites transition0 1, is way o-resonance
and hence exerts a much weaker force. Consequently, the balance between the two waves
is broken in favor of wave1and the resulting force exerted by both waves is non-zero
and directed towards the right. The conclusions are inverted at point=, where
the total force is non-zero and directed towards the left. We obtain a restoring force,
proportional toin the vicinity of= 0, which traps the atoms around= 0. Such a
trap is called a magneto-optical trap or MOT.
For the sake of simplicity, we only considered a one-dimensional model, but the
extension to three dimensions is possible. Note in particular that the eld created by
Figure 5: Principle of the magneto-optical trap. The transition used is a= 0 = 1
transition, excited by two laser waves1and2propagating in opposite directions along
theaxis, with polarizations (+) and () with respect to the quantization axis.
A magnetic eld gradient is applied along theaxis, the magnetic eld being equal
to zero at= 0. Wave1, which excites the transition0 +1, is resonant for this
transition at point=+and the radiation pressure force it exerts on the atoms is then
maximal. At that point, wave2, which excites the transition0 , is o-resonance for
the corresponding atomic frequency, and, consequently, exerts a much weaker force. The
two forces are unbalanced, in favor of the wave1force. The resulting force is non-zero,
directed towards0. The conclusions are reversed at point=, where the resulting
force is non-zero, directed towards0. Finally at= 0both waves are o-resonance
by the same amount, and the resulting force is zero. We obtain a restoring force that
traps the atoms around= 0.
2035

COMPLEMENT A XIX
two circular coils centered around theaxis, placed on each side of point= 0and
carrying currents in opposite directions, is zero at= 0and exhibits non-zero gradients
along theand axis. The detuning towards the red of the laser frequency with
respect to the atomic frequency also has the advantage of providing a Doppler laser
cooling eect. The magneto-optical trap is nowadays a basic tool of cold atom physics
9
.
Other trapping methods
Other methods for trapping atoms with light beams exist, for instance laser dipole trapping
methods (ComplementXX, ,).
3. Blocking recoil through spatial connement
We now assume the atom or the ion under study is subjected to an external potential that
traps it in a region of space. The energy spectrum of the external variables is no longer
a continuous spectrum (as would be the case for a free atom), but a spectrum including
a discrete part corresponding to the atomic bound states. Furthermore, because of this
external potential, the atomic Hamiltonian is no longer translation invariant, and hence
the total momentum is no longer a good quantum number. In this section we study how
the absorption and emission spectra of the atom are modied by its connement within
the potential, and how the recoil of the atom can be blocked in certain cases.
3-a. Assumptions concerning the external trapping potential
We will assume the external potential acts only on the external variables, and
not on the internal variables. This is the case for example for an atomic ion trapped
by electric and magnetic elds, which only act on the center of mass via the global
ionic charge, but does not act on the internal variables
10
. Figure
potentials for an atom or an ion in the internal statesor. The two potential curves are
identical, deduced from one another by a vertical translation of amplitude~0where0
is the frequency of the internal atomic transition. The spectrum of the vibrational
levels, of energies,,, ... is the same for the two potentials. The atomic
states are labeled by two quantum numbers: a quantum numberorfor the internal
variables; a vibrational quantum number,,, ... for the external variables. The
atomic transitions between the two internal statesandnow present a structure due
to the vibrational motion of the center of mass. The frequencyof the transition
is given by:
~ =~0+ (52)
The quantity gives the variation of the external energy of the atom during the
transition.
9
See 14-7 of [24] for a description of the rst experimental realizations of such a trap and for a
more quantitative study of its performances.
10
The ion is generally conned in the center of the trap, a region where the electric and magnetic
elds are very weak. It is then legitimate to neglect the Stark or Zeeman shifts of the internal states.
2036

MOMENTUM EXCHANGE BETWEEN ATOMS AND PHOTONS
Figure 6: The external potential trapping the ion is the same when the ion is in either
of its internal statesor, separated by an energy dierence~0. Consequently, the
spectrum of the vibrational levels of the center of mass in the external potential is the
same for both internal states.
3-b. Intensities of the vibrational lines
We showed in
Hamiltonian could be factored into three terms pertaining to the three types of variables,
the internal and external atomic variables, and the radiation variables see relations (C-
34) and (C-35) of that chapter. The part relative to the external variables is equal to
ext
n
exp(k
^
R)
ext
infor an absorption process, where
ext
n
and
ext
inare the external
nal and initial states of the transition, equal here toand . This leads to an
intensity of the vibrational line proportional to:
= exp(k
^
R)
2
(53)
The obey the sum rule (obtained by the closure relation on the states):
= exp(k
^
R) exp(k
^
R)= 1 (54)
It follows that the relative weight of the transition compared to all the
transitions starting fromis precisely equal to.
Another sum rule
The relative weightsobey another sum rule:
= ( ) =
~
22
2
= rec (55)
2037

COMPLEMENT A XIX
which indicates that the average energy gained by an atom going from a given levelto
another levelis equal to the recoil energy, whatever the value of. To prove relation
(55), we rewrite the sum overin (55
exp(k
^
R)
^
ext exp(k
^
R) (56)
where
^
ext=
^
P
2
2+(
^
R)is the external variable Hamiltonian. The only term
inextthat does not commute withexp(k
^
R)is the kinetic energy term. We can
replace the commutator appearing in (56) byexp(k
^
R)
^
P
2
2. We now develop
this commutator and use the closure relation on thestates, as well as relation:
exp(k
^
R) (
^
P
2
2
) exp(+k
^
R) =
1
2
(
^
P+~k)
2
(57)
This yields:
exp(k
^
R)
^
P
2
2
exp(+k
^
R)
^
P
2
2
= rec+ ~k
^
P = rec (58)
since, taking Ehrenfest's theorem into account (Chapter, ), the average value
of operator
^
P(equal todRd) in the stationary stateis zero.
3-c. Eect of the connement on the absorption and emission spectra
The absorption and emission spectra of an atom are signicantly modied by the
connement of its center of mass.
As we saw in , when a free atom has a well dened initial momentumPin,
the absorption of a photon with momentum~kplaces it in a well dened momentum
statePn=Pin+~k. The conservation of the global momentum means that there is
only one nal state,Pn, corresponding to an initial statePin, and hence a single
absorption line.
On the other hand, when the global momentum is no longer conserved because of
the external potential's trapping of the atom, we get several lines going from an initial
given stateto several possible nal states, whose frequenciesare given by
(52). One can then ask which of these lines is the strongest.
To answer that question, we go back to expression (53) for the relative weight of
the transition . In this equation, operatorexp(k
^
R)represents a translation
operator, in momentum space, of the quantity~k. The quantityis thus proportional
to the squared modulus of the scalar product of the vibrational wave function(r)and
the wave function(r)translated in momentum space by the quantity~k. Let us
assume the atom is trapped in a region of spatial extension, very small compared to
the wavelength= 2 of the incident photon. The momentum spreadof the wave
function is then larger than~since:
(59)
leads to:

~

~
=
~
2
(60)
2038

MOMENTUM EXCHANGE BETWEEN ATOMS AND PHOTONS
This means that, as long as the excited levels' energy is not too large, the translation in
momentum space of the wave function(r)by a quantity~kmuch smaller than its width
leaves that wave function practically unchanged, and it consequently remains orthogonal
to(r)for=. The strongest line in the absorption spectrum is therefore the line
with no change in the vibrational quantum number (a line sometimes called thezero-
phonon line), whose frequency remains unchanged, equal to0. A strong connement
suppresses the Doppler shift and the atom's recoil.
Comment on momentum conservation
One may wonder what happened to the momentum of the absorbed photon in the zero-
phonon transition. Remember that we have treated the trapping potential of the atom
as an external potential, which breaks the translation invariance of the problem under
study: the Hamiltonian no longer commutes with the total momentum, which is no longer
conserved. It is thus not surprising that we cannot follow what becomes of the photon
momentum. We can, however, describe the trapping potential, not as an externally given
potential, but rather coming from the interaction of the atom with another physical object
whose dynamics must be taken into account. A quantum treatment of that device and its
interaction with the atom permits introducing for the global system, atom + trapping
device, a Hamiltonian that commutes with the total momentum; it is the global system
that absorbs the photon momentum. As this momentum is microscopic, whereas the
mass of the device is macroscopic, the recoil velocity is so weak that the corresponding
frequency shifts are totally undetectable.
3-d. Case of a one-dimensional harmonic potential
We now assume the external trapping potential is harmonic, and we calloscthe
oscillation frequency of the atoms in this potential. The energiesof the vibrational
levels are equal to(+ 12)~osc, whereis an integer, positive or zero. The spatial
extension0of the ground state wave function= 0is equal to
~2 osc. To
characterize the connement, we introduce the dimensionless parameter
11
:
=0= 2
0
(61)
If 1, the atoms are conned in a region small compared to the radiation wavelength.
The square of the parameterhas a simple physical signicance since:
2
=
~
2
2 osc
=
rec
~osc
(62)
is the ratio between the recoil energyrecand~osc, which is the energy dierence
between vibrational levels in the potential well.
It is instructive to compute, as a function of, the intensities00and01of the
vibrational lines0 0and0 1. Assume the photon wave vectorkis parallel
to theaxis. Exponentialexp(k
^
R)appearing in equation (53) can be replaced by
11
This parameter is often called theLamb Dicke parameter, after the names of the physicists who
rst introduced the idea of recoil-free absorption in a trapped system. To get a historical overview of
the various studies on the suppression of recoil due to connement, the interested reader may consult
6-4-4 of [24] as well as the references cited in that .
2039

COMPLEMENT A XIX
exp(
^
). We now use the expression of operator
^
in terms of the annihilation and
creation operators of the harmonic oscillator associated with the external potential:
^
=
~2 osc(^+ ^) = 0(^+ ^) (63)
We then get, in the limit1, using (61):
exp(
^
) = exp(^+ ^)
= 1 +(^+ ^)
2
2
(^+ ^)
2
+ (64)
The series expansion (64) used in (53) yields, to order 2 in:
00= 1
2
(65a)
10=
2
(65b)
All the other transitions0 with>2have relative intensities of a higher order,
in
2
. The transition with no change in the vibrational state, and hence with no recoil,
is predominant for a strong connement. The transition0 1has a much lower
probability, by a factorrec~osc; when it occurs, it increases the atomic energy by a
quantity~oscmuch larger thanrec. The sum rule (55) shows that, on the average, the
energy gained by the atom remains equal torec.
3-e. Mössbauer eect
In 1958, Rudolf Mössbauer observed very narrow lines in the resonant absorption
spectrum ofrays by the atomic nuclei in a crystal. Building on the previous work of
Lamb [25] on the suppression of the recoil in the resonant absorption of slow neutrons
(and not of photons) by the atomic nuclei in a crystalline network, he attributed the
narrow spectral structures he observed to a suppression of the recoil. This suppression
can occur if, in the crystal phonon spectrum, there are frequencies larger than the recoil
frequency=rec~. The interest of the Mössbauer eect comes from the high value of
the frequency of the internaltransition, which can reach 10
18
Hz, or even much higher
frequencies. If the Doppler width and the recoil shift are suppressed by the connement,
and if the natural width remains of the order of 10
6
to 10
7
Hz (as for an optical transition),
the quality factor of thetransition (ratio between frequency and the spectral width of
the resonance) can reach values of the order of 10
12
. Such a resolution allowed measuring,
already in 1960 [26], and for the rst time in a laboratory, the gravitational shift predicted
by general relativity between the frequencies of an emitter and a receptor, both located
in the earth's gravitational eld but separated by an altitude of roughly twenty meters.
4. Recoil suppression in certain multi-photon processes
Until now, we only considered one-photon processes. We shall see in Chapter
there are multi-photon processes in which the atom goes from an internal stateto
another stateby absorbing or emitting several photons. During such processes, the
total energy and momentum must of course be conserved
12
.
12
We now consider again free atoms, without an external potential.
2040

MOMENTUM EXCHANGE BETWEEN ATOMS AND PHOTONS
Figure 7: Saturated absorption spectroscopy. This gure plots the absorption prole of
the probe beam when scanning the frequencyof the two laser beams. A narrow hole, of
width, appears in the middle of a much larger Doppler prole, of width.
Imagine a two-photon process, where the two photons have the same frequency
and opposite wave vectors+kandk. The total radiation momentum is zero in this
case. If the atom absorbs those two photons, its momentum does not change. Its external
energy is not modied, meaning there is neither a Doppler eect, nor a recoil energy.
This possibility can be extended to-photon processes as long as the sum of the wave
vectors of thephotons is zero:
k1+k2+k=0 (66)
This idea was proposed independently by two groups, in Russia [27] and in France
[28]. It led to signicant experimental advances in high resolution spectroscopy where the
line width is no longer limited by the Doppler width, but rather by the often much smaller
natural width. A particularly interesting example is the study of the transition1 2
of the hydrogen atom by Doppler-free two-photon spectroscopy [29]. The upper state2
of this transition is metastable, with a long lifetime (around120ms). Consequently, its
natural width is very small and the two-photon line very narrow, which allows extremely
precise measurements of fundamental constants such as the Rydberg constant. Note must
be taken however that the interaction with the laser radiation inducing the two-photon
transition leads to shifts of the energy levels
13
, proportional to the light intensity; these
must be taken precisely into account to determine the non-perturbed frequency of the
two-photon transition.
Doppler-free saturated absorption spectroscopy
Nonlinear eects also appear in experimental set-ups where the atom interacts with two
counter-propagating light beams, with the same frequency, one having a high intensity
(pump beam) and the other a weaker one (probe beam). Contrary to the two-photon
transitions considered in the previous paragraph, we assume here that the transitions
induced by each beam are one-photon transitions between the two atomic internal states
and; the laser frequencyis therefore close to the atomic transition0(and not close
to02).
We neglect here the recoil energy, in general very small in the optical domain compared
with the natural widthof the upper state. However, the Doppler shifts of the absorp-
13
See ComplementXX, 2-b
2041

COMPLEMENT A XIX
tion lines of the various atoms play an important role, as they are dierent for the pump
beam and the probe beam which propagate in opposite directions. The pump beam in-
teracts with an atom of velocitypumpif its apparent frequency pumpfor that atom
coincides with the atomic frequency0(within), i.e. ifpump= 0within. In
the same way, the probe beam interacts with atoms of velocityprobeif+ probe=0
within, i.e. ifprobe=( 0)within. When is dierent from0, we have
pump=probe: the two beams do not interact with the same atoms in the velocity dis-
tribution, so that the absorption of the probe beam is not perturbed by the presence of
the pump beam. However, this perturbation becomes important when= 0(within
), since the two beams interact with the same sub-set of atoms (those belonging to the
same velocity group along the beams' axis).
The high intensity pump beam lowers the dierence in populations between theand
levels of the atomic transition, and tends to equalize these populations. The absorption
of the probe beam is thus diminished when the two beams interact with the same velocity
group, i.e in the vicinity of= 0. When scanning the frequencyof the two laser
beams, the absorption of the probe beam varies according to a Doppler prole centered
around0, with width, in the middle of which (Figure) appears a hole with a much
smaller width. This method, calledsaturated absorption, allows the determination of
the atomic frequency0with a much better resolution than when using a single beam.
Conclusion
In this complement, we showed how the analysis of the momentum exchanges between
atoms and photons allows introducing and interpreting several important physical phe-
nomena. These phenomena include Doppler width, recoil energy, radiation pressure
forces, Doppler laser slowing down and cooling of atoms, suppression of the Doppler
eect due to connement or in two-photon transitions, and the Mössbauer eect.
Thanks to these various methods, spectacular improvements in the resolution of
spectroscopic measurements have been obtained. This led to high precision measurements
and improvements of atomic clocks, which now have a relative stability of the order
of10
16
. Placing such a clock in the international spatial station and comparing its
frequency with that of a similar clock on the Earth, one hopes to be able to test the
value of the gravitational shift predicted by general relativity with a precision better, by
a factor close to100, than all the other existing tests. Another conclusion we can draw is
that atom-photon interactions are useful tools for controlling and manipulating atoms.
We shall see in ComplementXIXhow the exchanges of angular momentum be-
tween atoms and photons allows controlling the angular momentum of the atoms, polar-
izing them via optical pumping. Such achievements have opened new elds of research,
such as atomic interferometry and the study of degenerate quantum gases.
2042

ANGULAR MOMENTUM OF RADIATION
Complement BXIX
Angular momentum of radiation
1 Quantum average value of angular momentum for a spin
1 particle
1-a Wave function, spin operator
1-b Average value of the spin angular momentum
1-c Average value of the orbital angular momentum
2 Angular momentum of free classical radiation as a function
of normal variables
2-a Calculation in position space
2-b Reciprocal space
2-c Dierence between the angular momenta of massive particles
and of radiation
3 Discussion
3-a Spin angular momentum of radiation
3-b Experimental evidence of the radiation spin angular momentum2051
3-c Orbital angular momentum of radiation
Introduction
Radiation angular momentum plays an important role in many situations, in particular
in atomic physics experiments. As will be explained in ComplementXIX, the exchange
of angular momentum between atoms and photons is the base of many experimental
methods, such as optical pumping, which illustrated for the rst time the manipulation
of atoms by light.
In Chapter , a spatial Fourier transform of the classical elds led to the
introduction of the eld normal variables"(k)and
"(k), which are the eld components
in a basis of transverse plane waves. Upon quantization, these normal variables became
the annihilation^"(k)and creation^
"(k)operators of a photon in a modek". Such a
plane wave basis is particularly useful for studying the radiation energy and momentum,
since the photons of modek"have a well dened energy~=~and momentum~k.
On the other hand, the expansion of the eld angular momentum in terms of the normal
variables"(k)and
"(k)is not as simple, since the photons of modek"do not have
a well determined angular momentum. The aim of this complement is to nd another
expansion better adapted to the study of the radiation angular momentum, and establish
a number of useful results.
In the classical description of radiation, the normal variable(k) =
""(k)"is
a vector function ofkpresenting a certain analogy with a wave function in the reciprocal
space, and which could be seen as the wave function of the radiation (in momentum
space). A physical quantity, such as the radiation total energy or the total momen-
tum, does appear as the average value in that wave function of a one-particle operator
2043

COMPLEMENT B XIX
representing the energy or the momentum of a photon. We will see other examples of
this analogy in this complement. As it is a vector function, this wave function can be
regarded as the wave function of a particle of spin 1 whose total angular momentumJ
would be the sum of the orbital angular momentumLand the spin angular momentum
S. We will present, in , a quantum mechanical calculation of the average values, in
the state of a spin 1 particle described by the vector wave function (k), of the orbital
and spin angular momentum of that particle. Returning to classical physics, in
will establish the expression for the total angular momentumJof free radiation; we
shall rst write it as a function of the elds in reciprocal space, then as a function of
the eld normal variables(k). The expression thus obtained has the same form as that
obtained in , provided we replace the (k)by the(k). This will lead us to the
expansion in terms of normal variables of not only the eld total angular momentum,
but also of its orbital and spin angular momenta. The physical interpretation of these
results is discussed in , which highlights, in particular, some important characteristics
of these two types of angular momenta.
1. Quantum average value of angular momentum for a spin 1 particle
We rst study a spin 1 particle that has a mass, and is therefore not a photon. Our
results will be useful as a point of comparison for the next paragraph's computations,
where we return to the electromagnetic eld.
1-a. Wave function, spin operator
The conclusions of this
ing the radiation angular momentum expressed in terms of normal variables. As normal
variables characterize the eld in reciprocal space, it is important here to describe the
state of the spin 1 particle in that same space. The particle state vector can be expanded
on a basisk , wherekrepresents the wave vector andthe spin state:
= d
3
(k)k (1)
with:
(k) =k (2)
In general, we choose for the statesthe eigenstates+ 10 1of thespin
component. Here, we shall choose another basis:
= (1
2)1 + 1
= (
2)1++ 1
=0 (3)
The action of theScomponents on these basis vectors can easily be computed.
We must use = (++)2and = (+ )2, as well as the action of
2044

ANGULAR MOMENTUM OF RADIATION
+ on the states+ 10 1(Chapter, relations (C-50)). As an example,
we obtain:
=
1
2
(++)
1
2
1 + 1=
~
222020= 0
=
2
(++)
1
2
1++ 1=
~
2220+20=~
=
1
2
(++)0=
~
22+ 1+21=~ (4)
These equations, and those similar for the action ofand, can be written in a more
compact way as:
=~ (5)
where are the indicesand is the three-dimensional completely antisym-
metric tensor
1
. Equation (5) also leads to:
=~ (6)
1-b. Average value of the spin angular momentum
Taking (1) into account, the average value ofis written:
=d
3
d
3
(k)k k (k) (7)
As does not act on the orbital degrees of freedom, described by the variablek, we
get, taking (6) into account :
k k=~ (kk) (8)
Inserting this result in (7) then yields:
=~d
3
(k)(k) =~d
3
(k)(k) (9)
(using the fact thatis antisymmetric). As thecomponent of the cross product of
two vectorsVandWis written:
(VW)= (10)
we get:
=~d
3
( ) (11)
1
By denition = +1if are or can be deduced from by an even permutation;
=1ifis deduced from by an odd permutation; nally = 0if two of the three indices
(or all three) are equal.
2045

COMPLEMENT B XIX
1-c. Average value of the orbital angular momentum
The orbital angular momentum is written:
L=RP (12)
Itscomponent acts in position space as:
= (RP)= ~ with = (13)
Going to the reciprocal space amounts to performing a spatial Fourier transform.
The operators which inrspace correspond to a multiplication byand a derivation
with respect to, respectively become, inkspace, the operators derivation with respect
toand multiplication by(multiplied by appropriate factors):
FT
=
~
=
FT
(14)
with the notation:
~
(15)
In reciprocal space, the action of thecomponent of the orbital angular momentum is
therefore:
~ (
~
) () =~
~
=~(krk) (16)
In the last equality of the rst line, we have moved
~
to the right of, which is allowed
since the presence ofmeans that all the terms with equal indicesandmust be
zero. To obtain the equality in the second line of (16), we use the anstisymetry of
under the exchange ofand, and relation (10) of the vector product;rkis the gradient
with respect ofk.
We nally compute the average value ofin the state . Taking (1) into account,
we get:
=d
3
d
3
(k)k k (k) (17)
As does not act on the spin degrees of freedom, we must have=in the matrix
element on the right-hand side, which yields, using the dierential form (16) for:
=d
3
d
3
(k)k k (k)
=~d
3
(k)
~
(k)
=~d
3
(k) (krk)(k) (18)
2046

ANGULAR MOMENTUM OF RADIATION
Finally, adding (11) and (18), we obtain the following expression for the average
value of the particle total angular momentum in the state :
J =~d
3
(k) (k)
spin
+ (k) (krk)(k)
orbital
(19)
2. Angular momentum of free classical radiation as a function of normal variables
We now show that the classical calculation of the radiation angular momentum presents
a certain analogy with the results of .
2-a. Calculation in position space
Relation (A-53) of Chapter
radiation (in the absence of particles):
J=0d
3
r[E(r)B(r)] (20)
Let us replaceBbyrAand use the triple product expansiona(bc) = (ac)b
(ab)c. We obtain, keeping the right order betweenrandA:
E[rA] = r (Er)A (21)
whereis the component ofE(r)on the,oraxis, labeled by the index.
Inserting (21) in (20) leads to:
J=0d
3
(rr) r(Er)A (22)
Consider rst the contributionJ
(1)
of the second term in (22). Its
(1)
component
is written:
(1)
= 0d
3
r(Er)A
= 0d
3
(23)
We now move to the right of, using:
= (24)
The contribution of the termto (23) yields:
0d
3
=0d
3
(EA) (25)
2047

COMPLEMENT B XIX
As for the contribution of the termto (23), we perform an integration by parts.
The contribution of the integrated term yields a zero surface integral if the elds decrease
fast enough at innity. We obtain for the contribution of the termto (23) :
0d
3
( ) = 0 +0d
3
(26)
In the last term of (26), we note the quantity
=rE (27)
which is equal to zero since the electric eld, in the absence of sources, is purely transverse
(and hence of zero divergence). This term therefore disappears.
Finally, the average value ofJis the sum of (25) and of the rst term of (22):
J=0d
3
E(r)A(r)
spin
+ (r)(rr)(r)
orbital
(28)
2-b. Reciprocal space
Expression (28) forJcan be rewritten as a function of the Fourier transforms of
the eldsE(r)andA(r). Using the Parseval-Plancherel equality (Appendix, )
and relations (14), we get:
J=0d
3~
E(k)
~
A(k)
spin
+
~
(k)(krk)
~
(k)
orbital
(29)
We now use expressions (B-22a) and (B-22b) of Chapter
~
E(k)and
~
A(k)as a function of the normal variables:
~
E(k) =
2()
[(k)(k)] (30a)
~
A(k) =
1
2()
[(k) +(k)] (30b)
where()was determined by relation (A-3) of Chapter :
() =
02~
(31)
Inserting (30) into (29), we get:
=
~
2
d
3
[(k) (k)] [(k) +(k)]
+ [(k) (k)]
~
[(k) +(k)] (32)
2048

ANGULAR MOMENTUM OF RADIATION
Each line in (32) contains four terms: two of these terms include eithertwice for the
rst one, ortwice for the second both will be shown below to be equal to zero; the
other two terms contain eitheronce oronce we show below that they are equal.
We nally obtain:
J=~d
3
(k)(k)
spin
+ (k) (krk)(k)
orbital
(33)
This expression has the same form as (19): the angular momentum is the sum of a spin
term and an orbital term involving spatial derivatives. This result conrms that the
normal variable(k)can be regarded as the wave function in reciprocal space of the
photon eld, and that the photon is indeed a spin 1 particle. This result also gives the
explicit expressions of the radiation spin angular momentum (rst term in the bracket
of (33)) and orbital angular momentum (second term).
For a massive particle, we know (Chapitre, ) that in spherical (or cylin-
drical) coordinates, the action of theangular momentum component corresponds to
a derivation with respect to the azimuthal angle:
}(
~ ~
) =
}
(34)
This result simply comes from a calculation of partial derivatives; it is thus also valid for
a eld.
Computation of the various terms appearing in equation(32)
Consider, in the rst line of (32), the terms involving a product of twoor two, for
example:
(~2)d
3
[(k) (k)] (35)
Changingkintok, inverting the indicesand, and using= , we can show
that (35) is equal to its opposite, and hence must be zero. The same approach, followed
for the term:
+(~2)d
3
[(k) (k)] (36)
shows that this term is equal to:
(~2)d
3
[(k) (k)] (37)
which is identical to the other term on the rst line of (32) containing oneand one
, provided we change the relative order in which the term(k)and(k)appear. In
classical theory, these quantities are numbers, and hence commute: their order does not
matter. It is however useful to keep track of that order in order to obtain an expression
still valid when, upon quantization, theandwill be replaced by the non-commuting
creation and annihilation operators.
This computation can be extended to the terms on the second line of (32). In addition to
changingkintok, we must also perform an integration by parts. The integrated term,
2049

COMPLEMENT B XIX
which yields a surface integral, is zero if the elds tend to zero fast enough at innity.
Added to it is a contribution that shows that the terms containing twoor twoare
equal to their opposite, and hence equal to zero. On the other hand, the integration by
parts shows that the two terms containing oneand oneare equal if the order of the
andcan be switched. In the case where the order of theandis not taken into
account, we obtain expression (33).
2-c. Dierence between the angular momenta of massive particles and of radiation
In spite of the strong analogy between equations (19) and (33), we should not
forget an important dierence between the two angular momenta, arising from the fact
that the normal variables(k)are transverse. Maxwell's equationrE= 0for the free
eld does require(k)to always be perpendicular to the wave vectork:
k(k) = 0 k (38)
while the wave function (k)of a massive particle in the reciprocal space is not neces-
sarily perpendicular tok. Another dierence, of course, is that the norm of this wave
function does not have any particular physical meaning (it can arbitrarily be put equal to
unity), while changing the norm of the normal variables of the eld changes its amplitude.
3. Discussion
3-a. Spin angular momentum of radiation
The spin angular momentum, rst term on the right-hand side of (33), can be
written as:
()=~d
3
(k) (k) =~d
3
[(k)(k)] (39)
Instead of using the components(k)and(k)on a basis of three vectorse,e,e
independent of the wave vectorkdirection, we can choose a basis of three vectors"1(k),
"2(k),"3(k) ==k, including the unit vectoralongkand two other vectors"1(k)
and"2(k), orthogonal to each other and to, and forming a right-handed reference
frame. As the normal variables(k)and(k)are transverse, their components on
are zero. In addition, we introduce the two complex linear combinations ofe1(k)and
e2(k):
"+(k) =["1(k) +"2(k)]
2
"(k) = + ["1(k)"2(k)]
2 (40)
corresponding to right and left circular polarizations with respect to thekdirection. The
transverse normal variables(k)and(k)can be expanded on these two vectors:
(k) =+(k)e+(k) +(k)e(k)
(k) =
+(k)e
+(k) +(k)e(k) (41)
2050

ANGULAR MOMENTUM OF RADIATION
Using these two expansions, we compute the cross product(k)(k). Since:
"
+"+=
1
2
("1"2)("1+"2) =
2
("1"2"2"1) =
""=
1
2
("1+"2)("1"2) =
2
("1"2"2"1) =
"
+"=
1
2
("1"2)("1"2) = 0 =""+ (42)
we get:
S=d
3
+(k)+(k)~ (k)(k)~ (43)
The form of this expression, diagonal with respect to the spin variables, has
a clear physical signicance: to each plane wave with wave vectorkand a right (left)
polarization with respect tok, correspond photons of momentum~kand spin angular
momentum+~(~) along the directionofk. Upon quantization, when the normal
variables(k)and(k)are replaced by creation and annihilation operators, expression
(43) becomes:
^
S=d
3
^
+(k)^+(k)~^(k)^(k)~ (44)
Operator^
+(k)creates a photon with momentum~kand spin angular momentum
+~along the directionofk; operator^+(k)annihilates that photon, and operator
^
+(k)^+(k)corresponds to the number of photons in that mode. An analogous deni-
tion applies to the second term of (44) with a change of sign for the angular momentum.
Helicity
These results, which arise from the transverse character of free radiation, lead us to intro-
duce the so-called helicity. It is the projection of the photon spin angular momentum
onto the directionof the wave vectork, equal to+1for photons with a right circular
polarization with respect to, and1for photons with a left circular polarization. The
transverse character of the free radiation eld forbids the photons to have zero helicity.
Note also that helicity is apseudoscalar: upon reection in space, the polar vector
changes sign whereas the spin vectorS, an axial vector, remains unchanged. Conse-
quently, the scalar product ofandSchanges sign (as opposed to a scalar).
3-b. Experimental evidence of the radiation spin angular momentum
Consider a plane wave of wave vectorkand polarization". Using the expressions
for the electric eldE(r)and the vector potentialA(r)given in Chapter , one can
easily compute the two terms of equation (28) yielding, in position space, the radiation
spin angular momentum and orbital angular momentum
2
. The result is that the radi-
ation orbital angular momentum second term of (28) is always zero, whatever the
2
These are the two terms that, transformed into the reciprocal space, and after the introduction of
the normal variables, yield the two terms on the right-hand side of equation (33); by comparison with
the expression of the angular momentum of a spin 1 particle, these terms have been interpreted as the
two components of the radiation angular momentum.
2051

COMPLEMENT B XIX
polarization"is. As for the spin angular momentum rst term of (28) it is zero for a
linear polarization, but dierent from zero for a circular polarization, with opposite signs
for the right and left circular polarizations. This validates, in the simple plane wave case,
the general conclusions of the previous paragraph.
Such a result suggests sending a linearly polarized light beam through a quater-
wave plate. Assuming the plate transforms the incident linear polarization into a right
(left) circular polarization, the incident photons have a zero spin angular momentum
before they go through the plate, and equal to+~(~) as they come out of the plate.
The radiation spin angular momentum thus changes as beam goes through the plate, and
this must be accompanied by a change, in the opposite direction, of the plate's angular
momentum. Suspending the plate by a thin torsion ber, one should observe a rotation of
the plate induced by the incident radiation, in a direction opposite to that of the circular
polarization of the outgoing beam. This experiment, suggested by A. Kastler [30] was
performed by R. Beth [31], conrming the existence of angular momentum transfer.
Comment
A paradox arises when computing, again for a plane wave, the angular momentum of
the radiation, given by equation (20). In a plane wave, the Poynting vector(r) =
E(r)B(r)is always parallel to the wave vectorkat any pointr, and for any polarization.
The integral over the entire space ofr(r)must then be zero. As the orbital angular
momentum is also zero, it seems that the spin angular momentum should also be zero,
whatever the polarization is. This paradox arises because innite plane waves do not exist
in the physical world: any real light beam has a nite spatial extension. The authors of [32]
(see also [33]) show that the circular polarization at the center of the beam changes when
the eld amplitude changes around the edge of the beam. Taking this eect into account
quantitatively conrms the result obtained above, namely that the beam spin angular
momentum is equal to the sum of the angular momenta~of that beam's photons.
3-c. Orbital angular momentum of radiation
An important dierence between the radiation orbital and spin angular momenta
is clearly seen in expression (28): the denition of orbital angular momentum involves
a reference point O, since the vectorr, dened with respect to that point, explicitly
appears in the expression for the orbital angular momentum. This is not the case for the
spin angular momentum which, for this reason, is sometimes called intrinsic angular
momentum. There are actually at least two cases where the choice of the point O is
obvious, cases that we now analyze.
. Multipolar waves
When studying the radiation emitted or absorbed by an atom or a nucleus between
two discrete states, a natural choice for analyzing the exchanges of angular momentum
between the system internal variables and the photons is the center of mass of that atom
or that nucleus. In the next complementXIX, we study for example the exchange of
angular momentum between photons and the internal variables of an atom in a particular
case: an electric dipole transition, in the long wavelength approximation. Consequently,
the expressions describing the photon absorption only involve the radiation polarization
variables, and hence only the photon spin angular momentum; the radiation orbital
2052

ANGULAR MOMENTUM OF RADIATION
angular momentum does not actually play any role
3
.
There are other transitions, especially for atomic nuclei, where the variation of the
internal angular momentum between the two transition states is larger than or equal to
2; the photon spin angular momentum, equal to 1, can then no longer ensure the con-
servation of the angular momentum. Radiation states having a total angular momentum
larger than 1 must come into play, which implies a contribution from the radiation
orbital angular momentum. Waves corresponding to such states are called multipolar
waves.
The simplest way to build multipolar waves having a total angular momentum
characterized by the quantum numbersand is to associate a spherical harmonic
()with a spin= 1; the spherical harmonic is an eigenfunction ofL
2
and
with eigenvalues(+ 1)~
2
and~; the spin= 1has three eigenstates , with
= +101, isomorphic to the three polarization statese+=(e+e)
2,e,
e= (e e)
2. We therefore obtain avector spherical harmonic:
Y
1() = 1 ()e (45)
which is an eigenfunction ofJ
2
,L
2
,, with eigenvalues(+ 1)~
2
,(+ 1)~
2
,~.
In this equation, the rst term on the right-hand side is a Clebsch-Gordan coecient
(Chapter, ),can take one of the three values= 1= =+ 1and
= +.
A diculty is that the vector spherical harmonics are not all transverse functions
and hence cannot be used as a basis of normal transverse functions for expanding the
radiation eld. For a given value of, one can nevertheless build linear superpositions of
vector spherical harmonicsY
1
()with= = 1that are transverse and have,
in addition, a well dened parity=1. Each vector spherical harmonic, which only
depends on the directionofk, can also be multiplied by( 0), hence yielding a
function that is also an eigenfunction of the energy, with eigenvalue~0. These functions
form a possible basis of normal transverse variables for expanding the eld; they are
characterized by four quantum numbers: the energy~0, the total angular momentum
, and the parity. They are called electric (for=1) or magnetic (for= +1)
multipolar waves. As this book is limited to the study of electric or magnetic dipole
transitions, we do not give here the general expressions for multipolar waves. More
details can be found in complement BIof [16] and in [34].
. Beams with cylindrical symmetry around one axis
It often happens that the beams under study have a cylindrical symmetry. This
is the case, for example, for Gaussian beams propagating along aaxis, and whose
transverse sections are circular. If the reference point O is taken on the axis, the beam
symmetry causes the orbital angular momentum to be necessarily along theaxis and
to have the same value regardless of the position of O along this axis. If the reference
point O is taken outside this axis, the orbital angular momentum will change, but not
its component, which exhibits an intrinsic character.
3
This orbital angular momentum may, however, come into play during the angular momentum
exchanges with the atom's external variables for certain types of light beams, the Laguerre-Gaussian
light beams ( XIX).
2053

COMPLEMENT B XIX
A particularly interesting case concerns the Laguerre-Gaussian beam (LG) whose
eld has anexp()dependence with respect to the azimuth anglethat denes
the direction of a point in the plane perpendicular to the beam axis. The cylindrical
symmetry is preserved since a rotation of the beam of an angle0around theaxis
yields the same eld to within a global phase factorexp( 0). Relation (34) then
shows that thecomponent of the orbital angular momentum of each photon of the
LG beam is}.
Consider an LG beam propagating along theaxis, with awave vector along that
axis. The phase at a point with cylindrical coordinatesis that ofexp(+ ).
For an ordinary Gaussian beam (for which= 0) the surfaces of constant phase are, in
the vicinity of the focal point, planes perpendicular to theaxis. When= 0,and
must both vary for the phase to remain constant, following the relationd+d= 0.
The surfaces of constant phase are therefore helicoidal surfaces spanned by a half-line
perpendicular to theaxis, starting from this axis, and which rotates of an angle2
whenincreases by. It is not surprising that under such conditions the eld has a
non-zero orbital angular momentum. Note also that the eld must be zero on theaxis
(otherwise its phase would vary discontinuously upon crossing that axis).
In conclusion, we showed in this complement that there are two types of radiation
angular momenta, the spin angular momentum and the orbital angular momentum, and
we studied their properties. The photon can be viewed as a spin1particle, except for
the fact that it only has two (instead of three) internal states, with respective heliticity
+1and1. We shall see in the next complement how the interactions between radiation
and atoms permit transferring angular momentum from the rst to the others.
2054

ANGULAR MOMENTUM EXCHANGE BETWEEN ATOMS AND PHOTONS
Complement CXIX
Angular momentum exchange between atoms and photons
1 Transferring spin angular momentum to internal atomic
variables
1-a Electric dipole transitions
1-b Polarization selection rules
1-c Conservation of total angular momentum
2 Optical methods
2-a Double resonance method
2-b Optical pumping
2-c Original features of these methods
3 Transferring orbital angular momentum to external atomic
variables
3-a Laguerre-Gaussian beams
3-b Field expansion on Laguerre-Gaussian modes
Introduction
Angular momentum exchanges between atoms and photons are at the base of several
experimental methods that played an important role in atomic physics and laser spec-
troscopy. The aim of this complement is to analyze the selection rules appearing in the
photon absorption and emission processes by an atom; they express the conservation of
the total angular momentum of the system atom + radiation during these processes.
We shall mainly focus on the transfer of angular momentum to the atomic internal
(electronic) variables (). The polarization of the eld, characterizing the spin angular
momentum of that eld (ComplementXIX), then plays an essential role. For the ab-
sorption process, we will establish the selection rules relating the eld polarization to the
variation of the magnetic quantum numbercharacterizing the projection of the total
internal angular momentum of the atom on a given axis. Two important applications of
these selection rules will be described in : the double resonance method, and optical
pumping. We shall see that the proper choice of the polarization of the exciting light
beam, and of both the direction and polarization of the detected light emitted by the
excited atoms, allows controlling the atomic Zeeman sublevels that can be populated
and detected by light. We shall emphasize in
transfer of the radiation orbital angular momentum to the atomic external variables will
be briey addressed in .
2055

COMPLEMENT C XIX
1. Transferring spin angular momentum to internal atomic variables
1-a. Electric dipole transitions
We shall limit ourselves to the case where the transitions between the atomic
internal states are electric dipole transitions. As seen in Chapter ), the
interaction Hamiltonian between atom and radiation can then be written in the form:
^
=
^
D
^
E(
^
R) (1)
where
^
Dis the operator associated with the atomic electric dipole moment and
^
E(
^
R)
is the radiation transverse electric eld operator at point
^
R, the position of the atom's
center of mass. Note that all the results established in this complement are still valid
for magnetic dipole transitions; one must simply replace
^
Dby the atomic magnetic
dipole moment operator
^
Mand
^
Eby the radiation magnetic eld
^
Boperator. For the
transitions where one photon is absorbed, the expressions (B-3) and (B-4) of Chapter
of the elds
^
Eand
^
Bcan be replaced by the part containing only destruction operators,
called positive frequency component (cf. ) and denoted
^
E
(+)
and
^
B
(+)
. For the transitions where one photon is emitted, these elds can be replaced
their components
^
E
()
and
^
B
()
containing only creation operators, called negative
frequency components.
Replacing
^
E
(+)
and
^
B
(+)
by their plane wave expansions, the internal variables
only appear in the scalar products of
^
D, or
^
M, with the polarization vector"of the plane
wavek. We shall assume in
waves, or linear superpositions of plane waves with wave vectors having directions very
close to an optical axis (paraxial approximation). These waves are supposed to have the
same polarization", which means that the beam angular aperture must be suciently
small. The transitions between the internal atomic states
1
andwe shall consider are
thus entirely characterized by the matrix elements"
^
D.
1-b. Polarization selection rules
We start with the simple case of an atom with a single electron, and a transition
between a ground statewith an orbital angular momentum= 0and an excited state
with an angular momentum= 1. We assume the radiation has a right circular polar-
ization, noted+, with respect to an axis noted: this means that the radiation electric
eld rotates around that axisfollowing the right-hand rule, at the angular frequency.
We have:
"=
1
2
(e+e) (2)
Since
^
D=^r, whereis the electron charge and^rits position with respect to the
nucleus, we have:
"
^
D=
2
(+) =
2
sinexp() (3)
1
In this complement and as is usually done in the literature, we shall notethe ground state and
the excited state (instead of using our previous notation ofandfor the two atomic levels).
2056

ANGULAR MOMENTUM EXCHANGE BETWEEN ATOMS AND PHOTONS
where are the spherical coordinates of the electron with respect to the nucleus.
Starting from a ground state with a magnetic quantum number= 0(dened with
respect to theaxis) and with no dependence on, the excitation with+polarized
light creates an excited state wave function that now varies withasexp(). This wave
function is an eigenfunction of the component
^
= (~) of the orbital angular
momentum, with eigenvalue+1(it corresponds to the state= +1). In a similar way,
an excitation with apolarization, for which"=
1
2
(ee), transfers the atom into
the state=1. Consider nally an excitation with apolarized light, for which
"=e: the electric eld has a linear polarization parallel
2
to theaxis. We then have
"
^
D==cos, with nodependence; the excitation brings the atom to the state
= 0. To sum up, the polarization selection rules for a transition= 0 = 1
are given by:
+ = +1 = 0 =1 (4)
The previous results can easily be generalized to any transition going from a ground
statewith angular momentum to an excited statewith angular momentum,
for an atom with any number of electrons. One simply has to use the Wigner-Eckart
theorem (ComplementXand exercise 8 in ComplementX). The dipole operator
^
D
(sum of the dipole operators for each individual electron) is a vector operator whose three
spherical components
^
Dwith= +101are equal to :
^
+1=
1
2
(
^
+
^
) (5a)
^
0=
^
(5b)
^
1=
1
2
(
^ ^
) (5c)
The Wigner-Eckart theorem (ComplementX) states that
^
D =
^
1; (6)
where the last term on the right-hand side is a Clebsch-Gordan coecient (Chapter X,
) and the rst term is a reduced matrix element independent of,and.
The Clebsch-Gordan coecient is dierent from zero only if:
a triangle can be formed with,1and, which means thatis equal either to
, or to1, the transition= 0 = 0being forbidden;
= +.
As the three values of(= +101) correspond to the three polarizations+
respectively, the selection rules (4) for any given transition can be generalized
to (see Figure):
+ = + 1 = = 1 (7)
2
Because of the transversality of the eld, the light beam must then propagate in a direction per-
pendicular to theaxis.
2057

COMPLEMENT C XIXme=mg1 me=mg
mg
me=mg+ 1
+
Figure 1: Selection rules for an electric or magnetic dipole transition. The magnetic
quantum number increases by one unit for an excitation with a+polarization, re-
mains unchanged for apolarization and decreases by one unit for apolarization.
1-c. Conservation of total angular momentum
We saw that, when an atom absorbs a photon having a polarization+with respect
to aaxis, the component of the atom's angular momentum along that axis increases
by one (in~units). The conservation of the total angular momentum means that the
absorbed+photon must have an angular momentum+~along theaxis.
This result can also be obtained from the study of the radiation angular momentum
presented in ComplementXIX, as we now show. Although the interaction Hamiltonian
pertaining to the atomic internal variables does not depend on the photon wave vector
3
k, one can always choose akwave vector parallel to the atomic quantization axis (since
these two directions must be perpendicular to the polarization vector of the circular
wave). Now we saw in ComplementXIXthat in a plane wave (or in a beam of plane
waves with a small angular aperture) a photon with polarization+with respect to
the wave vector has a total angular momentum (actually reduced to its spin angular
momentum) equal to+~and parallel to its wave vector. The total angular momentum
is indeed conserved.
2. Optical methods
The polarization selection rules show that it is possible to selectively excite a Zeeman
sublevel of an atomic excited state. In a similar way, we shall see below that the ob-
servation of the emitted light in a given direction with a specic polarization allows
determining from which excited sublevel the light was emitted. Such a selectivity in
excitation and detection is the base of the optical methods for Hertzian spectroscopy, as
will be illustrated below with two examples.
2058

ANGULAR MOMENTUM EXCHANGE BETWEEN ATOMS AND PHOTONS
Figure 2: Energy diagram and Zeeman structure of the6
1
S0 6
3
P1transition of the
even isotopes of mercury at 253.7 nm (Fig.a). A static magnetic eld, applied along
theaxis, lifts the degeneracy of the excited state, which is split into three equidistant
Zeeman sublevels:=10+1. A resonant light beam, propagating along theaxis,
with a linear polarizationparallel to theaxis (Fig.b), selectively excites the atoms
into the= 0sublevel (upward arrow on Fig.a). When they are in the excited state,
the atoms are transferred from the= 0state to both states=1by a resonant
radiofrequency eld (oblique small double arrows on Fig.a). A detector D placed along
theaxis, just behind an analyser that only transmits light with+polarization along
theaxis, detects the light emitted by the atoms (Fig.b). Consequently, the detector is
only sensitive to the light emitted from the= +1sublevel (wiggly arrow on Fig.a); it
yields a signal proportional to the population of that sublevel.
2-a. Double resonance method
We now explain the principle of the method taking as an example the even isotopes
of mercury, for which the rst theoretical predictions were made by Brossel and Kastler
[35]; the rst experimental evidences were obtained by Brossel and Bitter [36]. Since for
these isotopes the nuclear spin is zero, and as in a mercury atom all the electron shells
are lled in the ground state, the energy diagram is particularly simple. The ground
statehas a zero angular momentum (= 0), and the rst accessible excited states, an
angular momentum = 1. In the presence of a static magnetic eldapplied along the
axis, the three Zeeman sublevels=10+1of the excited state undergo Zeeman
shifts proportional toand to the applied eld, so that the energy of thestate is
written:
=0+ (8)
3
In the long wavelength approximation, the radiation wave vectorkno longer appears in the inter-
action Hamiltonian for the atomic internal variables. This wave vector only appears in the part of the
interaction Hamiltonian,exp(k
^
R), pertaining to the external variables.
2059

COMPLEMENT C XIX
where0is the excited state energy in the absence of the eld,is the Landé g-factor
of that state, andthe Bohr magneton. The ground state= 0 = 0, whose
energy is chosen to be zero, is not aected by theeld (Figurea).
In the double resonance method, the atoms are selectively excited by a resonant
light beam with polarizationinto the sublevel= 0. For example, the exciting
beam propagates along theaxis, with a polarization"perpendicular to theaxis,
and parallel to theaxis, hence with apolarization (Figureb).
If the atoms were left alone, with no perturbation while in the excited state during
its radiative lifetime= 1(of the order of 1.5 10
7
sec), they would remain in that
sublevel for the entire time they stay in the excited state. On the other hand, if they
are subjected to a resonant radiofrequency eld
4
that is strong enough to make them go
from = 0to=1during the excited state lifetime, the two states=1
will be equally populated.
Is it possible, observing the light emitted by the atoms as they go back to the
ground state by spontaneous emission of a photon, to determine the sublevelfrom
which the light was emitted, and hence obtain a signal proportional to this sublevel
population? This problem must be analyzed with more care than the absorption process
for the following reason. Once the atom has reached the excited sublevel, it can
emit in any direction with all possible polarizations, which are not necessarily the three
basic polarizations+,or. We are going to show that one can place the detector
in a specic direction, far from the atom, and put in front of it a polarization analyzer
suitably chosen so as to be able to determine from which sublevel the detected photon
was emitted.
To demonstrate this result, it is important to rst study the oscillations of the
atomic dipole associated with a transition . Let us assume the detector is
placed on theaxis where the atom is located.
(i) For the transition= 0 = 0, as the dipole oscillates along theaxis,
it does not emit along that axis. The detector does not receive any fraction of the light
emitted by an atom in the= 0state.
(ii) On the other hand, the dipole associated with the transition= 0 =
+1, which rotates around theaxis following the right-hand rule, in a plane perpendicular
to that axis, emits along that axis light with a+polarization; this light yields a signal
on the detector equipped with an analyzer selecting right circular polarization. This
analyzer will, however, block the light emitted by an atom in the=1sublevel,
which has a left circular polarization.
To sum up, placing a detector in a well chosen direction, preceded by an analyzer
selecting a suitable polarization, one can block out all light except the one coming from
a specic uniquesublevel, and get a signal proportional to that sublevel population.
The principle of this double resonance experiment is to selectively excite the atoms
in the= 0sublevel, and to detect, by observing the+light emitted along the
axis, the variations of the number of atoms transferred into the= +1sublevel
by a radiofrequency eld with angular frequencyclose to the Zeeman frequency
= ~. Observing the variation of the emitted light as one scansfor a
xed value of the magnetic eld, or as one scans the magnetic eld for a xed, one
4
The atoms are thus submitted to two resonant excitations: an optical resonant excitation that brings
them from = 0to = 0; a radiofrequency resonant excitation that brings them from= 0to
=1. This is why this method is called double resonance method.
2060

ANGULAR MOMENTUM EXCHANGE BETWEEN ATOMS AND PHOTONS
can optically detect the magnetic resonance in the excited state.
Comment
What happens if the light emitted by the atom is not observed along theaxis, with a
right circular analyzer, but with a detector placed along theaxis, behind an analyzer
selecting a linear polarization parallel to theaxis, and hence perpendicular to the
axis, a polarization called sigma
5
? The light emitted along theaxis by the atom in
the = 0state has a linear polarization parallel to the oscillating dipole, and hence
parallel to theaxis. It is blocked by the analyzer that only lets through the orthogonal
polarization. On the other hand, whether the atom is in the= +1or =1
sublevel, the rotating dipole in theplane (following the right-hand or left-hand rule)
emits in that plane light with a linear polarization perpendicular to theaxis; that light
can be detected by the sigma polarization detector and it yields a signal proportional
to the sum of the populations of both sublevels= +1and =1. Actually, the
resonant radiofrequency eld excites a linear superposition of these two Zeeman sublevels.
It was later discovered that the waves emitted from these two= +1and =1
states gave rise to interference, hence modulating at the frequency2 the detected
sigma light intensity
6
. The detector signal therefore contains a component modulated at
the frequency2, on top of a continuous component (in steady state), proportional to
the sum of the two populations of the Zeeman sublevels. This continuous component was
the signal used in the rst double resonance experiment.
The shape of the magnetic resonance line can be exactly computed; it leads to
analytical expressions in excellent agreement with experimental observations. The center
of the resonance line yields the Landé g-factor of the excited state, i.e. the magnetic
moment of that state. The resonance width, extrapolated to zero radiofrequency intensity
to eliminate radiative broadening, yields the naturalwidth of the excited state.
Calculation of the line shape
Broadband excitation with apolarization prepares, in a quasi-instantaneous way, the
atom in the= 0excited state. In the steady state,0atoms per unit time are excited
into that state. Each atom then evolves, because of its interaction with the radiofrequency
eldB1, and its state becomes a linear superposition of the three sublevels=10+1.
Let us assume the radiofrequency eld is a rotating eld, which allows introducing the
associated rotating reference frame (ComplementIV) where the atom's evolution actually
becomes a simple rotation aroundB1. Using the rotation matrix for a spin 1, one can
nd the expression for(= 0 = +1), the probability for the atom, initially
in the= 0state, to be found after a time t in the= +1state [38]. Because of the
radiative lifetime= 1of the excited sublevels, this probability is reduced by a factor

. Consequently, in steady state, the number of atoms transferred to the= +1
state is equal to
7
:
+1=0
0
(= 0 = +1)

d (9)
Using the expression for(= 0 = +1), one nally obtains:
+1=
0
2
2
4
2
+
2
+
2
(
2
+
2
+
2
)(4
2
+
2
+ 4
2
)
(10)
5
This set-up was the one used in the rst double resonance experiment [36].
6
Such modulations, called light beats, were observed in 1959 [37].
7
A similar computation can be carried out for1.
2061

COMPLEMENT C XIX
Figure 3: Principle of optical pumping for a= 12 = 12transition. The reso-
nant absorption of a photon with
+
polarization selectively excites the=12
= 12transition. Once it has reached its excited state, the atom falls back, through
spontaneous emission, into the=12states. If it falls in the= +12state,
it can no longer absorb an incident photon, and remains in that state (since there is no
transition
+
originating from the= +12state where the atoms accumulate). In
addition, any change in the population dierence between the=12sublevels can
be detected by a change in the incident beam absorption, since this absorption is only
possible starting from the=12sublevel.
In this expression,is the Rabi frequency associated with the radiofrequency eldB1,
proportional to that eld's amplitude;= is the dierence between the fre-
quency of the RF eld and the Zeeman frequencyassociated with the gap between
the Zeeman sublevels, which is proportional to the static eld0. The resonance is plot-
ted by scanning either the frequency of the RF eld, or the static eld, which amounts to
scanning.
2-b. Optical pumping
Optical pumping, proposed by A. Kastler [39], extends to atomic ground states
the essential ingredients of the double resonance method. It also opens the possibility
of achieving, in a steady state, large population dierences between Zeeman sublevels in
the ground state.
We shall explain this method's principle in the simple case where the ground state
(as well as the excited state) has only two Zeeman sublevels=12(or=12
for the excited state). The ideas introduced for that example remain valid for more
complex transitions where theandstates have a higher degeneracy.
The principle of optical pumping is illustrated in Figure. Atoms are excited by
a resonant beam with
+
polarization, propagating along theaxis (left-hand side of
the gure). A static magnetic eld is also applied along that same axis. The absorption
of a resonant photon with
+
polarization is selective, meaning it can only excite the
=12 = 12transition, during which the magnetic quantum number
increases by one unit. Once in the excited state, and after an average time= 1,
the atom falls back, through spontaneous emission, to the=12states. The
probabilities of the various possible transitions are proportional to the square of the
Clebsch-Gordan coecients. If the atom falls into the=12state, it can reabsorb
a+photon. After a certain number of cycles, it will eventually fall in the= +12
state, where it can no longer absorb an incident photon. It will remain in that state (since
no
+
transition originates from the= +12state) and the atoms will accumulate in
2062

ANGULAR MOMENTUM EXCHANGE BETWEEN ATOMS AND PHOTONS
that sublevel. The cycles of absorption of a+photon starting from the=12state
followed by a spontaneous emission a photon transferring the atom in the= +12
level can be considered as an optical pump that empties the=12sublevel to
ll the= +12sublevel, hence the name optical pumping.
Balance of angular momentum exchanges
During an optical pumping cycle, the atom gains an angular momentum+~since it goes
from the state=12to the state= +12. The radiation looses one unit~
of angular momentum as a+photon is absorbed. Since the total angular momentum
of the system atom + radiation is conserved, the angular momentum of the radiation
emitted by the atom going by spontaneous emission from the state= +12to the
state= +12must be zero.
In order to directly demonstrate that the radiation does not lose, on average, any angular
momentum during this transition, one must take into account that, as it goes from the
= +12state to the= +12state, the atom actually emits a spherical wave
in all directions, with all possible transverse polarizations; it is therefore not correct to
say that the atom emits a single photon with apolarization, which would imply the
atom only emits photons with a wave vector perpendicular to theaxis. The correct
way is to calculate the total (orbital and spin) angular momentum of the spherical wave
emitted by the atom going from= +12to = +12. We give a brief outline of
the computation
8
as this involves expanding the eld no longer onto plane waves but onto
multipolar waves (ComplementXIX, ). Those waves are eld modes characterized
by four quantum numbers: wave number(or energy), parity equal to+1or1, total
angular momentum (which must be an integer), and nally thecomponent of the
angular momentum on the quantization axis(which varies by unit steps between
and+). For an electric dipole transition such as the one studied here, the parity equals
1and the total angular momentumequals1. It can be shown thatis equal to0
when the atom goes from to=, and is equal to1when the atom goes from
to= 1.
The correct language to describe the spontaneous emission into the ground state is as
follows: for a= 12 = 12transition, an atom in the= +12state can fall
back either into the state=12by spontaneous emission of an electric dipole photon
= +1, or into the state= +12by spontaneous emission of an electric dipole
photon = +0. This correlation betweenand is due to conservation of total
angular momentum, arising from the rotational invariance of the interaction Hamiltonian.
The nal state of the system after the spontaneous emission is thus an entangled state, a
superposition of the=12 = +1state and the= +12 = 0
state, with coecients weighted by the Clebsch-Gordan coecients of the two atomic
transitions, coming from the dipole matrix elements. This computation also yields the
speed with which the two ground state sublevels repopulate, the branching ratio being
given by the square of the Clebsch-Gordan coecients.
Light plays a double role in these experiments. As we just saw, it polarizes the
atoms by accumulating them in a ground state sublevel; it also permits the optical
detection of the atoms' polarization. As the atoms can only absorb the incident+
light if they are in the=12sublevel, the absorption of that light yields a signal
proportional to the population of that sublevel. Any change in the population dierences
8
The interested reader will nd more details on multipolar waves properties in Complement BIof
[16] and in [34
2063

COMPLEMENT C XIX
between the=12and = +12sublevels, induced by a resonant radiofrequency
eld or by collisions, can therefore be detected by a change in the absorbed light.
2-c. Original features of these methods
We now review some of the original features of these optical methods, to understand
their prominent role in the development of atomic physics.
At the time it was suggested, the double resonance was among the rst methods
to extend the magnetic resonance techniques to atomic excited states. These tech-
niques had been developed for ground states or metastables states with very long
lifetimes, using essentially atomic or molecular beams: Stern-Gerlach type exper-
imental set-ups were used to select atoms in given internal states; the ipping of
the spins by a RF eld would change the trajectories of the atoms or molecules in
a detectable way, hence allowing, in most cases, the monitoring of the magnetic
resonance (see for example reference [40]). These techniques could not be extended
to the excited states because of their very short lifetimes.
An interesting feature of these optical methods is their selectivity, both for the
excitation and the detection. This selectivity comes from the light polarization,
and not from the light frequency. The width of the spectral sources used at the
time, and the Doppler width of the spectral lines of the atoms contained in a
glass cell were considerably larger than the frequency dierences between optical
lines going from the ground state Zeeman sublevels to the excited states Zeeman
sublevels; it was thus out of the question to try to excite or detect a single Zeeman
component of the optical line.
The measurements of the Zeeman or hyperne structures in the excited states by
the double resonance method are high resolution measurements. The structures
under study are not determined by the measurement of the dierence between
two optical line frequencies, but by a direct measurement of the structure. In
the radiofrequency or microwave domain, the Doppler width is negligible and the
measurement resolution is only limited by the natural width.
The optical methods are highly sensitive methods. A radiofrequency transition
between two sublevels of the excited or ground state is not detected by the loss or
gain in energy of the radiofrequency eld, but via an absorbed or reemitted optical
photon, which has a much higher energy than a RF photon, and whose polarization
depends on which sublevel the atom is in. It is therefore possible to detect magnetic
resonances in a very dilute medium, such as a vapor.
Very high polarization ratios in the ground state, up to 90%, can be achieved
by optical pumping. Such ratios are considerably larger than those expected at
thermodynamic equilibrium: because of the very weak Zeeman shifts between the
ground state sublevels, and the high temperature at which the experiments are
conducted, the Boltzmann factorsexp(~ )are all very close to 1. Note
in addition that optical pumping may easily result, by a suitable choice of the
polarization, in a larger population for the ground state Zeeman sublevel having
the higher energy. This is one of the rst examples of a method for achieving a
population inversion, an essential condition for obtaining a maser or a laser eect.
2064

ANGULAR MOMENTUM EXCHANGE BETWEEN ATOMS AND PHOTONS
More details on the optical methods and their applications can be found in the
well documented work [24] and the references suggested therein.
3. Transferring orbital angular momentum to external atomic variables
3-a. Laguerre-Gaussian beams
In optics or atomic physics experiments, one often uses Gaussian beams, linear su-
perpositions of plane waves with wave vectors nearly parallel to an optical axis (paraxial
approximation). If all the plane waves forming the Gaussian beam have the same polar-
ization", the eld phase in planes perpendicular to the beam axis does not depend on
the azimuth anglethat determines the direction around the beam axis. New types of
Gaussian beams have recently been realized
9
, calledLaguerre-Gaussian(LG) beams, for
which the eld has an azimuthal dependence inexp , where=12, in planes
perpendicular to the beam axis. We already mentioned the existence of such beams in
of ComplementXIX. We now show how the absorption of photons from such
beams can transfer to the atomic center of mass a non-zero orbital angular momentum
with respect to the beam axis.
3-b. Field expansion on Laguerre-Gaussian modes
The LG modes form a possible basis for expanding any eld. They are charac-
terized by three quantum numbers: the wave number, the number of nodesin the
radial direction, and the integer numbercharacterizing the phase dependence on the
azimuth angle. We assume the polarization"to be uniform in the beam. We now
place an atom in that beam, and use the LG modes basis"(r). Instead of being
written
ext
n
exp(k
^
R)
ext
in, the matrix element pertaining to the external variables
of the interaction Hamiltonian in this basis is now written :
ext
n (
^
R)
ext
in (11)
This relation shows that the initial wave function
ext
in(R)of the atomic center of mass
is now multiplied by the function(R)characterizing the mode. The phase factor
exp()is, in a manner of speaking, imprinted on the initial wave function. Equation
(11) means that the transition amplitude, concerning the external variables and induced
by the interaction Hamiltonian, is equal to the scalar product of(R)
ext
in(R)and the
nal center of mass wave function
ext
n
(R).
Imagine that the initial external state of the atom has a zero angular momentum
with respect to theaxis, i.e. that
ext
in(R)does not depend on the angle. The
absorption of a photon from such an LG beam, with quantum numbers , gives
to the product(R)
ext
in(R)adependence given byexp(). This means that
in its nal state, the center of mass must have an orbital angular momentum~with
respect to theaxis, since= (~) . The LG beam has transferred to the atom's
center of mass an orbital angular momentum~. It is important to note that the
transfer's eciency, described by the matrix element (11), will only be signicant if the
9
The main method used to achieve such beams is to numerically design and fabricate holograms,
then use them to diract a Gaussian beam. For a review of the properties and applications of these new
types of beams, see reference [41].
2065

COMPLEMENT C XIX
spatial extent of the initial and nal wave functions are well adapted to the geometrical
characteristics of the LG beam. In the vicinity of the focal point, the width of the beam
is of the order of its waist0, i.e. of the order of a few microns, much larger than the
atomic wave packets, of the order of nanometers at normal temperatures( 300).
This explains why the orbital angular momentum transfer to the atoms' centers of mass
became operational only when atoms could be cooled down to very low temperatures,
in the microkelvin, or even nanokelvin range. The matter waves thus obtained, in Bose-
Einstein condensates for example, can have spatial extensions of the order of a few
microns. The transfer of orbital angular momentum by phase imprint can then be
used to generate quantum vortices (see ComplementXV, ) in matter waves,
where the atoms rotate in phase around an axis. This method was actually used to
create such vortices in a Bose-Einstein condensate of trapped atoms [42].
2066

Chapter XX
Absorption, emission and
scattering of photons by atoms
A A basic tool: the evolution operator
A-1 General properties
A-2 Interaction picture
A-3 Positive and negative frequency components of the eld
B Photon absorption between two discrete atomic levels
B-1 Monochromatic radiation
B-2 Non-monochromatic radiation
C Stimulated and spontaneous emissions
C-1 Emission rate
C-2 Stimulated emission
C-3 Spontaneous emission
C-4 Einstein coecients and Planck's law
D Role of correlation functions in one-photon processes
D-1 Absorption process
D-2 Emission process
E Photon scattering by an atom
E-1 Elastic scattering
E-2 Resonant scattering
E-3 Inelastic scattering - Raman scattering
Introduction
In this chapter, we will use the results established in the previous chapter to study
some elementary processes concerning the absorption or emission of photons by atoms.
Knowing the Hamiltonians describing the atomic energy levels and the radiation, as well
Quantum Mechanics, Volume III, First Edition. C. Cohen-Tannoudji, B. Diu, and F. Laloë.
© 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

CHAPTER XX ABSORPTION, EMISSION AND SCATTERING OF PHOTONS BY ATOMS
as their interactions, we can now focus on solving Schrödinger's equation that governs
the evolution of the system atom plus eld. Our objective is to compute the probability
amplitude for that system to go from a given initial stateat timeto a certain nal
state at a later time.
In quantum mechanics, the evolution of the system's state vector between the
instantsandis controlled by the evolution operator(), which is the basic tool
for computing the amplitudes of the various processes studied in this chapter. This is
why we start in (), which will
be useful for the forthcoming computations.
The rst processes we shall study in
by an atom undergoing a transition between two discrete states. We shall rst consider
monochromatic incident radiation, and then a broadband excitation. We will then show
in
emission, which is also predicted in a semiclassical treatment, and spontaneous emission,
which requires a quantum treatment of the radiation. We will make a connection with
the method used by Einstein to reestablish Planck's law (giving the spectral distribution
of the black body radiation), and deduce the absorption and emission coecients. The
role of correlation functions (pertaining to the atomic dipole and to the incident eld) in
the computation of transition probabilities is discussed in .
An important example that involves not one, but two photons, is the scattering of
a photon by an atom: during that process, an incident photon is absorbed and a new
one is created either by spontaneous or induced emission. This process is studied in .
When the frequency of the incident photon is close to the atomic transition frequency,
the scattering is said to be quasi-resonant. Its description requires a non-perturbative
treatment that will be developed, based on the results of XIII.
In this entire chapter, we shall only consider cases where the atomic levels are discrete;
a case where those levels include a continuum will be treated in ComplementXX.
Notation:In Chapter , it was important to distinguish between the classical
quantities and the corresponding quantum operators, so that the latter were de-
noted with hats. In the present chapter, this distinction becomes less important,
and we will come back to a more standard and simpler notation, without the hats;
for instance, the annihilation and creation operators will be denotedand,
instead of^and^.
A. A basic tool: the evolution operator
The unitary evolution operator(0)has been dened in ComplementIII; it yields
the state of a quantum system at instantknowing the state of that system at a previous
time0:
()=(0)(0) (A-1)
It is a unitary operator:
(0)(0) = (A-2)
2068

A. A BASIC TOOL: THE EVOLUTION OPERATOR
If()is the system Hamiltonian,(0)obeys the dierential equation:
}
d
dt
(0) =()(0) (A-3)
with the initial condition(00) =. The integral equation:
(0) =
}
0
()(0) (A-4)
is equivalent to the dierential equation together with its initial condition. If the Hamil-
tonian is time-independent, the evolution operator is simply:
(0) =
( 0)}
(A-5)
A-1. General properties
In this entire chapter, we use the evolution operator to express the probability
amplitude()for the system, starting from the initial stateat instant, to
be found in the stateat time:
() = () (A-6)
Consider the total Hamiltonian:
= + + =0+ (A-7)
where0= + is the non-perturbed Hamiltonian (sum of the isolated atom
Hamiltonian and the free radiation Hamiltonian) and is the interaction
Hamiltonian (between the atom and the eld). The evolution operators0andasso-
ciated respectively with0andread:
0(0) =
0( 0)~
(A-8a)
(0) =
( 0)~
(A-8b)
These operators are related through the integral relation:
(0) =0(0) +
1
~
0
d 0() (0) (A-9)
To demonstrate this relation, we take the derivative of each side (taking into ac-
count the derivative with respect to the integral's upper bound, which appears in addition
to that of the function to be integrated):
~
(0) =00(0) +0()(0) +
1
~
0
d 00() (0)
(A-10)
that is, taking into account the relation0() =and relation (A-9):
~
(0) = (0) +0 0(0) +
1
~
0
d 0() (0)
= [+0](0) =(0) (A-11)
2069

CHAPTER XX ABSORPTION, EMISSION AND SCATTERING OF PHOTONS BY ATOMS
The operator dened in (A-9) therefore obeys relation (A-3) with the total Hamiltonian
; since, in addition,(00) =, this operator is an evolution operator.
Inserting expression (A-9) for(0)into the integral on the right-hand side of
that same expression (A-9), we obtain an expression of that operator containing a double
integral in time. Reiterating this process several times, we get the series expansion of
the evolution operator in powers of:
(0) =0(0) +
1
~
0
d 0() 0(0)+
+
1
~
2
0
d
0
d 0() 0() 0( 0) + (A-12)
The-order term of this series is a succession of+ 1non-perturbed evolutions, each
described by0, separated byinteractions.
The evolution operator also obeys another integral equation, symmetric to (A-9),
which will be useful for what follows:
(0) =0(0) +
1
~
0
d() 0(0) (A-13)
Its demonstration is similar to that of (A-9).
Finally, we can also insert expression (A-13) forin the integral of (A-9), replacing
byandby. We then obtain:
(0) =0(0) +
1
~
0
d 0() 0(0)+
+
1
~
2
0
d
0
d 0() () 0( 0) (A-14)
Contrary to (A-12), the right-hand side of (A-14) only has three terms, and not an innity.
It is, however, the perturbed evolution operator, and not0, that appears in the last
term on the right-hand side of (A-14), in between the two interaction Hamiltonians.
This form of the evolution operatorwill be used in .
A-2. Interaction picture
For the following computations, it will often be useful to write Schrödinger's equa-
tion in the interaction picture. Let()be the Schrödinger state vector. Setting:

()=
0
(0)()= exp [( 0)0~]() (A-15)
the new state vector

()obeys the time evolution equation:
~
d
d

()=

()

() (A-16)
where

()is dened as:

() =
0
(0) 0(0) (A-17)
2070

A. A BASIC TOOL: THE EVOLUTION OPERATOR
The evolution operator

(0)in the interaction representation yields the evolu-
tion of

():

()=

(0)

(0) (A-18)
From equations (A-1) and (A-15), we can deduce the relation between evolution operators
in the two points of view:

(0) =
0
(0)(0) = exp [( 0)0~](0) (A-19)
In addition, insertion of (A-18) into (A-16) shows that:
~

(0) =

()

(0) (A-20)
which leads to the following series expansion for

(0)(perturbation expansion):

(0) =+
1
~
0
d

() +
1
~
2
0
d
0
d

()

() + (A-21)
The great advantage of the interaction picture is that the state vector only evolves
under the eect of the interaction since

()is the only operator appearing on the
right-hand side of (A-16). We shall see that this point of view also allows expressing the
transition probabilities in terms of time correlation functions of dipole and eld operators,
i.e. as average values of products of physical quantities taken at two dierent instants,
and evolving freely (under the eect of only0). Finally, when trying to calculate
the transition probability between two eigenstatesand of0, with respective
energiesand, it is often convenient to use the interaction picture since, according
to (A-19), the transition amplitude is of the form:
(0)= exp [( 0)0~]

(0) (A-22)
=
( 0)}
(0) (A-23)
As the phase factor
( 0)}
disappears from the probability (modulus squared of
the amplitude), it can be ignored; this allows simply replacing the evolution operator
by

and directly using the more compact expansion (A-21).
In this entire chapter and its complements, we describe the interaction between
atom and eld by the electric dipole Hamiltonian=DE(R)introduced in
of Chapter . For the sake of simplicity, we assume that0= 0. In the interaction
picture
1
, this operator becomes:

