Quantum Mechanics by G. Aruldhas .pdf

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Quantum
Mechanics

’ G. Aruldhas

QUANTUM MECHANICS

SECOND EDITION

PHI Learning Drive Med
New Bety-110001
2008

(© 208 y Ph Leumig Pirate Und. Now De. AN revere. No
esl mt box may be eroded ny im. by mimar oa tet

ar ra of hi boc ae vee so mth euler
Ent Pining (Second Eon) November, 2008
Paar by Ase K Geen PI Le Pre Unies 497, Carr

‘heute Da 000" sed Pied by Aapamal or Pes, 8359,

To
My Parents
Gnanasigamoni & Rajammal
soho taught me the merits of discipline
and the reward of education

Contents

Preface si
Preface tothe Firs Elion E
1 Origin of the Quantum Theory 1-24

LA Limitations of Classica Physics 1
12 Planck’s Quantum Hypothesh 4

13 Einstein's Theory of Photoleie Effect 5
14 Compton Effect” €

15 Quantum Theory of Spocifl Heat 7

1 Bohr Model of Hydrogen Atom 7

17 Existnce of Stationary Suns 70

LS Wikon-Sommerfld Quantization Rule 17
19 Elliptic Oris of Hydrogen Atom 12

110 The Harmonic Osilator 1

LLL The Rigid Row 15

112 Panicle in a Box 16

113 The Correspondenes Principle 17

LIS The Stem-Gertach Experiment 17

LIS inadequacy of Quantum Theory 18

Worked Examples 19
Review Questions 22

Problems 23

Wave Mechanical Concepts 25-57
21 Wave Nature of Panicles 23

22 The Unceraint Principe 27

V Contre

23 The Principe of Superposition 32
24 Wave Packet 32

25 Time-Dependent Seuodinger Equation 34

26 Interpretation of the Wave Fonction — 37

2.7 Ehrenfew's Theorem 40

28 Time-lndependent Schrüdinger Equation — 42
29 Stationary Stes 43

2.10 Admissibility Conditions on the Wave Function — 45

Worked Examples 45

Review Questons 54
Problems 38

3 General Formalism of Quantum Mechanics 58-98

34 Linear Vestor Space 58
32 Lincar Operaor 02

33 Eipenfunctions and Eigenvalues 64

34 Mermiian Operator 64

35 Posts of Quantum Mechanics 66

36 Simultansous Measurahiiy of Observablss 77
37 General Uncerainty Relation 72

38 Dira Notation 74

39 Equations of Marion 25

310 Momentum Representation — 79

Worked Esamples 8
Review Questions 95
Problems %

99-140

41 Square We Poentia with Rigid Walle 99
42 Square Well Potential with Finke Walle 102

43 Square Potential Barker 105

44 Alpha Emission 109

AS Bloch Waves in a Periodic Potemsl 110

16 — Kronig-Penney Square-Well Perodie Potetal IL
37 Linear Harmonie Osilitor: Schrdinger Method 114
1% Linear Harmonie Oscilltor: Operator Method 119
49 The Free Panicle 123

Worked Examples 124

Review Questions 138
Problems 139

conan vit
Three-Dimensional Energy Elgenvalue 141-176
Problems
5.1 Panice Moving in a Spherislly Symmetrie

Potential 747

System of Two Interacting Parcs 147.
Rigid Rotor 149

ydrogen Atom 130

Hydeogsnic Orbials 136

The Pre Panicle 158

Tur Dimensional Square-Well Potential 160
The Deweron 162

Worked Esamples 163

Review Questions V4
Problems 175,

Heisenberg Method 177-192

61 The Meisnberg Method 177
52 Mauix Representan of Wave Function 178
63 Mari Representation of Operator 128

64 Properties of Matix Elements 179

65 Schningee Equation in Matrix Form — 180

66 Eigenvalue Problems 180

67 Unity Transformations — 182

63 Linear Harmonie Oilltor: Matix Method — 185

Worked Examples 189

Review Questions 191
Problems 191

‘Symmetry and Conservation Laws. 193-205

71 Syameuy Transformations 193
72 Translation la Space: Conservation of
Linear Momentum 194
13. Translation in Time: Conservation of Energy 197
74 Rotation in Space: Conservation of Angular
Momentum 198
75 Space Inversion: Paty Conservation 200
746 Time Reversal 207

Worked Examples 203
Review Questons 205
Problems 205

MI canons

8 Angular Momentum 206-236

81 The Angular Momentum Operators 2065
#2 Angular Momentum Commutation Relations 207
83 Eigenvalues and Eigenfunctions of L? and L, 209
84 General Angular Momentum 209

85 Eigenvalues of J? and J, 210

6 Angular Momentum Matrices 213

87 Spin Angular Momentum 216

88 Spin Vectors for Spin (2) System 277

89 Addition of Angular Momenta 218

Worked Esamples 224

Review Questions 234
Problems 235

9 Time-independent Perturbation Theory 237-283
91 Base Concepts 237.
92 Nondegenerute Energy Levels — 238
93. Auharmanie Oslo: Frsonder Comcetion 241
94 The Ground State of Helium 242
95 Eifer of Electric Feld on the Ground State of
Hydrogen 244
96 Depensrte Energy Levels 247

97 fect of Electric Feld on the # = 2 State of
Hydrogen 248
98 Spin Om Incraction 250

Worked Examples 232
Review Questions 261
Problems — 262

10 The Variation Method 264-279

10.4 The Variational Principle — 264
102 Rapligh-Rive Method 265

103 Variation Method for Excited States 266
10.3. The Iclhmann-Feyaman Theorem 267
105 The Ground State of lam 208

106 The Ground Site of the Deuteron 270

Worked Branples 272
Review Questions 278
Problems 279

"

2

13

WKB Approximation 280-293
UL The WKB Method 280

112 The Connection Formulas — 282

113 Validity of WKB Method 284

MA Bator Penctation 284

115 Alpha Emission 287

116 Bound States in a Potential Well 288

Worked Examples

Review Questions
Problems 293

Time-Dependent Perturbation Theory 294-319
121 Inodetion 294
122 Firs Order Perturbation 206
123 Harmonie Deruration 297.
124 Transitions wo Continuum States 298
125 Absorption and Emission of Radiation 299
Tintin’ À and B Coctliciots 304
Selection Rules 306
Ray Scattering 208.
Raman Scaterias 310

Worked Examples 313,
Review Questions 318
Problems 319

Many Electron Atoms 320-347
13.4 Indstinguishable Panic 220,

132 Pauli Principle 323

133 Inclusion of Spin #24

134 Spin Functions for Tuo-Electrons 326
135 Spin Functions for Thee-Eletons 326
136 The Helium Atom 127

137 Central Fild Approximation 330
138 Thomas-Fermi Model of the Atom 330
139 Manse Equation 332

1310 Hlanroe-Fook Equation 335

Worked Examples 337

Review Questions 6
Problems 345

14

15

Scattering 348-378

catering Cross-Section 349
Seauering Amplitude 33

Paral Waves 351

Seating by a Contra Potential

Pamial Wave Analysis 392

Signiicam Number of Pal Waves — 356

Scaring by an Anractive Squareovell Poental 357
Breit Wigner Formula 358

Seatesing Lengih 360

Expression for Phase Shifts — 361

Integral Equation 362

‘The Boro Approximation — 364

Seatering by Screened Coulomb Potential 365,

Validity of Bom Approximation 356

Laboratory and Centre of Mass Coordinate Systems — 367

Worked Examples 370
Review Questions — 376
Problems 376

Relativistic Wave Equations 379-409

151 Kicia-Gordon Equation — 379
152 Interpretation ofthe Klein-Gordon Equation — 380
153° Panicle in a Coulomb Field 382
154 Dirac's Equation for a Free Pako 384
185 Dirae Mareos 385
1546 Covariant Form of Dirac Equation 387
157 Probability Density 388
188 Plane Wave Solution 389
159 Negative Energy States 397
15.10. Spin ofthe Dire Panicle 392
ISL Magnetic Moment ofthe Electron 394
15.12 Spin-Orbi Interaction 396
15.13 Radial Equation for an Electron in a
Central Potetial 198
15.14 The Hydrogen Aum 401
15,5 Lamb Shit 404

Worked Examples 404
Review Questions 408
Problems 409

16

17

Elements of Field Quantization 410-441

161 Concepts of Classical Mechanics 410
162 Classical Field Equation Lagrangian Form 412

163 Classical Field Equation—Hamiltonian Formolaion 415
16.4 Quantization of the Fila 418

165 Quanizaion of the Schrédinger Equation 418

166 Relativo Fields 424

167 The Klein Gordon Field 427

168 The Dirac Field 430

169° Classical Theory of Elecwmmagnete Fields 435

16.10. Quantization of Flecuomapuct Field 428

Review Questions 440

Problems 441

Chemical Bonding 442-478

17 BormOppenkeimer Approximation — 442
172 Molecular Orit Method 443

173 MO Treatment of Hydrogen Molecule Ion 444

74 MO Treatment of Hydropen Molecule 448

17 Disiomie Molecular Otbials 457

176 Electronic Configuration of Diatomic Molecules 454
177 Valence Bond Method 457

178 The Valence Bond Treatment of Hy 457.

173 Refinements of Simple MO and VB Approximations 461
17.10 Directo Bonds — 163

VA Mybridizaon 464

1712. Semi Empieical xletron MO Methods — 468

17.13 The Macks! MO Method 468

17.1. tekst Treatment of Beren 469

Worked Examples 471
Review Questions 477
Problems 478

Appendix A Some Useful integrals 479-480
Appendix B Coordinate Transformation 481-482
Appendix Dirac Delta Function 483-485
Bibliography 467-488
Answers to Problems 489-407

Index

499-504

Preface

The aim ofthis rovised socond edition 56 to bring in now material, worked
examples and problems tht makes the book more broad based and useful
These have been done keeping in mind the dual goals of te fst edition, i.
10 help students to build a borough conceptual understunding of Quantum
Mechanics and to develop a more positive and realistic impression of the
subject.

A now chapter om Fick quantisation is added as most of ths Universos
have is as a par ofthe course. Classical cd equation Doth nthe Lagrangian
and llamo form, quantization of non relativo Schrdinger equation nd
relais aa, spinor and vector fields have been discussed. The mechanism
that olds the atoms together in a molecule are discussed in the chapter on
Chemical Bonding, e includes the molecular sbi, valence bond and Hückel
methods along with hybridization and Muckel’s ueaunent of benzene molecule.
New sections on Rayleigh catering and Raman scaring bave Reon added to
the chapter on time dependent perturbation theory.

Learning how to approach and solve problems ás a hase par of any
physics cours, since it helps the understanding ofthe subt, Additional
worked examples and problems, ninety four in all, string the various
concepts involved have also been included in most ofthe chaptes. solutions
manual is available from the publisher for the use of teachers

"The autor is pratefl to Prof. V.K. Vaidya, Pro. V.U, Naya, Prof. CS.
‘Menon and Prof, VS ayakumar fo thee constant encourgement and sport
‘The help recive rom Asia LR. and Divya PS. during the preparation of
‘he manuscript acknowledged. also express my gratte to those who have

given meaning 10 my my wife Myrtle and members of my Family
Vinod and Anitha, Mano) and Bn, Ann and Suresh ad grand children. Finally
"expres my sincere thanks tothe publisher, PIU Leaming for thir unfailing
co-operation and for the meticulous processing af the manusr.

"Above all 1 thank Lord Jesus Christ who has given me the wisdom,
might and puidance al rough my lite

6. ARULDI

Preface to the First Edition

"The concept and formulation of quantum mechanic are not clementary inthe
sense that they aro easily understood. They are hase on the outcome of
considerable theoretical research supported by experimental cvidence. The
quantum mechanical approach o physical problems cannot be explained or
Expresed comprehensive in simple non mathematical terms. Often it he
gap becom Ihe high hcoretical weatmons and he descriptivo accounts found
in many of the texts that makes the subject a difficult one for students, The
present hook, Quantum Mechanic, is expte to bridge this gap. Simple and
lea mathcmaticlccwigucs have heen wet leidos the physical concep

"The book has origina rom a seres of Leurs on quantum mechanics
‘whic the autor had piven for a number of years a the posteraduat level in
Afferent universos in Kerala nd as such the material 1 thoroughly classe

cl vis designed as a 1cibook ot only or postgraduate students of physics
and chemistry but also for students offering an advance course in quanum
‘mechanics. Emplasis is given fr giving the students a thoeouph understanding
ofthe basic principles and thee aplication o various physical and chemical
problems Details of mathematical steps ar provided wheter found necessary
Physical ideas contained in the resul have Bcn discussed, Every eff has
een taken tu make the book explanatory, exhaustive and user friendly.

In Chapters 1-14, the non-oativistic areas of quantum mechanics bave
Icon dealt with whereas the elite asp is discussed in Chapter LS.
CChaper 1 serves as an introduction 1 quantum theory Winging out the historical
vents that led to the development of quantum ideas and is subsequent
proprssive advances, It also discusses the inadequacies of quantum theory In
Chapter 2, wave nature of marr, the uncertainty principle, the timo dependent
and time-independent Schridinger equations are invoduced along with a

discussion om the physical significance of the wavefuneton. Chapter 3, the
ore ofthe ook, deals with the general formalism of quantum mechanics. I
presents the asie ideas of veer space, Herman operators, postulates of
Quantum mechanics, momentum representation and equations of motion
Chapters and $ discuss the energy cigenvalus and cigcofuncions of cain
simple potentials on the basis of Schädinger method. The Iicienherg's
formulation of quuntum mechanle based on mates is presented in Chapter 6,
The diferent types of symmewies and th related conservation laws ae the
suce of discussion im Chapter 7. À detailed chapter on angular momenum
(Goth ori and sin), à topic of fundamental importance im physics, i alo
included

(Chapters 9-12 tea the important approximation methods, tb variation,
WKB, time-indepondent and time-dependent perturbation methods. Th
application of these techniques 10 study he round tate ol two clecton am,
Sark effect in hydrogen, the deuteron ground tte, alpha emision. spin-orbit
interaction, transition probability. emission and absorption of radiation and
selection nes i weatd in the related chapters. The information provided by
many eecran atoms is of considerable importance for the understanding ofthe
structure of molecules and their properties, The searing phenomenon Is very
important as it represents one of the best methods of staying the properties
of toms, nuclei and interaction of elementary panicles among themselves
Even the concept of an atom isthe oulcome ofa scateringeaperimen

The book conelodes with chapter on relvistic wave eqstion which
accounts for election spin, electon mapneie moment, the concept of bole and
‘any other interesting phenomena,

Considerable anio devoted to worked examples inthe text, Mone
‘han hundred examples ranging fom simple plug-in type to Faly complicated
nes have been fully worked out. About 150 problems, given at the end of
respective chaptrs, ar useful forthe understanding ofthe basic concepts and
Applications. Answers to these problems ar alo provided atthe end of th
text. The review questions may serve as he bass fr self study, cas discussion,
sssignment, et

Tes with dep sense of gratitude and pleasure that | acknowledge my
indcbtcdnes to my student for al the discusion and questions hey have
raised. Iam likewise indebted to Dr. A.D. Damodaran, Dr. V.K. Vaidyan
Dr. C.$. Menon, Dr. V. Ramakrishnan and Profesor Jose Davidson for thet
suppor and cooperation. Last, but nt Las, acknomledgs my gastado to my
‘ste Myc Gra and children or he encouragement, cooperation and academie
environment they have provided thoughout my care.

Finally, expres my sincere tanks to the publishers, PHL Learning for
‘he meticulous processing o he manuscript both during editorial ad production
stages

à ARULDHAS

Nini

Origin of the Quantum Theory

1.1. LIMITATIONS OF CLASSICAL PHYSICS

The firs scientific temps to study the nature f light radiation vas at of Sie
Jaane Newton who proposed the corpuscular theory of Tigh. According 1 this
theory. ight consists of in pote clastic parcs, called corpuscles which
Lave in al icons insight ies with the velocity of it. I could not
explain phenomena such a inefeence fraction and polarization I pedited.
thatthe velociy of igh is more in a denser medium than that in a race
medium

“The theory tht Hight is propagaed as a wave through à byposbetcal
else medium called ether was developed by Huygens, Young and Fresa,
“They considered light propagation of mechanical energy. I go etliche
a5 could explain most of the experimentally observed phenomena including
the veloiyof ight in ferent media. A completely diferent concept ranting
the ature of ight was proposed by Maxwell athe second al ofthe nineteen
(entry. Acconding to hin, light consists of eeewomagactic waves wih thsi
lec and magnetic Held in planes perpendicular 10 the direction of
propagation. The electric and magactc fies associated withthe wave are
povemed by Maxwell's equation. Predictions of Maxwell were confimed
«xperimenally hy Ter The clctromagnctc tory of light was eccived wel
Thought had failed to explain plenomena such as photocicti cc and
emision of light

This, towards the end of he nineteenth century thee was a seso of
‘complstion among the physicists as thy thought that classical physics was.
‘capable of espliningalobscrabl phenomena Then came arc of important

2 Overton Mesta

experimental discoveries starting with X-rays in 1895, radioactivity in 1896
nd lecron in 1897, which could not he cxplinod, on the basi of clasial
physics. In addition, lo of experimenta observation stating with blck-body
radiation and opal pee accumulated could nt be explained bythe elasial
on.

Black-body Radiation Curves
‘The spectral energy density u, of blackbody radiation fom a back-body
cavity depends ony on frequency ad temperature Tot the cavity. Based on
thermodynamic arguments, Wien artompted to explain the observed spectral
energy density verses fiequeney y curves. The apreemen with experimental
resul was good only in the high-fequeney repion (Figure 1.1). Treating
radiation inside a black body as standing lecromagncic waves, Rayleigb and
Jeans estima ths number of modes of vibration per unit volume in the
Frequency range Vand y Vand evaluated the spectral energy density u, by
taking its produc with the average energy of an oscillator of frequency v.
“Their expresion agrcd with experiment only at low frequence. The mets,
based on clasica) coros has alld give a single formula that could agree
vith he experimental black-body radiation curve or he entre frequency range.
“There is no wonder tat this disagreement led to a complete revision of our
ideas of physics.