() =

D()

E(R) (A-24)
where:

D() = exp (0~)Dexp ( 0~) (A-25a)

E() = exp (0~)Eexp ( 0~) (A-25b)
1
Here, the atom's external degrees of freedom are treated classically. The atom is supposed to be at
rest at pointR, meaningRis not modied when going to the interaction picture.
2071

CHAPTER XX ABSORPTION, EMISSION AND SCATTERING OF PHOTONS BY ATOMS
A-3. Positive and negative frequency components of the eld
In the present chapter, we assume the system is contained in a cubic box of volume
3
. The transverse electric eld is given by relation (B-3) of Chapter . It is a linear
combination of annihilationand creationoperators, and can be expressed as the
sum of two terms:
E(R) =E
(+)
(R) +E
()
(R) (A-26)
where:
E
(+)
(R) =
~
20
3
12
"
kR
E
()
(R) =
~
20
3
12
"
kR
=E
(+)
(R) (A-27)
OperatorE
(+)
(R), obtained by keeping only the annihilation operatorsin the expan-
sion of

E(R), is called
2
the electric eld positive frequency component. As for the
operatorE
()
(R), it is the negative frequency component. These two operators are
not Hermitian, and do not commute. In a product of eld operators, the order is said
to benormalif the creation operators are to the left of the annihilation operators, as in
E
()
E
(+)
; the order is said to beantinormalfor a product in the inverse order, as in
E
(+)
E
()
.
In the interaction picture, the annihilation and creation operators become:
() = exp (0~)exp ( 0~) =
() = exp (0~)exp ( 0~) =
+
(A-28)
(these equalities can be veried by computing the matrix elements in the Fock state
basis, eigenvectors of0, and using the fact that the only action of operatoris to
annihilate a photon in the mode). The positive and negative frequency components of
the eld are thus:

E
(+)
(R) =
~
20
3
12
"
(kR )

E
()
(R) =
~
20
3
12
"
(kR )
=

E
(+)
(R) (A-29)
Suppose now that we wish to study the lowest order process of photon absorption
by atoms. To compute the action of

()on the system's initial state, we can keep
2
The positive frequency component annihilates a photon, the negative frequency component creates
one. Furthermore, in the Heisenberg picture, we shall see that the free evolution of the positive component
goes as , and that of the negative one, as
+
. This labeling as positive frequency may seem
somewhat counter intuitive, but is widely accepted.
2072

B. PHOTON ABSORPTION BETWEEN TWO DISCRETE ATOMIC LEVELS
only the terms in

()that annihilate one photon, i.e. the Hamiltonian terms contain-
ing(). This amounts to keeping only the positive frequency component of the eld
operator, and using the simplied interaction Hamiltonian:

()

D()E
(+)
(R) (absorption) (A-30)
In a similar way, to study the lowest order photon emission process, we can use the
expression:

()

D()E
()
(R) (emission) (A-31)
B. Photon absorption between two discrete atomic levels
We start with monochromatic radiation ( ), and will study later the broadband
radiation case (). The base of the computation is the study of the transition rate
between stationary states of the non-perturbed Hamiltonian0; ComplementXXwill
present a more detailed study in terms of wave packets propagating in free space, built
from coherent superpositions of stationary states.
B-1. Monochromatic radiation
B-1-a. Probability amplitude (absorption)
We callandtwo discrete levels with respective energiesand, and set:
=~0 (B-1)
where02is the atomic eigenfrequency, assumed to be positive (thelevel has an
energy higher than thelevel). For the sake of simplicity, we shall ignore the external
variables, which amounts to considering the atom as innitely heavy and at rest
3
.
We assume the radiation is at the initial time= 0in a state
in=
containingphotonswith wave vectork, polarization"and frequency2; it is a
monochromatic radiation. The initial state of the system atom+radiation is written:
in=; with energyin=+~ (B-2)
We are trying to compute the probability amplitude for the atom to absorb a photon
and be in the excited stateat instant= . The nal state of the system must then
be:
n=; 1 with energyn=+ ( 1)~ (B-3)
As mentioned above, wheninand nare eigenstates of0, it is easier to
carry out the calculation in the interaction representation. In the expansion (A-21) of

,
the lowest order term that can link those two states is the rst order term in. Calling
3
ComplementXIXshows how taking into account the external variables and momentum conser-
vation allows introducing the Doppler eect and the recoil eect in the absorption and emission of a
photon. It also shows how the connement of the atom in a region of space by a trapping potential
allows controlling those eects.
2073

CHAPTER XX ABSORPTION, EMISSION AND SCATTERING OF PHOTONS BY ATOMS
the duration of the interaction, the probability amplitude for the system, initially in
the stateinat time0= 0, to be at time= in the nal statenis:
n

()in=
1
~

0
d n
0~ 0~
in
=
1
~
n in

0
d
(n in)~
(B-4)
or else, taking into account the relationn in=~(0 ):
n

(0)in=
1
~
n in

0
d
(0 )
= n in
1
~(0 )
(0 )
1
(B-5)
which leads to:
n

(0)in=
~
n in
(0 )2
2 sin (0 ) 2
(0 )
(B-6)
The absorption probability is the squared modulus of that expression:
n

(0)in
2
=
1
}
2
n in
24 sin
2
(0 ) 2
(0 )
2
(B-7)
Taking relations (A-24) and (A-27) into account, the matrix element of, appearing
in the amplitude (B-5), can be written:
n in=
~20
3"D
kR
(B-8)
so that the probability becomes:
n

(0)in
2
=
2~0
3
"D
24 sin
2
(0 ) 2
(0 )
2
(B-9)
It is proportional to the number of incident photons, i.e. to the incident intensity in
the statein, as well as to the squared modulus of the atomic dipole matrix element
between the statesand(sinceDis an odd operator, the absorption of the photon
can only occur between two states of dierent parity). This probability is an oscillating
function of the time.
B-1-b. Energy conservation
The presence in the denominator of (B-9) of the factor( 0)
2
means that, the
closergets to0, the larger the absorption probability will be. A photon absorption
is said to be resonant when the absorbed photon energy is exactly equal to the energy
the atom must gain to go fromto(energy conservation). The width of the resonance
described by (B-9) is of the order of= 1or, in energy, of the order of=~.
This is consistent with the time-energy uncertainty principle: in a process extending over
a length time, the energy is only conserved to within~.
2074

B. PHOTON ABSORPTION BETWEEN TWO DISCRETE ATOMIC LEVELS
Actually, when the variable is the angular frequency, one may consider the
function within brackets in (B-6) to be an approximate delta function
()
(0 ),
having a non-zero width of the order of1. As relation (10) of Appendix
II shows that the integral overofsin[(0 )2](0 )is equal to, and since
(0 )is proportional to the dierence in total energy between the nal and initial
states, equation (B-6) can be written (ignoring the phase factor):
n

(0)n 2 n in
()
(n in) (B-10)
B-1-c. Limits of the perturbative treatment
The lowest order perturbative treatment inthat we have used cannot remain
valid for arbitrarily long times. To understand why, imagine for example that the res-
onance condition=0is satised. Sincesin()tends towardwhengoes
to 0, the absorption probability predicted by (B-9) becomes proportional to
2
, which
gets very large as the time intervalincreases. Now a probability can never be larger
than one; it is therefore obvious that expression (B-9) is no longer valid for long times.
The same is true ifgets close to0without being strictly equal to it: it is then quite
possible for the oscillation amplitude of the right-hand side of (B-9) to be larger than
one. The previous results can thus only be used for short enough times, ensuring the
validity of the perturbative treatment.
We shall use in ComplementXXthe dressed atom method to get a more precise
treatment of the coupling eects between a two-level atom and a single mode of the eld.
The coupling intensity will be characterized by a constant1called the Rabi frequency.
At resonance, one shows that the probability amplitude presents a Rabi oscillation
4
insin 1. The quadratic behavior in
2
found above for the absorption probability
at resonance is simply the rst term of the series expansion ofsin
2
(1)in powers of
1. We shall also discuss the extent to which relation (B-9) can be used far from
resonance.
B-2. Non-monochromatic radiation
We now study the absorption and emission processes when the radiation is no
longer monochromatic. The transitions are still between two discrete atomic levelsand
; the case of a continuum of atomic levels is discussed in ComplementXX.
B-2-a. Absorption of a broadband radiation
We now assume the initial state of the system atom+radiation to be of the form:
in=;
in (B-11)
The atom is still in the internal state of lower energy, but the radiation is now in a
state
inwhere photons occupy several modes with dierent frequencies. The radiation
frequency distribution is characterized by a certain spectral band. We shall see below
(very last part of ) the conditionmust satisfy for the results obtained in
4
We shall also show how this Rabi oscillation is modied when taking into account the width of the
excited level due to spontaneous emission.
2075

CHAPTER XX ABSORPTION, EMISSION AND SCATTERING OF PHOTONS BY ATOMS
this section to be valid. The calculations will be carried out in the same way as in ,
while focusing more on the time correlation properties of the incident eld.
We introduce the notation:
D= with: =e (B-12)
whereeis the unit vector
5
(a priori complex) parallel to, andthe (real) modulus
of that vector. In the interaction picture (with respect to the free atom Hamiltonian),
this equality becomes:

D()=
0
D
0
=
0
(B-13)
. Transition probability
Let us insert expression (A-24) for

()into (A-21). The rst order term in

()
yields the probability amplitude for the system, starting at= 0from the state;
in,
to be found at time= in the state;
n
. Taking (B-13) into account, we obtain:
n

()in=
1
~

0
d

D()
n

E(R)
in
=
~

0
d
0
n
(+)
d
(R)
in (B-14)
where we have included only the positive frequency component of the eld, which is the
only one involved since
n
contains less photons than
in(we are dealing with an
atomic absorption process); in this equality and the following, we shall use the convenient
notation:
(+)
d
(R) =e

E
(+)
(R);
()
d
(R) =e

E
()
(R) (B-15)
where

E
(+)
(R)and

E
()
(R)are the eld operators in the interaction picture dened
in (A-29); the atom is supposed to be xed at pointR. The corresponding transition
probability
abs
()is obtained by squaring the modulus of amplitude (B-14), and then
summing over all possible radiation nal states. Replacing in (B-14) the integral variable
by, we obtain:
abs
() =
2
~
2

0
d

0
d
0( )
() (B-16)
where()is dened as:
() =
n
in
()
d
(R)
n n
(+)
d
(R)
in
=
in
()
d
(R)
(+)
d
(R)
in (B-17)
5
The matrix elements of the three components ofare three a priori complex numbers. We
set=
2
+
2
+
2
and introduce the vectore= . It is a unit vector sinceee= 1;
we have=e =eD.
2076

B. PHOTON ABSORPTION BETWEEN TWO DISCRETE ATOMIC LEVELS
This function characterizes the role of the incident beam in the absorption process under
study. It is the average value in the initial state of the product of two eld operators,
arranged in normal order (), and taken at two dierent times.
Changing the integral variablefor= (the jacobian of this change of
variables is equal to unity), equation (B-16) becomes:
abs
() =
2
~
2

0
d

d
0
(+) (B-18)
The transition probability is therefore proportional to the time integral of the Fourier
transform
6
with respect toof the function(+), limited to the time interval
[ ].
. Excitation spectrum
If the radiation initial state
inis an eigenstate of the free radiation, the function
()depends only on the dierence. For example, if
inis a Fock state
7
:
in= 1 (B-19)
where the populated modes have the polarization", inserting expansions (A-29) for the
eld operators leads to:
( ) =
~
20
3in in
=
e"
2 ( )
=d ()
( )
(B-20)
where:
() =
~
20
3
e"
2
( ) (B-21)
If the radiation is initially in the Fock state (B-19), the value ofis simply, the
number of photons in that state. On the other hand, if the radiation is in a statistical
mixture of such states (see note),represents the corresponding statistical average.
Expression (B-20) thus appears as the value at of the Fourier transform
of a function()of, which depends on the initial photon populations. As
~is the average energy of modewith frequency2, the function()actually
gives the variation of the energy density of the radiation as a function of frequency; this
function is also referred to as the spectral distribution of the incident radiation (excitation
spectrum). This distribution can have, a priori, any shape, but the energy density often
presents a single peak of width, with no other particular structure. Its Fourier
transform( )is then a function of width1; as a result, when
6
The components of the eld
(+)
d
vary in e. This is why the Fourier component ofat 0
appears in (B-18).
7
Instead of a Fock state as the initial state, one could choose a statistical mixture of such states with
arbitrary weights; this would not signicantly change the following calculations and conclusions.
2077

CHAPTER XX ABSORPTION, EMISSION AND SCATTERING OF PHOTONS BY ATOMS
Figure 1: The function to be integrated in expression (B-16) yielding the absorption
probability is signicant only in a band along the rst bisector, of width1, where
is the width of the incident radiation spectrum. If 1, the area of the domain
actually contributing to that integral increases linearly with the diagonal of the square,
and hence with(and not with its square).
becomes large compared to1, the correlation function( )goes to zero and
can be neglected. We shall see that in such a case the probability becomes proportional
to, which naturally leads to introducing a probability per unit time. Our reasoning
will be similar to that of ComplementXIIIfor a classical perturbation, but here the
radiation is treated quantum mechanically.
. Probability per unit time
The integration domain of the double integral in (B-16) is plotted in Figure. As
( )goes to zero as soon asis large compared to1, the portion of that
domain where the function to be integrated is not negligible is a band of width1,
along the rst bisector; the width of this band is very small compared to the domain
extension if 1. To make use of that property, we again replace the integral
variableby= (the associated Jacobian for this change of variables in the
double integral is equal to1):

0
d

0
d=

0
d

d (B-22)
In the second integral, the values ofthat actually yield a non negligible contribution
are of the order of the correlation time1of( ). If we assume 1,
we can replace the limits of that second integral byand+. Inserting then (B-20)
into (B-18) yields the integral:

0
d
+
d
+
d
(0)
() = 2 (0) (B-23)
2078

B. PHOTON ABSORPTION BETWEEN TWO DISCRETE ATOMIC LEVELS
since the summation overdyields the function2(0 ), which allows integration
overd; we get a function independent ofwhose integration is proportional to.
We nally obtain:
abs
() =
2
~
2

2
(0) =
2
~
2

2
~
20
3
e"
2
( 0)(B-24)
This means that
abs
()increaseslinearlywith, which leads us to dene an
absorption probability per unit time (absorption rate):
abs
=
abs
()

= 2
2
~
2
(0) (B-25)
(remember that, for monochromatic radiation, we found an absorption probability in-
creasing not linearly but as the square of). This absorption rate is proportional to
the radiation energy density at the atomic transition frequency0. Formulas (B-21)
and (B-25) give the dependence of the absorption rate on the various parameters of the
incident radiation (populations of the modes, polarization").
Our calculation is perturbative since it was carried out to the lowest order in.
It is therefore only valid for timessuch that
abs
() =
abs
1, i.e. such that
1
abs
. In addition, we saw above that the linear variation of
abs
()with
is obtained only if 1. These two inequalities are compatible if
abs
.
The approximation to lowest order is thus valid only if the broad band radiation has a
spectral width large compared to the absorption rate.
B-2-b. A specic case: isotropic radiation
The previous calculations can be pushed a step further when the radiation is
isotropic, meaning when, the average number of photons in mode, depends only
on, and neither on the direction of the wave vectorknor on the polarization". The
results we shall obtain for this specic case will be useful for later computations (
4) on the spontaneous emission rate, as well as for the isotropic radiation at thermal
equilibrium.
Consider the limit of (B-25) when the volume
3
containing the system goes to
innity. The summation over the indexcan be replaced by an integral:
3
(2)
3
2
dd
"k
(B-26)
where, for isotropic radiation, a sum is taken over two linear polarizations perpendicular
to àk. We assume that the vectoreis real as well, and choose anaxis that is parallel
to it
8
. We rst calculate, for a given direction ofk, the sum of the quantities
2
for the
two polarizations. We take two polarization vectors,"1and"2, both perpendicular to
k, and perpendicular to each other. Imagine the rst one is in the plane containingk
and theaxis, so that1= sin, whereis the angle betweenkand theaxis; the
8
Ifeis complex, it is easy to see that the contributions of its real and imaginary part simply add
in (B-24). The sum of these two contributions then gives (B-27) again.
2079

CHAPTER XX ABSORPTION, EMISSION AND SCATTERING OF PHOTONS BY ATOMS
second is necessarily perpendicular to that plane, which means2= 0. We then have
"k
2
= sin
2
. The integral over all the directions ofkis then trivial:
d sin
2
= 2
0
dsin
3
=
8
3
(B-27)
We are left with the integral of the modulusofk. Using (B-25), (B-26), (B-27) and
changing the variableinto the variable=, we nally obtain:
abs
=
23
0
30~
3
(0) (B-28)
C. Stimulated and spontaneous emissions
C-1. Emission rate
We now assume that the atom is initially in the upper state, and that the radiation
is in the initial Fock state (B-19); we study the emission processes where the atom falls
back into the statewhile emitting a photon. We still assume that the spectrum of the
incident radiation is broad, so that its correlation function tends to zero in a time that
is much shorter than. The computations are then similar to the one we just did for
the absorption process, but with a certain number of changes. First of all, we must now
use expression (A-31) for the interaction Hamiltonian, the one that contains the negative
frequency component of the eld operator (with solely creation operators). Secondly,
concerning the electric dipole operator, we must only keep the term connectingto,
which amounts to replacing (B-13) by the complex conjugate expression:
~
D()=
0
=e
0
(C-1)
The correlation function (B-17) of the eld in normal order is now replaced by the
correlation function in antinormal order():
() =
in
(+)
d
(R)
()
d
(R)
in (C-2)
For the radiation state (B-19), this function is given by:
( ) =
~
20
3
e"
2 ( )
=
(+ 1)~
20
3
e"
2 ( )
(C-3)
In the present case, it is nowthat is present in (C-3), and not as in (B-20).
Sinceanddo not commute, this leads to an important dierence compared with
expression (B-20) for( ): the number of photonsinitially populating the
modeis replaced by+ 1. It is the quantum character of the eld (non-commuting
operators) that is responsible for these essential dierences between the absorption and
emission processes.
2080

C. STIMULATED AND SPONTANEOUS EMISSIONS
The emission rate is obtained by computations similar to those that led to the
absorption rate (B-25) (with a change of sign for0and for the). We obtain:
em
=
em
()

= 2
2
~
2
[+ 1]~
20
3
e"
2
( 0) (C-4)
This result diers from the absorption rate only by the replacement ofby +
1. This formula gives the general expression of the emission rate as a function of the
population of the modes and of their polarization". We shall now discuss its two
components, one proportional to, and one that does not depend on.
C-2. Stimulated emission
We rst consider the terms in (C-4) that contain, i.e. the contribution of the
modes containing initially at least one photon. These terms correspond to an emission
induced by the incident radiation; their rate is proportional to the incident light intensity.
For this reason it is calledstimulatedemission. Its rate is the same as the absorption
rate, since the terms depending onare identical in (B-24) and (C-4):
stimem
=
abs
(C-5)
In particular, for isotropic radiation, we get a result identical to (B-28):
stimem
=
23
0
30~
3
(0) (C-6)
A photon resulting from stimulated emission is created in the same modeas the
photons inducing that emission: the numberof photons in that mode goes fromto
+ 1. The added photon has the same energy, same direction and same polarization as
the initialphotons. If the incident radiation is coherent, one can show that radiation
emitted by stimulated emission has the same phase as the incident one. This results
in a constructive interference eect (in the direction of the incident radiation) between
the radiation emitted by the induced dipole and the incident radiation, hence leading to
an amplication eect. For this phenomenon to occur, the atomic populations must be
inverted, meaning that the probability of occupation of the upper levelmust be larger
than that of the lower level. However, if this is not the case, the interference is destruc-
tive, which explains the attenuation of an incident beam by the absorption process. The
coherent amplication by stimulated emission of an incident beam propagating through
atoms with an inverted population plays an essential role in laser systems. The word
laser is an acronym of Light Amplication by Stimulated Emission of Radiation.
C-3. Spontaneous emission
If all theare equal to zero, the radiation is initially in the vacuum state; the
absorption rate (B-25) is then equal to zero. On the other hand, the emission rate (C-4)
is not, because of the term1in = + 1. It follows that an atom, initially
in the upper stateand placed in a vacuum of photons, has a non-zero probability per
unit time of emitting a photon and falling back into the lower state. This is called the
spontaneous emissionprocess.
2081

CHAPTER XX ABSORPTION, EMISSION AND SCATTERING OF PHOTONS BY ATOMS
The term1appearing in (C-4) is the same for all the modes, as opposed to the
term that only exists for the modes already containing photons. As this term1does
not favor any particular direction or polarization, the situation is similar to that where
the radiation is isotropic; the calculations leading to (B-28) remain valid provided we
replace(0)by1. This leads to the rate of spontaneous emission:
spontem
=
23
0
30~
3
(C-7)
This result could also be obtained by Fermi's golden rule (see Chapter ,
b) in the following way. The system atom+radiation starts from the initial state; 0
(atom in state, photon vacuum). This discrete state is coupled to an innity of nal
states; 1, atom in statewith one photon in mode. The golden rule enables one
to compute the probability per unit time of the transition from the discrete initial state
towards the continuum of nal states, which is simply the spontaneous emission rate:
emspont
=
2
~
; 1 ; 0
2
(~ ~0) (C-8)
Using; 0; 1=(e")(~20
3
)
12
, for a real polarization, as well as equa-
tions (B-26) and (B-27), we get the same result as (C-7).
This equation shows that the spontaneous emission rate increases as the cube of
the atomic transition frequency; this explains why spontaneous emission is negligible in
the radiofrequency domain, becomes important in the optical domain, and even more so
in the ultraviolet or X ray domain. This
3
0factor has two origins: on one hand, the
square of the
12
factor appearing in the electric eld expression, on the other, the
2
factor present in the density of nal states.
The spontaneous emission rate
spontem
is also called the natural width of the
excited state, and noted. The inverse ofis the radiative lifetime of the excited
state, which is the average time necessary for the atom to undergo radiative decay:
=
spontem
=
1
(C-9)
It is instructive to compare the rate
spontem
to0. It follows from (C-7) that:

0
=
22
0
30~
3
(C-10)
The dipoleis of the order of0whereis the electron charge and0the Bohr radius.
This yields the quantity
2
(30~)which, within a factor43, is the ne-structure
constant 1137multiplied by
2
0
2
0
2
. Since00is of the order of the electron
velocity in the rst Bohr orbit, which istimes smaller than the speed of light,
2
0
2
0
2
is of the order of
2
, so we nally have:

0
3
(C-11)
The natural width of the excited level is therefore much smaller than the atomic tran-
sition frequency: the atomic dipole may oscillate a great number of times before these
oscillations are damped. Typically, in the optical domain,is of the order of10
7
to10
9
s
1
whereas02is much larger, of the order of10
14
to10
15
s
1
.
2082

C. STIMULATED AND SPONTANEOUS EMISSIONS
C-4. Einstein coecients and Planck's law
Let us now assume the radiation is at thermal equilibrium at temperature(black
body radiation). In this case, it is current practice to use another notation for the
various absorption and spontaneous or stimulated emission rates: the Einsteinand
coecients from his 1917 article. In that article, he introduces for the rst time the
concept of stimulated emission:
abs
=
stimem
=
spontem
= (C-12)
One can then write the change per unit time of the populationsand of the
andlevels due to the various absorption and emission processes:
_=
_= + + (C-13)
As an example, on the right-hand side of the rst line of (C-13), the rst term describes
how levellls up whenabsorbs a photon, the second term how it is emptied by
stimulated emission towards, the third one how it is emptied by spontaneous emission.
Similar explanations can be given for the second line.
In a steady state, there is a balance between the various processes, and we have:
_= _= 0 (C-14)
We then get from the rst equation:
=
+
(C-15)
Now, according to the Boltzmann distribution law,must be equal to
(~0 )
.
Relations (B-28), (C-6) and (C-7) then show that, at equilibrium, the populations
and obey the relation:
=
(~0 )
=
+
=
(0)
(0)+ 1
(C-16)
which means that:
(0)=
(~0 )
1
(~0 )
=
1
(~0 )
1
(C-17)
Multiplying the average energy per mode~0(0)by the mode density8
2
0
3
in
the vicinity of0, yields Planck's law for the energy density per unit volume of the black
body radiation, as a function of the frequency0=02:
(0) =
8
3
0
3
1
(0 )
1
(C-18)
In other words, when an ensemble of two-level atoms reaches Maxwell-Boltzmann equi-
librium through absorption and spontaneous or stimulated emission of radiation, that
radiation must necessarily obey Planck's law. This is the essence of the argument used
by Einstein to establish this law
9
.
9
Einstein could not reason in 1917 in terms of the quantum theory of radiation, which was not
available. The heuristic introduction of theandcoecients illustrates his remarkable intuition.
2083

CHAPTER XX ABSORPTION, EMISSION AND SCATTERING OF PHOTONS BY ATOMS
D. Role of correlation functions in one-photon processes
The probabilities associated with one-photon processes can be expressed in terms of atom
and eld correlation functions.
D-1. Absorption process
Let us write again the probability amplitude for the system, starting at= 0
from the state;
in, to end up at time= in the state;
n
after undergoing
a one-photon transition see (B-14):
n

()in=
1
~

0
d

D()
n

E(R)
in (D-1)
When dealing with an absorption of radiation by the atom, the number of photons in
the nal state
n
is lower than in the initial state
in; only the positive frequency
component of the eld

E
(+)
, which can destroy photons, yields a contribution to the
matrix element
n

E(R)
in. Let us start with a radiation initial state containing
photons with a single given polarization"in; only modes with this particular polarization
are involved in the matrix element. If the polarization is not linear (circular for instance),
"inis complex, and we must replace it by its complex conjugate"
inis all negative
frequency components of the eld. Moreover, since"
in"in= 1, the"inpolarization
components are obtained by a scalar product of"
inby the eld. We then have:
n

()in=
1
~

0
d "in

D()
n"
in

E
(+)
(R)
in (D-2)
The probability
abs
()of the absorption process is then:
abs
() =
1
~
2

0
d "
in

D()
in"in

E
()
(R)
n

0
d "in

D()
n"
in

E
(+)
(R)
in (D-3)
To obtain the probability of nding the atom in any nal stateother than, whatever
nal state
n
the radiation is in, we must sum this result over all possibleand
n
states. This yields two closure relations, one in the atom state space
10
, the other in the
radiation state space. This leads to the following result:
abs
() =
1
~
2

0
d

0
d a()() (D-4)
with the denitions:
a() = "
in

D()"in

D() (D-5)
10
In the summation over, the initial atomic statecan be included since the matrix element of the
atomic dipole in that state is zero (because of a parity argument).
2084

E. PHOTON SCATTERING BY AN ATOM
and:
() =
in"in

E
()
(R)"
in

E
(+)
(R)
in (D-6)
The two functions we just dened correspond, respectively, to the correlation function of
the atomic dipole and to that of the electric eld expressed in normal order.
In the more general case where the eld initial state includes several polarizations,
(D-1) must now include the matrix elements:
=

()
n

(R)
in (D-7)
with:

() =e

D()and

(R) =e

E(R) (D-8)
whereeis the unit vector of each of the three= axes. Probability (D-4) then
becomes:
abs
() =
=
1
~
2

0
d

0
d
a() () (D-9)
with the following denitions of the9dipole correlation functions, and the other9eld
correlation functions (the vectorseare real) :
a() = e

D()e

D()
() =
ine

E
()
(R)e

E
(+)
(R)
in (D-10)
We thus get a correlation tensor, which slightly complicates the equations, but does not
change the essence of the results.
D-2. Emission process
For the photon emission processes, spontaneous or stimulated, we can make sim-
ilar computations. The main dierence is that the

E
(+)
and

E
()
operators must be
interchanged, which yields antinormal instead of normal eld correlation functions; fur-
thermore, we must use the more general formula (D-9) instead of (D-4) since an emission
process does not favor any specic polarization.
E. Photon scattering by an atom
We now consider a photon scattering process where the initial state includes an atom in
stateand an incident photon, and where in the nal state the atom is still in state,
but the incident photon has been replaced by another one. This is a two-photon process,
since one photon disappears, and is replaced by another one.
2085

CHAPTER XX ABSORPTION, EMISSION AND SCATTERING OF PHOTONS BY ATOMS
E-1. Elastic scattering
As in , we shall treat classically the center of mass of the atom, supposed to be
xed at the origin of the reference frame (see however the comment on page ). The
initial stateinof the system atom+radiation at timecorresponds to an atom in
statein the presence of a photon in mode, with wave vectork, polarization"and
frequency(we assume the radiation to be monochromatic):
in=;k" with energy in=+~ (E-1)
At the nal time, the scattering process has replaced the initial photonk"by a new
photonk"with wave vectorkand polarization". The nal state of the system is:
n=;k" with energy n=+~ (E-2)
Conservation of total energy requires=. This scattering thus occurs without
frequency change and is called for that reason elastic scattering. We shall study in

dierent process is called Raman scattering.
As the electric dipole interaction Hamiltonian (A-24) can only change the photon
number by one unit, the system must go through an intermediate state often called a
relay state where the atom is in stateand the radiation in a state dierent from its
initial state. The lowest order term of the series expansion (A-12) that contributes to
the scattering amplitude is of order two.
E-1-a. Two possible types of relay state
There are two possible types of relay states: those corresponding to processes we
shall label(), where the photonk"is absorbed before the photonk"is emitted;
and those corresponding to processes labeled(), where the photonk"is emitted
before the photonk"is absorbed. In the rst case, the relay state is the state
rel
=
; 0, whereis an atomic relay state and0is the radiation vacuum, since the photon
k"present in the initial state has been absorbed; the energy of this relay state is
rel
=. In the second case, the relay state is
rel
=;k";k", since the
photonk"has been emitted before the photonk"was absorbed: the energy of
this relay state is
rel
=+~+~. Figures
representing these same two processes.
In Figure
represents an absorption, whereas a downwards arrow represents an emission. The ad-
vantage of this representation is to directly show the energy dierence between the initial
state and the relay state, equal toin rel
=+~ for the()processes, and
toin rel
= ~ for the()processes: this dierence is simply the dis-
tance between the height of the dashed line and the height of the line representing the
atomic relay state. In particular, these two lines coincide for the()processes when
+~=, i.e. when the absorption of the incident photon is resonant for the
transition (resonant scattering, which will be studied later on).
In Figure, an incoming arrow represents an absorption, whereas an outgoing one
represents an emission. Reading the diagram from bottom to top, one clearly sees which
atomic state and photons are present in the initial state, the relay state and the nal
2086

E. PHOTON SCATTERING BY AN ATOM
Figure 2: First diagram representation of the scattering processes labeled()and()
in the text; these processes have a dierent chronological order for the absorption of
the incident photon and the emission of the scattered photon. The full horizontal lines
represent the atomic levels, and the upwards arrows represent absorption processes and
downwards arrows, emission processes; the horizontal dashed lines clearly show the energy
dierences that will appear in the denominator of the transition amplitude expression.
state. For the()processes, no photons are present in the relay state, whereas both
incident and scattered photons are present in that state in the()processes.
E-1-b. Computation of the scattering amplitude
The computation of the scattering amplitude is of the same type as the calcula-
tions already presented above. In addition, it is almost identical to the computation of
the two-photon absorption amplitude explained in detail in XX.
Consequently, it will not be explicitly carried out here, but we shall merely highlight the
dierences with the computation of that complement. The reader interested in more de-
tails may want to read that complement before continuing with this paragraph. Relations
(13) and (14) of that complement are written here:
n

()in=2 (in) +(in)
()
(n in)
=2 (in) +(in)
()
(~ ~) (E-3)
where we have introduced the probability amplitudes:
(in) =
rel
nDE
()
relrel
DE
(+)
in
in rel
=
~
20
3
"D "D
+~
(E-4)
2087

CHAPTER XX ABSORPTION, EMISSION AND SCATTERING OF PHOTONS BY ATOMS
Figure 3: Another possible diagram representation of the scattering processes()and().
The state of the atom is shown on the vertical full line. As for the photons, the incom-
ing arrows (wiggly arrows pointing towards the vertical full line) represent absorption,
whereas outgoing ones (wiggly arrows leaving the vertical full line) represent emissions.
The bottom to top reading of each diagram shows the chronological succession of the states
of the system atom + photons.
(in) =
rel
nDE
(+)
relrel
DE
()
in
in rel
=
~
20
3
"D "D
~
(E-5)
The two amplitudes(in)and (in)correspond to the two types of relay states
considered above. In (E-3), the delta function expressing the energy conservation is
proportional to
()
( )sincen in=~( ). To write (E-4) and (E-
5), we have replaced in relation (13) of ComplementXXthe interaction Hamiltonian
byDE
(+)
orDE
()
, depending on whether it pertains to an absorption or
an emission process. In the numerator of the fractions on the right-hand side of (E-4),
operatorE
(+)
(which absorbs the incident photon) acts before operatorE
()
(which
creates the scattered photon); this is to be expected for an()type process. The order
of the two operatorsE
(+)
andE
()
is reversed in (E-5), as expected for a()type
process. The
coecients on the second lines of these equalities come from the
plane wave expansion (A-27) of the electric eld;is the edge of the cubic cavity used to
quantize the eld. We could have added a factor
, whereandare the photon
numbers in the initial and nal states; for the sake of simplicity, we have assumed that
== 1.
2088

E. PHOTON SCATTERING BY AN ATOM
E-1-c. Semiclassical interpretation
Elastic scattering can also be explained by a semiclassical treatment, where the
quantum treatment only applies to the atom; the incident wave is described as a classical
eld of frequency. This wave induces, in the atom, an oscillating dipole at the same
frequency. This dipole radiates into the entire space a eld oscillating at that frequency.
The semiclassical approach also enables a simple interpretation of the absorption
of the incident beam as resulting from a destructive interference, in the direction of the
incident eld, between that eld and the eld scattered by the induced dipole. One can
also use such a description to account for the amplication of an incident beam by an
ensemble of atoms whose population has been inverted, meaning atoms for which the
population of an excited level is larger than the population of a lower energy level. The
scattered eld then has the opposite phase of that it would have without population
inversion, so that the interference becomes constructive.
E-1-d. Rayleigh scattering
Assume that the frequencyof the radiation is much smaller than all the atomic
frequencies ~. One can then ignoreand in the denominators on the
second lines of (E-4) and (E-5). The onlydependence of the scattering amplitudes
comes from the prefactor
, equal tosince=. The scattering cross section
involves the product of the squared modulus of that amplitude, proportional to
2
, by
the density of the radiation's nal states at frequency=, also proportional to
2
.
The scattering cross section therefore varies as
4
, much higher for blue light than for
red light.
One usually calls Rayleigh scattering the elastic scattering when
~. It explains the scattering of the visible solar light by the atmospheric oxygen
and nitrogen molecules, which have much higher resonant frequencies, in the ultraviolet
domain. This rapid variation with frequency of the Rayleigh scattering cross section is
a reason for the sky being blue.
E-2. Resonant scattering
Assume now that the frequencyof the incident photon is very close to the
frequency:
= ( )~ (E-6)
of a transition between a stateand a statehaving a higher energy. The absorption
of the incident photon is then resonant for the transitionand the amplitude (E-4)
becomes very large when the statebecomes the atomic relay state it even diverges if
the resonant condition is exactly satised. In that case, one can neglect all the()type
processes; in addition, even if there are other possible atomic relay states,,.., one
can keep only the term involving.
To avoid the diculties related to the divergence of (E-4) when= , it is
convenient to use the exact expression (A-14), which only involves three terms, instead of
an innity as in expansion (A-12) for the evolution operator. Only the last of those three
terms plays a role, since it can destroy a photon and create another one. A computation
similar to that leading to relation (6) of ComplementXX, but using (A-14) instead of
2089

CHAPTER XX ABSORPTION, EMISSION AND SCATTERING OF PHOTONS BY ATOMS
(A-12), then yields:
n()in
=
1
~
2
d d n0() () 0()in (E-7)
Note that it is now()that appears in the middle of the matrix element in (E-7),
and not0(). Starting fromin=;k", the rst interaction Hamiltonian
of (E-7) brings the system to the state; 0. In a similar way, if the system starts from
nthe second interaction Hamiltonian of (E-7) brings it to the state; 0. Expression
(E-7) can then be written:
n()in=
1
~
2
d d;k" 0() ; 0
; 0(); 0; 0 0();k" (E-8)
Had we used (A-12) instead of (A-14), the central matrix element of relation (E-
8) would be; 00(); 0= exp( ( ). In our case, using (A-14) leads
to; 0(); 0which is the probability amplitude for the system, starting from
the state; 0at time, to still be in that same state at time. The calculation of
that amplitude appears in the study of the radiative decay of the excited state through
spontaneous emission of a photon, hence the decay of a discrete state; 0coupled
to a continuum of nal states;k". Those states represent the atom in statein
the presence of a photon with any wave vectorkand polarization". Now we showed
in ComplementXIII() that it was possible to obtain a solution of Schrödinger's
equation yielding the amplitude; 0(); 0at long times (and not only at short
times, as for the perturbative solution). This solution is written:
; 0(); 0= exp [(+ )( )~] exp [( )2] (E-9)
where is the energy shift of the state; 0due to its coupling with the continuum of
nal states
11
, andthe natural width of the excited state(the inverse of the radiative
lifetime of that state). We shall assume from now on that the shiftis included in
the denition of the energyof the state. Starting from the more precise expression
(A-14) instead of (A-12) thus leads to a very simple result: we just have to replace, in
all the computations of the scattering amplitude,by ~2.
~2 (E-10)
Once this replacement has been made, we get, keeping only the amplitude (E-4)
and a single relay state, the following scattering amplitude:

(in) =
}
20
3
"D "D
~( +2)
(E-11)
This resonant scattering amplitude no longer diverges when= ; asis scanned
around, it exhibits a resonant behavior over a range equal to.
11
This shift is related to the Lamb shift of the excited atomic states.
2090

E. PHOTON SCATTERING BY AN ATOM
E-3. Inelastic scattering - Raman scattering
We now consider a scattering process where, as before, an incident photon is ab-
sorbed and another emitted, but the nal atomic stateis now supposed to be dierent
from the initial atomic state.
E-3-a. Dierences with elastic scattering
Figure Raman scattering, where
the energy of the scattered photon is dierent from that of the incident photon
12
. The
initial state of the scattering process is, as before, the statein=;k", with energy
in= +~; the nal state, however, is nown=;k"where=, with
energyn= +~.
Figure 4: Raman scattering: an atom in stateabsorbs an incident photon, with energy
~; a photon~is then spontaneously emitted by the atom, which ends up into a nal
statedierent from the initial state.
Conservation of total energy requires:
+~= +~ (E-12)
If , the Raman scattering is called Raman Stokes scattering; the energy
of the scattered photon is lower than that of the incident photon. If, the Raman
scattering is called Raman anti-Stokes scattering; the energy of the scattered photon is
higher than that of the incident photon. As we assumed here that the mode (k") was
initially empty, the scattered photon is emitted spontaneously. The process is then called
spontaneous Raman scattering. We shall study later the case where (k") photons
are initially present, a situation resulting in stimulated Raman scattering.
Equation (E-12) shows that the angular frequency of the scattered light is dierent
from that of the incident light by a quantity= ( )~, equal to the frequency
of the atomic transition. This means that Raman light spectrum provides infor-
mation about the eigenfrequencies of the scattering system; this is the base ofRaman
12
This gure only shows a type () process, where the incident photon is absorbed in the rst place.
2091

CHAPTER XX ABSORPTION, EMISSION AND SCATTERING OF PHOTONS BY ATOMS
spectroscopy. The systems under study are often molecules and the states,are vibra-
tional or rotational sublevels of the ground state, which means that thefrequencies
belong to the microwave or infrared domain. Instead of measuring directly these fre-
quencies using spectroscopic techniques in a frequency domain where detection might be
a problem, it is sometimes more convenient to illuminate the medium with an optical or
ultraviolet frequency beam, and to measure thefrequency via the frequency shifts
of the Raman scattered light. Raman spectroscopy developed considerably once laser
sources became available, yielding much higher signal intensities. The detection condi-
tions have also been greatly improved by focusing the incident laser light on very small
volumes. The analysis of the scattered light spectrum is now a very powerful tool for
analyzing the chemical composition of any scattering media, since each molecule can be
identied by its specic vibration-rotation eigenfrequencies.
To keep things simple, we have limited our discussion to Raman scattering by
atoms or molecules in a dilute medium, where each scattering entity acts individually.
Raman scattering is also used in condensed media such as liquids, crystals, surfaces, etc.
and provides valuable information on the dynamics of these structures.
E-3-b. Scattering amplitude
The computation of the Raman scattering amplitude is very similar to the one
leading to (E-3), (E-4) and (E-5), and yields:
n

()in=2 (in) +(in)
()
(n in)
=2 (in) +(in)
()
(+~ ~)(E-13)
where:
(in) =
rel
nDE
()
relrel
DE
(+)
in
in rel
=
~
20
3
"D "D
+~
(E-14)
(in) =
rel
nDE
(+)
relrel
DE
()
in
in rel
=
~
20
3
"D "D
~
(E-15)
When the photon frequencyis close to thetransition frequency, Raman scat-
tering becomes resonant and amplitude (E-14) can become very large. As we did for the
resonant elastic scattering, to avoid the divergence of (E-14), we just have to replace
by 2, whereis the natural width of thestate.
E-3-c. Semiclassical interpretation
As in , let us consider the dipole induced by the incident eld on the scatter-
ing object. When that object is a molecule which vibrates and rotates, its polarizability
2092

E. PHOTON SCATTERING BY AN ATOM
changes with time, and is modulated by its rotation and vibration frequencies. The
dipole's oscillations induced by the incident eld have an amplitude modulated at the
rotation and vibration frequencies of the molecule. The Fourier spectrum of the dipole's
motion contains lateral bands at frequencies shifted from the incident eld frequency;
these frequency shifts are equal to the molecule's rotation and vibration frequencies.
This semiclassical interpretation accounts for the essential properties of the Raman scat-
tering spectrum.
E-3-d. Stimulated Raman scattering. Raman laser
We now assume the Raman photon appears in a mode that is not initially empty, as
= 0photons already occupy the mode (k"). Similarly, we assume several photons
(for example), initially occupy the mode (k"). To compute the Raman scattering
amplitude; ; 1+ 1, we must include the factor
(+ 1)in
expressions (E-14) and (E-15). Stimulated emission now comes into play: the factor
+ 1appearing in the probability expresses the fact that the initial presence of
photons in mode (k") stimulates the emission probability of a Raman photon in that
mode.
Consider now (right-hand side of Fig.5) the inverse scattering process symbolized
by; 1+ 1 ; . The corresponding scattering amplitude is simply
the complex conjugate of the previous one, meaning that the probability of these two
processes are equal. If we start with the same number of atoms in stateand state, the
number of photons created in one of the processes is equal to the number of photons that
Figure 5: The left-hand side of the gure represents a stimulated Raman process where
an atom in levelabsorbs a photon, and ends up in stateafter the stimulated
emission of anphoton. The right-hand side of the gure shows the inverse process,
where an atom starting from stateabsorbs anphoton, and falls back in stateafter
the stimulated emission of anphoton. These two processes have, a priori, the same
probability. However, if the population of state(shown as a large dot on the left-hand
side of the gure) is higher than that of the state(shown as a small dot on the right-
hand side of the gure), the number of processes resulting in the stimulated emission of
anphoton is bigger than the number of processes where that photon is absorbed. The
radiation at frequencyis thus amplied by stimulated emission, which allows creating
a Raman laser at this frequency.
2093

CHAPTER XX ABSORPTION, EMISSION AND SCATTERING OF PHOTONS BY ATOMS
disappear in the inverse process. What will happen now if we start with an ensemble of
atoms where the populations of statesandare not equal? If, for example, levelhas
a lower energy than level, the relaxation mechanisms leading to thermal equilibrium
tend to create a larger population inthan in. The number of scattering processes
; ; 1+ 1is then larger than the number of the inverse processes
;1+ 1 ; , leading to an amplication of the number ofphotons.
This amplication mechanism is the basis ofRaman laseroperation. This type of laser
is dierent in two major ways from lasers involving a transition between an upper level,
populated by a pumping process, and a lower level (with no relay level). First of all, they
do not require a population inversion; the atomic media can be at thermal equilibrium,
since the stimulated Raman scattering at the origin of the amplication starts from the
atomic state with the largest population. They do require, however, a high intensity
radiation at frequency, furnished by another laser called the pump laser. Secondly,
thefrequency of the Raman laser oscillation can be scanned by changing the pump
frequency, whereas lasers using a two-level system necessarily oscillate at a frequency
very close to the atomic transition, and have thus a very small tuning range.
Conservation of total momentum
If the positionRof the atomic center of mass is no longer treated classically, and placed at
the origin as we have done until now, we must keep the exponential functionsexp(kR)
andexp(kR)in the interaction Hamiltonian describing the absorption of an
photon and the emission of anphoton. The matrix element of the product of those
two operators must be taken between an initial state of the center of mass, with momentum
~Kinand a nal state with momentum~Kn. This yields a(KnKink+k)function
expressing the global momentum conservation in a Raman process: the momentum of the
atom increases by the quantity~(kk)during that process. It often happens that
the two atomic statesandare two sublevels of the same electronic ground state,
so that the frequency = ( )~falls in the microwave domain; it is then
much smaller than the frequenciesand, which are optical frequencies. The energy
conservation equation (E-12) then shows that the moduli of the two wave vectorskand
kare practically the same. If the two wave vectorskandkhave opposite directions,
the momentum gained by the atom during a Raman process is equal to~(kk)2~k.
The interest of such a Raman process is to couple two statesand, energetically very
close to each other, by transferring to the atom in one of the two states a very large
momentum2~k, equal to twice that of an optical photon. On the other hand, if the
two statesandwere to be coupled directly by absorption of a single photon in the
microwave domain, the momentum transfer would be much smaller. This possibility of
coupling two sublevels of the ground state (hence having long lifetimes) by transferring a
large momentum to the atom in one of these two states, has interesting applications, in
particular in atomic interferometry.
To remain concise, in this chapter we have not treated a certain number of inter-
esting related problems. Among them are multophotonic processes, photoionization, the
dressed atom method that facilites the study of light shifts, or the use of photon wave
packets. All these subjects are treated in the complements of this chapter.
2094

COMPLEMENTS OF CHAPTER XX, READER'S GUIDE
AXX: A MULTIPHOTON PROCESS: TWO-
PHOTON ABSORPTION
In a multiphoton absorption process, an atom
simultaneously absorbs several photons. This
complement focuses on the simplest case where
two photons are absorbed, while presenting gen-
eral ideas that also apply to processes involving
a larger number of photons. Monochromatic and
broad band excitations are successively consid-
ered. The very short time the system spends
in the relay state violating energy conservation
is proportional to the inverse of that energy
mismatch.
BXX: PHOTOIONIZATION In a photoionization process, a photon can
remove an electron from an atom, which then
becomes an ion (photoelectric eect). This
complement studies this process by using a
quantum theory of radiation that no longer
couples two discrete atomic states but rather
a discrete (ground) state to a continuum of
(excited) states. Two important cases are
considered: a quasi-monochromatic incident
radiation, and a broad band excitation. In that
second case, this study provides a justication
for Einstein's equation of the photoelectric eect.
Lastly, we consider the case where the radiation
eld is so very intense that the atomic ionization
no longer occurs through the absorption of one
or several photons, but rather by the tunnel eect.
CXX: TWO-LEVEL ATOM IN A MONOCHRO-
MATIC FIELD. DRESSED ATOM APPROACH
The dressed atom approach is a powerful tool for
describing and interpreting higher order eects
that appear as a two-level atom interacts with
a quasi-resonant radiation. It is valid both in
the weak coupling domain (low eld intensity)
and the strong coupling domain (very high eld
intensity). An essential parameter is the ratio of
the Rabi oscillation (characterizing the coupling
with the eld) and the natural width of the
atomic levels. This general approach allows, in
particular, a full understanding of the various
properties of light shifts.
DXX: LIGHT SHIFTS: A TOOL FOR MANIPULAT-
ING ATOMS AND FIELDS
Using light shifts has become a basic tool in
atomic physics, as it allows manipulating atoms
and photons. A number of applications of such
methods are briey described in this complement:
laser trapping of atoms by dipole forces, mirrors
for atoms, optical lattices, Sisyphus cooling,
and one by one detection of photons in a cavity.
2095

EXX: DETECTION OF ONE- OR TWO-PHOTON
WAVE PACKETS, INTERFERENCE
Just as for a massive particle, on can build wave
packets for a photon by the coherent superposition
of states, each having a dierent momentum; this
leads to a description of radiation propagation in
free space, and we have the possibility to model
the arrival of a photon on an atom. We obtain a
description of the photon absorption or scattering
processes that is more realistic than that given
in Chapter , where the incident radiation is
described by a Fock state having a well dened
number of photons (and hence without any
spatial propagation). We introduce for the
photon a function that is not its wave function,
but rather yields the probability amplitude that
it might be detected at a given point. Absorption
and scattering of wave packets are studied, as
well as the one- or two-photon detection signals;
the case of two entangled photons (parametric
down-conversion) is treated at the end of the
complement.
2096

A MULTIPHOTON PROCESS: TWO-PHOTON ABSORPTION
Complement AXX
A multiphoton process: two-photon absorption
1 Monochromatic radiation
2 Non-monochromatic radiation
2-a Probability amplitude, probability
2-b Probability per unit time when the radiation is in a Fock state
3 Discussion
3-a Conservation laws
3-b Case where the relay state becomes resonant for one-photon
absorption
In the process studied in this complement the atom absorbs, not one, but two
photons of energy~, to go from a discrete levelto another discrete higher energy
level
1
; this process is schematized in Figure. For the moment we shall ignore the
external degrees of freedom and suppose the atom to be innitely heavy
2
. Conservation
of total energy then requires:
=~0= 2~ (1)
The calculation of the transition amplitude will explicit the role played by this conser-
vation law.
1. Monochromatic radiation
Studying this process involves the computation of the transition amplitude:
n()in (2)
The initial state at timeof the system atom+radiation is:
in=; with energy in=+~ (3)
It describes an atom in statein the presence ofphotons in mode, with wave vector
k, polarization"and frequency(we assume the radiation is monochromatic). After
the absorption process, at time, the photon number is lowered by two units, going
fromto 2. The nal state of the system is:
n=; 2 with energy n=+ ( 2)~ (4)
The atom-radiation interaction is described, as before, by the electric dipole Hamil-
tonian (A-30) of Chapter, which lowers the photon number by a single unit. This
1
In ComplementXX, we shall see how multiphoton processes can also make an atom go from a
discrete state to a nal state belonging to a continuum of states (photoionization).
2
When the atom's mass is nite, there are interesting physical eects arising from the conservation
of total momentum; these will be studied later on (cf. ; see also XIX).
2097

COMPLEMENT A XX
Figure 1: During a two-photon transition, the atom goes from stateto stateby
absorbing two photons of energy~. The horizontal dashed line represents the energy
half-way between theandlevels. A third level, the relay level, is also involved in
the transition; its energy is not necessarily between those of levelsandand it is not
shown in the gure. However, we assume that the energy of that relay atomic levelis
so far from the dashed line that no one-photon resonant transition can occur between
and.
means that in the expansion (A-21) of Chapter

(), the lowest term that
gives a non-zero contribution to the transition amplitude (2) is of order two, hence con-
taining two operators. The rst brings the system atom+radiation from the initial
statein= to an intermediate relay state
rel= 1 with energy rel=+ ( 1)~ (5)
whereis any atomic state; the second operatorbrings the system from this relay
staterelto the nal staten= 2. One must of course sum over all accessible
intermediate states. Nevertheless, to keep the computation simple, we shall only take
into account a single intermediate state (the summation of the probability amplitudes
over several such states does not pose a serious problem). If we insert relation (A-21) of
Chapter nand the ketin, the rst two terms on the right-hand
side yield zero, and the third one becomes
3
:
n

(0)in=
1
~
2
rel
n relrel in

0
d
0
d
n~ rel( )~ in~
(6)
Let us write explicitly the argument of the exponents appearing in the integral on
3
We assumed that the two absorbed photons were identical. If the two absorbed photons1and2were
dierent, either by their energies, their wave vector directions, or their polarizations (albeit satisfying
the conservation of energy~1+~2= =~0), this would lead to a situation similar to that of
. Two types of processes should then be considered, those where photon1is rst
absorbed, photon2next, and those where the photons are absorbed in the inverse order (cf.Figure).
2098

A MULTIPHOTON PROCESS: TWO-PHOTON ABSORPTION
the second line:
[+ ( 2)~][+ ( 1)~] ( )[+~] (7)
The terms inand cancel out, leaving the expression:
[ 2~][ +~] ( ) (8)
Choosingand= as the integral variables, the second line of (6) becomes the
double integral:

0
d
[ 2~]~
0
d
[ ~]~
(9)
or else:

0
d
[ 2~]~
~
[+~ ]
1
[+~ ]~
(10)
We are dealing with situations where the frequency2of the photons is close
to the two-photon resonance expressed by the conservation of energy (see relation (1)).
In addition, we assume that the process is a real direct two-photon transition, and not
successive one photon absorptions. In other words, we assume levelcannot absorb
a photon in a resonant way and get to the intermediate level; this means that the
energyof that intermediate levelmust be very dierent from the half-way energy
(+)2shown as the horizontal dashed line in Figure. It is easy to see that the
rst term1in the bracket of (10) does correspond to a two-photon resonant absorption,
since its probability amplitude is written:
~
[+~ ]

0
d
[ 2~ ]~
2~
2
[+~ ]
()
( 2~) (11)
(the signsimply means we have ignored an irrelevant phase factor), with:
()
() =
1
sin(2~)
(12)
The function
()
tends towards a delta function when , as shown by relation
(10) in Appendix II; it expresses an energy conservation satisfying condition (1), within
~. On the other hand, the second term in the bracket of (10) introduces, in the sum
over, an exponential
[ ~]~
that oscillates rapidly as a function ofwhen
condition (1) is (exactly or approximately) satised
4
. This term yields a non-resonant
contribution, hence negligible. Its physical signicance (sudden branching of the coupling
between atom and eld) will be discussed in comment (ii) below. We shall ignore it for
the moment because of its non-resonant character. This leads us to:
n

(0)in=2
n relrel in
+~
()
( 2~)(13)
4
In that case,
[ ~]~ [+ 2]2~
, whose exponent is necessarily large since we
assumed that is very dierent from the half-sum ofand.
2099

COMPLEMENT A XX
Comparison with relation (B-10) of Chapter
the probability amplitude of a one-photon absorption process and that of a two-photon
transition. We go from the rst to the second by substituting the variable in the function
()
by the one relevant for the two-photon energy conservation written in (1), and by
replacing the matrix elementn inby
5
:
n relrel in
in rel
=
~
20
3
( 1)
"D "D
+~
(14)
This means that we just have to replace, in relation (B-6) of Chapter
amplitude, the matrix elementby a product of matrix elements divided by an energy
dierence.
Comments
(i): Characteristic time of the intermediate transition
The transit of the physical system through the intermediate relay state occurs without
energy conservation, since it involves a dierence= +} with the initial
energy. Mathematically, this results in the presence, in the second time integral of (9),
of an oscillating term; the larger the energy dierence, the more rapid the oscillation.
Once that integral is performed, we obtain the bracket appearing in (10), multiplied by
a prefactor. This bracket starts from zero at time= 0, then oscillates as a function of
the intermediate time. After a time= , which corresponds to one oscillation
period, its average value over time equals one, precisely the value we have used for the
computation of the probability amplitude.
The transit through the relay state brings in a characteristic time= , after which
the modulus of the integral over timeno longer increases. The larger the departure
from energy conservation, the shorter that time is (this short transit through such a
relay state is sometimes referred to as a virtual transition). The integral over time
thus behaves completely dierently from the integral over, which, at resonance,
increases linearly with time as shown from (11) and (12); this latter integral over
may accumulate contributions over much larger times. The limitation of the times
that actually contribute to the probability amplitude has a natural interpretation in the
context of the Heisenberg time-energy uncertainly relation }.
(ii): Physical meaning of the term left out of the transition amplitude
We have left out the second term in the bracket of relation (10). Its origin is nevertheless
interesting, as it arises from the sudden branching of the coupling between atom and eld
at time= 0, as assumed in the computation. To see this, we can use a model where
the interaction Hamiltonianis replaced by an operator(); the time dependence
of the function()allows introducing an adiabatic turning on of the coupling. It can
be shown that the term we had ignored does disappear when turning on the interaction
very slowly.
A more rigorous description can be obtained by describing the eld as a wave packet
propagating in space (ComplementXX), and overlapping the atom only during a limited
time. In that case, the interaction Hamiltonian only acts during that overlap time, even
5
We use for expression (A-24) of Chapter, as well as expression (A-27) for the electric eld.
2100

A MULTIPHOTON PROCESS: TWO-PHOTON ABSORPTION
though the operator itself is time independent. For a wave packet with a progressive
wave front, this approach shows that the term we have ignored does not even come into
play.
As for the transition probability, the computation is the same as for a one-photon
transition. We get for the transition probability
(2)
():
(2)
() =
~
2~0
3
2
[( 1)]
"D "D
+~
2
4 sin
2
(02) 2
(02)
2
(15)
At short times, it is proportional to the square of()
2
.
Finally, if several relay states are involved in the two-photon transition, one must
sum expressions (13) and (14) over all the possible intermediate statesrel(taking into
account the fact they each have dierent energiesrel); on the right-hand side of (14),
this amounts to summing over all accessible intermediate atomic stateswith energy
. Interference eects between amplitudes associated with dierent intermediate states
may then appear in the probability (15).
2. Non-monochromatic radiation
Consider now what happens when the initial state of the systemincontains photons
of dierent frequencies. We are going to show that, just as for one-photon transitions,
the two-photon transitions can lead to a transition probability per unit time; however,
this probability involves a higher order correlation function (order 4 instead of 2).
2-a. Probability amplitude, probability
The computation of the probability amplitude is similar to that discussed in ;
it is based, as before, on the expression of

to second order in. We will carry out
the calculation so as to highlight the properties of the time correlation functions of the
incident electric eld. The radiation initial state is
in, its nal state
n
, and
rel
is its intermediate state when the atom is in the relay state. The two-photon transition
is described by the sequence of the following states for the system atom+photon:
;
in ;
rel ;
n (16)
corresponding to the transition amplitude to the lowest order:
;
n

(0);
in=
1
~
2
0
d

D()
n

E
(+)
(R)
rel
0
d

D()
rel

E
(+)
(R)
in (17)
(Ris the atom's position). By analogy with relation (B-13) of Chapter, we set:

D()=
( )}
with: = e

D()=
( )}
with: = e (18)
2101

COMPLEMENT A XX
We then get:
;
n

(0);
in=
~
2

0
d
( )}
ne

E
(+)
(R)
rel
0
d
( )}
rele

E
(+)
(R)
in (19)
Expression (A-29) of Chapter

(+)
(R)is a
sum of modes' contributions, each including the exponentialassociated with its
eigenfrequency. Let us focus on the contribution of modein

E
(+)
(R)and modein

E
(+)
(R). It involves the time integrals:

0
d
( )}
0
d
( )}
(20)
with an exponent containing:
[ }]+ [ }]
= [ } }]+ [ }] ( ) (21)
reminiscent of result (8) obtained for monochromatic excitation. The computation is
then very similar to that of , assuming that the relay state is not half-way between
levelsand, and that the frequency distribution of the incident photons does not
include any of the resonance frequencies for the one-photon transitionsand .
We make the usual change of variable= , and, in the integral over, we only
keep the upper boundary contribution, as we did going from (10) to (13):
0
d
[ }]}
}
[+~ ]
(22)
(we discussed in comment (ii) at the end of
contribution, and why it is justied to ignore it). In addition, we assume the width of
the frequency spectrum of the incident photons to be small compared to the one-photon
resonance detuning+~ ; consequently, the denominator of (22) does not vary
signicantly in relative value, and can be replaced by the constant value+~ex ,
whereex2is the central excitation frequency. We notethe distance from the
resonant absorption of a photon in the intermediate state:
=
+~ex
}
(23)
The replacement of the integral overbyyields an exponential depending
on the variable, with argument[ } }]}. Each summation over
the modesandwith the exponential factors reconstructs the electric eld

E
(+)
(R), which leads to:
;
n

(0);
in=
~
2

0
d
0}
ne

E
(+)
(R)
relrele

E
(+)
(R)
in (24)
2102

A MULTIPHOTON PROCESS: TWO-PHOTON ABSORPTION
where0was dened as0= ( )}. Finally, the summation over all the radiation
intermediate states
rel
yields a closure relation and we obtain:
rel
;
n

(0);
in=
~
2

0
d
0}
ne

E
(+)
(R)e

E
(+)
(R)
in (25)
This result is similar to the probability amplitude for a one-photon transition, written
on the second line of (B-14) in Chapter, provided we make the substitution:
ne

E
(+)
(R)
in
~
ne

E
(+)
(R)e

E
(+)
(R)
in (26)
We then follow the same line of reasoning as for a one-photon transition. In equality
(B-17) of Chapter, we must now substitute:
()
ine

E
()
(R)e

E
()
(R)
e

E
(+)
(R)e

E
(+)
(R)
in (27)
which is then inserted in (B-18) to yield the transition probability. This probability is
given by the Fourier transform, at the angular frequency0of the atomic transition, of a
correlation function of the eld in the initial state, and which involves four eld operators
(4-point correlation function). This function is in general dierent from a product of
correlation functions involving two eld operators (those determining the absorption
probability of a single photon). This means that measurements of two-photon transition
probabilities yield access to characteristics of the quantum eld that are dierent from
those measured in single photon transitions.
2-b. Probability per unit time when the radiation is in a Fock state
Let us assume now that the radiation is initially in a Fock state such as that
described by (B-19) in Chapter. In (25), we replace the positive frequency components
of the electric elds by their expressions (A-27) of Chapter. Only the occupied modes
in state
innow come into play, since each annihilation operator yields a factor equal
to the square root of the mode's initial population; the other modes give a zero result.
We then consider two modes,1and2, initially occupied in the incident radiation.
They yield two contributions (Figure) to relation (25): in one of them (term=1
and=2), the photon1is absorbed rst and brings the atom from the ground state
to the relay state, then photon2completes the two-photon transition and brings the
atom to level; in the other (term=2and=1), the order of the two absorptions
is inverted. These two contributions interfere in the probability: once the amplitude
modulus is squared, four terms arise from the cross contributions of the two modes (to
which we must add two non-crossed contributions==12where only one mode is
involved).
2103

COMPLEMENT A XX
Figure 2: Two diagrams schematizing a two-photon transition with a multimode source
where two modes1and2are initially occupied. In the left-hand side diagram, photon
1is absorbed rst, bringing (in a non-resonant fashion) the atom from the initial sate
to the relay state; photon2then completes the (resonant) two-photon transition. In
the right-hand side diagram, the order of absorption of photons1and2is inverted.
These two diagrams describe probability amplitudes that interfere when computing the
two-photon transition probability.
The same line of reasoning as in , and summarized in Figure
of that chapter, can be followed here. We assume that the 4-point correlation of the eld
goes rapidly to zero when is larger that a value1that is small compared to
. One can then show that the transition probability becomes proportional to, and
that the two-photon transition probability per unit time can be written:
(2)
=
(2)
()

=
2
~
2
~
2
1
4
2
0
6
2 (+ 1)(~) (~)
(e") (e")
2
(+ 0) (28)
The delta function at then end of this expression obviously expresses total energy con-
servation: for the atomic transition to occur, the sum of the energies of the absorbed
photons must be equal to the energy of the transition. As expected, the probability
includes the photon populations that satisfy this condition. A general property is that,
for=(photons absorbed from two dierent modes), it is the average value
of the product of the mode populations that appears in the two-photon transition prob-
ability, and not the product of the average values which dierent in general (they are
nevertheless equal in the special case of a Fock state of the radiation). For=(two
photons absorbed from the same mode, as in ), it is the average value( 1)
that comes into play; this value equals zero if only one photon is present in the mode, as
obviously one single photon cannot induce a two-photon transition.
2104

A MULTIPHOTON PROCESS: TWO-PHOTON ABSORPTION
Comment: There also exists3- ,...,-photon transitions, corresponding to an energy
conservation relation =~. The corresponding transition rates are propor-
tional to eld correlation functions of order6, ..,2.
3. Discussion
Even though the transition amplitudes for one- and two-photon absorption processes are
similar, the second type of processes has a number of specic features we now discuss.
3-a. Conservation laws
. Total energy conservation
As we just saw, the function
()
( 2~)appearing in (13) expresses
the conservation of total energy. When the atom absorbs two photons to go from state
to state, its gain in energy equals the sum2~of the energies of the two
absorbed photons.
. Total momentum conservation
In the computation leading to the two-photon transition amplitude, the external
variables have been ignored. To take them into account, we must assume the atomic
center of mass is at pointRand keep the exponentialsexp(kR)appearing in the
operatorsE
(+)
(R)in the two interaction Hamiltonians; as these two exponentials
are multiplied by each other, they yield the operatorexp(2kR). We must also include in
the initial and nal states the quantum numbersKinandKncharacterizing the initial
~Kinand nal~Knmomenta of the center of mass. We then get in the transition
amplitude an additional term:
Knexp(2kR)Kin=(KnKin2k) (29)
which shows that the atom's momentum increases by2~kwhen it absorbs two photons.
Comment:
Imagine the atom is excited by two light beams 1 and 2, having the same frequency
2but propagating in opposite directions. The previous computations must then
be generalized to the case where the two photons absorbed by the atom belong, one to
beam 1, and the other to beam 2. The momenta+~kand~kof these two photons
are then opposite and the total momentum gained by the atom during the transition
is zero. As the Doppler eect and the recoil eect are linked to the variation of the
atomic momentum during the transition (see ComplementXIX), it follows that the
two-photon absorption line does not present any Doppler broadening nor any recoil shift.
Such a situation presents many advantages for high resolution spectroscopy, and is used
for example in the study of the two-photon transition between the1and2states of
the hydrogen atom.
2105

COMPLEMENT A XX
. Conservation of total angular momentum and parity
Expression (14) appearing in the two-photon absorption amplitude is a product
of two matrix elements of a component of the atomic electric dipole which is a vector
operator and an energy denominator. In a rotation of the atom, this expression will
be transformed as the product of two vectors, since the energy denominator is rotation
invariant. Vectors are irreducible tensor operators (ComplementX, exercise 8) of or-
der= 1. Consequently, expression (14) may be expanded
6
as a sum of components
with total angular momentum= 012. Using the Wigner-Eckart theorem (Comple-
mentX), we can show that the two-photon absorption amplitude between two levels
with quantum numbers and (whereand are the components ofFon
theaxis) is dierent from zero only if:
=210
=210 (30)
In addition, the electric dipole operator appearing in (14) is an odd operator, as
it is proportional to the electron position operator. Consequently, the initial and nal
states of the two-photon transition must have the same parity, and a parity inverse to
that of the relay state.
These selection rules can be applied to the1 2transition of the hydrogen atom,
that occurs between two states having the same parity and a total angular momentum
dierence equal to1at most (the12spins of the electron and the proton are taken
into account). For electric dipole transitions, a two-photon transition1 2is allowed,
whereas it is forbidden for a one-photon transition.
3-b. Case where the relay state becomes resonant for one-photon absorption
In the denominator of expression (14), we have the quantity:
~=+~ (31)
which is the dierence between the energy of the atomic state, increased by~, and the
energy of the relay sate. Ifgoes to zero, we get a divergence and the expressions
we have obtained become meaningless. In the computation, we did explicitly assume
that the intermediate level was not resonant for a one-photon absorption, so that this
divergence should not occur. Let us examine, however, what would be involved if
were to go to zero. As the resonance condition for the two-photon transition is written
= + 2~, the condition= 0means that the atomic relay level
7
is exactly
6
The product of two irreducible tensor operatorsand can be decomposed as the product of
two kets with angular momentum and, hence involving Clebsch-Gordan coecients. This means
that, according to the general results of Chapter
irreducible tensor operators can be decomposed as the sum of other irreducible tensor operators of order
, where varies between and+. In the particular case where= = 1, we get
three possible values= 012.
7
We assume here that the relay stateis discrete. If it belongs to a continuum, the sum over this
relay statein (14) becomes an integral over. An adiabatic branching calculation then introduces a
fraction1(+~+ )with 0, which can then be expressed in terms of(+~ )and
(1(+~ )), whereis the Cauchy principal value. This calculation yields, after integration
over, functions ofin= + 2~which have no reason to diverge in the vicinity ofin n.
2106

A MULTIPHOTON PROCESS: TWO-PHOTON ABSORPTION
half-way in energy betweenand, the initial and nal atomic levels. This means that,
starting from level, a resonance would occur for both a two-photon and a one-photon
process.
Is it possible to study the case whereis zero or very small, while avoiding the
divergence of the two-photon absorption? A method to overcome this diculty is to note
that statehas a nite lifetime, because of spontaneous emission of photons from that
state. As we did before in , when studying the resonant scattering
of a photon by an atom and where similar divergences appeared, we can show that it is
legitimate to replace the energyof stateby ~(2), where= 1is the
natural width of level. The denominator of the transition amplitude no longer goes
to zero and the divergence disappears. The transition amplitude still varies signicantly
over an interval of widthwhen varies around+~.
Replacingby (~2)leads to valid results only if the matrix elements
; 1 ;and; 2 ; 1, characterizing the coupling of the eld
with the atom for the and transitions, are small compared to~. If this
is not true, we cannot limit the computation to the lowest eld order. We must then
diagonalize the Hamiltonian of the global system atom + eld within the subspaceof
the states which, in the absence of coupling, are very close to each other
8
. When no
relay state is resonant, the subspaceis two-dimensional; it is spanned by the two states
;and; 2. When one relay statebecomes resonant, we must include the
state;1in the subspace which then becomes three-dimensional. To study the
dynamics of the system, we must diagonalize the matrix:
+~ ; ; 1 0
; 1 ; + ( 1)~ ; 1 ; 2
0 ; 2 ; 1 + ( 2)~
(32)
This general treatment allows taking into account simultaneously the one- and two-
photon transitions.
Concluding this complement, let us emphasize that the two-photon transitions
involve a physical process dierent from the mere succession of two one-photon absorp-
tions. We stressed in the discussion of , and in particular in its two comments, the
dierence between populating the nal state, which is cumulative in time and conserves
the energy, and a transit through an intermediate relay state, which can only last a very
short time, limited by the non-conservation of energy. It is also noteworthy that the
two-photon transition amplitude can take a form very similar to that of a one-photon
transition; the only major change is the replacement of the matrix element to rst order
in the interaction, by a second order matrix element, divided by an energy defect factor
in the relay state. These concepts can be generalized to higher order processes: similar
techniques can be used to evaluate three-, four-, etc.. photon transition amplitudes.
8
Such a description of the atom + radiation interactions is called the dressed atom method (see
for example Chapter VI of reference [21]). In ComplementXX, this method is applied to the problem
of a two-level atom interacting with a strong eld. The eigenstates of the total Hamiltonian restricted
to the subspaceare called the dressed states.
2107

PHOTOIONIZATION
Complement BXX
Photoionization
1 Brief review of the photoelectric eect
1-a Interpretation in terms of photons
1-b Photoionization of an atom
2 Computation of photoionization rates
2-a A single atom in monochromatic radiation
2-b Stationary non-monochromatic radiation
2-c Non-stationary and non-monochromatic radiation
2-d Correlations between photoionization rates of two detector
atoms
3 Is a quantum treatment of radiation necessary to describe
photoionization?
3-a Experiments with a single photodetector atom
3-b Experiments with two photodetector atoms
4 Two-photon photoionization
4-a Dierences with the one-photon photoionization
4-b Photoionization rate
4-c Importance of uctuations in the radiation intensity
5 Tunnel ionization by intense laser elds
All the atomic processes of absorption, emission or scattering of photons studied in
Chapter
discrete levels, atoms also have continuums of energy levels. The most directly accessible
one is the simple ionization continuum, which corresponds to the loss of a single elec-
tron by the atom (ionization). This continuum starts at an energy thresholdabove
the ground state energy, and extends over all energies larger than this threshold. This
energyis called the ionization energy. The aim of this complement is to study the
photoionization process where incident radiation takes the atom from its ground state
to a state belonging to the ionization continuum.
Once the atom's electron has reached the ionization continuum, it can travel an
arbitrary distance from the remaining ion; it has been ejected from the atom by the inci-
dent radiation. Such a process is reminiscent of the photoelectric eect where radiation
ejects an electron from a metal. This is why we shall review in
the photoelectric eect to underline its analogies with photoionization.
We shall then use quantum theory to compute, in , the probability per unit
time for the incident radiation to photoionize an atom. We shall assume that the inci-
dent radiation spectrum is entirely above the ionization threshold, so that no resonant
absorption can bring the atom to a discrete excited state. Since the emitted electron can
be amplied in a photomultiplier, the atom can play the role of a photodetector. In the
case where only one photodetector D is used (), the computations are very similar
to those exposed in Chapter; the only dierences arise from the continuous character
2109