> Bena Cane, — Wis Gun Ra ee

Optical Spectra
[ach chemical clement showed characteristic emision spectrum consisting of
diet Tins. Comparatively, simple spectrum was observed for hydrogen
som. In 1885, 1. Balmer aranged them in the form of a series and had
Suggested the following empica formula forthe wavelength:

tg ote Guante There 3

page) omas an

het, R isthe Rydberg constan for hydrogen. Laer, ote spect series
er also discovered (Table 11) and toy were found lo obey the formula

=a) a2

here
melde name AD 08 D am

“The regular found inthe special ines indicate ha shee must he some
gonoral mechanism in the emission of Tight. The classical here fala lo
ive the correct mechanism responsible for Abe spectral series in hydrogen.

Table 1.1. The Hogan Speci Saree

‘Goan munter

Name of serie Umersmaieem UTE Region
Tynan ares PET) 234 Uravier
Bracket sees mos neon Ancares
und eis nas fran In
Photoelectrie Ettect

When light is incident on certain metallic surfaces, clecuons are release.
These ate called photoelectrons and he phenomenon is called photoelectric
efec. Some ofthe important conclusions srived fom a detail investigation
fF he phenomenon ate lcd Below

1. The energy distribution of the photoelectrons is independent of the
intensity of ths incide ih

‘The maximum kinetic energy ofthe photoelectrons from a given metal

is found 10 be proportional 1 dhe Frequency ofthe incident radiation.

3. Fora given metal, phooslecuons ate not emited be frequency ofthe
incident lights below a rin thesbold valu, whatever be the intensity
oF inciden ight

4, Fora given frequency, the number of photocecrons emitted is direct
proportional tothe intensity of incident ih.

5. There seems 0 be no time lag etwcon the onset of irradiation and the
resulting photocutent.

made to explain thes aspects on the hast of
lass eas, Ihe foret conclusión Sond unexplained

Specific Heat of Solids
Tn soi, the atoms vibrate shout hei equilibrium positions in cir aioe
sites. Based on the lw of cquipaniton af energy. the average energy of a
Simple harmonic ose i KT, where isthe Boltzmana constant and is
the absolute temperature, An stom cn vibrate about thee mutaly perpendicular
ircctiom, the average energy of an atom is MT, Then the energy pet gram
or is SNKT = BR N Bing the Avogadro's number and R, ih zas consan
“This leads 10 à value of C, = 3K, which is known as the Dolo and Peis
law. Ar dinar temperatures, solid generally obey this relation, which is
isc on classical hooey. However, hen the temperatures homer the scie
heat decretos and gocs to zero when Y => K. That, he classical ides ald
10 explain the variation of specific heat of solids with temperatures.

"Thus, by 1900 sient were convinced that a number of experimentally
observed phenomena could ot he cxplincd onthe basis of classical physics
and contain new revolutionary das ac nse to understand things har.

1.2. PLANCK'S QUANTUM HYPOTHESIS

‘The problem that confonted Max Planck was a theoretical explanation forthe
black-hody radiation eurves. As alcady meationed, the Wien's formula agreed
we with experimental rule at high fequencis whereas the one dec 10
Rayleigh and Jeans agro at low frsquencis. In 1900, Planck modified Wien’s
formula in such away tht it fited wih the experimental curves precisely and
then he looked for a sound theoretical basis forthe formula. He assumed that
the atoms of the walls of the black body” Rehave like tiny clcctromagnetie
esciltors cach wih a characteristic frequency of oscillaten. The oscillators
mit elctromapnei energy into the cavity and absorb clectromapneti energy
fiom it Planck then Holly put forthe following suggestions regarding the
atomic oscillators:

1. An oscillator can have energies given by

aah, = 0.123, us

her vise oscillator que and constan known as Planck's
Constant Its value i 6526 x 10. In ather words, the osciltor
energy is que.

2. Oscillators can absorb or emit energy any in discret nis ale quant,
That iv

a,

ed sae, ner emits nor absorbs cr

An

Ww as

An osiltr, in a quant

‘The average energy € oF an oscillator can De evaluated om the hası of
‘canonical distribution formule

Yer bean

Sone exp Enr

Yew ı-menn

wv

PDT us
The quantization condition, Eq. (13), thus invalides the theorem of
‘exuipation of energy whieh is based on classical pbysis. tis known tha the
mer ofosilaters per anit volume inthe frequency range Y and Ys dv is
(Savio) dv. With the above expression for average energy & the spectral
energy density, e piven by

ar

A ÁÑ 16

2 Sp WAT ENT ww

Equation (1.6) the Planck's radiation formula which reduces he Wien's
or Rayligh-Jeans law according a

wv be
Beto Mec

Planck’ explanation ofthe blackbody radiation curves in 1900 provided.

the crucial stp in the development of quantum ideas in physics whieh was put

on a firm bass y the pionscring work of Eimtin, Bohr and others in cr

years. The concept hat energy i quantized was so radical that Planck himself

‘vay telucint to accept it This can be seen fom his owe words “my fue

‘Mtempes oft te elementary quam of action (tht ste quantity) somebow

o Ihe classical theory continued or a number oF yeas ad they cost mo a
sat deal of for.

13 EINSTEIN'S THEORY OF PHOTOELECTRIC EFFECT

Einstein succeeded in explaning potoelectie effect on the basis of quantum
ideas He assumed eleciromapoetic radiation aves tough space in discrete
quant called photons as during the emission and absorption processes. The

energy of a photon of froqueney vis hy. When light photon of energy vis
inciden on a metallic surface, par o energy A, 8 wd o Fae the «lero

fiom the metallic surface and the other pat appears as Kinetic energy
ofthe photoelectrons. The conservation of energy requires

Were Ay refered o as the work Junction, depends on the nature of the
‘miter. The frequency Y, is called the thresh frequency, This teaion
‘ccounted forthe <xpcimenal observations regarding photnslctc cf

"Though Planck quamived the energy of an esilator, he holcwcd that
light travels through space as an elecomagoeti wave. However, Einstein's
photon ypothesissuppests that it travels hrouph space nt like à ware but
Tike a pate The photon hypothesis is thus o dirt confit withthe wave
thoory of light. Millikan who veri Einstein’ bypoess cll a "bol, not
10 say reckless, hypotesis

In ie photon picture, each photon transports a linea momentum p = EIc
«= hie, where E is the photon energy. This conclusion can also be deduced
From the thoory of laity. The ati expression for energy Es given hy

Bech ene!

he isthe velocity of light and m, i the fest mass ofthe panico. Since
the Fest mass of ths photon is ur,

£
caw o poke

Ez as
“This equation contains in it bo particle concept £ and p) and wave concept
{and A), Confirmation of panicle mature of radiation was provided by Compton
fte. Compton effet is discussed inthe next section

1.4 COMPTON EFFECT

‘The spectrum of Xe1ays scatered from a graphite Nock contains intensity
peaks a wo wavelengths ne a the same wavelength as the incident ration
and the other ata longer wavelength 2. Assuming Ue incoming X-ray beam
a an assembly of photons of energy hc, Compton was able 10 show that

Mira

oy us

‘wtere m, is he rest muss of electron, is de velocity of light and 9 isthe
Scattering angle The factor Amy) is called Ihe Compr waselengih. The
Compton shit A2 vais Between zero (for 9 = 0, corresponding o à grazing
collision) and 2hlinge) (for @ = 180", coresponding 10 head-on collision).

unm They 7

{Agreement of Bg, (19) withthe experimental reis conims the partite
rare of radiation

15 QUANTUM THEORY OF SPECIFIC HEAT

Einstein explained the anomaly in the specific het of solis using quantum
ideas, He replaced the 3N degrees of freedom of N atoms ofthe solid by 3
citons, al having the sam frequency, Quantizarion ofthe energy ofthe
reiten Heads to the following expression for th vibrational pocii heat

nf) ae teat

Gen tint)

ir

(22) er

7) [open Y
tre 0 = Ax defines the Einstein temperature. At ordinary temperatures
>> y, the expression for C, appmaches the Dong and Pt low At lo
rempart, Pe 8,

m

ant (2) op[-&
am (2) 00 (-2) um
ich nt green wih the experimental 7 dependa:

Dee improve the ich's mod y sami te quence
vibrato forthe AN oscila and dismi tem a Halu
Fan. cata een Y was abe met. Debye dived he following
Baw for C, when AD > |

122% ary
EJES am

“The agreement hotweon the Debye model and the experimental observations I.
god

15 BOHR MODEL OF HYDROGEN ATOM

Rotberlocd, based on the results of ascatering experiment, was the frst 10
propose the nuclear model of ihe atom, In this modal, he postive charge is
‘confined to a very small sphere called the nucieus and the sIscwons move
‘wound it, This model of the atom is highly unstable as a moving electic
charge edits energy. Consequently, the elton inthe Rutheford model of
the atom spirals int the mucus, Moreover failed to plan Me observed
sharp spectrl nes of atoms, The concep ofthe nuclear model of he atom
twas thus available for Bob in 1913.

‘The emission sper ins of hydrogen were groupsd into Several srs
which fit the empirical formula, Ug. (12). A theoretical explanation of the
hydrogen spectrum, based on quantum ideas, wa fist formulated by N. Bebe

1913. He based is arguments on two assumprons, now knowa as Bohr
postulates:

Postulate 1 An electron moves only in certain allowed cirevlar orbits, which
are stationary sates in he sense that no radiation is emited, The condon for
Such states is thatthe orbital angular momentum of the cseton equals an
integral multiple of (= 4/21, called modified Planck's constant Thertor,

123, a

Postulate2 Emission or absorption of ration occurs only when the electron
makes à Mansion om one stationary slate 10 amor. The radiation has à
elite frequency Yan ven by the condition

-£, a

mor = nh

‘Consider a hydrogen atom in which the icon of mass mowss with senc
$ in a circular orbit of radius y conte nis mucous. For simplicity, the
fucks is assumed tobe atest. The Coulombic action between the electos
and the proton provides the necessary centripetal for, That is

as

ln is expression eis the eleewonic charge measure in coulombs an 4, is a
constant called the perninei of vacuum. The experimental val oF is
885 x 10 CIN Im 2. For conveniones, we have writen

Ae = koto? Nic ? (1.16)

‘The kincie energy (7) of the clean is thon given by

re bot = am

Prom Eq. (1.13), he velocity ofthe clean in its mi ori is

q E cas)

Substituting this value of v, in Bq (115), we pet

a)

‘The sate for which = 1 i ale ths ground tte while tte or which n> 1
ao called the exited states. Ths ras ofthe obit in he ground ats is called
the Bohr radius and is usually denoted by a

la Stunts, we get

ES (120)

eon proton system is

fen
“he tou eneay of Hydrogen atom inthe mh state is
=
a
123 ua
ES)

Substtton of Eq. (12) in Eq. 1.1) gives the frequency ofthe spectra ine
When the elccron drops from the mth to nth stats
foment)

= monzı das

For hydrogen He systems (It, Li?, Be}, Ihe ener is piven by
Lime ne
ETES

us

where Zi Ihe atomic number o he sytem. Bohr model was highly successful
in explaining the spectrum of hydrogen and hydrogen like atom,

“Thee electronic length sales sed in numerical calculations ae the Bohr
radius a, Ea (1.203 the Compton wavelength À, = Mm athe fie structure
constant a defined by

aa

10 Cuarto Vochanes

lem he Dob formula, Eg. (123) is expressed in tems of Rydherg constant
2, defina by

aa

For solving mumeral probleme, often we qui the folowing lion
‘onoeting these paris
2 peat
A en Pa
Te Rydbery consta for an tom with a cios of fine mass is
denne by, Thon in he expression fr Rp 0 as eps hy m
mas of em

1250)

1.7 EXISTENCE OF STATIONARY STATES

Einstein's photoeecsie equation proved unambiguously that electromapnetic
radiation interacts with mater like an assembly of dinero quant of ner.
Im 1914, Franck and Hort repored an ingenios experiment 10 prove that
‘mechanical energy i also absorbed hy atoms in discrete

The experimental setup consists of an electrially ested fi
(aod) along the axis ofa clinical pid which is surounded by a colector
(node), The whole stp i placed in a quart ebamber filled with mercury
‘vapour: refer to Figure 1.2, Elcetons from the filament ac accclratd 10
the grid by a positive potential VA small retarding potential Y, (Y, << Y)
between the pd and anode reas the accelerated electons. The electrons
collected by the anode give rise o a current which is measured hy a
micromstor The plot of collector curent veros the ccclrating potential Y,
is shown as in Figure 1209). When V, is increased from zero o à eitial
Potential Y, the accelerated clectons make only elastic collisions with the
atoms of mercury. However, when V, = Va the clctrons make inelastic
‘collisions near the grd and give te eli kinctie nergy to mercury atom
After losing their energy, he electrons are unable to overcome the reading
potential Y, leading 10 a sharp fll in current. Elecuons which have not
made inclastic collisions reach the anode giving small cute. The fst
drop occurs at 49 V and a spectral lime appears simultancously in the
emission spectrum of the mercury vapour at 253.6 am, the value
corresponding to a photon energy of 4.9 eV.

When the potential is increased fur, the region where the clcrons
reach the ecal energy of 49 eV moves closer o the flame. Ate losing
the energy to mercury vapour by initie colision, the electron picks up
ray on their way 10 the grid resulting in an inrease of erent, À second
laste collision occurs car he grid, when V, = 2V, and a second curent

tino he Quantum Tory 11

o o

Figure 12. (2) Experimental arrangement of Franck and Hers,
0) Bot ot ealeriorcunen! veras accelerating olga.

minimum occur a shown in Figure 120). A similar behavinur is found at
integral multiples of Y That is, the mercury atoms absorb mechanical energy
in quanta of 43 eV. Ths, the currence or minima in the / verses Y, curve
‘can be explained oy by the existence of stationary sta In the atom,

1.8 WILSON-SOMMERFELD QUANTIZATION RULE

In 1915, Wilson and Sommerlchd proposed independent} a more general
quantization role in which the Hamilton equations of motion ae first solved
in the independent variables 9. 9. «9, And Pe ys» The stationary
States an toss for which the action integral a any perde motion equals an
integer ics

Gnienenpe: cosa 126

er engen over one pc af moon. la clar ai. he
Sn agus omer La mo conan of mol. ee rl
einen
m Ñ
$ mar do ant o mure am

whic is Bote’ quantvation rue, Eg. (1.13). Inthe following sections, we
Shall apply the general quantization sue to some cases of intros

12 Custom Mechanics

19. ELLIPTIC ORBITS OF HYDROGEN ATOM

‘The simplest fore that is associated with point paris ie ie mutual comal
force acting along the in joining the two, Consider the two parcs ofthe
hydrogen atom with the nucleus of charge Ze fixed atthe origin and the
electron of mass m moving relative tthe nucleus (eer o Figure 13. As per
sal mechanics, the frst ntgrals ofthe equations of motion ofthe system

m 2 L (a consant) am
ud
da const am
Figure 1.4 Paramotor of th eine
where

ve = u

Land E are pla monito nd ol energy fie system respecte
“he mal women p, = m? ande aga oma Zp hy ı
nan and Gr. dr=ah am
where and, I Since ps oma.
ro $40 «th où marrant um

From Eg, (129, we have

a

CEE

= mi am +
\

and

“The positive va
‘ths negative val 0 the othe Ral oF the elipcal path, Theo

ofp, comespords 1 the increase OFF 0M Fy 19 Fax

gin he Quart Theory 18

{Combining this result with Eqs. (131) and (132, we get

me [m
Are VE

het the pricipal quantum number, n

(k +n) = ah a)

+ A Solving for £, we have

3 us

Bia aa am

whic is same as Eq (122. I may be mentioned here that no restriction is
made to crea orbits while deriving Ea. (135)
de

An points À and a, €
= a

and this happens for r

Wien Dm 0, om Eg (LA we got

PL EE 136)
QE ne
nd ths sum oF the soon oF this equations given by
tan they Boe
lan + ln * CRE
A Faas Fas = 2 he semi-major axis ofthe clips is
a am

GE

Substitution of th vals o E gives the leat ofthe semi-major axis ofthe
sh obit

a 038

here a is the Bobe adios.

ite orbits of hydrogen atom was fit solved by Sommerfeld as an
example of the general quantization role. The energy is dependent on the
principal quantum number which i the sum of 4 an, Therefore, all bits
for which £1, = has the same Vale, will have the Som energy. States of
‘motion corresponding woth same energy ate sid to be degenerate, Among all

14 Quantum Vochanes

bis having the same vals of ons will be à circular ono for which, = 0,
L = m. This explains why the energy expression is the same as the one by
Bohr theory for crear orbits The case = 0 was ruled out by Sommerfeld
as I eoesponds 10 motion ofthe electron along a sight line tough the
rocks, Electron obits for = 1,2, 3 ae shown in Figure 1

Figure 14 Sommer leeren ote for m= 1.2.5

1.10. THE HARMONIC OSCILLATOR

‘The displacement x with time FOF a harmonic oscille of frsqueney Y, is
ven by

= sin ayy) a)
The force constant and frequency ny ae related by the expresion:
w LE or k= arme um
2e Vm

“Therefor, the potential energy is

em stay)

ve tie

rin tte Gata Theory 15

‘The him energy de

Bm

And th total energy is
E=Te Vamo am

According to the quantization rule

fr

‘When x completes one ey, changes by period T= 1/4 Hen, subsitming
the valves of dd and ds, we pet

à fia ce

On solving, we get

mt ent |; assy
Sting the vale of in Ei. (LAD we have
Bw aby mo 97 0.1.2 a

‘That is, according to old quamum theory. the energies of a linear harmonic
oscillator are integral multiples of A, =o.

1.11. THE RIGID ROTATOR

A rig rormor consists of 1o mass pons comected hy a massless rod. As
the rotator is rigid, potential energy i ero. A rigid roter restrict 0 move
ina plane i described by an angle cooninate 0 The momentum conjugate 10
the angle coordinate py = 10 = for where 1 is the moment of icra, is a
constant of mation. The quantization rule neduces to

th

Ino-m ane core 049

“Thats, the angular momentu py i a integral map of Ar. With is
vale oF py = lathe energy of à Clasica! stator E = 1072 reduces to

Tea... u

16 aston Vector

To describe a rigid rotator in space, one requires the two coordinates Hand à
ofthe polar coordinates. Application of the quaniation rte gives the same
expression as the one in Eg, (1.45) forthe total angular momentum and

ES

for the component of angular momentum along the z-axis. The ner is
independent ofthe quantum number mand is piven by Eq. (1.16). Each energy
level fs (2 + 1) fold dsgencrats

1.12 PARTICLE IN A BOX

Conde a panico of mass m moving inside roctamgular or of idos a,
and © without be influence of any force. Let us assume tha the colision wit
th walls ofthe box is perfectly clas. The cartesian axes are taken along the
sedges of the box. The linet momentum p,,p, and p fe constant of motion
and change sign during collision with Ihe walls Applying the rule of
‘quantization:

da ae] am

»

We got

The total enerry

E,

ana ee

In the case oa cubical box of side a

= Lites 7 a

“The eneray of the sytem is thus quantize.

rin lo Quantum Theo 17

1.13. THE CORRESPONDENCE PRINCIPLE

"The correspondence principle of Bohr stats hat he quantum physics reduces
to classical hysic at large quantum number. To explain the principle, Bar
ad a simple lation based on the ine spesra of one electron atom.