COMPLEMENT B XX
of the nal atomic state. Another interesting situation occurs when two detectors D1
and D2 are placed in the radiation eld at pointsR1andR2() and when we focus
on the correlations between their signals. For example, we shall compute the probability
per unit time to observe a photoionization atR1at time1and another one atR2at
time2.
One may wonder if a quantized radiation theory is needed to quantitatively account
for the photoionization processes. Could a semiclassical theory suce to describe the
ionization of one or several quantized atoms by a classical eld? In other words, can the
photoelectric eect be explained without photons []? This question will be discussed
in .
The atom can also be photoionized by the absorption of a number of photons
, larger than one. These processes are called multiphoton ionization and play an
important role in experiments using high intensity laser sources. In , we shall give
an idea of how to compute the rates of those processes for= 2. We shall then briey
mention in
occurring when the incident radiation electric eld becomes of the order of the Coulomb
eld between an atom's electron and the nucleus.
1. Brief review of the photoelectric eect
In 1905, Albert Einstein [43] introduced for the rst time in physics the concept of
light quanta, which we now call photons. Considering the great analogy between
certain statistical properties of black body radiation and those of an ideal gas of particles,
Einstein proposed the idea that radiation was in fact composed of discrete quanta, each
having an energy. In view of the successes of the wave theory of light, this return to
a particle description seemed totally unrealistic for most physicists at the time. Energy
quantization had indeed been introduced a few years earlier by Max Planck to account
for the spectral distribution of black body radiation, but it was the exchanges of energy
between matter and radiation that were quantized, not the radiation itself.
1-a. Interpretation in terms of photons
In that same 1905 article, Einstein used the concept of light quanta to give a new
description of the photoelectric eect. In this process, an electron is ejected from a metal
irradiated by light. Einstein postulated that the energyof a light quantum from the
incident beam was absorbed by an electron in the metal, hence allowing it to escape from
the metal. This escape requires an energy at least equal to that of the binding energy
of the electron in the metal. The frequencyof the light beam must therefore be
larger than a threshold value given by=. If , no electron can be ejected.
If , the energy surplus provides the electron with a kinetic energy
kin=
2
2. This interpretation leads to Einstein's equation:
1
2
2
= if (1)
giving the kinetic energy of the ejected electron as a function of. It means, in particular,
that the kinetic energy of each electron depends only on the frequency of the light beam,
2110

PHOTOIONIZATION
Figure 1: Photoionization of an atom. Stateis the ground state, stateone of the
discrete states. The continuum of states belonging to the continuous part of the spectrum
(ionization continuum) starts at an energyabove the ground state. Energyis
called the ionization energy. As the atom in stateabsorbs a photon with frequency
such that , it goes into a state, which is part of the ionization continuum.
The electron is ejected, and its kinetic energykin(when it is far enough from the ion
formed as it left the atom) is equal to the dierence betweenand.
and not on its intensity
1
(which, on the other hand, determines the number of ejected
electrons per unit time). Equation (1) also tells us that ifis varied and one plots
the variation of
2
2as a function of, one should get a straight half-line with slope
, starting from the abscissa axis at point. All these predictions generated a
certain skepticism and it was not until several years later (1913) that an experimental
conrmation of the predictions of equation (1) was obtained by the work of R. Millikan
and H. Fletcher on the photoelectric eect [44].
1-b. Photoionization of an atom
Figure of an atom, one of the statesfrom the
discrete part of the spectrum, and the continuum part of the spectrum which starts at
a distanceabove the ground state. The origin of the energies is often chosen at the
beginning of the continuum, and hence the discrete states have a negative energy. When
it is in the positive energy states, belonging to the continuum and called scattering
states, the electron is no longer bound to the nucleus, although it is still attracted to it.
Consider an atom in the ground state. A photon with energyhas dierent
ways to bring energy to the atomic electron. If, this photon can be absorbed
only ifcoincides with the frequency= ( )of a transition between state
1
A classical theory would tend to predict that the higher the light intensity, the more energy could
be furnished to the electron, thereby increasing its acceleration.
2111

COMPLEMENT B XX
and statebelonging to the discrete spectrum. This is the absorption process between
two discrete states already studied in Chapter. On the other hand, if , the
atom can always absorb the photon and end up in a state within the continuum. The
electron is no longer bound and can move away from the remaining ion formed once
the electron has left. When the electron is far enough from the ion for their Coulomb
interaction energy to be neglected, its kinetic energy is given by:
1
2
2
= if (2)
This is the photoionization process of an atom. Equation (2) is the generalization for an
atom of equation (1) introduced by Einstein for the photoelectric eect in a metal.
2. Computation of photoionization rates
Let us now see how to adapt the computations of Chapter
photoionization rate.
2-a. A single atom in monochromatic radiation
We start from expression (B-7) of Chapter
leaving at= 0the initial statein=;(atom in statein the presence of
photons), ends up at timein the nal staten=; 1(atom in state
with an energy~0above, and one photon less in mode) :
n

(0)in
2
=
1
~
2
n in
2
4 sin
2
[(0 )2]
[(0 )2]
2
(3)
In Chapter, we used this expression for studying the case where statesandare
both discrete states. It is nevertheless still valid whenbelongs to a continuum; its
interpretation, however, is dierent. Whereas the probability of nding the atom in
a discrete nal state makes sense, from a physical point of viewn, when dealing with a
continuum we must compute the probability of nding the atom within a non-zero energy
interval. We must then sum probability (3) over states.
Asvaries,0= ( )~varies, and so does the matrix element of.
However, for large enough, the variation of thematrix element is much slower
than that of the ratio in the right-hand side of (3). This ratio is the square of a diraction
function, whose maximum equals
2
for0= and whose width is of the order of
1. The area under this function is thus of the order of
2
(1) = . Compared
to functions of0with slow variations over an interval of the order of1, this function
behaves, within a proportionality factor, as the product(0 ). It follows that
the sum overof (3) is proportional to, meaning we can dene a probability per unit
time for the atom to reach the continuum, i.e. a photoionization rate.
The proportionality factor between a diraction function and a delta function is
given by relation (11) in Appendix II, which is written:
lim

sin
2
[(0 )2]
[(0 )2]
2
=(
0
2
) = 2~( ~) (4)
2112

PHOTOIONIZATION
Inserting (4) into (3), summing over, and dividing by, nally yields the photoion-
ization rate:
1

n

(0)in
2
=
2
~
n in
2
( ~)
=
2
~
n in
2
=+~
(+~) (5)
where(+~)is the density of states in the continuum around the energy+~.
This expression is just a consequence of the Fermi golden rule (Chapter , )
applied to the coupling between the discrete level;and the continuum; 1.
It is also reminiscent of expression (C-37) of Chapter
probability per unit time between a discrete atomic state and a continuum, with an
excitation induced by a classical wave described by a time-dependent sinusoidal function.
This point will be discussed further in .
2-b. Stationary non-monochromatic radiation
We now consider a single atom interacting with non-monochromatic radiation,
described by a spectral distribution(). We rst assume that the eld statistical prop-
erties are time-invariant. We shall consider the case of non-stationary radiation in .
. Field and atomic dipole correlation functions
In , we obtained the expression for the probability
abs
()
for the atom to go, through the absorption of a photon, from stateto any statedif-
ferent from, after a time. This probability is given by relation (D-4) of that chapter
as a double integral of a sum of products of eld and atomic dipole correlation functions.
We rst examine the correlation functions of the atomic dipole. As in relation
(B-13) of Chapter, we write the matrix elements of this dipole, in the Heisenberg
picture (with respect to the Hamiltonian of the free atom), as:

D()=D = (6)
with:
= e (7a)
whereeis the unit vector parallel to the vectorD. Sinceee= 1, we have:
=e =eD (7b)
For the sake of simplicity, we shall assume the unit vectoreto have the same
direction for all the states.
Comment:
This vectoreis, a priori, not the same for all the statesrelated toby matrix elements
ofD. The direction of that vector actually depends on the rotation symmetry properties
2113

COMPLEMENT B XX
of the two states
2
. One can sort the dierentstates by categories having the same
symmetry (hence the same direction fore) and for which the computations presented
hereafter are valid. One must then add all the ionization probabilities calculated for each
category.
Since the eld always appears in a scalar product withD, assuming thatehas the
same direction for all statesimplies that only the scalar products
(+)
(and
()
) of
the elds withe(and withe) appear in the correlation functions of the eld.
The calculation is then very similar to that of , and we obtain:
abs
() =
1
~
2

0
d

0
d ( )( ) (8)
where:
( ) =
2 ( )
(9)
( ) =
ine

E
()
(R)e

E
(+)
(R)
in
=
in

()
(R)

(+)
(R)
in (10)
In this last equality,
inis the radiation initial state. As this state is stationary, its
properties are invariant under time translation; consequently, the correlation function
depends only on the dierence.
The atomic correlation function (9) can also be rewritten as:
( ) =d
~
()
( )
(11)
where:
~
() =
2
( ) (12)
The quantity
~
()represents the spectral sensitivity of the photodetector atom, that
is the variation withof the transition intensities from the ground stateto a levelin
the ionization continuum, at an energy~above. We shall assume here that the width
of the function
~
()is much larger than the bandwidth of the incident
radiation.
(13)
2
The ground stateis an eigenvector of the angular momentum component along the quantization
axis, with eigenvalue}; for the state, the eigenvalue is}. Thetotransition thus corre-
sponds to a variation= . If= 0, symmetry arguments show thateis parallel to;
if=1,eis in the plane perpendicular to:e= (e e)
2; we note that the complex
conjugate of this vector appears in the matrix element (7b).
2114

PHOTOIONIZATION
Such a condition denes what we shall call a broadband photodetector.
The eld correlation function (9) has already been calculated in of Chap-
ter 1 , or a statistical
mixture of such states with weights(1 ) see equations (B-20) and (B-21) of
Chapter. For the problem we are studying now, i.e. stationary non-monochromatic
radiation, we can use the same assumption for the radiation initial state.
. Photoionization rate
To transform equation (8), it is useful to study in more detail thedependence
of( ). We assume that the spectral distribution is centered around a non-zero
valueex, and that this distribution is entirely above the ionization threshold. Since the
eld

(+)
varies in e, and the eld

()
in e, we can then write:
( ) =
ex( )
( ) (14)
where( )is an envelope function whose Fourier transform is a function centered
at= 0and of width. This envelope function varies very slowly over time intervals
short
3
compared to1. For=, i.e. for= 0, equation (14) leads to:
(0) =(0)
=
in

()
(R)

(+)
(R)
in= (15)
whereis the radiation intensity (which is time-independent since the radiation state is
supposed to be stationary). We shall see in the next section how to generalize our results
to non-stationary radiation.
Let us go back to the double integral of (8) and assume that the integration interval
satises the condition where = 1is the detector correlation time.
In the plane, the function to be integrated in (8) is dierent from zero only in a band
along the rst bisector (Figure ), of widthvery narrow compared
to. If we change the integral variablesandto the variablesand= ,
we can neglect the variation of()since, according to (13), 1, and use
(15) to rewrite (14) as:
( ) =
ex( )
( )
ex( )
(0) =
ex( )
(16)
The double integral of (8) is easily performed with the new variablesand. Using
expression (3) for( ), the integral overof a function that no longer depends
on this variable introduces a simple factor. We are then left with the integral over
which leads to:
abs
() =
1
~
2

+
d
ex
()
=2

(ex)
(17)
3
This is not the case for( ), because of the exponential
ex( )
that varies a lot over
time intervals of the order of1, sinceex.
2115

COMPLEMENT B XX
As this probability is proportional to, we can dene a photoionization rate:
phot=
1

abs
() =
2
~
2

(ex) (18)
This rate is proportional to the incident intensityand to the spectral sensitivity
~
(ex)
of the photodetector, evaluated at the radiation central frequencyex.
2-c. Non-stationary and non-monochromatic radiation
For non-stationary radiation, the initial radiation state is no longer a Fock state or a
statistical mixture of Fock states; it is rather a linear superposition of such states, creating
wave packets such as those described in ComplementXX. The radiation correlation
function()is no longer a function of the single variable, but depends on
bothand. One can still assume that the frequencies appearing in()are
centered aroundexin an interval of width, which permits generalizing expressions
(14) and (15) to:
() =
ex( )
() (19)
and
4
:
() =
in
()
(R)
(+)
(R)
in= () =(R) (20)
where(R)is the intensity at pointRand at time.
Using the expansions inandfor the eld operators appearing in (10), we
get an expression for()that generalizes equation (B-20) of Chapter
stationary elds:
() =
~
20
3
(e )(e )
in in
(kR )(kR )
(21)
For xed values ofkand, the summation overof (21) represents a wave packet
of central frequencyex, and whose envelope passes by a pointRin a time interval of
the order of1. Ifvaries over a time interval 1, the variation of
the envelope can be neglected. Similar conclusions are valid for a summation overof
relation (21), with xed values forkand.
Let us now go back to the double integral in (8). As the phenomenon now depends
on, the integration interval will be taken betweenand+ (instead of between0
and). We assume thatsatises the condition :
1

1

(22)
which is possible when (13) is taken into account. Since 1, we can neglect
in (19) the variation of()whenandvary in the integration domain; we
therefore replace()by:
() =
in
()
(R)
(+)
(R)
in=(R) (23)
4
From now on, we simplify the notation by omitting the bar over operators in the Heisenberg picture,
since the explicit time dependence is sucient to indicate this point of viewn.
2116

PHOTOIONIZATION
Using this equality in (19) we get, in the integration interval betweenand+ :
()
ex( )
(R) (24)
The computations are then quite similar to those carried out above for a stationary eld:
the integral overleads to aterm; the integral over= is equal to:
+
d
ex
() = 2
~
(ex) (25)
We call(R)the probability that a photodetector atom, placed atR, will undergo
a photoionization between timesand+ . We get the result:
(R) =
in
()
(R)
(+)
(R)
in (26)
where= 2

(ex)~
2
is a factor characterizing the photodetector sensitivity at the
radiation central frequencyex. The atomic photoionization rate is thus a signal that
constantly follows the time variations of the incident radiation intensity, written in (20).
2-d. Correlations between photoionization rates of two detector atoms
The previous computations can be generalized to analyze other experiments where
two detector atoms are placed atR1andR2, and where we study correlations between
the photoionizations observed on those two atoms at times1and2. More precisely,
let us call(R22;R11)12the probability to detect a photoionization atR1
between1and1+ 1and another one atR2between2and2+ 2. Computations
very similar to those performed above, and which will not be explicited here (for more
details, see Complement AIIof [21]), lead to:
(R22;R11) =
2
in
()
d
(R11)
()
d
(R22)
(+)
d
(R22)
(+)
d
(R11)
in (27)
It is easy to understand why two operatorsE
(+)
preceded by two operatorsE
()
appear in (27). The double photoionization rateis computed from a probability that
is the modulus squared of a probability amplitude for a photon to be absorbed atR11
and another one atR22. This amplitude must contain a product of two operatorsE
(+)
.
Its conjugate must contain two operatorsE
()
arranged in inverse order. We therefore
should nd in (27) two operatorsE
()
followed by two operatorsE
(+)
with dierent
orders ofR11andR22.
There is a great analogy between the simple and double photoionization rates
and given by equations (25) and (27) and the correlation functions1and
2studied in Chapter . The functions1and2give the probability densities of
nding a particle atr11for1, or a particle atr11and another one atr22for
2. Note, however, that1and2give the probability of nding one or two particles
at specic points, whereas we are now dealing with the probability of photoionization
of atoms placed at specic points. The eld operators appearing in (26) and (27) are
the positive or negative frequency components of the electric eld, since these are the
operators describing the emission or absorption of photons.
2117

COMPLEMENT B XX
3. Is a quantum treatment of radiation necessary to describe photoionization?
In a semiclassical treatment of photoionization, the radiation eld is described as a
classical eld, while the atom follows a quantum treatment. The atom-eld coupling
is then a time-dependent perturbation that can induce transitions between a discrete
atomic state, such as the ground sate, and a state, part of the ionization continuum.
Does such a treatment yield the same results as those obtained in a quantum treatment?
We are going to show that, while this is often the case, this is not always true.
3-a. Experiments with a single photodetector atom
In the simple case of an oscillating monochromatic classical eld, with frequency
, the transition probability per unit time takes a form that is reminiscent of the Fermi
golden rule, valid for a constant perturbation coupling a discrete state to a continuum
5
.
In the more general case where the classical eld is non-monochromatic but sta-
tionary, one can follow the same line of computations that led to equation (8). This
shows that the transition probability from a discrete stateto any state of the contin-
uum can still be expressed as the integral of the product of two correlation functions:
one,( ), for the atomic dipole, another,( ), for the radiation. In both
cases, the eld appears in the transition probability only via a correlation function. For
the classical case, the quantum average value (10) must be replaced by the product of
the negative and positive frequency components of the classical eld:
( ) =
()
d
(R)
(+)
d
(R) (28)
Note that in this relation, in order to distinguish the quantum elds from the classical
elds, these latter elds are written with curly letters; the subscriptmeans that the
eld has been projected onto the polarization unit vector dened in (7a).
Expression (28) has been obtained for a perfectly known classical eld. Another
possibility is that the classical eld is only known in a probabilistic sense, as is the case
with a classical statistical mixture of elds, with given probabilities. The transition
probability (8), where the correlation function

( )is replaced by( ),
must then be averaged over all the states of the statistical mixture, which amounts to
replacing (28) by:
( ) =
()
d
(R)
(+)
d
(R) (29)
where the bar above the product of the two elds symbolizes the statistical average. It
seems that for all signals involving one single photodetector atom, the quantum predic-
tions are identical to those of a semiclassical theory, using a classical eld with the same
correlation function as the quantum eld. In particular, for a stationary eld, the Fourier
transform of the eld correlation function is simply the spectral distribution()of that
eld. For a quantum eld, this property was established in Chapter
(B-20). For a classical eld, this property is a consequence of the Wiener-Khintchine
theorem. It follows that the photoionization probability of an atom is the same, whether
it is computed with a stationary quantum or classical eld, as long as they both have
the same spectral distribution.
5
See for example , and relation (C-37
2118

PHOTOIONIZATION
Comment
The equivalence of the predictions of the two theories is also valid for a non-stationary eld.
The semiclassical theory predicts that the photoionization rate at timeis proportional to
the classical eld intensity
()
d
(R)
(+)
d
(R), which now depends onsince the eld
is no longer stationary. We shall see in ComplementXXthat a similar result is obtained
in quantum theory: the photoionization probability at timeof an atom receiving a
one-photon wave packet is again given by the modulus squared of a function, which can
be considered to be the photon wave function, evaluated at=.
3-b. Experiments with two photodetector atoms
The same line of computations that led to equation (27) can be followed for a clas-
sical eld. It leads to an expression similar to (27), where the eld quantum correlation
function is replaced by the statistical average of the product of two negative frequency
components and two positive frequency components of the classical eld:
()
d
(R11)
()
d
(R22)
(+)
(R22)
(+)
(R11) (30)
As classical elds commute with each other, expression (30) can be rewritten as:
(R11)(R22) (31)
where:
(R11) =
()
d
(R11)
(+)
(R11) (32)
is the classical eld intensity at pointR1at time1and with a similar equation for
(R22). Correlations between the photoionization rates of the two photodetector atoms
thus involve, in the semiclassical theory, the product of the intensities arriving on both
photodetectors. The correlation function of the eld amplitude is replaced by the corre-
lation function of the intensity.
. Situations where a semiclassical treatment is adequate
A rst situation where quantum and semiclassical predictions agree is the case
where the eld state is a coherent statedescribed by the set of classical normal
variables(Chapitre , ). Each modeis in a coherent state, meaning
the state is an eigenket of the operatorE
(+)
(R)with eigenvalue
(+)
( R)
equal to the classical eld corresponding to the set of classical normal variables. In a
similar way, the brais an eigenbra ofE
()
(R)with eigenvalue
()
( R).
For a coherent state of the eld, the ratewritten in (27) therefore becomes: égal à:
(R22;R11)
=
2 ()
d
(R11)
()
d
(R22)
(+)
d
(R22)
(+)
d
(R11)
=
2()
( R11)
()
( R22)
(+)
( R22)
(+)
( R11)(33)
The quantum result fordoes coincide with the semiclassical prediction. The same
conclusion holds when the state of the quantum eld is a statistical mixture of coherent
states with statistical weights().
2119

COMPLEMENT B XX
Another situation where the quantum and semiclassical predictions agree is the
case of a thermal eld. In the quantum description, Wick's theorem (ComplementXVI)
allows expressing the four-point correlation function appearing inas the sum of
products of two-point correlation functions. In a similar way, in the semiclassical theory,
the thermal eld is a Gaussian random eld, and here again, the classical four-point
correlation function is the sum of products of two-point correlation functions. Provided
we use the same two-point correlation functions in both theories, their predictions agree.
Comment
The interferometric analysis of the electric eld of the light emitted by the stars (to mea-
sure their angular diameter) is confronted with the problem of atmospheric uctuations,
which introduce a random phase shift between the two arms of the interferometer. The
analysis of intensity correlations is much less sensitive to these uctuations. As dier-
ent parts of the stars emit incoherent waves, the total eld received from the star is
Gaussian, and the result we just established shows that analyzing intensity correlations
allows obtaining two-point correlation functions and hence the same information as the
one contained in a eld correlation measurement. The validity of such a method, based
on intensity correlations, was experimentally demonstrated in 1956 by Robert Hanbury
Brown and Richard Twiss [46].
. Situations requiring a quantum radiation treatment
An example of a situation where a quantum treatment of the radiation becomes
essential is shown in Figure (2). A one-photon wave packet is emitted by atom; this
wave packet is described by(cf. ComplementXX). It then goes through a beamsplitter
LS that divides it into two wave packets: a transmitted wave packetand a reected
wave packet, which then arrive on two detectors1and2.
Figure 2: Atom A emits a photon described by a wave packet. This wave packet goes
through a beamsplitter that divides it into a transmitted wave packetand a reected
wave packet, which then arrive on two detectors1and 2. The quantum and
semiclassical predictions concerning the correlations between the signals detected on1
and2are signicantly dierent (see text).
2120

PHOTOIONIZATION
In a quantum description of the radiation, the radiation stateafter crossing the
beamsplitter is still a one-photon state, described by the linear superposition of the two
one-photon wave packetsand, that is:
= + (34)
In the expression (27) for the rategiving the probability of a photoionization at
time1of detector1atR1, and a photoionization at time2of detector2atR2,
appears the squared norm of the ket:
(+)
d
(R22)
(+)
d
(R11) (35)
whereis given by (34). The rst operator
(+)
d
(R11), which destroys a photon,
yields the vacuum0when it acts on statethat contains only one photon. The
second destruction operator
(+)
d
(R22)will then yield0when acting on the vacuum.
The two detectors1and2cannot both undergo a photoionization. This result was,
a priori, obvious: a single photon cannot produce two photoionizations.
If, on the other hand, the radiation emitted by the atom is described classically,
the two wave packetsandare classical wave packets which can ionize the detectors
1and2they encounter.
Single photon sources are not easy to fabricate. An experiment close to the sit-
uation in Figure 47]. Instead of the atom A in Figure, it
uses as a light source atoms emitting pairs of photons in a radiative cascade: the atom
emits a photon of frequencygoing from a stateto a state, then a photongoing
from stateto a state. If we callthe radiative lifetime of state, photonis
emitted after photonin a time window having a width of the order of. Imagine
we add to the experimental set-up of Figure
that detects thephoton and can trigger a departure time: after each detection of a
photon, detectors1and2are activated, but for a short time interval of the order
of. The probability of detecting a single photonduring that time window is much
higher than in a time window of the same length, but not triggered by the detection
of aphoton. This trigger method provides an equivalent of a single photon source,
and was used to observe that a singlephoton could not simultaneously excite both
detectors1and2.
. Resonance uorescence of a single atom. Photon antibunching
Another experiment clearly shows the need for a quantum description of radiation:
the study of the second order correlation functionof the uorescent light emitted
by a single atom or ion, and excited by a resonant laser beam.
Imagine the emitting object A in Figure
6
. Submitted to
the resonant laser excitation, the ion emits a series of photons which enter the set-up
of Figure. The distances between the beamsplitter and the detectors1and2are
equal, so that the two wave packets associated with each photon arrive at the same time
on1and2.
6
Ion trapping is now a well mastered technique. The results presented in Figure
on a single
24
Mg
+
ion [48]. The rst experimental evidence for photon antibunching in the uorescent
light from a single atom were obtained on a sodium atomic beam at very low intensity, with an observation
volume small enough for the probability of its containing more than one atom to be negligible [49].
2121

COMPLEMENT B XX
For a continuous laser excitation with a constant intensity, the statistical properties
of the uorescent light are invariant under time translation; consequently, the quantum
correlation function(R22;R11)characterizing the photoionizations detected on
1and2depends only on=2 1. It shall be noted
(2)
(). Figure
variations of
(2)
()as a function of, for increasing values (from bottom to top) of
the laser intensity. This gure shows that
(2)
()is zero for= 0; in other words, the
detected photons are antibunched in time (one cannot detect simultaneously a photon
at1and another one at2).
The quantum interpretation of that result is as follows. Each photon emitted by
the ion is detected either by1, or by2. Right after the emission of a photon, the
ion is projected into the ground stateof the transition excited by the laser.
This means it cannot immediately emit another photon as it must rst be re-excited by
the laser, and that takes a certain time. This is why
(2)
()is zero for= 0. Actually,
after it emits a photon, the atom starting fromwill oscillate between the stateand
the excited stateat the Rabi frequency characterizing the atom-laser coupling, and
proportional to the laser eld amplitude. The Rabi oscillations explain the oscillations
of
(2)
()that appear in Figure
increases.
Figure 3: Intensity correlations in the resonance uorescence of a single ion excited by
a laser. The gure shows the time correlations
(2)
()between the signals from the two
detectors1and2as a function of the delaybetween two detections. The three curves
correspond to increasing intensities (from bottom to top) of the laser beam exciting the
resonant uorescence of the trapped ion. It shows that
(2)
()is zero for= 0and, for
small positive values of, it increases with(gure adapted from [48]).
Let us examine now the predictions of a theory that classically treats the eld
emitted by the ion. We established above that the correlations between the photoioniza-
tion rates of the two detectors are described by the correlation function
()(+)of
2122

PHOTOIONIZATION
the classical intensity(). For a stationary eld, this classical correlation function
(2)
cl
depends only on:
()(+) =
(2)
cl
() (36)
In addition, writing that
(()(+))
2
>0, we get:(())
2
+((+))
2
>2()(+)
that is, taking into account the eld stationarity and relation (36):
(2)
cl
(= 0)>
(2)
cl
() (37)
The semiclassical theory therefore predicts that
(2)
cl
()should not be an increasing func-
tion ofin the vicinity of= 0. This is contradicted by the experimental results shown
in Figure, and hence proves that the uorescent light emitted by a single ion excited
by a resonant laser beam cannot be described as a classical eld.
The radiation quantum theory is thus essential to account for all photoionization
experimental results. This remains true even though the simple photoelectric eect
observed on a single photodetector can be described by a semiclassical theory (without
photons).
4. Two-photon photoionization
4-a. Dierences with the one-photon photoionization
We now consider a two-photon absorption process similar to those studied in Com-
plementXX, but where the nal stateof the two-photon absorption process is now
part of the atomic ionization continuum. This continuum starts at an energy(ion-
ization energy) above the energy of the ground state(Fig.). This process is called
two-photon photoionization.
The photoionization process transforms the atom into an ion and an electron,
which moves away. When the distance between the electron and the ion is large enough,
their Coulomb interaction energy becomes negligible and the electron energy is just its
kinetic energy. Total energy conservation tells us that this kinetic energy is equal to:
kin= 2~ (38)
If we plot the variations ofkinas a function of, we get a straight half-line with slope
2~, which starts from the abscissa axis at point2~. This result is a generalization of
the photoelectric law established in 1905 by Einstein.
The previous result clearly shows that it is not necessary for the incident photon
energy~to be larger than the ionization energy for the atom to undergo photoioniza-
tion. Figure ~is lower thanwhereas2~is larger than. This
result can be generalized: if~ , with= 12 1, but if~ , we
have a-photon photoionization. The kinetic energy of the photoelectron, once it is far
enough from the ion, is equal tokin=~ .
2123

COMPLEMENT B XX
4-b. Photoionization rate
We rst assume the radiation is monochromatic, and use expressions (13) and (14)
of ComplementXXfor the two-photon absorption probability amplitude, whose mod-
ulus squared yields the probability. As the nal statenow belongs to a continuum,
we must sum this probability overand use Fermi's golden rule to compute the pho-
toionization probability per unit time. Since the modulus squared of equation (14) is
proportional to(1), whereis the number of incident photons, the photoioniza-
tion rate increases as the square of the incident radiation intensity (for1). In a
similar way, it can be shown that a-photon ionization increases as theth power of the
incident radiation (for1).
Consider now the case of non-monochromatic stationary radiation. As in section
, we assume the radiation spectral density is centered around a frequencyexwith
a widthmuch smaller than the spectral bandwidthof the detector. The eld
correlation function( )that appears in the two-photon absorption probabil-
ity is still given by relation (27) of ComplementXX. The two operators

E
(+)
(R)
appearing in this equation have a predominantly exponential
ex
time dependence.
Similarly, the two operators

E
()
(R)have a predominantly exponential
+ ex
time
dependence. With the same reasoning as in , which led to (14), we set:
( ) =
2ex( )
( ) (39)
where( )is an envelope function with a much slower dependence in,
and on a time scale of the order of1. In the double integral of (8), the correlation
Figure 4: Two-photon photoionization. The atom goes from stateto state, which is
part of the ionization continuum, through the absorption of two photons, with energy~.
The unbound electron produced at the end of that process leaves the atom with a kinetic
energy equal, when the electron is far enough from the atom, tokin= 2~ .
2124

PHOTOIONIZATION
function( )is dierent from zero only for where is the
correlation time of the atomic dipole, much shorter than1. We can therefore, as in
, take=in the envelope function( )dened in relation (39) which
yields. We obtain for the eld correlation function appearing in the two-photon
ionization rate:
( ) =
2ex( )
ineE
()
(R)eE
()
(R)
eE
(+)
(R)eE
(+)
(R)
in (40)
Note that the average value appearing in this equation is independent ofsince the
radiation is supposed to be stationary.
4-c. Importance of uctuations in the radiation intensity
Even when the radiation is monochromatic, i.e. when only a single modeis
populated, its intensity can take dierent values, spread out around an average value;
the only case where the radiation intensity is well dened is when the radiation is in
a Fock state. If we only consider stationary monochromatic radiation, the most
general state is a statistical mixture of Fock stateswith weight().
As an example, if the modeis in thermal equilibrium at temperature, the
probability of its containingphotons is:
() =
1
(+12)~
(41)
whereis the Boltzmann constant andthe partition function given by:
=
(+12)~
(42)
From these two equations one can easily compute the average valueof the number of
photonsin this mode, as well as the average value
2
of
2
, for radiation at thermal
equilibrium (see demonstration below). In particular, we can show that:
2
= + 2
2
(43)
According to the previous results, the two-photon photoionization rate is proportional
to(1)=
2
. If the radiation has a well-dened intensity, i.e. if it is in a
Fock state, we have
2
=
2
and the photoionization rate is proportional to
2
if 1. On the other hand, if the radiation is in thermal equilibrium with the same
average value, the photoionization rate is, according to (43), proportional to2
2
,
i.e. twice as large as for a state with no dispersion in intensity but same average value.
An intensity uctuation, keeping the average valueconstant, considerably increases
the photoionization rate. This result is to be expected for a nonlinear phenomenon: the
values ofabove the average valuecontribute much more than the values below.
2125

COMPLEMENT B XX
Demonstration of relation(43)
This calculation has already been explained in XV. We
briey recall its principle. We note=~ and eliminate the irrelevant factor
2
from the partition function by setting:
() =
2
() (44)
The function()is easily computed, since:
() = 1 ++
2
++ +=
1
1
(45)
We also know that:
= () =
1
()
=
1
()
d()
d
(46)
This leads to:
=
1
1
(47)
A similar calculation permits computing
2
using the second derivatived
2
()d
2
,
which leads to relation (43), the equivalent of relation (42b) of ComplementXV.
If the radiation is no longer monochromatic, but still stationary and at thermal
equilibrium, we can use the eld correlation function (40) and Wick's theorem (Comple-
mentXVI) to rewrite this function as the sum of products of second order correlation
functions:
( ) =
2ex( )
ineE
()
(R)eE
(+)
(R)
in
ineE
()
(R)eE
(+)
(R)
in
+
ineE
()
(R)eE
(+)
(R)
in
ineE
()
(R)eE
(+)
(R)
in (48)
We thus get a sum of two terms, each proportional to the square of an average intensity.
5. Tunnel ionization by intense laser elds
As high power lasers became available, studies of multi-photon ionization processes led
to the discovery of many new physical phenomena. In particular, when the instantaneous
laser eld becomes of the order of the Coulomb eld binding the electron to the nucleus,
ionization no longer results from a multi-photon ionization process, but from a tunnel
eect. The laser eld, yielding a potential that varies linearly as a function of the electron-
nucleus distance, lowers the Coulomb potential suciently to allow the electron to escape
2126

PHOTOIONIZATION
Figure 5: Eective potential seen by an electron undergoing tunnel ionization. The elec-
tron leaves the ion by tunneling through the potential barrier, sum of the ionic Coulomb
potential and the linear potentialassociated with the laser electric eld, assumed
to be linearly polarized along theaxis.
via a tunnel eect (see Fig.5). Once the electron has left the ion, it is accelerated by the
laser eld. As the oscillating laser eld changes sign, the acceleration produced by the
laser eld is inverted, the electron comes back toward the ion and emits, as it passes close
to the ion, a bremsstrahlung radiation (braking radiation). It can be shown that the
frequency of this radiation is an odd harmonic of the laser frequency. The order of this
harmonic is very high and can reach several hundred. As the fraction of the period of the
laser eld where the electron can escape by the tunnel eect is very small, the electron
wave packet that leaves the ion has a very short time extension. The bremsstrahlung
radiation it emits when it comes back to the ion also extends over a very short time,
expressed in tens of attoseconds (one attosecond is equal to10
18
sec). The interested
reader can nd an up to date review of these developments in Chapters10and27of
reference [24].
2127

TWO-LEVEL ATOM IN A MONOCHROMATIC FIELD. DRESSED-ATOM METHOD
Complement CXX
Two-level atom in a monochromatic eld. Dressed-atom method
1 Brief description of the dressed-atom method
1-a State energies of the atom + photon system in the absence of
coupling
1-b Coupling matrix elements
1-c Outline of the dressed-atom method
1-d Physical meaning of photon number
1-e Eects of spontaneous emission
2 Weak coupling domain
2-a Eigenvalues and eigenvectors of the eective Hamiltonian
2-b Light shifts and radiative broadening
2-c Dependence on incident intensity and detuning
2-d Semiclassical interpretation in the weak coupling domain
2-e Some extensions
3 Strong coupling domain
3-a Eigenvalues and eigenvectors of the eective Hamiltonian
3-b Variation of dressed state energies with detuning
3-c Fluorescence triplet
3-d Temporal correlations between uorescent photons
4 Modications of the eld. Dispersion and absorption
4-a Atom in a cavity
4-b Frequency shift of the eld in the presence of the atom
4-c Field absorption
Introduction.
The probability amplitude for an atom, subject to monochromatic radiation, to absorb
a photon and go from a discrete stateto another discrete statewas calculated in
. We used, however, a perturbative treatment limited to lowest order
with respect of the interaction Hamiltonian. The predictions of such an approximate
calculation are, a priori, only valid for times that are suciently short for the higher
order corrections to remain negligible. This complement presents another approach to
atom-photon interactions, called the dressed-atom approach, which does not have those
limitations. It considers the atom and the mode of the quantum eld it interacts with
as a single quantum system. As this unied system is described by a time-independent
Hamiltonian
1
, one can study its energy diagram to obtain very useful information.
1
This would obviously not be possible in a classical description of the radiation (even with a quantum
treatment of the atom), since the eld varies sinusoidally with time, and its coupling Hamiltonian with
the atom is time-dependent.
2129

COMPLEMENT C XX
This will allow us to improve the results of Chapter. The dressed-atom method
yields a non-perturbative description of the physical processes under study, and hence
remains valid even for intense elds. It provides new insights into several important
physical phenomena: atoms' behavior in an intense electromagnetic eld, spectral dis-
tribution of the light spontaneously emitted by an atom in such an intense eld, time
correlations between the emitted photons, origin of the forces exerted on an atom by
radiation with a space-varying intensity.
As is usual in the literature, and as we already did in ComplementXIX, we shall
notethe atomic ground state andthe atomic excited state (instead of using our
previous notation ofandfor the two atomic levels). For the same reason, rather than
keeping the notationof Chapter
we shall use, that refers more explicitly to a laser beam, which can have a very high
intensity.
We start in
2
. We assume
the frequencyto be close to resonance with the atomic frequency, noted0= (
)~, but far enough from all the other atomic transition frequencies. The radiation-
atom interaction will be characterized by a frequency called the Rabi frequency. It
is the equivalent, in this radiation quantum treatment, of the precession frequency of a
spin turning around a classical radiofrequency eld in a magnetic resonance experiment
(ComplementIV). Establishing the energy diagram of the unied atom-photon system
will enable us to study both the weak coupling regime (Rabi frequency small compared
to the natural widthof stateor to the detuning 0between the eld and
atomic frequencies) and the strong coupling regime (Rabi frequency large compared to
the natural width and to the detuning).
The weak coupling regime is studied in . We show that the ground state's
energy undergoes a light shift, by an amount that is proportional to the eld intensity,
and whose value as a function of the detuning follows a Lorentzian dispersion curve. The
ground state also undergoes a radiative broadening, proportional to the eld intensity,
and which can be interpreted as a probability per unit time of leaving the ground state
through the absorption of a photon.
We focus in
satesandappears, although damped because of the radiative instability of. The
energy diagram of the dressed-atom allows interpreting phenomena that are specic to
the strong coupling regime, such as the uorescence triplet and the temporal correlations
between the photons emitted in the lateral components of that triplet.
The atom-eld coupling perturbs, not only the atom, but also the eld. We show in

presence, are actually perturbations of the eld, just as the light shifts and radiative
broadening are perturbations of the atom.
1. Brief description of the dressed-atom method
Let us call laser mode the quantum eld mode that is populated by photons with
frequency. In the absence of coupling between the atomand this laser mode, the
Hamiltonian of the total system+is equal to+, where is the atomic
Hamiltonian, andthe Hamiltonian of the radiation in laser mode. The energy levels
2
A more detailed description can be found in Chapter VI of [21].
2130

TWO-LEVEL ATOM IN A MONOCHROMATIC FIELD. DRESSED-ATOM METHOD
of+ are labeled by two quantum numbers:orfor the atomic internal state
3
,
for the photon number in the only radiation mode that is not empty and contains
photons of frequency.
1-a. State energies of the atom + photon system in the absence of coupling
Consider the two states and 1, whose energies with respect to the
photon vacuum state are:
=+~ 1=+ ( 1)~ (1)
Their energy dierence is:
1=~ ( )
=~( 0) =~ (2)
where:
= 0 (3)
is the frequency detuning between the eld frequencyand the atomic transition
frequency0= ( )~. For a resonant eld, i.e. when=0, the two states are
degenerate. Here we consider very small detunings (in absolute value) compared to0:
0 (4)
Consequently, even if the eld is not exactly resonant, the two statesand
1can be grouped in a two-dimensional multiplicity():
() = 1 (5)
which is far from all the other states of the system atom + eld. There is an innity of
other multiplicities, for values ofgoing from1to innity. As an example, Figure
shows the three multiplicities(1),()and(+ 1); there are1others with
lower energies, and an innity with higher energies. Each multiplicity is separated from
the next one by the distance~, and the spitting between the two levels inside one
multiplicity equals~. The multiplicity(1)corresponding to= 1, includes the two
states1and0; on the other hand, the state0is isolated.
1-b. Coupling matrix elements
The couplingbetween the atom and the modeis proportional to the product
of the atomic dipole momentDand themode component of the radiation electric
eldE. We can choose the origin of the coordinates so as to be able to write
kR
=
whereRis the position of the atomic center of mass. We then get for:
=
~20
3
D("+") (6)
3
The atomic external degrees of freedom are treated classically by assuming that the atom is xed
at pointR.
2131

COMPLEMENT C XX
Figure 1: Energy levels of the system atom + photon in the absence of coupling. Only
three adjacent multiplicities(1),()and(+ 1)are shown in the gure; many
more exist below or above, corresponding respectively to smaller or larger values of.
Each vertical arrow links a pair of states having an energy dierence indicated next to it.
The only non-zero matrix elements of the odd operatorDare those betweenand
. The annihilation and creation operatorsandchangeby1. It follows that
is coupled to 1and + 1, whereas 1is coupled to and
2. The two states and 1of the multiplicity()are thus coupled
to each other by the matrix element:
1 =
~
2
(7)
whereis the Rabi frequency dened as:
=
1 (8)
with:
1=
2
~~20
3
" D (9)
We assume1is real and positive. If this is not the case, it suces to change the relative
phase
4
of the ketsand, which modies the phase of the matrix elementD;
a suitable choice of that phase will make1real and positive.
As operator changes both the photon number by one unit and the atomic
internal state, it does not have matrix elements between the kets of multiplicity()
and those of multiplicities( 1). On the other hand, it can couple these kets with
those of multiplicities( 2)and, to higher order, to those of multiplicities even
4
Such a phase change will aect the non-diagonal elements of the density matrix, leaving the physical
predictions unchanged.
2132

TWO-LEVEL ATOM IN A MONOCHROMATIC FIELD. DRESSED-ATOM METHOD
further away. However, the distances (in energy) between these multiplicities and()
are of the order of2}(or of a multiple of that energy), whereas we have assumed that
the interaction matrix elements are very small compared to}. The multiplicities other
than()have therefore energies too dierent from those of()to play a signicant
role. We shall ignore their non-resonant coupling, which has a negligible eect for a
quasi-resonant excitation.
1-c. Outline of the dressed-atom method
At the beginning of section , we have described the quantum states of the system
+(atom + laser mode) in the absence of coupling; we showed they can be grouped
into multiplicities(), with= 0,1,2, ... well separated from each other when
condition (4) is satised. As an example, Figure ()for
an atom with two levelsandincludes the statesand 1separated by
an energy}. Relation (7) tells us that these two states are coupled by an operator
describing the interaction betweenand, and that the corresponding matrix elements
equal}2. We also discussed why, for a quasi-resonant excitation, the couplings
between dierent multiplicities were negligible, so that one can separately study each
multiplicity().
. Dressed states and energies
The rst step in the dressed-atom approach is to study the energy levels of the
system+inside a multiplicity(), taking into account the couplingrestricted
to(). We must diagonalize the Hamiltonian++inside this sub-space. For
a two-level atom, the restriction of that Hamiltonian to(), noted
()
, is represented
in the base of the kets written in (5) by a Hermitian22matrix equal to:
()
=
+~ ~2
~2 + ( 1)~
= [+ ( 1)~]+~
2
2 0
(10)
whereis the identity operator in().
Because of the coupling created by the non-diagonal elements2, the states
and 1whose non-perturbed energies are separated by}(left-hand side
of Figure) are transformed into two states+()and ()with energies}
(right-hand side of the gure):
}= [+ ( 1)~] +
~
2
~
2

2
+
2
(11)
The new states+()and ()are linear superpositions of the initial states; they
are called dressed states, and their respective energies}the dressed energies. This
complement will show that a great number of interesting physical phenomena occurring
when an atom is coupled to a laser mode can be interpreted in terms of these dressed
states and their energies.
2133

COMPLEMENT C XX
Figure 2: Energies of the states of the system+within(), in the absence of
coupling (left-hand side of the gure), and in the presence of a coupling of intensity
}2between the two initial states (right-hand side of the gure).
. Rabi oscillation
Let us consider rst a particularly simple application of the dressed-atom method.
Imagine, for example, that the system is, at time= 0, in the state:
(= 0)= in= (12)
and let us try to nd the probability that it will be found at a later timein the state:
n= 1 (13)
We are dealing with the evolution of a system with two levels coupled by a static per-
turbation~2. This problem was studied in detail in . We must
rst expand the initial state on the states(), and multiply each of them by an
exponential whose argument is proportional to its energy}:
()=
+
+() +()+ () () (14)
The probability amplitude of nding the system in state1is then:
1()
2
+() 1+()
+
2
() 1() (15)
where we introduced the Bohr frequency:
=+ =

2
+
2
(16)
In (15), the signsimply means that we omitted a global phase factor
(++)2
,
with no physical signicance. The probability of nding the system at timein the nal
state (13) is therefore:
() = 1()
2
=
+
2
+
2
+
2 2
+
+ + +c.c. (17)
2134

TWO-LEVEL ATOM IN A MONOCHROMATIC FIELD. DRESSED-ATOM METHOD
where theare the scalar products:
= () and = () 1 (18)
and c.c. means the complex conjugate.
We see that the probability()is an oscillating function of time, with a frequency
that is the only Bohr frequency

2
+
2
of the system within the multiplicity.
This frequency can obviously be expanded to all orders of the perturbation
2
, but
the result we obtained is not perturbative. The oscillation we found concerns the total
system formed by a two-level atom placed in a monochromatic radiation eld that can
be intense and resonant: starting from state, the atom absorbs a photon and goes
to state 1; it then comes back to stateby stimulated emission of a photon,
and so on.
1-d. Physical meaning of photon number
The situation is dierent depending on whether the atom is placed in a real cavity
or in free space.
If the atom is placed in a real cavity, as in some experiments, the eld modes are
the cavity eigenmodes. Such a situation will be discussed in ; the photon number
then has a perfectly clear physical meaning. The volume
3
, appearing in the modal
expansion of the elds, and which is found in expressions (6) and (9) above, is simply
the volume of the cavity containing the photons.
If the atom is in free space, the volume
3
, introduced to obtain discrete modes,
is simply used in the computation, without any precise physical meaning. On the other
hand, the energy density in the vicinity of the atom, proportional to~
3
, does
have a physical meaning. Provided we keep
3
constant, we can changeand
3
arbitrarily without changing the coupling between the atom and the eld; this is because
the coupling is characterized by the Rabi frequency, which depends on
3
. In
that case, the photon numberdoes not have an intrinsic physical meaning.
Imagine, for example, that the eld is in a coherent state (ComplementV).
The values ofare then distributed around an average valuein an interval of
width=
, very small in relative value compared to, but very large in
absolute value. If bothand
3
go to innity, keeping the ratio
3
constant, the
Rabi frequencywill barely change in relative value even whenvaries over a large
interval around. The frequencycan thus be replaced in (10) by a constant
(which does not depend on):
(19)
This will be done in what follows and in .
1-e. Eects of spontaneous emission
We have ignored, until now, all the eld modes others than the laser mode. How-
ever, when the atom is in the excited state, it can spontaneously emit a photon in
another mode. This means that, in addition to the atomand the laser mode, we
must take into account the systemincluding all the modes that, initially, did not
contain any photons. Asis a very large system (sometimes called reservoir for that
2135

COMPLEMENT C XX
reason), the coupling eects between+andmust be described by a so-called
master equation; this equation describes the evolution of the density operator+of
+under the eect of the coupling with(see part D of Chapter VI in [21]). Though
we shall not introduce this master equation here, we shall merely discuss the physical
interpretation of the results it leads to.
As the frequency spectrum of the reservoirhas a widthof the order of the
optical frequency, its associated correlation time1is much shorter than all
the other characteristic times of the problem. It is, in particular, much shorter than the
radiative lifetime:
=
1

(20)
whereis the natural width of the excited state; it is also shorter than the inverse of
the Rabi frequency, which yields the characteristic time of the coupling to the laser mode.
This means that, when the system+is in the state 1and a spontaneous
emission occurs, it lasts for a time interval too short for the atom to have sucient time to
couple with. The system then goes quasi-instantaneously from state1, which
belongs to(), to state 1in the lower multiplicity( 1) see Figure.
As a consequence (see C-3 of Chapter III in [21], as well as ComplementXIII), the
evolution within()can still be described by the same equations as above, provided
we simply add an imaginary term to the energyof the excited state:
= }

2
(21)
This means that, to describe the evolution of the system+within
5
()while
taking into account spontaneous emission processes, we must replace the Hamiltonian
()
written in (10) by the eective non-Hermitian Hamiltonian:
()
e
= [+ ( 1)~]+~
2
22
(22)
Because of the imaginary term2appearing in the matrix, the two eigenvalues of
()
e
also have an imaginary part: the two dressed states are now unstable as a result
of spontaneous photon emission, which can occur in any of these states.
Within a constant factor, given by the term proportional toon the right-hand
side of (22), the eigenvalues}ofeare obtained by diagonalizing the 2-dimensional
matrix that follows on the right-hand side of (22). The exact solution is written
6
:
=

4
+
2
1
2

2
++

2
2
(23)
We now discuss the physical meaning of these results in two limiting cases.
5
The coupling with the reservoir induces other important eects leading to transitions between
dierent multiplicities. This is what happens, for example, in the uorescence phenomenon studied in
3-c.
6
For brevity, we use a slighlty incorrect mathematical notation, since the square root sign must in
principle be applied on a real and positive number. What we mean with the square root sign written
on the right-hand side of (23), is either one of the two complex numbers whose square is equal to the
complex number under the root sign.
2136

TWO-LEVEL ATOM IN A MONOCHROMATIC FIELD. DRESSED-ATOM METHOD
2. Weak coupling domain
We start with the weak coupling domain, which is more directly related with the results
of Chapter.
2-a. Eigenvalues and eigenvectors of the eective Hamiltonian
Consider rst the case where the non-diagonal coupling~2between the two
non-perturbed states of()is small compared to the dierences between the energies
of these two states (including the imaginary term associated with the natural width of
). As this dierence is complex, we must take its modulus:
+

2
(24)
This inequality is satised if:
or R (25)
The weak coupling domain is thus obtained for low light intensities, or large frequency
detunings.
For weak coupling, we can apply perturbation theory to obtain the energy correc-
tions for the statesand 1, to order 2 in. Starting from (22), we obtain
in this way the correctionto the energy of state:
=~
(2)
2
+2
=~ ~
2
(26)
where:
=
4
2
+
2

2
and =

4
2
+
2

2
(27)
A similar calculation yields for the correction to the energy of state1:
1=~+~
2
(28)
We can write the approximate eigenvalues of the eective Hamiltonian (22) in the form:
+ +
2
+

2
+
2
+ (29)
which coincides with an expansion in powers ofof the exact result (23).
Perturbation theory also allows computing the eigenstates ofeto rst order in
. The state
, which tends towards whengoes to zero, is written:= +
(2)
+2
1 (30)
This means that stateis contaminated by state1. A similar computation
for state
1, which tends towards 1whengoes to zero, yields:1= 1
(2)
+2
(31)
2137

COMPLEMENT C XX
Figure 3: Non-perturbed states (left-hand side of the picture) and perturbed states (right-
hand side) in the()multiplicity. The coupling, characterized by the Rabi frequency
, shifts stateby a quantity~(representing the light shift of the ground sate
); its wave function is contaminated by the unstable wave function of state1,
meaning that the ground state also becomes unstable as shown by its radiative broadening
~. State 1is shifted in the opposite way, compared to; its width is reduced
from~to~( ).
2-b. Light shifts and radiative broadening
The real parts of and 1represent shifts in the energy levels induced
by the coupling with the light and called for that reason light shifts. The imaginary
part of represents a radiative broadening of state, which becomes unstable
under the coupling eect. The imaginary part of 1describes a reduction of the
radiative broadening~of state 1.
Figure and 1
in the()multiplicity. They are separated by the gap~; ifis positive, state
is above state 1; conversely, ifis negative, state 1is now above state
. The thickness of the line representing state1symbolizes its natural width
~. The right-hand side of the gure represents the states perturbed by the interaction
with light. The two states and 1repel each other, meaning that they
undergo light shifts of opposite signs. The~shift of is positive ifis above
, i.e. ifis positive; it is negative ifis negative. The stablestate is also
contaminated by the unstable state1, which makes it unstable as shown by its
radiative broadening~. An atom in statecannot stay there indenitely: it will leave
that state with a probability per unit time equal to, which can be interpreted as the
photon absorption rate of an atom in state. Conversely, due to the contamination
of the unstable state 1by the stable state, state 1becomes less
unstable and its width is reduced from~to~( ).
2138

TWO-LEVEL ATOM IN A MONOCHROMATIC FIELD. DRESSED-ATOM METHODg
g

Figure 4: Plots of the light shift~(dashed line curve) and of the radiative broadening
~(solid line curve) as a function of the detuningbetween the laser frequency and the
atomic frequency.
2-c. Dependence on incident intensity and detuning
The shifts~and radiative broadenings~given by equation (27) are all pro-
portional to
2
, hence to the number of incident photons, meaning to the light intensity.
Their variations with the detuningfollow respectively Lorentzian dispersion and ab-
sorption curves (Fig.).
When the detuning is very large (in absolute value) compared to the natural width
of( ), we can neglect
2
compared to4
2
in the denominators of expressions
(27), which yields:
=

2
4
=

2

4
2
(32)
This leads to:
=

(33)
For large detunings, the light shifts are thus much larger than the radiative broadenings.
2-d. Semiclassical interpretation in the weak coupling domain
In this weak coupling domain, the atom responds linearly to the incident eld; the
results we just discussed can be interpreted semiclassically, in terms of a dipole induced
by the incident eld (see for example [50]). This dipole has a component in phase with
the eld and a quadrature component, related to the eld by a dynamic polarizability
().
The quadrature component absorbs energy from the eld. It varies with the de-
tuningas an absorption curve; it is responsible for the absorption rate associated with
the radiative broadening. The in-phase component of the dipole yields a polarization
energy. Its variation with the detuning follows a dispersion curve. It is responsible for
the light shift, just as the Stark shift results from the interaction of a static electric eld
with the static dipole it induces. This is why this light shift is often called a dynamic
Stark eect.
2139

COMPLEMENT C XX
2-e. Some extensions
We now discuss some direct, important extensions of the previous study.
. Non-monochromatic incident radiation
Imagine the radiation state is now a Fock state12 , or a statistical
mixture of such states; the radiation spectral distribution is then described by the func-
tion(). To second order perturbation theory (weak coupling domain), the processes
that come into play in the light shifts and radiative broadenings are stimulated absorp-
tion and re-emission of photons. When several modes contain photons and the radiation
state is a Fock state, the photon must be re-emitted by stimulated emission in the same
mode it was absorbed from (otherwise the matrix element describing the second order
coupling would be zero). This means that the eects of the dierent eld modes can be
added independently; we then get forand:
d
2
()
4
2
+
2
d
2
()

4
2
+
2
(34)
. Degenerate ground state
Assume the ground statehas a non-zero angular momentumand therefore
contains several Zeeman sublevels; one can then show [51] that the sublevels of
having a well-dened light shift and radiative broadening are obtained by diagonalizing
the Hermitian matrix whose elements are:
"D "D (35)
where the statesare the sublevels of. The eigenstatesof this matrix, with
eigenvalues, undergo light shifts proportional toand radiative broadenings pro-
portional to(whereandare the shifts and broadenings for a two-level atom).
Reference [52] studies the symmetry properties of matrix (35), and discusses the
equivalence between the light shifts and the eect of ctitious magnetic and electric elds
acting on the ground multiplicity of the atom. We shall simply focus here on the simple
case of a= 12 = 12transition such as, for example, the hyperne component
= 12 = 12of the6
1
06
3
1transition of the 199-isotope of mercury (=
253.7 nm). It is on such a transition that light shifts were observed for the rst time [53].
The left-hand side of Figure +andof this transition,
that link respectively=12to = +12and = +12to =12. If
the beam polarization is+, level=12has a non-zero and well dened light
shift, since the absorption and re-emission of a+photon can link sublevel=12
only to itself. On the other hand, level= +12is not shifted because there is no
+optical transition starting from= +12. We get opposite conclusions for a
polarization of the light beam: the light shift of sublevel= +12is well-dened and
sublevel=12is not shifted. Now, by symmetry, the Clebsch-Gordan coecients
(ComplementX) for the+andtransitions are equal; the light shifts have the same
2140

TWO-LEVEL ATOM IN A MONOCHROMATIC FIELD. DRESSED-ATOM METHODmg=1=2 mg= +1=2
me=1=2 me= +1=2
+
mg= +1=2
mg=1=2
+

Figure 5: The left-hand side of the gure represents the= 12 = 12transition,
and the light beam polarizations that can induce transitions between its Zeeman sublevels.
The diagram on the right-hand side plots in its center the ground state energy levels in
the absence of any light beam (Zeeman levels in a static magnetic eld); the two lateral
extensions depict their light shifts by a non-resonant light beam with polarization+(on
the right), or(on the left). In the rst case, the light selectively shifts the sublevel
=12, in the second case, it shifts the sublevel= +12. This is why, depending
on whether the beam polarization is+or, the variation in the gap between the two
Zeeman sublevels changes sign.
value for a+excitation of sublevel=12and for aexcitation with same
intensity of sublevel= +12.
In the presence of a static magnetic eld, there is an energy gap between the
two atomic sublevels=12(Zeeman eect). The right-hand side of Figure
shows that a non-resonant light excitation changes this gap by the same amount, but
in the opposite directions
7
depending on whether it has a+orpolarization. The
ground state magnetic resonance line, detected by optical methods using a resonant
beam (ComplementXIX, ), is thus shifted when a second non-resonant beam is
applied; this shift has opposite directions, depending on whether that beam has a+or
polarization. As relaxation times can be very long in the ground state, its magnetic
resonance line is very narrow, which allows detecting very small light shifts, of the order
of a few Hz. This is how the existence of light shifts were demonstrated in 1961, when
laser sources were not yet available in laboratories []. With laser sources, one routinely
observes shifts of the order of 10
6
Hz, and even more.
3. Strong coupling domain
We now examine how the previous results are modied in the strong coupling regime.
3-a. Eigenvalues and eigenvectors of the eective Hamiltonian
A strong coupling regime means that the non-diagonal element~2of the eec-
tive Hamiltonian written in (22) is large compared to the dierence between two diagonal
elements:
and (36)
7
We assume the detuningis large compared to the Zeeman splitting.
2141

COMPLEMENT C XX
For the sake of simplicity, we shall only consider the resonant case (= 0). Equation
(23), which yields the eigenvalues of the 22 matrix of (22) for any value of, then
becomes:
=

4
1
2

2

2
4
(37)
where, as we did above, we use the concise notation for the square root of a number that
is not always positive (see note).
As long as 2, the last term on the right-hand side of (37) is purely
imaginary. The same is true for the two eigenvalues, which are equal to:
=

4
1
2
2
4

2
(38)
If, in addition, , a limited expansion of (38) in powers ofyields+=
2and =( )2; as expected, we conrm the results of the previous
for the weak coupling regime. Asincreases, while remaining lower than2, the
eigenvalue+increases whereasdecreases, but their sum(++)remains constant
and equal to2. Whenreaches the value2, both eigenvaluesare equal to
4.
As soon asgoes beyond2, the last term in (37) becomes real. The two
eigenvalueshave opposite real parts and the same imaginary part, equal to4;
the two dressed levels now always have the same width4. As the coupling becomes
strong ( ), the energies are equal to:

4

2
(39)
and the eigenvectors tend toward symmetric and antisymmetric linear combinations of
and 1:
()
1
2
[ 1] (40)
Such states can no longer be considered to be a result of a light mutual contamination of
the non-perturbed states of multiplicity(). They are actuallyentangled, and hence
impossible to consider as products of an atomic state and a eld state. These states of
the global atom + eld system are commonly calledatom-eld dressed states.
3-b. Variation of dressed state energies with detuning
The solid lines in Figure ()as a
function of}. The energies are dened with respect to the energy of the non-perturbed
state 1, chosen to be equal to~0, and represented by a horizontal line with
ordinate~0. Compared to this energy, statehas an energy equal to~which
varies, as a function of~, as a straight line of slope unity. This line intersects the
horizontal line representing the energy of1at a point with abscissa~0.
At resonance (= 0), the two dressed states()are separated by an
energy~, since we assume we are in the strong coupling domain where . Let
2142

TWO-LEVEL ATOM IN A MONOCHROMATIC FIELD. DRESSED-ATOM METHOD Energy
Figure 6: Energies of the dressed states()(solid lines) and of the non-perturbed
states and 1(dashed lines) as a function of~. The energies are dened
with respect to the energy of the non-perturbed state1, chosen to be equal to~0.
As~varies, the energies of the dressed states follow a hyperbola whose asymptotes are
the straight lines representing the energies of the non-perturbed states (anticrossing).
us rst completely neglect. Leaving resonance, and as the detuning becomes larger and
larger (in absolute value), one nally reaches regions whereis larger than, which
corresponds to a weak coupling regime. Varying the detuning, one then continuously
goes from a strong coupling to a weak coupling region. The energies of the dressed levels
()follow a hyperbola whose asymptotes are the energies of the non-perturbed
states and 1(Figure). As they come close to their asymptotes, the
dressed states become very close to the non-perturbed corresponding states, and the
distance between the hyperbola and its asymptote is simply the light shiftdened in
(27).
To take into account the natural widthof the excited level, one should add a
width to the dressed levels shown in Figure. Far away from the anticrossing center, close
to the asymptotes, the width would be for the dressed states that are close to the
horizontal asymptote, andfor the dressed states that are close to the asymptote with
slope one. Following one hyperbola branch continuously, the width will progressively
change from one of these values to the other, and take the value4at the center of the
anticrossing.
Another interesting phenomenon occurs when the system continuously follows one
of the hyperbola branches. Imagine it follows the lower branch, from left to right, for
instance because the excitation frequency is slowly varied. If the transit is slow enough
to neglect any non-adiabatic transition to the other dressed state, i.e. to the other
hyperbola branch, one continuously goes from stateto state 1. This is
2143

COMPLEMENT C XX
another convenient way to go fromto: instead of applying a resonant eld during
the time necessary for the Rabi oscillation to bring the system fromto(pulse),
one slowly scans the eld frequency through resonance, from a lower to a higher value.
Note however that this scanning cannot be too slow, since it must occur on a time scale
that is too rapid for the dissipative processes to be able to change the atomic internal
state. Such a transit is often referred to as an adiabatic fast passage, as it must be slow
enough to remain adiabatic and fast enough to avoid any dissipation during the transit
time. The dressed-atom approach allows clearly specifying the conditions for transferring
the atom from one level to another.
3-c. Fluorescence triplet
With the dressed-atom approach, we can also simply explain the spectrum of the
lines spontaneously emitted by an atom subjected to intense radiation. When studying
elastic scattering in , we showed that, when the exciting radiation
had an intensity low enough to allow a perturbation treatment, the radiation emitted
spontaneously by the atom had the same frequency as the exciting radiation. We now
show that the situation is dierent in the case of an intense excitation radiation: new
frequencies appear in the light emitted by the atom
8
.
We assume the exciting radiation to be resonant and intense, so that the two
dressed states()of multiplicity()are separated by an energy interval~
(Figure). These two states are linear superpositions of the statesand 1;
consequently, they both have a non-zero projection onto1. Similarly, the two
states(1)of multiplicity(1)are linear superpositions of the states1
and 2; they both have a non-zero projection onto1. The lines emitted
spontaneously by the atom are those that link two energy levels between which the atomic
dipole operatorDhas a non-zero matrix element. SinceDdoes not change the photon
numberand can linkto, the matrix element 1D 1is non-zero; each
of the two states()can be linked viaDto each of the two states(1). The
four radiative transitions represented by the curly arrows in Figure
and correspond to three distinct frequencies: frequency+ for the+()
( 1)transition; frequencyfor both the+() +( 1)and the
() (1)transitions; frequencyfor the() +(1)
transition. We get a frequency triplet for the spontaneously emitted light, which was
rst predicted by Mollow [54] by using a semiclassical treatment.
Autler-Townes doublet
Imagine that one of the two atomic states we considered until now, for example, is
connected via an allowed transition to a third state, meaning thatD is non-
zero. Let us also assume that the radiation frequencythat is resonant for the
transition, is completely o-resonance for thetransition; consequently, it does not
perturb state, so that the sates can be considered as eigenstates of the total
Hamiltonian, even if they are slightly shifted. The two states(), which both have a
non-zero projection onto, can therefore be connected viaDto . This means
that, because of the presence of an intense radiation exciting thetransition, the
transition is split into two lines separated by~, called the Autler-Townes
doublet [55].
8
This is not Raman scattering as we assume there are no other atomic states exceptand.
2144

TWO-LEVEL ATOM IN A MONOCHROMATIC FIELD. DRESSED-ATOM METHOD
Figure 7: Radiative transitions between one of the two states()of multiplicity
()to one of the two states( 1)of multiplicity( 1), whose energy is
lower than that of()by the quantity~. We assume the exciting radiation to be
exactly at resonance, so that the energy interval between the two statesof each
multiplicity is equal to~.
3-d. Temporal correlations between uorescent photons
We now study the characteristics of the radiation spontaneously emitted by an
atom that interacts continuously with the electromagnetic eld of a laser.
. Radiative cascade of the dressed atom
We saw that a dressed atom, spontaneously emitting a photon, goes from the
multiplicity()to the one just below,(1), located at an energy distance}. We
shall not study here the precise evolution of the physical system as it leaves multiplicity
(), which requires the master equation, already mentioned in . Our discussion
will remain qualitative, but the interested reader will nd a more detailed approach in
D of Chapter VI in [21].
Once it reaches( 1), the atom can spontaneously emit a new photon, which
brings the dressed atom to( 2), and so on. The series of photons spontaneously
emitted by the atom in continuous interaction with the laser radiation can be viewed as
a radiative cascade of the dressed atom descending its energy diagram.
This image of a radiative cascade permits studying the time correlations between
the photons emitted by the atom. As we shall see, the observed correlations depend on
the spectral resolution of the photodetectors used.
2145

COMPLEMENT C XX
. High spectral resolution photodetection
With a high enough spectral resolution, one can observe the time-correlations
between photons emitted in the two side-bands of the triplet. Suppose we place lters
in front of the photodetectors, so that each can receive only one of the components of
the uorescent triplet, centered at frequencies,+ and . This means
that the spectral resolution of the apparatus is better than the splitting frequency
of this triplet, but it does not imply that it is lower than the natural widthof each
components. If we callthis spectral resolution, we then have:
(41)
Imagine that at a given time, a detector registers a photon emitted for example in
the lateral band centered at+, as the system undergoes the transition +()
( 1)(curly arrow on the left-hand side of the Figure). The next photon is
emitted as the system, starting from ( 1), undergoes either the ( 1)
( 2)transition, emitting a photon of frequency, or the ( 1)
+( 2)transition, emitting a photon of frequency. This means that a
second photon with the same frequency+ as the rst one, cannot be emitted right
after the rst one.
If the frequency of that second photon is, the system ends up in state ( 2);
from that state, it can emit either a third photon with frequency, or a third photon
with a lower frequency . If, on the other hand, the frequency of that second
photon is , the system ends up in state +( 2); from that state, it can
emit either a photon with a higher frequency+ , or a photon of frequency.
As opposed to the second photon, the third photon may thus have the same frequency
+ as the rst one. Following the same line of reasoning one can argue that, if the
rst photon has a frequency, this cannot be the case for the second photon; one
must wait until the third photon to eventually obtain the same frequency. This
means that, if photons with only the two extreme frequenciesare selectively
observed, the detected emission processes will necessarily alternate in time (but these
events may be separated by any number of photon emissions at the central frequency
).
Comment:
Taking a Fourier transform to return to the time domain, relation (41) imposes a limit
to the temporal resolutionof the detection system:1. This means that it is
not possible to measure the exact time at which a photon is emitted with a precision of
the order of the Rabi precession period.
. Photodetection with high temporal resolution
We now study the opposite case where the detectors have a temporal resolution
better than the Rabi precession period. This allows a precise determination of the time
at which the photon is emitted, but one can no longer distinguish the frequencies of the
three triplet components.
We have seen above () that the elementary spontaneous emission processes
have a correlation time that is very short compared to all the other characteristic times
2146

TWO-LEVEL ATOM IN A MONOCHROMATIC FIELD. DRESSED-ATOM METHOD
of the problem (because of the large spectral widthof the empty modes' reservoir).
A spontaneous emission from()thus corresponds to a very short quantum jump
taking the system from state 1in()into state 1in( 1). Once
the atom has reached this second state, it cannot emit a second photon right away, since
no spontaneous emission can occur from a ground state. A certain time must elapse for
the atom-laser interaction to bring the system from state1to the state 2
it is coupled with, and from which another photon can be spontaneously emitted. The
system then falls back to state2, and the previous process repeats itself (with an
value lowered by one unit). It therefore becomes clear why one observes a temporal
antibunching of the photons emitted by a single atom, as they must be separated by a
time interval at least of the order of1; this antibunching was already referred to in
of ComplementXX.
4. Modications of the eld. Dispersion and absorption
We now study how the eld is modied by its interaction with the atom.
4-a. Atom in a cavity
The atom-eld interaction does not solely perturb the atom; it also changes the
eld. In order to study this eect, it is convenient to imagine the atom being placed in
a real cavity assumed to be perfect, meaning that its losses can be ignored (they occur
on a time scale much longer than all the other relevant times of the experiment). As
opposed to what we did before, we shall keep thedependence of the Rabi frequencies
given by equations (8) and (9), since in a cavity the photon numberhas a physical
meaning ().
Figure ()of the system atom + eld for low
and increasing values of the photon number, starting at= 0. Multiplicity(0)
Figure 8: Energy levels of the system atom + eld for low values of the photon number
(in angular frequency units, meaning the energies are divided by~). States and
1of()undergo opposite shifts, proportional to. State0is not shifted.
2147

COMPLEMENT C XX
contains a single state0. Multiplicity(1)contains the two states1and0.
Multiplicity(2)contains the two states2and1, and so on.
We shall assume we are in the weak coupling regime, so that we can use the
perturbative results of 0
is not shifted as it is not coupled to any other state
9
. States1and0of(1)
undergo opposite light shifts, respectively+~and~, whereis given by equation
(27), where we have replacedby1, the Rabi frequency for= 1(see (8)). Setting:
0=
2
1
4
2
+
2
(42)
the light shifts of states1and0are, respectively,+~0and~0. According to
(8), the squares of the Rabi frequenciescharacterizing the atom-eld coupling in the
multiplicities(), are proportional to; this means that states2and1of(2)
undergo light shifts respectively equal to+2~0and2~0. More generally, states
and 1of()undergo shifts respectively equal to+~0and ~0.
A similar reasoning can be applied to the radiative broadening. It shows that the
radiative broadenings of statesand 1of()are respectively equal
10
to
+ 0and 0, where0is given, according to (27), by:
0=
2
1

4
2
+
2
(43)
4-b. Frequency shift of the eld in the presence of the atom
Consider the left column in Figure. The gap between the perturbed energies of
states1and0is equal to~(+0); the gap between the perturbed energies
of states2and1is equal to~(+ 20 0) =~(+0), and so on. As the
light shifts of the statesare proportional to0, increasing linearly with, the
perturbed levels in the left column of Figure
in the presence of the coupling; the distance between consecutive levels simply goes from
~to~(+0). In other words, the presence in the cavity of an atom in its ground
statechanges the eld frequency fromto+0. As the light shifts of the levels
in the right column of Figure
presence in the cavity of an atom in the excited statechanges the eld frequency from
to 0.
The atom-eld interaction thus shifts the eld frequency inside the cavity by a
quantity that changes sign, depending on whether the atom is in the internal stateor
. Let us assume this interaction lasts a time, as will be the case if an atom is introduced
into the cavity and takes that time to traverse it. Compared to the free oscillation in
the absence of the atom, the eld oscillation will be out of phase by an amount; this
phase shift is equal to= +0if the atom is in state, and to= 0if the atom
is in state.
This change in the eld frequency is a phenomenon similar to that described by the
real part of the refractive index: a light beam going through an atomic media changes
9
There is actually a coupling between state0and state1, but is highly non-resonant; we shall
ignore it since, as stipulated above, our computation is to zeroth order in .
10
State0does not undergo any radiative broadening.
2148