Classical, during the perodic motion ina stationary ort, the frequency
ofthe light cited bythe atom is equal 0 is Frequency oF revelation v, From
gs (117) and CLIS), we got

a0)

ET

For the transition (+ 1) >, ih frequency of the emited radiation () is

sven by Bq. (1.28) as
= Baim! fant) u,
OS

Tis hear that vv, ven e, as rogue by
principe

correspondence

1.14. THE STERN-GERLACH EXPERIMENT

‘The concep o veto tom model was induced to explain erin experimental
observations inthe atomic spectra of complex atoms. This atom mode] has the
two special features! spatial quantization and the electron spin, Direct
experimental confirmation of these fe
experiment

‘The magnetic moment fan atom arises due to orital and spin motions
oF the econ. In a uniform magnetic il, the magnetic moment vector
Experience a torque which tends so tur the dicton ofp and hence that af
‘gular moimentum withthe magnetic il. This makes the vector Jo precess
around the field Keeping a fied anple © with the fel Ifthe magnetic field
is nonuniform the atom with the magnetic moment experiences an additional
traslaory motion. Tence. i an atom with a magnetic momen ds shot in the
direction through a magnetic field whieh increases along be zas, twill
be deleted inthe pose or negative zalirectin, The sift will depend on
the val of, IF J takes continuous vals the eam would Spread out into
à continuous Hand in th x2 plano. Tasted, thor is space quamzation cach
nom will ete tb magnetic field in particular quantum state defined with
the magnetic field direction asthe axis. Consequeny, th beam wl sp into
Separate houms and produce sores of distinct spots or sharp lines, one for
ch possible vale off

In ter experiment, Stem and Gerlach passed a collimated beam of silver
toms though an inbomogencous magnetic field produced by a specially

18 aston Vector

designed electromag (Figure 15). While pasing through the magnetic cd,
the hcam spits mo two parts, omo traveling upwards and o other downwards
producing two spots on the seen,

Figure 15 The Stem-Gerach esperar,

In the ground sas the outer eletrn of silvr atom i in an state (5)
‘Therefore, it has no oral angular momentum and consequendy no orbital
masse moment. The observation of 2 spots suggests that ll electron have
an ntinsi magnstic moment of the sams magnitude with two possible
úoriemarions—paral or antpaale tn he magnetic Held. The origin of is
magnetic moment can be understood if its assumed tat the elcron ofthe
silver atom hasan inrinsic spin angular momentum. Ifthe spa o the eleewon
ds, he component of spin can have 2s + Y orieniatons. As two pots have
Pen observed esporimtlly 2 + 1=2 or = 12. That the spin of electron
is 12, Ths, spin, fist detected inthe Stem-Celach experiment, has become
an “observable

1.15 INADEQUACY OF QUANTUM THEORY

‘The quantum hcory developed by Max Planck, Einstein and Bahr was fou
to ho adequate in caplamin certain phenomena such a black-body radiation,
photeleti effect, harmonic oss, gid outr, spectral ins of hydeogen
som, ee. However, it was inadequate to explain number of cases. Some of
them at:

1. The general quantization rule is valid only for periodic systems. Hence
the quantum theory 8 no applicable to non peidie caes.

2. Even inthe case of hydrogen spectrum, the theory could explain only
the broad fears,

3. Application ofthe quantum tory to hellum atom, hydrogen molcul
ie. ed o results conte 10 experiments

4. Pied to account the concept of half od integer quantum numbers
which are necessary for explaining Zeuman oc and ne structure of

rin lo Gata Theory 19

ctrl lines. In other words, it fied o account for the concept of
spin

5. I fled to give a method for cvaloing transition probables and
intenses of spectra ins

6. The theory was unsuccessful in explaiaing the dispersion of light.
‘These shortcomings suggest thatthe quantum theory i na à very general one

and the defects canot be removed by extensions and modification, Therefore,
one has to Took for a more basi theory to account forall phenomena,

CORRE D AMPLES ——

EXAMPLE 1.1 The work function of barium and tungsten are 2.5 <V and
42 eV, respectively. Chock whether these materials are useful in a photocell
which is 10 be used 10 detect visible it.

Warelengih (2) of visible light is in the range 4000-70008.
he 6.626 10 x 3x 10°

‘000 x 10° x 1.6 x 10
= 3106 eV

Energy of 4000-4 tight =

70 10 16310
“The work funciono tungen i 42 eV which is mor thant cora range
of vial gh ence baum I the only mal nf Fr A pupas
EXAMPLE 12. Verify Eg 0123) forthe nery of the hydrogen atm inthe
ih sats,

From Ea. (1.29), we have

Energy of 7000-À ight

ne

OA x 10 (LG x 10 CN
P6826 10

Sen

AO
21208210)
TES

1356

E

20 Quantum Vector

EXAMPLE 1.3 Calculate the maximum wavelength that hydrogen in is
ground stat can absorb, What would he the next maximum wavelength?

‘Maximum wavelength comesponds to minimum energy. Hence the jump.
ftom ground sat t first cxchod ste gives the maximum À.
Energy of the ground state = 136 eV

Energy of the first excited state =

Energy of the m = 3 state

‘Maximum wavelength comesponds the energy = 13.6~ 3:
Maximum waelongth|
©. he (6626 10" 55) (3.0% 10" mA)
vÜB-E 102% 10x10 °F

= 120m

‘The next maximum wavelength coresponds to jump from ground state 1 the
second excited stat, This requires an energy = 136 LS = 12.1 eV. This
comesponds to the wayclegth:

he (6626 1039) x 0 x 10" ma)
DIET

103 mm

EXAMPLE LA À hydrogen atom in a sate having a binding energy of
85 eV makes a wanton 1 à te with an excitation energy of 102 eV.
Caleulate the energy ofthe emited photon.

is the

citation energy of à st ay aiterence betwen that tte

and the ground sta

Excitation energy of the given state = 102 eV

Energy of the state baving excitation energy

102 eV =-136 + 102 =-34 ev

Energy ofthe emited photon daring transition from OBS eV to 34 eV
85 (34) = 255 eV

{Lt the quantum muse of - 085 eV state bem and that of 34 V state
be m. Then

138 2005 on

BE usa or med

Te nation is rom n = 410 m = 2 state

rin lo Ghani Theo 21

EXAMPLE LS Dstermine the ionization energy of the Ho ion, Also eats
‘he minimum frequency a photon must have to cause nition

Energy of à hydrogen lke atom inthe ground state

“The ground sae energy of Tle on = 4 x 13.6 eV

Tonizaion energy of ie on

“The minimum frequency of a photon hat can cause nization

E Sax 16x02

610 He.
CRETE

EXAMPLE 1.6 Calculate he velocity and frequency of evolution of the
electron ofthe Bol hydrogen atom in is ground state

From gs 1.17) and (1-18), the velocity of Me electron of a hydrogen
om in is ground state

A NO gt 2218 x 106 ms
Beh” 26AS 10) 62610

rom Eq, (1.50), the frequency of eve

fom of the cccron in the groun

OA x10 )0.6 x 107)

2 = 655 100 Ha
388% 10 776.026 x 10

EXAMPLE L7 What poteatldiference must be applied to stop the fastest
hoteleros emited by a suce wa clecuomapneticradiioo of frequency
1.5 10" 11 i allowed 1 al on i, The work Function ofthe surface is 8 SV,

Energy of the photon = hy = (6526 x 10°%) (LS 1005

652610 "510%
16x10”
= em
orgy ofthe fastest clacton = 6212 - 50 = 1212 eV

ene, te potential ifrene regule to sp the ists electo i 1212 V
EXAMPLE LA Xrays with 2= LD A are card fom a metal lock The
scaterd ration aro viewed a 90° 10 the inciden direcion. alte the
Compton sit

CCompron shit 44 = E

os)

6.526% 10 "4 = c0s90)
CIA

22 ou

242 x 102m
= 002 À

EXAMPLE 19 From a sodium surface, light of warclengih 31254 and

3650 À causes emission of cleans whose maximum kinetic energy is 2.128 ev.

and 1395 eV, respectively. Estimate Planck’ constant and work funcion of
sodium.

hehe
MM Kine enery

27%

he
Mans x 16% 10%
a

he
a TL
3650 x10 A

“(a )
io ® (3108 ~ 3650,
Therefor,
0.533 «1.6 « 10°" x 10" x 3125 x 3650

S25 x3 x 10

533 x 16% 10

16 x 10%

rom the fst equation, the work function

he 6.176 0 x 3310
PRET TT
S24 x 16% 101

S24 eV

EEN QUESTIONS > K——

How classical physic fica 10 account for he spectral distribution of
energy density ina black body?

Explain photocictrc effet, Define th term: work funcion, threshold
frequency and cutoff wavctenth

3. Briefly outine the mechanism by which photocictrons ae emited.

4. List ou the basic experimeml results ofthe phone phnomena,

5. Wei the expresion for the velocity of photoelectron ented when
radiation of wavelength A is incident on à photosensitive sures, the
(hreshld wavelength being Ay

6. What is Compton effet? Explain its sign

= 2128 1.6%

rin lo Qua Theory 23

7. Explain the assumptions of Planck with regar 0 cavity radiation and of
Einstein with regard to phowslsiie efect

8. How particle nature frdíain was confirmed by the photoelectric sect
and Compton efec

Explain tb postulates of Bohr with regard to hydrogen stom,

10. Apply Bobr's theory to singly ionized hllum atom, What relationship
‘ut Detwocn this spectrum and the hydrogen spccrun?

11, Can a hydrogen atom absorb photon whose enerpy exceeds its binding
ner?

12. Whit are the different posites of special line emission for the
Iydrogen atom when the eecron is excited 10 the n = 5 state?

13. State and explain the general quantization rulo, Explain how i eas to
Bohrs postulate regarding stationary stats

14. Sketch Sommerttd electro ori or n=,

15. Explain the significance of Stern-Gerluch experiment

ii —

Light of wavelength 2000 Á falls on a metalic surface. Ifthe work
ution ofthe surface is 4,2 eV, what isthe Kino energy ofthe atest
hotelsiran mited? Also callate he stopping potential ad hrshold
wavelength for the metal.

‘What i the work Function of a mal, ithe threshold wavslngth for it
ds 580 nm? I High of 475 nm vasclengt fall on ths mal what i

3. low much energy is required 10 remove an clecwon fom the n= 8 sate
of à hydrogen stom?

4. Calculate the frequency ofthe radiation that just onizes a normal hydrogen

5. A photon of waveleni 4 À sus an clecoa at rest and scared at
an angle of 150° 10 ús orginal direcion. Find the wavelength of the

photos ater collision,

‘When ation of wavelength 1500 A is incident on photocel electron

are emilted. Ifthe topping potential is 44 vol, calcule the work

ution, threshold frequency and heshold wavelength,

If a photon has wavelength equal to dhe Compton waveleagb of the

patie, show thatthe photons energy iy equal o tb rest energy of the

panico

8. Xrays of wavelength LA A are sentered from a Block of carbon. What
willbe the wavelength o scattered X-rays at () 180", Gi) 90”, i) 0

24 Quantum Vector

[Determine the maximum wavcleng that hydrogen in is ground state

can abro, Whit would ho the mov wavclengih that would work?

10. Consruct the energy level diagram for douby-onized liom,

11. What isthe potential energy and kinetic energy of an electron in the
around sate ofthe hydrogen atom?

12. Show that dhe magnitude oF the potential energy of an clctron in any
Bohr orbit of dhe hydrogen atom is vice the magnitude o is Kine

‘nergy in that orbit. What isthe Kinctic energy of the electron inthe

2 = 3 ori. What is its potential energy inthe = 4 orbit?

Nr

Wave Mechanical Concepts

2.1 WAVE NATURE OF PARTICLES

The theory tradition rave in space in the form of waves got established
asi succes explained the optical phenomena lis rfection, refraction
intrterene, diffraction and polarization, However, to explain phototetric
effet and Compton effet one needs the parle or copuscuar nature of
radiation. Thus, radiation possesses wave-parle dual. Sometimes it behaves
Tks à wave and a times Ike a parie

Matter Waves

The dual nature of radiation prompted Louis de Bropi to extend tw material
particles also, Te reasoned hat 6) nature 3 strikingly symmstric in many
‘nays, Gi) our observable universe is composed emily of radiation and mater
(Gi if light bs a dual nature, perbaps mater has also, Since matter is composed
of pales, his reasoning suggested that one should lok for à wavesike
ehaviour for mater, In other words, de Brogie asumed tha a ways is
associated with a pari in motion, called matter wave which may be regardod
as localized with the parce. Again be sugested at he wavelength of mater
Wave he given by te same relationship, namely
ala

asked en
rar]

cr mis the mass and y isthe vl ofthe particle. This elation is often
refereed to as the de Broglie relation

=

26 aston Vechner

Electron Diffraction Experiment

‘The concept of wave nature of material panicles was independently tested by
Davisson and Germer and by G.P. Thomson, Thomson's experimental
arrangements analogous to Lau X-ray diffraction method. Th arrangement
{shown in Figure 2.1) consists of a discharge tube in which a eam af
cleetons from a cathode C is aveerated by a potential diference ranging
from 10,000 40 $0,000 vols, The elctons collated by the tube A fll on a
‘hin gold film o thickness ofthe order of 10 cm. The apparatus is evacuatd
to avoid collision of electrons with the molecules ofthe gas. The diactd
beam i allowed 1 fll on a Nvrescnt screen oF on «photographie pate
IP. The plotograp of the dise beam lus a system of coneentie ing. as
shown in Figure 2.10). Measuring the radii of the rings and the distance
hetweon the fil and photographic plate, the angle of difracion 8 can ho
obtained. Knowing the distance between atomic planes d. the wavelength of
the difrcted beam can be calculated, The experiment cleuly demonstaes the
wave natur of electron as diffraction pat can only e produced hy waves

pump pump E
A |
4 — vo
DIT
|

Figure 24 a) Thomson apparaus lor electron diracion, (1) iactan
pate of beam of sitions by hn ol ol

In the acceleration proces the electron behaves lke a particle, inthe
<itaction process i hchavs Ike à wave and in the detector Ht Dehaves ike
à parie The elcctron which show wave aspect in ons pata he expriment
showed parle aspect in two other parts ofthe same experiment, That is for
à complet description of physical phenomena, both particle and wave aspects
‘of material particles ac required. once, th new theory which we ae looking.
For must account forthe dual nature of radiation and mat.

Standing Wave of an Electron in the Orbit
An clon orbiing around a nucleus at a distance ie a bounded onc and
therfore the motion is repeseted by a standing wave. Only coin definite
number of wavelengths an now exis in an omit, otherwise the wave aller

Ware Mechanical Corps 27.

travelling once round he orhit ily out of phase with the previous one
Mathematical,

a

a

tthe nin oer on compi chin. Sua A= Min)
ERES
Gro

whic ia form ofthe general quantization rule, Fo circle orbits, de = 7 a

2)

nei es

pa

mh ouh où mar = E,

es

ich is Mohs quantization rule, Thus, the de Droghs relation gives the
uantizaion ruc in reined way which was arir induced as an a Joc
hypoibesis. Th standing wave pates of le clecton in an orbit ae lutte
in Figure 22.

Figure 22 Standing wave patoms ofan lee in an ot or

22 THE UNCERTAINTY PRINCIPLE

[As per classical des, i is posible to determine all dynamical varias of a
system to any desired depre of accuracy, This principi of determinism isthe
backbone of easial physi,

Posiion-momentum Uncertainty
The position of a plane wave is completly indeterminate as iti of infinite
extent, Therefore, when waves are assigned to particles in motion an
indeterminacy anses automatically in the formalism because an electron wave
of definite requeney is mot localized Heiser analysed thi ndctermiacy
and proposed that no 190 canonically coupe quantities can be measured
simultaneously For the canonically conjugate variables and, mathematically,
the principle i stated a

Ar dp,

‘The unceniny relation can be ilastated by te singles experiment
discussed below

Singlet experiment, Considera beam of monoenergtcclovtrons of speed
19 moving along te y-axis. Let us uy o measure the position x ofan electron
and is velocity component U, in the vetcaldiccion (axis) To measure x,
Wwe ins a seen Sy bic has a si of width Ax (Figur 2.3) I an electron
passes though this lt is vertical poston is known 10 this accuracy This can
be improved by making the slit namowe.

AS the electron has a wave naar, i ill undergo difaction a be sit
giving the pater asin Figure 23. Just at he time of reaching the lit, the
elec v, of the electron is zero. The formation ofthe diffraction pate
Sows tha te electron has developed velocity component», after rosin the
sli. For the fist minimum, the theory of dit

es

Figure 23. station of Heisenberg uncraiiy pénis eurent
4
snaso= 4 a
re es

where À is dhe wavelength ofthe elect beam. Let be the time of transit
from 0 toa and tbe the value of v, at . Then

ab

CES en

From Bas (26) and (27), we pet

>
el

Replacing À hy Mm, and taking yy a8 a rough mcasure of uncertainty Av,
A ae

Mor a
mar, aa

es

ich is the desired relation. Im the same way, we have

yt, ah and Aap, a h es
AS the product of craintes i a universal constant, dhe more precise we
termine one variable, the less accua it our determination OF the hor
variable

Before the inuoduction ofthe slit, the electrons travelling along they
si had the finie value of ero fr p,. By inoducing the sit we measured
the xcoondinato ofthe particles 10 an accuracy Ax, but This measurement
inuoduced an uncertainty into the p, values of the particles. Tus, the act of
measurement introduced an uncontrollable disturbance in the system being
measured which is a consequence ofthe wave particle dual,

"Though ih result is hasi ona particular eerimentl se up, saver
general one sine iti independent ofthe parle mass and constante of the
Apparatus used. Heisenberg assumed it to be a fundamental law of ature
"Therefore the new mechanics we a looking for must abandon the deterministic
‘model and allow only for probable values for dynamical variables

‘The new mechanics was formulate independently by Werner Helsenber
(1925) and Erwin Schrödinger (1926), Heisenberg based his mechanics on
matrix methods whercas Scbródingor used th idea of wave nature of clon.
Introducing slave ideas, DAMM. Disc generic quantum mechanics

Uncertainty Relations for other Variables

Unceaity rations can ls be banc for paso canonically conjugate
variables. Fora tee parle moving along x-axis, energy £ is given by

Pay, = 0,00,
Theor,

a
ae Op, or agar
Sap, a.

Mence
AE Ash 2.10)

“This equation indicates that fa system maintains a particular state fr time A,
its energy is uncertain at least by AE = WAY

"Ths uncertainty fr the pai of variables, component of angular momentum
along th direction perendicula othe plan of the ori (L) o patio and
the anpular position (9) can be obtained follows

30 custom Vector

eue

On differemisting, we get

Therefore,

Using Bq. (20), we have
en

These uncertainty relations are very useful in explaining number of
‘observed phenomena whic the classical physes filed. We shall now consider
Some of them,

Applications of Uncertainty Relations.
Ground state energy of hydrogen atom. The classical expression forthe
total energy of the electron in the ground state i given by

LL em
here a i the radius ofthe first ori. Let the uncenainty inthe poston of
ths elscnon Arh ofthe order ofa, More comet, the product of ucenantcs
is given by ho Therefore,

ann u met an

Taking the momentum o Be ofthe order of la, we get

Gree
For the ground sit, the energy E has 0 he minimum. For this, dda must
be zero. Denoting the minimum rads by a, we have

eu

ae
a

a = e es)

Varo Machel Coup 31

With this value o a, Bq. (215) becomes

aa aE

hich isthe ground stats energy of hc hydrogen atom

Ee

Width of spectral lines. Spectral ins have finite wid dueto varios facts
One such factor isthe natural broadening which is a diet consequence of
Inerainty principle. Atoms romain in the excited state for a Finite timo
alls he fe im: before making a transition. Tonos Ur wl be an ungeraity
in time of the onder of = Then

1

rAE=h or Ave st em

» = em

For most of the states, the life time € = 10-1. Hence, Av = 10° Hz. This
sprcading is experimentally observed when the pressure fs very low.

Mass of meson. Yukawa proposed that nuclea forces are doe to an exchange
of mesons, Uncertainty principle may be used to derive a relation between
mass of meson (m) andthe range (7) of nuclear free. When one nuccon
verte foros on the ost, a meson is ee. During transit is positon is
vncertin by an amount ry, Use ofthe uncertainty elation gives

ra h or ape À eo
For relativistic panicle p = me. Taking the unccriimy in momentum to ho
‘ofthis order, we o

Lo mat e

% ne
For ry = 18 x 10 cm, m = 200 my, whete m, is the eset mass
Nonexistenee of electron in the nucle. For an elecon to ext inside à
Duels, he unccrtany in its position must be at east oF the ande af Zr Fy
Bein the radius of the cleus, The uncertainty in the electron moment is
thea

a ex

For a typical mucus 1, = 10- m. Hence

A a 528 x 10" kgs!
ET

we

32 stam Vector

The momentum of the electron must at last e of tis ore, The Kin
energy of the ccm
FL (52810

2m ~ 29.110)
In fietay, the eneray of the emited electron is usually of the order of few
MeV. Therefore ane does not expect the clecwon to Rs a const of the
rues

957 Mev em)

23 THE PRINCIPLE OF SUPERPOSITION

‘Two or more waves can travers the sme space iadependatly of one another.
Hence de tual displacement at any point due to number of waves i simply
the vector sum ofthe displacements produced hy the individual waves. This is
own as th principle of sperposton. This principle i vey important inthe
diferent branches of physics, namely opie, acoustics, electrical engineering,
«se. Superposition principle allows us to analyse complicated wave motion
as a combination of number of simple harmonic motions. Though the linear
superposition Of waves i important in pics, sound, ct, ts unknown inthe
classical theory of parties. Since ware motion is assigned to particles in
motion, we tentatively extend it to matter waves also. The concept of
Superpsition of states would allow us the constuction of wave packs

24 WAVE PACKET

We have alway sn that mater exhibits nave ko behaviour under suitable
conditions. When the momentum ofa particle is well defined, the wave can be
of infinite extent. Therefore, free panicle moving along «ans with a well
defined momentum is described by an Infinite plane wave #6.) given by

Hea) = Ay exp is - 001 em

here ie wave vector & = 2/2 and is the angular frequency. In the ease
of electromagnetic waves, the electromagnetic field varies in space and time
Sound waves can be described by the pressure variation in space and inc. In
ther words to derbe wave motion. one requires a qua which varios in
Space and tine In analogy with thes, 1 describe mater waves associated with
Prices in motion, one requires à quanty which varies in space and tine
‘This variable quamy, cll the ware fame, Wort, must De argo in
regions where the panic is ikly 10 e found and small nthe region where
itis ess kel tobe found. Tha sc wave function of a patil in conformity
withthe vacertiay principle must be localized in à small region around it
‘The wave fonction ofthe mater wave whichis confincd to 4 small pion of
space as in Figure 24 ered as a ware packer or wave group.

Ware Mecha Cong 33

Figure 24 Roprotenaion of à wave packet

‘Mathematically, a wave packet can be constructed by the superposition
of an infinite umber of plane waves with sigh difcing values.

LT

Jatoespfis - at em

[As the parie is localized, we ae interested in the superposition which cad
10 a wave group which tasis without change of shape. This possible when
A is zero everywhere except forthe small range of k valves

(urdu), merca 0m

AAs lies in avery small interval, expanding 6%) as a power series in (kh)
about fy, we have

Neplecting Biber onder terms and writing

i=, mt (SP) =

we have

wens Jens imma aay

[Adding and subtracting lr wo the exponent, we get

ven

tuent - a] em
so
nan 0
ron] awenfw-rnfe- ela am
win

34 Custom Vector

[equation (2.27) represent à plane wave with propagan constant & and
angular frequency @, modulatod hy PU.) which depends on x and #
through 1 = Ge. follows that be wave packet moves with the group
velocity

an
Lars
il the envelope ofthe wave packet (doute ine In Figure 2.4) moves with
the group velocity e, ih individual waves of the packet ass with velocity
y als phase von or wave velocity. is the wave group that caries the
hey and what we measure experimentally i the group velocity.

In can easily be proved that the group velocity of the wave packet 0, is.

the same asthe velocity oF the material panicle. Consider the relations

0

B= ho sd p= tk
‘The grp velocity
do de
da ue em
at oe,
Fora fice nonrlatvisie parle
ey le
Ewe ce Bak ew can
an dp =D
Fora reatvite parce, we have
Tieren,
dE ep _ émane OI) ds

dE men)
“Thus, the velocity of a panicle and the group velocity of the coresponding
wave packet ae the sume, The phase velocity

© Em

Ep we

es

has no physical significance and is mot a moasurahle quant.

25 TIME-DEPENDENT SCHRODINGER EQUATION

‘The nature of the wave function YC) for localiza and nontocalize free
parices have been discussed inthe previous section. However, for detailed
Study of systems, we require the equation of motion for "PG. which vas

Ware Mechanical Cong 35,

Formula hy Ervin Schrdnger in 1926, Sebring oquaton fundamental
fone in quantum mechanics as Newton's oquation in classical physics,
‘One-dimensional Equation for a Free Particle

‘The wave function of a Jocalied fre particle isthe one given in Bg. (2.23).
For a fre paco the casa expression for energy 6

Z em
opting py A and Ey ho, e
w
ne es
Suing hi ss of in ta. 029
wen Jae aso
tenia VD wih wpe ve St
YA aan
2-2 rane em

Ditfreaing V2 unis respect I 4, we got

: pr
Tree es
Gaming Es (37) an 23, ws bs
ran „mar
mar a am
ic anna Senge eun o pa

Operators for Momentum and Energy
“To obtain the operators for momentum and ener, Eg. (2.39) may be writen

My > nd) na ,
(o2)eun- ¿alfaro ean

From a comparison of Es. (2.38) and (2.40), lt may he concludd hat the
energy E and momentum p can be considered as the differential operators

a a
End and pnd ean

‘operating on the wave función 4x): Equation (2.39) i obtained even if he
‘operator fr pis taken as ¿Vin place of ha THe choice o the negative
sign i very significant which is explained in Seco 27.

Extension to Three Dimensions

‘The one dimensional treatment given above can es he extended 10 three
dimensions. The Ihree-dimensional wave packet ean be weiten as

Yo,

Tamer ~ ea, at, ak a

Procscdig on similar Inc a inthe anc dimensional case we got the thse
dimensional Schrodinger equation fora free particle as

ave) |
atte

‘An analysis similar tothe one made for one-dimensional system leds o the
following operators for energy and momentum.

Wye can
Even ea

a
eine an > ea
3 » am

Inclusion of Forces.
Modification of the fee particle equation 1o system moving in potential
Ver. can easily be done. The classical energy expresion for such a syst
is given by

ps

= Fave (245
LE + ve easy

Schrödinger then made the right guess regarding the operators for and / as
ror ad rr En

Replacing E, p. © and 1 in Eq. (245) by their operators and a
‘operator equation to opera on the wave function Wr, we get

Ware Mecha Conapts_ 37.

ich isthe rime dependent Schrdinger equation fora panico of mass m
moving in a potential Vir), The quantity inthe square racket in Eg. 2.47)
is the operator for the Hanilonian ofthe system. ln gener ts solution wil
be complex because ofthe presence a / in te equation, The equation cannot
Petite invariant a8 le contain fst derivativo intimo and second
derive in space coordinar

26 INTERPRETATION OF THE WAVE FUNCTION

Probability Interpretation
‘The wave function Wr, D has no physical existence sine it can be complex
Also it cannot be taken a a direct measure ofthe probability a (rn) since the
probably is ral and omncgativ. However, Wr, ) must in same way De an
Andex ofthe presence of he parce a (FA universally acc spl tal
interpretation was suggested by Born in 1926. He interet the product of
Vr.) and its complex congupite W* a the postion probability density PA,

Pee = (6,0 Hee = HEP as
‘The quam |Wr. 9 Pars thon the probably of Finding the system at time
Lin the small volume element dr sumounding the point, When I, DR dr

is intprated over the entire space one should pet he taal probability, whieh
is unity. Therefore

Siret 21 am

Pr Dg, (249) 10 he Finite, Ye) must end o zero sufficiently rapidly as
1 4, one, one can mulópls VF.) by a constan, say N, so hat NY
satsis the condition in Eq. (249). Then

Parar eso

“The constant is called the normalization constant and EA. (2.30) the
normalisation condition. Since tb Schrüdinger equation sa linet diferent
‘vation, NY solution of i, The wave Functions for which the integral in
(249) dost nor converge will he tested depending on the nature of the
functions.

Probability Current Density
‘The probability of Finding a sytem, described by a wave function Wee 1 in
à fie volume Vin space is piven by J" cr and this changes asthe wave

function evolves in time, To study this, consider the Schrdinger cquaión and
its complex conjugar form, as under:

[

Se sole

es

es

lr, the pora Vis assumed to be real, Muliplyng Eg. (251) by and
1g (232) by Y from Ltt and subracting one fom the other

aes gee Jor reas
eM ow). Ev we
Lee Aly ore vr] es
a in

Integrating, we get

in

E [vor wor jar

¿jura

[ww - wer] es»

Te we have a localized wave packet, which isthe almost at case,
Y and We “> 0 ás y > and the right-hand side of Eg, (234) vanishes.
‘Thea

“That i, the
By de

:nalizadon itera i consta in tine.
y 2 veto Je), called the probability current densi.

de

2 (eww ww) 256)
and substi in Eg. 2.5), we gt

are.)

+(e) = as
VO es

Ware Mecha Conapts_ 39

Equation (2,59 i the equation of cotinuiy Fo probability, which is analogous
to the equation of continuity in hydrodynamics and eletobynamies, With tis
definition of probability eurent density, Eq. (237) becomes a quantum
‘mechanical probability conservation equation. I may be note that if is el,
the vector Jr) vanishes, Weng the itera form of Eq. (2.87) over finite
volume Y and using Gauss theorem, we get

a
Brunn foie fya esp
a soc a i a iy
o e manor pally
sss ati ls wt Ge meh ym o
‘ain ae wih ns py oe eye foe ees
een

Expectation Value

“The definition of prohabiiy density immediatly allows the calculation of the
‘expectation value of the position vector ofa particle, Considera lage number
‘of measurements of the postion vector fof a parle made when it in a
Pair stats. Ensure hat the particle has the samme wave Function Wr.)
before each measurement. The average of all the diferent values is the
“expectation value” (e) af the poston coordinate, Ax Wr, DE represent the
probably with which the value r occur in the measurement, () can be

(0 = fer ar = fera es»

In hi finition, the wave function Y is a normalized onc, Th necesi for
sandwiching y between and would be made clar in the Later par ofthis
section. Based on similar arpuments the expectation value ofa function of e
may Be orion a

(ye) = [Vias nar es

Ltt mohiplying the time dependent Sebring equation, Eg. (247), by Y
and intersting from > 10%, we et

Jv (od)eare fol Bo jee Juve

(12) - (Le). es,

40 Quart Mechas

ere,

= (E)+m am

which is the quantum analogue of the classical energy expression, That is
sandwiching of the operator between Y? and Y is consistent with the
Schrödinger equation. Here, we have assumed thatthe average motion af the
ave packet to he the same as the classical motion of the panicle. This
requirements fre jsi y the rentes sore (ero Section 27)

From the above discussion, we can reasonably take the expectation value
of any dynamical variable A whose operator is Ay, 48

a) = [raver a
I he wave faction isnot normal
WA, dr
to ate ae
Tv

Since the space coordinates have bee integrated out, he expectation value is
à function of time oly

2.7 EHRENFEST'S THEOREM
Comespondence between the motion of a classical parc and the motion of
a wave packet was worked out by Eluentes in 1927. In the limit when the
ave packet associated with a parle reduces 10 a point one expees the
panic tohehave lke a classical one. Consider the one dimensional motion of
A particle of mass m. The time derivative of (x) is given by

0.2
a “à

Joe Jerre 00

Suhsiing the values of 21/21 and 3¥°/8¢ from Eqs. 2.51) and (252), we

DRE u

Integran the second term by parts evice and using the condition that °F and
its derivative tends to en a5 à = (or a localized packet), we have

ef MMe je ae

E

Varo Machel Coops 41

ence

Ben
cta

rs,

ee

In the imi the wave packet reduces a pot, (x) = x and = (p,) = pa and
Eg. (267) reduces 10 ie classical equivalent

mi 2
Ken as
“The time rate of change of (p,) is given by
a at
gona tel
E [aview
a] Pr a
An ati AVAL and DW. we Bt
2 (2) aw ov]
co dé Ox
yet vr
Mer)
aw ay
. - area

[rl Here

42 Quantum Mechas

“The fs term om ih ight sido can csi e negated out and is equ to zero
since We and Wax ate zero as à -> 20. Hence

0. (ey am

Nowon's second
law of motion, Generalizing Bas. 2.67) and (70), we obtain

ae) de)
cnt ana (69 em
[Equation (271), called the Esrefes's theorems, are the quantum equivalent
‘ofthe equations of motion of a classical parce. Reduction ofthe ware packet
to a point means the violation of uncunaimy princpl, enc, uncertainty
principi lim the equivalence of quantum and clasica mechanics.
AE AMV, instead of ~inV, is selected a he operator fr p, we would have
obtained

a)
4) - er)
m
ich is not the quantum analogue of Newton's second law of mation, The
selection of the operator —¡AV for the dynamikal variable p is thus jst.

28 TIME-INDEPENDENT SCHRODINGER EQUATION

The time-dependent Schrdinger equation, Eq, 2.47), describes the evolution.
of quantum systems using time-dependent wave fonction Wer.) It completely
neglects the time dependence of the operators. More about these spc will
be discussed in Section 3.9. I the Tlamiltonian operator dows not depend on.
lime, the variables r and of de wave function We) can be separated ato
two functions yee) and 40,

VD = yo 60) em

Substtting this value of Wir, 1) in Eg. (247) and dividing throughout by
engin we get

dou (Rp
nt 160 2 1 A am
‘a à wal zu re a

“Ths left ide ofthis square is a function of im and igh side a function of
space coordinates. Since rand rare independent variables, each side must be
‘equal to à constant, say E. This gives ise o the equations

le

= em

am
oo da

Varo Mecha Cons 43.

Hr vole en em

Solution of Eg. (2.74) is sraightorward and is giv

by

#0 = con (2) aro

Where Cis a constant, The equation for y Eq. (275). the sim independent
Schrldinger equation or simply Schrstinger equation. Since WE) determines
the amplitude ofthe wave Function VF, cal the ampliado equation.
Equation 2.72) now takes the form

vr

E
vos (8) em

“The constant € is included in the normalizaon constant for yl).

Significance of the separation constant, E, can be understond by
¿ittrentating Wr.) im Eq 0.72) with respect te and makiplying y ih.
Then

DD «eve em

Multipying both sides of Ea. (2.78) by Y” from left and imeprating over the
space coordinates fom = 10 =. we get

Jr(ug)voos

As the left ide of Eg. 0.79) ls the expectation value ofthe energy operator,
the constant £ isthe energy ol the system. The same can be Understood fom
Bu. 29 2

E e)

m
AT
2m + MO

is Me operator associated with the Hamitonian of the system,

29 STATIONARY STATES

‘The solution of the time-independent Schrdinger equation. Eq. 2.75). can be
obtained if the explicit form of Ve) is known. 1 is found tat in peneral à
sytem has à set of well defined energy valves By for = 1, 2,3, with
corresponding wave functions Ye. ncluding the time dependent par, the
save funtion ofthe system i

48 Guanta Mechas

IE

von vse 00 (8) i

we eso

“The probably density le. iin von y
Por = nf = MON = cons in me 282)

het #= 1,2,3... The sates for which the probably density is constant
ims av called stationary states, The time dependent factor exp (1,111)
of such sates are governed by the energy E, ofthe panic.
TW can be sen that in Stationary stats, the expectation value of an
‘observable whose operator docs not depend on mc expiily sa constant in
time, Then

= Jo [at en (Jae

= constant in time es

Its obvious rom Tg, (2.56) that for stationary sats the probability current
density Jr.) i aso constant in time. As 27 = 0 or such sae, from the
‘equation of cominuty or probability, Eg. (257), we have

CHEN es)
A stationary state i a bound one if he corresponding wave function Vi) or
probability density IC) vanishes an infin That is, fr Bound sass

ñ

vi =0 ess

N. Bohr was the Fist to postulate the existence of stationary states. Th
are the tats on which physical measurements re performed. Spel ransiions
ae induced between such sates, Owing to these reasons, solution of the Ume-
ndependent Schrödinger equation for differnt systems i of fundamental
importance in quantum mechanics.
In stationary stes, the probability density is constant ia tne. A panico
is in a superpsition state (o stationary state) Hike

286)

Ware Mecha Coop 45.