TWO-LEVEL ATOM IN A MONOCHROMATIC FIELD. DRESSED-ATOM METHOD
its propagation velocity without any changes in its frequency. In a cavity, the eld
wavelength cannot change as it is xed by the boundary conditions on the cavity walls,
and hence by the cavity size. The phase shift compared to the free evolution cannot
accumulate in space but must accumulate in time (resulting in a frequency change of the
eld). Note that if one varies thefrequency of the eld around the atomic frequency
0, the sign change of the light shift0is reminiscent of the sign change of the real part
of the refractive index in the vicinity of an atomic resonance
11
.
4-c. Field absorption
Consider an atom in its ground statein the presence, at time= 0, of an
mode of the eld in a quasi-classical coherent state(ComplementV). The state of
the total system reads:
(0)= =
=0
!
2
2
(44)
The time evolution in the presence of coupling changes the energies of the states
to:
~
=~(+0 02) (45)
and the state of the system at timebecomes:
()=
=0
!
2
2
exp[(+0 02)]
exp[(+0 02)] (46)
The atom is still in the presence of a quasi-classical coherent state. However, compared to
the free eld evolution of that state in the absence of coupling, the atom-eld interaction
has introduced a phase shift0(as already discussed above) as well as a decrease in
amplitude
02
, resulting in an attenuation of the amplitude of the eld. This is
reminiscent of the radiation absorption described by the imaginary part of the refractive
index.
To sum up, we showed that the atom-eld coupling produces light shifts and ra-
diative broadening of the atomic levels, corresponding to the well-know eld dispersion
and absorption phenomena in optics.
Conclusion.
In conclusion, we showed for many various situations that the dressed-atom approach
brings strong clarications while keeping the calculations simple. Considering the atom
and the eld mode with which it interacts as a quantum system described by a time-
independent Hamiltonian allows introducing true energy levels for the global system; this
leads to a new, broad overview of the stimulated absorption and emission of photons.
As an example, this approach makes it very clear how the atom-photon coupling
changes the energy diagram of the dressed atom at high eld intensity; this leads to a
11
This eect is sometimes referred to as anomalous dispersion.
2149

COMPLEMENT C XX
very simple interpretation of the new frequencies appearing in the atomic uorescence
spectrum in the strong coupling domain. As the energy diagram of the dressed atom
is a succession of multiplicities separated by an energy equal to a photon energy, the
spontaneous emission of a photon is viewed in this approach as a quantum jump of
the dressed atom from one multiplicity to the one just below (radiative cascade). This
approach allows a simple calculation of the delay function yielding the distribution of the
time intervals between two successive quantum jumps; this permits studying the time
correlations between uorescent photons. Let us also mention that this delay function
allows simulating the temporal evolution of an atom, hence obtaining individual quantum
trajectories, which can be used to get an averaged atomic evolution. Several experimental
applications of the dressed-atom method are presented in the next complement.
2150

LIGHT SHIFTS: A TOOL FOR MANIPULATING ATOMS AND FIELDS
Complement DXX
Light shifts: a tool for manipulating atoms and elds
1 Dipole forces and laser trapping
2 Mirrors for atoms
3 Optical lattices
4 Sub-Doppler cooling. Sisyphus eect
4-a Laser congurations with space-dependent polarization
4-b Atomic transition
4-c Light shifts
4-d Optical pumping
4-e Sisyphus eect
5 Non-destructive detection of a photon
Light shifts studied in XXexhibit a number of important
properties leading to numerous applications; these will be briey discussed in this com-
plement.
As these light shifts are proportional to the laser intensity, their magnitude can be
space-dependent if the laser intensity is not homogeneous in space. These shifts can be
used to create either potential wells () to trap atoms once they are cold enough (laser
trapping), or potential barriers () reecting atoms (mirrors for atoms). A particularly
interesting example involves periodic optical potential wells created at the nodes and
antinodes of a laser standing wave in an o-resonant condition (). This situation is
reminiscent of that encountered by electrons trapped in the periodic potential of a crystal
lattice. Neutral atoms trapped in optical lattices can thus serve as models for condensed
matter problems.
For low enough values of the detuningbetween the laser frequency and the atomic
frequency, and if the ground state has several Zeeman sublevels, non-dissipative eects,
such as light shifts, can coexist with dissipative eects, such as optical pumping between
Zeeman sublevels. We explain in
can lead to new cooling mechanisms, such as Sisyphus cooling, allowing the atoms to
reach temperatures much lower than with Doppler cooling.
Finally, we show in
highly detuned cavity allows determining the number of photons present in the cavity,
by performing measurements on the atoms at the cavity exit, without absorbing any of
the cavity photons.
1. Dipole forces and laser trapping
When the light intensity varies in space, as with a focalized laser beam or a standing
wave, the light shifts also become space-dependent. If the detuningbetween the laser
frequency and the atomic frequency is large compared to the natural widthof the
excited level, it is then justied to ignore the dissipation due to spontaneous emission, on
2151

COMPLEMENT D XX
the characteristic time scales of the experiment. The light shift~(R)of ground state
depends, as does the light intensity, on the positionRof the atomic center of mass; it
can therefore be considered as a potential energy(R) =~(R)that aects the atomic
motion. This potential has the same sign as the light shift, and hence depends on the
sign of the frequency detuning.
The potential(r)gives rise to a force:
Fdip(R) =rR(R) (1)
called the dipole force, or sometimes the reactive force ( 11-4 in [24]). It is dif-
ferent from the radiation pressure forces studied in XIX, which
come from momentum exchanges as the atom absorbs photons that are spontaneously
reemitted. The dipole forces introduced here arise from the spatial variations of the
light shifts undergone by the dressed-atom levels. One could say they are caused by the
redistribution of photons between the dierent plane waves composing the laser beam
1
:
the atom absorbs a photon from one plane wave and re-emits it, by stimulated emission,
in another plane wave; this process changes the atom's momentum, and hence giving rise
to a force.
Comment:
As is the case for light shifts, the intensity of the dipole forces, as a function of the
frequency detuningbetween the laser frequency and the atomic frequency, follows a
dispersion curve. In addition, the light shifts of the two dressed levels in multiplicity
()have an opposite sign for a given detuning; the dipole force thus changes sign
from one dressed state +()to its associated state (). When the detuningis
not too large, and if spontaneous emission processes can occur, the dressed-atom radiative
cascade can lead to sign changes of the dipole force, as the atom goes from states ()
to ( 1); this is the origin of the uctuations of the dipole forces.
An important application of dipole forces is the implementation of laser traps.
Consider rst a laser beam detuned toward the red (0) and focalized at point O.
The light shift, zero outside the laser beam, is negative inside the laser beam; it increases
in absolute value as one gets closer to the focal point, where it reaches its maximum
value. This creates a potential well that could trap a neutral atom; this will indeed
happen if the atom's kinetic energy, of the order of, is lower than the depth0of
the potential well. This is why these laser traps have been built only since the 1980's,
once atomic cooling techniques (ComplementXIX, ) allowed slowing down atoms to
temperatures of the order of a microkelvin [56].
Comment:
The trapping forces involved in laser traps are of the order of an atomic dipole multiplied
by the gradient of a laser eld. They are much weaker than the forces exerted by a static
electric eld on a charged particle. This explains why laser traps for neutral atoms are
much shallower than ion or electron traps. There exist, however, other types of traps for
1
A single plane wave does not have an intensity gradient, and cannot exert a dipole force. These
forces, due to intensity gradients, require the presence of several plane waves with dierent wave vectors.
2152

LIGHT SHIFTS: A TOOL FOR MANIPULATING ATOMS AND FIELDS
neutral atoms, using dierent physical mechanisms (for a short review, see for example
XIX, and Chapter 14 of reference [24] ).
2. Mirrors for atoms
A laser detuned toward the blue ( 0) gives rise to repulsive potentials. Imagine
for example that the laser wave propagates within a bloc of glass, and undergoes total
internal reection at the boundary between the glass and the vacuum (Figure-a). An
evanescent wave appears outside the glass, with an amplitude decaying exponentially in
a direction perpendicular to the boundary, becoming negligible over a distance of the
order of the laser wavelength. This evanescent wave creates a potential barrier of height
0, which reects atoms
2
arriving with an energy 0(Figure-b). This set-up can
be used as a mirror for neutral atoms [57].
Figure 1: (a) A laser beam traveling within a block of glass (shaded in grey in the gure)
undergoes total internal reection at the boundary. Outside the glass, an evanescent
wave appears. (b) If the laser is detuned toward the blue (0), this evanescent wave
creates a potential barrier of height0. Atoms falling on this barrier with an energy
lower than0are reected by the barrier and turn around.
3. Optical lattices
When laser beams form a standing wave, the light intensity is modulated in space, with
a periodicity2: the intensity is zero at the nodes, and maximal at the antinodes.
This creates periodic potential wells, located at the antinodes of the wave for a negative
detuning ( 0), and at the nodes for a positive detuning (0). Figure
a two-dimensional optical lattice created by two standing waves, along two orthogonal
axes
3
.
The study of optical lattices is interesting for several reasons, in particular because
the motion of a neutral atom in an optical lattice is reminiscent of that of an electron in
2
Atoms falling on a solid surface would stick to it, rather than being reected.
3
The frequencies 1and 2of the two standing waves are in general suciently far apart for
the interference terms between the two waves to have a negligible eect on the atom's motion; the
potentials created by the two waves can then be independently added. This requires1 2to be
large compared to all the characteristic frequencies of the atom's motion, such as it's vibrational motion
inside a well; in that case, the interference terms oscillate too fast to have a signicant eect.
2153

COMPLEMENT D XX
Figure 2: Schematic representation of a two-dimensional optical lattice: placed in a
superposition of two standing laser waves along two orthogonal axes, the atom is subjected
to a potential periodic in space, represented by the undulating surface in the gure. These
periodic potential wells, located at the antinodes of the standing waves for0, and
at the nodes for 0, form an optical lattice. The spheres above the surface indicate
the positions where the atoms can be trapped.
a crystal lattice. Granted the order of magnitudes involved are quite dierent, since the
spatial period of an optical lattice is of the order of a micron, whereas the period of a
crystal lattice is of the order of a fraction of a nanometer. Nevertheless, optical lattices
oer a large number of possibilities not available to crystalline lattices:
One can easily change the intensity of the laser waves forming the standing waves,
hence modifying the depth of the potential wells; this allows controlling the tunnel
eect between adjacent wells. This method was used to explore the transition
between a deep well regime where the atoms are localized at the bottom of the
wells, and a shallow well regime where the atoms' wave functions are delocalized
over the entire lattice [58].
One can abruptly switch o the trapping laser beams (which obviously cannot
be done for a crystal lattice) and study the resulting behavior of the liberated
atoms. Studying the expansion velocity of the clouds of atoms yields information on
their velocity distribution, and hence on their temperature (time-of-ight method).
Studying their spatial distribution and the possible appearance of a diraction
pattern allows determining whether the matter waves trapped in distinct potential
wells of the optical lattice were coherent or not.
One can use two dierent frequencies1and 2for the two laser waves coun-
terpropagating to form the one-dimensional standing laser wave. This leads to a
standing laser wave, moving with constant velocity if1 2is xed, or with
an acceleration if1 2increases linearly with time. In this latter case, the
atom experiences a constant inertial force in the rest frame of the standing wave; its
motion is then similar to that of an electron in a crystal lattice periodic potential,
subjected in addition to a static electric eld. The motion of such a particle, in
a periodic lattice and subjected to a constant force, is predicted to be oscillatory,
2154

LIGHT SHIFTS: A TOOL FOR MANIPULATING ATOMS AND FIELDS
following the so-calledBloch oscillations; the experimental observation of such os-
cillations is facilitated in an optical lattice, as the atom's relaxation time can be
much longer than the oscillations' period [59].
Cold atoms trapped in an optical lattice are a model system for simulating a
number of situations encountered in solid state physics. Cold atom studies involve inter-
actions between atoms much weaker than the Coulomb interactions between electrons.
Furthermore, they can be controlled thanks to resonance eects occurring as atoms col-
lide with each other.
Note nally that optical lattices are a good example highlighting the importance of
light shifts. One may wonder if it might not be simpler to shift atomic levels by Zeeman
or Stark eects in static magnetic or electric elds, rather than using an o-resonance
light beam to produce a light shift. The advantage of the light shifts is that they can
be used to form potentials varying over very short distances, of the order of an optical
wavelength, which is much more dicult to attain with static elds.
4. Sub-Doppler cooling. Sisyphus eect
We described, in XIX, a cooling mechanism for the atoms, based
on the Doppler eect, and called for that reason Doppler cooling. We computed the
friction and diusion coecients associated with that mechanism and showed that the
lowest temperaturethat could be reached by Doppler cooling was of the order of
~ (whereis the natural width of the atoms' excited states, andthe Boltzmann
constant). Actually, the rst measurements of the temperatures reached by laser cooling,
and based on the time-of-ight method [60], showed that temperatures much lower than
could be obtained; furthermore, their dependence on the detuningbetween the laser
beams' frequency and the atomic frequency did not follow the prediction of the Doppler
cooling theory. This implied the existence of other cooling mechanisms for the atoms,
leading to temperatures lower than the Doppler limit; as expected, these mechanisms
were called sub-Doppler mechanisms. One of them, called the Sisyphus eect, will
be described in this section.
The theory of Doppler laser cooling, exposed in XIX, does
not take into account several important characteristics of laser cooling experiments.
In most experiments performed in three-dimensional space, the polarization of the
laser eld cannot be uniform. The spatial variations of this polarization should not
be ignored.
The atoms under study have several sublevels, in the lowerstate and in the
excitedstate. The two-level atom approximation of XIX
is therefore not sucient.
As there are several sublevels in the lower state, one should include the eects
of the optical pumping between these sublevels, eects whose characteristic time
constants (pumping times) are longer than the lifetime1of the excited state.
As the detuningbetween the laser beams' frequency and the atomic frequency is
dierent from zero, one must take into account the light shifts of the lower level,
which can take on dierent values for the dierent sublevels.
2155

COMPLEMENT D XX
Before describing the Sisyphus eect, we rst show on a simple example how these
dierent eects come into play.
4-a. Laser congurations with space-dependent polarization
Laser congurations with a space-dependent polarization do not necessarily involve
three pairs of laser beams counterpropagating along the,and axes. They can
be achieved in one-dimension, and are easier to study, as long as the two counterpropa-
gating laser waves have dierent polarizations. As an example, Figure
laser waves propagating in opposite directions along theaxis and having linear orthog-
onal polarizationseande. The polarization of the total eld changes from right-hand
circularly polarized (+with respect to the quantization axis) to left-hand circularly
polarized () in planes separated by a distance4, and is linear at45of the
and axes, half-way between these planes.
Figure 3: Laser conguration with a space-dependent polarization: two laser waves prop-
agate in opposite directions along theaxis, having linear orthogonal polarizationse
ande.
4-b. Atomic transition
Many of the laser cooling experiments use transitions between a lower state
with angular momentumand an excited statewith an angular momentum equal to
=+ 1. Here, we shall consider the simplest possible case= 12, where the lower
state contains only 2 sublevels
12. We then have= 32and the excited state has
4 sublevels
12and
32.
4-c. Light shifts
Consider rst a point in space where the laser eld polarization is+(with respect
to the quantization axis). We saw in XIXthat photons with
a+() polarization have a spin angular momentum+~(~) along theaxis.
Conservation of total angular momentum in the photon absorption process leads to the
selection rule = +1(1)for the absorption of a+() photon, whereand
are the magnetic quantum number of the states involved in the transition. Figure
represents the 2 transitions
12 +12and
+12 +32( = +1) that can
be excited by the laser eld. The numbers13and1shown next to these 2 transitions
are the squares of the Clebsch-Gordan coecients of these transitions (ComplementX);
they indicate that the
+12 +32transition is 3 times more intense than the
12
2156

LIGHT SHIFTS: A TOOL FOR MANIPULATING ATOMS AND FIELDS
Figure 4: Transition1232. The oblique upwards arrows show the transitions excited
at a point where the laser eld polarization is+, the vertical downward arrow indicates
the spontaneous emission transition from sublevel
+12toward sublevel
+12. The light
shifts of states
12and
+12are noted}3and}. At a point where the laser
polarization becomesinstead of+, the shifts of the two sublevels are interchanged,
by symmetry.
+12transition. As the detuningbetween the laser beams' frequency and the atomic
frequency is negative in a laser cooling experiment, both states
12have a negative
light shift, but with a modulus 3 times larger for state
+12than for state
12. These
light shifts are written in the gure as~and~3, whereis positive.
At a point in space where the polarization is, the previous results are inter-
changed. It is now the
12 32transition that is 3 times more intense than the
+12 12transition, yielding light shifts equal to~and~3for the states
12and
+12, respectively.
Finally, at a point where the polarization is linear, the two light shifts are identical
for symmetry reasons, and proportional to the square of the Clebsch-Gordan coecient
(equal to23), indicated in the gure for the
+12 +12transition. Consequently,
they are both equal to2~3.
This means that as one moves alongaxis, the positions of the 2 Zeeman sublevels
12and
+12oscillate, with opposing phases, between the values~and~3
(taking the energy of the unperturbed ground state equal to zero).
4-d. Optical pumping
Let us focus on a point where the laser eld polarization is+and there is an atom
in state
+12. The atom can absorb a+photon and end up in state
+32. From this
state, it can only fall, by spontaneous emission, back to its initial state
+12; optical
pumping ( XIX) does not lead, in this case, to any population
change. On the other hand, if the atom is initially in state
12and absorbs a photon
+that brings it to state
+12, it can then fall back, by spontaneous emission, into state
+12; optical pumping takes place from the least shifted sublevel
12towards the most
shifted sublevel
+12. A comparable situation is found at a point where the laser eld
polarization is. Optical pumping can only occur from the least shifted sublevel
+12
toward the most shifted sublevel
12. As for a point where the laser eld polarization
2157

COMPLEMENT D XX
is linear, since the Clebsch-Gordan coecients of the
12 12and
+12 +12
transitions are equal, as are those of the
12 +12and
+12 12transitions,
optical pumping cannot favor one of the the populations of the 2 sublevels
12and
+12. To sum up, optical pumping can only transfer population from the least shifted
sublevel to the most shifted sublevel, with a maximum eciency at points where the
laser eld polarization is circular.
4-e. Sisyphus eect
We now show how the correlations between the light shifts and the optical pumping
eects studied in the last two sections can reduce the atom's kinetic energy, and hence
cool it down.
Figure
12and
+12, shifted by the light. Let us assume the atom starts from the bottom
of a potential valley, at a point where the laser eld polarization is+, and is initially
in its most shifted state
+12. As it moves toward the right, it climbs a potential well,
and looses some kinetic energy. If the optical pumping time is long enough, it will have
time to reach the top of the hill, where the laser eld polarization is; it then has
a high probability to undergo an optical pumping cycle and be transferred to the most
shifted sublevel, which is now sublevel
12. The whole cycle we just described can
repeat itself, and each time the kinetic energy of the atom is lowered by a quantity of the
order of the maximum energy dierence between the two sublevels in Figure, equal to
(23)~. The atom is facing a situation similar to that of the hero of Greek mythology,
Sisyphus: it must endlessly climb a potential hill since it is sent back to the bottom as
soon as it reaches the top, hence the name Sisyphus eect given to this mechanism.
The temperature reached by such a mechanism can be estimated by a simple
Figure 5: Principle of Sisyphus cooling: an atom in state
+12moving from a point
where the laser eld polarization is+must climb a potential hill of height2~3,
which decreases its kinetic energy. When it reaches the top of the hill, where the laser
eld polarization is, it has a strong probability to fall back, by spontaneous emission,
to the state
12. As the cycle repeats itself, the atom is for ever climbing potential hills,
like the hero Sisyphus in Greek mythology. Its kinetic energy diminishes constantly.
2158

LIGHT SHIFTS: A TOOL FOR MANIPULATING ATOMS AND FIELDS
argument. The atom's kinetic energy decreases during each Sisyphus cycle, until it is
low enough for the atom to be trapped at the bottom of a potential well. At that
point its kinetic energy must be of the order of~. The ultimate temperaturethat
can be reached by Sisyphus cooling is expected to be~ . In laser cooling
experiments, the laser intensities are generally low and the atomic transitions are not
saturated; consequently,~ ~and hence . This explains why the measured
temperatures can be two orders of magnitude lower than the Doppler temperature, and
reach values of the order of10
6
, opening the way to numerous applications.
All these qualitative predictions have been conrmed by more quantitative models,
see ([61]) and ([62]). Experiments have conrmed the theoretical predictions, in particu-
lar those concerning the dependence ofon the various experimental parameters, such
as laser intensity and detuning [63].
5. Non-destructive detection of a photon
Consider now an experiment where the atoms of a beam cross, one after the other, a
cavity containing radiation whose quantum state is described by a Fock state; the number
of photons in the cavity mode is xed, equal for example to0(radiation vacuum) or1
(single photon). The atoms are prepared in a coherent superposition of the two states
and:
in=
1
2
(+) (2)
While each atom interacts with the photons, its levels undergo light shifts resulting in
dierent phases for the two atomic states as the atom crosses the cavity; note, however,
that if the detuningbetween the laser frequency and the atomic frequency is suciently
large, no photon will be absorbed or emitted. As the atom exits the cavity, the radiation
state is the same initial Fock state, whereas the atomic state is modied by this phase
factor. The nal atomic state can be written (within a global phase factor of no physical
signicance):
n=
1
2
(+ ) (3)
The phaseis simply the integral over time of the energy dierence between the dressed-
atom levels that come into play as the atom crosses the cavity. It is given by the energy
diagram of the dressed-atom.
Figure XXshows that the gap between states0and0
is reduced from~0to~(0 0)by the light shifts. For a cavity with no photons
(= 0), when the atom exits the cavity, the coherence between its statesandhas
been dephased by:
0= 0()d (4)
where0()is obtained by replacing in (42) of ComplementXXthe Rabi frequency
1by a function of time that accounts for the motion of the atom in the cavity mode,
where it is subjected to a time-dependent light intensity; remember that we assumed the
2159

COMPLEMENT D XX
detuningbetween the atomic frequency and the laser frequency to be large enough for
no real photon absorption by the atom to occur
4
. If now the cavity contains one photon
(= 1), Figure 1and1is reduced by the
light shifts to~( 20 0), i.e. to~( 30). When the atom exits the cavity, the
coherence between its statesandis now shifted by three times the amount obtained
in (4). This means that an atom, traversing the cavity in a superposition of statesand
, keeps in the phase of that coherent superposition a trace of the number of photons
present in the cavity; this occurs without any photon absorption (since the detuningis
too large). To sum up, if= 0, the state of the atom at the cavity exit is:
n(= 0)=
1
2
(+
0
) (5)
whereas if= 1, this state is:
n(= 1)=
1
2
(+
1
) (6)
with1= 30.
How can we make use of this trace left on the atom by the possible presence of a
photon in the cavity, and determine if this cavity contains zero or one photon? The time
taken by the atom to cross the cavity can be adjusted by changing the atom's speed.
Imagine that this time is tuned so that0 1=; this means that the two states
(5) and (6) are now orthogonal. As the atom exits the cavity, we can apply to it a2
laser pulse adjusted to transform(= 0)into. That same pulse will transform
(= 1)into the state orthogonal to, that is to. This means that measuring
the atomic state after this2laser pulse allows concluding that= 0if the atom is
found in state, and that= 1if the atom is found in state. The measurement can be
repeated several times by sending a stream of atoms, one after the other, and applying
to each of them the same procedure; one can measure several times in a row the same
value, which proves the number of photons in the cavity did not change during the
measurements. As opposed to photoionization where a photon is absorbed giving rise to
a photoelectron (ComplementXX), this method is non-destructive: the presence of the
photon is detected without it being absorbed. This experiment, generalized to the case
where the photon number is larger than one, is described in more detail in reference [64].
Conclusion.
For a long time, light shifts have been considered as an interesting physical phenomenon
without specic applications, and even as an undesired perturbation for high resolution
spectroscopy, since they modify the atomic transition frequencies one is trying to measure
with the highest possible precision. These shifts must be taken into account to extract
from the measurements the non-perturbed frequencies of atomic transitions; most of the
4
We also assume that the eld variation encountered by the atom as it crosses the cavity is slow
enough for non-adiabatic transitions fromto, or fromto, to be highly improbable. We also
suppose that the natural widthof the excited state, and the timethe atom takes to cross the
cavity, are small enough for 1, meaning spontaneous emission from statedoes not have time to
occur while the atom crosses the cavity.
2160

LIGHT SHIFTS: A TOOL FOR MANIPULATING ATOMS AND FIELDS
time, several measurements at dierent light intensities must be performed to extrapolate
the results to zero light intensity.
This complement clearly shows how much the situation has changed, by presenting
the large variety of experimental methods using light shifts of atomic energy levels, and
their great number of applications. These methods were implemented more than 20
years after these shifts were theoretically predicted and experimentally demonstrated;
this illustrates the long term practical impact of fundamental research. These methods
allow acting both on the internal and external atomic variables; they also permit using
atoms as a very sensitive non-destructive probe for the properties of a eld composed
of only a few photons. These methods made it possible to trap atoms in a standing
laser wave, or to obtain periodic lattices of neutral atoms trapped in such a wave. It
also led to laser cooling methods that allowed reaching temperatures previously totally
inaccessible for atomic gases, millions of times lower than the lowest temperatures found
in the interstellar or intergalactic space of the Universe.
2161

DETECTION OF ONE- OR TWO-PHOTON WAVE PACKETS, INTERFERENCE
Complement EXX
Detection of one- or two-photon wave packets, interference
1 One-photon wave packet, photodetection probability
1-a Photoionization of a broadband detector
1-b Detection probability amplitude
1-c Temporal variation of the signal
2 One- or two-photon interference signals
2-a How should one compute photon interference?
2-b Interference signal for a one-photon wave packet in two modes
2-c Interference signals for a product of two one-photon wave
packets
3 Absorption amplitude of a photon by an atom
3-a Computation of the amplitude
3-b Properties of that amplitude
4 Scattering of a wave packet
4-a Absorption amplitude by atom B of the photon scattered by
atom A
4-b Wave packet scattered by atom A
5 Example of wave packets with two entangled photons
5-a Parametric down-conversion
5-b Temporal correlations between the two photons generated in
parametric down-conversion
Introduction
In Chapter
states that, in the absence of interaction, had a well dened energy; before the interaction,
such states do not evolve in time, as if the photon were not propagating in space. As an
example, in the scattering process of a photon by an atom, the chosen initial radiation
state is a photon with momentum~kand energy~=~, which spreads over the
entire space; similarly, the nal state is also a photon with momentum~kand energy
~=~. The interaction was turned on at time, which allowed computing the
probability amplitude for the atom + photon(s) system to go from one state to the other
betweenand. This is clearly a phenomenological approach: what actually happens
is that the interaction operator remains constant but only comes into play to change the
state vector when the atom is in the presence of radiation. A more realistic description of
the process should involve the propagation of wave packets, with the incident radiation
being described by a wave packet initially very far away from the atom, but going towards
it. Their interaction then gives rise to a scattered wave packet moving away towards
innity, while the incident wave packet, modied by the interaction, also continues on
its way.
2163

COMPLEMENT E XX
Note however that introducing a wave packet for a photon cannot be done by the
standard method used for a massive particle. As already pointed out at the end of
Chapter , a photon does not have a position operator. One cannot obtain its spatial
wave function by projecting its state vector onto the eigenvectors of that operator, and
then squaring this wave function's modulus to get the probability of nding the photon
in any given region of space. One could then imagine using the spatial variations of the
electric and magnetic elds to infer the photon localization. But for radiation states with
exactly one, two, etc. photons, the average value of theses elds at any point in space
is zero (it is the sum of zero average value creation and annihilation operators in each
mode). Consequently, for a single photon, this average value cannot be directly used for
building a wave packet localized in space. This is why we shall use another approach: we
shall assume the photon interacts with detectors, well localized in space, and compute
the probability of its detection by these apparatus. This will lead us to introduce an
amplitude for the photon detection (by photoionization) at a given point, which presents
close analogies with the spatial wave function of a massive particle in non-relativistic
quantum mechanics.
We start in
function(r)that allows localizing a single photon in space through its probability of
being absorbed by a broadband detector. This leads to the concept of wave packets, even
though the average value of the electric eld remains zero throughout the entire space.
In the perturbative computations, one can also introduce initial and nal radiation states
that are wave packets, described by linear superpositions of photon states with dierent
momenta and energies.
In , we show how the detection amplitude(r)allows studying light inter-
ference phenomena involving one or two photons. These phenomena are interpreted in
terms of interference between the transition amplitudes associated with dierent paths
leading the quantum eld from a given initial state to a given nal state. Starting rst
() with a general discussion of the interference signals, we then focus in
interference involving one photon being simultaneously in two modes of the eld. Finally,
we examine interference involving two photons in the simple case where the system is
described by the product of two one-photon wave packets (
In , we replace the broadband detector by an atom with discrete energy levels.
Without having to assume that the coupling between the atom and the eld is turned
on abruptly (which is hard to justify from a physical point of view), a number of results
of Chapter
aspect of the absorption phenomenon. In , we extend this method to study the
scattering of photons by an atom. Here again, we will conrm the results of Chapter,
while enriching our understanding of the temporal aspects of the physical process.
Finally, in , we consider real two-photon wave packets that are entangled
wave packets. Parametric down-conversion is an example of a situation leading to strong
temporal correlations between two entangled photons. Such correlations are impossible
to understand in terms of a classical treatment of the radiation.
In this entire complement, we have limited our studies to one- or two-photon wave
packets, but the computations can be extended to wave packets containing a larger
number of photons
1
.
1
Or even an undetermined number, as in a coherent state.
2164

DETECTION OF ONE- OR TWO-PHOTON WAVE PACKETS, INTERFERENCE
1. One-photon wave packet, photodetection probability
A one-photon wave packet is described by a state, where the photon number is
precisely equal to one (eigenstate of the photon number operator, with eigenvalue equal
to 1). We build this wave packet
2
as a linear superposition of states with dierent
momenta~k:
=d
3
(k)
k
0=d
3
(k)k (1)
Stateis not stationary (it is not an energy eigenstate). It is of the type considered
in , an eigenket of operator
^
(total number of photons) with an
eigenvalue equal to one. It is assumed to be normalized:
d
3
(k)
2
= 1 (2)
which allows interpreting(k)
2
as the probability density for the photon momentum to
be equal to~k.
1-a. Photoionization of a broadband detector
Imagine we place an atom playing the role of a photodetector at pointrin the
radiation eld described by state (1). According to relation (26) of ComplementXX,
the probability(r)dfor observing a photoelectron emission between timesand
+dis given (to the interaction's lowest order) by:
(r) =
()
(r)
(+)
(r) (3)
whereis a constant depending on the photodetector sensitivity.
()
(r)and
(+)
(r)
are the negative and positive frequency components of the electric eld operator appear-
ing in its plane wave expansion as given by (A-7) in Chapter :
(+)
(r) =
()
(r)=
d
3
(2)
32~20
(k)
(kr )
(4)
with:
= k= (5)
whereis the speed of light. Remember that expression (3) was established in the
interaction representation, where the state vectorevolves only under the eect of
the atom-radiation interaction; the operators evolve freely only under the eect of the
atomic or radiation Hamiltonians (i.e. without mutual interaction). However, as we are
performing a computation to lowest order, we can consider in (3) thatis actually
2
For the sake of simplicity, we ignore in this complement the degrees of freedom of the radiation
polarization, which do not play a signicant role in the eects under study. This amounts to assuming
that all the vectorskappearing in (1) have almost the same directions and that they are all associated
with the same polarization".
2165

COMPLEMENT E XX
constant. The annihilation operator
(+)
(r)in (3) acting on the one-photon state
yields the vacuum, and we can rewrite (3) as:
(r) =
()
(r)00
(+)
(r)=0
(+)
(r)
2
(6)
that is:
(r) =
d
3
(2)
32~20
(k)
(kr )
2
(7)
For a massive particle of mass, the probability to nd it at pointrand at time
is given by the squared modulus (r)
2
of its wave function (r). This wave function
is the Fourier transform of the probability amplitude(k)that a measurement of the
particle's momentum gives the value~k. For a free particle, this probability amplitude
(k)has a time variation in. Equality (7) is thus reminiscent of this probability
for a massive free particle; however, in view of the
factor (proportional to) in
front of(k)in the integral of (7),(r)is not proportional (at a given instant) to
the modulus squared of the spatial Fourier transform of(k) =(k) . This means
that, limiting ourselves to one-photon states, we can indeed consider the function(k)
appearing in (1) as a wave function in momentum space, since(k)
2
is a probability
density for the photon momentum. However, the probability to detect a photon at point
rand at timewith a photodetector is not simply the modulus squared of the Fourier
transform of that wave function in momentum space(k) . This conrms that
it is not possible, for a photon, to introduce a spatial wave function that is exactly
equivalent to that of a massive particle.
1-b. Detection probability amplitude
The right-hand side of (6) is the squared modulus of the function:
(r) =0
(+)
(r) (8)
which plays an important role in all the computations to follow; it has the dimensions of
an electric eld. For the wave packet written in (1), its expression is:
(r) =
d
3
(2)
32~20
(k)
(kr )
(9)
It should not be confused with the average value in stateof the operator
(+)
(r)
written in (4), since that average value is zero, as we mentioned above. Nor is it, as
already pointed out, a wave function for the photon in position space; it is a probability
amplitude for the detection (and not the presence) of the photon at pointrand time.
When we mention, in this complement, the space time wave packet associated with the
photon in state (1), we will always be referring to the amplitude (8).
Note, however, that in the particular case where the function(k)in momentum
space is well centered around a valuewith a dispersionvery small compared to
, one can neglect in (7) thevariation of
and replaceby; the integral in
(7) will therefore involve the spatial Fourier transform of(k) . This approximation
will often be used in what follows.
2166

DETECTION OF ONE- OR TWO-PHOTON WAVE PACKETS, INTERFERENCE
Comments:
(i) In this section, we have ignored the radiation polarization, assuming that all the plane
waves in relation (1) are associated with the same polarization vector". When this is
not the case, the detection amplitude becomes a three-component vector function. Its
component along any given axis yields the detection probability amplitude by a pho-
todetector preceded by a polarization analyser letting through only the light polarized
linearly along that axis. This vector detection amplitude is similar to the wave function
of a spin 1 particle, which also has three components.
(ii) In this complement, we shall study only wave packets containing a well dened photon
number, 1 or 2, which allows directly generalizing the computations of Chapter. Wave
packets can, however, be built many dierent ways, without exactly xing the photon
number. It often happens, for example, that one wishes to reproduce a classical eld for
which each eld modekhas a given amplitude(k); it is then natural to use a state
where each quantum mode is in a coherent state with eigenvalue(k). In that case, only
the average photon number is xed, not its exact value.
1-c. Temporal variation of the signal
When the photodetector is placed atr= 0, it delivers a signal that is given,
according to (7), by:
(r=0) d
3
(k)
2
(10)
This signal is proportional to the squared modulus of the Fourier transform of
(k).
Let us assume that()is a real positive function of, and that()
2
is a function of
centered at=, with a width. If this widthof the wave packet is very small
compared with the average wave number, one can replace
by, and the signal
becomes proportional to the modulus squared of the Fourier transform of(k). At time
= 0, and since we assumed all the()to be positive, all the waves forming the wave
packets are in phase and the signal observed on the photodetector is maximum; it is zero
for= , and takes on signicant values only during a time interval 1
around= 0. This signal describes, in a way, the passage of the wave packet at the
detector's position.
To study the detection probability at a pointr=0, we just have to replace
by
(kr )
in the integral of relation (10). As an example, imagine we have
a one-dimensional wave packet, all the wave vectorskbeing parallel to theaxis
(= = 0); the exponential reduces to
( )
. The phenomena observed at a
point in space of coordinate= 0are thus deduced from those observed at= 0by
a simple time shift equal to: the wave packet moves along thedirection with
velocityand without any deformation.
2. One- or two-photon interference signals
We now discuss in terms of wave packets what happens in light interference experiments
involving one or two photons.
2167

COMPLEMENT E XX
2-a. How should one compute photon interference?
In non-relativistic quantum mechanics, a particle with a non-zero massis de-
scribed by a wave function(r)whose squared modulus(r)
2
yields the probability
density of nding the particle at pointrand time. In a Young's type interference ex-
periment, the wave function, after going through the two slits pierced into a screen, is
a linear superposition of two wave functions1(r)and2(r)originating from the
two slits. These two waves overlap in a region of space where the probability density of
nding the particle at pointrand time, which is equal to1(r) +2(r)
2
, contains
a term2Re1(r)
2(r)oscillating in space and time; this results in interference
fringes.
However, we recalled in
function for a photon that would be strictly analogous to(r), and whose squared
modulus would yield the probability density for the presence of the photon at a given
point. This led us to dene an amplitude(r)in (8), whose squared modulus yields the
probability density for photodetecting the photon at pointrand time. We are going to
show in
as an example, we shall study the fringes appearing in the single photodetection signal
(r)observed on a one-photon wave packet after it goes through a screen pierced with
two slits. As already underlined, it is important not to confuse(r)with the average
value of the electric eld in the quantum state under study which in any case is zero
in a one-photon state. In classical electromagnetism, the electric (or magnetic) elds
directly interfere; in quantum electromagnetism, one must reason in terms of probability
amplitudes.
For eld states containing at least two photons, the double photodetection signal
(r r )is dierent from zero. To interpret it in the simplest possible case,
we assume (in ) that the radiation is described by a tensor product of two one-
photon wave packets
3
. We will show that interference fringes observable oncan also
be interpreted in terms of products of detection probability amplitudes; these fringes
result from interference between transition amplitudes associated with two dierent paths
leading the eld from its initial state (where it contains two photons) to the vacuum.
Here again, one should reason in terms of interference not directly between average values
of electric or magnetic elds, but between paths.
2-b. Interference signal for a one-photon wave packet in two modes
We start with the simplest photon interference experiment, the well-know Young's
double slit experiment, but in a case when only one photon at a time passes through the
screen pierced with the two slits. The state vector of this single photon is then the sum
of two components associated with the passage through one or the other of the two slits.
When the photon reaches the interference region, these two components are associated
with two dierent radiation modes.
3
A simple example of a two-photon entangled state, which is not a product of two one-photon states,
will be studied in .
2168

DETECTION OF ONE- OR TWO-PHOTON WAVE PACKETS, INTERFERENCE
. One-photon wave packets, after passage through a two-slit screen
We now focus on the radiation state after it passed the two-slit screen. This state
is described by a one-photon wave packet, which, in the interaction representation, is of
the form:
=11+22 with:1
2
+2
2
= 1 (11)
In this expression, the ket1describes the wave packet emerging from the rst slit;
as in (1), it can therefore be written with a function1(k)that is peaked around the
valuek1. The ket2describes the wave packet emerging from the second slit, and its
function2(k)is peaked around the valuek2. Since before going through the slits the
two wave packets came from the same source, they must be centered around a common
frequency= 1= 2; consequentlyk1andk2have the same modulus, but their
directions can be dierent. We shall nally assume that the wave packets emerging from
the two slits arrive at the same time in the interference region (meaning the optical paths
along the two trajectories are equal) and that each wave packet is suciently long for
the frequencyto be well dened.
As in (8), for each wave packetwe introduce a detection amplitude(r):
0
(+)
(r)=(r)where:= 12 (12)
In the interference region, we assume the two modes
4
to be close to plane waves with
wave vectorsk1andk2. We then set:
(r) =(r)
(kr )
(13)
where the function(r)has a much slower space and time variation than the expo-
nential
(kr )
.
. Calculation of the single photodetection signal
We assume that the eld is contained in a box of volume
3
; we use a complete
orthonormal set of eld modes, with wave vectorsk, which includes bothk1andk2.
Relation (B-3) of Chapter
electric eld can be written
5
as:
(+)
(r) =
}20
3
(kr )
(14)
with= . When this operator acts on the ket (11), all theterms lead to a zero
result, except for the= 1and= 2terms. For these two terms, we have12=0,
so that (11) and (12) lead to:
(+)
(r)= 11(r) +22(r)0 (15)
4
Another possibility would be to use Gaussian wave packets with the same focal point, having in the
vicinity of that point plane wave structures with wave vectorsk1andk2, and lateral extensions very
large compared to the wavelengths2 1and2 2.
5
For the sake of simplicity, we ignore polarization variables of the eld.
2169

COMPLEMENT E XX
The probability for detecting the photon at pointrand timeis proportional to the
square of norm of this ket, written as:
()
(r)
(+)
(r)=11(r) +22(r)
2
(16)
The equality includes square and cross terms. The square terms can be written, taking
(13) into account:
(r)
2
= (r)
2
(17)
and they vary slowly as a function ofrand. The crossed terms are expressed as:
121(r)
2(r) +c.c.=121(r)
2(r) exp[(k1k2)r)]+c.c. (18)
and exhibit spatial modulations characteristic of interference phenomena (c.c. stands for
complex conjugate).
. Discussion
Relation (16) shows that the photodetection signal is the squared modulus of the
sum of two amplitudes,11(r)and22(r), which interfere. Amplitude11(r)is
the amplitude for detecting at pointrand timethe photon in mode1; it is equal to
the amplitude1of nding the eld in state1, multiplied by the amplitude1(r)for
detecting the photon at pointrand timewhen the eld is in state1. The amplitude
22(r)is interpreted in a similar way. During the detection process, the eld goes
from statewritten in (11) to the vacuum state0following two possible paths: the
photon is absorbed either while in mode1, or while in mode2. As nothing allows
deciding which path the system followed, the two corresponding amplitudes interfere.
This conrms what we stated above: in the quantum theory of radiation, the interfer-
ence fringes observed on a photodetector signal are associated with the interference, not
between two classical electromagnetic waves, but rather between two transition ampli-
tudes corresponding to dierent paths (leading the system from the same initial state to
the same nal state).
2-c. Interference signals for a product of two one-photon wave packets
Let us generalize this type of interpretation, in terms of transition amplitudes,
to interference experiments involving two photons and where one measures correlations
between signals coming from two photodetectors.
. State vector for the two photons
We now assume the eld contains two photons, and can be described as the product
of two wave packets such as the one written in (1):
12=d
3
1(k)d
3
2(k)
kk
0 (19)
Is it possible to observe spatial and temporal modulations on the signalsand
coming from one or two detectors placed in that eld? We are going to show that the
2170

DETECTION OF ONE- OR TWO-PHOTON WAVE PACKETS, INTERFERENCE
answer to that question is no if one considers only single photodetections, but yes if
one takes into account the correlations between double photon detections.
We assume the two wave packets in (19) to be well separated: there exists no
overlap between the domain1where the function1(k)is dierent from zero and the
domain 2where the function2(k)is non-zero. Let us introduce the matrix element
that generalizes relation (8) to the two-photon case:
0
(+)
(r)
(+)
(r)12 (20)
We now insert in that expression the plane mode expansion (14) for both electric eld
operators. To yield a non-zero result, the annihilation operators appearing in these elds
must act on a mode that, according to (19), contains one photon. This means that
either the mode selected in
(+)
(r)belongs to the1domain and the one selected in
(+)
(r)to the2domain, or the inverse. In the rst case, the scalar product of the
vacuum bra and the modes that came into play yields the detection amplitude2(r)
associated with the second wave packet, multiplied by the detection amplitude1(r)
associated with the rst one. In the second case, the wave packets are inverted. The
nal result is:
0
(+)
(r)
(+)
(r)12=1(r)2(r) +2(r)1(r) (21)
where the functions1and2are the detection amplitudes associated, as dened in (8),
with the two individual wave packets included in12.
. Single photodetection signal(r)
To get the single photodetection signal, we rst compute the result of the action
on state (19) of the eld positive frequency component
(+)
(r). As we argued above,
to yield a non-zero result, this operator must destroy a photon, either in a mode for
which1(k)is non-zero, or in a mode for which2(k)is non-zero. In the rst case, the
summation over all the modes involved reconstructs the function1(r)multiplied by
the vacuum ket associated with these modes; the modes of the other wave packet remain
unchanged. In the second case, the two wave packets exchange roles and it is now the
function2(r)that is reconstructed. This leads to:
(+)
(r)12=
1(r)d
3
2(k)
k
0+2(r)d
3
1(k)
k
0 (22)
The probability per unit time to detect a photon at pointrand timeis the square of
this ket's norm. The terms in1(r)
2
and2(r)
2
contain the square of the two
wave packets' norms, each equal to one; these terms do not oscillate, neither in space,
nor in time. The cross terms are the only ones that could yield spatial and temporal
modulations; they contain, however, the scalar product of the two wave packets, which
is zero since we assumed the wave packets were orthogonal (there is no overlap between
the two1and2domains). This means that, when the eld is described by state (19),
no interference fringes are observable in the signal of a single photodetector.
The interpretation of this result is similar to the one we gave before. The system
can follow two paths: either an absorption of a photon from the rst wave packet, or an
2171

COMPLEMENT E XX
absorption of a photon from the second. However, as opposed to what happens when
the system started from the initial state (11), the nal state of the eld is not the same
for these two paths: if a photon from the rst wave packet has been absorbed, the nal
state includes a photon from the second wave packet. Consequently, the two nal states
associated with the two paths are orthogonal, and observing the eld's nal state one
could (in principle) determine which path the system has followed; this is why the two
amplitudes cannot interfere.
Comment:
One could consider other states for the two modes, each containing several photons, as
for example states1 2where each mode is in a coherent state, characterized by a
classical normal variable,1for modek1,2for modek2. We then know that state
1is an eigenstate of operator
(+)
(r)with an eigenvalue value
(+)
cl
(1r)equal
to the positive frequency component of the classical eld in modek1, corresponding
to the classical normal variable1(Chapter , ). A similar result is valid for
state2:
(+)
(r)=
(+)
cl
(r) = 12 (23)
This leads to:
(+)
(r)12=
(+)
cl
(1r) +
(+)
cl
(2r) 12 (24)
The probability of detecting a photon at pointrand timeis equal to the square of the
norm of ket (24). It is proportional to:
(+)
cl
(1r) +
(+)
cl
(2r)
2
As this is the squared modulus of the sum of two classical elds, it is the usual interference
signal of classical elds. As opposed to what we found before for the radiation state (19),
when the two modes are in coherent states, the one-photon detection signal exhibit
interference. This is an illustration of the quasi-classical character of coherent states.
. Double photodetection signal(r r )
Assuming, as above, the eld initial state is given by (19), we now focus on the
probability(r r )(per double unit time) that a detector, placed atr,
detects a photon at timeand that another detector, placed atr, detects a photon at
time. This probability is proportional to the correlation function (ComplementXX,
):
12
()
(r )
()
(r )
(+)
(r )
(+)
(r )12 (25)
Since12contains only two photons, we can insert in the middle of this expression the
projector onto the vacuum state, which leads to the squared modulus of expression (21).
We obtain:
(r ;r ) 2(r )1(r ) +1(r )2(r )
2
(26)
2172

DETECTION OF ONE- OR TWO-PHOTON WAVE PACKETS, INTERFERENCE
In addition to the square terms2(r )1(r )
2
and1(r )2(r )
2
, which
have a slow variation withr,r,,, we get cross terms:
1(r )
1(r )2(r )
2(r ) +c.c.
which do have spatial and temporal modulations. If, for example, the rst wave packet
is centered around the valuesk1and1, and the second one around the valuesk2and
2, these modulations are of the form:
exp(k1k2)(rr)(1 2)( )+c.c. (27)
This result is not in contradiction with the fact that the probability of detecting a photon
atr,, or atr,varies slowly with these variables: once a rst photon has been
detected atr , the probability to detect another one atr varies sinusoidally
withrrand .
(i)Discussion
The amplitude whose squared modulus appears on the right-hand side of (26) is
the sum of two amplitudes associated with two possible paths leading the system from
the initial state12(containing two photons) to the same nal state0(where all
the modes are empty). Along the rst path, with amplitude2(r )1(r ), the
k1mode photon is absorbed atr and thek2mode photon is absorbed atr .
Along the second path, with amplitude1(r )2(r ), the opposite happens: the
k2mode photon is absorbed atr and thek1mode photon is absorbed atr . As
explained before (), interference occurs between the dierent transition amplitudes
associated with two possible paths leading the system from the same initial state to the
same nal state, as long as there is no way one can determine which path is actually
followed.
(ii)Another interpretation
The photodetection signal (25) can also be written in the form:
(+)
(r )
(+)
(r ) (28)
with:
=
(+)
(r )12
= 1(r )d
3
2(k)
k
0+2(r )d
3
1(k)
k
0 (29)
where, in the second line, we used relation (22). Signal (28) can be interpreted as the
probability of detecting a photon when the eld is that statewhere the photon has
a probability amplitude1(r )to be in the wave packet with amplitude2(k), and
a probability amplitude2(r )to be in the other wave packet with amplitude1(k).
This situation is quite similar to that encountered in , where we showed that the
photodetection probability of a photon in state (11) exhibits modulations.
In other words, we started from a state12with no coherence. It is the detection
of a rst photon that introduces the state (29) where a second photon is now in a
coherent superposition, the coherence arising from the fact that the detected photon
can come either from the rst wave packet, or from the second. The coecients of the
2173

COMPLEMENT E XX
superposition (29) depend on the pointrand the instantwhere the detection of the
rst photon occurred. In this description of the phenomena, it is the rst detection that
introduces quantum correlations between the two modes, and the dependence of these
correlations onrandexplains why the probability of the second detection oscillates
as a function ofrrand .
3. Absorption amplitude of a photon by an atom
We now replace the broadband photodetector by an atom with two discrete levels, a
ground leveland an excited level. This atom is placed atr=0, and interacts with
the same wave packetas that written in (1). We propose to compute the probability
amplitude for the atom, initially in state, to absorb the incident photon and be found
in stateat time.
3-a. Computation of the amplitude
The initial and nal states of the process under study are:
in=; n=; 0 (30)
since the absorption of the photon transfers the radiation from stateto the vacuum
0. According to relation (B-4) in Chapter, the amplitude we are looking for is, to
rst order in:
n

( )in=
1
~
d n

()in (31)
where the bar above the operators indicates they are expressed in the interaction picture,
with respect to the non-perturbed Hamiltonian.
The interaction Hamiltonianis given by
6
:
=
(+)
(r=0) (32)
The matrix element of

()appearing in (31) equals:
n

()in=
0
0
(+)
(r=0) (33)
In this equality,= ,0= ( )~is the frequency of the atomic transition,
and
(+)
(r=0)the electric eld positive frequency component in the interaction
representation. Using notation (8) for the matrix element on the right-hand side of (33),
we get:
n

()in=
0
(r=0) (34)
which allows rewriting the absorption amplitude (31) as:
n

( )in=
~
d
0
(r=0) (35)
6
As we have ignored the radiation polarization degrees of freedom, we also ignore here the vector
character of the atomic dipoleD. Operatorappearing in (32) is actually the projection ofDonto
the radiation polarization vector.
2174

DETECTION OF ONE- OR TWO-PHOTON WAVE PACKETS, INTERFERENCE
The quantity (r=0)that appears in this expression can be considered to be the
interaction Hamiltonian between the atomic dipoleand a classical eld(r=0).
The eld(r=0)thus appears as the classical eld that would yield the same
transition amplitude between the two atomic statesandas a quantum eld, when
the radiation is in a one-photon wave packet described by(k).
3-b. Properties of that amplitude
Let us return to the wave packet in innite space (1) and use relation (8) to get
the probability amplitude for the detection of the photon at pointr=0. Describing
the eld operator (4) with an integral over d
3
instead of d
3
, and using it in (1), the
commutation of operators(k)and(k)leads to a(kk)function. This leads to:
0
(+)
(r=0)=(r=0) =
(2)
32
d
3
}20
() (36)
where=. As()is centered around an average wave vector, assumed very
large compared to the widthof(), amplitude (36) can be written as:
(r=0) = () (37)
which is the product of a carrier wave of frequency= by an envelope(). This
latter function has avariation that is slower than the preceding exponential; it can,
for example, follow a bell-shaped curve, centered at= 0and with width 1.
Inserting (37) into (35), we get:
n

( )in= d
(0 )
1() (38)
where1()is the instantaneous Rabi frequency dened as:
~1() = () (39)
Equation (38) allows understanding the behavior of the absorption amplitude of
the photon whenincreases fromto+. As long as , both functions()
and1()are zero; the incident wave packet has not yet reached the atom's vicinity and
no photon absorption can occur. Asincreases from2to+2, the wave packet
crosses the atom, and the integral in (38) becomes larger. When +, the wave
packet has left the atom; the absorption amplitude remains constant and equal to:
n

(+ )in=
+
d
(0 )
1() (40)
This expression yields the probability amplitude for a photon to have been absorbed once
the wave packet has crossed the atom. This conrms the results of Chapter, but in
the present approach we did not have to articially introduce any initial or nal times
for the process.
Let us evaluate an order of magnitude for the amplitude (40). Assume rst that
=0(resonant wave packet). The integral in (40) is then of the order of
max
1,
2175

COMPLEMENT E XX
where
max
1is the maximum value reached by the Rabi frequency when the atom is at
the center of the wave packet, and the envelope()takes on its largest value. When
=0(o-resonance wave packet), the absorption amplitude is weaker. According to
(40), this amplitude is actually the Fourier transform of1()at frequency0 .
This result simply expresses energy conservation: for the incident photon to be absorbed,
its frequency must be equal to the atomic transition frequency. However, as the eld
envelope varies over time intervals of the order of, the photon average frequency does
not have to be strictly equal to the atomic frequency; the two frequencies must be equal
to within 1.
4. Scattering of a wave packet
We now study a process involving two atoms: a wave packet impinges on an atom
placed atron theaxis; after interacting with it, the wave packet is scattered in all
directions, and then interacts with a second atomplaced atr. The incident wave
packet, propagating along thedirection, is described by the function(k) . As
before, we have two main goals. The rst one is, while assimilating atomwith a
device for measuring the photon scattered by, to conrm the interpretation of(r)
as a detection amplitude of a photon at pointr. The second goal is to study the time
dependence of the scattering process itself.
We shall rst study the spatial and temporal dependence of the scattered wave
packet, in particular when the central frequencyof the incident wave packet is close to
the resonant frequency0of the scattering atom. We shall then compute the probability
amplitude for the scattered wave packet to have excited at timethe atomfrom its
ground stateto its excited state. As in , we will associate with this amplitude a
spatial wave packet describing the passage of the scattered wave packet by pointr.
4-a. Absorption amplitude by atom B of the photon scattered by atom A
We rst consider a photon with a wave vectorkparallel to theaxis, and a
frequency= . We are looking for the probability amplitudekkfor this
photon to be scattered by atom A located atrfrom statekto statek. This amplitude
is given by relation (E-3) of Chapter, where we only take into account the resonant
processes (we assume the incident photon frequency to be fairly close to the atomic
resonant frequency):
kk= n

(0)in
=2 (in)
()
(n in)
()
( ) (41)
In this relation,(in)is obtained from relation (E-4) in Chapter
assume that the radiation is contained in a box of volume
3
):
(in) =
}
20
3
"D "D
+~
(kk)r
(42)
where we have assumed that only one levelcontributes, which explains why the sum
overhas been suppressed; this is correct if the frequency of the radiation is close to the
resonance frequency of onetransition, but far from all the other resonances. Note
2176

DETECTION OF ONE- OR TWO-PHOTON WAVE PACKETS, INTERFERENCE
that we have added to the right-hand side an exponential factor that comes from the
spatial dependence of the electric eld: in this complement, as in Chapter, we treat
the atom's positionrclassically, but we no longer assume the atom to be placed at the
coordinate origin. Expression (42) is a product of two matrix elements of the interaction
Hamiltonian, one for the absorption of thekphoton, the other for the emission of thek
photon, divided by a common energy denominator. The function
()
( )simply
expresses energy conservation, within}, for the elastic scattering process, as was the
case, for example, in . We assume that the interaction
timeis suciently long for this function to be assimilated to a real delta function
( ).
The coecientintroduced in the second equality (41) is proportional to expres-
sion (42); it contains the product of two matrix elements, which depends on the polar
anglesandof the vectorkwith respect to the direction ofk. We characterize this
dependence by a function( ), with:
=k=k (43)
As we assumed the frequencyof the incident photon to be close to resonance, we can
use the results of
(E-11) of that chapter, we write the energy denominator in the form0+2,
whereis the natural width of the excited state of the scattering atom. Amplitude
(41) then becomes:
kk=
( )
0+2
[(kk)r]
( ) (44)
whereis a coecient proportional to.
We now move to the next stage, the interaction of atom B with thekphoton.
As in (33), it is described by a matrix element (here again, we must add an exponential
factor to account for the fact that atom B is not at the coordinate origin, but at point
r):
0
0
(+)
(r);k
kr (0 )
(45)
We are now looking for the amplitude at timeof the complete process, scattering by
A of thekphoton, with an amplitude given by (44), and absorption by B, with an
amplitude given by (45). Consequently, we multiply these two amplitudes and sum the
product over all the possiblekvectors for the scattered photon, and over the linear
combination of stateskforming the incident wave packet.
4-b. Wave packet scattered by atom A
To study the properties of the wave packet scattered by atom A, we successively
carry out the two summations.
. Summation over all possible directions of the scattered photon
Let us start with the summation overk. Taking into account the function(
)appearing in (44), the summation over the modulus ofkleads to:
== (46)
2177

COMPLEMENT E XX
Regrouping thekdependent terms in (44) and (45), we nd that the summation over
the directions ofkintroduces the angular integral:
d( )
k(rr)
(47)
The summation over the polar anglesof the exponentials describing the phase shift
betweenrandrof all the plane waveskyields a spherical wave centered atr:
d( )
k(rr)
( )
with =rr (48)
whereand are the polar angles of vectorrrwith respect to the directionk
of the incident photon. The right-hand side of (48) is reminiscent of a classical result in
collision theory see for example relation (B-12) of Chapter . This means that the
sum of all the plane waves scattered by atom A located atrhas the structure of an
outgoing spherical wave with the same wave numberas thekwaves it is composed
of. The amplitude of this spherical wave varies as1, which ensures that the outgoing
energy across a sphere of radiusand surface4
2
does not depend on.
The fact that the polar anglesand appearing on the right-hand side of (48)
are those of vectorrrcan be understood by stationary phase arguments. The phase
factor
k(rr)
associated with the scattered wavekis equal to
cos
, where
is the angle betweenkandrr. Since 1, this phase factor has a very rapid
variation with, except in the vicinity of points wherecosis stationary with respect
to, i.e. when= 0for the outgoing wave. The angular integral (47) therefore gets
most of its contribution from values of the angles that are close to the polar angles
and of vectorrr.
Taking (48) into account, we deduce from (44) and(45):
k
; 0

();kkk ( )
1
0+2
kr (0)
(49)
where we have replacedandby.
. Summation over the energies
The initial state of the scattering process is a superposition of stateskmultiplied
by(k). We consider a one-dimensional wave packet, propagating along theaxis.
With a proper choice of the coordinate origin, we can assume atom A is located atr= 0,
which amounts to replacingrby0. As before, we assume()is real, so that at= 0,
the wave packet is centered at the positionr= 0of the scattering atom A.
We now multiply (49) by(), integrate over, and over the timefrom to.
The amplitude of the absorption by atom B, at time, of the photon scattered by atom
A is therefore proportional to:
d
0
d()( )
1
0+2
(50)
2178

DETECTION OF ONE- OR TWO-PHOTON WAVE PACKETS, INTERFERENCE
where we have replaced the integral variableby=.
Let us compare (50) and (35). The integral overappearing in the integral over
in (50) can be interpreted as the classical eld scattered at timeand at a distance
from point O along an axis with polar angles:
di( ) d()( )
1
0+2
(51)
It will be useful for what follows to regroup the two exponentials of (50) and to set:
~= (52)
We then get:
di( ) =d()( )
1
1
0+2
~
(53)
which means that the wave packet scattered along the directionmoves at velocity
, and that its amplitude decreases as1.
. Spatial and temporal dependence of the scattered wave packet
We now assume that the frequency widthof the incident wave packet is much
smaller than its average frequency:
(54)
but we do not make any hypothesis concerning the relative values ofand. The
factor
appearing in the previous relations can now be replaced by, and comes
out of the integral. We can also neglect the variations withof( )over the
interval where the function( 0+2)
1
varies signicantly. The scattered
elddi(~)can then been seen as the temporal Fourier transform of the product of
two functions()and( 0+2)
1
. This eld is the convolution of the Fourier
transforms of these two functions of. Taking into account (36) and (37), the Fourier
transform of the rst function is:
() (~) =
~
(~) (55)
For the second function, we get:
1
0+2
0
~
(~)
~2
(56)
where(~)is the Heaviside function, equal to1for~0, and to0for~0. This leads
to:
di(~)
1
+
d ()
0(~)
(~)
(~)2
(57)
2179

COMPLEMENT E XX
. Study of two limiting cases
Two interesting cases occur when the widthof the incident wave packet is
either very large or very small compared to the natural widthof the excited state of
atom A.
limit
The incident wave packet passes through a given point in a time1that is very
short compared to the radiative lifetime1of the excited state. The envelope()of
the incident wave packet is dierent from zero only during a time interval much shorter
than the characteristic times of the Fourier transform of( 0+2)
1
. We can thus
set= 0in the last two terms of (57), which yields:
di(~)
1
+
d
(0 )
()
0
~
(~)
~2)
(58)
The term in the rst bracket is proportional to the excitation amplitude of the scattering
atom by the incident wave packet. The second bracket describes a free oscillation at the
atomic frequency0, starting at time~= 0and damped over a time2.
The physical meaning of this result is as follows. The incident wave packet spends a
very short time near atom A, and hence excites it in a percussive manner before moving
away with velocity. Once the incident wave packet is gone, the atomic dipole thus
excited oscillates freely at frequency0, until it is damped by spontaneous emission. This
situation is the analog of the percussive excitation of an oscillator in classical mechanics.
limit
We can now replace in (57) the function()by(~)as~cannot be larger, in
modulus, than1. This is because of the presence of the last exponential term in (57)
and the fact that, when , the envelope of
~
(~)varies very slowly over that time
interval. One can then rewrite (57) in the form:
di(~) (~)
~
+
d
(~) 0(~)
(~)
(~)2)
(59)
Let us make the change of variable=~ in the integral over. Taking (56) into
account, we see that this integral is actually the Fourier transform of
0
()
2
calculated at, which is( 0+2)
1
. This leads to:
di(~) (~)
~ 1
0+2
(60)
The physical meaning of this result is as follows. When , the wave packet
takes a long time passing atom A, whose dipole undergoes forced oscillation at the fre-
quency. It thus emits radiation at the same frequency, with an amplitude that follows
adiabatically the slow variation of the envelope(~)of the incident wave packet; this
explains the rst term in (60). The second term describes the linear response of the
dipole with eigenfrequency0and damping time2to an excitation of frequency.
In this case, the oscillator's amplitude follows adiabatically the excitation's amplitude.
2180

DETECTION OF ONE- OR TWO-PHOTON WAVE PACKETS, INTERFERENCE
5. Example of wave packets with two entangled photons
In , we considered two-photon states that were tensor products of two one-photon
wave packets; the two photons described by these states were not entangled. There
obviously exist a number of two-photon states that cannot be described in the form of
a product of two one-photon states, and which thus describe entangled photons. In this
last section, we shall focus on such an example where the entangled photons appear in
an optical nonlinear process, calledparametric down-conversion. This process has the
advantage of producing pairs of photons bunched in time. Detecting one photon of the
pair at a given timeallows predicting the second photon will be detected to within a
very short time interval.
5-a. Parametric down-conversion
Computations involved in parametric down-conversion are similar to computations
we already discussed. We shall simply outline the general ideas allowing a physical un-
derstanding of the process, without going into details that would unnecessarily lengthen
the present complement.
. Description of the process
In , we studied the elastic scattering of a photon by an atom.
Figures
an incident photon with angular frequencyis absorbed and anphoton emitted
while the atom goes back to its initial level. Energy conservation of the total system
atom + photon requires that=. In the present complement, we study a nonlinear
scattering process during which, as before, an incident photon of angular frequency0
is absorbed by an atom in state, but where there are now two photons with angular
frequencies1and2that are emitted. At the end of the scattering process the atom
returns to state; energy conservation now requires that0=1+2. Figure
two possible representations of such a process, analogous to those of Figure.and.
in Chapter.
Several temporal orders are possible for the absorption and emission processes. For
example, Figures.and.of Chapter
absorption and emission occurring in the scattering of one photon. For the three-photon
process considered here, including one absorption and two emissions,3! = 6possible
temporal orders should, a priori, be considered; Figure
possible orders.
. Scattering amplitude
The principle for calculating the scattering amplitude of parametric down-conversion
is similar to the one that led us to formulas (E-3) to (E-5) of Chapter, but involves
now three, instead of two, interactions with the eld; two relay states (instead of one)
come into play. As an example, for the process represented in Figure, we must consider
the following states:
initial state:;0, with energyin=+~0
rst relay state:; 0, with energyrel 1=
2181

COMPLEMENT E XX
Figure 1: An incident photon, with angular frequency0is scattered by an atomic system
in the initial state. At the end of the scattering process, the atomic system has returned
to state, while two new photons have appeared with angular frequencies1and2.
Energy conservation requires that0=1+2. In the left hand side of the gure, the
absorption (emission) processes are represented with upwards (downwards) arrows; in the
right-hand side, these processes are shown with incoming (outgoing) wiggly arrows that
also symbolize the photon propagation.
second relay state:;1, with energyrel 2=+~1
nal state:;12, with energyn=+~1+~2
The probability amplitude associated with this process is obtained by generalizing rela-
tion (E-4) of Chapter. Within a constant and non-signicant factor, it is the product
of a function(1+2 0), imposing energy conservation
7
, by the following expression:
}
20
3
32
012
"
2D "
1D "0D
(+~2 )(+~0 )
(61)
where"0,"1and"2are, respectively, the polarizations of the photons of frequencies0,
1and2. Compared to relation (E-4) of Chapter, expression (61) now contains
three (instead of two) matrix elements in the numerator, and two (instead of one) energy
denominators containing the dierences in energy between the initial state and either the
relay 1 or the relay 2 state.
Six similar amplitudes can be written, generalizing equations (E-3) to (E-5) of
Chapter
sion processes. Once they have been added, one must also sum these amplitudes over all
7
As in , we assume the total interaction timeto be suciently long for the function
()
to be assimilated to a real delta function.
2182

DETECTION OF ONE- OR TWO-PHOTON WAVE PACKETS, INTERFERENCE
the atomic relay statesand. All the contributions to the total amplitude contain the
same function(1+2 0).
The nal state of the system atom + radiation at the end of the scattering
process is the sum over1and2of the components thus obtained, with the condition
1+2=0. It can be written:
= (62)
where:
=
k1k2
(1+2 0)(k1k2)k1k2 (63)
with12=k12. This state cannot be written as the product of two eld states; it is
therefore entangled (see Chapter ).
The function(k1k2)characterizing the eld state is the result of the1and2
dependence of the scattering amplitudes, as well as of the density of nal states appearing
in the summations over the continuums
8
1and2(summation over the moduli of the
two vectorsk1andk2). We assume here that the energies of all the relay states are
far away from any resonance, so that the(k1k2)dependence onk1andk2does
not present any kind of narrow structure. In other words, all the energy dierences
= in relin the denominators of the scattering amplitudes are of the order of
a fraction of an optical frequency. We mentioned in comment (i) at the end of
ComplementXXthat the time spent in a relay state during the scattering process is,
according to the time-energy uncertainty relation, of the order of~. This means
that the times separating the emission of the two photons1and2cannot dier by more
than a few optical periods, i.e. a few tens of femtoseconds. This qualitative argument
shows that the two photons1and2are emitted quasi-simultaneously.
Comment:
If the interaction Hamiltonian appearing in the three matrix elements of the scattering
amplitude is the electric dipole Hamiltonian, and if the atomic states have a well dened
parity, this atomic parity changes with each interaction. After the three interactions, the
parity is therefore changed, which forbids the nal atomic state to be the same as the
initial state. Consequently, the parametric down-conversion process we just studied can
occur only when the atomic states do not have a well dened parity. Such a situation is
encountered when the atomic Hamiltonian is not invariant upon reection. This happens
for example when the atom is inserted in a crystal where the local crystalline eld, which
has the symmetry of an external electric eld, is not invariant upon spatial reection.
5-b. Temporal correlations between the two photons generated in parametric
down-conversion
We now compute the double detection signal from the two photons generated in
parametric down-conversion. The experiment we analyze is schematized in Figure. An
incident pump beam, with frequency0, propagates along a direction with unit vector
8
Two continuums of nal states come into play in this problem, but the condition1+2=0
reduces it to one.
2183

COMPLEMENT E XX
Figure 2: A pump beam with angular frequency0, propagating along a direction with unit
vectoru0, impinges on a nonlinear crystal placed in O. The parametric down-conversion
process generates two beams of frequencies1and2, with1+2=0. Diaphragms
allow xing the directionsu1andu2of these two beams. The two detectors1and2
register the arrivals of the photons and permit studying their temporal correlations.
u0, and impinges onto a nonlinear crystal O containing atoms performing the conversion.
Two diaphragms placed in front of the two detectors1and2allow selecting two
directions, with unit vectorsu1andu2, for the two beams generated by parametric
down-conversion.
We focus on the temporal, rather than spatial, aspect of the phenomenon. For the
sake of simplicity, we assume that the three eld states appearing in Figure
waves, innite in the two transverse directions. The only variable characterizing the
modes involved is therefore the longitudinal component of the vectorkor, equivalently,
the frequency. The incident photon is described by a wave packet, characterized in
frequency space by a real function(0), centered at
0
and of width0
0
. The
center of the incident wave packet arrives at crystal O at time= 0, and passes through
it in a time of the order of:
1= 10 (64)
The two-photon wave packet generated by parametric down-conversion is described by
an expression similar to (63), in which we now use the variables12=k12; this wave
packet depends, to a certain extent, on1and2via the function(12).
. Double photodetection signal(r1;r2+)
We rst compute the absorption amplitude of the two photons
9
, one at timeby
detector1located atr1, the other at time+by detector2located atr2:
0
(+)
(r2+)
(+)
(r1) =
}
20
3
0 12
(0)
12(12)
[k2r2 2(+)][k1r1 1]
(1+2 0) (65)
9
The signal is the squared modulus of this amplitude.
2184

DETECTION OF ONE- OR TWO-PHOTON WAVE PACKETS, INTERFERENCE
In this equation,k1andk2are the wave vectors of the two photons propagating freely
alongu1andu2. We note1and2the distances between O and1, O and2, and
dene1=1and2=2as the times taken by the photons to travel these two
distances. We have:
k1r1=
1
1=11
k2r2=
2
2=22 (66)
so that (65) can be rewritten in the form:
0
(+)
(r2+)
(+)
(r1) =
}
20
3
0 12
(0)
12(12)
2[2 ] 1[1]
(1+2 0) (67)
We now replace the two variables1and2by a single variableby setting:
1=
0
2
+
2=
0
2
(68)
Condition1+2=0is then automatically satised, so that the delta function ap-
pearing on the second line of (67) is no longer necessary. The summation over1and
2becomes a summation over, and the function(12)is replaced by a function
(). If we assume, for the sake of simplicity, that1=2=, we nally obtain:
0
(+)
(r2+)
(+)
(r1)
}
20
3
0
(0)
0[ ]2 0[]2
(
02
+)(
0
2
)() (69)
. Discussion
Thedependence of the double photodetection signal is given by the summa-
tion overon the second line of (69). Going to the continuous limit, it is there-
fore the Fourier transform of
(
02
+)(
0
2
)(). Nevertheless, the product
(
02
+)(
0
2
)varies very slowly withand can be taken as constant, as can
be the state densities introduced when replacing the discrete summation by an integral.
We also saw above ( ) that the variation ofas a function of1and2, hence the
variation of(), is very slow as long as no resonant (or quasi-resonant) relay states are
involved in the scattering process. We must thus take the Fourier transform of a func-
tion ofthat has a very large width, of the order of a fraction of the optical frequency.
This means that the double photodetection signal is dierent from zero only if the two
photodetections are separated by a time interval of the order of a few optical periods. In
other words, the two detections are always quasi-simultaneous.
Consider now the summation over0in the rst line of (69). We are going to
see that thedependence of the signal involves time scales much longer than those
2185