‘he probability density

ree.) = (ef = Dial yee

| em

ñ

Tene. if a particle is in à superposition sat, in general
density depends on tine.

+ probahiity

2.10 ADMISSIBILITY CONDITIONS ON THE WAVE FUNCTION

Inthe preceding sections, we have discussed the Schradinger equation and
interpretation of the wave function, However, we hase not formulated the
onions to be satisfied y a wave function. A physical system is descihad
by the probability density IV, and the normalization negra, Eq. 249).
For the probability density 10 be unique ad the total probability to be nity,
‘he wave function mist be Finis and single vals at ser point in saco. The
probably eurent density. Bq. (256), anıher importan parameter of the
probably interpretation, comas Y and VW. Hence Ÿ has to be comiauous
ad VS? mus be finite. The Schuidinger equation bas the term VA. For V4
La xt VW must be continuous. Por a ware function Wr, ) o be accepable,
SVG and VF must Bs finie, single valu and comio at all pits in
space

ED APES ——

(AMPLE 2.1 Calculate the de Broglis wavelength ofan cccron having
kinetic enerpy of 1000 eV. Compare the result with the wavelength of Kays
havia the same enerpy

inci en T= 2 = 1000 &V = 16 x 103

and
a. 6626 x 10%
BOAT x 10 o IN
= 039 x 10m 039 À
For Kıays

he
En = À

46 Guanta Mechanics

ere,
= OI 13,49 à
16x10 °
Mence
Wavelength of rayo E]

remain el Ks _ LAS
‘Broglie wavelength of elecuon ~ 039A

‘AMPLE 22. Determine the de Broglie wavslngth of an cecon hat has
been accelerated though a potential difference of 100 V.

‘he energy ane bythe electron = 100 ev
A
ear E 0e
Thee Ken
p= [2(0.11 x10 "kg}(100 x 16x10 PI”
5.299 10% igs!
Hence
a ome

125107

asa

D 53x10 "ke ms

EXAMPLE 23 Electron scaring experiment gives a valu of 2 x 10 m
forthe radius of «nucleus. Estimate the order of energes of electrons used for
the experiment. Use relativistic expressions.

Torclscton setting experiment the de Broglie waveleng of cctons
used must he ofthe order of 10. m the diameter of to atom, I 738 he
Kinetic cocay, tb tlavistic expression for momentum is given by

vale

Ware Machel Cononpe_ 47.

. (6626 x 10%
rw

ELSE
= 3637 10
On skin, we get
T= 0m = 68.19.11 x 10 CS x 10981
= M0 x 10 = 310 Mev

EXAMPLE 24 An election has speed of 500 m/s with an accuracy of
(2.0046. Caleulate the certainty with which we can cat the position ofthe
ecto,

Momentum, p = mu = (0.11 x 10° 197500 ms)

and
Percomags accuracy = 22 x 100 = 0.008
v
0004 x 9.11 x 10 e 500 gai
008 9x 10 7 EURE ETES
ee 100 ió
Menos
6626 x10 x

LO gm
TS

“The position of e elton cannot be measured to accuracy ss than 0.036 m.

EXAMPLE 28 The average lifetime of an excited atomic tate is 10°95. I
the secta ine associated with he decay ofthis state is 6000 A, estate the
width of the line

We have
r= 10 5, A 6000 x 10° m = 6 x 107 m
he
Eto of A6 = Maa
2
Multiplying bo ide by A, we get

AAA
E

Mers,

2, 36x

rm
NT

48 Guanta Mechanics

EXAMPLE 26 An lern in then =2 state hydrogen remains oro on
the average of about 10 X, before making transition tom = 1 state Estima
the uncertainty inthe energy of the n= 2 state, What fraction of he transition
energy i his? What isthe wavelength and width of this ie ia the spectrum
(oF hydrogen atom?
6626 x10 is
#34 10%

= 6626 x 10 © = 4.14 «1070

Energy of the m= 2 > n= À

ad
Fraction, AE AMOO”
em
Worle,
y ICAA À ag mann
ET mateo
Win of the ine nthe spect is aid as
an Ma
ë

AA = 406 x 10 x 1218 x 1077 = 495 x 101 m = 495 x 10%

EXAMPLE 27 Normalize the wave function ys) =A expan’), A and a
‘constants over the domain = € à € =.

Using the result

Jenene an

E
Va
we pet

OS ep Gar)

sie
Pi

Ware Machel Compte 49.

EXAMPLE 248 A parie constrained to move along sais in Ihe domain
OS x € L has a wave funcion y) = sin (nm), where n isan img.
Normalize the wave function and evaluate the expectation value of its

‘The normalization condition gives

« DAC

pra ta

Then
it
"7
The normale ware feat is
By BEX
a

to)

EXAMPLE 29 Give the mathematical representation ofa spherical wave
traveling outward from a point, and evaluate its probability cument density.

‘The mathcmatical representation of a spherical wave tavelling outwards
ftom a point is given by

A
vo= 2

here Aisa constant and Ks the wave vector. The probably eurent den is

n
Een)

-2 (242)

50 Quantum Vechner

EXAMPLE 2.10 The time-ndependent ave function of parco of mass
Im moving in a potential VO)
4)

ing a constan Find the energy ofthe system,

ts
mn (E

On ditfremiating with respect to x, we get

yen) = exp

On solving, we pet

DOC
EXAMPLE 211 For a panicle of mass m
the wave equation

Lae a my

For at À

Ware Machel Coup 51

‘Show that plane wave slain ofthis equation i consistent th he lave
ner momentum rolasionshi.
For plane waves, we have

¥en ape — 0]

Substituting this solution in the given wave equation, we obtain

Multipiying by PA? and writing 409 = E and kA =p, we pet
Ba cp? à att
which isthe relaie energy-momentum relationship.

EXAMPLE 212 An electron of est mass m, is aeslrted by an extremely
high potential of Y vol. Show hat its wavelength e

—
UA ON

Energy gained by the electron in the potential = Ve
EN

‘The reais expression or kinetic energy m6

Tatin the to and rcaranging,

Ter Las
Je

D Les me = met | Ve (We +2m 0)
ea Wet me

cle Wes 2m

52 aston Mechanic

wey
m

ae bogie sek 2

A me em
im Ves mae ee Age

— ——
Weer am?

EXAMPLE 2.3 Show that dhe phase velociy of reluivistic electron is

if ee] ere Ast de Brople wevkngt

For oise scan

EXAMPLE 214 Find the probably curren density Kr. 9 associated with
the chuged partite of charge e and mass m in a magnetic field of vector
potential A which is cal

"The Hamilton of the system is

hip Say = Ev vars tan A

Te ine do tig sui
Bowes wae mare avr Pont

de

Ev ma al ave E

Ware Machel Cones 53.

Mutiplyng the frst equation hy W from et and the comptes conjugate
‘equation by Y and sharing, we Rave

orgy waters aw -
ar vr ve or war PA

LOA AA A OE

¿IE
Fa)
¿a gen pate
ne Mice nn Lv ros

+ Lr OA

cron wows Lor]

tig pity u day co J by
inne ee vun Loren

‘Te equation reduces to

a
PAUL

whic isthe familiar equation of continuity for probability.

EXAMPLE 215 Using tims independent Schrdinger equation, ind the
potential Vi) and energy E for which the wave funcion

vo Ser. tg constants

is an eigenfunction. Assume that Vis) 0. as x > =.
Differentiating the wave Function with respect 19 x

54 Quantum Vechnier

EN
2m

whic gives the operator equation

E-v)=

When x > =, Yo > 0, Hence

a
En

mint) 2n |

REVIEW QUESTIONS

1. Explain de Broglis hypotesis. Why tbe wave nature of mater is not
apparen in our daily observations?

For a complete description of physical pbenomena. both parce and

wave aspects of material particles are require, Comment

3. Mae the uncertainty principe onthe as

4. Explain the uncertain principle. How does it account forthe natural ine
‘vidi of spectral Toes?

5. Does the concept of Bohr orbits violate the uncerainty principle?
6. Prove he noncxistnos of elec inthe nucleus on he basis af uncerainy
inp,

7. Consider the standing wave ofan electon in an orbit and obtain Bobs
quantation rue

8 What is a wave packet? How is it represented analytically and
siagrammateally?

9. Prove ha the velciy of a pail andthe velocity ofthe comesponding
wave packet ae the same,

vo

2

of single sit esperma

Ware Mecha Coop 55.

Caine ths probaly nro of
Show that ay ce in probably ia à regia is acompaic by an
Duo of probability arse it sure
12, Sta the quantum analogue of the classical energy expression E =
(pany VO) and expan ts inane.
ls the tne dependent Seine equation reiviically invariant?
Explain
14, Schrüdnger equation ofa fee pace results when £ is cepas by
‘naan, by mF) or QD la E = p EM) and he rein
eqeraoe cation opera om the wavefunction. Why ENS no
cpu?
Esplin the sign of Chen
I the detain fo te expectation vals ofan observable the operator
acte with the bvevable i sandwiched between Y" nd y. Why?
17. What at stationary tas? a sanar st, show tha he probity
rent density is constant in tine
18, Define probability cument density. Wha is value when the wave
function a?
19. Outline the various admissibility conditions on the wave funtion of à
20. plinth rai of superposiin of states. Prove thatthe probably
density of a pue in à soperposion state depends on time.
21, Dire demonstrate the superposition proper oF the Schrdinger
uation
. Distinguish bewsen group velocity and phase sc
. Vacersiny principle limits te equivalence of quantum and clasica
mechanic: comment.

—\_ —

1. What i the de Brople wavelength of an elecuon aeclerated from rest
by a potential of 200 V?

2. Evaluate the ratio of the de Broglis wavelength of electron 1 tha of
proton when (a) both have the sume Kinetic energy (0) the electro Kinetic
nes is 1000 eV and that of proton is 100 eV.

3. Proton foam is used 0 obtain information about the sis and shape of
somie mule. I the diameter of reli is of the onder of 10 m. what
is the approximate kinetic eneray to which protons are to be aceeerted?
Use relativo expressions

5.

there

PR

56 Quantum Vechner

4, Estate the velo of neutrons ci for the say of ncutrn traction
cof ental structures, if the Imerstomie spacing inthe ental i oF the
order of? A. Also, estimate te kine coergy of neutrons corresponding
{0 this velo

5, Esiate the energy of lets neck for the study lc diffraction
crystal structures, if the interatomic spacing inthe crystal is ofthe
fonder of 2 À

6. A bullet of mass 0.03 kg is moving with a velocity of 500 ms). The
speed is measured accurate to 0.02%, Calculate the uncerainty in à
Abo, comment on the res

7. Warelengih can be determined with an accuracy of on in 10% What is
the uncertainty inthe postion of a 10-A photon wen is wavelength is
simoltneously measured?

8. If th position of a SASV clean is located within 2 A, what isthe
percentage uncetiny in is momentun?

9. The uneceanty i the velocity oa panico is cual to is velocity. Show
thatthe uncersinty im is location i ts de Brolie wavclenth

=,
vue eau Be)

who A an ae consta. id pl frm fhe pen Vo,

11, The tag weve a a pue à y) ca sf
ecm Cha al nme nd D ere

Cala he poy are Sty fr Te

Show th he hv vl for puto wih eas males

pact Oo vy of ight at opt acho reg

12. Show tat he vaig «pale tak m, wi Kae
tray Tr gen ye al

n.

he
[Famer

14, An electron moves wit a constant velocity Lx 10 ms fhe velocity
ls measured 0 a precision of 0.1 percent, hati the maximum precision
‘with which is positon could he simultancously measured

18. What isthe rato of the kinetic energy of an electron to that of a proton
if ther wavelengths are equal?

16, Calcula ths probability eurent density) For Ihe wave function

vor = a) exp HO (0) m6 ral

Ware Machel Cononpe 57.

17. Find the form of the potential for which 1), a constant, sa solution of
the Schrdinger equation.

18, Obs the form ofthe equation of continu for probability, if the potential
in the Schrodinger equation is ofthe form Ve) = Var) = Wi, where
V, and Y, ae real.

19. For a one dimensional wave function of the form,

YO. A exp D

ho date prob cure desiy an he ten as:

YO and ya) be he normale ground and fra exc
cigenfuncions of à linear harmonie oscillate. At some instant of tine
‘AY, By. A and Bare constata, the wave function ofthe osilter
‘Show that (x) in general different fom zoo.

21. Waves on the surface of water trav with a phase velocity u, = (RATER
where is be acceleration due to gravity and Ais the wavelength of the
‘wavs, Show thatthe group velocity of a wave packet comprised these

waves is 9/2

[NUEZ

General Formalism of Quantum Mechanics

used on the ware nature of mater we have developed the Sehuodinger
uation (hich isthe hasi squatio of quantum mechanics), the physical
Significance ofthe wave fonction and the uncorinty principle in Chapter 2.
Amore systematic presentation ofthe mathematical formalism of quantum
mechanics along with a set of basic postulates will be discussed in this
chapter.

3.41 LINEAR VECTOR SPACE

Vector in a Three-dimensional Space
A veeto in tos dimensions is a physical quantity having both magnitude and
dicton. A vector of magnitude 1 is called a unit vector, To reprsent an
ssbirary vector, we introduce unit vectors say ey ©, ande, along the positive
Aircton or a right-handed system OXYZ of bros mutually perpendicular acs.
‘Then any vector a can ho expressed as

AS on

et a. y, y are salas. Given a vector te seals a, ay a, in Ea
(6:1) ar unique determined, The unit sectors ey, &, €, are ah to form à
Basis forthe set ofall vector in os dimensions, Further, sings hose are
unit vectors along mutual perpendicular directions, we say Mat € y, €;
form an orthonormal basis. The sels a,, ay ay in Eg. Gul) are the
component of in th basis (@, ey €. We may represent a as à column

oneal Female of Quart Meche 59)
(a
a 62)

“The components of are 1, 0 0; tase of e, ae, I, 0 and thse of ey re
0,0, 1 Henee, oa expressing them as column Vector, we have

jf =

“The totality of vector rs dimensions is call a three-dimensonal space

arr =

in his space, the sealer productor inner produc o and b. donot by ab
FD) is de

os)

Vectors in an ndimensional Space
We now geocrlize these concepts to an mdimensinal real space, withthe
vector €, 6, forming an orthonormal hass, À vector a can Be expressed
in this onhanormal ass as

“The totaly of all n-dimensional vectors is called an m-dimensionel space, or
simply an mxectr space, In this space, the inner producti defined

m= Yan an

tthe vectors ar compe, that i, fhe components he vers aro complex
numbers the inner prue s dined as

(ab)

Sen En

her, for any complex number
thar for any sector a,

nots the conjugate. Thus, we observe

0

= aa Él

is real. We then define the norm (AD of a vector by
Na a 6.10)

A vector whose norm is unity said o he normalisc. Thus, for a normalized
vector a we ave

a= Sera

\ om

“Two vectors a and b aro sai to o orhnzonal if
(by =0 cu

Using te concept of inner produc, we say thatthe vector à, form an
nthonormal set if and only if

Wei, ise, as
here 8, isthe Aroncher de defined by
no

ah aa a

In an dimensional space à 62 07 # NN Ay Ay 9, an sid to BS

linearly dependent if there exit seat cj Cp € mall Bro such that
“The vecto are incay independent if moscas o the form Eq, (3.18) exists
less cy = € eo

À sc inary independent sectes 2, … pans an dimensional
space and any vector $ which les in this space can he expressed in dh form

619

= Sen si

“This so of vectors is complet hen is mo her vector which Falls in his sc
of nca independent secos

The generalization of these concepts 10 an infinke-dimensiona space,
called vector space is suaghorward

oneal Foal of Quariem Mechanin_ 61

Hilbert Space
In the vector space we considered, the unit vector ey. €. €. form the
‘onhonormal basis. Alternately, a space can be defined in which a set of functions
ALO. 640), Ao, — form the ormonormal unit vector of the coordinate
Stem. The corresponding infinie dimensional inca space is called fant
space. In quantum mechanis, very often we deal with complex. funcions and
the comesponding function space is called the Hilbert space.

‘Orthogonal Functions

We shall now consider some ofthe import definitions regarding othoponal
functions,

(9 The inner product or salar product of wo funcions FU) and GL)
defined in the interval a << b, dented as (F, G) 0€ (FG), is

o + rivas om

The notation (F<) for the sear product of functions Fs) and Gls) is
sometimes fered a8 backer notation.

(i) Two funetions Fa) and Gla) are orthogonal if their inner product is

w= [rana sn

(ii) The nom of à funcion is define by square ret of mer product OF the
Sanction with ist

”

ene

{i iret « eu

Gs) function i normalized is mor is iy

620)

en fran «1 oa

where te jteral on the righthand side is called the normalization
integral

(6) Functions that ate orthogonal and normalized aro called orthonorml
funcions, From Eqs. (318) and (3203). we get
083 jan, ca

(Y A set of funcions ELLO. FF is linearly dependen if a
‘sation of the type

Lane =o 0)

exists, where the cS are not al zero. Otherwise they are Heard
Independent
ui À set of lineal

independent funcions Fy, Es) is complete if
Function whi falls inthe st Of Yncarl- independent

‘The expansion sheore states that any funcion 6 (2) define in ihe same
interval can he expanded in terms ofthe st of nca independent Funcion

oo = ario om

‘The complete set ned nat be orhonarmal, Tower, i is convenient o use
‘orthonormal sis. In such a css, Ihe coccion in Bg. (323) ao given by

Er) Gas

“The expansion ofa function ia terms fa complete oonornal sto functions
is of fundamental importance in quantum mechanics

32 LINEAR OPERATOR

An operator can be define asthe ul y which a dternt funtion is obtained
fiom any given function. Therefore, in

ge) = A 70
the operator À operating on A) gives be function (0). So, in
TOS

th operator squats th function fs). The operator A dierent the Faction
fs) With respect to if

sed
ay = Aft = Lo

fies the relation
‘Anon + Ago 625

‘An operator ls sid 0 be linear if su
Ale, AO + ea

oneal Formal of Quart Moss 63

lahore ey and aro constant. In
a
a = af + eg] = a CD ah

the operator (da) nca, The operator which squares a function is not nca
À les 440 + ey KOM = Les hl + OO?
+R + uhh,
vanrafh
Linear operators ae he most important ones in quantum mechanics and

therefore we shall consider only such operators.
The sum and difference of operators A and B are defined by:

PATO 620
Addition is commutative

Adio i asocia:
Armed
“The produc of two operators À and Bi dnd by:
Alb fo = Ático!
A + Ope) = dds Ayn
Comm o open À and, dete y AB, defined as
ii om,

Multiplica

In follows at
th 327)
W AB {Go = BA sor), toa is LA BI = 0. À and À ae said 10 commute. I
Ais à BA = 0, À an B are sido anticommate, The aticommatstr of À with
B is usually denoted as (4, 1. An operator canbe applied in sucesson on
the same function. This is writen a follows

Kin

att ignity aa
and

CRC
“Thats, the powers ofthe same operator commute. The inverse aperaor À
ds dfined hy the relacion

8280)

64 stam Vector

Aa
An operator comunas with is inverse since

16,49 = A

As an example of commutator, consider the operators and (dt)

[tajo ma afr

1 629)

We

ence

630)

For convenience, we hal denote an operator associated with a dynamical
‘variable by the same letter without cap in the est ofthe book.

23 EIGENFUNCTIONS AND EIGENVALUES

‘fen an opera A operating on a funcion apis ts Function by à constan

Ayia) = ayia) 03»
Where a is a constant with respec to x. The function YO) i called the
lgenfonction of the operator A comesponding 10 the eigenvalue a. la

we,

ke 0)
= 6x)

ii an cigonfuneton of the operator ds comesponding to the cigonvals k
say be pointed out hee that formulation of quantum mechanics is dominated
by ideas connected with te solution of such eigenvalve eu
In general, we have a sc of values a for which cigonfancion oxi
Dong ih st by a runing index m, we can writ the cienvalu equation as

AS 63

As AV.) is unique, we cannot have different a, associat with asile Y.

lowevcr, we may have a given associated wi à large number Y, „Now
the cigenvaluc is sad o he degenerate. 1 al he eigenvalues of an operator
ae dire itis said 1 havea discrete spectrum, The continous eigenvaloes
allowed for certain opero frm a continuous spectra: In certain eases the
Sigenvalucs may be disrete over a cena rang and continuous over the os

3.4 HERMITIAN OPERATOR

Let us consider wo arity funcion 1) and y,(0)- The operators sad
10 be Mermiian it

oneal Formal of Quart Moss 65

Trap. are (ave de® ii pa) 030
N ! )

1 Drache notation
Wr AV) = (AY eo) = War AY a

A opera is sid 10 he anieHermilkn if
(Ye AY) == Aue Wed == Wy AVR 63

‘Two important tbeorems regarding Mermitan operators which we use
‘hroughout quantum mechanis ac the following

‘Theorem 3.1 The eigenvalues of Hermitian operators are rel
Proof Considera Herman operator A. Is eigenvalue equation be
LEA
Taking ier product with y, we ga
AV = ave YO)
Since À is Hermitian, we have

635

AY) = AY 0) = 08 (he 0) 030
follows from Eqs. (335) and (3.36) hat a, = a8 which is possible nly when
1418 tal. Real cigenvaluc of Hermitian operators play a very important role
in quantum mechanics

Theorem 3.2 Any two sigenfunctons ofa ermita operator dat belong 10
diferent eigenvalues are orthogonal

Proof Let y, and y, be the cigenfunctions of the operator À
corresponding o the eigenvalies a, and a, respectively. Then

Ay = AY, = 4M, m
From Eg. (337), we obtin
AV) = Ye Y

Since operator A is Hermitian

BY We) = (Yop Ye) Y We) = 2 Yd
=) We Wed =O
AS 4, # a we have
Laer) 638)

Hence, the eigenfunctions y, and Y, ae ortogonal

Schmidt Onthogonalization Procedure
When eigenvalues ae degenerate, the above arguments fil. However, one ean
form ortogonal linear combinations from the funcions belonging to different
cigenvalues Le ÿ and y be two normalized eigenfunctions of the Hermia
operator À having the sue eigenvalue a. Then

Aye ay A,
“The linear combination of y, and y is given by

ay, 62)

CETTE Ga
here e, and, are constats Als,
+ ey) = av + ew) ean

that is i lso an eigenfunction of he operat A withthe sume eigenvalue a.
‘We asomo that

Ay, = AC

oY) =O aod (Ys YO = ca
Then
fe VD 2 0 WS cal Y) 0
and
Ge gies erden + Y=! Gad

Te constants ae assumed 10 be real From Eqs. 0.13) and (344), we have

27)
2

er

” + cas
av

“This y sa normalied cigonfunction of the operator A comesponding tothe
sigemvaloc a. I is onbogonal to Y This procedure is a case of Schmidt
“rth genalizaion procedure for systems having two-fold degemeracy. Similar
procedure can he Followed for higher order degencrae eas

35 POSTULATES OF QUANTUM MECHANICS

Postlaes are not unknown in physics, They often serve asthe ass of physical
theories. Thoupb they canon be proved, one ean prove the conclusions derived
fon the basis of the postes, The suecess or fare of he postulates depends
on the strength ofthe experimental results. There ar diferent ways of stating
‘he asi postulates of quantum mchanics, ut he following formulation sins
10 be satsactory

oneal Formal of Quart Moss 67

Postulate 1: Wave function

‘The sate of a system having n degrees o freedom can be completely specified
by a function Y of coordinates q q. , and timo # whichis all the wave
funcion où sate function o sa vector of he system, W and is derivatives
must he comicas, finite and single valud over th domain of the variables
‘of Y. All possible information about the sytem can be derived from tis wave
funcion.

“The wave funcion Y as such i ot an observas, but in some way iis
role 10 ho presence ofthe parc, Is physical irprtaion has alrady
been discussed in Section 25, The representation in which the wave function

à function of coordinate and time is called the coordinate representation
Inthe momennum representarion, the wave functions are functions of the
Momentum components and time, The deals worked ou in this hack ar in
the coordinate representation

Postulato 2: Operators
“To every observable physical quant there comesponds a Terman operate
or matrix. The operators in quantum mechanics ae derived from the Poisson
bracket of the corespondin pair ol elascal variables according 10 the rule
10. #1 = inter) 646)
where Q and R ar the operators scctcd or he dynamical variables 4 and r
and Lg e] isthe Poisson racket of q and
Some of the important operators associated with observables in the
coorinate representation we piven in Table 31

Table 9. Civic obnenvabnn ard ther quantum macharicl operators

‘Oper cortar

berate lassi fam presen
Ces Er
Pain of coa Siwy

a?
Be ® al

‘Operators representing some ofthe other dynamical variables tke the
following form:

Kinetic energy operator. For a particle of mass av and momentum p, the
Kinetic energy

ras
Tah ep} +e)

ARA E
CES
Hamiltonian operator. For à parce of mass m moving in a potential
Vex. 2 the Mami

am

m- Es vo
a HD

E vere Gas

Angular momentum components. The angular momentum

L=rxo=lx y à G49
In», pl

here the operators representing the components of L ar:

60

‘We shall nex sc some ofthe portant commutation lations of position
and momentum operators Its obvious from Eq. (330) that

a
[a] «uen os

voici En) as
ws

bop d= ts pl= bris brise te 9 )=0 0
“The shove commutation relations can be combined int à single one as
DENT 6s

The following two com

ation relations con aso he proved
1a, o 05

When observables ae sepecsentd by matics, he condition for a matic A 10
be Hermitian is

wrt

Ae asso)
het A! i formed by uaaspsiag the mais A

Postulato 3: Expectation Value

When a system is in à state described by a wave funcio Y. the expectation
value of any observable is given by

we raver es

het À in the ineprl isthe operator asocited wi ile observable À. In
Lg (36), Me wave function is assumed 10 he normalized. W he wave
funtion i not normalized

wm os

free

As already discus in Section 2.6, the sandwiching of the operator hetwocn
Y and Y à à necesi,

Postulate 4: Eigenvalues
“The posible values which a measurement of an obssevable, whose operators
A. cam give ac the egenvalues a, ofthe equation

AY = ay, Palo 0%

“The eigenfunctions (y) form a comple sc of independent functions,
The normalized cigentunctions of an operator A be Ye 1 = 1,2,

belonging tothe eigenvalues à = 1, 2... The expectation valve of the

observable A when i is in a sit Y, is given by

= Jriam de= fria, ar x)

Ta, whos e caerme pro to determin the veloc fan
nemabl A na pal te, o vals we eget e nese à
i eme ln le won. ie ena ol an operat we e al
cima mesa quie. Tens goma ays ge à

TO Custom Vector

real number. We have ae scon thatthe cigenvals of nition operators
are real Therefore operators associated with physical quantiics must he
Hermitian,

‘When the wave fncion oa pe no an igenonctn, e cun be
resi ac comimos y fon a complete A Expanding
Si eme of yon we Be
o- Dew, ac
Wht he esis a pen by
Iwan. ira ce

In such a situation

a

Eco frame
DE
-Ehfa 6

YE 0, is the probably for occurrence of the sigenvaluc a in a measurement
OF the observable A

{= Son am

Since 4 ate constants, from Eqs. (3.62) and (3.63), we have

a ter 06)
Hence the coefficients ,. €. are called the probably amplitudes. I the
system ds in one of the cons, say Ye Eg. 61) gives

¡amis

Consequenty.

of aul

eoee, a, (369) can be writen as

a) = Yaw = Elafa

665)

oneal Formal of Quant Mechanin_ 71

Thats when he system iin an sigestte the probably for Ihe occurence
‘ofthe alas a, in a measurement of Ais uni, I cher words, we will cortan
be geuing the value a inthe measurement

‘Aa important eigenvalue equation is that of the Manitonian operwoe
ven by

[Evo juan env aso

rs the eigenfunction of the Hamiltonian operator corresponding to the
Sigenvalus E, This equation is the ime-independent Setrodinger equation
Which we have alway derived in Section 27.

Postulate 5: Time Development of a Quantum System
“The fisfou postulates describe the concept of a quantum system at a given
instant of time wheres the fl one deals withthe time development of a
system. The timo development can he tue sysematialy with the help of
“equations of motion which could he differential equations of he physical
arabes descrbing the system. The se vector or wave Function (1)
Thich describes the sate of system as flly as posible may be Brot into
the piro

Postulate: The tino development of a quantum system can o descibd by
the evolution of state function in time by the time dependent Sebréingce
‘equation

MELON
à

here Hi he Hamiltonian operator of the system which i independent of

“This procedure of considering the ste funcion depends on coordinates

and time and the operator lo be independent 0 time is called the Sebring

picture or Schrodinger representation. As tine development of a quantum

System is an important topic, sti considered in detail ter in Scion 39)

lowever, ii mentioned here to completo the discussion regarding the
postes

nv D Gen

3.6 SIMULTANEOUS MEASURABILITY OF OBSERVABLES

We have been discussing the measurement of one observable at a time 1 wo
observables ae simultaneously measurable in a particular state of a given
system, then the state function is an eigenfunction of bs the operators, Two
serves ac sai o be compatible, i thsi operators have a common sct of
igentancions, The following two theorems indiat the connection Reeves
compatible obssrvalss and commuting operuos

72 Custom Vector

‘Theorem 33. Operators having common se of eigenfunctions commute.
Proof Consideroperaors and Bwil he common seo igenfoncions

wi =

AY ave and By, = by, 8.68)
Thon

Aby, = Alby) = DAV = aby, 8.0)
and

Bay, = By) = By, = apy, 670

Since AD, = BAY, A communes with ens theres

‘Theorem 34 Commuting operators have common set of eigenfunctions.
Proof Consider two commuting operators A and B. The eigenvalue
uation for A be

Mundo ieh, om
Operating both sides fom left by B
bay, = aby,
Since B commutes with A
AB) = aka) em

“That is, By isan cigefuncion of À with ho sume cigenvalc a, IFA bas only
nondegenerate cigemalues, By, can dile fom Y, only hy a makiplicatve
constant sb,

by, = by, em

In other words y, is a simultaneous eigenfunction of both A and B.

3.7 GENERAL UNCERTAINTY RELATION

‘The general uncertainty relation follows ftom what we have discussed so far

a is chape The uncerainry (AA) in à dynamical variable À is defined as
the root mean square deviation fom the mca re mean lies expectation
vale, Therefore,

(aa = (A (A) Gas)
= 240A) + (AP)

= 4-0 Gran

‘Consider two Hermitian opertors, A 2d. Let their commutator is piven by.

PE 625)

In is convenient 10 define 4 new operator by
Re A 4 in 670

oneal Formal of Quart Moshi 73

her mr isan apiteary real mumher. The inner product oF RY with sf must
ba peat than or equal to zero. That i,

tev ny) = [wmv a mB are 0

J

Since A and B are Memmitian Ey. 0.77) becomes

Has moyar= io Joya + moyae20 0m

or in + immperzo ema
[vc + anyar2o
mon
KR = m{C) + me (87) 2 0 (3.78)
icq in Eg 7) mun ol Ios te ac fin. To ad de
hf hhc hd ef (27)
Ure wih pc der

0
Om me am
Eiminaing m hetucen Eqs. 078) and (379) we et

wr E 65)

“This equation ls very general and is valid for any two operators À and B
obeying Lg. (3.75).
Let us now evalua 1

commutator of (A = (A) with (2 - (8),

LA = A). Be GB] = = (AD (8 (0) ~ B= (0) (A = (Ad
AB BA
sic Ga
Hence in Eq, 380), A can be replaced by A — (A) and B by B (8),

(ar as > À

Using Bq, (374). we have

or

nr cay? 2 À

74 Custom Vector

ayan: © ox

“This isthe general uncertainty relation which express the imitation tothe
accuracy with which ne can hope to measur the vales of wo incompatible
‘observables If he operators A and 8 commute, C = 0 and thea

(a am =0 a

Wand B = py the commutator of operators spring
in and ih inequaliy in Bg (332) takes the familiar

In tho case where A
x and p, pies [ep]
form

a
can ay = À ss
hic was introduced as Meisenberg's uncersaty relation (Section 22)

3.8. DIRAC'S NOTATION

‘The ste of a system can be represented by a vector called sate vector inthe
vector space. Die introduced the symbol |), called the ker vector or simply
Le to denots a sate vector which will take different forms m diferent
representations. To distinguish the Let vectors corresponding to different tats,
a label is introduecd inthe Ket. Thus, the state vector corresponding o Y.)
is denorcd hy the hot la). Correspanding to every vector, la) i defined à
comjegate veto La)“ for which Dira asc the rotation (al which i called à
bra vector ox simply bra. The conjugate ofa ket vector is aba vector and vice
vera. A seaar in the Ket space becomes its complex conjugate inthe bra
space. The ba Let notation i a distoned form ofthe bracker notation. Thus
the bracket symbol (1) is dir to (Hand |) in dhe Dira notation. The
words "bra" and “ke” were derived from the word brackst by dropping the
tesoro

Operation by an operator A on a Ke vector produces another ke veto.
Ala) = id) 0

Operation on a bea vector rom the Fight by À gives another br vector
= 00

In ters of bra and ke Vectors, the definition of the inner product of the state
vecto Y, and y, as the form.

te wo = [e n= cay a
a a at
Dre om

oneal Formal of arin Meche 75.

‘The quality sign holds only a) = 0. The het a) is said 10 he normalizd
“

(ala) = 1 659)
ets la) and 1) are ortogonal i
a) = 0 6.90)
‘The orthonormaly relation ix expressed us
aja) = 8, 69
Ta this notrion, the condition for an operator to he min is
CEE 69

‘Compared 10 convention

ation, Dias notation is compact.

39 EQUATIONS OF MOTION

‘The motion of a physical system can be systematically studied only withthe
hp of equations of motion. I he state à known ata particular tims, they
allow the determination ofthe stat a a previous or future time. AS the stake
fa physical system is describe as fully as posible by a state vector inthe
vector space, the equation of motion could be an equation forthe sate vector
Sate vector as such not an obscrvable, But the cxpecaion valo of a
dynamic variable (4 sn observable quay. Therefor. th variation with
time of (4) can be considered as a equation of motion. The definition o (A)
Eg. (56), suggests thatthe variation wih time of (A) may be due to one of
‘the following stations

1. The state vector changes with time but the operator remains constant
(Schnödinger representation or Schrdinger pierre,

2. The operator changes with time while the state vector remain constant
Ueisenborg representation ot Hesenbers pere.

3. Boal tate vector and operator change wih time (interaction representa
sion où interaction picture)

Schrodinger Representation
We ae very familar withthe wave mechanical approach to quantum mechanics
and therefore its appropriate to stat with te Schrodinger representation. As
already stated (Section 3.) inthis picture the state vectors are time-dependent
kei I) and the operators ar constants in time. The equation of mation is
th an equation for). the subscripts is 10 indicate Schedinger picture
“The Ket iy) vatios in accordance with the time-dependent Sebriinger
equation

alv) mp) vo

76 Custom Vector

[As the Hamiltonian Hi independent of time, Bg. (393) can o iterated 10
fe

mn ap (mon aon

the operator xp (iff) is dined ax

o (20). $ came 300

Equation (3.94) reveals thatthe operator exp -iH/h changes the ket IV.)
into ket I). Since Hs Hermitian and srs is operator is unary and

‘he nom of th het romains change. The Terman adit o E. (398)
=i to] = moin = OU 690)

sone ston is
wor note) son

Nest we consider th time derivative of expectation valve ofthe operator À,
“The time desivative of (A) is piven by

dat ,
Kan cola lo) ap

here A, is te operator representing the observable A. Replacement of the
factors

4 a
Ivo) ma Lyin)
ins Es. (899 and (3.96) gives
ent o
La) = Loan) + wolle)
days jan
day Eta (2) em
AE 4, hs co expli dependence an me. we gut
a
ada) um

If the operator A, commutes withthe Manitonan, tis a constant ia te.