COMPLEMENT E XX
characterizing the variation withof the signal. To show this, we replaceby0on
the right-hand side; after going to the continuous limit, we get:
d0(0)
0( )
(70)
which is the Fourier transform of the incident wave packet. This packet arrives at point
O at= 0, and for the entire packet to pass that point, it takes a certain time;
this time intervalis much longer, in general, than the time characterizing the
dependence of signal. Relation (70) thus indicates that both detectors yield (almost
simultaneously) a signal at any timewithin a time intervalcentered around=;
this time corresponds to the arrivals at1and2of the photons generated at O by
the incident wave packet. But once a photon is detected by one of the detectors, the
other photon is detected practically at the same instant by the other detector. Such a
temporal correlation could not be predicted by a semiclassical treatment.
These results remain valid when the parametric down-conversion process is pro-
duced, not by a single incident photon described by a wave packet, but rather by a con-
tinuous laser excitation. The two beams generated by the parametric down-conversion
process then contain a series of pairs of photons, that are detected at the same instant;
they are referred to astwin photons.
Suchtwin beamscan excite two-photon transitions in a much more ecient way
than ordinary beams. This is because, in the absence of resonant relay states in the two-
photon absorption process, an argument similar to that presented above shows that the
two absorptions must be separated by a very short time interval (the two photons must
interact quasi-simultaneously with the absorbing atom). The two incident photons must
impinge on the atom at exactly the same time, which can be the case for twin beams
(with ordinary beams, one can only observe two-photon absorptions due to accidental
coincidences)
In practice, radiation parametric down-conversion is often performed, not on an
isolated atom, but rather on atoms or molecules in a solid. It is then imperative to take
into account the interference between beams generated in dierent parts of the solid,
and identify the conditions for getting a constructive interference. The refractive optical
index of the medium in which the beams propagate then plays an important role, which
leads to the so calledphase matching condition. This discussion, outside the scope of the
present complement, is treated in detail in quantum optics books [65] [66].
2186

Chapter XXI
Quantum entanglement,
measurements, Bell's
inequalities
A Introducing entanglement, goals of this chapter
B Entangled states of two spin- 12systems
B-1 Singlet state, reduced density matrices
B-2 Correlations
C Entanglement between more general systems
C-1 Pure entangled states, notation
C-2 Presence (or absence) of entanglement: Schmidt decomposition2193
C-3 Characterization of entanglement: Schmidt rank
D Ideal measurement and entangled states
D-1 Ideal measurement scheme (von Neumann)
D-2 Coupling with the environment, decoherence; pointer states
D-3 Uniqueness of the measurement result
E Which path experiment: can one determine the path
followed by the photon in Young's double slit experiment?
E-1 Entanglement between the photon states and the plate states
E-2 Prediction of measurements performed on the photon
F Entanglement, non-locality, Bell's theorem
F-1 The EPR argument
F-2 Bohr's reply, non-separability
F-3 Bell's inequality
Quantum Mechanics, Volume III, First Edition. C. Cohen-Tannoudji, B. Diu, and F. Laloë.
© 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

CHAPTER XXI QUANTUM ENTANGLEMENT, MEASUREMENTS, BELL'S INEQUALITIES
We discuss in this last chapter an essential concept of quantum mechanics, entan-
glement. This will highlight a number of aspects of quantum mechanics that have no
equivalent in classical physics.
A. Introducing entanglement, goals of this chapter
Consider two physical systemsand, each having a state spaceand; they can
be grouped into a total system+, whose state space is the tensor product.
If we assume systemis described by a normalized quantum statebelonging to
, systemby a normalized quantum statebelonging to its space state, the
ketdescribing the total system is the tensor product:
= (A-1)
In this case, each of the three physical systems,, and+is described by a state
vector, which is the most precise possible description in quantum mechanics.
The situation is dierent when the state vector of the global system is no longer a
simple product. Let us note,, .., orthonormalized quantum states belonging to
the state spaceof the rst system, and,,..., orthonormalized quantum states
belonging to the state spaceof the second. We can then build products dierent from
(A-1) for the state of the global system, for example:
= (A-2)
Now, we can also, in view of the superposition principle, form any linear combination
ofand, which will no longer be a simple product:
= + (A-3)
In this relation, the complex coecientsandcan take on any value, as long as they
obey the normalization condition:
2
+
2
= 1 (A-4)
We shall assume, however, that neither of these coecients is zero so that (A-3) is not
reduced to a simple product:
= 0 (A-5)
A state such as (A-3), which contains a coherent superposition of two (or more) com-
ponents, each component being a product, is called an entangled state. The general
property associated with these states is called quantum entanglement. It expresses the
fact that the quantum state of each subsystem is, in a way, conditioned by the state of
the other.
In ComplementIII, we introduced the concept of a density operator, which pro-
vides a more general description of a physical system than a state vector. The density
operator of the total physical system+, whose state vector is known, is simply the
projector onto :
+= (A-6)
2188

A. INTRODUCING ENTANGLEMENT, GOALS OF THIS CHAPTER
whose trace is equal to one:
Tr += = 1 (A-7)
When a physical system can be described by a state vector, it is said to be in a pure
state. Its density operator obeys the relation:
[+]
2
= + (A-8)
and thus:
Tr[+]
2
= 1 (A-9)
Under such conditions, we can choose to describe the total physical system either by its
state vector , or by the density operator+. We are going to show that this is no
longer the case for the two subsystemsand, for which only the density operator can
be used.
Imagine, for example, that we are only interested in measurements performed on
subsystem. We saw in IIIthat, when the total system is
entangled as in (A-3), there generally does not exist any state vector belonging to
that allows computing the probabilities of measurements performed solely on. Instead,
one must necessarily use a density operatorobtained by taking a partial trace (taken
over the state spaceof the non-observed system we recall in
compute the matrix elements of a partial trace):
=Tr + (A-10)
Like operator (A-6), this operator is Hermitian, non-negative, and its trace is equal to
one; it is however not the projector onto a single state vector. When the system is
described by the entangled state (A-3), this density operator is given by:
=
2
+
2
(A-11)
which is actually the sum of two projectors. Consequently, subsystemis in state
with a probability
2
, and in statewith a probability
2
: as opposed to the state
of+, the quantum state ofis not known with certainty, but only with a certain
probability. The density operatorcan be called a statistical mixture, underlying
the fact that the results of measurements performed onare predicted by computing
averages on the (non-observed) properties of. We then get the inequality:
Tr[]
2
1 (A-12)
where the equality occurs only if one of the two coecientsoris zero; the equivalent
of relation (A-9) is, in general, not satised by. Inequality (A-12) expresses the
fact that, as the quantum state ofis known only in a statistical way, the quantum
description ofis less precise than the description of the total system+. This
discussion can be easily generalized to the case where is the superposition of, not
only two components as in (A-3), but of three or more.
We nd ourselves in a situation that might look rather surprising, as it does not
have any equivalent in classical physics. We do know that a perfect classical description
2189

CHAPTER XXI QUANTUM ENTANGLEMENT, MEASUREMENTS, BELL'S INEQUALITIES
of the total system+automatically implies that each of its two subsystems is also
perfectly described. This is because a complete description of the state of the total
system is simply the collection of all the complete descriptions of its subsystems. As an
example, a perfect classical description of the solar system is simply the knowledge of all
the positions and velocities of the planets, satellites, and all their constituent particles.
In quantum mechanics, things are drastically dierent: the most precise description
of the total system by a state vector (pure state) does not imply, in general, that its
subsystems can be described with the same precision. This dierence radically changes
the usual relation between the parts and the whole of a physical system. Schrödinger,
who rst introduced in 1935 the words quantum entanglement commented on this new
concept [67]: As far as I am concerned, I would not call this propertyonebut ratherthe
characteristic trait of quantum mechanics, the one that enforces its entire departure from
classical lines of thought. By the interaction, the two representatives [the quantum states]
have become entangled... Another way of expressing the peculiar situation is: the best
possible knowledge of a whole does not necessarily include the best possible knowledge of
all its parts, even though they may be entirely separate and therefore virtually capable
of being `best possibly known' i.e., of possessing, each of them, a representative (state
vector) of its own.
In a general way, when the state vector of a global system is not a simple product,
and quantum entanglement occurs, the quantum predictions for observations on part of
the system can become rather unexpected. This chapter will discuss a number of special
physical eects related to quantum entanglement. As a general introduction,
the simple case of two spin-1/2 systems entangled in a singlet state. This example is
generalized, in , to any physical system, and we present some of the properties of
entangled quantum states. The relations between entanglement and quantum measure-
ments is discussed in , using in particular the ideal measurement scheme proposed by
von Neumann. In , we describe an experiment where one tries to observe interference
fringes of a particle going through a two-slit plate while determining at the same time
which slit the particle went through; if this were possible, one would face a contradiction.
However, a partial trace operation on an entangled state of the particle and the plate
allows proving the coherence of the quantum formalism and illustrates an aspect of com-
plementarity. Finally,
non-locality, in the framework of the general Einstein, Podolsky and Rosen argument,
and of Bell's theorem.
B. Entangled states of two spin-12systems
We rst discuss a very simple case, which will prove useful for the rest of the chapter:
each of the two systemsandis a spin12; each state spaceis then spanned
by the two eigenstatesof the spin component on theaxis. We assume these
two states are entangled in a singlet state, such as the one written in relation (B-22) of
Chapter X:
=
1
2
: + : : : +
=
1
2
+ + (B-1)
2190

B. ENTANGLED STATES OF TWO SPIN- 12SYSTEMS
(on the second line, we have simplied the notation, assuming that the rst index in the
ket refers to spin, and the second to spin).
B-1. Singlet state, reduced density matrices
In the basis of the4kets++,+,+, taken in that order, the
matrix representing the density operator+is written:
(+) =
1
2
0 0 0 0
0 11 0
01 1 0
0 0 0 0
(B-2)
It is easy to check, performing a matrix product, that(+)
2
= (+), hence that
relation (A-9) is veried: this means the total system is in a pure state.
As indicated in 5-b of ComplementIII, the matrix representing the density
operatoris obtained by taking a partial trace, i.e. by adding the matrix elements of
(+)that are diagonal with respect to the quantum numbers of the second spin (this
amounts to summing over the states of the non-observed spin):
= + (B-3)
This leads to:
() =
1
2
1 0
0 1
(B-4)
We then get:
()
2
=
1
4
1 0
0 1
(B-5)
and thus Tr()
2
= 12; this means that spinis not in a pure state. By symmetry,
the same result would obviously be found for(). Note that after taking the partial
trace, all the non-diagonal elements (coherences) of (B-2) have completely disappeared.
When they are considered as an isolated system, each of the two spins is in a completely
depolarized state, and measuring itsspin component (or any of its spin component,
for that matter) will yield the results+orwith the same probability12. At the
level of each individual spin, the minus sign that characterizes the entanglement of the
state vector (B-1) becomes irrelevant; on the other hand, we are going to show that
this entanglement yields very strong correlations between the results of measurements
pertaining to both spins.
B-2. Correlations
Imagine now that we perform simultaneously measurements on both spins, the
rst one along a direction in the plane, making an anglewith theaxis, the
second along a direction in that same plane, making an anglewith. The results
2191

CHAPTER XXI QUANTUM ENTANGLEMENT, MEASUREMENTS, BELL'S INEQUALITIES
we are about to obtain will be important for the discussion of Bell's theorem in .
Relations (A-22) of Chapter equal to zero and= ) yield
the expressions for the eigenvectors of the measurements in the state spacesand:
+= cos
2
++ sin
2
=sin
2
++ cos
2
(B-6)
In the space of the two-spin states, the ket corresponding to a double+result of the
measurement is written:
+ += cos
2
cos
2
+++ cos
2
sin
2
+
+ sin
2
cos
2
++ sin
2
sin
2
(B-7)
This means that, when the system is in the singlet state (B-1), the probability amplitude
for obtaining this double result is:
+ + =
1
2
cos
2
sin
2
sin
2
cos
2
=
1
2
sin
2
(B-8)
The probability of the result(++)when measuring the components of both spins along
theanddirections is therefore:
++( ) =
1
2
sin
2
2
(B-9)
One can redo the same calculations for the three other possible pairs of results,
(+),(+)and(). This does not present any diculty but it is easier to note
that changinginto+exchanges the two eigenkets of (B-6), and hence the results
+andfor the rst spin; the same operation can be done with the second spin.
We then make these changes in (B-9) and get the probabilities for the4possible results
in the form:
++( ) = ( ) =
1
2
sin
2
2
+( ) = +( ) =
1
2
cos
2
2
(B-10)
When both spins are measured, strong correlations between the results appear
1
.
These correlations are the direct consequence of the entanglement present in the singlet
state vector (B-1).
1
The probabilities cannot, in general, be factored. As an example, relation (B-10) shows that
++ +is dierent from+ . This means that the ratio of the probabilities of obtaining
for the rst spin the result+or the resultdepends on the state of the second spin, clearly showing
correlations.
2192

C. ENTANGLEMENT BETWEEN MORE GENERAL SYSTEMS
C. Entanglement between more general systems
The concept of entanglement is obviously not limited to the singlet state of two spin-12
particles. We now study how to characterize the presence of entanglement when the total
system is in a pure state.
C-1. Pure entangled states, notation
We consider two quantum systemsandbelonging, respectively, to state spaces
(with dimension) and(with dimension). The normalized state vector
describing the total system+belongs to the tensor product space , with
dimension. Some of the states can be written as a simple product:
= (C-1)
where and are any normalized kets ofand, respectively. In such a case,
the two physical subsystemsandare not entangled; each of them, as well as the total
state, can be described by a state vector (pure state). On the other hand, the majority
of the states cannot be factored this way, and must necessarily be written as a sum
of products (the singlet state studied above is such an example); the two subsystems
andare then entangled.
It is not always obvious to guess from the expression of any given state vector
if it can actually be written as a simple tensor product. This ket has, in general,
components, and is expressed as:
=
=1=1
(C-2)
where the as well as theare orthonormalized kets. Now if we expand the kets
and , appearing in the tensor product (C-1), onto the ketsand as
=
=1
and =
=1
(C-3)
we obtain for a ket of the type (C-1):
=
=1=1
(C-4)
It is not obvious at all, just from the knowledge of the coecientsof , to know
if they can be factored into an expression of this type, leading to a product as in (C-1).
We present in the next section a systematic method for asserting if this factorization is
possible and actually performing it.
C-2. Presence (or absence) of entanglement: Schmidt decomposition
It can be shown (see the demonstration below) that any pure state describing
the ensemble of the two physical systemsandcan be written in the form:
=
(C-5)
2193

CHAPTER XXI QUANTUM ENTANGLEMENT, MEASUREMENTS, BELL'S INEQUALITIES
where the are a set of orthonormal vectors in the state space of the rst system,
and the another set of orthonormal vectors in the second state space. This ex-
pression is the Schmidt decomposition of a pure state, also called the biorthonormal
decomposition.
Whether the state of the total system is entangled or not, it always has a
corresponding density operator, written in (A-6). By taking partial traces, each of the
subsystems can be described by the density operators:
=Tr + ; =Tr + (C-6)
Performing these partial traces using (C-5), we get two symmetric expressions:
= (C-7)
and:
= (C-8)
This means that, when the total system is in a pure state, the two partial density oper-
ators always have the same eigenvalues
2
. In the particular case where they are all zero
except one, each of the two subsystems is in a pure state and state can be factored:
no entanglement exists in the total system. Most of the time, however, several eigenval-
ues are non-zero, in which case()
2
is obviously not equal to, and the same is true
for; entanglement is then present in the pure state .
Demonstration of relation (C-5):
As the two operatorsand are Hermitian, non-negative and have a trace equal to
unity, their corresponding matrices can be diagonalized to yield real eigenvalues included
between0and1. We call the normalized eigenvectors of(the indextakes on
dierent values, whereis the dimension of subsystem) andthe corresponding
eigenvalues, all positive or zero (but not necessarily dierent. Similarly, the eigenvectors
ofare noted (wheretakes ondierent values,being the dimension of the
second subsystem), andthe corresponding eigenvalues. The two partial density
operators can then be expanded as:
=
=1
and =
=1
(C-9)
with0 , 1.
State can then be expanded on the basis of the tensor products: :,
that we shall simply note assuming the rst ket represents a state ofand the
second a state of; we callthe components of in this basis and get:
=
=1=1
(C-10)
2
Note that this is not necessarily the case if the total system is described by a statistical mixture
rather than a pure state. As an example, we can assume that+equals a tensor product ,
where and can be chosen arbitrarily, and hence have dierent eigenvalues.
2194

C. ENTANGLEMENT BETWEEN MORE GENERAL SYSTEMS
We now introduce the ket
, belonging to the state space(this ket is not necessarily
normalized), as:
=
=1
(C-11)
Expansion (C-10) for now simply becomes:
=
=1
(C-12)
We also know that the matrix elements of the partial traceare:
= (C-13)
Now expression ( ) for leads to:
=
(C-14)
Inserting this result into (C-13), we are only left with terms for which=and=,
which yields:
=
= (C-15)
This means that:
= =
(C-16)
Now in the basis we have used, we know thatis diagonal and given by expression
(C-9); the comparison with (C-16) shows that we must necessarily have:
= (C-17)
For non-zero eigenvalues, this relation shows that one can dene a set of orthonormal
vectors belonging to the state space of systemas:
=
1
(C-18)
For all the values of the indexassociated with eigenvaluesequal to zero, that same
relation shows that the kets
are zero.
Replacing in (C-12) the
by, we complete the demonstration of equality
(C-5), and of relations (C-7) and (C-8) which follow directly.
2195

CHAPTER XXI QUANTUM ENTANGLEMENT, MEASUREMENTS, BELL'S INEQUALITIES
C-3. Characterization of entanglement: Schmidt rank
The number of non-zeroeigenvalues, i.e. the number of non-zero terms in (C-5),
is called the Schmidt rank of and noted. When = 1, the state of the total
system is not entangled, the two subsystems each being in a pure state. When= 2,
we are in the case studied in = 3
we get a more complex entanglement, etc. This gives us a criterion for deciding whether
a general ket (C-2) is entangled or not. We just have to compute a partial density
operator for one of the subsystems and the number of its non-zero eigenvalues (for the
sake of simplicity, we shall obviously choose the subsystem whose state space's dimension
is the smallest). If that number equals1, the eigenvector associated with that non-zero
eigenvalue becomes one of the factors of the decomposition, which makes it easy to nd
the other. If that number is greater than1, however, the decomposition into a single
tensor product is no longer possible.
In a way, entanglement is symmetrically shared betweenand. For example,
it is not possible for one of the two subsystems to be in a pure state and the other in a
statistical mixture. The rankmust be lower than the smallest of the dimensionsand
of the state spaces ofand; to allow a high rank entanglement, the two subsystems
must thus have state spaces with high enough dimensions.
Comment:
When all theeigenvalues of(and of) are distinct, the Schmidt decomposition
is unique. This is because decompositions (C-9) and (C-8) ofon the projectors onto
its eigenvectors must then necessarily coincide; the set of eigenvectorsis identical
to that of the. In this case, the eigenvectors of the partial density operators directly
yield the unique Schmidt decomposition. On the other hand, when certaineigenvalues
are degenerate, this is no longer true. As an example, we saw that for the singlet state (B-
1), the two partial density matrices have two eigenvalues, both equal to12; this singlet
state, decomposed in (B-1) into products of eigenvectors of thespin components,
can equally well be decomposed into products of eigenvectors of the spin components on
any spatial direction. There are an innity of possible Schmidt decompositions for that
state.
D. Ideal measurement and entangled states
Entanglement also plays an essential role in any quantum measurement process, as it
generally appears while the measured systemand the measuring apparatusinteract.
Furthermore, we shall see that it even propagates further and brings the environment of
the measuring apparatus into play.
D-1. Ideal measurement scheme (von Neumann)
Von Neumann's quantum measurement scheme proposes a general framework that
allows characterizing the quantum measurement process in terms of entanglement ap-
pearing (or disappearing) in the state vector describing the total system+. The
two systemsandare initially described by a factored state 0; however, as they
2196

D. IDEAL MEASUREMENT AND ENTANGLED STATES
interact during a certain time, they reach an entangled state . After the measure-
ment, we assume they no longer interact, imagining for example they have moved far
away from each other.
In the state space of, with dimension, the physical quantity measured on
is described by an operatorwhose normalized eigenvectors are the ketswith
eigenvalues(that we shall assume non-degenerate, to simplify the notation):
= (D-1)
Initially, state0ofis any linear combination of the:
0=
=1
(D-2)
with complex coecients, having only the constraint that the sum of their squared
moduli be equal to1(normalization condition). As for the measuring apparatus, we
assume it is, initially, always in the same normalized quantum state0. The initial
state of the total system is then:
0= 00 (D-3)
D-1-a. Basic process
We start with the particular case where the measurement result is certain and
where the systemis initially in one of the eigenstates associated with the measurement:
0= . In that case:
0= 0 (D-4)
Once the measurement is done,stays in the same state, but the measuring appa-
ratus reaches a statedierent from0and which depends on: this is a necessary
condition for the result to be experimentally accessible. This is because the position of
the pointer used for the reading of the result (a needle in a macroscopic apparatus,
the recording of the result in a memory, etc.) must necessarily depend onto allow for
the acquisition of the data. It is also natural to assume that the dierent states
are orthogonal to each other, since the pointer necessarily involves a large number of
atoms whose dierent states will allow a macroscopic observer to read the result. The
measurement process for the total system can be summed up, in this simple case, as
follows:
0= 0= = (D-5)
whereis a normalized state of. At this stage, no correlation or entanglement
has appeared between the measuring apparatus and the measured system. This is what
happens in the simple case where the measurement result is certain.
In the general case, the initial state of systemis a superposition (D-2) of eigen-
states. In this case, state (D-4) must be replaced by the linear combination, with
the same coecients:
0= 0 (D-6)
2197

CHAPTER XXI QUANTUM ENTANGLEMENT, MEASUREMENTS, BELL'S INEQUALITIES
As Schrödinger's equation is linear, we get:
0= = (D-7)
which now becomes a state in which the measuring apparatus is entangled with the
measured system. The states ofand are therefore strongly correlated: when
the pointer's position is associated with one of the state vectors, the state of the
systemmust be described by the ketassociated with a denite eigenvalue of.
After the measurement, one can no longer attribute a state vector (pure state) to
the system, which can only be described by a partial density operator. As the states
are normalized and orthogonal to each other, this density operator is given by:
=Tr =
2
(D-8)
This relation was to be expected: it simply states that the system has a probability
2
of being in the stateassociated with the measurement result, which is in
agreement with the Born rule for the usual probabilities. This useful formula sums up in
a simple way a number of characteristics of the quantum measurement postulate. Note,
however, that at this stage, all the possible results are still present in the partial trace,
as they are considered possible, even after the measurement. Nothing at this point tells
us that only one result is actually measured when the experiment is performed, nor that
the squares of the coecientscan be interpreted as classical probabilities associated
with mutually exclusive observations. The evolution predicted by Schrödinger's equation
cannot, by itself alone, explain the uniqueness of the results observed at the macroscopic
level. This is why von Neumann introduced the postulate of the state vector's reduction
(also called the wave packet reduction or wave packet collapse,cf.Chapter III, B-
3-c); more detail on this point will be given in .
D-1-b. Dynamics of the entanglement process
A simple interaction Hamiltonian between systemsandcan explain the ap-
pearance of entanglement between these systems, and lead to relations (D-5) or (D-7).
As an example, imagine this interaction Hamiltoniancan be written as:
= (D-9)
where is the operator (acting only on) already introduced above,an operator
acting only on, anda coupling constant. We shall also assume that, in the state
space of, there exists a Hermitian conjugate operatorof the operator:
[ ] =} (D-10)
This commutation relation means thatgenerates the translation operators with re-
spect to. In other words, the action of
}
on any eigenvectorof:
= (D-11)
leads to a translation byof the eigenvalue:
}
= + (D-12)
2198

D. IDEAL MEASUREMENT AND ENTANGLED STATES
whereis any real number see relation (13) of ComplementII.
We assume that0(state of the measuring apparatus before the measurement)
is a normalized eigenstate ofwith eigenvalue0, and we ignore
3
any other source
for the evolution of the total system other than the interaction betweenand. The
evolution operator between the time= 0before the measurement, and the time=
when the interaction is over, is:
(0) =
}
(D-13)
Its action on the ket (D-4) yields:
(0) 0(0)= 0(0+ ) (D-14)
where the variables in parentheses in the states of the measuring apparatus
4
refer to the
eigenvalues of. This means that the statesintroduced in (D-5) are the kets:
=(0+ ) (D-15)
These relations show that, as far as the measuring apparatusis concerned, the eigen-
value ofhas been shifted by a quantitythat depends on the eigenvalueof.
The observable therefore plays the role of a pointer's position in the measuring
apparatus (measuring needle), which yields the measurement result once the two systems
have interacted. As for the observable, it pertains to the systembeing measured
by the pointer's position.
If now the initial state is in a coherent superposition as in (D-6), the state after
the interaction (D-7) is written:
= (0+ ) (D-16)
which is a biorthonormal decomposition such as the one obtained in . If, initially,
the systemis not in an eigenstate of, the interaction with the measuring apparatus
changes its state into a statistical mixture (D-8). On the other hand, if the system is
initially in an eigenstate of, it will stay in the same eigenstate after the measurement:
the measurement process does not change its state. The measurement is then said to be
a quantum non-demolition measurement, or QND measurement.
D-2. Coupling with the environment, decoherence; pointer states
We now examine under which conditions the interaction and entanglement process
we have considered constitutes a good measurement. A rst obvious condition to be
satised is for the statesof the measuring apparatus to store the information about
3
To avoid this hypothesis, the computation could be performed within the interaction point of view
(exercise 15 in ComplementIII) with respect to free evolutions of bothand; this would somewhat
complicate the results. However, as we focus here on the dynamics induced by their mutual interaction,
we shall keep the computations simple and assume that these free evolutions have a negligible eect
during the durationof the interaction.
4
Needless to say, a measuring apparatus is macroscopic and has many other degrees of freedom apart
from the pointer's position. For the sake of simplicity, these other degrees of freedom have not been
introduced in the notation.
2199

CHAPTER XXI QUANTUM ENTANGLEMENT, MEASUREMENTS, BELL'S INEQUALITIES
the measurement result in a robust way, and prevent it from being destroyed as
continues to evolve on its own. This condition will be fullled ifis a constant of the
motion ofor, in other words, ifcommutes with the Hamiltonian of. In
addition, the measuring apparatus cannot remain isolated from its environment, even at
a microscopic level. In view of its function, the apparatus must be able to interact and be
correlated with a measurement recording device, and even with the observer collecting
the measurement result; it is, by denition, an open apparatus, which can interact
with the outside world. In any case, for this apparatus to be completely isolated, would
require that none of its atoms, electrons, etc., be in interaction and correlated with any
of the environment particles, which is obviously impossible to achieve with a macroscopic
apparatus.
This means that, as far as the coupling betweenand the environmentis con-
cerned, an entanglement phenomenon occurs, reminiscent of the one discussed forand
. We must determine which basis, in the state space of, will lead for the entangled
state+to a biorthonormal decomposition similar to (D-16). The computation of
the partial density operator ofis similar to the computation that yielded (D-8) for the
entanglement betweenand: it is in the basis of this biorthonormal decomposition
that the partial density operator of(which plays the role of) remains diagonal;
with another basis, the density matrix will have in general non-diagonal elements. As,
furthermore, the entanglement continues to propagate further and further into the envi-
ronment, it is necessary that the relevant basis ofremains constant in time. It is thus
important to nd this privileged basis.
Depending on the circumstances, the coupling between a measuring apparatus and
its environment can take on various forms, in general complex due to the large number of
degrees of freedom involved; several time constants come into play. Dierent models have
been proposed to account for this coupling and the dynamics it produces. Without going
into any details we shall make a few general remarks. The measurement process involves
a whole chain of amplication betweenand the macroscopic pointer, which can be com-
posed of mesoscopic or macroscopic objects sensitive to the environment. Entanglement
propagates along that chain via local interactions: the interaction potentials are diagonal
in the position representation, and have a microscopic range. Consequently, they cannot
couple quantum states corresponding to macroscopically dierent positions of the objects
concerned; the branches of the state vector corresponding to dierent spatial positions
propagate independently. This means that the coupling with the environment tends to
favor the basis of states where the positions of the dierent elements of the measuring
apparatus, including in particular its pointer, occupy well dened positions in space.
The corresponding preferred basis in the state space of the measuring apparatus, in
which its density matrix remains diagonal over time, is called the basis of the pointer
states. In this basis, and only in this one, dened by pointer localization criteria, the
entanglement withis prone to destroy the coherences (non-diagonal elements of the
density matrix), without changing the diagonal elements (meaning the positions of the
pointer's particles).
To sum up, several conditions are necessary for a device to be considered as an
acceptable measuring apparatusfor a physical quantity of. In the rst place, the
coupling betweenandmust be capable of transferring the right information from
one to the other. The transferred information must then be conserved over time, while
continues its own evolution, and is coupled with the environment. Obviously, these are
2200

D. IDEAL MEASUREMENT AND ENTANGLED STATES
necessary conditions. In practice, an eective measuring apparatus must be conceived
taking into account many other imperatives, such as high sensitivity, or strong protection
against unavoidable external perturbation.
D-3. Uniqueness of the measurement result
As mentioned above, nothing in the dynamics associated with Schrödinger's equa-
tion can explain the uniqueness of the results observed at the macroscopic level. This
is not surprising as (D-8) is a direct consequence of Schrödinger's equation, which is
incapable of stopping on its own the endless propagation of the von Neumann chain,
as we shall now discuss.
D-3-a. The innite von Neumann chain
Let us go back to the ideal measurement scheme of . After the measurement,
the state of+is the entangled state (D-7), a superposition of components associated
with all the possible measurement results. One may wonder if, using a second measuring
apparatus2to observe, one might be able to resolve this superposition and obtain
a unique result. In fact, the same entanglement process that occurred betweenand
will occur again, leading to a nal state:
= (D-17)
where the ketsrepresent the states of the second measuring apparatus2, orthog-
onal to each other for dierent values of. Adding a third measuring apparatus3will,
obviously, only continue further the entanglement's progression, each additional appa-
ratus playing the role of an environment for the previous one. This chain of measuring
apparatus may continue all the way to innity without permitting at any stage the reso-
lution of the superposition, and the demonstration of the uniqueness of the measurement
result. This is called the von Neumann chain (and the logical problem it poses is called
the von Neumann's innite regress).
The well-known Schrödinger's cat paradox involves a similar situation. The
systemis supposed to be a radioactive nucleus in a superposition of two states,1
where the nucleus is still in the excited state, and2where it has disintegrated, emitting
a particle. The kets,, etc. represent the states of the measuring apparatus that
can detect this particle, and then trigger a mechanical system killing the cat in the case
of positive detection. The last of these ketscharacterizes the cat, which can therefore
be in state1where it is still alive, or in state2where it is dead. Schrödinger points
out the absurdity of a physical description involving a cat that can be at the same time
both in an alive and a dead state.
As we just discussed, the uniqueness of the measurement results cannot be proven
with Schrödinger's equation; this equation merely predicts that the pointer of a measuring
apparatus, and any other macroscopic object, can become superpositions of states located
at points very far away in space. Because of the linearity of Schrödinger's equation,
nothing prevents the dierent components of the state vector from propagating further
and further away, without this innite chain of entanglements ever reducing into a single
one of its components. It is precisely to solve this problem that von Neumann introduced
a specic postulate: the postulate of the reduction of the state vector (Chapter
3-c) which forces the uniqueness of the measurement result.
2201

CHAPTER XXI QUANTUM ENTANGLEMENT, MEASUREMENTS, BELL'S INEQUALITIES
D-3-b. Postulate of reduction of the state vector
The postulate of reduction (or collapse) of the state vector is also called the projec-
tion postulate, or the postulate of collapse of the wave packet. As we saw in Chapter
( , this postulate states that, once the measurement has been performed, one must
suppress the summations appearing in (D-7), (D-8), and (D-17): one only keeps, among
all the terms, the component=corresponding to the measurement result actually
observed. After the measurement, the state vector becomes again a simple product from
which entanglement has disappeared;is once again in a pure state. This means that
the entanglement, initially created by the measurement, disappears once the result has
been recorded.
This postulate, as ecient as it may be, is somewhat dicult to interpret. Using
this postulate amounts to considering that the state vector can evolve under the inuence
of two dierent processes: a normal continuous evolution, obeying Schrödinger's dier-
ential equation, and a sudden discontinuous evolution upon measurement, governed by
the von Neumann reduction postulate. Obviously, this duality immediately introduces
the question of the limit between these two evolutions: from which time on, exactly,
should we consider that the measurement has been performed? In other words, how far
does the coherent superposition (D-17) propagate? Which physical processes constitute a
measurement, as opposed to those leading to a continuous Schrödinger evolution? These
dicult questions were the motivation for introducing other interpretations of quantum
mechanics. As an example, there are non-standard interpretations where Schrödinger's
equation is modied by the adjunction of a small stochastic term. This term is chosen
so as to be totally negligible at the microscopic level, while coming into play at a certain
macroscopic level; its role is to suppress all the macroscopically dierent components of
the state vector, except one. Both Schrödinger and von Neumann dynamics are then uni-
ed into a single equation for the evolution of the state vector. Many other interpretations
have been proposed: additional variables, modal interpretation, Everett, all suggesting
dierent solutions for the problem. The interested reader may consult reference [68].
E. Which path experiment: can one determine the path followed by the
photon in Young's double slit experiment?
Let us now return to a question already discussed in ComplementI. In Young's double
slit experiment where the photon may follow two dierent paths to reach the detection
screen, is it possible to observe interference fringes between these paths and simulta-
neously obtain information as to which path the photon followed? Figure
plementI, reproduced here in the above Figure, shows an interference experimental
set-up using a plate pierced with two slits1and2; this plate is mobile in a direction
perpendicular to the incident photon. As it receives momentum transfers1and2
that will be dierent, depending on whether the photon goes through1or2, one could
naively imagine observing interference while knowing through which slit the particle went
through. However, using the momentum-position uncertainty relations applied to this
mobile plate, we showed that the interference fringes were blurred as soon as the momen-
tum transfers1and2were suciently dierent to provide this information. The
reason is that if we want to be able to distinguish these two momentum transfers, the
momentum uncertainty of the mobile plate must be less than the modulus of12.
A simple calculation then shows that when this condition is met, the uncertainty in the
2202

E. WHICH PATH EXPERIMENT: CAN ONE DETERMINE THE PATH FOLLOWED BY THE PHOTON IN
YOUNG'S DOUBLE SLIT EXPERIMENT?
Figure 1: Young's double slit experiment using a plate, mobile along theaxis, and
pierced with two slits1and2. A photon, emitted by a sourceassumed to be far away
at innity, reaches the detection screen at point. Thecomponent of the momentum
transferred by the photon to the platedepends on whether it goes through slit1or slit
2.
position of the plate must necessarily be larger than the fringe spacing, which blurs out
the fringes. It is impossible to know which of the slits the photon went through without
destroying at the same time the interference pattern.
We shall take the analysis a step further and consider the entanglement between the
plate and the paths followed by the photon. This should allow us to envisage intermediary
situations where partial information on the particle's path can be obtained.
E-1. Entanglement between the photon states and the plate states
Consider the path1followed by the photon if it goes through1and arrives
aton the detection screen (Figure). We call1the photon state when it follows
that path and transfers a momentum1to the mobile plate as it goes through1. In
that case, after the photon's transit, the state of the plate is:
1= exp(1~)0 (E-1)
where 0is the initial state of the plate andexp(1)the momentum space trans-
lation operator, by a quantity1. The state of the global system photon + plate,
along the path1, is therefore1 1. A similar reasoning would yield the result
2 2along the path2. As Schrödinger's equation is linear, the state of the
global system after the photon has crossed the plate is:
= 1 1+ 2 2 (E-2)
2203

CHAPTER XXI QUANTUM ENTANGLEMENT, MEASUREMENTS, BELL'S INEQUALITIES
which clearly shows the entanglement between the photon
5
and the plate.
E-2. Prediction of measurements performed on the photon
Measurements performed only on the photon after its crossing the plate can be
predicted from the reduced density operator, which is the partial trace over the plate
variables of the global system density operator . The matrix elements ofare
obtained via the standard calculation of a partial trace (ComplementIII, ) and
lead to the operator expression:
= 1 1+ 2 2+ 1 2 21+ 2 1 12 (E-3)
(in this equation we did not write the factors of1 1and 2 2, which are the
scalar products11and 22, both equal to1if the state0is normalized).
The interference between the two paths is described by the terms in1 2and2 1,
which are multiplied by the scalar products21and 12.
Two extreme cases then appear. If the two states1and 2are very close
to each other, the two scalar products are practically equal to1and the interference
terms in (E-3) are barely modied by the presence of the factors21and 12:
the interference is then quite visible on the detection screen. In that case, however, the
states1and 2are too close to give any information as to whether the photon went
through1or2. In the other extreme case where1and 2are very dierent from
each other, their scalar product is practically zero: the interference terms disappear from
(E-3), but one can, in principle, determine which of the two slits the photon went through.
The present calculation allows studying intermediate situations where the scalar products
21and 12take on values included between0and1. They describe how the
contrast of the fringes diminishes when21and 12continuously decrease from
1to0.
Actually, these scalar products can easily be computed from (E-1) and the equiv-
alent relation for2. This leads to:
21= 0exp [(12)]0 (E-4)
Using the expression of12(noted1 2in ComplementI) and equations
(6) and (7) of that complement, we can show that21and 12are equal to the
overlap integrals between the plate initial wave function, and that same wave function
translated in momentum space by the amountwhereis the fringe spacing.
F. Entanglement, non-locality, Bell's theorem
We now present two important theorems, the EPR (for Einstein, Podolsky and Rosen)
theorem, and Bell's theorem, which are related, the second actually being a logical con-
tinuation of the rst. The EPR theorem was presented in an article published in 1935
by these three authors [69], and is one of the episodes of the famous discussion between
Einstein and Bohr concerning the foundations of quantum mechanics (in particular dur-
ing the Solvay conferences). Einstein's position was that the entire physical world had
5
All the conclusions of this section remain valid for Young's interference type experiments performed
with a massive particle instead of a photon.
2204

F. ENTANGLEMENT, NON-LOCALITY, BELL'S THEOREM
to be expressed in the general framework of relativity, where the concept of space-time
events is fundamental. Bohr had a dierent point of view, and considered that quantum
theory demanded abandoning a description of microscopic events in space-time terms,
while of course conforming to the actual predictions of relativity.
F-1. The EPR argument
The EPR theorem can be stated as follows: If all the predictions of quantum
mechanics are correct (even for systems made of several remote particles) and if physical
reality can be described in a local (or separable) way, then quantum mechanics is nec-
essarily incomplete: some `elements of reality' exist in Nature that are ignored by this
theory .
To demonstrate their theorem, Einstein, Podolsky and Rosen imagined an experi-
ment where two physical systems, originating for example from a common source S and
described by an entangled quantum state, are then measured in remote regions of space.
Historically, EPR developed their argument for correlated particles whose position and
momentum are measured. It is however simpler to present an equivalent version of the
argument concerning spins and discrete results, a version initially proposed by Bohm
(and often called for that reason EPRB).
F-1-a. Exposing the argument
Imagine that two spin 1/2 particles are emitted by a source S in a singlet state
(B-1), which is an entangled state where the spins are strongly correlated. The particles
then move towards two remote regions of space, without their spins interacting with the
outside world; the initial spin entanglement remains unchanged. In these remote regions
of space, the particle spin components are measured along a direction dened by angle
for the region on the left, and by anglefor the region on the right (Fig.). One often
calls Alice and Bob the two observers who perform the measurements in the two dierent
laboratories, which can be very far away from each other. Alice chooses the direction
freely, which denes her measurement type. With a spin12, she can only obtain
two results, that we will note+1or1, whatever measurement type was chosen. In a
similar way, Bob chooses the directionarbitrarily and obtains one of the two results
+1or1. In the thought EPRB experiment, one assumes for simplicity that the two
spins, once they have been emitted by the source, will only interact with the measuring
apparatus (without having any free evolution, as was the case above). Standard quantum
mechanics then predicts () that the distances and instants at which the measurements
are performed do not play any role in the probability of obtaining the dierent possible
double results.
To keep things simple, let us assume Alice and Bob limit their choices to a nite
number of directionsandfor their respective measurements. It may then happen,
by chance, that their chosen directions are parallel. Now if the anglesandare
chosen to be equal (parallel measurement directions), relations (B-10) indicate that the
results will be always opposite for the two measurements: each time Alice observes1,
Bob observes the opposite value1. This remains valid even if the measurements are
performed at points greatly separated in space, whatever the choice=, and even if
the two observers operate in totally independent ways, in their own regions of space; for
example, they could make their choice at the last moment, even after the emission and
2205

CHAPTER XXI QUANTUM ENTANGLEMENT, MEASUREMENTS, BELL'S INEQUALITIES
Figure 2: In an EPRB experiment, a source S emits pairs of particles in a singlet spin
state (entangled quantum state). These particles then propagate along theaxis, to-
wards two remote regions of spaceand, where Stern-Gerlach apparatus are used by
observers Alice and Bob to measure the components of their spin along directions perpen-
dicular to. For the rst particle, the measurement direction is dened by angle, and
for the second, by angle. Each measurement yields a result+1or1, and one looks
for correlations between these results when the experiment is repeated a great number of
times.
propagation of the pair of particles.
Let us assume Alice performs her measurement along a directionbefore Bob
starts doing his own measurement. When Alice nishes her measurement, it becomes
certain that, should Bob decide to chose a directionparallel to, he will observe the
opposite result; the result is certain in that particular case. Such a certainty can only
come from the fact that the particle measured by Bob possesses a physical property that
determines this certain result; this property (called element of reality by EPR) will
inuence the way that particle interacts with the measuring apparatus inand will
determine the result. On the other hand, the particle propagating towards Bob cannot
be inuenced by events occurring in Alice's laboratory. This means that this physical
property we are discussing existed before the measurement performed by Alice.
The reasoning is obviously symmetrical and establishes that, before any measure-
ment, the particles already possessed physical properties that determined the outcome of
the future measurements. As the directionchosen by Alice was random, it means that
the particles must possess enough properties to determine the results for any analysis di-
rections chosen by the observers. Now quantum mechanics does not predict the existence
of such properties, as it only gives a description of the particles via a singlet state vector,
which always predicts a totally random result for the rst measurement. Furthermore,
there exists no quantum state for which all the spin components on arbitrary directions
can be simultaneously determined (the corresponding operators do not commute with
each other). This means that quantum mechanics accounts only partially for the physical
properties of the system; it is therefore incomplete.
F-1-b. Assumptions and conclusions
Let us discuss in more detail the logical structure of the EPR argument.
(i) It starts by assuming that the predictions of quantum mechanics for the proba-
2206

F. ENTANGLEMENT, NON-LOCALITY, BELL'S THEOREM
bilities of measurement results are correct. The argument thus assumes that the perfect
correlations predicted by this theory are always observed, whatever the distance between
the two measuring apparatus.
(ii) Another essential ingredient of the EPR argument is the concept of elements
of reality dened with the following criterion [69]: if, without in any way disturbing a
system, we can predict with certainty the value of a physical quantity, then there exists
an element of physical reality corresponding to this physical quantity. In other words, a
certainty cannot be built on nothing: an experimental result known beforehand can only
be the consequence of a preexisting physical quantity.
(iii) And last, but not least, the EPR argument brings in the notion of space-time
and locality: the elements of reality they discuss are attached to regions of space where
the experiments take place, and they cannot suddenly vary (and certainly not appear)
under the inuence of events occurring in a very distant region of space. Einstein wrote
in 1948 [70]: Physical objects are thought of as arranged in a space-time continuum.
An essential aspect of this arrangement of things in physics is that they lay claim, at
a certain time, to an existence independent of one another, provided these objects are
situated in dierent parts of space. To sum up, one can say that the basic conviction of
EPR is that regions of space contain their own elements of reality (attributing distinct
elements of reality to separated regions of space is sometimes called separability), and
that their time evolution is local one often refers to local realism in the literature to
qualify the ensemble of the EPR hypotheses.
Basing their argument on these hypotheses, EPR show that, for any chosen values
ofand, the measurement results are functions:
(i) of the individual properties of the spins the particles carry with them (the EPR
elements of reality);
(ii) and of the orientations,of the Stern and Gerlach analyzers (which is
obvious).
It follows that the results are given by well dened functions of these variables, meaning
that no non-deterministic process occurs: a particle with spin brings along all the nec-
essary information to yield the result of a future measurement, whatever the choice of
the orientation(for the rst particle) or(for the second). This implies that all the
components of each spin must have simultaneously well determined values.
F-2. Bohr's reply, non-separability
Bohr rapidly replied [71] to the EPR article presenting their argument. In Bohr's
view, the only physical system to be considered is the entire experimental set-up, includ-
ing the measured quantum system and all the measuring apparatus, which are treated
classically. It is thus meaningless to try and select among this ensemble subsystems hav-
ing individual physical properties. The physical system Bohr considers is a whole that
one should not attempt to separate into parts. This is often called the non-separability
rule. In other words, Bohr considers that spatial separation does not lead to separability.
It is not the EPR reasoning that Bohr criticizes, but he considers that their start-
ing assumptions are not relevant in the framework of quantum physics. From Bohr's
point of view, the EPR criterion for elements of reality contains an essential ambiguity
when applied to quantum phenomena. Along the same line, more than ten years later
(in 1948), Bohr made his point of view explicit [72]: Recapitulating, the impossibility
of subdividing the individual quantum eects and of separating a behavior of the objects
2207

CHAPTER XXI QUANTUM ENTANGLEMENT, MEASUREMENTS, BELL'S INEQUALITIES
from their interactions with the measuring instrument serving to dene the conditions
under which the phenomena appear implies an ambiguity in assigning conventional at-
tributes to atomic objects, which calls for a reconsideration of our attitude towards the
problem of physical explanation . It is thus the very need for a physical explanation
involving such a subdivision that is questioned by Bohr.
Bohr refutes Einstein's basic idea, namely that one can attribute distinct physical
properties to two objects located in very remote space-time regions. He believes that
quantum non-separability applies, even in such a situation. It is understandable that
Einstein was unwilling to abandon concepts that are the pillars of special and general
relativity (gravitation).
F-3. Bell's inequality
In 1964, more than thirty years after the publishing of the EPR argument, an
article by Bell shed an entirely new light on the question [73]. This article, in a way,
took up the EPR argument from the point at which its authors had left it. Taking at
their face value the existence of the EPR elements of reality, and using the same local
realism considerations, Bell showed that there is actually no way to complete quantum
mechanics without changing its predictions, at least in some cases. This means that
one must either accept that certain predictions of quantum mechanics are sometimes
incorrect, or abandon certain EPR hypotheses, however natural they may seem.
F-3-a. Bell's theorem
Following Bell's idea, let us assume thatrepresents the elements of reality
associated with the spins;is, actually, just a concise notation that could represent a
multiple component vector, so that the number of elements of reality contained inis
totally arbitrary. One can even include incomponents that play no particular role in
the problem; the only important hypothesis is thatmust contain enough information
to yield the results of all the possible spin measurements. For each pair of spins emitted
in the course of the experiment,is xed.
Another commonly used notation for the two measurement results isand,
not to be confused with the small lettersandused for the parameters of the two
measuring apparatus. As expected,andare functions not only of, but also of
the measurement parametersand. However, locality requires thathas no inuence
on result(since the distance between the two measurements' locations is arbitrarily
large); conversely,has no inuence on result. We shall note()and()the
corresponding functions, which can take on two values,+1or1. Figure
the experiment we are discussing.
To establish Bell's theorem, it is sucient to take into account only two directions
for each individual measurement; we shall then use the simpler notation:
() () (F-1)
and:
() () (F-2)
For each emitted pair of particles, asis xed, the four above numbers have well dened
values, which can only be1.
2208

F. ENTANGLEMENT, NON-LOCALITY, BELL'S THEOREM
Figure 3: Source S emits particles toward two measuring apparatus located far away
from each other, each being set up with its own measurement parameter, respectively
and; each apparatus yields a1result. The oval under the source symbolizes a
uctuating random process, which controls the particles' emission process, and hence
their properties. Correlations between the measured results are observed; they are due
to the common random properties the particles have acquired upon their emission by the
uctuating process.
Consider then the sum of products:
() = + + (F-3)
that can also be written as:
() =( ) +(+) (F-4)
If=, the above expression reduces to2 =2; if= , it reduces to
2=2. In both cases, we see that=2.
If we now take the average value
of()over a large number of emitted pairs
(average over), we get:
=++ (F-5)
where
denotes the average value overof the product ()(), and a
similar notation has been used for the3other terms. As each()value can only be
2, we necessarily have:
2
+2 (F-6)
This result is the so called BCHSH (Bell, Clauser, Horne, Shimony et Holt) form of Bell's
theorem. This inequality must be satised by all sorts of measurements yielding random
results, whatever mechanism creates the correlations, as long as the locality condition is
obeyed:does not depend on the measurement parameter, anddoes not depend
on.
This means that any theory that ts in the framework of local realism must
lead to predictions satisfying relation (F-6). Realism is necessary since we used in the
demonstration the concept of EPR elements of reality to introduce the functionsand
2209

CHAPTER XXI QUANTUM ENTANGLEMENT, MEASUREMENTS, BELL'S INEQUALITIES
; locality is also essential as it forbidsto depend onand, conversely,to depend
on.
The simplicity of this demonstration is such that the inequality may be expected
to remain valid in many situations. This actually happens any time the observed correla-
tions can be explained by uctuations having a past common cause; they are then referred
to as classical correlations. In such cases, each time the experiment is performed, this
common cause is xed, and the4numbers,,andtake on well-dened values
(even though they are, a priori, unknown) all equal to1; the numberis therefore
also well dened, equal to2or+2. Whatever thevalues found in a random series
ofmeasurements, it is mathematically impossible for the sum of thesevalues to
be greater than2or smaller than2. Consequently, the average value obtained by
dividing this sum bynecessarily obeys (F-6): the mere existence of these4numbers
is sucient to obtain the inequality.
F-3-b. Contradictions
It would seem natural for any reasonable physical theory to automatically lead
to predictions satisfying this inequality. Now, surprisingly enough, this is not the case
for quantum mechanics and, furthermore, this contradiction has been experimentally
conrmed.
. Contradictions with quantum mechanics predictions
Relations (B-10) allow computing the average value of the product of the1results
obtained in the measurements of the two spins along directions making an anglewith
each other. This average value is given by (we write
^
and
^
to emphasize that these
letters now denote operators, not numbers):
^
()
^
()=+++ + +=cos (F-7)
This expression is the quantum equivalent of the average value over thevariable of
the product of results()()in a theory with local realism. To get the quantum
equivalentof the combination of the four products of results as they appear in (F-3),
we must compute the same combination of average values of these products of results,
which yields:
=
^
()
^
()
^
()
^
()+
^
()
^
()+
^
()
^
()
=cos+ cos cos cos (F-8)
Imagine now that the four directions are in the same plane, and that the vectors,
arranged in the ordera,b,aandb, all make a45angle with the preceding vector
(Fig.); all the cosines are then equal to1
2, except forcos that is equal to
1
2. We then get =22; exchanging the directions ofbandb, we get
= 2
2. In both cases, BCHSH inequality (F-6) is violated by a factor2, i.e. by
more than 40 %. In spite of the seemingly simple cosine variation of expression (F-7),
we just showed that no theory with local reality is able to account for it, as this would
violate inequality (F-6). This means that the EPR-Bell argument leads to an important
quantitative contradiction with quantum mechanics, proving it to be a theory that does
not comply with local realism in the sense of EPR.
2210

F. ENTANGLEMENT, NON-LOCALITY, BELL'S THEOREM
Figure 4: Position of the four vecteurs,,andleading to a maximum violation of
BCHSH inequalities for two spin-12particles in a singlet state. These vectors dene the
spin components to be measured, alongorfor the spin on the left, and alongorfor
the spin on the right. This means that the entire experiment needs four dierent set-ups.
The only pair of vectors leading to a negative correlation between the two measurement
results is (,), as the angle between the corresponding directions is larger than 90.
How is it possible to encounter such a contradiction and how come such an appar-
ently awless argument does not apply to quantum mechanics? Several answers can be
given:
(i) Had Bohr been aware of Bell's theorem, he would very likely have rejected the
existence of the4preexisting numbers,,,. If these numbers do not exist, the
argument of
would have considered Bell's theorem as mathematically correct in probability theory, but
totally irrelevant in quantum mechanics, as being improper for the quantum description
of the experiment under study.
Even if he had accepted to reason about these numbers, as unknown variables to
be determined later as is often the case in algebra, would the inequality have survived?
The answer is no, still reasoning with Bohr's logic. As already mentioned in ,
Bohr's point of view is that the entire experiment must be considered as a whole. One
cannot distinguish two separate measurements that would be performed, each on one of
the particles: the only true measurement process concerns the ensemble of both particles
together. A fundamentally indeterminist and delocalized process occurs in the whole
region of space containing the entire experiment.
The functionsandthen both depend on both the measurement parameters,
and must be written as()and(); this immediately forces an abandon of locality.
Instead of the2numbersand, we now have4numbers,=(),=(),
as well as=()and =(); the same is true forand, which must
be replaced by4numbers. We now must deal with a total of8numbers instead of4. The
demonstration of the BCHSH inequality is then no longer possible and the contradiction
2211

CHAPTER XXI QUANTUM ENTANGLEMENT, MEASUREMENTS, BELL'S INEQUALITIES
disappears.
(ii) One may prefer a more local point of view for the measurement process, which
allows keeping the concept of a measurement on a single particle. To avoid the contradic-
tion with quantum mechanical predictions, one must then consider that it is meaningless
to attribute four well dened values,,,to each pair. Since only a maximum
of two of them can be measured in a given experiment, we should not be able to talk
about these four numbers or argue about them even as unknown quantities. A well know
phrase of Péres [75] very clearly sums up this point of view: unperformed experiments
have no results. Wheeler [76] expresses the same idea as he writes: no phenomenon is
a real phenomenon until it is an observed phenomenon..
. Contradictions with experimental results
The question was: does Bell's theorem allows pointing out very particular sit-
uations where quantum mechanics is no longer valid? Or, on the contrary, are the
predictions of quantum mechanics always valid, which immediately entails that certain
hypotheses leading to the inequalities must be abandoned? A great number of experi-
ments have been performed from 1972 on; they all conrmed the predictions of quantum
mechanics, measuring, sometimes with great precision [77], the violation of Bell's in-
equalities.
After a moment of doubt, it now seems well established that quantum mechanics
yields perfectly correct predictions, even in situations where it implies a violation of Bell's
inequalities. However plausible they might look, one must abandon at least one of the
hypotheses that led to these inequalities.
Conclusion
The concept of quantum entanglement is quite essential; it leads to situations where
certain types of correlations, totally impossible in classical physics, can be produced and
observed. These situations can occur even when the observations are performed in regions
of space arbitrarily remote from one another. A fundamental idea of quantum mechanics,
without any classical counterpart, is that the most precise description of a whole does
not necessarily entail an equivalently precise description of its parts. This means that
there exists no theory both local and realistic, for describing a system containing two
remote and entangled particles (it would contradict quantum mechanics).
Entanglement also plays an essential role in the measurement processes, and comes
into play at dierent levels: entanglement between the measured systemand the
measuring apparatus, betweenand the environment, between two environments
and, and so forth. We also discussed how entanglement determines the contrast of
the fringes observed in an interference experiment where a particle has to cross a plate
pierced with two holes.
In addition to these important aspects, entanglement also plays an essential role in
quantum computing: one seeks to take advantage of the parallel evolution of the various
entangled branches of the state vector to perform computations. This domain of research
has undergone intense development in recent years, but is too extensive to be treated in
the present volume. The reader may want to consult specialized books on the subject, as
for example that of D.Mermin [78]. Entanglement also plays a central role in quantum
cryptography, whose aim is to fabricate devices for secure quantum key distribution that
2212

F. ENTANGLEMENT, NON-LOCALITY, BELL'S THEOREM
cannot be spied on, as any eavesdropping is detectable; a review on this subject can be
found in the article by N.Gisin, G.Ribordy, W.Tittel and H.Zbinden [79].
There still remains the fact that, in the presence of entanglement, and in partic-
ular during a measurement process, the standard interpretation of quantum mechanics
may present some diculties. Schrödinger's evolution equation does not predict the
uniqueness of the measurement result observed in the macroscopic world. To obtain
this uniqueness in the framework of the theory, one can introduce an ad hoc postulate,
such as the von Neumann postulate of reduction of the state vector. It then raises the
question of where to set the border: when exactly should one stop using the continuous
evolution of Schrödinger's equation and impose the reduction of the state vector? How
can one reconcile the intrinsic irreversibility of this ad hoc postulate with the reversibility
of Schrödinger's equation?
Another open question concerns the status of the state vector. We have used it
throughout this book as a mathematical tool, good for computing probabilities, but what
does it really represent? Does it directly describe physical reality? Or does it simply
give information about physical reality? A number of quantum mechanics interpretations
have been proposed (see reference [68]) that discuss this fundamental diculty.
2213

COMPLEMENTS OF CHAPTER XXI, READER'S GUIDE
AXXI: DENSITY OPERATOR AND CORRELA-
TIONS; SEPARABILITY
AXXI: This complement introduces von Neu-
mann statistical entropy associated with a
density operator, discussing its properties and
establishing some important inequalities it must
satisfy. Also discussed are the dierences between
classical and quantum correlations (arising from
quantum entanglement eects). The concept of
quantum non-separability is introduced.
BXXI: GHZ STATES; ENTANGLEMENT SWAP-
PING
GHZ states provide an example of conict
between quantum mechanics and the usual
concept of local realism. The contradiction is
even stronger than for Bell's inequalities, as it is
expressed as an opposition in signs. Entanglement
swapping allows entangling particles without
them ever having to interact with each other.
CXXI: MEASUREMENT INDUCED RELATIVE
PHASE BETWEEN TWO CONDENSATES
When two Bose-Einstein condensates overlap,
their relative phase is a priori totally undeter-
mined. However, such a phase may appear, in-
duced by a measurement process sensitive to that
phase. As measurements proceed, this relative
phase will progressively acquire a more precise
value.
DXXI: EMERGENCE OF A RELATIVE PHASE
WITH SPIN CONDENSATES; MACROSCOPIC
NONLOCALITY AND THE EPR ARGUMENT
This complement is an extension of the previous
one, studying the case where the two conden-
sates are formed of particles with spins. The
same phenomenon occurs: the emergence of a
relative phase, but in a context where the EPR
argument is harder to refute because of the
macroscopic character of the measured quantities.
Furthermore, situations may arise where Bell's
inequalities are violated, which proves that the
measurement induced phase between the two
condensates is of a non-classical nature.
2215

DENSITY OPERATOR AND CORRELATIONS; SEPARABILITY
Complement AXXI
Density operator and correlations; separability
1 Von Neumann statistical entropy
1-a General denition
1-b Physical system composed of two subsystems
2 Dierences between classical and quantum correlations
2-a Two levels of correlations
2-b Quantum monogamy
3 Separability
3-a Separable density operator
3-b Two spins in a singlet state
In Chapter , we mainly considered global systems+described by a state
vector (pure state). This complement will examine what happens when these global
systems are described by a density operator (statistical mixture); we shall study the
correlations quantum mechanics predicts in that case between the two subsystemsand
. We start by introducing, in , the concept of statistical entropy, which yields a useful
measure of their degree of correlation. We then analyze, in , the dierences between
classical correlations (introduced at the probability level) and the quantum correlations
(which can arise from the coherent superposition of state vectors). Finally, in , we
will come back to the important concept of separability already introduced in
Chapter .
1. Von Neumann statistical entropy
The statistical entropy introduced by von Neumann permits, in a straightforward way,
to distinguish between a pure state and a statistical mixture; in the latter case, it also
yields a measurement of the statistical character of the information known about the
physical system. It is also a useful tool for studying in a quantitative way the amount of
correlation between two physical systems.
1-a. General denition
With any density operator, we associate a statistical entropyby the relation:
= Trln (1)
whereis the Boltzmann constant. Asis Hermitian, this operator can be diagonal-
ized. Notingits eigenvalues, we get:
= ln (2)
2217

COMPLEMENT A XXI
Since all theare included between0and1, we necessarily have:
0 (3)
where the equality occurs only ifhas one eigenvalue equal to1, all the others being equal
to zero. The entropy associated withis therefore equal to zero only if this operator
is a projector, and hence corresponds to a pure state. On the other hand, whenever
describes a statistical mixture,is dierent from zero. It takes on its maximum value
when the density operatorhas equal populations in all the system's accessible states,
i.e. if it is proportional to the identity operator in the state space. To prove this, let us
vary eachby an amount d, and impose a zero variation for the sum overof (2),
while maintaining constant the sum of allusing a Lagrangian multiplier. We then
get:
d d= [1 +Log+]d= 0 (4)
For this expression to be zero for any dmeans that all the ln, and hence all the
themselves, must be equal.
One can associate a concept of information, or rather a lack of information, with the
entropy. When the physical system is in a pure state, that state provides the maximum
information on the system, compatible with quantum mechanics. In this situation, there
is no lack of information and= 0. On the other hand, when the system is spread over
several pure states with comparable probabilities, a large value ofmeans that a lot of
information about the system is lacking.
Comment:
The statistical entropycharacterizes the populations of the density matrix(
plementIII), but not the corresponding eigenvectors. Moreover, the same density operator can
in general be obtained from several dierent statistical mixtures of pure states (cf.comment at
the end of 4-a III); the value of the entropy does not distinguish between these
dierent mixtures.
A statistical mixture of several density operators can only increase the entropy of
the system. Imagine, for example, that the density operatoris actually the combination
of several density operatorswith probabilities(all positive, and whose sum over
is equal to1), written as:
=
n
(5)
Notingthe entropies associated with:
= Tr ln (6)
we can write
1
:
(7)
1
This properties is often called entropy concavity.
2218

DENSITY OPERATOR AND CORRELATIONS; SEPARABILITY
Demonstration:
In XV, we showed that, whenandare two density operators
with traces equal to1, one always has the following inequality:
Trln Trln (8)
(the equality occurring only if=). We can then write:
=Trln= Tr ln
Tr ln=
1
(9)
which establishes relation (7).
1-b. Physical system composed of two subsystems
We now compare the entropies +,and associated, respectively, with
the total density operator+and with the partial density operatorsand. We
are going to show that+ = +when the systemsandare not entangled,
and that+ +otherwise.
. Pure state
Imagine rst that the total system is in a pure entangled state. We have seen that,
in that case, the two subsystemsandare not described by pure but by statistical
mixtures of states, so that:
= Tr ln 0
= Tr ln 0
(10)
As the entropy+associated with a pure state is zero, it follows that:
+ + (11)
(the equality corresponds to the special case where the pure state is a product,
without entanglement, and where the Schmidt rank is equal to1; see Chapter refch21,
).
We can also use the Schmidt decomposition for , which yields relations (C-7)
and (C-8) of Chapter , to get:
= ln= (12)
Both entropies of the two subsystems are thus always equal whenever the total system
is in a pure state.
2219

COMPLEMENT A XXI
. Statistical mixture
When the total system is described by a density operator+not necessarily
corresponding to a pure state, its entropy+may not be equal to zero. We are going
to show, however, that this entropy+always remains lower or equal to the sum of
the entropies of each subsystem, meaning that relation (11) remains valid in this more
general case; this property is referred to as the entropy subadditivity. The equality in
(11) is obtained solely in the case where+is a product:
+= (13)
which corresponds to the case of two subsystems, separately described by statistical
mixtures, while remaining uncorrelated. The dierence+ +yields an estimate
of the loss of precision between the quantum description of the total system, and the
separate quantum descriptions of the two subsystems.
Demonstration:
According to inequality (8), we can write:
Trln Trln( ) (14)
We note the eigenvectors ofwith eigenvalues, and the eigenvectors of
with eigenvalues. Let us now compute the trace of the right-hand side of (14) in
the basis of the tensor products of the eigenvectors of the two operators, with
respective eigenvaluesand; we get:
TrLog( )= Log( )
= Log( )
= Log() + Log() (15)
Let us now choose= +. The rst term on the right-hand side can be written as:
+ ln() = ln()
= ln
=Tr ln (16)
The second term on the right-hand side of (15) yields a similar expression, where
replaces. Finally, inequality (14) can be written as:
Tr +ln+ Tr ln+Tr ln (17)
and leads to (11). The equality occurs if and only if (14) becomes an equality, i.e. if
+is equal to the product (13).
2220

DENSITY OPERATOR AND CORRELATIONS; SEPARABILITY
2. Dierences between classical and quantum correlations
Quantum mechanics oers more possibilities than classical physics for describing corre-
lations between physical systems. We now briey discuss such examples.
2-a. Two levels of correlations
The concept of correlation is not, intrinsically, a quantum notion, and it is well
known in classical physics. It is then based on probabilistic calculations, which results
in the linear weighting of a certain number of possibilities. In this classical context, one
introduces a distribution yielding the probability for the rst system to occupy a certain
state, and the second system, another state; the two systems are correlated when this
distribution is not a simple product. If, on the other hand, the distribution turns out to
be a product, the two systems are not correlated; a measurement on one of the systems
does not change the information about the other. In particular, this is what happens if
the states of the two systems, and consequently that of the total system, are perfectly
well dened, a case where the notion of correlation between the two systems becomes
irrelevant. This means that the notion of correlation between two classical systems is
closely linked to an imperfect denition of the state of the total system.
In quantum mechanics, things are totally dierent. To begin with, even if a physical
system is perfectly well dened by a state vector, many of its physical properties are not
so precisely dened: during several realizations of the experiment, their measurement can
provide uctuating results. These results can nevertheless be correlated: as an example,
we saw in 12particles
are completely indeterminate but perfectly correlated. Such correlations appear directly
at the level of the state vector itself, which can be written as a linear superposition of
states where the spins have various orientations. The correlations are therefore related
to the the quantum mechanical superposition principle; this is totally dierent from
the combinations of probabilities, which are quadratic functions of that state vector.
Letting correlations appear directly at the probability amplitude level, one has access to
a level that is, in a way, a step ahead of the linear weighting of classical probabilities,
and maintains the possibility of quantum interference eects. Note, however, that the
existence of this level of combinations does not exclude classical probabilities from coming
into play. One can also assume, in quantum mechanics, that the state of the total system
is only known in a probabilistic way, so that the two probability levels may coexist. To
sum up, it is clear that the concept of quantum correlations covers many more possibilities
than correlations in classical physics
2
.
2-b. Quantum monogamy
Another purely quantum property is that, if a physical systemis strongly entan-
gled with a physical system, it cannot be strongly entangled with another system.
Such a property does not have any equivalent in classical physics, where, obviously, noth-
ing prevents correlating a third systemwith two othersand, all the while keeping
2
We shall introduce, in , a criterion (negativity of the coecients of the total density operator
expansion into a sum of products) for conrming the quantum nature of the correlations between two
subsystems.
2221

COMPLEMENT A XXI
their initial correlation. This quantum property is often referred to as entanglement
monogamy.
Let us assume, for example, that two spins are in a state of the same type as the
singlet state (B-1) of Chapter :
=
1
2
: +;:+ :;: + (18)
(the singlet state is obtained for=). How can we add an additional spin without
destroying the correlation between the rst two? One could imagine the three spin state
to be written as:
= :=
1
2
: +;:;:+ :;: +;: (19)
where is any normalized state for the third spin. This ket obviously conserves the
same entanglement between spinsandas in state (18), but the third spin is then
totally uncorrelated with the rst two.
Another possibility is to choose as a state vector:
=
1
2
: +;:;:1+ :;: +;:2 (20)
The density operatordescribing spinsandis obtained by taking the partial
trace (ComplementIII, ):
=Tr (21)
Computing the matrix elements of this partial trace shows that:
=
1
2
11 : +;: : +;:
+ 22 :;: +:;: +
+ 12 :;: +: +;:
+ 21 : +;: :;: + (22)
One can then distinguish several cases:
If1=2, we nd again (19), and the third spin is not entangled with the
rst two. The density operatoris then written:
=: +;: : +;:+ :;: +:;: +
+ :;: +: +;:+ : +;: :;: + (23)
which is simply the projector onto state (18); it conserves all the entanglement of spins
and.
The opposite case is when1and2are orthogonal, so that becomes
a so called GHZ state (Greenberger, Horne and Zeilinger;cf.ComplementXXI:
=
1
2
: +;:;: ++ :;: +;: (24)
2222

DENSITY OPERATOR AND CORRELATIONS; SEPARABILITY
where, when one goes from the rst component to the second, the three spins switch
from one state to the orthogonal one. The second line in (22) then cancels out and the
partial tracebecomes:
=
1
2
: +;: : +;:+:;: +:;: + (25)
which is a statistical mixture of two possibilities, with probabilities12: the two spins are
either in the state: +;:, or in the state:;: +. The quantum coherences
between these two states (terms dependent on the phase) have totally disappeared.
The correlation between the two spinsandis then of a classical nature
3
, and no
entanglement comes into play.
In the intermediate situation where1and2are neither parallel nor orthog-
onal, we see from (22) that a certain coherence remains (non-diagonal elements). The
more parallel1and2are, the more the partial density operator resembles that of
the two initial spins which remain entangled, whereas the third one becomes less and
less entangled with the rst two; conversely, the more orthogonal they are, the more the
initial spins lose their correlation, which becomes entirely transmitted to the three spin
level.
This is actually a general property: when two physical systems are maximally
entangled, a principle of mutual exclusion makes it impossible to entangle them with a
third system. Mathematically, this property is expressed by the Coman-Kundu-Wooters
inequality [80].
3. Separability
From Bohr's point of view ( ), one must give up the notion of
separability. Even when two physical subsystems are well separated in space, it does
not infer that they have, each of them separately, their own physical properties (as
EPR assumed); only the total system, including the measuring apparatus, can have such
properties. On the other hand, we saw in
ways for introducing correlations between two systems: a classical way (by assuming they
have given probabilities to be found in such or such individual correlated states), and a
quantum way (by assuming entanglement directly at the level of a common state vector).
We also know that, even though there are situations where quantum mechanics predicts
violations of Bell' inequalities and hence of local realism, there are many others where
its predictions obey these inequalities; a violation is, in a way, the signature of an ultra-
quantum situation. It is thus interesting to look for a criterion allowing a distinction
between these two types of correlations.
3-a. Separable density operator
Consider a total system described by a density operator+and composed of
two subsystemsand. Let us assume that+can be expanded as a series of
density operatorsand pertaining to each of the two subsystems, with real and
3
The density operator is separable in a sense that will be dened in , and therefore cannot lead
to violations of Bell's inequalities.
2223

COMPLEMENT A XXI
positivecoecients whose sum equals to one, and can be assimilated to probabilities:
+= (26)
with:
0 1and = 1 (27)
Intuitively, one can guess that the correlations contained in+must then be
of a classical nature. The total system is, with a probability, described by a density
operator that is a product, without correlations, of density operators each describing one
of the subsystems. The correlations between these subsystems are therefore introduced
in a classical way, even if nothing prevents each subsystem from exhibiting strongly
quantum individual properties.
Any density operator that can be decomposed as in (26) with positivecoef-
cients is, by denition, said to be separable [81,]. On the other hand, if any
decomposition of+such as (26) necessarily includescoecients that are not real
and positive, the density operator+contains quantum entanglement and is said to
be non-separable. When the total system+is separable, correlation measurements
between physical properties of the two subsystemsandcan never lead to violations
of Bell's inequalities. These violations are thus a sure sign of the non-separability of the
density operator.
Demonstration
To show this, let us assume we perform two simultaneous measurements on the systems
and, the rst one depending on the measurement parameter, and the second, on
the measurement parameter. We note()the projector acting in the state space
ofand corresponding to the measurement result(this projector is the sum of the
projectors onto the eigenvectors associated with that measurement). In a similar way,
we note()the projector in the state space ofcorresponding to the measurement
result. When the total system is described by the density operator (26), the joint
probability of obtaining both resultsand is:
( ) = () () (28)
with:
() =Tr ()
() =Tr () (29)
As all the numbers appearing on the right-hand side of (28) are positive, this equality
has a natural interpretation in classical physics, which is the framework of our present
argument. We are dealing with two levels of probabilities. At one level, the total system
is prepared, with probability, in a state where the two subsystems are uncorrelated.
At a second level, for each value of, the individual states of the subsystems are only
known in a statistical way via the probability()of a result, and the probability
()of a result.
2224