Genet Formal of Ovarto Mochi 77

Helsenberg Representation
In dis representation, the time dependence is completely taken up by the
openton, This leaves the state vectors fixed in time, Let us se the unitary
Operator exp (UN to change the sate vector and opertos ofthe Schrlinger
picture. The state vector iy) and the operator A, ae defined a under

vo = on (wo) Guy
mi
2 (22) eu

Suhsiting Ihe vals of ID) from Eg. (394) in Eg. (3101), we get

ll mon ro

"Tias, een iA, does not depend on time explicitly Ay gnerlly depends
om time From Ls. (3.103), (3.101) and (3.102), e obvias that at = O the
state vectors and operators ate the same inthe two representations. Since H
‘commutes with exp (HUM). it follows from Eg, (3.102)

2) = # won
D 7

“That is, the Hamiltonian isthe same in both Heisenberg and Schridinger
representations.
Ditferciaton of Ay with respect 10 timo gives

lija) er oc
ll) 6105

By via of Eq, (3.102) the hir term on he ight sides 94/9 Remembesing
that commutes with exp (27071) we have from Eq, (3.105)

de
16 Ay has no xii nn En

POUR (8.106)

6.1063)

78 Custom Vechanie

ich ds the Heisenberg equation of motion that replace the Schrdinger
uation of motion

Interaction Representation

Another representan introduced by Dirac, cll interaction presentation
is cry useful in problems involving perturbations, Let the Hamiomian ofthe
system consists of two pats

men an
het H does not depend explicitly on time and H may depend on time. There
ds no trict rule for he division of A im MD and HH may represa the
Hamitonian Fr a rlatvey simple system and 7 some additional interaction.
ich is dependent on time. The iteration per is defined by the equations

en (ron 2.108)

m

Diteretiing Lg. 8.108) wih esp 1 timo and mulipling y ih, we get

rt o

vers
IO)

Ñ

Since 4° commutes with exp (IPA) sing the above result Eg. 3.110) ean
e written a6

un oo) = m) any
re

in’
o

Genet Formal of Cuarto Meche 79

Picture isthe same as the Ieisenberg picture Difeentiaion of (3.109) with
resp 1 timo gives

e oy
Mya] Be su
a al] + 5 sua
From Es, 2111 and (MU), evident that ate str changes in
acces wih Y ras the dail varies change I ateos
‘ihn te ecc pcre pa ofthe tine dende is sie 1
the ve seco and att te mama vale

3.10. MOMENTUM REPRESENTATION

As we have discussed in are sections, the wave Function ia Function ofthe
coordinates and time. In such a case, we have the coordinate or position
representation. la certain ease, itis convenient to work in the momentum
representation in which the ste function of a system 94. Di taken as a
funtion of the momentum and time. In the coordinate represemation. the
operator for the coowinate r is simply £ and the operator for p is HV.
However inthe momentum representation, the momentut pis represented by
the operator ps andthe coordinate i cprsented hy à diferent operator.
Since p = Eh, the momentum space is equivalent 10 à ÉS AS in which the
operator for kis tel, Relations inthe momentum representation equivalent
10 the ones in the coordiat represetaion can easily be derive,

Probability Density
For one-dimensional system, the Fourier representation of W(x, given by

A)

=

Jrunsocioa aus

Changing the varahle from £10 we get

wan joomla ous
aun

ta
Jrecnmeoae [ajeno 2o Jarro (Par

Changing the order of integration, we have

Jevactfounn Jou: 72 =p] ae

here Sip” p) isthe Dirac dle function (Aprons II)

O Jeunes ann

In follows thatthe probability density in the momentum representation is
op. 0F.

‘Operator for Position Coordinate
‘The expectation valve of the postion coordinate x is

= far 0 x ox oar

a

a ne a
[anew (IL

- Jon (#2)2, 00% 94

“The integrated term vanishes since iy, 1) = 0 tp = &=. Hence

à Ja evn noo) à Jour oxen (ir

a leJewnoo( o } Tenoft

Integration by parts gives,

00 Jon a zu foul yor rus

o-aje

vale [22 I

oneal Foal of Cuanto Mechanin_ 81

jowr(udlonne ana
E ay

“The operator associat with the coordinate inthe momentum representation
is then 13. Generalizing, the operator associated withthe vector F in
the momentum representation is Hp, where Wp isthe gradient in the p
space

Operator for Momentum

We shall next investigate the operator for momentum in the momentum
rypresemtaion, In the coordinar representation

= Fri of wb Jet nr

a jaje non (e j Benno (te

= joerg Jour ra q fel zion Ja

2000 4 foun ans’ - nur

flee. noan ero

‘Thus d

operator für momentum in dhe momentum space is p it,

Equation of Motion

“The equation of motion in the momentum space can casly be obtained by
fterentiating Eq. (3.117) wäh respect to me and muliplying dy Ih

quires oo

Replacing in} inside the integral by

a
Dr

BE

Me (E a

e

Using Bas. G.116) and (AID) we get

nant
200.0 aus
ivan fern! Lepr
+ fronds for[ qu near
Lon + Jveur.n ar sy»

Loro + V4) oH
mo + van

vo joo HE)

ich isthe equation of motion I Ihe momentum representation.

oneal Foal of Cuarto Meche 83

A DARE EXAMPLES Jo

EXAMPLE 31 A and £ are wo operators defined by
ay

Aye) = WO + x and By

Ben)
(Chock for cir fincas
‘An operator 0 i sid to be linear if
Oleg + € OF + 60.560)
For the operator A, A fof + E400 = 6,500 + 650) + x
CARO OA KO) = + HO + eux + ex

‘The above two righthand sides are not equal. Hence operator À is ot linear
Again forthe operator B,

BH + EL + 40e AGO 4040

a 2404 6 40 + 22,40 426,109)

= La fo) +2, 0 + Lf) + 26,609
¿BRO + AL
ens operators na.

EXAMPLE 32 Prove that the operators

and £ are ermiian

Consider the integral

ES

Integrating it hy parts and noring that Y, and Y, are zero atthe end points

(meda ros a

84 aston echan

Br] Jota [hen

Hence,& is Hera. The integrated terms in he above equations are zero

since Wf and y, ae 210 atte end ponts.

EXAMPLE 3.3 Show hat the catesan coordinates (7,1) and de artesian

components of angular momentun (Ly, LL) obey the commutation relations
on

her ll ae a cycle permutation of 1, 2, 3.
O Whe AV = (hy how

wa

Ten"

=

AE

Menee (by)

Ag

6 the nav =“

2
À

“Thorton,

My a= 0

EXAMPLE 34 FA and B are Hermitian operators, show that (AB + BA) is
Hermitian and (AB = BA) is not Hern,

Since A u 8 as Herm, we ave
(Yu AVA) = Ya Ys eI) = (Ys)
fuse + na wa = [nam de fyzeny, de
DOTE
jon + mores yas

Hence (AB + 34) à Hermia. Now

Jus an an y,de= Jura 0 yiyde== fan - my vide
once (AR = BA) wo ein

EXAMPLE 35 IF x and p, ae Ihe coordinate and mamentum operators
rove that

p= nite
PRET PAPER PATENTS

CRETE TETE |
TN
ÉTAT)

Continuing, we get

Evaluate (Ax) (Ap)

For evaluating (Ax) (Ap), we require the values of (o), (ED. (7) and (3)
Since y i symmarcal about «= 0,6)

2 mu re (22)

=0 Ge integrand is odd)

co) = cam wi foe} oe

wiege

(refe to Appendix A)

an (ap)

EXAMPLE 37 Show thatthe linear momento isnot quantized
“The operaor for component of linear momentum is ih. Let ya) be

it igenfontion coresponding to the cigenval a. The cigemaluc equation

ado mayo or

Integrating, we get

nsec (fae)
ts Gea amarante ER SE
BE oa lc eta

EXAMPLE 38 Can we measure the kinetic and potential energies of a
panico simutancously with array precision?

‘Operator or kinetic energy, 7

Operator for potential energy, V = Vo)

E ven-v

. simultancous
‘measurement of bath is ot possible. Simultaneous measurement is possible if
EXAMPLE 49 If the wave function for a system isan eigenfunction ofthe
pero associnted withthe observable A, show that (A) = A.

igenfuncions and eigenvalues of operator associated with the observable
Abe y and cc respectively.

ds

waryae

= [var tapar
jwartwar

oneal Formal of Cuarto Meche 87

= efva ya:

efyyar
-«
a
aye fay ane
CA
Heace (4) = (A.

EXAMPLE 3.10 The wave function y ofa sytem is expressed a a linear
combiaation of normalized eigenfunctions 9, ¿= 1.2, … of te operator a
Lo observable Aas Ya 9, Show that (A "bare 9 = 6,
En

Given hat -
PAN

‘hen :
= [rivas

dou

22
= Xe

Since the 9% ae orhonormal.

EXAMPLE 3.11. The Tamiltonian operator of a systom is
ae"
Show that A exp (222) is an eigenfunction o and determine the cipenvalue
Aso, evaluate N by normalizaton ofthe Function,

u=

} a a coma

= re |

Gé cé] cv (2)

Mence eigenvalue of His 3. The normalization condition gives

wf

‘The normalized función és

EXAMPLE 3.12 Obtain the form of the wave function for which the
uncenainty product (Ax) (p) = #72.
We have sen thatthe th

‘This implies that
(A + imán

‘The unceriimics (AA) and (AB) are defined by
A (BP = (8 - «BP

1A. BU =14 = (A) 8 - (Bi
Ice, the product (AA)? (A? will be minimum when
LA = (a) + ima - (apy =

Menttying x with A and p with B

le (D + imp = GDL Y= 0 and = Ae

Intepratng and replacing Ap with MBA. we get

CONS
|

ih a NC
cap =P, ode
(ar) [S|

sa A
EXAMPLE 313. Find the eigenfunctions and nature of cigenvalus ofthe
operator

v= New

ce

lus equation is given by

wa, oy

act de tar

Combining this equation withthe Fist one,

Ps du
y eau
Ra Tv ae

Its solution is

nce fee where y and ae constant,

For wo be a pial acceptable function, VA mus be iman. sy
foarte Ow =0. eee ed ce neq, Omega.

e, pale ed

A

Determine the momento function Di)
From Eg. 117)

n= gig joe
se run ku

ap

here

Since the integrated term is zo,

et

ona Forman of Qui Machanien 91

re = (- 7 -)
ar
Win is vale or

OO The ae a Di

à

av as au a) Coro rutas

and Ap... ence evaluate the uncersimty product (Ay)

ce Jue ae

since the intgrand is an od función of

PR (inf Le FR,
0°? Ci Le ae
ci

ty" eta ree
(Ey eee PEL preto
rr la)
The vncensiny put
a

EXAMPLE 3.6 Tor a one dimensional bound panico, show that

Y need mor ho à stationary stat

diw
o Learn

i) Af panico is in a stationary site a a given time, that it wil always
remain in a stationary state.
(6) Consider the Schrodinger equation and is complex conjugate form:

neo LEE eves)

Es
ro, coje

Malin the frst equation by WS and the second by W from Kft and
solractng the Second fom the fst ons, we Mave

oft

negra over
EN av
[zona Y |
aT an ay"
ale =]

Since the state is a Bound one W = 0 as x — os. Hence the sight hand sido
‘ofthe above equation is zero. The integrated quantity will be a function of time
Only. Hones

oneal Formal of Quart Moss 93

4 Ty,
[roo tuna
(i) La Brie na ann = 0. is in
ah e npn snd Bs ey aras, Then
ANGRY
Using. 90, ne

You

(vaa

Operating from left by M and using the commutbiity of A with

co (tw te

im

EXAMPLE 3.17 The solution ofthe Schrdinger equation for a fic parce
of mass m in one dimension is Pr, At

vo

Even

vao-10(-5)

Find the prohabilty amplio in momentum space a
From Ea 8,117)

26.0=

Tier, the otber term having sin(ps/h) reduces 1 zero. Using standard integral,

m2)

%p.0)

94 Custom Vector

‘The Schrdnger sqation in

momenta spas

a. ie

dan Lo
manne Lao

a
Janne

In ean easily be written as

ES

2)

2nh )

EXAMPLE 3.18 Wte the time dependent Schrdiaper equation for a free
Panicle in the momentum space and obtain the form of the wave funcion.

‘The Schrdinger equation inthe momentum space is:

ED
a Espa
CAS
ip!
Fon
ono
EE
Bw
tner
rg
Lo cons
Car

©.

When 4 = 0, 0) = A Hence

een)
anne]

ssh is ih form of wave Function in momentum space

oneal Formal of Quart Mochi 95

EE QUESTIONS Jam

Le What is Milben space?
2. When do you say two Functions ars omhonommal
us and explain the expansion theorem
Win examples explain linear operator.
Define the commutator of two operators. Evaluate the commutators of

[x] anû [4.700]

6. What ae eigenfunctions and eigenvalues of an operon?

7. Define a Hermia operator. Show thatthe eigevals of Hermia
pear ae ral.

A Show ia igntunctions of «Termin operator belonging eon
amas ar onogonl

Outine Schmid oxtogonaizaion procedure fora doubly degenerate

Distinguish tenen esos and momentun representations. What

se the operators for odia and moment the to presents?

11, Explain the comozion Retucen operators in quantum mechanics and
Poison bracket of he coespoding clasica! vale.

12. Outline the iron postie of quantum mechanics.

13. Te cgenvahes of an operar ar the oly experimental measurable
niin, Comment

14. Only Demi operator ar socio with physical quant. Why?

15. The wavefunction ola system is expressed a a ner combination of
«igentncions ofthe epsratr anna with an babe A. Why the
icon the inca combination ar cal the probability ampliados?

16. Define the uncruiny (AA) inthe measurement of a amical variable
Sats and explain the gel unteralıy aon,

17. Outline Dit’ bra and Ket notation,

18. plan Schrödige it, Oban the ine derivative ofthe expectation
vale ofan observa in

19. Explain Heisenberg ponte Ob be uation of maton for a0 operator

A he Mamitnia A) = H® + AO, show tat he state vector changes

in accordance with #7) whereas the dynamicl variables change in

fccordance wit Hin the merci pe:

Dininguish between Heisenbeg and Schringse peas Show tht the

sate vectors and operator are be same in bo the pictues a 4

so.

®

a.

2.
2

Daducs he equation of motion in the momentum represemation
I simultaneous measurement of energy and momentum with arbitary
Precision possible?

24. Can we measure the Kinetic and potential energies of a panicle
simolancously with abivary precision?

——— —

. Prove the follow

Iam.c)+I1.c1.Al+[ic.n.0
2. Evaluate the commutators

22) «(ère
o RS} ol
Show that he asian near momento components (fy) andthe

asian components of angular momentum (ly. E) Shey the
commutation elos: He, A] = par Wy Pal =O. whete RL me
freee put of 1,3

4. Prove th following commun BON: ) [LP = 8 E 0
Where y the radu vector and pe the Incar omentum,

If operators A and B are Herman, show that A, BI i Herman

6, What relation must exist hetwoen Iermitian operators A and 2 in ander
hat AB is Hermitian?

7. For the angular momentum components L, and L,, check whether
LL, + LL is Norman
Show that the commvtaor

Mer E

re His the Uamitonian operator
9. Prove tat

4) sin 9
‘san cignfunction af the Laplacian operator and determine the cipnvalu.

10. Check whether the operator ibe fe Mormon

11. 18 is a Hermitian operator and y is its cigenfuncion, show that
D @) = [lay dr and 6 A 20.

12. Prove the following commutation relations in the momentum
representation:

oneal Formal of Quart Moss 97

CIE

13. Show hat he Function exp (#8 simuttancously an eigenfunction of
the operators NV and HAUT and Find the eigenvalcs.

14. The normalized wave function af a parce is #0) = A exp (ax - i

where A, 0 and Dare constants, Evaluate the expectation value of is

15. Two normaized degonsrate eigenfunctions. yi(s) and Yi) of an
observable satisfies the condition

Jia
iba al Fad a nomad car combination of y andy, ta
à oem Y, Ye
16. Te grana sate wave funcio af à parle of mass len y
capa) mier gemas arn, Whats pol
in ic de panic moves?

17. IF an operator A s Hermitian, show thatthe operator B = iA is ami
Hermitian

18, Find the cigenvalcs and cigenfunction of the operar

ES

19. An operator A contains time as a parameter. Using time dependent
Schrüdinper equation forthe Hamionian 4 and dhe Hermitian propeny

of show tha
aa) à gs
ES
a AN Ny

20. Find the valve of the operator products

fee) a)

21. Evaluate the commutator (i) Lx, pi] Gi) (a, pi]

22, By what tem dhe oporto (xp? + pa?) and Lap, + pa" dtr?

2

28, Th Lapa wann opor Ll des by 10) er sd

© Is the operator Linear? Gi) Evaluate Lei > à

24. The operator eis defined hy the equation

Ara eee
uns

‘Show that e? 27, where

aa sd by FAO JD

25, Find the Hamiltonian operator of charged particle in an electromagnetic
field described by the vector potential À and sealar potential 9

Aga Tt

One-Dimensional
Energy Eigenvalue Problems

The problem of Finding the energy cigenvalues and sigentuncions of the
Hamilton (energy operator) is very important since they play à very erucal
role inthe understanding of atomic, molecular and crystal states, chemical
nding. optical, electrical and ether propeic of substances In his chapter
we shall apply the Meas developed sn fart the Solution of some simple
‘one-dimensional energy eigenvalue problems. In each ease, we solve the
time-independent Schringer equation

LE dv
m

+ MY = Evi a

Lo vain energy cigensalucs E and the energy eigenfunctions. The solution of
these simple models may also emphasize the difference between classical and
‘quanto descriptions of systems

4.1 SQUARE-WELL POTENTIAL WITH RIGID WALLS

Considera panicle of mass m moving in a one-dimensional potential specific
in Figure 41

ve fe casco >
we 42)
la. fala

100 Cuarta Macs

‘The time independent Schrödinger equation fo de region |x <a fs

Be Er un
tos ston i
Y= A sin) + Ba un

As Via) = 0 at = te, Y (20) = 0. Application of his boundary condition
ives

A sin ha) +B cos (ha) = 0 as
und
A sin (ha) + B eos (ha) = 0 as
= vn =

Figur A. The int pora well and the igotuncion of 1,2,9 stats

From Bas. (45) and (40), we get
cos (a) =O, À sin (ko) an

‘The solution A = 0, B = 0 lads to the physically unacceptable solution y = 0.
"The other possible caos are A 20 820 and 420, B= 0. The fis condition
ives

sos da)

ay Earl Probs. 101

mr
EN
“The cigontwnction comespanding to this energy sigemvalus is

nes, as

VD = B ou 49
“The second comdiion lad 0

EEE = 24,6, 4.10)

Sma
With the eigenfunction

D = A sin EE, win
vun =

“The quantum number a = 0 isnot included as i coresponds 10 the wivial
solution yx) = 0. The normalization ofthe wave function gives À = B = Wa,

‘Thus, confinement ofthe parle inthe box led to the quantization ofthe
‘nergy which is given by

Ara
ON

1.2.34, a

set of eigenfunctions are

om (4.13)
Loya

“These cigefoncions for the first states ae ls illustrated in Figure 4.
From Eg. (413) des evidem tha the wave functions coresponding 10 ed
‘quantum numbers are symmeti wih esp! 1 the operation — =x vec
those for even quantum numbers are antsymnetc
‘The energy and the wave function of the ground sate are
FY aoa = co
u win +

ony

“Thats, any panicle coatined in a ox must have a certain minimum energy
al the sero pint energy which sa manistain ofthe nca principle.
“This is understandale since an uncortimy of ore a in position implis an
uncenainty of order 2/04) in momentum which in ur gives a minimum
Kinetic energy of Am)

102 Quart Macs

8 particle trappe in he potential well V(s) 2 0 for OS x <a and

Meo = sober, Re nr soc manon a
eH nd pue ain, potage tty

42 SQUARE-WELL POTENTIAL WITH FINITE WALLS.

Considera particle of mass m moving inside a potential wel with faite barns
of bight Y, igure 4.2)

We re
weh sare en
lg =

“The wave equation in region 2 i he same as Ey. (4.3) with solucion piven
by Dg. (44) which lads o symmotie (n= 3.3.5, .) and antisymmctic
(12 24,6, 2.) wave functions, In regions Y and 3

Eee Ey a
E wo

lts convent to conser the two diffs

ses: () E € V and (i) E> Ya

vo vo
ie eg 2 Regan 3

o
Figure 42 (a) The sauarowol portal win rte walls () tho wavefunctions.

way ame Probans 103.