DENSITY OPERATOR AND CORRELATIONS; SEPARABILITY
We now show that if a relation of the type (28) is veried for all the measurement
parametersand, with any positive probabilities()and (), and with
positive values for all the, Bell's inequalities are always satised. Let's assume both
resultsand can take on the values1.
To start with, we assume that the physical properties of each pair of systemsand
depend on a classical random variable; this variable takes on a series of values,
corresponding to each term in the summation overappearing in (28), each with the
probability. This means that this summation overcan be interpreted as an average
value over the random variable.
We then assume that the properties of the classical systemdepend on a dierent
random variable, which determines the result of the measurement performed on.
As an example, one can imagine thatis regularly distributed on the segment[01];
resultis a function of, and takes the value+1on a fraction of the segment of length
(= +1), and the value1on the rest of the segment. This function models the
probability written on the rst line of (29); the measurement result is thus a function of
, of the measurement parameter, and of(which replaces). Finally, we introduce
the random variable, which determines the result of the measurement performed on
, with a distribution modeling in a similar way the probability written on the second
line of (29), for any value ofand any choice of the measurement parameter.
If we now regroup the three variables,and, as being the three components of
a single variable, we reproduce the exact same hypotheses stated at the beginning of
: the measurement results are functions, the rst one ofand
and the other one ofand. The same reasoning then leads to Bell's inequalities. Note
that, at no point in this classical reasoning, did we have to consider the ensemble+
as a whole; it was thus to be expected that Bell's inequality would be established in this
case.
3-b. Two spins in a singlet state
Let's go back to the example of two spin-12particles in a singlet state:
=
1
2
+ + (30)
In the basis of the4kets++,+,+, taken in that order, the matrix
representing the density operator+is written:
(+) = =
1
2
0 0 0 0
0 11 0
01 1 0
0 0 0 0
(31)
This matrix density(+)has non-diagonal elements between states+ and+,
as, for example:
1 =+ + + (32)
To obtain such a non-diagonal term by a sum of products such as (26), will require:
1 = + + (33)
2225

COMPLEMENT A XXI
This demands introducing at least one termthat contains partial density operators
and, both having non-diagonal elements. Now each of these two operators is a
positive-denite operator. This means, forfor example, that it must have popula-
tions (diagonal matrix elements)+ +and in the two individual spin
states, and the same is true for. The correspondingterm will necessarily intro-
duce in(+)populations in the4states++,+,+, ; it will then
be impossible to cancel those populations by adding other products of density operators
(whose populations are positive) with positivecoecients. Consequently, this den-
sity operator(+)is non-separable, and this is why it can lead to violations of Bell's
inequalities.
2226

GHZ STATES, ENTANGLEMENT SWAPPING
Complement BXXI
GHZ states, entanglement swapping
1 Sign contradiction in a GHZ state
1-a Quantum calculation
1-b Reasoning in the local realism framework
1-c Discussion; contextuality
2 Entanglement swapping
2-a General scheme
2-b Discussion
Greenberger, Horne and Zeilinger (GHZ) showed in 1989 [83,] that violations
of local realism even more spectacular than violations of Bell's inequalities could be
observed on systems containing more than two correlated particles. These violations
involve a contradiction in sign (and hence a violation of 100 %) for perfect correlations
between measurement results, as opposed to inequalities violated by 40% for imperfect
correlations. Observation of these violations requires creating an initial entanglement
between three particles or more, as will be discussed in . Another example, involving
the entanglement of more than two particles, is the entanglement swapping method,
explained in . This method highlights a surprising property of entanglement: the
possibility of entangling together two quantum systems, without them ever having to
interact with each other.
1. Sign contradiction in a GHZ state
We consider a system composed of three spin-12particles, as it is the simplest case for
explaining how the GHZ contradiction can appear.
1-a. Quantum calculation
The three-spin system is described by the normalized quantum state:
=
1
2
++++ (1)
In this equality, the statessymbolize the eigenstates of the spin components along the
axis in an reference frame; to simplify the notation of the three particle ket, the
spins are not numbered: the rst sign corresponds to the state of the rst spin, the second
to that of the second, and similarly for the third spin. The numberstands for either
+1, or1. We now look for the quantum probabilities of measurement results of the
components of each of the spins123of the three particles along two possible directions:
either along thedirection, or along the perpendiculardirection (Fig.).
We start with the measurement of the product123. As we now show,
is an eigenvector of this operator product, with eigenvalue, which means that the
2227

COMPLEMENT B XXI
measurement result is certain. The action of the rst operator
1
is written:
3 =
1
2
[+(3) +(3)]
=
1
22
[+(3) +(3)+++]
=
1
2
[ ++++] (2)
The second operator then yields:
23 =
1
2
[+(2) (2)]3
=
1
2
[++ + ] (3)
Finally, the product of the three operators yields:
123 =
1
2
[+(1) (1)]23
=
1
2
[++++ ]
= (4)
The probability of observing the resultis:
(123= ) = 1 (5)
whereas the probability(123=+)of observing the other result is zero.
As the three spins play the same role, it is clear that is also an eigenvector
of the two operator products123and123, with eigenvalues. The
corresponding probabilities are therefore:
(123= ) = 1
(123= ) = 1
(6)
It is thus certain that the three products take on the value.
We now consider the measurement result of the product of the three spin compo-
nents along the sameaxis. We use again (2), but (3) is now replaced by:
23 =
1
2
[+(2) +(2)]3
=
1
2
[++++ ] (7)
and (4) by:
123 =
1
2
[+(1) +(1)]23
=
1
2
[++++ ] (8)
1
The Pauli matrices are dened in ComplementIV; the operators= obey the relation
= 2.
2228

GHZ STATES, ENTANGLEMENT SWAPPING
Figure 1: Schematic set-up for observing the GHZ contradictions. Three spins, initially
in state written in (1), are measured in three dierent regions of space. In each of
these regions a measuring apparatus is placed, with a setting enabling the local observer
to choose between two possible spin component measurements, either along, or along
. Whatever choices the three observers make, the results given by the three apparatus
are=1,=1and=1.
This shows that is also an eigenvector of the operator product123, but this
time with the eigenvalue+. It follows that:
(123=+) = 1 (9)
One can conclude, with certainty, that the measurement result of this product will be
equal to+.
Quantum mechanical measurement of a product of commuting operators:
Three operators such as1,2and3, acting on dierent spins, commute with each
other; they form a CSCO (Complete Set of Commuting Observables) in the state space
of the three spins. One can thus build a basis of eigenvectors123common to the
three operators, labeled by the eigenvalue1=1of1, the eigenvalue2=1of2
and the eigenvalue3=1of3. Any vector can be decomposed onto this basis
as:
=
123
(123)123 (10)
The action of the operator product123on any ket is therefore to simply multiply
each of its component(123)by the product123. Now the vector written in
(1) is an eigenvector of that operator product, with the eigenvalue. The uniqueness of
decomposition (10) then means that the only non-zero(123)coecients are those
2229

COMPLEMENT B XXI
for which:
123= (11)
Suppose we measure, in a rst experiment, the component1of the rst spin. The
result1=1is random. After the measurement, the projection postulate leads to a
state that depends on this result, obtained by keeping in (1) only half of the components
those that correspond to the observed1value. The components of the projected
state vector still obey relation (11), where1is now xed. Similarly, if we continue the
experiment and measure2for the second spin, the result2=1is also random, but
the components of the new projected state vector still obey that same relation. As now
1and2are both known, the same is true for3, whose value is determined by the rst
two measurements.
To sum up, the results observed for each spin component measurement uctuate from
one experiment to another, but these uctuations are correlated and the product of the
three results remain constant. One can obviously do the same analysis for the other sets
of operators considered above,1,2and3for example.
1-b. Reasoning in the local realism framework
Let us leave, for a moment, standard quantum mechanics and examine what a local
realistic theory (in the EPR sense of these words) would predict in such a situation. As
we are in a particularly simple case where the initial quantum state is an eigenvector of all
the observables coming into play (all the results are certain), one could expect nothing
particular to happen. On the contrary, we now show that a complete contradiction
appears between local realism and the predictions of quantum mechanics.
The local realism argument we present is a direct generalization of that used to
obtain Bell's inequalities in . We rst notice that the perfect
correlations imply that the measurement result of a spin component along(or)
of any particle can be deduced from the results of measurements performed on other
particles, at arbitrarily large distances. The EPR argument then requires the existence
of elements of reality corresponding to these two component directions, that we shall
note =1for the rst spin,=1for the second, and nally=1for the
third. According to the EPR argument, for each experiment (i.e. for each emission of
a group of three particles), these six numbers have well determined values, even though
they are a priori unknown. These numbers are simply the results that shall be obtained,
should measurements be performed later on. As an example, a measurement on the rst
spin will necessarily yieldif the chosen analysis direction is along, orif it is
along, independently of the type of measurements performed on the other two spins.
To have an agreement with the three equalities (5) and (6) imposes that:
=
=
=
(12)
Now, in the logic of local realism, the same values of,andcan also be used for an
experiment where the three spin components are measured along the samedirection:
the result should simply be the product . As the squares of the numbers
2
,
2230

GHZ STATES, ENTANGLEMENT SWAPPING
etc., are always equal to+1, we can obtain that product by multiplying the lines of (12),
which yields:
= (13)
That is where the contradiction shows up: equality (9) predicts that the measurement of
123must always yield the result+, which has the opposite sign! There could not
be a greater contradiction between local realism and quantum mechanical predictions.
1-c. Discussion; contextuality
Compared with the violations of Bell's inequality, the GHZ contradiction seems far
more spectacular, since a 100 % contradiction is obtained with 100 % certainty. From an
experimental point of view, however, the necessity to bring into play three remote and
entangled particles is a complex challenge.
To easily identify the three spins (deciding which measurement pertains to spin
noted, to spin noted, and to spin noted), and to be sure the three measurements
are performed far from each other, let us assume the spins each occupy a dierent region
of space. When the spatial variables are taken into account, the ket (1) can be rewritten
more explicitly in the form:
=
1
2
1 :2 :3 : 1 : +; 2 : +; 3 : ++1 :; 2 :; 3 : (14)
where are three orbital states whose wave functions do not overlap. They can for
example be entirely localized in separate boxes where the measurements are performed.
One then assumes that none of the particles will be left unmeasured and that each of
them is separately observed. The experimental procedure is to rst choose, for each box,
a component or, then perform the three corresponding measurements in each
box, obtain the three results, and , and nally compute their product.
A rst necessary verication is to perform a large number of experiments and
measure successively the three products, and , to be sure that
the perfect correlations predicted by quantum mechanics are indeed observed (it is an
essential step for the EPR argument, which infers from it the existence of6separate el-
ements of reality). One then measures the product and, if quantum mechanics
is right, one will observe a sign opposite to the EPR prediction. This means that the
value obtained in a measurement of1(for example) depends on theorcompo-
nents measured on the other spins; this remains true even if the corresponding operators
commute with1. This leads us to the general concept of quantum contextuality: in
an experiment where several commuting observables are measured, one must take into
account, according to Bohr's prescription, the ensemble of the experimental set-up (the
whole context of the system to be measured); it would not be correct to reason as if these
measurements were independent processes.
Experimental tests of GHZ equalities have been performed [85,]. These ex-
periments require three particles to be placed in the quantum state (14), which is not
an easy task. Nevertheless, using elaborate quantum optics techniques, the correctness
of quantum mechanical predictions has been veried in such a case, with experiments
involving 3 or 4 entangled photons, as well as with NMR (Nuclear Magnetic Resonance)
techniques.
2231

COMPLEMENT B XXI
2. Entanglement swapping
We now describe the entanglement swapping method, which enables entangling parti-
cles coming from independent sources (i.e. having no common past) through a quantum
measurement process.
2-a. General scheme
Consider two sources12and34each creating a pair of entangled photons (Fig.
2). The rst one creates a photon with momentumk1and another one with momentum
k2, whose polarizations are entangled in states(horizontal polarization, in the plane
of the gure) and(vertical polarization, perpendicular to the plane of the gure). In
a similar way, the second source creates a photon with momentumk3and another one
with momentumk4, whose polarizations are entangled in the same way. The initial state
describing the two pairs is the tensor product of two states, each describing two particles:
=
1
2
[k1;k2+k1;k2][k3;k4+k3;k4] (15)
While the two photons emitted by a given source are strongly entangled, no entanglement
exists between the two pairs of photons, emitted by each of the two sources. It is useful
to introduce the four dierent states pertaining to the wave vectorsk,k:

()
=
1
2
[k;k +k;k ]

()
=
1
2
[k;k +k;k ] (16)
with, here again,=1. These states (often called Bell states in the literature, hence
the superscript) form an orthonormal basis of the state space associated with particles
and. One can show that (the computation is straightforward but a bit tedious and
will not be detailed here):

14
(+1)

23
(+1)

14
(1)

23
(1)
=;;;+;;; (17)
and that:

14
(+1)

23
(+1)

14
(1)

23
(1)
=;;;+;;; (18)
(to simplify the notation, it is implicitly assumed, on the right-hand side of both equa-
tions, that the order of the particle's momenta is alwaysk1,k2,k3andk4). We can
then write state (15) in the form:
=
1
2

14
(+1)

23
(+1)

14
(1)

23
(1)
+
+
14
(+1)

23
(+1)

14
(1)

23
(1)
(19)
Figure
the particles with momentak2andk3undergo a measurement in which they interfere.
This is achieved by sending these two particles to a beam splitter BS, followed by two
2232

GHZ STATES, ENTANGLEMENT SWAPPING1 2 3
4
BS
Figure 2: Schematic diagram of the entanglement swapping method. Two sources S12
and S34each emit a pair of entangled particles, with wave vectorsk1andk2for the rst
one,k3andk4for the second. These sources are independent. A beam splitter BS is
inserted in the path of particlesk2andk3; it is followed by two detectors Dand Dthat
measure the particle number in each of the exit channelsand. This measurement has
the eect of projecting the state vector, hence bringing the two particlesk1andk4into
a totally entangled state, even though these particles have never interacted.
detectors Dand Dmeasuring which exit channel were followed by the particles. If the
two particles exit through two dierent channels, the corresponding eigenvector for this
measurement result is the state
23
(1)
; this is because, as we show below, the three
other states
23
(+1)
,
23
(+1)
and
23
(1)
correspond to situations where the two
particles exit through the same channel. The measurement thus projects state (19) onto
the last of its four components. The net result is that if the two particles with momenta
k2andk3exit through dierent channels (which happens one out of four times), the
two particles with momentak1andk4reach the state
14
(1)
. This means that the
two non-observed particles reach a totally entangled state though they can be arbitrarily
far from each other. It is worth noting that the initial entanglement concerns the two
particlesk1andk2, and, separately, the two particlesk3andk4. Performing a suitable
measurement on a particle of each pair, one projects the two remaining particles into a
strongly entangled state, even though they never interacted at any stage of the process.
Demonstration:
Let us show that
23
(1)
is an initial state of two interfering particles that will lead to
their exiting through dierent channels. We introduce for that purpose the two creation
operators
k2
and
k2
in the individual state with wave vectork2and polarization
or, as well as the two operators
k3
and
k3
in the individual state with wave
vectork3and polarizationor. The state
23
(1)
can be written:

23
(1)
=
1
2
k2 k3 k2 k3
0 (20)
As the particles go through the beam splitter, their polarizations are not modied, but
2233

COMPLEMENT B XXI
their wave vectors are. In terms of creation operators, this leads to the unitary transfor-
mations:
k2
1
2
k2
+
k3
k3
1
2
k2
+
k3
(21)
where thefactors come from the phase change in a light beam as it undergoes internal
reection. Similar equalities are obtained for thepolarization, so that:
k2 k3 k2 k3
1
2
k2
+
k3 k2
+
k3
k2
+
k3 k2
+
k3
(22)
As creation operators in dierent modes commute with each other, this operator is equal
to:
k2 k3 k2 k3
(23)
so that state
23
(1)
is transformed, after the beam splitter, into:

23
(1)
1
2
k2 k3 k2 k3
0 (24)
This shows that if the state before crossing the beam splitter is
23
(1)
, the two
photons are still in two dierent exit channels after the crossing.
If now the state before crossing the beam splitter is
23
(+1)
, we must replace (22) by:
k2 k3
+
k2 k3
1
2
k2
+
k3 k2
+
k3
+
k2
+
k3 k2
+
k3
=
k2 k2
+
k3 k3
(25)
which means that the two photons always exit the beam splitter through the same chan-
nel. In the same way, for the state
23
(1)
, we get the operator:
k2 k3 k2 k3
1
2
k2
+
k3 k2
+
k3
k2
+
k3 k2
+
k3
=
k2
2
+
k3
2
k2
2
k3
2
(26)
It shows again that for each term the photons exit through the same channel. The state

23
(1)
is therefore the only one that will lead to the photons exiting through dierent
channels.
2234

GHZ STATES, ENTANGLEMENT SWAPPING
2-b. Discussion
In classical physics, it is also possible to obtain correlations between two objects
initially totally independent, by sorting objects with which each of them is correlated. To
underline the fundamental dierence with entanglement swapping, we now discuss such
a classical experiment. Imagine that two independent sources emit pairs of correlated
objects, numbered1and2for the rst source,3and4for the second, as in Figure.
Each time the experiment is performed, each source emits two classical objects sharing a
common property (such as, for example, the same color, or opposite angular momenta,
etc.). The two sources are nevertheless totally uncorrelated (the objects emitted by two
dierent sources present no correlations between their colors, their angular momenta,
etc.). If, however, one selects particular experiments where particles2and3present a
certain correlation (for example identical colors, or else parallel or antiparallel angular
momenta, etc.), it is clear that the particles1and4will also be correlated, even if they
never interacted in the past and if they are very far apart. It is a mere consequence of the
selection performed in a classical probability distribution, and could be called classical
correlation swapping.
Note, however, that this selection remains purely classical; no entanglement can
be produced by this method. Should a Bell experiment be performed on the objects1
and4, the correlations obtained will necessarily obey Bell's inequalities since we are in a
classical physics context. The entanglement swapping method, however, allows creating
by selection a true entanglement leading to strong violations of Bell's inequalities. This
method is a way of producing stronger correlations than classical correlation swapping,
and has been demonstrated in several experiments [87,].
Conclusion
The two examples we discussed illustrate the variety of situations where quantum entan-
glement produces signicant physical eects, even when the entangled quantum systems
are arbitrarily far from each other. In each situation, it is essential to follow the basic
rules of quantum mechanics, and perform the computations with a global state vector,
including all the physical systems under study. Any attempt to perform separate compu-
tations in dierent regions of space, and then add correlations using classical probability
calculations, will necessarily lead to predictions ignoring numerous non-local quantum
eects, in contradiction with experimental results.
2235

MEASUREMENT INDUCED RELATIVE PHASE BETWEEN TWO CONDENSATES
Complement CXXI
Measurement induced relative phase between two condensates
1 Probabilities of single, double, etc. position measurements
1-a Single measurement (one particle)
1-b Double measurement (two particles)
1-c Generalization: measurement of any number of positions
2 Measurement induced enhancement of entanglement
2-a Measuring the single density(1). . . . . . . . . . . . . . .
2-b Entanglement between the two modes after the rst detection
2-c Measuring the double density(21). . . . . . . . . . . .
2-d Discussion
3 Detection of a large number of particles
3-a Probability of a multiple detection sequence
3-b Discussion; emergence of a relative phase
Introduction
Referring to Bose-Einstein condensation (ComplementXV, a system of identical bosons,
all occupying the same individual stateis called a condensate. It is described by
a Fock state such as the one given by (A-17) of Chapter, where all the occupation
numbers are zero, except for, whose valuecan be very large:
:=
1
!
0 (1)
The operatoris the creation operator of a particle in the individual state, and
0is the vacuum state (for which all the occupation numbers are zero). In a similar
way, a double condensate is described by a Fock state where1particles are in the
individual stateand2particles in the individual state; its normalized state is
written as:
0= :1;:2=
1
1!2!
1
2
0 (2)
We shall focus on the case where the individual statesandare states with well
dened but opposite wave vectors:
0=+:1;:2=
1
1!2!
+
1 2
0 (3)
In such a state, while the occupation numbers are perfectly well dened, the relative
phase between the two condensates is completely undetermined; we will conrm this
2237

COMPLEMENT C XXI
Figure 1: The left-hand side of the gure represents two groups of particles prepared in-
dependently. The rst one is composed of a large number of particles,1, all in the same
individual state with momentum+along theaxis, and propagating towards the right;
the second group includes2particles, in the other individual state with oppositemo-
mentum, propagating towards the left. Each of these groups of independent particles is
in a condensate. The right-hand side of the gure shows that, after a certain time, the
two condensates overlap in space; this allows measuring the positions of the particles in
the overlap region. For clarity, the computations are limited to one dimension, taking
into account only thecoordinate.
The rst position measurement is totally random, but as measurements continue, there
appear a periodic bunching of the observed positions, progressively forming a sharper
fringe pattern. These fringes result from the emergence of a relative phase between the
two condensates, which can only be a consequence of the position measurements, as it
was totally absent at the beginning of the experiment.
If the whole process is repeated from the beginning, fringes appear with a position generally
dierent from the rst experiment: the phase appearing in each new experiment is totally
independent of the one observed in previous experiments.
later () by showing that measuring the position of a single particle with such a state
does not lead to any observable interference fringes.
Now, recent experiments [89] have shown that when the positions of many particles
are measured, interference fringes can indeed be observed in the region where the two
condensates overlap (Figure). This remains true even if the condensates have been
created in a totally independent way. This fringe pattern corresponds to a well dened
value of the relative phase of the two condensates; one may then wonder about the
origin of this observed phase. The object of this complement is to study the mechanism
responsible for the emergence of this relative phase. We will show that it results from
the successive detections of particles, which progressively modies the initial state: as
more and more particles are detected, it produces a progressively increasing entanglement
between the two condensates, dening their relative phase in a more and more precise
way.
2238

MEASUREMENT INDUCED RELATIVE PHASE BETWEEN TWO CONDENSATES
During the course of one experiment, the position of the fringes is determined.
However, should one repeat the experiment, preparing the condensates in exactly the
same way, a new relative phase will progressively appear during the successive particle
detections; its value is, in general, completely dierent from the one previously obtained.
This means that if one averages observations over a large number of independent succes-
sive experiments, the fringes will be blurred and eventually completely disappear. The
emergence of the phase is clearly observable only in the course of one specic experiment.
We rst compute, in , the probability of measurements concerning the positions
of one, two, and more particles; we will show that these probabilities are proportional to
spatio-temporal correlation functions of eld operators of the various particles. Starting,
in , from two condensates in an initial state described by a simple juxtaposition of
two Fock states, we will see how the successive particles' position measurements create
an increasing entanglement between the two condensates. A more general study of the
system's evolution is presented in , showing, in particular, how this growing entan-
glement leads to a better and better denition of the relative phase between the two
condensates. The computations presented in this complement are limited to the case
where the number of measured positions remains small compared to the total number
of particles in the condensates. ComplementXXI, will go a step further and relax this
hypothesis.
1. Probabilities of single, double, etc. position measurements
As we start the successive measurements of the particles' positions, we begin by com-
puting the probability of nding a rst particle in an interval
1
of innitesimal width
around position=1, then a second particle in the interval of widtharound posi-
tion=2, etc. The computations we present here are valid for any state0of the
identical particle system. They are, actually, the equivalent of those encountered in the
general study of correlation functions in ; nevertheless, we will go
through them again in the specic context of the present complement. The results will
be applied, in , to the particular case where0is a double Fock state.
1-a. Single measurement (one particle)
With a measurement of the position yielding a result included in the interval
1= 1

2
1+

2
, we can associate the Hermitian operator:
(1) =
1+

2
1

2
d () () (4)
where ()is the eld operator destroying a particle at point, and ()its Her-
mitian conjugate, creating a particle. The average value of(1)yields the average
particle number in the interval1. In what follows, we shall, most of the time, assume
thatis small enough compared to the other dimensions of the problem to justify the
approximation:
(1) (1) (1) (5)
1
To keep the notation simple, we consider a one-dimensional problem and note1,2,..,, the
particles' positions. Generalizing to three dimensions only requires replacing all theby the vectorsr.
2239

COMPLEMENT C XXI
Operator(1)is a symmetric one-particle operator of the type described in
relation (B-1) in Chapter. It can also be written as:
(1) =
=1
1+

2
1

2
d : : (6)
which is the sum over all the= 1,2, ..,particles of the projectors into the interval
1of the positions of each of them. As all these projectors commute with each other,
and since they each have eigenvalues1and0, the eigenvalues of(1)are equal to0,
1,2, ... Now, ifis small enough, there can be no more than one particle in the
interval1; this means that the only accessible eigenvalues are0and1, so that(1)
becomes the projector associated with the measurement of a particle's presence in the
interval1:
[(1)]
2
=(1) if0 (7)
Suppose now the system is in state0. The probability
1
of nding a particle
in the innitesimal interval1of lengthis:
1
(1) =0(1)0
= 0 (1) (1)0 (8)
Right after the detection of this rst particle, the system is now, according to relation
(E-39) of Chapter

0=
1
1
(1)
(1)0 (9)
1-b. Double measurement (two particles)
Let us now focus on the probability
12
(21)of detecting a rst particle in
an interval of widtharound point1, then a second one in an intervalaround point
2; we assume the system does not have time to evolve in between the two measurements.
We start by computing the conditional probability
2
12
(21)of detecting a
particle in the interval2

2
2+

2
noted2, knowing that a particle has been
detected in the interval1

2
1+

2
already noted1. This probability equals:
12(21) =
0(2)
0
=
1
1(1)
0(1)(2)(1)0 (10)
where, in the second line, we have used (9); the projector(2)may be obtained by
replacing1by2in expression (4). We assume that the two detection intervals do
2
Note the dierent notation used for the conditional probability
12
(21), with a fraction bar
between the variables, and the simple probability (a priori probability)
12
(21)of obtaining the
two results. These two probabilities are related by expression (14).
2240

MEASUREMENT INDUCED RELATIVE PHASE BETWEEN TWO CONDENSATES
not overlap in space, so that all the operators appearing in(1)commute with those
appearing in(2). We then have, taking (7) into account:
(1)(2)(1) = [(1)]
2
(2)
=(1)(2) (11)
Ifis small enough, we can replace(1)and(2)by their expressions (5); this
leads to:
(1)(2) =
2
(1) (1) (2) (2)
=
2
(1) (2) (2) (1) (12)
where, in the second line, we again used the fact that eld operators dened in non-
overlapping regions of space commute with each other. Inserting this result in (10), we
get:
12(21) =

2
1(1)
0 (1) (2) (2) (1)0 (13)
Now, the probability of detecting a particle at point1, then a particle at point
2, is the product of the probability
1(1)of detecting a particle at point1and the
conditional probability
12(21)of detecting a particle at point2knowing that a
particle has been detected at point1:
12
(21) =
1
(1)
12
(21) (14)
Taking (13) into account, this leads to:
12(21) =
2
0 (1) (2) (2) (1)0
=2121 (15)
where21is the non-normalized state:
21= (2) (1)0 (16)
The probability we are looking for is simply the squared norm of the ket obtained by
destroying in the initial state a particle at point1, and a second one at point2,
multiplied by the widthof the innitesimal measurement interval.
1-c. Generalization: measurement of any number of positions
The previous computations deal with simple and double density measurements;
we now generalize them to measurements of higher order densities. From now on and to
simplify the notation, we shall omit thesubscript in the probabilities.
To compute the probability associated with a triple measurement, we start from
the expression of the state vector right after the detection of the second particle at2.
Taking (10) into account, and similarly as for (9), this normalized state is written:

0=
1

0(2)
0
(2)
0=
1
(21)
(2)
0 (17)
2241

COMPLEMENT C XXI
or else, if we insert (9) and use (14):

0=
1
(21)(1)
(2)(1)0
=
1
(21)
(2)(1)0 (18)
The probability of the third measurement at3, knowing that the rst two mea-
surements gave results at1and2, is thus:
(321) =
0(3)
0
=
1
(21)
0(1)(2)(3)(2)(1)0 (19)
As before, we consider that the position measurement zones do not overlap, so that all
the projection operators commute with each other:
(321) =
1
(21)
0(3)(2)(1)0
=

3
(21)
0 (1) (2) (3) (3) (2) (1)0 (20)
In the second line, we assumedwas small enough to use the approximate relation (5).
As the law of conditional probabilities indicates that the probability of the three
measurements at1,2and3is given by:
(321) =(21)(321) (21)
we simply get:
(321) =
3
0 (1) (2) (3) (3) (2) (1)0 (22)
which is a direct generalization of (15).
The same line of reasoning allows showing that the probability associated with the
measurement ofpositions is proportional to the average value in the system's state
of a product of2eld operators and arranged in normal order, and evaluated
at1,2,.... The probabilities are therefore equal to the spatio-temporal correlation
functions of the eld operators arranged in normal order (and multiplied by).
2. Measurement induced enhancement of entanglement
We have reasoned until now in a general way, without specifying the initial state0
of the system under study. We now assume we are dealing with a double condensate,
as in (3), and for simplicity we shall take1=2=(actually the computation that
follows only requires the hypothesis1 2). We thus haveparticles occupying the
individual statewith a well-dened momentum}, and an equal number of particles
occupying the statewith opposite momentum:
0=+:;:=
1
!
+
0 (23)
2242

MEASUREMENT INDUCED RELATIVE PHASE BETWEEN TWO CONDENSATES
We propose studying the interference signals that may occur in the single and double
counting rates measured on such a state. We shall need the probabilities calculated above
as well as expressions (A-3) and (A-6) of Chapter
() =
1
12
+ +
() =
1
12
+ + (24)
where(or) and(or) are the annihilation and creation operators of a particle
in mode(or), and whereis the edge of the box used to normalize the plane waves.
The dots on the right-hand side of these formulas stand for the other terms present in
the eld operator expansions of these operators. Because of the choice of the initial state
(23), these additional terms do not play any role in the following calculations, as will be
shown below.
2-a. Measuring the single density (1)
Relation (8) now becomes:
(1) = +:;: (1) (1)+:;: (25)
Using expressions (24) for the eld operators, and the fact that the cross terms
and have a zero average value in the double Fock state (23), we get:
(1) =

+:;: + +:;:=
2
(26)
This means that there is no interference in the single density measurement signal. This
was to be expected since the initial double Fock state includes no phase that could help
determine the eventual position of such fringes.
2-b. Entanglement between the two modes after the rst detection
Relation (9) yields the ket
0, right after the rst measurement. It can be written
as:

0=
1
(1)
1+

2
1

2
d () ()0
=
1
(1)
1+

2
1

2
d + + + +0(27)
Taking (23) into account, and for an innitesimal, we get:

0 20+
(+ 1)
2 1
:+ 1;: 1
+
2 1
: 1;:+ 1+ (28)
2243

COMPLEMENT C XXI
where+stands for the components of
0where a particle occupies an individual state
other thanand; these components do not play any role in what follows. Relation
(28) shows that the entanglement of state
0has increased as a result of the detection
of the rst particle. This state now contains a linear superposition of the initial state and
two additional states of the global system,1 : +;:and: +; 1 :;
the coecients of this superposition, and in particular their relative phase, depend on
the point1where the rst particle has been detected.
2-c. Measuring the double density (21)
We now compute the probability(21)associated with a double density mea-
surement. Relations (15) and (16) show that the probability is the squared norm of the
ket:
21= (2) (1): +;: (29)
Inserting in this equality the rst relation (24), the terms symbolized by the dots
disappear (as they involve annihilation operators yielding zero when acting on the+
and states, the only initially populated states). We obtain:
21=

2
+
2 1
+
1
: +;: (30)
or else:
21=

( 1)
(1+2)
2 : +;:
+
(1+2)
: +; 2 :
+
(2 1)
+
(1 2)
1 : +; 1 : (31)
The squared norm of this state vector yields the probability:
(21) =

2
2
2( 1) + 4
2
cos
2
[(2 1)] (32)
The presence of the cosine of(2 1)reveals the existence of a spatial dependence,
contrary to what happened for(1): once a rst particle is detected at1, the most
probable positions2for the second detection are those for which(2 1)is a multiple
of. In other words, fringes appear in the double density measurement.
2-d. Discussion
One may wonder which objects interfere in the double counting signal. They
are not waves but transition amplitudes associated with two dierent paths leading the
system from the initial state (23) to the same nal state1+; 1 , where
each of the two modes has lost one particle. In the above computation, the rst path
corresponds to the term
2 1
, where one particle with momentum+
disappears as it is detected at1and the particle disappears as it is detected at
2; the second path corresponds to the term
2 1
where it is now the
2244

MEASUREMENT INDUCED RELATIVE PHASE BETWEEN TWO CONDENSATES
particle with momentumthat disappears as it is detected at1and the+particle
that disappears as it is detected at2.
The double counting signal observed on a double condensate is very similar to the
double photodetection signal obtained, in of ComplementXX, in the study of
a product of two one-photon wave packets. In both cases, the signal spatial dependence
comes from a quantum interference between the amplitudes of two dierent paths between
the same initial and nal states. The dierence between the paths comes from a dierent
switching between one of the two components of the initial state and one of the two
components of the nal state.
3. Detection of a large number of particles
We now extend the previous reasoning to the case where any numberof particles' po-
sition measurements are performed; we shall limit ourselves to the case whereremains
much smaller than the total particle numberof each condensate.
3-a. Probability of a multiple detection sequence
Generalizing relation (22) allows writing the probability( 21)for de-
tecting a particle at1, a particle at2, .. a particle at, in the form:
( 21) = : +;: (1) (2) ()
() (2) (1): +;: (33)
As before, we use relations (24) to replace the eld operators and their adjoints by linear
combinations of annihilation operators, and creation operators,. We then
get:
( 21) =

: +;:(
1
+
1
)
( + )( + )
(
1
+
1
): +;: (34)
. Simplifying hypothesis
When several annihilation operators act successively on the right on the initial ket,
each of them introduces a varying factor
; this factor depends on the numberof
particles already annihilated by the other operators. In the same way, when the creation
operators act on the left on the initial bra, they also introduce varying factors. To keep
things simple, we shall ignore these variations, assuming that the total detection number
is always very small compared to the total particle numberin each individual state:
(35)
One can then replace all the factors
by: (36)
2245

COMPLEMENT C XXI
Apart from multiplication by a xed factor
, the only eect of each operator
is to vary by one unit the occupation number; this result does not depend on the pre-
vious actions undergone by the state vector (all the operators commute, once the above
approximation has been made). One can then freely move the annihilation and creation
operators in the product of operators appearing in (34). Regrouping all the operators
associated with the same values of, we get the operator:
() = ( + )( + )
= + +
2
+
2
(37)
and expression (34) becomes:
( 21) =

: +;:
=1
(): +;: (38)
We are then left with the computation of the average value in the initial state of
a product of operators(). Expanding each of them according to the second line of
(37), we get the sum of4products, most of them having, nevertheless, a zero average
value in the double Fock state0. This is because the only products having a non-zero
average value are those for which the repeated eect of the annihilation operatorsis
exactly balanced by the eect of an equal number of operators(the particle number
in the individualstate is then also constant, since the total number of particles must
be conserved).
Consider then one of those non-zero products. Still using approximation (36), the
contribution of each operator()will be one of the three factors(), with
= 01and:
0() = 2
1() =
2
(39)
The contribution of0leaves the particle numbers unchanged, the contribution of+1
replaces a+particle by aparticle, and nally that of1performs the opposite
substitution. Relation (38) then becomes:
( 21) =

=1 =01
() with= 0 (40)
where, when we expand the product in the right hand side of the equation, we retain
only the terms for which the sum of allvanishes:
=
=1
= 0 (41)
This constraint simply expresses the conservation of the particle number in each individ-
ual state.
2246

MEASUREMENT INDUCED RELATIVE PHASE BETWEEN TWO CONDENSATES
. Simple expression for the multiple detection probability
An easy way to impose constraint (41) on all theis to introduce a Kronecker
delta0, and write:
( 21) =

12
0
=1
() (42)
where the summation over theis now free of constraint. We can then use relation:
2
0
d
2
= 0 (43)
which amounts to multiplying each term()in (40) byexp()and summing over
d. We then get:
( 21) =

2
0
d
2
=1
() (44)
Each summation overthen yields the quantity:
0() ++()+()
=[2 + exp(2+) + exp(2 )] (45)
which can be written as:
2[1 + cos(2+)] = 4cos
2
+
2
(46)
Finally, we obtain the following simple analytical expression for the multiple de-
tection rate:
( 21) =
4
2
0
d
2
=1
cos
2
+
2
(47)
3-b. Discussion; emergence of a relative phase
Relation (47) allows understanding how the successive measurements enable the
progressive emergence of a relative phase.
. Detecting the rst particles
Let us start with the very rst detection at=1. Equation (47) yields the
probability of such an event:
(1)
2
0
d
2
cos
2
(1+
2
) (48)
The term in cosine appearing in the integral yields the fringes that one expects from the
interference between two waves, with wave vectors+and along theaxis, and a
2247

COMPLEMENT C XXI
phase shift. However, the summation overindicates that the interference pattern
must be averaged over all the possible values of, uniformly distributed between0and
2: this means that the fringes are completely blurred out.
The double detection rate at1and2is obtained by keeping the terms= 1and
= 2in (47). It is equal to:
(21)
2
0
d
2
cos
2
(2+
2
) cos
2
(1+
2
) (49)
As the rst detection has already occurred,1is xed in this equation, and the product of
the two cosines yields the probability of nding the second particle at2. But the integral
over d, which yields the2dependence of the probability, is no longer over a phase
uniformly distributed between0and2, because of the presence of thecos
2
(1+2)
associated with the rst detection; the blurring of the fringes is not as radical as before.
For this second detection, the functioncos
2
(1+2)actually plays the role of an
1dependent phase distribution; the two detections are no longer independent. This
conrms the qualitative discussion of .
This mechanism can be generalized to higher order measurements. As an example,
the triple detection rate at1,2and3is equal to:
(321)
2
0
d
2
cos
2
(3+
2
) cos
2
(2+
2
) cos
2
(1+
2
) (50)
Once the rst two detections have been made at1and2, the relative phase distribu-
tion that comes into play for the third detection is the product of two cosine functions
cos
2
(2+2) cos
2
(1+2) and no longer a single one as was the case before. As the
product of two cosine functions yields a sharper curve than a single cosine function, the
relative phase is better dened for the third detection than for the second. The process
continues in the same way with the following detections, and the phase is more and more
precisely dened. This means that it is the rst detections that determine the positions
of the fringes appearing in the following detections, each of them contributing to a more
and more precise denition of the relative phase distribution.
This argument is of course only valid for a given experiment. If one performs a
new experiment with the same experimental conditions, the rst detections will not, in
general, happen at the same places as in the rst experiment. Consequently, after a large
number of detections, a fringe pattern will appear, shifted with respect to the pattern
observed in the rst experiment. Finally, if one adds up the positions measured in a large
number of successive experiments, the fringes average to practically zero, and one gets a
quasi-uniform position distribution.
. Emergence of a well-dened relative phase after a large number of detections
After a large number of detections,, the relative phase distribution for the(+
1)detection is given by the product of a large numberof cosine functions, yielding
a very narrow phase distribution, centered at a value. One can then replace in (47)
all the[1 + cos(2+)]by[1 + cos(2+)], so that the probability becomes a
product: the detections are now independent, the interference pattern becomes stable
with a sharper and sharper contrast. These predictions have been conrmed by numerical
simulations based on equation (47).
2248

MEASUREMENT INDUCED RELATIVE PHASE BETWEEN TWO CONDENSATES
Narrowing of the relative phase distribution
The narrowing of the relative phase distribution can be explained by an analytical cal-
culation. Let us assume that afterdetections this distribution can be approximated
by a Gaussian curve centered at, and with a width:
()
( )
22
(51)
After the(+ 1)detection, the new distribution will be given by:
+1()
( )
22
[1 + cos(2+1+)] (52)
As the function1 + cos(2+1+)is much broader than +1(), it can be expanded
in powers of in the vicinity of= where the distribution()takes on
signicant values:
1+ cos(2 +1+) = 1 + cos(2+1+)
( ) sin(2+1+)
1
2
( )
2
cos(2 +1+) (53)
One can also expand()in the vicinity of=:
( )
22
= 1
( )
2
2
(54)
We then multiply (53) and (54) and obtain an expansion for+1():
+1()1 + cos(2+1+)( ) sin(2+1+)
( )
2
1
2
cos(2 +1+) +
1
2
1 + cos(2+1+) (55)
We note that +1()depends on the position+1of the(+ 1)detection. We
can obtain an average value for +1()by weighting +1()by the probability
[1 + cos(2+1+)]for the(+ 1)detection to occur at= +1, and integrate
+1over a spatial period of the interference pattern:
+1() =
22
0
d+1[1 + cos(2+1+)] +1() (56)
Since:
cos(2 +1+) =sin(2+1+)
=
cos(2 +1+) sin(2+1+) = 0 (57)
and:
cos
2
(2 +1+) =
1
2
(58)
we nally obtain:
+1()
3
2
1( )
21
2
+
1
6
3
2
( )
2
)
2
+1 (59)
2249

COMPLEMENT C XXI
where:
1
2
+1
=
1
2
+
1
6
(60)
Equation (60) shows that +1 , meaning that the distribution curve becomes
narrower after each detection. One can easily iterate equation (60) to obtain:
1
2
+
=
1
2
+
6
(61)
whereis a positive integer. This shows that if1
2
, the width of the relative
phase distribution decreases as1
.
A similar computation can be made to study the position of+1()'s maximum when
increases. One nds that the center of the relative phase distribution is shifted by a
quantity proportional to1.
Finally, it is interesting to note the link between the uncertainty on the relative
phase (which decreases as the detection numberincreases) and the uncertainty on
the dierence+ between the numbers of particles in thecondensates
(which, on the contrary, increases). At the beginning of the experiment (before the rst
detection), we have+= =. After the rst detection, we saw in
of the system contains a linear superposition of states+= 1and = 1:
the dierence+ between+and is no longer xed and equal to zero, but
can take on several values0,2. After the second detection, the state of the system
contains a superposition of states having always the same value of++, but values
of+ that can be equal to0,2, .. and so on. Afterdetections, the values of
+ spread out between2and+2. This result is an illustration of the fact
that the relative phase and the dierence between the particle numbers between the two
condensates are conjugate quantities.
Conclusion
This complement illustrates how successive measurements on a system having compo-
nents on two individual states, each withparticles, can build up (from zero) a relative
phase between these components; for this to happen, the measurements must depend on
the relative phase between those two components. Mathematically, relation (47) shows
that the results obtained for an ensemble ofposition measurements (with ) are
exactly the same as if an initial well dened phasehad existed from the beginning of
the experiment, even though it was totally unknown (and could have taken on any value
uniformly distributed between0and2). The measurements did indeed introduce en-
tanglement and its associated relative phase, but the quantum predictions are equivalent
to those obtained by assuming that the measurements only reveal a preexisting phase
(as in quantum theories with so-called additional variables).
The process we have discussed is, however, of an essentially dierent nature: it is
indeed each individual measurement that contributes to a better and better denition
of the relative phase for the measurements to come; these will occur at pointswhose
probability distributions depend on the results of all the previously performed measure-
ments. We shall see in ComplementXXIthat if, instead of measuring a fraction of
2250

MEASUREMENT INDUCED RELATIVE PHASE BETWEEN TWO CONDENSATES
all the particles, they each undergo a measurement, the phase properties can no longer
be understood as that of a classical preexisting (but unknown) phase; these properties
clearly become quantum, as shown by the possibility of violations of Bell's inequalities.
2251

EMERGENCE OF A RELATIVE PHASE WITH SPIN CONDENSATES
Complement DXXI
Emergence of a relative phase with spin condensates; macroscopic
non-locality and the EPR argument
1 Two condensates with spins
1-a Spin12: a brief review
1-b Projectors associated with the measurements
2 Probabilities of the dierent measurement results
2-a A rst expression for the probability
2-b Introduction of the phase and the quantum angle
3 Discussion
3-a Measurement number 2. . . . . . . . . . . . . . . . .
3-b Macroscopic EPR argument
3-c Measuring all the spins, violation of Bell's inequalities
This complement continues the discussion of ComplementXXIon the measure-
ment induced emergence of a relative phase between condensates, but in a more general
case. We established inXXIthat as more and more measurement results are obtained,
their number still remaining smaller than the total particle number, the relative phase
of the two condensates becomes better dened. It soon reaches a classical regime where
it is (almost) perfectly determined. This necessarily comes with large uctuations of the
numbers of particles occupying the two individual states (or more precisely of their dif-
ference), as required by the uncertainty relation between phase and occupation numbers.
In the present complement, a rst important dierence is that we no longer assume that
the number of measurements remains small compared to the total particle number. This
will enable us to follow the evolution of the phase properties during the whole series
of measurements, including the last moments when the number of particles remaining
to be measured is just a few units. For these few remaining particles, the uctuations
in the dierence of the occupation numbers is necessarily limited to a few units, mean-
ing that the phase can no longer be precisely determined. The phase then comes back
to a quantum regime, where one can no longer interpret the measurement results in
a classical context (preexisting but totally unknown phase). Another dierence with
ComplementXXIis that we now assume the two condensates correspond to dierent
individual spin states. Instead of position measurements yielding continuous results, we
can now perform measurements on the spin directions, which yield discrete results. This
will make it easier to discuss the quantum eects, which can lead to violations of Bell's
inequalities (Chapter , ). Another advantage of dealing with spins is that we
can go back to the EPR argument (Chapter , ) in a case where the elements
of reality, introduced by EPR, are macroscopic and have, in addition, a simple physical
interpretation (spin angular momentum).
2253

COMPLEMENT D XXI
1. Two condensates with spins
We now assume that the two individual states populated in the condensates are the two
states corresponding to two dierent internal states noted, but to the same
orbital state:
= (1)
If(+)and()are the creation operators associated with these states, the state
0of the system formed by the juxtaposition of the two condensates can be written as:
0=
1
!
(+) () 0 (2)
which replaces relation (23) of ComplementXXI; the total particle number is2.
By commodity, we will often call spin states the two states, and reason as if
they were indeed the two accessible states of a spin-12particle. This is just a manner of
speaking: according to the spin-statistic theorem (Chapter , ), bosons cannot
be half-integer spin particles. The system we consider is actually an ensemble of bosons
that have access to only two internal states; these can be, for example, the two= 0
and= 1states of a spin equal to1, or not necessarily spin states.
1-a. Spin 12: a brief review
For the reasoning that follows, it may be useful to recall a few relations (Chap-
ter, ) concerning a spin12(with no orbital variables). As pointed out above, we
are dealing with a ctitious spin, whose operators act on any two internal states, noted
by pure commodity. Operator, associated with the rst Pauli matrix (Comple-
mentIV), is the dierence between the projector onto the state+and the projector
onto the state:
=++ (3)
whereas operatorsandare expressed as a function of the two non-diagonal operators
+ and +as:
=+ + +
= + + + (4)
As for the ctitious spin component along a directionin the plane, making an
anglewith theaxis, the corresponding operator is written:
= cos + sin = + + + (5)
Its eigenvalues are=1, and its eigenvectors can be expressed as:
=+1=
1
2
2
++
2
=1=
1
2
2
++
2
(6)
2254

EMERGENCE OF A RELATIVE PHASE WITH SPIN CONDENSATES
as can be easily checked by applying operator (5) to these relations. The projector onto
the ket with eigenvaluecan thus be written:
() =
1
2
[1 +]
=
1
2
1 + + + + (7)
1-b. Projectors associated with the measurements
For an ensemble of identical bosons having orbital variables, we note (r)the
two eld operators associated with the two internal states. The operator associated
with the total particle density at pointris the sum of the local densities,
+(r) +(r)
corresponding to the+spin state, and (r) (r)corresponding to thestate:
(r) =
+(r) +(r) + (r) (r) (8)
As for the operator associated with thespin component along thequantization
axis, relation (3) indicates that it is the dierence:
(r) =
+(r) +(r) (r) (r) (9)
According to (5), the operator associated with a measurement performed along a direction
of the plane making an anglewith theaxis is written as:
(r) =
+(r) (r) + (r) +(r) (10)
The measurements we are interested in pertain both to the position of the particles
and their spin: for each measurement, the position is measured in an innitesimal volume
centered at pointrand, when measuring the direction of the spin along the
direction, we obtain the result=1. By analogy with (7), the projector associated
with such a measurement can be written:
(r) =
1
2

d
3
(r) + (r)

2
(r) + (r) (11)
where(r)and(r)are given by (8) and (10). Operator(r)projects both
the orbital variables onto this small domain and the spins onto the eigenstate of the
component along the axis, with eigenvalue=1.
2. Probabilities of the dierent measurement results
Consider now a system of2bosons in the state (2). Spin measurements are performed
in a series of regions of space, which cover the whole extension of the orbital wave
function(r)without overlapping. The measurements are supposed to be ideal, so that
every particle is detected. The regions are supposed to be suciently small to obtain
a negligible probability of double detection in any of them. Those where a particle is
actually detected are centered atr(with= 1,2, ..,2), as illustrated in Figure.
2255

COMPLEMENT D XXI
Figure 1: Two condensates, each havingparticles, the rst one with+spins, and the
other withspins, share the same orbital wave function(r), represented by the oval
in the gure. Measurements of the transverse direction of the spins are performed in2
non overlapping regions of space, centered at pointsr(with= 1,2, ..,2). In each
region, the measurement is performed along a transverse direction (perpendicular to the
quantization axis), dened by an angle, which may depend on; the corresponding
result is=1.
In each of these regions, one performs a measurement of the spin component along a
transverse direction (perpendicular to the quantization axis), dened by the angle,
and the measurement result is=1. We now calculate the probability of getting a
series of results=1in those2regions.
2-a. A rst expression for the probability
The associated projectors(r)all commute with each other, since they con-
tain eld operators (and their adjoints) at dierent points in space (we assume that all
measurements are done simultanously, or separated by a very short time). The proba-
bility2of a result is therefore the average value in the state0of the product of
projectors:
2( ) =0
2
=1
(r)0
=

2
2
0
2
=1

+(r) +(r) + (r) (r) +
+(r) (r)
+ (r) +(r)0 (12)
where symbolizes the ensemble of the variables(11 ). As the oper-
ators commute, we can also move all the eld operators (r)towards the right, and
their adjoints (r)towards the left. We now introduce a basis(r)for the wave
2256

EMERGENCE OF A RELATIVE PHASE WITH SPIN CONDENSATES
functions, its rst element being the wave function of the populated states (1). This
basis allows expanding the eld operators according to relation (A-14) of Chapter ,
and we can write:
(r) = (r) (13)
where is the annihilation operator of a particle in the individual state. Now
the only term that actually plays a role in this expansion is the= 1term: all the other
= 1terms yield zero when acting on the state0, which only contains particles in
the orbital state1(r) =(r). One can simply replace (r)by(r). The same
is true for the adjoint of the eld operators which, acting on the bra placed on their
left, can only destroy particles in a state previously populated; consequently, we can also
replace (r)by(r). Once these replacements have been performed, we get
an expression that can be written, in a symbolic way, as:
2( ) =

2
2
0:
2
=1
(r)
2
+ ++
+
+ + +:0 (14)
The two dots surrounding the product overin this symbolic writing express the following
convention, which originates from the rearrangement of the operators (r)and (r)
mentioned above: in each of the4
2
terms of the product of sums, all the annihilation
operators are regrouped towards the right, and all the creation operators
towards the left. To obtain the probability we are looking for, we have to compute the
average values in the state0of4
2
products, in normal order (ComplementXVI,
), of creation and annihilation operators in the two states=1.
The situation now becomes very similar to that leading to relation (37) in Com-
plementXXI. The computation that follows is indeed similar, except for the fact that
we no longer use the approximation (35) of that complement (number of measurements
small compared to the particle number): we now assume that all the particles are mea-
sured. Actually, most of the terms of the product overappearing in (14) have a zero
average value in the state0. The only relevant terms are those that contain exactly
annihilation operators+andother annihilation operators, in which case
their action yields the vacuum, hence a normalized ket; if this is not the case, the result
is zero. For a similar reason, they must also contain exactlycreation operators
+
andother , otherwise the result is zero. All these non-zero terms have the same
average values, since the product of operators in the normal order introduces each time
the same factor
!!
2
, that is(!)
2
; we also get the product of2coecients
, which can take one of these4values:
+1+1= 11= 1 (15)
and:
+11= 1+1= (16)
2257

COMPLEMENT D XXI
Now+1+1corresponds to a term associated with a particle destruction in+,
followed by a particle creation in that same state. In the same way,11corresponds
to an annihilation-creation in the individual state. Finally,+11corresponds to
an annihilation in statefollowed by a creation in state+, and the opposite
for 1+1. All the non-zero terms therefore correspond to products of2numbers
such that the sum of theand the sum of theare both zero; this condition
automatically ensures the conservation of the particle number in each individual state.
The nal result is:
2( ) =

2
2
(!)
2
2
=1
(r)
2
=1
with== 0(17)
where, in the right hand side, we retain only the terms satisfying the double condition:
== 0
== 0 (18)
2-b. Introduction of the phase and the quantum angle
Because of the summation constraints, expression (17) is not easy to handle. This
is why we introduce two delta functions0and 0, which obey:
0=
+
d
2
0=
+
d
2
(19)
This amounts to multiplying in (17) each by
(+)
and integrating over the
two anglesand. This enables us to write the probability in the form:
2( ) =

2
2
(!)
2
+
d
2
+
d
2
2
=1
(r)
2 (+)
(20)
where the summations overandare now independent, thanks to relations (19)
that automatically ensure the constraints are obeyed. For each value of, each sum
contributes the factor:
+1+1
(+)
+ 11
(+)
++11
()
+ 1+1
()
= 2 cos (+) + 2cos ( )
(21)
2258

EMERGENCE OF A RELATIVE PHASE WITH SPIN CONDENSATES
We nally make the change of variables
1
:
=+
= (22)
This leads to the simpler expression:
2( ) =
()
2
(!)
2
+
d
2
+
d
2
2
=1
(r)
2
[cos +cos ( )] (23)
The following discussion is entirely based on this result. For reasons that will be explained
below,is called the phase, whereasis called the quantum angle.
Comment:
It will be useful in what follows to note that the right-hand side of the above equality
stays the same if we make the change:
+
d
2
2
+2
2
d
2
(24)
To show this, we can decompose the integral over din a sum=1+2, where1
is the integral between2and+2, and2, the integral between2and32(as
the period of the function to be integrated is2, any integration domain covering the
entire circle is equivalent). Now2is just equal to1. This is because the function to
be integrated is multiplied by(1)
2
= 1when one changesto= as well as
to= . Consequently, changing the integration variables,to,allows
giving to2the same integration domain as1.
3. Discussion
Let us examine rst the case where the number of measurements is negligible compared
to the particle number in each condensate; this will enable us to compare the results
obtained with those of ComplementXXI.
3-a. Measurement number 2
We rst recall a general property of quantum mechanics concerning compatible
observables (Chapter, ). When several operators,,, etc. commute with
each other, one can build a basis with their common eigenvectors. The scalar product
of these eigenvectors and the system state vector yields the probability amplitude for
nding the system in each of these eigenvectors. If the eigenvalues are non-degenerate,
the squared modulus of this amplitude yields the probability of nding the corresponding
eigenvalues upon a series of simultaneous measurements associated with all the operators
1
The Jacobian of this change of variables is equal to2, and this factor should be introduced in
the denominator. Nevertheless, since the integrated function is periodic, this factor can be taken into
account by reducing the integration domain of the two variablesandto the interval,+, which
reduces the area of the integration domain by a factor2.
2259

COMPLEMENT D XXI
,,, etc. If they are degenerate, we just have to sum the probabilities over all the
orthogonal eigenkets. This is the rule we have followed until now in this complement.
Imagine now that we ignore the measurement results associated with one or several
operators of the series, for exampleand; the probabilities of the measurement results
we still consider relevant are then simply the sum over the possible results associated with
the ignored measurements (sum of the probabilities of exclusive events). One can also
imagine another situation where the quantities associated withandare actually
never measured. The possible series of eigenvalues of the measured operators are then
less numerous than in the previous case (since a smaller number of measurements is
performed), which increases the degree of degeneracy of these eigenvalues. As for the
eigenvalues ofand, even though they do not correspond to actual measurements, they
can still be used as indices to distinguish between the dierent orthogonal eigenvectors
associated with measurement results of operators,, etc. Consequently we still have
to sum the probabilities on these dierent eigenvalues, just as we did in the case where
these measurements were ignored. This means that quantum mechanics yields the same
probabilities whether we assume that the measurement results ofandare ignored
or have never been measured.
Let us now compute the probabilityof obtaining results1 when perform-
ingmeasurements on the spins. As we already know the probability (23) corresponding
to the case= 2, we can consider that all the2measurements have been performed,
but that we ignore the results of2 among them. As we just discussed, this amounts
to summing in (23) the probabilities of the two possible results for each of these2
ignored measurements, i.e. the probabilities associated with two opposite values of the
. It follows that in the product overin (23),cos ( )will disappear from all the
2 terms, leaving onlycos . We get forthe following expression (omitting from
now on the numerical factors, which are not relevant for our discussion):
( )
+2
2
d
2
[cos ]
2
+
d
2
=1
(r)
2
[cos +cos ( )] (25)
In this expression, the notation( )now stands forpairs of variables (),
instead of2as before. The integral over dcontains the functioncos to the power
2 ; if 2, this power is very high, and the function becomes a very narrow
peak centered at = 0. This allows us to write:
( )
+
d
2
=1
(r)
2
[1 +cos ( )] (26)
This result is very similar to the one obtained in relation (47) of ComplementXXI,
namely a product of two positive individual probabilities
2
:
() =
1
2
(r)
2
[1 +cos ( )] (27)
2
Thanks to the factor12, the sum of the two probabilities (=1) is normalized to the probability
of presence of a particle in the detection volume.
2260

EMERGENCE OF A RELATIVE PHASE WITH SPIN CONDENSATES
This product is then averaged over the angle, which can take on any value between
and+. As we show below, probability (27) is actually the probability of nding the
resultwhen measuring the component of a single spin along an axis with direction,
assuming that spin was initially polarized along a direction dened by the angle.
Demonstration:
We callthe spin quantization axis, and consider a spin polarized in the (transverse)
direction in the plane, making an anglewith theaxis. Relation (6) indicates
that its state is then:
=
1
2
2
++
+ 2
(28)
When measuring the spin component along an axis dened by the angle, the state
associated with the= +1measurement result is:
=
1
2
2
++
+ 2
(29)
Consequently, the probability of that result is:
=+1=
2
=
1
4
( )2
+
( )2
2
= cos
2
2
(30)
=
1
2
[1 + cos()] (31)
As for the probability of the=1result, it is simply the complementary probability,
obtained by changing the sign in front ofcos( ). In both cases, we get the probability
given by (27).
This means thatcan be interpreted as the relative phase between the two conden-
sates. In this case, the predictions of quantum mechanics are identical to the predictions
of a theory where the phase would be considered as a classical quantity, perfectly deter-
mined but as yet unknown at the beginning of the experiment. From such a point of view,
this phase would be revealed more and more precisely by the successive measurements,
instead of being created as assumed in the standard quantum mechanics interpretation.
Therein lies a link to the heart of the EPR argument.
3-b. Macroscopic EPR argument
The EPR argument was presented in . It is based on the
double hypothesis of reality and locality, as well as on the assumption that all quantum
mechanical predictions are correct. The conclusion of the argument is that quantum
mechanics is necessarily incomplete; to render it complete, elements of reality must
be added to it. In an EPRB experiment, involving two spins in a singlet state, these
elements of reality can be spin directions, well dened even before any measurements has
been performed.
Such an addition necessarily falls outside the framework of standard quantum
mechanics. Bohr was opposed to it; he argued that the concept of elements of reality
proposed by EPR could not be relevant for microscopic systems, since it was meaningless
to try and dissociate them, conceptually, from their experimental surrounding. As we
2261

COMPLEMENT D XXI
discussed in Chapter , this position is logically sound; it allows invalidating the
conclusions of the EPR argument. However, we are going to show that the double
condensates oer another context for applying the EPR reasoning, particularly interesting
as it involves macroscopic quantities (as well as the conservation of angular momentum).
These physical quantities can, in principle, be on our scale, thereby making it more
dicult to deny them an independent physical reality.
We consider a physical system in a quantum state similar to (2), where the two
internal states of the particles are eigenstates of the spin components along the
quantization axis, for example the= 0and= 1states of a spin= 1. For
the clarity of the discussion, we will assume that the orbital wave functions of each
condensate are distinct but overlap in two regions of space, as schematized in Figure.
In each of these two regions (which may be separated by an arbitrarily large distance), two
observers, Alice and Bob, perform measurements of the spin components along transverse
directions
3
,measurements for Alice,for Bob. For each measurement performed,
each of the observers chooses an arbitrary direction dened by an angle; Alice's choices
are completely independent of those of Bob, and vice versa. A rst series of measurements
(1 ) is performed by Alice in a rst region of space; right after that, Bob
performs another series (+ 1 +) in his own laboratory, located very far
away.
Now we saw that, as soon as Alice has measured the spins of a few particles,
the relative phase of the two condensates in the entire space is xed with a fairly good
precision (the larger the number of measurements, the better the precision). These
measurements also x the transverse direction of the spins. Alice cannot, however, decide
what this direction will be, as it is xed in a totally random way in the measurement
process. Standard quantum mechanics then predicts that when Bob will perform his
own measurements, it is practically certain (within a negligible error) that he will nd
the same relative phase. As he can perform a large number of measurements, he can
nd out, practically instantaneously, the spin direction created and observed by Alice.
The EPR argument underlines that, as no interaction had time to propagate from Alice
to Bob, it is not possible for this transverse orientation to have been created by Alice's
measurements: it necessarily existed prior any measurements.
What is new in our case, compared to the two-spin case, is that Bob's observa-
tions may concern an arbitrarily large number of spins; his experiment then amounts to
measuring the angular momentum direction of a macroscopic spin system, which has an
arbitrarily large angular momentum. As we are now dealing with macroscopic quantities,
one can no longer argue, as Bohr did, that the microscopic world is accessible neither
to human experimentation nor to human language description. In our present case, it
seems more articial to refuse, as suggested by Bohr, to consider separately the physical
properties of systems located in distinct regions of space. The EPR argument becomes
harder to refute. Reference [90] contains a discussion of this unexpected situation in
terms of conservation of angular momentum.
3
The longitudinal direction is the direction of the spin polarization in the initial state (2
transverse directions are all the perpendicular directions.
2262

EMERGENCE OF A RELATIVE PHASE WITH SPIN CONDENSATES
Figure 2: Scheme of an experiment on a double spin condensate, one in a+internal
state, the other in astate. The two condensates have distinct orbital wave functions,
overlapping in two regions of space where two observers, Alice and Bob, perform mea-
surements. These two regions can be separated by an arbitrarily large distance. Alice and
Bob measure the spins one by one, choosing each time a transverse component (perpen-
dicular to thequantization axis) dened by an angle. Whereas, initially, the two
condensates have no relative phase, the rst measurements performed by Alice create one.
The paradox is that this phase propagates instantaneously to the remote region where Bob
performs his measurements. This is reminiscent of the EPR argument, but in a more
striking case where the EPR elements of reality can be macroscopic.
3-c. Measuring all the spins, violation of Bell's inequalities
Imagine now that we measure all the spins; the selective eect around the = 0
value does not occur any longer. The interpretation in terms of probabilities of individual
events is no longer possible: the factor[cos +cos ( )]in (23) can sometimes
take on negative values, which rules out its possibility to be considered as a standard
probability. Actually, it is not unusual in quantum mechanics that purely quantum
eects arise through negative probabilities , as in this case. The angleis called the
quantum angle , which underlines that its role is to introduce such quantum eects,
such as non-locality eects and violations of the Bell's inequalities.
To prove that such violations occur for any value ofrequires using relation
(23), which involves many parameters (all the measuring angles, which are arbitrary);
it is easier to perform a numerical calculation as explained in the second reference of
[90]. Our objective here is to simply show that the phase does not always behave in a
classical way. This is why, without presenting the numerical calculation, we shall study
the behavior of expression (23) for the simple case of two measurements on two spins
(= 2and= 1). This will enable us to show that this expression does predict the
existence of violations of the inequalities, for certain cases (more general cases are treated
in the above reference). Clearly, in the case of two spins we could have carried out this
computation in a simpler and more direct way.
Using denition (A-7) of Chapter 2) can be
2263

COMPLEMENT D XXI
written in the form:
= (+)()0=
1
2
[1 :+; 2 :+1 :; 2 :+]
=1 :; 2 :
1
2
[1 : +; 2 :+1 :; 2 : +] (32)
This leads to an entangled spin state, very similar to the one considered in
Chapter . The only dierence is the+sign in the present spin state, instead of
thein the singlet state considered in that chapter, but this dierence is of no great
consequence (we shall come back to this point more precisely see note). Such a
state can be expected to lead to signicant quantum eects, as for example to situations
violating Bell's inequalities.
We can also go back to the general relation (23) to show that it indeed leads to
violations of Bell's inequalities in this simple case. For= 1, this relation becomes:
2(11;22)
+
d
2
+
d
2
[cos +1cos ( 1)] [cos +2cos ( 2)]
=
1
2
[1 +12cos1cos2+12sin1sin2] (33)
where we have used the fact that the average value on the circle of cosine squared or sine
squared is equal to12, whereas the average value of the product of cosine and sine is
zero. We obtain:
2(11;22)
1
2
[1 +12cos (1 2)] (34)
Normalizing to unity the sum of the4probabilities obtained for1=1and2=1,
we nally get:
2(+1+1) =2(11) =
1
2
cos
2 1 2
2
2(+11) =2(1+1) =
1
2
sin
2 1 2
2
(35)
These relations are very similar to equalities (B-10) of Chapter , if we make the
change
4
:
1 +
2 (36)
The angles1and2now play the role of the analyzer orientation angles in Figure
2 , ) that these equalities lead to
signicant violations of Bell's inequalities (by a factor
2), hence to marked non-local
quantum eects; such eects should therefore be expected in our present case.
4
This change results from the+sign in the spin state (32), instead of thesign in the singlet state.
2264

EMERGENCE OF A RELATIVE PHASE WITH SPIN CONDENSATES
In the general case wherecan take on any value, measuring the totality of
the spins may lead to strong violations of Bell's inequalities, provided the measurement
angles are judiciously chosen (for large, the angular domain leading to such violations
decreases as the inverse of the square root of that number [90]). Note however that
these violations will disappear as soon as certain spins are no longer measured (or their
corresponding results no longer taken into account, which amounts to the same thing).
Conclusion
This complement is an illustration of the limits of the phenomenon studied in the previous
complement: the process of successive measurements builds up a phase that has all
the properties of a classical phase, but only up to a certain point. If all the particles
are measured, and for certain particular choices of the measuring angles, in an ideal
experiment the phase should exhibit some distinctly quantum properties.
Furthermore, one could have expected that the extreme quantum properties, as for
example their non-local aspects discussed in , would only
concern systems with a small particle number, or in singlet spin states (these having
a special status among all the states accessible to a physical system). The present
complement shows that this is not at all the case: in principle, the same properties
should exist for systems composed of a very large number of particles, in a fairly simple
quantum state (a double condensate).
2265

FEYNMAN PATH INTEGRAL
Appendix IV
Feynman path integral
1 Quantum propagator of a particle
1-a Expressing the propagator as a sum of products
1-b Calculation of the matrix elements
2 Interpretation in terms of classical histories
2-a Expressing the propagator as a function of classical actions
2-b Generalization: several particles interacting via a potential
3 Discussion; a new quantization rule
3-a Analogy with classical optics
3-b A new quantization rule
4 Operators
4-a One single operator
4-b Several operators
In Chapter, we introduced the postulates of quantum mechanics using the
Hamiltonian approach, with quantization rules applied to conjugate Hamiltonian vari-
ables. It is however possible to introduce quantum mechanics and its quantization rules
in an entirely dierent way, starting from a classical Lagrangian, and using Feynman
path integrals. This approach provides an interesting insight to the relationship between
classical and quantum physics, reminiscent of the connections between geometric and
wave optics. Furthermore, this approach is preferable in a certain number of cases, in
particular for situations where we know the classical Lagrangian but not the conjugate
variables necessary to dene a Hamiltonian
1
.
This appendix is an elementary introduction to Feynman path integrals, and some
of their properties, without too much concern for mathematical rigor. We rst study in

sum of contributions coming from dierent classical histories (possible evolutions) of the
physical system. Once these results have been established, we discuss in
the inverse point of view, and start from these classical histories to derive the usual form
of quantum mechanics, its quantization rules, its propagators and, in , its operators.
For the sake of simplicity, we shall only consider an ensemble of particles interacting via
a position dependent potential; for the study of more general cases (vector potential,
commutative or non-commutative gauge invariance), the reader may consult references
[91], [92], or [93].
1. Quantum propagator of a particle
Consider a spinless particle. The propagator(r;r)of this particle is dened as:
(r;r) =r()r (1)
1
This happens, for example, if the Lagrangian does not depend on the time derivative of a coordinate
, in which case one cannot dene the conjugate momentum.
2267
Quantum Mechanics, Volume III, First Edition. C. Cohen-Tannoudji, B. Diu, and F. Laloë.
© 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

APPENDIX IV
where the ketsrare the eigenkets of the position operator, and()the evolution
operator between the initial timeand the nal time(ComplementIII). This prop-
agator gives the probability amplitude for the particle, starting at timefrom a state
localized at pointr, to be found at a later timeat pointr.
1-a. Expressing the propagator as a sum of products
We can split the time interval[]intoequal smaller segments by setting:
=
(2)
We therefore introduce1intermediate times1,2, ..,, ..,1(see Figure),
which are equal to:
=+ = 12 1 (3)
It will also be useful for what follows to set0=and =. We can then express
()as a product ofterms:
() =( 1) ( 1) (21)(10) (4)
Between each evolution operators, we introduce a closure relation in therbasis:
() =d
3
1d
3
2d
3
d
3
1
( 1)r1r1( 1 2)r2
r( 1)r1 r1(10) (5)
Inserting this equality in (1), the propagator is expressed as:
(r;r) =d
3
1d
3
2d
3
d
3
1
r( 1)r1r1( 1 2)r2
r( 1)r1 r1(10)r (6)
We now lettend towards zero (or, equivalently,towards innity). The number of
integrations over the d
3
will then tend towards innity but the matrix elements are
now those of the evolution operator over an innitesimal time.
1-b. Calculation of the matrix elements
We now compute the matrix elementsr( 1)r1of the evolution oper-
ators, with= 1+. The particle's Hamiltonianis the sum of the kinetic energy
and the energy associated with an external potential:
=
P
2
2
+(R) (7)
wherePandRare, respectively, the momentum and position operators of the particle;
is its mass.
2268

FEYNMAN PATH INTEGRAL
Figure 1: To express the evolution operator as a product of operators, one introduces
between the initial timeand the nal timea whole series of intermediate times1,
2, ..,, ..,1; an evolution operator is associated with each of thetime intervals.
Introducing at each timea closure relation then leads to a summation over all the
possible positionsrof the particle at that time.
. Free particle
For a free particle, the Hamiltonian is simply the kinetic energy. Introducing a
closure relation on the momentum eigenstates, we can write:
r
P
2
2}
r1=
1
(2)
3
d
3 k(rr1) }
2
2
(8)
We show below that this propagator can be computed exactly, whether the time interval
is innitesimal or nite; its expression is:
r
P
2
2}
r1=
2}
32
34 (rr1)
2
2}
(9)
Imagine for a moment that in the argument of the last exponential function the factoris
replaced by a simple minus sign (this amounts to switching to an imaginary time). Within
a numerical factor, the propagator of the free particle becomes a Gaussian function,
decreasing rapidly as soon as the distancerr1becomes large compared to its
width
2} .
Comment:
This result is not surprising since if one replaces, in Schrödinger's equation, the real time
by an imaginary time, one gets the diusion equation whose propagator is indeed a
Gaussian. The width of that Gaussian goes to zero as0: as expected, the shorter
the time, the smaller the distance covered by the particle. It is worth noting, however,
that this distance is not proportional to, but to the square root of this time; in other
words, during very short times, the particle may propagate much further than if it had a
constant velocity. This situation is characteristic of a random motion whose correlation
time and mean free path tend simultaneously to zero, in a way where the diusion
coecient remains constant: such a process lead to the classical diusion equation, and
its propagator therefore has the same property.
As for Schrödinger's equation itself (without switching to an imaginary time), when
the distancerr1increases, instead of a decrease of the propagator's modulus, we
2269

APPENDIX IV
get an oscillation that gets faster as the distance gets greater; these oscillations are also
faster asgets shorter. We discuss below how all these phases interfere, as well as the
particular role played by the phases associated with classical paths.
Demonstration of relation (9):
We modify the exponent of the function to be integrated in (8) to introduce a perfect
square:
}
2
k
2
+k(rr1) =
}
2
k+
}
(rr1)
2
2}
(rr1)
2
(10)
Making the change of variables:
q=k+
}
(rr1) (11)
leads us to:
r
P
2
2}
r1=
1
(2)
3
d
3 }
2
2 (rr
1)
2
2}
(12)
The integral that still remains between the brackets on the right-hand side is now a
number independent ofrandr1. As the three components of the vectorqyield the
same contribution, this number is the cube of the integral:
=
+
d
}
2
2
(13)
Following a classical procedure, we compute the square of that integralusing the polar
coordinates where=
2
+
2
:
2
=
+
d
+
d
}(
2
+
2
)2
= 2
0
}
2
2
=
2
}
1 (14)
Now, writing the number is mathematically meaningless: as , the imaginary
exponential oscillates indenitely between+1and1, around a zero average value.
We shall simply take this number equal to its zero average value
2
, so that:
=
2}
4
(15)
Replacing by
3
the integral over d
3
in (12), we get relation (9).
2
One can reach this same result by adding a small imaginary partto the coecient}
2
2of
2
in the exponent. Thanks to this imaginary part, which can be arbitrarily small, the oscillating term
does disappear.
2270

FEYNMAN PATH INTEGRAL
. Eects of the external potential
We must now compute the matrix elements of the exponential operator
}
whenis the sum of two non-commuting operators. We shall only consider the case
whereis innitesimal. In that case, the exponential of a sum of non-commuting
operatorsandcan be written:
(+)
= 1[+]
2
+ + +
2
2
2
+ 0
3
(16)
We can also expand the following product of exponentials as:
2 2
=1
2
2
2
8
+
1
2
2
2
+ 1
2
2
2
8
+ (17)
This leads to:
2 2
= 1[+]
2
2
+
2
2
+
2
2
+
2
2
+ +
2
2
+ 0
3
=
(+)
+ 0
3
(18)
Setting=P
2
2}and=(R)}we get, assuming thatis innitesimal
so that we can neglect the
3
terms:
r( 1)r1=r
(R)2} P
2
2} (R)2}
r1
=
[(r)+(r1)]2}
r
P
2
2}
r1 (19)
This result is simply the product of an exponential including the potential and a term
that is the propagator of the free particle. Taking (9) into account, we can write:
r(1)r1
=
2}
32
34 (rr1)
2
2} [(r)+(r1)]2}
(20)
The eect of the external potential is straightforward: it adds an exponential including
half the sum of the two potential energies associated, one with the position of the bra,
the other with the position of the ket.
. Final expression
Inserting (20) into (5), we get the following expression for the propagator:
(r;r) =
2
2}
32
d
3
1d
3
2d
3
d
3
1
exp
1
=1
(rr1)
2
2}
(r) +(r1)
2}
(21)
This equality is valid in the limit where0(or ) since, to establish (19), we
neglected the
3
terms.
2271

APPENDIX IV
Figure 2: A Feynman path is obtained by associating a positionrwith each time.
The path thus obtained is continuous, but looks in general like a zigzag (the velocity is
discontinuous at each time). Nevertheless, one can associate a classical action with
each of these paths, and get the quantum propagator by summing exponentials of the
actions along all these classical paths: while keeping the two end positionsrandrxed,
the summation is performed on all the intermediate positionsr1,r2,...,r,...r1.
2. Interpretation in terms of classical histories
The probability amplitude (6) is obtained by inserting, as many times as necessary,
the propagator matrix elements (20). It thus contains as many sums over intermediate
positions as there are intermediate times, and this number goes to innity as0.
This expression, somewhat complicated at rst sight, actually has a simple interpretation
in terms of classical paths the particles can follow between the initial and nal times.
2-a. Expressing the propagator as a function of classical actions
Let us go back to classical physics and consider for a moment all theras being
xed, and a particle that goes through these successive positions at the dierent times
. In between two consecutive times, we assume the particle keeps a constant velocity
equal to:
v=
rr1
(22)
This denes a classical path, with a linear interpolation for times between the discrete
instants(cf.Figure). The particle's Lagrangianis written:
=
1
2
v
2
(r) (23)
For any classical pathfollowed by the particle between the initial timeand the
nal time, the position and velocity are both functions of time, and so is the Lagrangian
(). The corresponding action is written (Appendix, ):
= d () (24)
2272

FEYNMAN PATH INTEGRAL
Let us compute this integral using Riemann's method by introducing, between the times
and,time intervals of length. We consider that during an innitesimal time
interval, the potential energy can be approximated by half the sum of its values at
each end of the interval. We then get:

=1
(rr1)
2
2
2
(r1) +(r)
2
(25)
This approximate equality becomes exact in the limit0.
On the right-hand side of this equality, we nd (to within a factor}) the argument
of the exponential appearing in expression (21) for the quantum propagator. This means
that this quantum propagator contains the exponential of an approximate value of the
classical action, multiplied by}. When 0(and hence ), the approximate
value becomes exact
3
, and the sums over d
3
1, d
3
2,.., d
3
1on the right-hand side of
(21) introduce a summation over all the paths going fromrtor:
paths
exp

}
(26)
For this summation over the paths to be meaningful, we must now choose a path den-
sity; we therefore assume that the number of paths in a path interval, determined by
the set of d
3
, is given by the product
32
d
3
1d
3
2d
3
d
3
1, whereis the
constant (inverse of a length):
=
4
2}
(27)
This allows us to write:
(r;r) =r()r=
paths
exp

}
(28)
In the limit0, the sum over the paths is no longer discrete. This is why, instead of
(28), one often writes:
r()r= [r()] exp

}
(29)
where the notation[r()]symbolizes the limit of a sum over the paths:
[r()] = lim
4
2}
32
d
3
1d
3
2d
3
d
3
1 (30)
3
Remember that this complement does not pretend to be mathematically rigorous. This would entail
a more careful study of the classical as well as quantum expressions, and of the eects of the simultaneous
limitsgoes to zero and(number of terms in the products) goes to innity, keeping in mind that
these approximations are done on functions that are arguments of exponentials.
2273

APPENDIX IV
2-b. Generalization: several particles interacting via a potential
The previous considerations can be directly generalized to a system with several
particles interacting via a potential depending on their positions. Since the system Hamil-
tonian is, as above, the sum of two non-commuting terms, we again use the approximate
expression (18) for the evolution operator. But rather than inserting the closure relations
for a single position, one must now use a basis involving the positions of all the particles:
each integral over d
3
is then demultiplied into as many integrals as there are particles
in the system.
We will not include here the case where the particles are charged and subjected to
a magnetic eld, which would include terms such asvA(r), whereA(r)is the vector
potential. Even though it is an interesting case, in particular for its relation to gauge
invariance, it will not be considered here for the sake of brevity. The interested reader
should consult the references given in the introduction.
3. Discussion; a new quantization rule
The path integral approach is particularly t for developing an analogy with classical
optics. In addition, it allows building new quantization rules. These two points will be
discussed successively.
3-a. Analogy with classical optics
Relations (28) and (29) allow making a link between two a priori unrelated quan-
tities: the classical mechanics paths with their associated actions, and the quantum
propagator. Knowing the wave function (r)at a given time, it is this quantum
propagator that enables computing that wave function at a later timeas follows:
(r) =r ()=r() ()
=d
3
r()rr () (31)
Taking into account denition (1) of the propagator, the above expression can be rewrit-
ten as:
(r) =d
3
(r;r) (r) (32)
The propagator is thus the kernel of the integral equation expressing the temporal prop-
agation of the wave function. Now, equality (28) shows that this propagator is equal
to a sum of exponentials of the actions corresponding to all the classical paths. There-
fore, whereas in classical mechanics a single path (or in certain cases a nite number
of paths) is selected by the stationarity condition of the action, in quantum mechanics
all the paths come into play to determine the propagation amplitude, each one with its
particular phase. In a manner of speaking, one can say that, in quantum mechanics, the
particle goes through all the possible intermediate positionsrand hence follows all the
possible histories between the two end points.
It is worth noting that all these histories (even highly unlikely histories involving
totally arbitrary positions) contribute with the same amplitude. On the other hand, the
2274

FEYNMAN PATH INTEGRAL
phases associated with the histories are dierent from each other and allow understanding
how the dierent histories come into play. This situation can be analyzed in terms
of stationary phase conditions. It is easy to understand that in the summation, the
histories corresponding to a stationary action will play a particular role since all the
neighboring histories will add their contribution in a coherent way. On the other hand,
in the vicinity of histories for which the action varies rapidly, the phase oscillates quickly
and the corresponding contributions will cancel out through destructive interference.
Consequently, classical histories play a privileged role, which becomes more prominent
as the phase oscillations become more rapid. In the limit where}0, these oscillations
become innitely rapid and only the classical histories prevail.
Finally, this situation reminds us of classical optics and the Huyghens-Fresnel
principle, where a light wave is computed as the sum of waves radiated from each point
of an intermediate surface, taking into account the phases linked to the propagation
along each path. In the geometrical approximation, where the wavelength tends towards
zero, the trajectories of the light rays correspond to paths having a stationary phase,
i.e. which does not change for innitely close paths. Geometrical optics is the analog of
classical mechanics, whereas Huyghens wave optics is the analog of quantum mechanics.
The Feynman integral path is therefore a useful tool to study the link between classical
and quantum mechanics, and in particular the semiclassical limit of quantum mechanics
(WKB approximation, etc.).
Comment:
The preceding analogy is well founded for a single particle. For a system withparticles,
the histories no longer propagate in the ordinary three-dimensional space, but in a3-
dimensional conguration space. The analogy with optics described above is no longer
as adequate, since in classical optics, electromagnetic waves propagate in ordinary3
space.
3-b. A new quantization rule
At this stage, we can invert the approach. Until now, starting from the rules of
Hamiltonian quantum mechanics, we deduced an equivalent expression for the propa-
gator, i.e. another way for nding the solutions of Schrödinger's equation. It is also
possible to consider this equivalent expression as the starting point and postulate that
the propagator is dened ab initio by a sum over all the classical paths, each con-
tributing an exponential exp(}). This yields another method for the quantization of
a physical system, which oers several advantages. First of all, as we just saw, it high-
lights the relation of quantum mechanics with classical mechanics, where only a single
classical path exists (or sometimes a nite number of paths), as opposed to an innite
number of possible paths in quantum mechanics. Furthermore, it is remarkable that the
probability amplitudes thus computed only depend on classical functions (involving only
numbers and not operators), the only explicit quantum component being the presence of
}in the denominator of the phase. We shall see in
be introduced in this approach. In addition, the expressions involving path integrals are
symmetric with respect to time and space, since both type of coordinates are integrated
2275

APPENDIX IV
in a similar way
4
. Reasoning directly in space-time makes it easier to include Einstein's
relativity, since one can replace the time dierentialby a proper time dierential. If
now the Lagrangianis a space-time scalar, so is the action, and the theory acquires rel-
ativistic invariance. Finally, Feynman's quantization method only requires the existence
of a Lagrangian, with its associated variational principle. Now all the physical systems
that have a Lagrangian do not necessarily have the conjugate variables permitting the
denition of a Hamiltonian. For such systems, the Feynman path integral method is
powerful and this is why it is so important in quantum eld theory.
4. Operators
Feynman paths also permit computing the matrix elements of operators in the Heisenberg
picture, where they are time-dependent. We shall mostly consider the simplest case where
the operators are functions of the position operatorR.
4-a. One single operator
Let us insert any operatorin the middle of the evolution operator (1) by
splitting the time interval[]into two adjacent intervals[]and[], with
. We get the expression:
r() ()r=(r ) (r ) (33)
with:
(r )=()r
(r )=()r=()r (34)
In this matrix element of, the ket(r )is obtained by the evolution until
the timeof a state localized atrat time; the bra(r )corresponds to the ket
(r )which, as it evolves betweenand, becomes a ket localized atr:
()(r )=()()r=r (35)
. Operator function of the position
We now assume operatoris a function(R)of the particle's position operator.
Inserting a closure relation on the positions, we can write the left-hand side of (33) as:
r()(R)()r=d
3
r()r (r)r()r
(36)
4
The sum over all the paths introduces an integral over all the positionsrin Figure, hence
dierentials of the three space coordinates; in addition, the integral over the times introduces a dierential
d= 1. The product of three space dierentials by a time dierential thus allows one to introduce
a dierential of space-time volume.
2276

FEYNMAN PATH INTEGRAL
Using the general relation (28), we can then write the two propagators appearing under
the integral as the sum over all the paths1or2:
r()r=
paths1
exp
1
}
r()r=
paths2
exp
2
}
(37)
where1is a path linking the initial positionrat timeto the intermediate position
rat time, and2the path linking thereafter the intermediate positionrat time
to the nal positionrat time. Ifcoincides with the intermediate time, relation
(36) becomes:
r()(R)()r
=d
3
paths1and2
exp
2
}
(r) exp
1
}
(38)
Now the product of the two exponentials yields a single exponential exp[}]associated
with the actionof a pathconsisting of the two paths1and2joined together end
to end atr. The sum over d
3
reconstitutes the ensemble of all paths going from the
initial positionrat timeto the nal positionrat time, the only dierence being that
now each exponential exp[}]is multiplied by the value(r)taken by the function
at the intermediate point at time.
We nally obtain:
r()(R)()r=
paths
(r) exp

}
(39)
where(r)is the value of(r)at positionrwhich the pathtraverses at time.
The matrix elements of the operator in the Heisenberg picture (special case=) are
thus given by the same summation over the histories as for the propagator, the only
dierence being that the contribution of each path is now multiplied by the value taken
by the operator at the positionrat the intermediate time.
As before, we can now invert the approach and consider relation (39) as the de-
nition of an operator in the framework of the Feynman path quantization method. Here
again, it is remarkable that this relation involves only classical functions, without any
operator.
. Velocity operator; canonical commutation relations
In order to dene an operatorWassociated with the particle's velocity at time
(we use the notationWto avoid any confusion with the potential), and taking (22)
into account, a natural extension of (38) leads to setting:
r()W()r
=d
3
paths2
exp
2
}
paths1
rr1
exp
1
}
(40)
2277

APPENDIX IV
where the paths1are all those going from the initial positionrto the intermediate
positionr(the preceding intermediate positionr1does depend on the path), whereas
the paths2are all those going fromrto the nal positionr. Introducing in the middle
of the left-hand side a closure relation on the ketsr, this relation becomes:
d
3
r()rrW()r
=d
3
r()r (r) (41)
with:
(r) =rW()r=
paths1
rr1
exp
1
}
(42)
This wave function is the result of the action of operatorWon the wave function at time
, equal to:
(r) =r()r (43)
Let us compare the sum over the paths1in relation (42) and in the relation (28)
used to build the propagator between the timesand. In these two equalities, the
actions are given by the summations (25) over the intermediate positions. Concerning the
contributions of the paths between the initial time and the intermediate time1(the
last time over which the summation runs), the two sums in (42) and (28) are identical.
Actually, their only dierence concerns the very last time interval (between1and
=), which in (42) is multiplied by the factor(rr1). Now this multiplicative
factor can also be found by taking the derivative of (28) with respect torsince, using
(25), we have:
rrr()r=
}
paths1
rr1
2
+
1
2
rr exp
1
}
(44)
Asris a nal xed point, the term inrron the right-hand side can be taken out of
the summation over the paths. It yields a contribution inrrwhich goes to zero
in the limit0. We are left with:
rrr()r=
}
paths1
rr1
exp
1
}
(45)
so that relation (42) becomes:
(r) =
}
rrr()r=
}
rr (r) (46)
This means that the action of the velocity operatorWis simply proportional to a deriva-
tive
5
with respect to the positionr, which is the variable of the wave function at the
5
The demonstration pertains to a wave function at the instantthat is issued from a wave function
localized at pointrat time. By linear superposition, it can be generalized to any wave function at
time, hence conrming the same derivation property.
2278

FEYNMAN PATH INTEGRAL
instant. In other words, ifP=Wis the particle's momentum operator, its ac-
tion on the wave function is(})times the gradient with respect to the position. We
have established a basic result of the usual quantum mechanics, starting from operators
introduced in the path integral approach.
The canonical commutation relations betweenRandPare easily derived, since:
[ ()] =[ ()] + () (47)
These commutation relations can also be considered as consequences of the path quan-
tization rules.
4-b. Several operators
Feynman postulates also permit introducing products of several operators, acting
at the same instant or at dierent times.
. Several operators at dierent times
The previous argument can be generalized to several operators(R),(R), etc.
acting at intermediate times,, etc. As before, we can split the evolution operator
into several parts corresponding to the successive time intervals, and insert position
closure relations at the intermediate times. Each operator introduces a factor dependent
on the corresponding intermediate position, and the time propagation is a sum over
histories between successive time intervals. For instance, for two operators, the same
reasoning followed above leads to:
r( )(R)( )(R)()r
=
paths
(r)(r) exp

}
(48)
where(r)is the value offor the positionrcrossed by the pathat time,(r)
the value offor the positionrcrossed by the pathat a later time. The result
is easily generalized to any number of operators. Note the order in which the operators
are arranged in the matrix element on the left-hand side: it corresponds to the order in
which the times,, etc. are arranged in the classical histories used to calculate the
actions. The quantum operators are automatically arranged in decreasing times from
left to right, even if(r)and(r)are numbers that commute in the right hand side
of relation (48).
. Position and velocity operators, symmetrization
Imagine now we want to introduce, for example, the operator corresponding to
the productRPof the position and momentum. We will then proceed as in (41) and
(42), however with an added precaution: should we multiply(rr1)byror by
r1(the order does not matter, since we are dealing with numbers). For the sake of
symmetry, we multiply by half their sum, so that in (42)(rr1)is replaced by:
1
r+r1
2
(rr1) =
1
2
[r(rr1) + (rr1)r1] (49)
2279

APPENDIX IV
On the right-hand side, we wrote the two terms so that the times are always decreasing
(or stationary) from left to right. Because of the order of theindices, the rst term
on the right-hand side introduces the operatorRW, whereas the second introduces
WR, i.e. the same operators but in the inverse order. This is an example of how the
path integral method leads quite naturally to a symmetrization of the operator order,
which automatically ensures the hermiticity of their product.
Conclusion
To be able to use two complementary approaches, the Hamiltonian method and the path
integral method, is often quite valuable in the study of numerous physical problems.
As an example, path integrals play a fundamental role in eld theory. They are for
a large part at the base of the implementation of symmetry groups (Abelian or non-
commutative) in this theory, which allows building a theory for elementary particles and
their interactions. There are, however, other cases where the path integral formalism
is very useful, as for example, in the computations of quantum interference with cold
atoms. Conceptually, the path integral approach can shed new light on the relations
between quantum mechanics and classical mechanics, as well as classical optics as we
saw in .
In this appendix, we only used path integrals as a method to compute the time
propagator of a quantum physical system, which involves imaginary exponentials of the
Hamiltonian. Path integrals can also be used in quantum statistical mechanics (Ap-
pendix), and involve real exponentials of the Hamiltonian (multiplied by the inverse
of the temperature). It is the basic tool for many numerical calculations; the interested
reader can consult Zinn-Justin's book [92], or reference [94] where, in particular, the
PIMC (Path Integral Quantum Monte Carlo) methods are described.
2280

LAGRANGE MULTIPLIERS
Appendix V
Lagrange multipliers
1 Function of two variables
2 Function of variables
When a functiondepends on non-independent variables (i.e. which are related
by constraints), its extrema (maxima or minima) can be found by the Lagrange multiplier
method. A brief summary of this method is proposed in this appendix. The rst part
concerns functions of two variables, and the second part will generalize the concept to
any number of variables.
1. Function of two variables
Consider rst a real function(12)of two independent variables1and2. We
assume the fonctionto be regular, continuous, dierentiable with continuous deriva-
tives. The extrema ofcorrespond to values of the variables for which the two partial
derivatives are zero:
(12)
1
= 0 ;
(12)
2
= 0 (1)
These two relations amount to stating that the gradient ofmust be zero:
= 0 (2)
Two equations with two unknowns1and2generally admit a nite number of solutions
(pairs of values for1and2); this number can even be zero if the functiondoes not
present any extrema.
Let us now look for the extrema ofwhen the variables are no longer independent,
but must obey a constraint:
(12) = (3)
whereis a constant and(12)a regular function (continuous, dierentiable, etc.).
When this constraint is satised, the pointwith coordinates1and2is forced to
follow a curve in the plane (solid line in Figure). Imagine we place the point close to
an arbitrary pointof the curve, and move it by varying slightly its coordinates by d1
and d2. Forto remain constant, d1and d2must necessarily obey:
d=
(12)
1
d1+
(12)
2
d2= d= 0 (4)
The pointtherefore necessarily moves along the tangent to the curve, i.e. perpendic-
ularly to its gradient, as shown in Figure. As for the variation of, it is given
by:
d= d (5)
2281

APPENDIX V
Figure 1: When the constraint(12) =is satised, the point, with coordinates
1and2, is forced to move along a curve in the plane, shown as a solid line. The
tangent to this curve is perpendicular to the gradientof the function, meaning
that any small displacement of pointalong the curve must be perpendicular to this
gradient. When the displacement starts from an arbitrary point, the vectorsand
are not parallel, and the functionvaries to rst order in d= d. However,
if the variation starts from a point, such as0, where the two gradients are parallel, the
functionis stationary. Geometrically, this parallelism means that the solid line is
tangent to a contour line of the surface representing the function(12).
As in general the vectorsand are not parallel, this scalar product is not zero.
The functionthus varies to rst order ind, meaning it is not stationary at that
point.
If, however, we start from a point0on the curve where the two gradients are
parallel (or antiparallel), condition (4) means that the variation (5) is zero, and station-
arity is attained. In such a case,moves (at constant) along a curve that is tangent
at0to a contour line of the surface representing the function. Geometrically, it is
easy to understand that a displacement along a contour line keepsconstant to rst
order. Algebraically, imposing the gradients to be parallel amounts to writing that there
exists a constant, called Lagrange multiplier, such that:
= (6)
which is equivalent to saying that the dierential of the functionis zero:
d( ) =( )d
=
[ ]
1
d1+
[ ]
2
d2
= 0 (7)
2282

LAGRANGE MULTIPLIERS
This means that one must simply replace the functionby the fonction with an
arbitrary Lagrange multiplierto obtain the stationarity ofwhen its variables obey
the constraint (3).
Whenis xed, we get as before two equations with two unknowns, so that the
variables1and2are determined. Inserting them into (3) yields a value for, which
is thus xed. If, however, the Lagrange multiplieris allowed to vary, the constant
becomes a function of, and can be adjusted by changing. As an example, when
studying the canonical equilibrium (Appendix, ), one maximizes the value of
the entropy(which plays the role of the function) while keeping the average energy
valueconstant. A Lagrange multiplieris then introduced to impose the stationarity
of ; changingallows controlling the value of.
2. Function of variables
We now consider a function(12 )ofsupposedly independent variables
1,2,... ,. The extrema ofare obtained by annulling thecomponents of the
gradient of(each component being the partial derivative ofwith respect to one of
the variables):
= 0 (8)
We get equations to determineunknown variables, yielding a nite number of
extrema.
Imagine now that thevariables are no longer independent, but linked by
conditions:
(12 ) = with= 12 (9)
Consider a pointin an-dimensional space, with coordinates12 . If these
coordinates satisfy theconditions (9), their innitesimal variations obey the
relations:
d= 0 with= 12 (10)
For all the functionsto remain constant, the displacementdof pointin the
-dimensional space must be orthogonal to all the gradients. Two cases are then
possible:
(i) either the gradientbelongs to the sub-space generated by the, in
which case the orthogonality condition (10) implies thatdis also orthogonal to.
Consequently the variation d= dis zero and the stationarity is ensured.
(ii) or the gradientis not contained in that subspace, and it possesses a non-
zero component orthogonal to that subspace. One can then choosedparallel to
and obtain a rst order variation of, while satisfying theconstraints.
In conclusion, the stationarity ofis equivalent to the condition that the gradient
be contained in the subspace generated by the. This amounts to stating that
there exist Lagrange multipliers(with= 12 ) such that:
=1 1+2 2++ (11)
2283

APPENDIX V
In an equivalent way, the stationarity condition can be obtained by annulling the dier-
ential:
d( 11 22 ) = 0 (12)
and then treating the variables12 as if they were independent.
When the Lagrange multipliersare xed, each component of relation (11) yields
an equation, so that we have as many equations as variables12 . One therefore
obtains for the functiona nite number of extrema linked by the constraints, yielding
xed values for the functions1,2, ...,. A variation in the Lagrange multipliers
will change the values of these functions, which can therefore be adjusted to a value that
has been chosen in advance.
2284

BRIEF REVIEW OF QUANTUM STATISTICAL MECHANICS
Appendix VI
Brief review of Quantum Statistical Mechanics
1 Statistical ensembles
1-a Microcanonical ensemble
1-b Canonical ensemble
1-c Grand canonical ensemble
2 Intensive or extensive physical quantities
2-a Microcanonical ensemble
2-b Canonical ensemble
2-c Grand canonical ensemble
2-d Other ensembles
In quantum mechanics, as in classical mechanics, it is not possible to describe a
system having a very large number of degrees of freedom (for example a system containing
a number of particles that is of the order of the Avogadro number) with highest precision.
Such a description would in particular include the value of many quantities, for instance
many particle correlations, which uctuate rapidly and are not necessarily of interest. A
less detailed and more probabilistic description must be used, in which the state of the
system in known only statistically. The system occupies one on a series of possible states,
with a certain probability. One then says that the system is described by a statistical
ensemble. The use of a density operator (ComplementIII) to describe the physical
system is particularly convenient in this case.
We do not attempt in this appendix to give a general introduction to statistical
mechanics and its postulates. We simply summarize a number of quantum statistical
mechanics results used in several complements. For example, most of the complements
of Chapter, as well asXVIIandXVII, use the concept of chemical potentialor
of grand potential; their interpretation in the framework of the dierent statistical
ensembles will be given in this appendix.
1. Statistical ensembles
Several statistical ensembles are commonly used to describe physical systems at equi-
librium. We shall focus here on the three main ones: the microcanonical, the canonical
and the grand canonical ensembles. The rst of these ensembles provides the general
setting for introducing the two others.
1-a. Microcanonical ensemble
Consider a physical system containingparticles in a box of volume. The
energyof the system lies within an interval:
2 + 2 (1)
2285

APPENDIX VI
with . The system is isolated from its surroundings preventing any exchange of
particles or energy. We notethe eigenstates of its Hamiltonian, whereis an
index reecting the possible degeneracy of each eigenvalueof this Hamiltonian.
. Density operator, entropy
The system is supposed to have the same probability of being in any state whose
energy falls within the interval (1); no state is favored over any other. The microcanonical
density operator of the system at equilibrium is then:
eq=
1
() (2)
where()is the projector onto the subspace containing all the accessible states:
() =
+

2
=

2
(3)
and whereis the microcanonical partition function dened as:
=Tr() (4)
What relation (2) means is that the occupation probabilities of the statesare
all equal to1. Relation (3) shows that each of the projectors onto a state
contributes one unit to the trace of relation (4); the partition functionis simply the
number of terms in the summation (3), i.e. the number of levels in the energy interval
(1). If()is the density of states, we can write:
=() (5)
As in XXI, we dene the entropyas:
= Treqlneq (6)
whereis the Boltzmann constant. The are eigenvectors ofeq, with an eigen-
value equal to1ifbelongs to the interval[

2
+

2
], and equal to zero
otherwise. Ifbelongs to the interval, we have:
eqlneq =
ln
(7)
Ifdoes not belong to the interval, sincelim0ln= 0, we get:
eqlneq = 0 (8)
We now multiply the two previous relations by the braand sum overandto
get a trace. Only the bras whose energy falls within the interval will yield a non-zero
contribution. As there areof them, we obtain:
Treqlneq=ln (9)
The equilibrium value of the entropy is therefore:
= ln (10)
2286

BRIEF REVIEW OF QUANTUM STATISTICAL MECHANICS
. Temperature, chemical potential
Suppose we now change the equilibrium energyby an innitesimal amount d,
keeping the volumeand the particle numberconstant. Since no work is exchanged
with the outside (the external walls are xed), this amounts solely to a heat change:
d=d=d (11)
where we have used the usual thermodynamic denition of the entropyd= d . In
the microcanonical ensemble, the temperature is thus dened as:
1
= (12)
where we have used the partial derivative notation to emphasize that the changes are
made keeping bothandconstant.
Let us now change the particle number, keeping the volume and the energy
constant. We then dene the chemical potential(which has the dimension of an
energy) as:
=
(13)
At xed temperature, the faster the entropy (which depends on the number of
accessible levels in the energy band) grows with the particle number, the larger
the absolute value of. The chemical potential plays an essential role in the grand
canonical equilibrium as we shall see (). The third partial derivative of(with
respect to the volume) will be determined in .
Comment:
Let us insert (5) in relation (10), but rst multiplying()by and dividing
by this same quantity (this has the advantage of providing dimensionless arguments for
the logarithmic functions). This yields:
=ln[() ] +ln

(14)
In a macroscopic system, the particle number is very large, of the order of the Avogadro
number. Let us see then what happens when the particle numbergoes to innity. We
assume that the energyas well as the volume are proportional to(thermodynamic
limit). We then expect the entropy to also be proportional to. This linear variation
of the entropy cannot come from the second term in (14): even if the energy interval
is proportional to, it will only yield a much slower logarithmic variation. Most
of the variation ofactually comes from the rst term of (14), and from the fact that
the density of states increases within an exponential way: as the exponent of()
contains, this variation is phenomenally rapid. In the limit of large systems, the rst
term in (14) largely dominates the second. This is why it is often said that the entropy
characterizes the density of states of a physical system (or more precisely the number of
its quantum energy levels in a microscopic energy interval, chosen here to be equal to
).
2287

APPENDIX VI
. Entropy maximization
We now choose foran arbitrary Hermitian density operator, with positive or
zero eigenvalues whose sum is equal to1. We denote its eigenvectors andits
eigenvalues (0 1) which obey:
= 1 (15)
We assume thatis restricted to the energy band (1): all thefor which= 0are
arbitrary linear combinations of the eigenvectorsobeying (1).
An entropy can be associated with:
= Trln (16)
where is the Boltzmann constant. We are going to show that among all possible
operators, the equilibrium one,eq, maximizes this entropy. We can write:
= ln= ln (17)
and therefore:
= Tr ln = ln (18)
Any variation of theresults in a variation ofwritten as:
d= [1 +ln]d (19)
However relation (15) requires the sum of the variations dto be zero. To write
that relation (19) is zero while taking into account this constraint, we use a Lagrange
multiplier(Appendix) and obtain the equation:
[+ 1 +ln]d= 0 (20)
which must be satised for any d. Canceling the corresponding coecients leads to:
ln= 1 (21)
This means that all the non-zeromust be equal. Operatoris therefore proportional
to the projector (3). Once we normalize its trace to 1, we get (2): the microcanonical
density operator corresponds to an entropy extremum. As all theare between0and
1, relation (18) shows that this extremum is positive. To nd out if it is a maximum or
a minimum, we consider anotheroperator, whose eigenvalues are all zero except one,
equal to1; its associated entropyis zero. Consequently the extremum ofobtained
for the microcanonical equilibrium is an absolute maximum.
This result proves an important theorem: the density operator that maximizes the
entropy is the sum of the projectors onto all the accessible states, with equal eigenvalues
(the probabilities of nding the physical system in each of these states).
2288

BRIEF REVIEW OF QUANTUM STATISTICAL MECHANICS
1-b. Canonical ensemble
We now consider a physical systemSno longer isolated but in contact with a
reservoirwith which it exchanges energy; for exampleSandcould be coupled
through a wall conducting heat but remaining xed so that no work can be exchanged
between them. Let us call, and the particle number, the energy and the
volume of the reservoir.
. Density operator
Assuming the reservoirto be much larger than the systemS, its temperature
remains constant as it exchanges energy withS. According to relation (12), this implies
that its temperature, dened as:
1
= (22)
is a constant. It will be characterized by the constant:
=
1
=
1
(23)
As relation (10) showed that= ln, whereis the number of the reservoir's
accessible levels in an energy bandaround, we deduce:
ln
= (24)
This means that this number of levels varies exponentially as a function of the energy
(keepingand constant):
(25)
The total systemS+is described by an equilibrium microcanonical density
operator. Its energy eigenvectors are the tensor product
1
of the energy eigenvectors
of the systemSand the energy eigenvectors of the system:
(26)
The microcanonical density operator ofS+, with a total energytot=+ is
given by:
S+=
1
S+
tot+

2
+ =tot

2
(27)
We get the density operatoreqof the systemSby taking a trace over the reservoir:
eq=Tr S+ (28)
1
We assume that the coupling betweenSandis weak, so that its contribution to the total energy
is negligible.
2289

APPENDIX VI
In (27), the trace overof each projector is just equal to one.
The density operatoreqis simply a sum of projectors onto the energy eigenstates
, multiplied by the number of levels ofwith an energytot within an
energy band. Relation (25) shows that this number of levels varies exponentially as
=
(tot)
. Omitting the proportionality factors1S+and
tot
, we get:
eq = (29)
whereis the Hamiltonian of the systemS. Normalizing the trace of, we obtain:
eq=
1
(30)
whereis the canonical partition function dened as:
=Tr (31)
These two relations dene the density operator ofSin the canonical thermal equilibrium.
Contrary to what happened in the microcanonical equilibrium, the energy of the system
Sis no longer restricted to a small interval, but may spontaneously uctuate outside
this energy band under the eect of the coupling with the reservoir.
The thermodynamic potential of the canonical equilibrium is dened by the func-
tioncalled the free energy:
= (32)
At equilibrium, whenis given by (30), this free energy is equal to:
= + Treqln = ln (33)
and we obtain:
= ln (34)
. Minimization of the free energy
Starting from an arbitrary density operatorof unit trace, let us show that its
associated free energy will be minimal whenis equal to its value at the canonical
equilibrium (30). We rst compute the variation of:
d=Tr + (1 +ln)d (35)
This variation is zero for any donly if the operator between the inner brackets is zero,
which means:
ln=1 (36)
This indicates that , which is the canonical equilibrium operator. Finally, if
we choose forthe projector onto a state having a large positive energy,will be zero,
arbitrarily very large, and consequentlywill be very large as well. It is thus clear
that the extremum of, which occurs whentakes the equilibrium value, is a minimum.
2290

BRIEF REVIEW OF QUANTUM STATISTICAL MECHANICS
1-c. Grand canonical ensemble
We now assume that the physical systemScan exchange not only energy but
also particles with the reservoir:Sandmust be coupled through an interface the
particles can cross. As above, this reservoir is supposed large enough for its temperature
to remain constant when it exchanges energy withS. We also assume it contains a
very large number of particles, which barely changes in relative value during the particle
exchanges withS. Its chemical potential therefore remains constant:
=
=constant (37)
. Density operator
As the particle numbersof systemSand of the reservoir are no longer
constant, their state spaces are now Fock spaces (Chapter). We must add indices
and respectively to the ketsand and there are now two summations
over these indices in the expression for the microcanonical density operatorS+written
in (27). The microcanonical density operator of the total system is then:
S+=
1
S+
tot+

2
+ =tot

2
+ =tot
(38)
wheretotandtotare respectively the energy and the particle number of the total
systemS+. The argument then follows the same lines as in . A partial trace
over the reservoir leads to the density operator ofS, which is a linear combination of
projectors:
(39)
with weights corresponding to the number of states of the reservoir in an energy band
centered around=tot , the number of particles in the reservoir being=
tot . Two reservoir variables change simultaneously, instead of one for the canonical
equilibrium. Asandremain constant in (24) and (37), the entropyvaries linearly
with respect to these variables:
=
0
+
1
=
0
+ ( ) (40)
where
0
is a constant that is of no importance in what follows. Using again relation
(10) to relate the reservoir entropy to the number of statesaccessible to this reservoir,
we get:
=
( )
=
(tot tot) ( )
(41)
2291

APPENDIX VI
whereandremain constant. The same argument as above shows that the trace over
the reservoir variables lead to the following density operatorfor the systemS:
eq=
1
(42)
with:
gc=Tr (43)
whereis total particle number operator ofS.
. Grand potential
The thermodynamic potential for the grand canonical ensemble is the grand po-
tentialdened as:
= (44)
Following the same demonstration as forin the canonical ensemble, we can show that
the equilibrium value of this potential is:
= ln (45)
If we letvary, we can show, as above, that this potential reaches a minimum whenis
equal to (42); a detailed demonstration is given in XV.
2. Intensive or extensive physical quantities
Take a macroscopic physical systemSat equilibrium, and divide it into two subsystems
of equal sizesSandS; one can imagine that a wall separatesSfromS. Certain
physical quantities associated withSor, taken separately, are half of what they were
forS: the volumes, the energies, the particle numbers, the entropies, etc. Such quantities
are said to be extensive. Inversely, other physical quantities do not change upon this
division: the particle number per unit volume, the temperature, the chemical potential,
etc. Such quantities are said to be intensive. In a general way, when a macroscopic
physical system of volumeis divided into several macroscopic parts of volumes1,2,
etc., the physical quantities measured in each part and which are proportional to their
respective volume are said to be extensive, and those which remain constant are said to
be intensive.
As for the ensembles studied above, their description involves a mixture
2
of exten-
sive and intensive variables:
(i) In the microcanonical ensemble, the three independent variables describing the
physical system at equilibrium are the three extensive variables,and the system's en-
ergy; the other physical quantities (temperature, entropy, chemical potential, etc.) are
2
including at least one extensive variable, otherwise the system's size would not be determined.
2292

BRIEF REVIEW OF QUANTUM STATISTICAL MECHANICS
considered functions of these variables. The thermodynamic potential is the entropy,
extensive and directly related to the logarithm of the microcanonical partition function.
(ii) In the canonical ensemble, the three independent variables include two exten-
sive variablesandas well as an intensive variable(or). The thermodynamic
potential is the free energy, an extensive function directly related to the logarithm of
the canonical partition function.
(iii) In the grand canonical ensemble, there is only one independent extensive vari-
able, the volume, and two intensive variablesand. The thermodynamic potential
is the function, extensive and directly related to the logarithm of the grand canonical
partition function.
For a macroscopic system, the three ensembles are generally considered equivalent.
The statistical descriptions are, however, dierent. In the canonical equilibrium, for
example, the energy is not restricted to an intervalbut can uctuate and take
on values outside this interval. However, for a macroscopic system, the uctuations in
energy are very small compared to its average value. Assuming that the system's energy
is conned within a xed bandis a valid approximation and allows taking for
^
the microcanonical energy:
^
= (46)
Another example, in the grand canonical ensemble, is the particle number, which uctu-
ates around its average value
^
. For a macroscopic system, the relative value of these
uctuations is in general
3
very small, and the average value
^
is practically equal to
the particle numberof the microcanonical or canonical ensembles:
^
= (47)
2-a. Microcanonical ensemble
Relations (12) and (13) give the partial derivatives of the entropywith respect
to the variablesand; we now compute that derivative with respect to the volume
.
Let us change the physical system volumeby a small quantity d, keeping the
particle numberconstant, and without any heat exchange (the system is surrounded
by isolating walls). The system, having an internal pressure, is doing the workd,
which means that its internal energy varies as:
d= d (48)
As there is no heat exchange, d= 0, and the thermodynamic relation d=dmeans
that the entropy does not change either:
d=
d+d= 0 (49)
3
There are exceptions to this rule: for a Bose-condensed ideal gas, the grand canonical uctuations
of the particles' number remain large for a macroscopic system. This is a very special system for which
the canonical and grand canonical ensembles are not equivalent for certain physical properties.
2293

APPENDIX VI
Inserting relation (12) in this result and multiplying by, we obtain:
d+d= 0 (50)
As relation (48) shows that the pressureis given byddand taking (10) into
account, we nally get:
=
=
ln
(51)
which denes the pressure in the microcanonical ensemble.
We already studied, in , the entropy changes due to variations of either
(keepingandconstant) or(keepingandconstant). The present calcu-
lation is the last step for obtaining the three partial derivatives of the microcanonical
thermodynamic potential, and we can express its total derivative as:
d=
1
d+dd
(52)
2-b. Canonical ensemble
For a macroscopic system, we just saw thatcould be replaced by the micro-
canonical energyin the denition (32) of the free energy. Taking the dierential of
(32) then leads to:
d=d d d (53)
Using (52) in thedterm of this equation, the dterms cancel out and we get:
d= d d+d (54)
This is the total dierential of the thermodynamic potential in the canonical ensemble.
This relation allows a physical interpretation of the chemical potential: it is the
gain in free energy when one particle is added to the system
4
, keeping constant the
temperature and the volume of the system. As for the pressure, it is given by:
=
(55)
or, using (34):
=
ln
(56)
which is similar to (51). We have obtained the pressure of the physical system as a
function of its volume and its temperature, i.e. its equation of state.
4
When the temperature is zero, the free energy is just the energy, andis the increase of energy
when one particle is added.
2294

BRIEF REVIEW OF QUANTUM STATISTICAL MECHANICS
To compute the average energy, we can use relations (30) and (31), which
yields:
=
1
Tr =
1
Tr (57)
where the partial derivative is taken keepingandconstant. We then have:
=
1
=
ln
(58)
.
2-c. Grand canonical ensemble
In a macroscopic system, as the particle number generally uctuates very little
in relative value, we can replacebyin the denition (44) of the thermodynamic
grand potential. This leads to:
d =d d d (59)
Using (54) in this relation, thedterms cancel out and we are left with:
d = d d d (60)
In this ensemble, the volumeis the only extensive variable. For a xed tempera-
ture and chemical potential, and for a large volume, we get a macroscopic system whose
energy, entropy and particle number are proportional to. We simply get:
= (61)
The grand potential divided byyields the pressure directly, without any partial deriva-
tive.
Taking (45) into account, the average particle number and the pressure obey:
=

=
ln
=

=ln (62)
Using these two equalities to eliminate the chemical potential, we get the particle
number in a given volume as a function of the pressureand the temperature(equation
of state for the physical system).
2-d. Other ensembles
We have studied the three most commonly used statistical ensembles, but there are
others, as for example the isothermal-isobaric ensemble. In this ensemble, the systemS
is coupled with a reservoir allowing exchanges of energy and volume, but not particles;
the numberremains xed. The only extensive variable is precisely this variable,
2295

APPENDIX VI
the other two, the temperatureand the pressure, being intensive. The thermody-
namic potential associated with this isothermal-isobaric ensemble is the Gibbs function
dened as:
=+ (63)
As before, we take the dierential of this function and note, using (52), that the terms
danddcancel out. We then get:
d=d d+d (64)
The functionis extensive. It increases as the particle numbergets larger (for xed
pressure and temperature), and for a macroscopic system it is proportional to the system
size:
= (65)
Varying both the temperature and the pressure of an ensemble ofparticles, we can
get the resulting variation of the chemical potentialby dividing (64) byand then
setting d= 0. This yields the Gibbs-Duhem relation:
d=
dd
(66)
This ensemble is particularly useful in the study of a two-phase equilibrium such as a
liquid and its vapor, both at the same pressure and temperature.
We have presented a brief review of the general principles of statistical mechanics.
For more details, the reader may consult, for example, the following references [95,,
97,].
2296

WIGNER TRANSFORM
Appendix VII
Wigner transform
1 Delta function of an operator
2 Wigner distribution of the density operator (spinless par-
ticle)
2-a Denition of the distribution, Weyl operators
2-b Expressions for the Wigner transform
2-c Reality, normalization, operator form
2-d Gaussian wave packet
2-e Semiclassical situations
2-f Quantum situations where the Wigner distribution is not a
probability distribution
3 Wigner transform of an operator
3-a Average value of a Hermitian operator (observable)
3-b Special cases
3-c Wigner transform of an operator product
3-d Evolution of the density operator
4 Generalizations
4-a Particle with spin
4-b Several particles
5 Discussion: Wigner distribution and quantum eects
5-a An interference experiment
5-b General discussion; ghost component
Introduction
In classical mechanics, it is possible to specify with an arbitrary precision both the po-
sitionrand the momentump(and hence the velocity) of a particle. If the state of the
particle is dened in a statistical way, one describes the classical particle by a distribution
cl(rp)in the phase space (Appendix, ) , which can be any positive function,
normalized to unity. This distribution can, for example, include correlations between
the particle's position and velocity. In quantum mechanics, the situation is dierent.
It is true that one often uses two representations, one in position space, the other in
momentum space, and that we go from one to the other via a Fourier transformation.
But in quantum mechanics these two representations are exclusive: in the position rep-
resentation, one loses all information on the particle's momentum, and conversely, in
the momentum representation one loses all information on the particle's position; conse-
quently, no information on an eventual correlation between position and momentum can
be obtained.
It is interesting to introduce a quantum point of view intermediate between these
two extremes, to keep at the same time information about position and momentum,
2297

APPENDIX VII
while obeying the general rules of quantum mechanics; these rules impose a limitation
on the information precision. The Wigner transformation oers this intermediate point
of view as it introduces a quantum mechanical function(rp)that allows computing
average values in the same way as with a classical distributioncl(rp). Historically,
this transformation was introduced
1
in 1932 by Wigner [98,] as he was working on the
quantum corrections to thermal equilibrium, but it turned out to be a much more general
tool. It yields very naturally semiclassical expansions in powers of}while studying, for
instance, the temporal evolution of a quantum system. We shall also show that it provides
a quantization method, leading in particular to correctly symmetrized expressions for
quantum operators starting from classical functions of the positions and momenta. There
are numerous domains in physics where the Wigner transform has proven useful, and
sometimes indispensable.
This appendix will show how to associate with any quantum density operator
aWigner distribution(rp), sometimes called a semiclassical distribution, and
we will discuss a certain number of its properties. In a similar way, to any operator
(observable) acting in the state space, we can associate a Wigner transform(rp)
that is a simple function ofrandp.
In classical or semiclassical situations (i.e. when the spatial variations of physical
quantities occur over large enough distances), the function(rp)possesses all the
properties of a classical distribution: it is a positive function that, multiplied by(rp)
and then integrated over the variablesrandp, does yield the average value of the
operator. In this case, we will show that the Wigner distribution simply describes the
ow of the probability uid (Chapter, ). As(rp)allows keeping track
of the correlations between position and momentum, this function is particularly useful
in a number of cases, such as in the theory of quantum transport, with its numerous
applications.
In the general case where the quantum eects are important (rapid spatial vari-
ations), the classical relations between the distribution and the average values are still
valid, meaning the Wigner distribution continues to be quite useful. This distribution
shows, however, a signicant dierence with a probability distribution: the function
(rp)can sometimes take on negative values. Furthermore, as we shall point out
later, it can sometimes present ghost components where it is dierent from zero at
points where the probability of nding the particle is zero. It is thus not possible to
interpret the product(rp)d
3
d
3
as the probability for the particle to occupy an
innitesimal cell d
3
d
3
of the phase space centered at(rp); moreover, when dealing
with innitesimal volumes such an interpretation would be in clear contradiction with
Heisenberg's uncertainty relations. Therefore, one should consider(rp)to be a
quasi-distribution, a tool for computing all the average values without being a real
probability distribution, even though we shall use the word distribution, following com-
mon usage.
This appendix introduces the tools necessary for the study of these dierent situ-
ations. We shall obtain, in particular, gradient expansions directly yielding expansions
in powers of}. We rst introduce in
operator. This form will be useful, in , for dening the Wigner distribution of a spin-
less particle density operator. Several of its characteristics will be studied, in particular
1
Wigner does mention in his article that the same transformation had already been used by L. Szilard,
but in another context.
2298

WIGNER TRANSFORM
when dealing with a Gaussian wave packet. We then focus in Wigner trans-
formof operators and show how it can be associated with the Wigner distribution of the
density operator, leading to computations similar to those corresponding to a classical
distribution. An important step will be the computation of the Wigner transform asso-
ciated with a product of operators. Generalization of these concepts to take into account
the spin, as well as the possible presence of several interacting particles, is discussed in
. Finally, we focus in
using it to analyze a quantum interference experiment. We will show, in particular, how
ghost components of the Wigner transform can appear, rapidly change sign, and are
the signature of quantum eects.
1. Delta function of an operator
Consider a Hermitian operatorhaving a continuous spectrum, and whose eigenvalues
are noted. We dene the operator(), depending on a real continuous parameter
, as:
() =
1
2
d
( )
(1)
It is, in a way, a delta function of an operator associated with the dierence between
operatorand the constant. We note the eigenvectors of; the index(assumed
to be discrete) accounts for the possible degeneracy of the eigenvalue. In a quantum
state dened by the density operator, the average value of this operator is:
()=
1
2
dTr
( )
=
1
2
dd
( )
=
1
2
dd
( )
(2)
The integral over dyields2( ), and we obtain:
()= =() (3)
where()dis the probability, in a measurement associated with operator, of nding
a result in the interval[+d].
2. Wigner distribution of the density operator (spinless particle)
Imagine now that the physical system under study is a spinless particle, described by
a density operator. Our purpose is to introduce a function(rp)yielding simul-
taneously information on the probability of measurement results on either the position
operatorR, or the momentum operatorP. As these are incompatible observables, we
should not expect to make highly accurate predictions concerning both types of mea-
surements: the precision is limited by Heisenberg's uncertainty principle. This function
2299

APPENDIX VII
will have to make a compromise between the two types of information, resulting from an
unavoidable quantum uncertainty. As already mentioned in the introduction, this func-
tion is actually a quasi-distribution, even though it is commonly called the Wigner
distribution.
2-a. Denition of the distribution, Weyl operators
By analogy with (1), we can introduce the Weyl operator, which will be noted
(rp). It is, in a way, a delta function of an operator, associated at the same time
with the operatorsRrandPp. We set:
(rp) =
1
}
3
d
3
(2)
3
d
3
(2)
3
e
[(Rr)+x(Pp)}]
(4)
where the integration variablehas the dimension of a wave vector (the inverse of a
length), andxthe dimension of a length. This operator is Hermitian, as can be shown by
changing the sign of the two integration variables. For each quantum state, the Wigner
distribution(rp;)is dened as the average value in that state of the Weyl operator:
(rp;) =(rp) (5)
If the system under study is characterized by a density operator()whose trace equals
1, this denition amounts to:
(rp;) =Tr()(rp) (6)
On the other hand, if the system is described by a normalized state vector , this
denition becomes:
(rp;) = ()(rp) () (7)
Note that the density operator as well as the state vector are dimensionless. Taking into
account the factors}introduced in (4),(rp)and (rp)have the dimensions of
}
3
; the product(rp)d
3
d
3
is thus dimensionless, as a probability should be. We
are now going to show that this distribution has numerous useful properties.
Later, we shall need the matrix elementsr1(rp)r2of the operator(rp)
in the position representation. As demonstrated below, they are written:
r1(rp)r2=
1
(2})
3
p(r2r1)}
r1+r2
2
r (8)
Demonstration:
The demonstration uses relation (63) of ComplementII, which expresses the exponential
of the sum of two operatorsandas a product of exponentials, provided both operators
commute with their commutator[]:
+
=
1
2
[ ]
(9)
2300

WIGNER TRANSFORM
We choose:
=(Rr) and =(Pp)x} (10)
The commutator of these two operators:
[] =x (11)
is a number, so that bothandcommute with their commutator. Inserting relation
(9) into (4), we get:
r1(rp)r2=
1
}
3
d
3
(2)
3
d
3
(2)
3
x2
r1
(Rr)(Pp)x}
r2 (12)
Now relation () of ComplementIItells us that the action of operator
Px}
is a
simple translation of the position eigenvalue byx, which means that:
(Pp)x}
r2=
px}
r2x (13)
The function to be integrated in relation (12) then becomes:
x2
r1
(Rr)(Pp)x}
r2=
x2(r2xr) px}
(r1r2+x)(14)
The delta function, integrated over d
3
in (12), leads to replacing in the exponentxby
r2r1, andr2xbyr1. We then get:
r1(rp)r2=
1
(2})
3
d
3
(2)
3

r
1
+r
2
2
rp
r
2
r
1
}
(15)
The integration over d
3
then yields a delta function, and we obtain relation (8).
2-b. Expressions for the Wigner transform
Denition (5) of the Wigner transformation may lead to various expressions for
the Wigner transform, depending on the representation used in the state space.
. Position representation
Using the position representation to calculate the trace appearing in (6), we get:
(rp;) =d
3
2d
3
1r2()r1r1(rp)r2
=d
3
d
3
R+
y
2
()R
y
2
R
y
2
(rp)R+
y
2
(16)
where, on the second line, we used the integration variablesR= (r1+r2)2andy=
r2r1. Using relation (8) in this expression, we get the function(Rr), to be
integrated over d
3
, which leads to:
(rp;) =
1
(2})
3
d
3 py}
r+
y
2
()r
y
2
(17)
2301

APPENDIX VII
This relation is often used as a denition of the Wigner distribution. Its integration over
d
3
(2})
3
yields a function(y), which leads to:
d
3
(rp;) =r()r=(r) (18)
We conrm a property of the classical distributions in phase space: the integral of the
distribution over the momenta yields the probability density(r)of nding the particle
at pointr.
In the particular case where the particle is described by a pure state see
relation (7) the denition of the Wigner distribution becomes:
(rp;) =
1
(2})
3
d
3 py}
r+
y
2
; r
y
2
;
(19)
where (r;)is the wave function in the position representation:
(r;) =r () (20)
. Momentum representation
As position and momentum play symmetrical roles in the argument, we expect to
nd a similar relation for the Wigner distribution, involving now the matrix elements of
the density operatorin the momentum representation. This is indeed the case, and we
are going to show that:
(rp;) =
1
(2})
3
d
3 qr}
p+
q
2
()p
q
2
(21)
This expression is the exact analog of (17); it can be considered as an alternative denition
of the Wigner distribution. As for the analogy of the property expressed by (18), it is
easy to show that:
d
3
(rp;) =p()p (22)
Just as for a classical distribution, the integral over the positions of the Wigner distri-
bution yields the probability density of nding a given momentump.
Demonstration:
Inserting in the matrix element of (17) two closure relations on normalized momentum
plane wavesqandq, yields the two scalar products:
r+
y
2
q=
1
(2})
32
q(r+
y
2)}
qr
y
2
=
1
(2})
32
q(r
y
2)}
(23)
and we can write:
(rp;) =
1
(2})
6
d
3 py}
d
3
d
3
q(r+
y
2)} q(r
y
2)}
q()q (24)
2302

WIGNER TRANSFORM
The summation over d
3
yields a delta function:
1
(2})
3
d
3
q+q
2
py}
=
q+q
2
p (25)
so that if we takeQ= (q+q)2andq=qqas integration variables, we obtain
(21).
In the particular case where the particle is described by a pure state , relation
(21) becomes:
(rp;) =
1
(2})
3
d
3 qr
p+
q
2
;
p
q
2
;
(26)
where
(p;)is the wave function in the momentum representation: (p;) =p () (27)
If the wave function is factored
2
:
(p;) =(;)(;)(;) (28)
relation (26) shows that the Wigner transform is also factored:
(rp;) =(;) (;) (;) (29)
with:
(;) =
1
2}
d
}
+2
;
2
; (30)
The other two components(;)and (;)are dened in a similar way. In
this particular case, one can reason independently for the three dimensions.
. Inverting the relations
We just saw that to each density operator corresponds a unique and well dened
Wigner distribution. Inversely, starting from this distribution, one can reconstruct the
corresponding density operator via its matrix elements. To take the inverse Fourier
transform of (17), we multiply this relation by
pz}
and integrate over d
3
:
d
3 pz}
(rp;) =
1
(2})
3
d
3
d
3 p(zy)}
r+
y
2
()r
y
2
(31)
On the right-hand side, the integral over
3
yields a function(2})
3
(zy), so that
this equality becomes:
d
3 pz}
(rp;) =r+
z
2
()r
z
2
(32)
2
Whether the wave function is factored in the momentum or position representation is equivalent.
2303

APPENDIX VII
Settingr1=r+z2andr2=rz2, we obtain the matrix elements ofin the position
representation:
r1()r2=d
3 p(r1r2)}
r1+r2
2
p;
(33)
Knowing the Wigner distribution(rp)thus denes the operatorin a unique way.
In a similar way, multiplying (21) by
pr}
, integrating over d
3
, then setting
p=p1p2andp= (p1+p2)2, yields the inversion relation in the momentum
representation:
p1()p2=d
3 (p2p1)r}
r
p1+p2
2
;
(34)
2-c. Reality, normalization, operator form
Let us take the Hermitian conjugate of relation (17). As the density operator is
Hermitian, the matrix element on the right-hand side becomes:
r+
y
2
()r
y
2
=r
y
2
()r+
y
2
(35)
Changing the sign of the integration variableythen yields again relation (17); the dis-
tribution(rp)is therefore equal to its complex conjugate, meaning it is real.
We now compute the integral of(rp)over the entire phase space. Summing
(18) over d
3
, we get:
d
3
d
3
(rp;) =d
3
r()r=Tr()= 1 (36)
where the last equality comes from the fact that the density operator has a trace equal to
one. The Wigner distribution of the density operator is thus a real function, normalized
to one in phase space, as is the case for a classical distribution.
Comment:
The density operator must obey a stronger constraint than having its trace equal to
unity. It is dened as positive denite, meaning that for any ket, we must have:
() 0 (37)
Now this condition is not merely equivalent to a positivity condition for the Wigner
transform. In fact, we shall see below that the Wigner distribution of a density operator
can become negative at certain points of phase space. However, the only Wigner dis-
tributions(rp;)acceptable for describing quantum systems are those that lead to
a density operator obeying constraint (37). To know if a distribution in phase space is
acceptable or not, is thus more dicult to decide in quantum mechanics than in classical
mechanics.
2304

WIGNER TRANSFORM
We now show that relations (33) and (34) can be written in a simple operator
form:
() = (2})
3
d
3
d
3
(rp;)(rp) (38)
In this relation,(rp;)is the Wigner distribution, hence a function of position and
momentum, but (rp)is the Weyl operator dened in (4). To prove relation (38), we
calculate the matrix element of this equality between the brar1and the ketr2to
verify that we indeed obtain relation (33). Taking (8) into account, the matrix elements
of the right-hand side are:
(2})
3
d
3
d
3
(rp;)r1(rp)r2
=d
3
d
3
(rp;)
p(r2r1)}
r1+r2
2
r
=d
3 r1+r2
2
p;
p(r2r1)}
(39)
which is equivalent to the right-hand side of (33).
2-d. Gaussian wave packet
A particular case where the computation can be completed is the one-dimensional
Gaussian wave packet, studied in ComplementI. Relation (1) of this complement yields
the normalized wave function()of such a wave packet in the position representation.
We slightly modify it to center the wave packet at an arbitrary non-zero position0, and
replace the variableby=}(setting in particular0=}0). We obtain:
() =
(2)
34
d
}
2
( 0)
2
4}
2
( 0)}
=
2
2
14
0( 0)} ( 0)
22
(40)
where the second equality corresponds to relation (9) of ComplementI(within an0
translation along). The wave functions(2})
12 }
correspond to plane waves
with momentum (normalized with respect to that momentum); looking at the rst
equality in (40) we recognize the wave function in the momentum representation:
() =
1
(2)
14
}
2
( 0)
2
4}
2
0}
(41)
The Wigner distribution (30) can then be written (to simplify, we temporarily
ignore the time dependence):
() =
(2)
32
}
2
d
2
4}
2( 0+
2)
2
+( 0
2)
2
(
0
)
}
=
(2)
32
}
2
2
2}
2
( 0)
2
d
22
8}
2
(
0
)
} (42)
2305

APPENDIX VII
The integral over dis a Fourier transform whose value can be obtained by replacing
by
2in relation (50) de l'Appendice. We get:
() =
(2)
32
}
2
2
2}
2
( 0)
2
2}2}2
2( 0)
22
=
1
}
2
2}
2
( 0)
2
2
(
0
)
2
2
(43)
Taking (40) and (41) into account, we see that the Wigner distribution is simply
the product of the probability densities in the position and momentum spaces:
() =()
2
()
2
(44)
This result is particularly simple and shows that the Wigner transform of a Gaussian wave
packet (40) contains no correlations between the variablesand. It can be factored
into two Gaussian functions, one concerning the momentum, the other the position. The
rst one is centered on the average momentum0, and has a width of the order of};
the second, on the average position0, with a width of the order of. These two widths
are within the boundaries imposed by Heisenberg relations. Note that, in this case, the
Wigner transform remains positive for all the values of its variables, as will also be the
case for the semiclassical situations we consider in .
Comment:
In the preceding paragraph, we ignored the time dependence of the wave packet. To take
it into account and assuming we are dealing with a free particle, one can multiply, in
(41),
()by , with:
=
2
2}
(45)
whereis the mass of the particle. This introduces, in the integral on the second line
of (42), an additional exponential
(})
whose eect on the Fourier transform is
to make the following substitution:
(46)
This simply corresponds to the motion of the particle with a velocity. Making
this substitution in (43), we nd that the Wigner distribution is still a product of two
Gaussian functions, but no longer the product of a function of momentum by a function
of position: correlations have appeared between the momentum and position variables.
2-e. Semiclassical situations
To what extent is it possible to consider the Wigner transform to be a true prob-
ability distribution? Relations (18) and (22) seem to be in favor of it, as they show that
integrating that distribution over the momenta (or over the positions) actually yields
a probability distribution of nding the particle at a given point (or with a given mo-
mentum). These two marginal distributions thus obtained by integration are both
probability distributions. But this is not sucient to ensure that the function(rp)
2306

WIGNER TRANSFORM
itself (before integration) has the same property. Actually, we already mentioned in the
introduction that it is not possible, in general, to interpret the product(rp)d
3
d
3
as yielding directly the probability for a particle to occupy an innitesimal cell d
3
d
3
of
phase space, centered at(rp). Such a probability distribution is meaningless in quan-
tum mechanics, as Heisenberg's relations forbid the existence of a quantum state dened
with an arbitrary precision both in position and momentum spaces.
There are, however, some simple cases, that we shall call semiclassical, where the
Wigner transform is very similar to a classical probability distribution. They correspond
to situations we will now study, where the physical quantities vary suciently slowly
in space compared to a scale we shall dene explicitly. In the following section, we
will consider more general situations, where the properties of the Wigner transform are
radically dierent. In particular, the Wigner transform can become negative, which
immediately excludes any interpretation in terms of probability density.
. Wave packet with slow spatial variations
Consider the wave function:
(r) =(r)
(r)
(47)
where(r)is the modulus of the wave function and(r)its phase. The probability den-
sity of presence is then[(r)]
2
, while relation (D-17) of Chapter
currentJ(r):
J(r) =
}
[(r)]
2
r(r) (48)
The matrix elements of the corresponding density operator are written:
r()r=(r)(r)
[(r)(r)]
(49)
We assume that, in the vicinity of each pointr, the wave function behaves locally as a
plane wave:
(r) (r)
[K(r)r+(r)]
in the vicinity of each pointr (50)
and that the two functions(r)andK(r), as well as the phase(r), vary slowly
in space: their variations are negligible over distances of the order of the de Broglie
wavelength2(r). Whenrandrare close enough, one can expand the argument
of the exponential in (49); the matrix elementsr()rof the density operator are
then written:
r()r (r)(r)
K(rr)
(51a)
whereKis dened as:
K=r(r=
r+r
2
) (51b)
2307

APPENDIX VII
. Density operator; link with the probability uid
To characterize a semiclassical situation in a more general way, we will now reason
in terms of a density operator, without restricting our study to a pure state as we did
earlier. To start, we assume there is no long-range non-diagonal order
3
:
r()r 0 ifrr (52)
whereis a macroscopic coherence length. For a pure state,would be determined by
the size of the domain where the wave function has a non-zero modulus(). For a
statistical mixture of states, we have a dierent situation: the phases of the various wave
functions may interfere destructively at shorter distances, so thatcan be much smaller.
Nevertheless, we shall assume thatremains larger than a few de Broglie wavelengths
2(r)and that whenrr., the non-diagonal matrix elements of the density
operator vary in a similar way as (51a):
r()r (rr)
K(rr)
ifrr. (53)
This expression is simply the generalization of (51a), only valid for a pure state;
the real function(rr)replaces the modulus product(r)(r). Both functions
(rr)andKare supposed to remain practically constant as the variablesrandr
vary by a quantity of the order of.
With these assumptions, the values of the integration variable giving a signicant
contribution to the integral in relation (17) correspond toy., so that we can write:
(rp)
1
(2})
3
d
3 py}
r+
y
2
r
y
2
K(r)y
(54)
where the integration domainis centered aty= 0and extends over a few coherence
lengths. As the functionis practically constant in this domain, and since(rr) =
r()r, we get:
(rp)
1
(2})
3
r()rd
3 [K(r)ypy}]
(55)
or else:
(rp)r()r
[pp0(r)] (56)
To write this expression, we have used the following denitions:
(p) =
1
(2})
3
d
3 py}
(57)
and:
p0(r) =}K(r) (58)
3
The concept of long-range non-diagonal order is introduced in ComplementXVI, ,
where, in particular, its relation with Bose-Einstein condensation is established. The present hypothesis
concerning the absence of long-range order prevents()from being the one-body density operator of a
system of condensed bosons.
2308

WIGNER TRANSFORM
The function
(p)is a momentum distribution centered atp= 0, with a width
}. It is normalized to unity as the integral of
(p)over the momenta yields a
function(y), which integrated over d
3
is equal to one. Note that in (56) this function
takes on its value for a momentum equal topp0(r), which means the momentum
distribution is centered at the valuep0(r). As this momentum value depends onr,
correlations between position and momenta are now introduced in(rp).
Expression (56) for the distribution(rp)can be interpreted as a classical
distribution in the probability uid phase space: it is the product of the local probability
densityr()rby a function of momentum
[pp0(r)]centered around the value
p0(r)dened in (58). Now thisp0(r)value is precisely the momentum value that, divided
by(to go from momentum to velocity) and multiplied by the probability density,
yields the uid probability currentJ(r). Note that the distribution keeps a certain
width aroundp0(r), of the order of}, as required by Heisenberg's uncertainty relation.
To sum up, in such semiclassical situations, the Wigner distribution directly reects the
spatial variation of the probability, and of its associated local current. It simply describes
the ow of a probability uid (III, ), as does the distribution in phase space
of an ensemble of classical particles forming a moving uid.
2-f. Quantum situations where the Wigner distribution is not a probability distribution
In the previous examples, the properties of the Wigner distribution are very similar
to those of a classical distribution. This is, however, not always the case: as surprising
as it may seem, the Wigner transform can, in general, become negative.
. Odd wave function
A very simple case oers such an example. In a one-dimension problem, imagine
that the system has an odd wave function, as is the case for example for the rst excited
state of the harmonic oscillator. We then have, according to relation (19):
(= 0= 0) =
1
2}
d
22
=
1
2}
d
2
2
(59)
which is obviously negative. As odd wave functions often occur in quantum mechanics,
we see that there exist numerous situations where the Wigner distribution has some
properties unexpected for a distribution. Strictly speaking, the term quasi-distribution
should always be used.
. Two-peak wave function
Imagine now the particle wave function is the sum of two wave packets, one local-
ized around= +, the other around=:
() =
1
2
( ) + (+) (60)
where the wave function()is normalized; the relative phase factoris arbitrary. For
the sake of simplicity, we assume that()is zero when and that it is even. We
2309

APPENDIX VII
also suppose that in our case, meaning that the two wave packets forming the
total wave function are well separated.
Let us compute the Wigner distribution at point= 0, therefore at a point where
the wave function()is zero. In one dimension, relation (19) is written as:
(= 0) =
1
4}
d
}
2
+
2
+
2
+
2
+ (61)
In this expression, the functionsare zero if their argument's modulus is larger than.
As an example,
2
is dierent from zero only if2, whereas
2
is
dierent from zero only if2; consequently their product is always zero. Actually,
in the product of the two brackets, only the crossed terms are non-zero, and we obtain
(with our assumption that the functionis even):
(= 0) =
1
4}
d
}
2
2
+ +
2
2
(62)
Changing the sign of the integration variablefor the second term in the bracket, we
can write:
(= 0) =
1
2}
dcos
}
+
2
2
(63)
In the limit where the widthbecomes very narrow, the squared modulus of the wave
functionin the integral behaves as a delta function( 2), and we get:
(= 0)
1
}
cos
2
}
+ (64)
This result illustrates two properties of the Wigner distributions that both seem
quite surprising. The rst is that the distribution is non-zero at point= 0, whereas
the probability of nding the particle at this position is strictly zero. The second is that
the distribution is an oscillating function of momentum, taking successively positive and
negative values, whereas a classical distribution always remain positive or zero. These
two properties are actually related: integrating the distribution over all possible momenta
yields zero, which is in agreement with relation (18) stating that the integral of the
Wigner distribution over the momenta yields the probability of the particle's presence at
each point. More details on the properties of a two-peak wave function will be given in
.
3. Wigner transform of an operator
Consider now any operatoracting in the particle state space. We dene its Wigner
transform(rp)in the same way as for a density operator, but without the prefactor
1(2})
3
that appears in front of the integrals in (17) and (21):
(rp) =d
3 py}
r+
y
2
r
y
2
=d
3 qr}
p+
q
2
p
q
2
(65)
2310

WIGNER TRANSFORM
To simplify the notation, this denition does not include a time dependence; one can,
however, directly replaceby()and (rp)by (rp;), without any other
change. The inversion relations (33) and (34) now become:
r1r2=
1
(2})
3
d
3 p(r1r2)}
r1+r2
2
p
p1p2=
1
(2})
3
d
3 (p2p1)r}
r
p1+p2
2
(66)
Taking the complex conjugate of relation (65) shows that the Wigner transform of a
Hermitian operatoris necessarily a real function. Similarly, the fact that the complex
conjugate of (66) is real indicates that it is a sucient condition for hermiticity.
As the prefactor1(2})
3
is no longer included in the denition (65), the equivalent
of relation (38) is now:
=d
3
d
3
(rp)(rp) (67)
We saw previously that the operator(rp)is Hermitian. The above relation then
allows building a Hermitian operatorfrom any real function(rp)of position and
momentum. In other words, we found a quantization procedure for any classical function,
often called Weyl quantization or phase space quantization [,,]. Starting
from two functions(rp)and (rp), whose product obviously commutes, this
procedure yields two operatorsandthat, in general, do not commute. Such an
operation, which introduces in phase space a non-commutative structure, is sometimes
referred to as geometric quantization.
3-a. Average value of a Hermitian operator (observable)
We can now compute the average value of operatorin the quantum state dened
by the density operator():
=Tr()=d
3
1d
3
2r1()r2r2r1 (68)
We are going to show that:
=d
3
d
3
(rp;) (rp) (69)
This relation is the exact analog of the relation one would obtain with a classical distri-
bution. It is the reason the Wigner transform of the density operator is referred to as a
quasi-classical distribution, or more simply as a distribution.
Demonstration:
Inserting in (68) the equalities (33) and (66) leads to:
=
1
(2})
3
d
3
1d
3
2d
3 p(r1r2)}
d
3 p(r1r2)}
r1+r2
2
p;
r1+r2
2
p (70)
2311

APPENDIX VII
We now replace the integration variablesr1andr2by the following variables:
r=
r1+r2
2
; r=r1r2 (71)
The summation over d
3
introduces a delta function:
(2)
3 pp
}
= (2})
3
pp (72)
which takes care of the integration over d
3
. We then nally obtain (69).
3-b. Special cases
In the special case in which the operatordepends only on the position operator:
=(R) and hence: r1r2=(r1)(r1r2) (73)
the rst line of (65) leads to:
(rp) =(r) (74)
The Wigner transform of the operator is then simply the function(r), which does not
depend on the momentump.
In a similar way, ifdepends only on the momentum operator:
=(P) and hence: p1p2=(p1)(p1p2) (75)
the second line of (65) leads to:
(rp) =(p) (76)
As a further illustration, let us nd an operatorwhose Wigner transform involves
both position and momentum, for example:
(rp) =rp (77)
Relation (66) yields its matrix elements:
r1r2=
1
(2})
3
d
3 p(r1r2)}
r1+r2
2
p
=
}
r1+r2
2
rr1
(r1r2) (78)
We recognize in this expression the matrix elements of the operatorP, equal to the
gradient of a delta function of the positions, multiplied by}. Note, in addition, that
r1is the result of the action of the position operator on the bra, whereasr2is the result
of the action of the position operator on the ket. This means that:
=
1
2
[RP+PR] (79)
We thus get a Hermitian operator, as expected since its Wigner transform is real. It
is however remarkable that building a quantum operator via the Wigner transforms
spontaneously introduces an arrangement of the operators' order leading to the necessary
symmetry. This property is quite general: starting from real classical functions, the
Wigner transform allows building operators symmetrized with respect to position and
momentum. This method is a real quantization procedure.
2312

WIGNER TRANSFORM
3-c. Wigner transform of an operator product
We are going to show that, in general, the Wigner transform associated with the
product of two operatorsandis not simply the product of the Wigner transforms
of each operator.
. General expression
Let us apply relation (65) to obtain the Wigner transform of a product of two
operatorsand. Inserting a closure relation on the ketszleads to:
[](rp) =d
3 py}
d
3
r+
y
2
zzr
y
2
(80)
We can then replace the matrix elements ofandby their expressions (66), which
leads to:
[](rp)
=
1
(2})
6
d
3
d
3 py}
d
3
1d
3
2
p1(r+
y
2
z)}p2(r+
y
2
+z)}
r+z
2
+
y
4
p1
r+z
2
y
4
p2 (81)
Instead of using the position representation, one can use the momentum representation;
we then must use the relations on the second lines of (65) and (66). A reasoning similar
to that used before leads to:
[](rp)
=
1
(2})
6
d
3 qr}
d
3
d
3
d
3(qp
q
2)x}(p
q
2
q)y}
x
q+p
2
+
q
4
y
q+p
2
q
4
(82)
Depending on the case, it will be easier to use either (81) or (82). These two expressions
are exact, but fairly complicated. They can be simplied, however, in a certain number
of cases.
. A few simple cases
As a rst example, imagine that operatoris simply the position operatorR
whilecan be any operator. Asis no longer dependent onp1, the integration over
d
3
1(2})
3
in (81) yields a delta functionr+
y
2
z; this allows integrating over d
3
to obtain:
[R](rp) =
1
(2})
3
d
3 py}
d
3
2
p2y}
r+
y
2
(rp2) (83)
For the term inr, the integral over d
3
of exponential
(p2p)y}
introduces a function
r(p2p)with the coecient(2})
3
. As for the term iny2, it yieldsrp2(p2p),
2313

APPENDIX VII
with the coecient(}2) (2})
3
. After integrating over d
3
2, we get:
[R](rp) =r (rp)
}
2
rp (rp) (84)
If we now reverse the order of the operatorsRand, the roles ofp1andp2are
interchanged in (81); the integration over d
3
2(2})
3
yields a functionr+
y
2
+z
and the integration over d
3
leads to:
[R](rp) =
1
(2})
3
d
3 py}
d
3
1
p1y}
(rp1)r
y
2
(85)
Compared to (83), the only change is the sign ofyin the nal bracket, so that we simply
obtain the nal result by changing the sign of the gradient on the right-hand side of (84).
This means that the Wigner transform of the commutator is:
[R](rp) =
}
rp (rp) (86)
Starting from (82), the same reasoning leads to:
[P](rp) =p (rp) +
}
rr(rp) (87)
This relation can now be iterated to obtain:
P
2
(rp) =p
2
(rp) + 2
}
prr(rp)}
2
r(rp) (88)
We then get the expression for the Wigner transform of the commutator of the momentum
squared and any operator:
P
2
(rp) =
2}
prr(rp) (89)
This relation will be useful for what follows.
. Gradient expansions
We now show that relation (81) can be expressed as a series expansion of higher
order derivatives of the two functionsand , of the form:
[](rp) = (rp) (rp) +
}
2
(rp) (rp)+ (90)
where we have used the classical denition of the Poisson bracket [103,] of classical
Lagrangian mechanics:
(rp) (rp)
=rr(rp)rp (rp)rr(rp)rp (rp) (91)
This shows that, to lowest order in}, the Wigner function of an operator product is simply
the product of the Wigner transforms of these operators. To rst order, a correction
must be added, which contains the Poisson bracket of the two Wigner transforms. It
2314

WIGNER TRANSFORM
is remarkable that purely quantum considerations bring in this classical Poisson bracket
denition; this explains why these results are well suited for the study of the classical
limit of quantum mechanics.
In (90), the expansion is limited to the contribution of the rst order derivatives of
the two functions. The following terms involve higher order derivatives and, consequently,
higher powers of}(the corresponding result is called the Groenewold's formula; see
for example [99]).
Demonstration:
Let us make in (81) the following change of momentum integration variables:
P=
p1+p2
2
q=p1p2 (92)
(despite the notation with a capital letter,Pis a classical variable, not an operator).
This leads to the new expression:
[](rp) =
1
(2})
6
d
3
d
3
d
3
d
3 (Pp)y}q(rz)}
r+z
2
+
y
4
P+
q
2
r+z
2
y
4
P
q
2
(93)
If the two Wigner transformsand vary slowly with position and momentum, we
can use the expansions:
r+z
2
+
y
4
P+
q
2
=
r+z
2
P+
y
4
rr +
q
2
rp +
r+z
2
y
4
P
q
2
=
r+z
2
P
y
4
rr
q
2
rp + (94)
Keeping only the rst term in each of these two expansions (zero-order term in the gra-
dient expansion), the integrals over d
3
and d
3
introduce the delta functions(Pp)
and(rz)respectively, each with a coecient(2})
3
. We then get:
[](rp) = (rp) (rp) + (95)
In this approximation, the Wigner transform of the product of two operators is thus
simply the product of the Wigner transforms.
We now take into account the rst order terms in the gradient expansion (94). The
rr term on the rst line contains a summation over d
3
modied by the presence of
yin the integral:
1
(2})
3
d
3 (Pp)y}
y=
}
rP(Pp) (96)
The integral over d
3
in (93) is now modied and leads to a derivation with respect toP
of the function to be integrated, a multiplication by the coecient}, and nally the
replacement ofPbyp. On the other hand, the integral over d
3
(2})
3
is unchanged
and leads to the replacement ofzbyr. The corresponding term is therefore written:
}
4
rp[(rp)rr(rp)] (97)
2315

APPENDIX VII
As for therp term on the rst line of (94), it can be handled in the same way. The
presence of the variableqtransforms(rz)intorz(rz), with a coecient},
where the sign change of this coecient comes from thezin the exponentq(rz);
the integral over d
3
is unchanged. This yields the term:
}
4
rr[(rp)rp (rp)] (98)
which, added to (97), leads to the contribution (the terms involving a double derivation
of cancel each other):
}
4
[rr(rp)rp (rp)rr(rp)rp (rp)] (99)
Finally, the terms coming from the second line of (94) are obtained by exchanging the
roles ofand , and changing the signs because of the opposite values ofyandq
in the of relation (94). We thus double the result (99), and nally obtain expression (90)
to rst order in the gradients.
3-d. Evolution of the density operator
The Schrödinger evolution of the density operator obeys the von Neumann equa-
tion:
}
d
d
() = [()()] (100)
Taking its Wigner transformation, this equation becomes:
}
(rp;) =
1
(2})
3
[](rp;) (101)
where, on the right-hand side, is written the Wigner transform associated with the com-
mutator of()and(); the factor1(2})
3
comes from the denition of the Wigner
distribution of the density operator, remembering that no such coecient appears in the
transform of an arbitrary operator. We already saw that the general expression of the
Wigner transform of an operator product is somewhat complex, and the same is of course
true for their commutator.
. Classical limit
If we only keep, as in (90), the rst order terms in the gradients, we see that the
zero-order terms disappear, and that the terms in()()and()()double up; in
addition, factors}on each side of the equations cancel out. Using this approximation,
we get:
(rp;) = (rp;) (rp;)+} (102)
where the Poisson bracket of(rp1;)and(rp1;)is dened in (91). As noticed
earlier in , the neglected terms are proportional to}, and vanish in the classical
limit}0. We nd in this limit, where the gradients of the Wigner transforms with
respect to position and momentum are small, the usual equations of classical dynamics.
2316

WIGNER TRANSFORM
. Particle in an external potential
An exact calculation can be made if the particle's Hamiltonian is simply the sum
of a kinetic energy and an external potential energy:
=
P
2
2
+(R;) (103)
whereis the mass of the particle. The contribution of the kinetic energy to the
right-hand side of (101) comes directly from relation (89):
kinetic
(rp;) =
p
rr(rp;) (104)
The evolution of the Wigner distribution induced by the kinetic energy operator is thus
given by a drift term just as in classical physics.
As for the contribution of the potential energy, the computation is very similar to
the one conducted at the beginning of , except that instead of dealing with the
operatorRitself, we are now dealing with a function(R)of that operator. Taking
=in relations (83) and (85), they become:
[(R)](rp) =
1
(2})
3
d
3 py}
d
3
2
p2y
r+
y
2
; (rp2;) (105)
and:
[(R)](rp) =
1
(2})
3
d
3 py}
d
3
1
p1y
(rp1;)r
y
2
; (106)
Finally, the evolution of the Wigner distribution(rp;)obeys the following
equation:
(rp;) +
p
rr(rp;) =
1
}
1
(2})
3
d
3
d
3(pp)y}
r+
y
2
; r
y
2
; (rp;) (107)
This is an exact equation. It contains all the quantum eects that play a role in the
particle's evolution. It obeys a local conservation law for the probability:
(r) +rrJ(r) = 0 (108)
where the local probability density(r)is dened in (18), and its associated current
J(r)is dened as:
J(r) =d
3p
(rp;) (109)
2317

APPENDIX VII
This can be shown by integrating (107) over d
3
, as the left-hand side then becomes
identical to the left-hand side of (108), just as in classical mechanics; as for the right-
hand side, the integration over d
3
introduces a function(y)that cancels the bracket
in the remaining integral.
When the external potential varies slowly enough, one can use in (107) the following
approximation:
r+
y
2
; r
y
2
;=yrr(r;) + (110)
The integration over d
3
(2})
3
then leads to a function(})p(pp)and we get:
(rp;) +
p
rr(rp;) =rr(r;)rp(rp;) + (111)
One recognizes here the Liouville equation of classical mechanics. The dots at the end
of the equation symbolize the possible contributions of terms containing higher order
spatial derivatives of the potential(r;). They come with a power of}increasing with
the order of the derivative. This means that they correspond to quantum corrections:
the faster the potential varies in space, the more terms need to be taken into account.
On the other hand, when the potential varies slowly, only keeping the classical evolution
term is a good approximation.
4. Generalizations
The above considerations can be directly generalized to particles with spin, or to an
-particle system.
4-a. Particle with spin
For a particle with spin, a basis in state space is formed by the ketsr, where
ris the eigenvalue of the position operator, andthe eigenvalue of the spin component
on the quantization axis. The matrix elements of the density operator are then written:
r()r (112)
For each value ofandwe can perform a Wigner transformation and dene, as in
(17), the functions:
(rp;) =
1
(2})
3
d
3 py}
r+
y
2
()r
y
2
(113)
As an example, for a spin12the two indicesandcan take on two dierent
values, noted. We thus dene four Wigner functions, which can be arranged in a22
spin matrix:
++
(rp;)
+
(rp;)
+
(rp;) (rp;)
(114)
It is easy to show that this matrix is Hermitian:
+
(rp;)=
+
(rp;) (115)
Such a matrix is frequently used when studying the quantum properties of spin polar-
ization transport in uids (spin waves for example).
2318

WIGNER TRANSFORM
4-b. Several particles
For two spinless particles, relation (17) is easily generalized to:
(r1p1;r2p2;) =
1
(2})
6
d
3
1d
3
2
p1y1} p2y2}
r1+
y1
2
r2+
y2
2
()r1
y1
2
r2
y2
2
(116)
Actually, any number of particles can be treated this way. Including the spin can be done
as in the previous section, but it rapidly leads to a great number of Wigner functions
(4forparticles each having a spin12).
The Wigner distribution for a system including a large particle numbertherefore
depends on6variables when the particles have no spin; when the particles have a spin
12, it is no longer a single distribution that must be studied, but rather4distributions
which are the matrix elements of a spin operator. In practice, one usually uses the Wigner
distribution of the one-particle density operator, resulting from the partial trace over the
1other particles, or sometimes the Wigner distribution of the two-particle density
operator.
5. Discussion: Wigner distribution and quantum eects
Knowledge of the Wigner distribution allows computing the average values of observables,
as seen from relation (69). It can be used to obtain the probability of any measurement
result, since this probability is simply the average value of the projector onto the eigen-
subspace associated with this result. We simply have to compute the Wigner transform
of this projector, multiply it by(rp;), and integrate the result over the two vari-
ables. From a practical point of view, all the information is contained in(rp;).
However, and as already underlined with the examples given in , that does not mean
we should attribute too much physical content to the Wigner distribution itself. Strictly
speaking, the Wigner distribution is rather a useful and powerful computation tool than
a direct representation of the physical properties of the system.
To highlight the behavior of the Wigner transform in a situations where quantum
eects are predominant, we now study an interference experiment.
5-a. An interference experiment
When the wave function of a particle goes through a screen pierced with two holes,
it is split into two coherent wave packets propagating in space, and interfering when
they overlap. Figure
propagate towards the regionwhere they will interfere. As they propagate in free
space, the Wigner distribution associated with the particle simply obeys relation (104),
which is just a classical equation of motion. What causes the interference eects in region
? To answer this question, we shall use relation (19), or its equivalent (26), which allow
computing the Wigner transform associated with the particle's wave function.
This wave function is now the sum of two components, 1(r)for the wave packet
emerging from the rst hole in the screen, and 2(r)for the wave packet emerging
from the second hole:
(r) = 1(r) + 2(r) (117)
2319

APPENDIX VII
Figure 1: The wave function of a quantum particle can be split into two coherent com-
ponents1and2, after passing, for example, through a screen pieced with two holes, or
through an interferometer. As long as the two wave packets do not overlap, the Wigner
distribution is the sum of three components, schematically drawn in ordinary space in
the gure: a rst one localized with wave packet1, a second with wave packet2, and
nally a third one (circled with dashed lines) remaining at mid-distance from the two
wave packets. This third component is called the ghost component: when measuring
its position, the particle can never be found in this component. The value of the ghost
Wigner distribution oscillates rapidly as a function of the momentump.
Later on, as the two wave packets1and2overlap, the three components are dierent
from zero in the same region of space; in addition, the momentum oscillations of the
ghost component slow down and even vanish. This component now plays an essential
role: as it is added to the terms1and2, it is responsible for introducing the density
oscillations producing the fringe pattern (schematized as horizontal lines in region). It
plays a virtual role as long as the wave packets are well separated, but an essential one
when they overlap, as it leads to quantum interference eects.
A similar situation has already been studied in . Inserting (117) into relation (19),
which is quadratic in , four contributions will come into play:
(rp;) =
1
(rp;) +
2
(rp;) +
12
(rp;) +
21
(rp;) (118)
In this equality,
1
(rp;)is obtained when we replace in (19) the functions (r)and
(r)by 1(r)and
1(r)respectively. The contribution
2
(rp;)is obtained by
replacing them by 2(r)and
2(r)respectively. Finally, the crossed contributions
12
(rp;)and
21
(rp;)come from replacing (r)by 1(r)and (r)by

2(r), and conversely. For example, relation (19) leads to:
12
(rp;) =
1
(2})
3
d
3 py}
1r+
y
2
;
2r
y
2
; (119)
whereas the equivalent relation (26) yields another expression as a function of the Fourier
transforms
1and
2. It can easily be shown that the two distributions
12
(rp;)and
21
(rp;)are complex conjugates of each other. Their sum is real, as is, consequently,
(rp;).
2320

WIGNER TRANSFORM
As an example, imagine that the two wave packets are Gaussian, as were the
wave packets studied in . We saw in ComplementIshows that a Gaussian wave
packet, as it propagates in free space, remains Gaussian at all times; its momentum
dispersion remains constant, while its spatial width changes with time. For the sake of
simplicity, we shall consider a one-dimensional problem and will not explicitly write the
time dependence. We assume one of the wave packet to be centered at+0, and the
other at0. Relation (41) then leads to:
1() =
1
(2)
14
2}
2
( 0)
2
4}
2
0}
2() =
1
(2)
14
2}
2
( 0)
2
4}
2
0}
(120)
(a factor1
2has been added to ensure the normalization of the total wave function;
we assume 0, so that the spatial overlap of the two wave packets is negligible, and
the squared norm of the sum is the sum of the squared norms). The same computation
as in
1
() =
1
2}
2
2}
2
( 0)
2
2
(
0
)
2
2
2
() =
1
2}
2
2}
2
( 0)
2
2
(+
0
)
2
2
(121)
As for the crossed contributions, the computation is slightly dierent. Since the two
lines of relation (120) have dierent signs in front of0, the product
1(+2)
2( 2)
contains the exponential e
2 0}
, whereas the product
1(+2)
2( 2)con-
tains e
2 0}
. The computation of (42) then becomes:
12
() +
21
()
=
1
2(2)
32
}
2
d
2
4}
2( 0+
2)
2
+( 0
2)
2
} 2 0}
+
2 0}
=
(2)
32
}
2
cos
20
}
2
2}
2
( 0)
2
d
22
8}
2} (122)
or else:
12
() +
21
() = cos
20
}
2
2}
2
( 0)
2 2
2
2
(123)
Finally, the total Wigner transform is:
() =
1
2}
2
2}
2
( 0)
2
2
(
0
)
2
2
+
2
(+
0
)
2
2
+ 2 cos
2 0
}
2
2
2
(124)
The rst two terms in the bracket are easy to understand: they are simply half the
sum of the Wigner transforms associated with each of the wave packet. Each of these
two terms is centered on the wave packet it corresponds to, that is at= 0. The
third term is the crossed term, which corresponds to an interference between the two
wave packets, and is centered at= 0, half way between them. In addition, this term
oscillates as a function ofwith a frequency proportional to the distance between the
two wave packets.
2321

APPENDIX VII
5-b. General discussion; ghost component
The distribution
1
(rp;)propagates as if it were the distribution of a free parti-
cle described by the single wave packet 1(r); the distribution
2
(rp;)corresponds
to the second wave packet, here again as if it were isolated. If these were the only contri-
butions, when the two wave packets overlap these two Wigner distributions would simply
add to each other, since they follow a classical evolution; no quantum interference eects
would result from this addition.
However, we saw that in (118) we must also include crossed terms (interference
terms) whose properties are radically dierent from the rst two terms. A rst signicant
dierence comes from their oscillations as a function of momentum, which necessarily
involves positive and negative values of the distribution. This is denitely a quantum
eect since a classical distribution must always be positive or zero. Another dierence
is that this crossed term in the Wigner transform propagates in a region of space where
the wave function is zero, and consequently cannot correspond to any probability of the
particle's presence; the integral over momentum of the last term on the right-hand side
of (124) is indeed zero (in the limit0 of well separated wave packets corresponding
to the assumption made for our computation). The sum
12
() +
21
()is
sometimes called the ghost component of the Wigner distribution (or sometimes, in
quantum optics, the tamasic component); when measuring the particle's position, it
can never be found in this component
4
. Its value is always real, but not necessarily
positive, because of its oscillations.
This means that, as long as the two wave packets1and2are well separated,
the Wigner transform associated with the particle is the sum of three independent com-
ponents: two components separately associated with each wave packet and propagating
with them; one ghost component, also propagating but remaining at mid-distance from
the two wave packets. However, when the wave packets meet in region, the three com-
ponents of the Wigner transform overlap in space. The ghost component, which has
a changing sign, combines with the other two components to modulate the particle's
probability of presence, hence producing the interference pattern predicted by quantum
mechanics. In a certain sense, one can say that the ghost component carries the quantum
eects associated with the particle.
Conclusion
Quantum mechanics and classical mechanics are two very dierent theories. It was not
obvious that, using the Wigner transforms, one could write the quantum equations in
a form so akin to the classical equations of a distribution in phase space. Furthermore,
we showed that any real classical function of position and momentum could be used
in this formalism to generate a Hermitian operator acting in state space. In the limit
where}0, the quantum equations of motion lead to the same Poisson brackets as
the classical equations; quantum and classical theories then show strong similarities.
Quantum eects, however, can manifest themselves in several ways:
- the evolution of the Wigner distribution can be signicantly dierent from the
classical evolution when the potentials vary rapidly on a scale on the order of}(de
4
It is also known as the empty component stressing the fact that this component contains no
particle.
2322

WIGNER TRANSFORM
Broglie wavelength), as higher order terms in the gradient expansion become essential.
- the Wigner transform is not always positive. We saw an example of this with the
ghost component in an interference experiment, which, in a manner of speaking, carries
the quantum eects to the usual components.
- whereas in classical physics any distribution in phase space, as long as it is posi-
tive and normalized, can be accepted, this is no longer the case in quantum mechanics.
The only acceptable Wigner distributions are those which correspond to a density op-
erator that is positive denite, a condition that is not expressed simply in terms of the
distribution.
The Wigner transformation is frequently used in quantum physics. We already
mentioned that it was introduced in 1932, while studying quantum corrections to ther-
mal equilibrium [98]. It probably plays an even more important role in the study of
transport properties where Boltzmann type equations contain simultaneous information
on particles' positions and momenta. Furthermore, the Wigner transform is also useful
for understanding and characterizing quantum eects, as its negativity in certain regions
of phase space is a sensitive indicator of the existence of such eects. One can even use
the Wigner transforms to introduce a phase space formulation of quantum mechanics
[100,], totally equivalent to the usual formalism in terms of state space and operators,
and which is a real quantization procedure. In a general way, the Wigner transformation
belongs to the class of the so-called Liouville formulations of quantum mechanics [105],
which have many uses.
Finally, there are many domains of physics (such as signal processing, in particu-
lar) in which the Wigner transformation is part of a larger class of mixed time-frequency
transformations. Numerous types of such transformations exist (such as sliding window
or envelope transforms, wavelets, etc.) chosen to best t the problem at hand. Even in
quantum mechanics there are other quasi-classical transforms, beside the Wigner trans-
form, as for example the Husimi or the Kirkwood transforms, or the Glauber transform
expressed in terms of creation and annihilation operators of the electromagnetic eld;
a review on that subject can be found in [99]. The Wigner transform still remains one
of the most useful transforms, allowing, in particular, analytical calculations for many
interesting cases.
2323

Bibliography of volume III
[1] Quantum mechanics, Volume I, Wiley
(1977).
[2] Quantum mechanics, Volume II, Wiley
(1977).
[3] Statistical mechanics, Pergamon press (1972).
[4]
condensates,Phys. Rev.A 74, 033612 (2006).
[5] Quantum theory of nite systems, the MIT Press (1986).
1678, ,
[6]
https://en.wikipedia.org/wiki/Density_functional_theory
[7] Quantum statistical mechanics, Benjamin (1976).
[8] Quantum liquids, Oxford University Press, 2006. , ,
[9] Phys.
Rev.108, 1175-1204 (1957).
[10]
gases inProceedings of the international school of physics Enrico Fermi, Course
CLXIV, Varenna, edited by M. Inguscio, W. Ketterle and C. Salomon, IOS Press
(Amsterdam), 2008; arXiv:0801.2500v1.
[11]
1926
[12]
1926
[13] 1and2, Dekker, 1969.
[14]
Press, 2016.
[15]
ford University Press, 2016.
Quantum Mechanics, Volume III, First Edition. C. Cohen-Tannoudji, B. Diu, and F. Laloë.
© 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

BIBLIOGRAPHY OF VOLUME III
[16] Photons and atoms, intro-
duction to quantum electrodynamics, Wiley (1997). , , , , ,
2007, , , ,
[17]
the electromagnetic eld,Phys. Rev.74, 1157-1167 (1948).
[18]
noenergetic atomic beam,Phys. Rev. Lett.49, 1149-1153 (1982).
[19] Opt. Comm.
13, 68-69 (1975).
[20] 10
14
laser uorescence spec-
troscopy on
+
mono-ion oscillator III (sideband cooling),Bull. Am. Phys. Soc.
20, 637 (1975).
[21] Atom-Photon Interactions.
Basic Processes and Applications, Wiley-Interscience (1992). , , , ,
2136,
[22] JOSA B, Optical Physics,6, Number
11 (1989).
[23] Advances
in atomic, molecular, and optical physics,37, 181-236 (1996).
[24] Advances in atomic physics. An
Overview, World Scientic, Singapour (2011). , , , , , ,
2152,
[25] Phys. Rev.55, 190-197
(1939).
[26] Phys. Rev. Lett.
4, 337-341 (1960).
[27]
absorption in a standing-wave eld in a gas,JETP Lett.12, 113-116 (1970).
[28]
tonique sans eet Doppler,J. Phys. (Paris)34, 845-858 (1973).
[29] Rev. Mod. Phys.78, 1297-1309 (2006).
[30]
Sciences physiques et naturelles de Bordeaux, Jan. 28 (1932).
[31]
Light,Phys. Rev.50, 115-125 (1936).
[32] States, waves and photons: a modern introduc-
tion to light, Addison-Wesley (1970), Chap. 9.
2326

BIBLIOGRAPHY OF VOLUME III
[33]
applications,Advances in Optics and Photonics, IOP Publishing3, 161-204 (2011).
2052
[34] Classical electrodynamics, 3rd ed., Wiley (1999). ,
[35]
excités ,C. R. Acad. Sci.229, 1213 (1949).
[36]
atomic energy levels. Application to Hg
3
1,Phys. Rev.86, 308 (1952). ,
[37]
structure of atomic energy levels,Proc. Phys. Soc.74, 789 (1959).
[38] Nuovo Cimento9,
43-50 (1932).
[39]
optique d'une inégalité de population des niveaux de quantigation spatiale des
atomes. Application à l'expérience de Stern et Gerlach et à la résonance magné-
tique,J. Phys. Radium11, 255-265 (1950).
[40] Molecular beams, Oxford University Press (1956).
[41] Optical angular momentum, IOP Publish-
ing (2003).
[42]
W.D. Phillips, Quantized rotation of atoms from photons with orbital angular
momentumPhys. Rev. Lett.97, 170406 (2006).
[43]
heuristischen Gesichtspunkt,Annalen der Physik17, 132-149 (1905).
[44]
Physical Review2, 109-143 (1913).
[45] Polari-
sation, Matière et Rayonnement, (Presses Universitaires de France), Jubilee volume
in honour of Alfred Kastler, p. 363 (1969).
[46]
on Sirius,Nature,178, 1046-1048 (1956).
[47]
correlation eect on a beam splitter: a new light on single-photon interferences,
Europhys. Lett.1, 173-179 (1986).
[48]
of the resonance uorescence of a single ion,J. of Mod. Optics44, 1999-2010 (1997).
2121,
2327

BIBLIOGRAPHY OF VOLUME III
[49]
rescence,Phys. Rev. Lett.39, 691 (1977).
[50] J. Opt. Soc. Am.
56, 1636 (1966).
[51]
tion expérimentale des nouveaux eets prévus,Ann. Phys.13, 423-461 and 469-
504(1962).
[52]
magnetic elds,Phys. Rev. A5, 968-984 (1972).
[53]
tique causé par l'excitation optique,C. R. Acad. Sci.252, 394-396 (1961). ,
2141
[54] Phys. Rev.
188, 1969-1975 (1969).
[55] Phys. Rev.
100, 703-722 (1955).
[56]
cally trapped atoms,Phys. Rev. Lett.57, 314-317 (1986).
[57] Opt. Com-
mun.43, 258-260 (1982).
[58] Nature415,
Quantum phase transition from a superuid to a Mott insulator in a gas of ultracold
atoms, 39-44 (2002).
[59]
of atoms in an optical potential,Phys. Rev. Lett.76, 4508-4511 (1996).
[60]
Observation of atoms laser coooled below the Doppler limit,Phys. Rev. Lett.61,
169-172 (1988).
[61]
polarization gradients: simple theoretical models,J. Opt. Soc. Am. B6, 2023-2045
(1989).
[62]
theory,J. Opt. Soc. Am. B6, 2058-2071 (1989).
[63] Europhys. Lett.,
Laser cooling of Cesium atoms below 3K,12, 683-688 (1990).
[64]
Raimond and S. Haroche, Quantum jumps of light recording the birth and death
of a photon in a cavity,Nature446, 297-300 (2007).
2328

BIBLIOGRAPHY OF VOLUME III
[65] Introduction to quantum optics, with contri-
butions from F. Bretenaker and A. Browaeys, Cambridge University Press (2010).
2186
[66] Quantum optics, Springer (1994).
[67]
Proc. Cambridge Phil. Soc.31, 555 (1935); Probability relations between separated
systems,Proc. Cambridge Phil. Soc.32, 446 (1936).
[68] Do we really understand quantum mechanics?, Cambridge University
Press (2012); second expanded edition (2019). , ,
[69]
physical reality be considered complete?,Phys. Rev.47, 777780 (1935);Quantum
Theory of Measurement, J.A. Wheeler and W.H. Zurek eds., Princeton University
Press (1983), pp. 138141. ,
[70] Dialectica2, 320324 (1948).
2207
[71]
complete?,Phys. Rev.48, 696702 (1935).
[72] Dialectica2, 312319
(1948); Science, New Series111, 51-54 (Jan20, 1950).
[73] Rev. Mod.
Phys.38, 447452 (1966); reprinted inQuantum Theory and Measurement, J.A.
Wheeler and W.H. Zurek editors, Princeton University Press (1983), 396402 and
in chapter 1 of [74].
[74] Speakable and Unspeakable in Quantum Mechanics, Cambridge University
Press (1987); second augmented edition (2004), which contains the complete set of
J. Bell's articles on quantum mechanics.
[75] Am. J. Phys.46, 745747
(1978).
[76] Quantum Theory and Measurement,
J.A. Wheeler and W.H. Zurek editors, Princeton University Press (1983), pp. 182
213.
[77]
ing the door on Einstein and Bohr's quantum debate,Physics8, 123 (2015).
[78] Quantum computer science, Cambridge University Press (2007).
[79] Rev.
Mod. Phys.74, 145-195 (2002).
[80] Phys. Rev.
A 61, 052306 (2000).
2329

BIBLIOGRAPHY OF VOLUME III
[81]
a hidden variable model,Phys. Rev.A 40, 4277-4281 (1989).
[82] Phys. Rev. Lett.77, 1413-
1415 (1996).
[83]
pp. 69-72 inBell's theorem, quantum theory and conceptions of the Universe, M.
Kafatos (ed.), Kluwer Academic Publishers (1989).
[84]
out inequalities,Am. J. Phys.58, 1131-1143 (1990).
[85]
tion of three-photon Greenberger-Horne-Zeilinger entanglement,Phys. Rev. Lett.
82, 1345-1349 (1999).
[86] 68].
[87]
glement swapping: entangling photons that never interacted,Phys. Rev. Lett.80,
3891-3894 (1998).
[88]
berg, R.F. Vermeulen, R.N. Schouten, C. Abellan, W. Amaya, V. Pruneri, M.W.
Mitchell, M. Markham, D.J. Twitchen, D. Elkouss, S. Wehner, T.H. Taminiau and
R. Hanson, Experimental loophole-free violation of a Bell inequality using electron
spins separated by 1.3 km,Nature526, 682-686 (2015).
[89]
Ketterle, Observation of interference between two Bose condensates,Science275,
637-641 (1997).
[90] Eur. Phys.
J. D33, 87-97 (2005); F. Laloë and W.J. Mullin, Non-local quantum eects with
Bose-Einstein condensates,Phys. Rev. Lett.99, 150401 (2007). , ,
[91] Quantum mechanics and path integrals, McGraw
Hill (1965).
[92] Intégrale de chemin en mécanique quantique: Introduction, CNRS
Editions et EDP Sciences (2003). ,
[93] Physique quantique, EDP Sciences et CNRS Editions (2013), tome II,
chapitre 12.
[94] Rev. Modern
Physics,67, 279-355 (1995).
[95] Physique statistique, Hermann
(1989).
[96] Statistical mechanics, Wiley (1963).
2330

BIBLIOGRAPHY OF VOLUME III
[97] Fundamental of statistical and thermal physics, McGraw-Hill (1965).
[98] Phys.
Rev.40, 749-759 (1932). ,
[99]
in physics; fundamentals,Physics Reports,106, 121-167 (1984). , ,
[100]
(1986); see in particular Chap. 16. ,
[101] Quantum mechanics in phase space,
World Scientic, Singapore (2005); with the same title,Asia Pacic Newsletters,
01, 37-46 (2012) or ArXiv:1104.5269v2. ,
[102]
Nuclear PhysicsB 425, 343-364 (1994).
[103] Mechanics, Course of theoretical physics, Vol. I,
C 42, Pergamon Press (1960) and Elsevier Butterworth-Heinemann (1976).
[104] Classical mechanics, Addison-Wesley (2001).
2314
[105]
in mock phase spaces,Physics Reports,104, 347-391 (1984).
2331

Index[The notation (ex.) refers to an exercise]
Absorption
and emission of photons,
collision with,
of a quantum, a photon, ,
of eld,
of several photons,
rates,
Acceptor (electron acceptor),
Acetylene (molecule),
Action,, ,
Addition
of angular momenta, ,
of spherical harmonics,
of two spins 1/2,
Adiabatic
branching of the potential,
Adjoint
matrix,
operator,
Algebra (commutators),
Allowed energy band,, ,
Ammonia (molecule),,
Amplitude
scattering amplitude,,
Angle (quantum),
Angular momentum
addition of momenta, ,
and rotations,
classical,
commutation relations,,
conservation,,,
coupling,
electromagnetic eld, ,
half-integral,
of identical particles, (ex.)
of photons,
orbital,,,
quantization,
quantum,
spin,,
standard representation,,
two coupled momenta,
Anharmonic oscillator,,
Annihilation operator,,,,
Annihilation-creation (pair), ,
Anomalous
average value, ,
dispersion,
Zeeman eect,
Anti-normal correlation function, ,
1789
Anti-resonant term,
Anti-Stokes (Raman line),,
Antibunching (photon),
Anticommutation,
eld operator,
Anticrossing of levels,,
Antisymmetric ket, state, ,
Antisymmetrizer, ,
Applications of the perturbation theory,
1231
Approximation
central eld approximation,
secular approximation,
Argument (EPR),
Atom(s),seehelium, hydrogenoid
donor,
dressed, ,
many-electron atoms, ,
mirrors for atoms,
muonic atom,
single atom uorescence,
Atomic
beam (deceleration),
orbital,, (ex.)
parameters,
Attractive bosons,
Autler-Townes
doublet,
eect,
Autoionization,
Average value (anomalous),
Azimuthal
quantum number,
Band (energy),
Bardeen-Cooper-Schrieer,
Barrier (potential barrier),,,
Basis
2333
Quantum Mechanics, Volume III, First Edition. C. Cohen-Tannoudji, B. Diu, and F. Laloë.
© 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

INDEX [The notation (ex.) refers to an exercise]
change of bases,
characteristic relations,,
continuous basis in the space of states,
99
mixed basis in the space of states,
BCHSH inequalities, ,
BCS,
broken pairs and excited pairs,
coherent length,
distribution functions,
elementary excitations,
excited states,
gap, , ,
pairs (wave function of),
phase locking, , ,
physical mechanism,
two-particle distribution,
Bell's
inequality,
theorem, ,
Benzene (molecule),,
Bessel
Bessel-Parseval relation,
spherical Bessel function,
spherical equation,
spherical function,
Biorthonormal decomposition,
Bitter,
Blackbody radiation,
Bloch
equations,, ,
theorem,
Bogolubov
excitations,
Hamiltonian,
operator method,
phonons, spectrum,
transformation,
Bogolubov-Valatin transformation, ,
1919
Bohr,
electronic magneton,
frequencies,
magneton,seefront cover pages
model,,
nuclear magneton,
radius,
Boltzmann
constant,seefront cover pages
distribution,
Born
approximation,,,
Born-Oppenheimer approximation,,
1177,
Born-von Karman conditions,
Bose-Einstein
condensation, , ,
condensation (repulsive bosons),
condensation of pairs,
distribution,,
statistics,
Bosons,
at non-zero temperature,
attractive,
attractive instability,
condensed,
in a Fock state,
paired,
Boundary conditions (periodic),
Bra,,,
Bragg reection,
Brillouin
formula,
zone,
Broadband
detector,
optical excitation,
Broadening (radiative),
Broken pairs and excited pairs (BCS),
1920
Brossel,
Bunching of bosons,
C.S.C.O.,,,,
Canonical
commutation relations,,,
ensemble,
Hamilton-Jacobi canonical equations,
214
Hamilton-Jacobi equations,
Cauchy principal part,
Center of mass,,
Center of mass frame,
Central
2334

INDEX [The notation (ex.) refers to an exercise]
eld approximation,
potential,
Central potential,,
scattering,
stationary states,
Centrifugal potential,,,
Chain (von Neumann),
Chain of coupled harmonic oscillators,
Change
of bases,,,
of representation,
Characteristic equation,
Characteristic relation of an orthonormal
basis,
Charged harmonic oscillator in an elec-
tric eld,
Charged particle
in an electromagnetic eld,
Charged particle in a magnetic eld,,
321,
Chemical bond,,, ,
Chemical potential, ,
Circular quanta,,
Classical
electrodynamics,
histories,
Clebsch-Gordan coecients, ,
Closure relation,,
Coecients
Clebsch-Gordan,
Einstein, ,
Coherences (of the density matrix),
Coherent length (BCS),
Coherent state (eld),
Coherent superposition of states,,,
307
Collision,
between identical particles, , (ex.)
between identical particles in classi-
cal mechanics,
between two identical particles,
cross section,
scattering states,
total scattering cross section,
with absorption,
Combination
of atomic orbitals,
Commutation,
canonical relations,,
eld operator,
of pair eld operators,
relations,
Commutation relations
angular momentum,,
eld, ,
Commutator algebra,
Commutator(s),,,,
of functions of operators,
Compatibility of observables,
Complementarity,
Complete set of commuting observables
(C.S.C.O.),,,
Complex variables (Lagrangian),
Compton wavelength of the electron,,
1235
Condensates
relative phase,
with spins,
Condensation
BCS condensation energy,
Bose-Einstein, , ,
Condensed bosons,
Conduction band,
Conductivity (solid),
Congurations,
Conjugate momentum, , , ,
1983, ,
Conjugation (Hermitian),
Conservation
local conservation of probability,
of angular momentum,,,
of energy,
of probability,
Conservative systems,,
Constants of the motion,,
Contact term,
Contact term (Fermi), ,
Contextuality,
Continuous
spectrum,,,,
variables (in a Lagrangian),
Continuum of nal states, , ,
1380
Contractions,
2335

INDEX [The notation (ex.) refers to an exercise]
Convolution product of two functions,
Cooling
Doppler,
down atoms,
evaporative,
Sisyphus,
sub-Doppler,
subrecoil,
Cooper model,
Cooper pairs,
Cooperative eects (BCS),
Correlation functions, ,
anti-normal, ,
dipole and eld,
for one-photon processes,
normal, ,
of the eld, spatial,
Correlations,
between two dipoles,
between two physical systems,
classical and quantum,
introduced by a collision,
Coulomb
eld,
gauge,
Coulomb potential
cross section,
Coupling
between angular momenta,
between two angular momenta,
between two states,
eect on the eigenvalues,
spin-orbit coupling, ,
Creation and annihilation operators,,
513,, ,
Creation operator (pair of particles), ,
1846
Critical velocity,
Cross section
and phase shifts,
scattering cross section,,,,
972
Current
metastable current in superuid,
of particles,
of probability,
probability current in hydrogen atom,
851
Cylindrical symmetry,(ex.)
Darwin term, ,
De Broglie
relation,
wavelength,seefront cover pages,,
35
Decay of a discrete state,
Deceleration of an atomic beam,
Decoherence,
Decomposition (Schmidt),
Decoupling (ne or hyperne structure),
1262,
Degeneracy
essential,,,
exchange degeneracy,
exchange degeneracy removal,
lifted by a perturbation,
rotation invariance,
systematic and accidental,
Degenerate eigenvalue,,,,
Degereracy
lifted by a perturbation,
parity,
Delta Dirac function,
potential well and barriers,85(ex.)
use in quantum mechanics,,,
280
Density
Lagrangian,
of probability,
of states,, , ,
operator,,
operator and matrix,
particle density operator,
Density functions
one and two-particle, (ex.)
Depletion (quantum),
Derivative of an operator,
Detection probability amplitude (photon),
2166
Detectors (photon),
Determinant
Slater determinant, ,
Deuterium,, (ex.)
Diagonalization
2336

INDEX [The notation (ex.) refers to an exercise]
of a22matrix,
of an operator,
Diagram (dressed-atom),
Diamagnetism,
Diatomic molecules
rotation,
Diusion (momentum),
Dipole
-dipole interaction, ,
-dipole magnetic interaction,
electric dipole transition,
electric moment,
Hamiltonian,
magnetic dipole moment,
magnetic term,
trap,
Dirac,seeFermi
delta function,,,,
equation,
notation,
Direct
and exchange terms, , , ,
1646,
term, ,
Discrete
bases of the state space,
spectrum,,
Dispersion (anomalous),
Dispersion and absorption (eld),
Distribution
Boltzmann,
Bose-Einstein,
Fermi-Dirac,
function (bosons),
function (fermions),
functions, ,
functions (BCS),
Distribution law
Bose-Einstein,
Divergence (energy),
Donor atom,,
Doppler
cooling,
eect,
eect (relativistic),
free spectroscopy,
temperature,
Double
condensate,
resonance method,
spin condensate,
Doublet (Autler-Townes),
Down-conversion (parametric),
Dressed
states and energies,
Dressed-atom, ,
diagram,
strong coupling,
weak coupling,
E.P.R., (ex.)
Eckart (Wigner-Eckart theorem),seeWigner
Eect
Autler-Townes,
Mössbauer,
photoelectric,
Eective Hamiltonian,
Ehrenfest theorem,,,
Eigenresult,
Eigenstate,,
Eigenvalue,,,,
degenerate,,
equation,,
of an operator,
Eigenvector,
of an operator,
Einstein,
coecients, , ,
EPR argument,,
model,,
Planck-Einstein relations,
temperature,
Einstein-Podolsky-Rosen, ,
Elastic
scattering,
scattering (photon),
scattering, form factor, (ex.)
total cross section,
Elastically bound electron model,
Electric
conductivity of a solid,
Electric dipole
Hamiltonian,
interaction,
2337

INDEX [The notation (ex.) refers to an exercise]
matrix elements,
moment,
selection rules,
transition and selection rules,
transitions,
Electric eld (quantized), ,
Electric polarisability
NH3,
Electric polarizability
of the1state in Hydrogen,
Electric quadrupole
Hamiltonian,
moment,
transitions,
Electric susceptibility
bound electron,
of an atom,
Electrical
susceptibility, (ex.)
Electrodynamics
classical,
quantum,
Electromagnetic eld
and harmonic oscillators,
and potentials,
angular momentum, ,
energy,
Lagrangian, ,
momentum, ,
polarization,
quantization,,
Electromagnetic interaction of an atom
with a wave,
Electromagnetism
elds and potentials,
Electron spin,,
Electron(s)
congurations,
gas in solids,
in solids, ,
mass and charge,seefront cover pages
Electronic
conguration,
paramagnetic resonance, (ex.)
shell,
Elements of reality,
Emergence of a relative phase, ,
Emission
of a quantum,
photon,
spontaneous, ,
stimulated (or induced),
Energy,seeConservation, Uncertainty
and momentum of the transverse elec-
tromagnetic eld,
band,
bands in solids, ,
conservation,
electromagnetic eld,
Fermi energy,
ne structure energy levels,
free energy,
levels,
levels of harmonic oscillator,
levels of hydrogen,
of a paired state,
recoil energy,
Ensemble
canonical,
grand canonical,
microcanonical,
statistical ensembles,
Entanglement
quantum, , , ,
swapping,
Entropy,
EPR, ,
elements of reality,
EPRB,
paradox/argument,
Equation of state
ideal quantum gas,
repulsive bosons,
Equation(s)
Bloch,
Hamilton-Jacobi, , ,
Lagrange, ,
Lorentz,
Maxwell,
Schrödinger,,,
von Neumann,
Essential degeneracy,,
Ethane (molecule),
Ethylene (molecule),,
2338

INDEX [The notation (ex.) refers to an exercise]
Evanescent wave,,,,,
Evaporative cooling,
Even operators,
Evolution
eld operator,
of quantum systems,
of the mean value,
operator,,
operator (expansion),
operator (integral equation),
Exchange,
degeneracy,
degeneracy removal,
energy,
hole,
integral,
term, , ,
Excitations
BCS,
Bogolubov,
vacuum,
Excited states (BCS),
Exciton,
Exclusion principle (Pauli), , ,
1463,
Extensive (or intensive) variables,
Fermi
contact term,
energy, , , ,
gas,
golden rule,
level, ,
radius,
surface (modied),
,seeFermi-Dirac
Fermi level
and electric conductivity,
Fermi-Dirac
distribution, , ,
statistics,
Fermions,
in a Fock state,
paired,
Ferromagnetism,
Feynman
path,
postulates,
Fictitious spin,,
Field
absorption,
commutation relations, ,
dispersion and absorption,
intense laser,
interaction energy,
kinetic energy,
normal variables,
operator,
operator (evolution), ,
pair eld operator,
potential energy,
quantization, ,
quasi-classical state,
spatial correlation functions,
Final states continuum, ,
Fine and hyperne structure,
Fine structure
constant,seefront cover pages,
energy levels,
Hamiltonian, , ,
Helium atom,
Hydrogen,
of spectral lines,
of the states1,2et2,
Fletcher,
Fluctuations
boson occupation number,
intensity,
vacuum,,
Fluorescence (single atom),
Fluorescence triplet,
Fock
space, ,
state, , , ,
Forbidden,seeBand
energy band,,,
transition,
Forces
van der Waals,
Form factor
elastic scattering, (ex.)
Forward scattering (direct and exchange),
1874
Fourier
2339

INDEX [The notation (ex.) refers to an exercise]
series and transforms,
Fragmentation (condensate), ,
Free
electrons in a box,
energy,
particle,
quantum eld (Fock space),
spherical wave,,,
spherical waves and plane waves,
Free particle
stationary states with well-dened an-
gular momentum,
stationary states with well-dened mo-
mentum,
wave packet,,,
Frequency
Bohr,
components of the eld (positive and
negative),
Rabi's frequency,
Friction (coecient),
Function
of operators,
periodic functions,
step functions,
Fundamental state,
Gap (BCS), , ,
Gauge, , , ,
Coulomb,
invariance,
Lorenz,
Gaussian
wave packet,,,
Generalized velocities,,
Geometric quantization,
Gerlach,seeStern
GHZ state, ,
Gibbs-Duhem relation,
Golden rule (Fermi),
Good quantum numbers,
Grand canonical, ,
Grand potential, , ,
Green's function,,, , ,
1789
evolution,
Greenberger-Horne-Zeilinger,
Groenewold's formula,
Gross-Pitaevskii equation, ,
Ground state,
harmonic oscillator,,
Hydrogen atom, (ex.)
Group velocity,,,
Gyromagnetic ratio,,
orbital,
spin,
H
+
2
molecular ion,(ex.),,
Hadronic atoms,
Hall eect,
Hamilton
function,
function and equations,
Hamilton-Jacobi canonical equations,,
1532, , ,
Hamiltonian,,, , , ,
1995
classical,
eective,
electric dipole, ,
electric quadrupole,
ne structure, ,
hyperne, ,
magnetic dipolar,
of a charged particle in a vector po-
tential,
of a particle in a central potential,
806,
of a particle in a scalar potential,
of a particle in a vector potential,
225,,
Hanbury Brown and Twiss,
Hanle eect, (ex.)
Hard sphere
scattering,,(ex.)
Harmonic oscillator,
in an electric eld,
in one dimension,,
in three dimensions,
in two dimensions,
innite chain of coupled oscillators,
611
quasiclassical states,
thermodynamic equilibrium,
2340

INDEX [The notation (ex.) refers to an exercise]
three-dimensional,,(ex.)
two coupled oscillators,
Hartree-Fock
approximation, ,
density operator (one-particle),
equations, ,
for electrons,
mean eld, ,
potential,
thermal equilibrium, ,
time-dependent, ,
Healing length,
Heaviside step function,
Heisenberg
picture,,
relations,,,,,,,
Helicity (photon),
Helium
energy levels,
ion,
isotopes,
isotopes
3
He and
4
He, ,
solidication,
Hermite polynomials,,,
Hermitian
conjugation,
matrix,
operator,,,
Histories (classical),
Hole
creation and annihilation,
exchange,
Holes,
Hybridization of atomic orbitals,
Hydrogen,
atom,
atom in a magnetic eld,,,
862
atom, relativistic energies,
Bohr model,,
energy levels,
ne and hyperne stucture,
ionisation energy,seefront cover pages
ionization energy,
maser,
molecular ion,(ex.),,
quantum theory,
radial equation,
Stark eect,
stationary states,
stationary wave functions,
Hydrogen-like systems in solid state physics,
837
Hydrogenoid systems,
Hyperne
decoupling,
Hamiltonian, ,
Hyperne structure,seeHydrogen, muo-
nium, positronium, Zeeman ef-
fect,
Muonium,
Ideal gas, , , ,
correlations,
Identical particles, ,
Induced
emission, , ,
emission of a quantum,
emission of photons,
Inequality (Bell's),
Innite one-dimensional well,
Innite potential well,
in two dimensions,
Innitesimal unitary operator,
Insulator,
Integral
exchange integral,
scattering equation,
Intense laser elds,
Intensive (or extensive) variables,
Interaction
between magnetic dipoles,
dipole-dipole interaction, ,
electromagnetic interaction of an atom
with a wave,
eld and particles,
eld and atom,
magnetic dipole-dipole interaction,
picture,, ,
tensor interaction,
Interference
photons,
two-photon, ,
Ion H
+
2
,
2341

INDEX [The notation (ex.) refers to an exercise]
Ionization
photo-ionization,
tunnel ionization,
Isotropic radiation,
Jacobi,seeHamilton
Kastler, ,
Ket,seestate,,
for identical particles,
Kuhn,seeThomas
Lagrange
equations, , ,
fonction and equations,
multipliers,
Lagrangian, ,
densities,
electromagnetic eld, ,
formulation of quantum mechanics,
339
of a charged particle in an electro-
magnetic eld,
particle in an electromagnetic eld,
323
Laguerre-Gaussian beams,
Lamb shift,, , ,
Landau levels,
Landé factor, , (ex.), ,
Laplacian,
of1,
of()
+1
,
Larmor
angular frequency,
precession,,,,,,
1071
Laser, ,
Raman laser,
saturation,
trap,
Lattices (optical),
Least action
principle of,
Legendre
associated function,
polynomial,
Length (healing),
Level
anticrossing,,
Fermi level,
Lifetime,,,
of a discrete state,
radiative,
Lifting of degeneracy by a perturbation,
1125
Light
quanta,
shifts, , , ,
Linear,seeoperator
combination of atomic orbitals,
operators,,,
response, , ,
superposition of states,
susceptibility,
Local conservation of probability,
Local realism, ,
Longitudinal
elds,
relaxation,
relaxation time,
Lorentz equations,
Lorenz (gauge),
Magnetic
dipole term,
dipole-dipole interaction,
eect of a magnetic eld on the lev-
els of the Hydrogen atom,
hyperne Hamiltonian,
interactions, ,
quantum number,
resonance,
susceptibility, ,
Magnetic dipole
Hamiltonian,
transitions and selection rules, ,
1098,
Magnetic dipoles
interactions between two dipoles,
Magnetic eld
and vector potential,
charged particle in a,,
eects on hydrogen atom,,
harmonic oscillator in a,(ex.)
Hydrogen atom in a magnetic eld,
1263,
2342

INDEX [The notation (ex.) refers to an exercise]
multiplets,
quantized, ,
Magnetism (spontaneous),
Many-electron atoms,
Maser,, ,
hydrogen,
Mass correction (relativistic),
Master equation,
Matrice(s),,
diagonalization of a22matrix,
Pauli matrices,
unitary matrix,
Maxwell's equations,
Mean eld (Hartree-Fock), , ,
1725
Mean value of an observable,
evolution,
Measurement
general postulates,,
ideal von Neumann measurement,
of a spin 1/2,
of observables,
on a part of a physical system,
state after measurement,,
Mendeleev's table,
Metastable superuid ow,
Methane (molecule),
Microcanonical ensemble,
Millikan,
Minimal wave packet,,,
Mirrors for atoms,
Mixing of states, ,
Model
Cooper model,
Einstein model,
elastically bound electron,
vector model of atom,
Modes
vibrational modes,,
Modes (radiation), ,
Molecular ion,
Molecule(s)
chemical bond,,,,,
883,
rotation,
vibration,,
vibration-rotation,
Mollow,
Moment
quadrupole electric moment, (ex.)
Momentum,
conjugate,,, , ,
diusion,
electromagnetic eld, ,
mechanical momentum,
Monogamy (quantum),
Mössbauer eect, ,
Motional narrowing,
condition, , ,
Multiphoton transition, , ,
Multiplets, , ,
Multipliers (Lagrange),
Multipolar waves,
Multipole
moments,
Multipole operators
introduction, ,
parity,
Muon,,,
Muonic atom,,
Muonium,
hyperne structure,
Zeeman eect,
Narrowing (motional), ,
condition,
Natural width,,
Need for a quantum treatment, ,
Neumann
spherical function,
Neutron mass,seefront cover pages
Non-destructive detection of a photon,
2159
Non-diagonal order (BCS),
Non-locality,
Non-resonant excitation,
Non-separability,
Nonlinear
response, ,
susceptibility,
Norm
conservation,
of a state vector,,
of a wave function,,,
2343

INDEX [The notation (ex.) refers to an exercise]
Normal
correlation function, ,
variables,,,,
variables (eld),
Nuclear
multipole moments,
Bohr magneton,
Nucleus
spin,
volume eect, ,
Number
occupation number, ,
photon number,
total number of particles in an ideal
gas,
Observable(s),
C.S.C.O.,,
commutation,
compatibility,
for identical particles, ,
mean value,
measurement of,,
quantization rules,
symmetric observables,
transformation by permutation,
whose commutator is},,
Occupation number, ,
operator,
Odd operators,
One-particle
Hartree-Fock density operator,
operators, , , ,
Operator(s)
adjoint operator,
annihilation operator,,,,
1597
creation and annihilation,
creation operator,,,,
derivative of an operator,
diagonalization,,
even and odd operators,
evolution operator,,
eld,
function of,
Hermitian operators,
linear operators,,,
occupation number,
one-particle operator, , , ,
1756
parity operator,
particle density operator,
permutation operators, ,
potential,
product of,
reduced to a single particle,
representation,
restriction,
restriction of,
rotation operator,
symmetric, ,
translation operator,
two-particle operator, , , ,
1756
unitary operators,
Weyl operator,
Oppenheimer,seeBorn, ,
Optical
excitation (broadband),
lattices,
pumping, ,
Orbital
angular momentum (of radiation),
atomic orbital, (ex.)
hybridization,
linear combination of atomic orbitals,
1172
quantum number,
state space,
Order parameter for pairs,
Orthonormal basis,,,,
characteristic relation,
Orthonormalization
and closure relations,,
relation,
Oscillation(s)
between two discrete states,
between two quantum states,
Rabi,
Oscillator
anharmonic,
harmonic,
strength,
Pair(s)
2344

INDEX [The notation (ex.) refers to an exercise]
annihilation-creation of pairs, ,
1874,
BCS, wave function,
Cooper,
of particles (creation operator), ,
1846
pair eld (commutation),
pair eld operator,
pair wave function,
Paired
bosons,
fermions,
state energy,
states,
states (building),
Pairing term,
Paramagnetism,
Parametric down-conversion,
Parity,
degeneracy,
of a permutation operator,
of multipole operators,
operator,
Parseval
Parseval-Plancherel equality,
Parseval-Plancherel formula, ,
Partial
reection,
trace of an operator,
waves in the potential,
waves method,
Particle (current),
Particles and holes,
Partition function, , ,
Path
integral,
space-time path,
Pauli
exclusion principle, , , ,
1481
Hamiltonian, (ex.)
matrices,,
spin theory,
spinor,
Penetrating orbit,
Penrose-Onsager criterion, , ,
Peres,
Periodic
boundary conditions,
classication of elements,
functions,
potential (one-dimensional),
Permutation operators, ,
Perturbation
applications of the perturbation the-
ory,
lifting of a degeneracy,
one-dimensional harmonic oscillator,
1131
random perturbation, , ,
sinusoidal,
stationary perturbation theory,
Perturbation theory
time dependent,
Phase
locking (BCS), ,
locking (bosons), ,
relative phase between condensates,
2237,
velocity,
Phase shift (collision),, (ex.)
with imaginary part,
Phase velocity,
Phonons,,
Bogolubov phonons,
Photodetection
double, ,
single, ,
Photoelectric eect, (ex.),
Photoionization, ,
rate, ,
two-photon,
Photon,,,, , ,
absorption and emission,
angular momentum,
antibunching,
detectors,
non-destructive detection,
number,
scattering (elastic),
scattering by an atom,
vacuum,
,seeAbsorption, Emission
Picture
2345

INDEX [The notation (ex.) refers to an exercise]
Heisenberg,,
interaction, ,
Pitaevskii (Gross-Pitaevskii equation), ,
1657
Plancherel,seeParseval
Planck
constant,seefront cover pages,
law ,
Planck-Einstein relations,,
Plane wave,,,,
Podolsky (EPR argument),,
Pointer states,
Polarizability
of the1state in Hydrogen,
Polarization
electromagnetic eld,
of Zeeman components,
space-dependent,
Polynomial method (harmonic oscillator),
555,
Polynomials
Hermite polynomials,,,
Position and momentum representations,
181
Positive and negative frequency compo-
nents,
Positron,
Positronium,
hyperne structure,
Zeeman eect,
Postulate (von Neumann projection),
Postulates of quantum mechanics,
Potential
adiabatic branching,
barrier,,,,
centrifugal potential,,,
Coulomb potential, cross section,
cylindrically symmetric,(ex.)
Hartree-Fock,
innite one-dimensional well,
operator,
scalar and vector potentials, ,
1960,
scattering by a,
self-consistent potential,
square potential,
square well,
step,,,,
well,,
well (arbitrary shape),
well (innite one-dimensional),
well (innite two-dimensional,
Yukawa potential,
Precession
Larmor precession,,
Thomas precession,
Preparation of a state,
Pressure (ideal quantum gas),
Principal part,
Principal quantum number,
Principle
of least action, ,
of spectral decomposition,,
of superposition,
Probability
amplitude,,,
conservation,
current,,,,,
current in hydrogen atom,
density,,
uid,
of photon absorption,
of the measurement results,,
transition probability,
Process (pair annihilation-creation), ,
1887
Product
convolution product of functions,
of matrices,
of operators,
scalar product,,,,
state (tensor product),
tensor product,
tensor product, applications,
Projection theorem,
Projector,,,,,, (ex.)
Propagator
for the Schrödinger equation,
of a particle, ,
Proper result,
Proton
mass,seefront cover pages
spin and magnetic moment, ,
Pumping,
2346

INDEX [The notation (ex.) refers to an exercise]
Pure (state or case),
Quadrupolar electric moment, , (ex.)
Quanta (circular),,
Quantization
electrodynamics,
electromagnetic eld,,,
of a eld,
of angular momentum,,
of energy,,,,
of measurement results,,,
of the measurement results,
rules,,,,
Quantum
angle,
electrodynamics, , ,
entanglement, ,
monogamy,
number
orbital,
principal quantum number,
numbers (good),
resonance,
treatment needed, ,
Quasi-classical
eld states,
states,,,
states of the harmonic oscillator,
Quasi-particles, ,
Bogolubov phonons,
Quasi-particle vacuum,
Rabi
formula,,, ,
formula),
frequency,
oscillation,
Radial
equation,
equation (Hydrogen),
equation in a central potential,
integral,
quantum number,
Radiation
isotropic,
pressure,
Radiative
broadening,
cascade of the dressed atom,
Raman
eect,,, (ex.)
laser,
scattering,
scattering (stimulated),
Random perturbation, , ,
Rank (Schmidt),
Rate (photoionization), ,
Rayleigh
line,
scattering,,
Realism (local), ,
Recoil
blocking,
eect of the nucleus,
energy, ,
free atom,
suppression,
Reduced
density operator,
mass,
Reduction of the wave packet,,
Reection on a potential step,
Refractive index,
Reiche,seeThomas
Relation (Gibbs-Duhem),
Relative
motion,
particle,
phase between condensates, ,
phase between spin condensates,
Relativistic
corrections, ,
Doppler eect,
mass correction,
Relaxation,, , , , (ex.)
general equations,
longitudinal,
longitudinal relaxation time,
transverse,
transverse relaxation time,
Relay state, , ,
Renormalization,
Representation(s)
change of,
in the state space,
2347

INDEX [The notation (ex.) refers to an exercise]
of operators,
position and momentum,,
Schrödinger equation,185
Repulsion between electrons,
Resonance
magnetic resonance,
quantum resonance,,
scattering resonance,,,(ex.)
two resonnaces with a sinusoidal ex-
citation,
width,
with sinusoidal perturbation,
Restriction of an operator,,
Rigid rotator,, (ex.)
Ritz theorem,
Root mean square deviation
general denition,
Rosen (EPR argument),,
Rotating frame,
Rotation(s)
and angular momentum,
invariance and degeneracy,
of diatomic molecules,
of molecules,,
operator(s),,
rotation invariance,
rotation invariance and degeneracy,
1072
Rotator
rigid rotator,, (ex.)
Rules
quantization rules,
selection rules,
Rutherford's formula,
Rydberg constant,seefront cover pages
Saturation
of linear response,
of the susceptibility,
Scalar
and vector potentials,,
interaction between two angular mo-
menta,
observable, operator,,
potential,
product,,,,,,
product of two coherent states,
Scattering
amplitude,,
by a central potential,
by a hard sphere,,(ex.)
by a potential,
cross section,,,
cross section and phase shifts,
inelastic,
integral equation,
of particles with spin,
of spin 1/2 particles, (ex.)
photon,
Raman,
Rayleigh,,
resonance,,(ex.)
resonant,
stationary scattering states,
stationary states,
stimulated Raman,
Schmidt
decomposition,
rank,
Schottky anomaly,
Schrödinger,
equation,,,,
equation in momentum representa-
tion,
equation in position representation,
183
equation, physical implications,
equation, resolution for conservative
systems,
picture,
Schwarz inequality,
Second
quantization,
harmonic generation,
Secular approximation, ,
Selection rules,,, ,
electric quadrupolar,
magnetic dipolar, ,
Self-consistent potential,
Semiconductor,,
Separability, ,
Separable density operator,
Shell (electronic),
Shift
2348

INDEX [The notation (ex.) refers to an exercise]
light shift,
of a discrete state,
Singlet, ,
Sinusoidal perturbation, ,
Sisyphus
cooling,
eect,
Slater determinant, ,
Slowing down atoms,
Solids
electronic bands,
energy bands of electrons,
energy bands of electrons in solids,
381
hydrogen-like systems in solid state
physics,
Space (Fock),
Space-dependent polarization,
Space-time path,,
Spatial correlations (ideal gas),
Specic heat
of an electron gas,
of metals,
of solids,
two level system,
Spectral
decomposition principle,,,
function,
terms,
Spectroscopy (Doppler free),
Spectrum
BCS elementary excitation,
continuous,,
discrete,,
of an observable,,
Spherical
Bessel equation,
Bessel function,,
free spherical waves,
free wave,
Neumann function,
wave,
waves and plane waves,
Spherical harmonics,,
addition of,
expression for= 012,
general expression,
Spin
and magnetic moment of the proton,
1237
angular momentum,
electron,,
ctitious,
gyromagnetic ratio,,,
nuclear,
of the electron,
Pauli theory,,
quantum description,,
rotation operator,
scattering of particles with spin,
spin 1 and radiation, , ,
system of two spins,
Spin 1/2
density operator,
ensemble of,
ctitious,
interaction between two spins,
preparation and measurement,
scattering of spin 1/2 particles, (ex.)
Spin-orbit coupling, , , ,
Spin-statistics theorem,
Spinor,
rotation,
Spontaneous
emission,,, , ,
emission of photons,
magnetism of fermions,
Spreading of a wave packet,,
Square
barrier of potential,,
potential,,,,
potential well,,
spherical well,(ex.)
Standard representation (angular momen-
tum),,
Stark eect in Hydrogen atom,
State(s),seeDensity operator
density of,, , ,
Fock, , , ,
ground state,
mixing of states by a perturbation,
1121
orbital state space,
paired,
2349

INDEX [The notation (ex.) refers to an exercise]
pointer states,
quasi-classical states,,,,
801
relay state, , ,
stable and unstable states,
state after measurement,
state preparation,
stationary,,,
stationary state,,
stationary states in a central poten-
tial,
unstable,
vacuum state,
vector,,
Stationary
perturbation theory,
phase condition,,
scattering states,,
states,,,,
states in a periodic potential,
states with well-dened angular mo-
mentum,,
states with well-dened momentum,
943
Statistical
entropy,
mechanics (review of),
mixture of states,,,,
Statistics
Bose-Einstein,
Fermi-Dirac,
Step
function,
potential,,,,
Stern-Gerlach experiment,
Stimulated
(or induced) emission, , ,
2081
Raman scattering,
Stokes Raman line,,
Stoner (spontaneous magnetism),
Strong coupling (dressed-atom),
Subrecoil cooling,
Sum rule (Thomas-Reiche-Kuhn),
Superuidity, ,
Superposition
of states,
principle,,
principle and physical predictions,
Surface (modied Fermi surface),
Susceptibility,seeLinear, nonlinear, ten-
sor
electric susceptibility of an atom,
electrical susceptibility,,e1223
electrical susceptibility of NH3,
magnetic susceptibility,
tensor, , (ex.)
Swapping (entanglement),
Symmetric
ket, state, ,
observables, ,
operators, , , , ,
1628, ,
Symmetrization
of observables,
postulate,
Symmetrizer, ,
System
time evolution of a quantum system,
223
two-level system,
Systematic
and accidental degeneracies,
degeneracy,
Temperature (Doppler),
Tensor
interaction,
product,,
product of operators,
product state,,
product, applications,
susceptibility tensor,
Term
direct and exchange terms, , ,
1634, ,
pairing,
spectral terms, ,
Theorem
Bell, ,
Bloch,
projection,
Ritz,
Wick, ,
2350

INDEX [The notation (ex.) refers to an exercise]
Wigner-Eckart, , ,
Thermal wavelength,
Thermodynamic equilibrium,
harmonic oscillator,
ideal quantum gas,
spin 1/2,
Thermodynamic potential (minimization),
1715
Thomas precession,
Thomas-Reiche-Kuhn sum rule,
Three-dimensional harmonic oscillator,,
841,(ex.)
Three-level system, (ex.)
Three-photon transition,
Time evolution of quantum systems,
Time-correlations (uorescent photons),
2145
Time-dependent
Gross-Pitaevskii equation,
perturbation theory,
Time-energy uncertainty relation,,,
345, ,
Torsional oscillations,
Torus (ow in a),
Total
elastic scattering cross section,
reection,,
scattering cross section (collision),
Townes
Autler-Townes eect,
Trace
of an operator,
partial trace of an operator,
Transform (Wigner),
Transformation
Bogolubov,
Bogolubov-Valatin, ,
Gauge,
of observables by permutation,
Transition,seeProbability, Forbidden, Elec-
tric dipole, Magnetic dipole,
Quadrupole
electric dipole,2056
magnetic dipole transition,
probability,, , ,
probability per unit time,
probability, spin 1/2,
three-photon transition,
two-photon,
virtual,
Translation operator,,,
Transpositions,
Transverse
elds,
relaxation,
relaxation time,
Trap
dipolar,
laser,
Triplet, ,
uorescence triplet,
Tunnel
eect,,,,,,
ionization,
Two coupled harmonic oscillators,
Two-dimensional
harmonic oscillator,
innite potential well,
wave packets,
Two-level system,,,,
Two-particle operators, , , ,
1756
Two-photon
absorption, (ex.)
interference, ,
transition, (ex.),
Uncertainty
relation,,,,,,
time-energy uncertainty relation,
Uniqueness of the measurement result,
2201
Unitary
matrix,,
operator,,
transformation of operators,
Unstable states,
Vacuum
electromagnetism,,
excitations,
uctuations,
photon vacuum,
quasi-particule vacuum,
state,
Valence band,
2351

INDEX [The notation (ex.) refers to an exercise]
Van der Waals forces,
Variables
intensive or extensive,
normal variables,,,,
Variational method, , , (ex.)
Vector
model,
model of the atom, ,
observable, operator,
operator,
potential,
potential of a magnetic dipole,
Velocity
critical,
generalized velocities,,
group velocity,,
phase velocity,,
Vibration(s)
modes,,
modes of a continuous system,
of molecules,,
of nuclei in a crystal,,,
of the nuclei in a molecule,
Violations of Bell's inequalities, ,
Virial theorem,,
Virtual transition,
Volume eect,,, ,
Von Neumann
chain,
equation,
ideal measurement,
reduction postulate,
statistical entropy,
Vortex in a superuid,
Water (molecule),,
Wave (evanescent),
Wave function,,,
BCS pairs, ,
Hydrogen,
norm,
pair wave functions,
particle,
Wave packet(s)
Gaussian,,
in a potential step,
in three dimensions,
minimal,,,
motion in a harmonic potential,
one-photon,
particle,
photon,
propagation,,,,
reduction,,,,
spreading,,,,(ex.)
two-dimension,
two-photons,
Wave(s)
de Broglie wavelength,,
evanescent,
free spherical waves,
multipolar,
partial waves,
plane,,,
wave function,,,,
Wave-particle duality,,
Wavelength
Compton wavelength,
de Broglie,
Weak coupling (dressed-atom),
Well
potential square well,
potential well,
Weyl
operator,
quantization,
Which path type of experiments,
Wick's theorem, ,
Wigner transform,
Wigner-Eckart theorem, , ,
Young (double slit experiment),
Yukawa potential,
Zeeman
components, polarizations,
eect,,,, , , ,
1261,
polarization of the components,
slower,
Zeeman eect
Hydrogen,
in muonium,
in positronium,
Muonium,
2352

INDEX [The notation (ex.) refers to an exercise]
Zone (Brillouin zone),
2353

Cohen-Tannoudji, Diu and Laloë - Quantum Mechanics (vol. I, II and III, 2nd ed.).pdf

About This Presentation

Slide Content

Tags

Categories

Download

Quick Actions

Statistics

Related Slideshows

Cohen-Tannoudji, Diu and Laloë - Quantum Mechanics (vol. I, II and III, 2nd ed.).pdf

About This Presentation

Slide Content

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 25

Slide 26

Slide 27

Slide 28

Slide 29

Slide 30

Slide 31

Slide 32

Slide 33

Slide 34

Slide 35

Slide 36

Slide 37

Slide 38

Slide 39

Slide 40

Slide 41

Slide 42

Slide 43

Slide 44

Slide 45

Slide 46

Slide 47

Slide 48

Slide 49

Slide 50

Slide 51

Slide 52

Slide 53

Slide 54

Slide 55

Slide 56

Slide 57

Slide 59

Slide 60

Slide 61

Slide 63

Slide 64

Slide 65

Slide 66

Slide 67

Slide 69

Slide 70

Slide 71

Slide 73

Slide 74

Slide 75

Slide 76

Slide 77

Slide 78

Slide 79

Slide 80

Slide 81

Slide 82

Slide 83

Slide 84