Case (E < Vy
Equation 4.16) can now be writen as

Le La
AA

(160,

vo ay

Where as postive, C and D ate constants. IF D is not zer. ys) = = as
5 = (cion 3), whic is physically unacesplaMle. Similar argument ads
10 C=O in region 1, Thun, the wave fonction for the different regios are

Be
vi = {Asin kx) + too ks) ws
ee a
where
419)
“The wave function in pion 2 ean citer he symmetric or anisymmerie about

the origin (Section 4.1), Tene, we can math cier the Spmmeti or the
anisymmenie wave funcion of region 2 tothe wave function of regions 1 and
5 at atime. Accowingly the solutions in tepion 2 may be divided int two
lypes symmetric funcion and amisyamnii function,

Symmetrie function in region 2,
its daivatine at x = a gives

The contiavty conditions on yand

Des =H cos (ka) and Da Bk sn (ha) (4200)
Ce = B cos ha) and Cao" = Bk sin ka) (4200)

From Ba. (420), it obvious that €
La tan (ha) = m (a)

Antismmetrie function in region 2, B = 0: Use of continuity conditions at
rade pied

Dan

Det = -Asin(ba) and Duett = Ak eos (ha) (228)
Ce A sin (ha) and Ca ® = Ak cos (ha) (8220)

From Eg. (4:2), we obtain C =D and
ha cot (ha) = an 623

“The energy eigenvalues are obtained by solving Eqs. (4.21) and (4.23)
apically or numerical. Defining

104 Quart Macnee

la=B and m=y
equations (421) and (423) take the form

Brn B= ron)

Boo B=-y 423)
‘The constants and 7 ae related by the equation

og yt amt

peor" a2)

hich ie equation of a ice in Broplane with ados Cm V1 02)". A
apical procedure à flowed to solve Eg. (1213) To pet he slat, e
Curve representing the varon of an, aginst is pote (continuous
une) lon wih eles of eat Qn 3) or different values af yp?
(Fig 43). As land ya have only postive values, heiten fe two
ares in the ft quran gives the cer levels dor nm 1.3, 5 The
inrsection the ce wit he curs of Ba 4219) be By By Than

=the (425)
2m
A similar construction for the solution oF E, (4239) (dshed eves) gives the
energy lve form = 2, 4 6,
For a given panico, te number of bound stes depend on the ight and
width ofthe potential through the factor Vr. From Figure 4.3 it follows hat
there will be

ne
2
Figure 43, Graphical soliton of Eas. (421) an (4:23) fr 4 values
Va. Comino came et tan ts)» an Dashed coro a
a ot (a) = as. Tho dot show the. merci.

ay Egal Prins 105,

One intersection, if 0 < radins < À
Tro ines, Ir E ers <a

Thee merci, rain < 3
nd 0 on Thais the willbe

rs

One ne level ofthe fst pe orme or 0 < Mo < A

Py AO
eve He

Two (one symmee and one antsymmenie) for

cyte RH
UA

ae
"Tire (2 symmatc and one anisgmmei) for “EL

and so on

"The fire bound state eigenfunctions are represented in Figure 42,
Im region 2, the wave functions are similar to those in a potential wel with
infinie walls. However, they have tail in regions 1 and 3, The points x= La
aus te classical turing points forthe well, Wave funcions bave tal beyond
the classical tuning points mean a finite probabiliy forthe particle o be
‘tide the well nother word, he quantum mechanical parce has nelle
+ Teak into th Clasica Forbidden regions.

Case (ME E> Y,

In the sep E > Vp the quantity Va — Eis negative which makes a imaginary
Conscqucntiy he solution of Eq. (16) fr ys) sinusoidal i regios 1 and
3, too. Mens the probability density is disibuled over all space and the
Paice is not bound

43 SQUARE POTENTIAL BARRIER

‘The potential funcion of the ype in Figure 4: allows exact solution forthe
uation of motion. The potential Vir) is dined y

0 xe
worl, oeree ws
lo ro

Consider steam of parles of mass m approaching the square barr
From Left Classical, if the energy the panico E < Y ts always reflect
ores its tansmitod i E> Vy However, quantum mechanically can he

108, Cuarta Machin

w
vl
Incident wave,
"Transmite wang
flected wave
Region] Resion2 Resions
o a z

Figure 44. Roprsenaton of one dimonsinal square barr

sen that tere ls always a fis probability for a panic to penetrate or Jak
"rough the Barr and continue ls Forward motion even HF E < Vy. This
pnomenon,clld quen mechanical nel, posible cause ofthe
ave sauce of mater

Case (li B< Vy

‘The Schrdinger equation forthe tec regions of the premi! are

uan
428)
(429)

whose solutions ane respectively
yee Act 430)
= Ben à Cees “m
y= Det 432)

“The plane wave present he incident pails avclling rom Iw igh
and Ae represents the steam of rflecte partes taveling fom tight 10
let, Frsimplcty the amplitud ofthe incident wave is taken as one nrepion
3, we expect only waves travelling from left to right. In region 2, the
‘spony increasing function 2 is also an accepable wavefunction ike
the exponentially decreasing Ce, since the bier is of iio extent,

One Dmansinal Egy Een Pr 107.

“The propi density o the incident fected and transmied waves
are 1, LAP and IDP, respectively. Conscquemly, the transmission coctfcio
T= WP and the reflection coefficient R = UP. The two ate connected by the
relation + T= L AS we are mal interested i the wansinission and reflection

Couficions, the dctrinaton ofall the constants are not costa. The
Continuity Conions em dh wave funcions and tit Fist dias a.
dares pe
ES ao
Mia = aB- ac ano
wd
Dem à Cote = De ea
abe ae (435)
On solving Eu (430). we pot
es
(435)
From gs. (433) and (4.38), we have
DR +) a
Mes Bae «so
Sabana oft ya o A, and Cin Eq (433) vcs
u et _ am
EB si (20) D wih Cd
Now
DO ME am
La °F sin (aa) + AC cos (an)
(+ Cu) a
Y sinh? (cna) + or cost (ar) m
For brad high Dar
a >> À nd si (ma) = cosh aa) = Lee
ne lore a

CASE EE

108 Cuarta Machin

Subsitaión ofthe vals of and

? in a, (640) gives

AGE je

aan

An illustration ofthe wave fonction in the tc regions i give in Figure 4.

II remit wae

Figur 45, Hurt ofthe wavs funcion nthe vo region
avtospondieg 1o E < V, Th feetes wave ln glo 1 ard
the oxporarialy increasing wave in son 2 am O! show,

‘The phenomenon of hrrcr pencrrion has numeros applications in
physics. The mos important ne he explanation o alpha emission hy nucki

When the encrpy of the parle is greater than the height of the Dari, the
vas fection inthe region D € © alo comes pomme and i given by

verses, pre Ben

“The constant A (in Eq. 4.30) and F are finite as ection can occur at x = 0
and x = a, Figure 46 Musics the waves (without eMccted ones) in the
Siren regions.

MMA

Region! | Region? | Regions

reo sea

Figure 4 Huston of tho wave Autor in the two
gions cormspondng to E> Y

ay Egal Probs. 109

4.4 ALPHA EMISSION

lever, when they ar outside the clous here exists song Coulon
repulsive force between the residual nuclcus andthe alpha paricle Because
both ate positively charged. Figure 4.7 shows the variation of th potential

withthe alpha patel distance from the conte of the muleus. Ih
muckar charge is Ze the repulsive Coulomb potential heyan the
clar potential is 22+ The radis (rf he muceaalpha panico ation
is approximately equal to nuclear radins. If i the pont st which the alpha
Partick: energy E exceeds the Coulomb potential,

we

ua

vn

ou repulion,

panicle energy E

Nackar

Figure 47 Tre variation of pote! wi Ih pare stance

Tren the potential inthe 7, < € 7, as one-dimensional square barter,
we get
15500)
LLOSA

% a

643)

110 Cuarta Machin

As the teal Parr is wot a square, more rigorous

Tap - | Fr
HA

Fat of ht 10" times per second, The probability thatthe particle ponernos
its the wall give hy Ihe transmision cotfcion 7. Therefore,

ws)

(a7)

Wasa 405 8)
where A and B are constants. Equation (4.48) is Geiger-Nual la. Ths, the
baer penetration could explain the phenomenon of cdecay

45 BLOCH WAVES IN A PERIODIC POTENTIAL
As the next example we shall conser the motion of an eleeron in a one-
dimensional period potential. A one dimensional metal crystal consisting of
A number of stationary positive ions provides a feine potential of period
(igure 4.8). Thais
Vere nd) = M, y ww)
For discussion, consider a crystal Hat with N fons in the form of a
closed loop. The Schrdinger equation a point x and (+ d) is th

Ame

NAAN, LL

Figure 48 Onedmersana cyst iat along wih the pre part

ay Eerie Probs. 111

di 2m
Leven ye a
HO) ER evenly =o uso
and
dard 2m
moy +4) 20 aso
a EVO «

respectively. Here we have wed Eq. (449). Since yls) and pur +) satisfy
the same equation, the two can ie only hy a muliplictive constan, say 2

+ d= ay and yore M) a ASI
Since the laico is in the form of a ring

VE ND = eye vi 42)
Hence,
Ala O ee WD)
Therefore,
2 eee) 655
I means that
ye) = uo us
where
Mr a kB pw osha. (39

“The justification for Eq (4.54) can emily be done by replacing» by (x + 4)

es d= et ued)
ot Aun)
yo)
eme
av

which is Eq. (4:10) Equation (4.4) with he condition in Eg, (4.88) called
the Bloch theorem, Thais, the solution of Schrdinger equation ofa perodic
potential wil have he form ofa plane wave modulated by a function baving
"he periodicity ofthe ate, Functions of the typ a in a. (454) ar somtimes
refered to as Mach functions.

46 KRONIG-PENNEY SQUARE-WELL PERIODIC POTENTIAL

In the preceding section, we have considered a one-dimensional metal crystal
consisting of a numberof postive ions providing a periodic potential. The
approximation ofthe price potential in Figure 4.8 is known asthe Kris
Penney potential which i lutte in Figure 49. The width ofeach Well he
‘and that ofeach barier be D. The period of the potential d = a +. Then

112 Cuarto Machin

Vis + nla + D = Vex

ná) = Vo) (450)

Me have already seen that in apoio potential of the form in Eq, (456) the
One-elecron wave functions are the Bloch fonctions in Eg. (453), where ws)

has the periodicity of the lic, In the region O < x < a VU) = O and the
Schrödinger equation takes the form
Hye 8 a
am
Figure 42 The Krcrg-Penrey poroci potential
ves

According o Bloch thsorm, th ola

theme
wire Oeree u
meet) baren way
Sutton o (49) nT (45) 08.40) (as
Eat o = on
ed o won
For EC), ane ln of fr
warner un
Wh his aa of (8) E, (41) toas to
mem. Dérée 68)

CETTE ET]

ay Een Prbans 113.

ones the solution of a, (86) ie

A) = A exp Wk, Rid exp IH + eh O <a (465)
In the same way
1969) = Corp Ik, - Abs] + Dexp Fy + il. -b € x € 0 14.66)

The wave functions and thee derivatives most be continous at = 0,
“That is

du
a

a
ee UM

ho

“Those conditions give
Arb=c+D ass

and
ik, OA kB = (DC + (oh

As the potential is pero, the value of the wave funcion atx = a must be
‘equal to that atx =. Newco

Ag HDs ya um
Wi thes conditions in a, (465) and Lg, (460). we have
À exp lik, Aal +8 exp ic, + Bad
xp 1 = DL + D exp Ik, + BI
eos

and
ik, =) A exp Fit = hea] 4, + HB Pi, + Kal
= exp E = HL ~ hy + HD exp (Us + 200
aa)

Pr a nontrivial solution of Eqs. (468) and (470) the determina of the
‘coefficients of A, B,C and D should vanish, This gives the relation

AE sity sin) + co iP) cos Uo) = cos 0) 4.70)
AS Ky and k, a functions of encre the Left side i funcion of energy
"The right side of Eg. (471) can have values only between +1 and 1
wheres the lft side cam have values outside this range. Therefore, Ex. (4.70)
can be satisfied only for values of E for which the letsand side remains
Ictwcon +1 and -1. Tati na sta, only certain energy hands ae allowed
forthe elecwon and dere are regions of energy which ac Forbidden. In her
words, in a one-dimensional erst the period of the potential together
withthe condition lx + Nd) = wx) 1d to the concept of energy bands

114 uri Machin

‘The transcendental Eg. (471) can he solved graphically. For that the let

side of Eq (871) is plonsd as a fonction of EN (continuous curse) and the

Timing lines of eos i) = +1 (roken lines) are lso dea (Figure 1.10. The

‘enetay ranges fr which cos (4) is between -1 and +1 ae the allowed ones
nthe Bigs, those are AB, CD, BF, et

mn

bergers

Forbidden egos
Figure 440 Graoheal evaluation of erergy vais inthe Krong-Pemey model

4.7 LINEAR HARMONIC OSCILLATOR: SCHRODINGER
METHOD

‘The problem of lincar harmonie oscillator so importance since many systems

Finest can be approsimated to its pot energy U = Lk, kis the

force constant, a continuous function of the cnordinate x and Ierefore is

compltcly diffrent trom systeme we considered so far where th pati is

Constant over à reson,

wove Equation
Wits he fore constant expression aE (LD), poten V is gen by
y= Lange Emus 7
vo langen um
“Tein indepen Singer ution fh ir armani car is
th
Bl
Ls meet va wih aw vray a a new params tnd
y

us

CRES us

ay Een Prbans 115.

Im terms of these quanto, i, (4.73) reduce 10

PY ay
das
Fa v

o ws

‘Asymptotic Solution

We shal investigate fist the solution of Eq (4:75), when y => =. When y is
ver lago, 2= 3° = 3° and By. 4.75) becomes

¥

wo)

1 asymptotic solutions are
ne am
since substnuion of Eq (47) in Eq, (176) gives

ER Aue yee gy
ey
e

Out of the mo asymptotic solutions, ©" is no accepuable as it diverges
when I => =. The exact solution of Eg. (475) may be write as

ven (478)

where AG) is a function of y and the product 7H) tends 10 zero as
blo.

Series Solution

Substitution of Eg. (4.78) in Eq (476) gives

PHO) ya yao um

Which is known as the Hermie equation. We shall look for A), à series
Solution ofthe type

ne Ter as
Balon (40) when sbi ia Ee (7) we have

Fle vu da ria) = 0 sn

116 Quant Machin

For the validity ofthis equation, cocficint of each power of y must vanish
separately. Cosficiont of y when equated 1 zero gives the recurenes ration

Mela
CE)

3 um)

“This formula allows the caution of ll even cosficions in tems of a and
the odd coefficient in terms of ay, Equation (180) will have only edd
oc ify = 0 and even cocficiots ia, = 0. Thus we have two
independent solutions for Eg. (480) and à nca Combination of he two wil
be the most general solution The two solutions are:

HG) = a + ap + ag à 65)

und
HO) = ay + 432 + pt) us

Energy Eigenvalues
When Æ > = ia Ey. (4.82), we get

2 5
2 ws

Consider the Taylor serie expansion of exp (2)

on (= Zi

Lan um

“Ths rato of th cosficins OF the succcsiv terms in Eg. (486) is

tna (0) 12
CN Gye Me

here K s lage. Therefore, fr large values of. y = exp (312110) tends
te Behave Tike exp 072), if the seres ven, and y xp (9/2) H the seis is
fa: which is not acceptable. This unraitie solution can he avoid if the
Series in Bq. (480) terminates after finite numberof terms In such a situation,
Ye) will end to 210 433 > = because ofthe factor exp (3). The Series
an o torminatc by selecting A in sucha way that Qk + 1 > A) santos for
Lam Thus one of the series Becomes a polynomial and the other can he
lumina by seing the fut coefficient 1 zero. Substitution of the value of
Aves

us

way Egal Prins 117.

‘The energy value of an oscillator ass on quantum copy is
ha

From the two expresions ii evident tht te quantum mechanical energy
valo higher dan the quantum theory value by os whichis he energy
Densesed bythe lower ste =. The onilitor bave this energy even
e aloe zero. Ths energy Of {called the zerompninr energy wich is
{manifestation funca principle. £ = 0 means hat otk positon and
inomentom at wel defined which violin of uncertain principle Jan
Be soma tht he mim eyo an car viu Vii nc
principe à Ro. Figure 4.) he the cry values a pen by a
RES]

Energy Eigenfunctions
When À 2 2n + 1, Eg. (479) reduce to

EAN
a

“The solutions of Eg, (4.89) ae e Hermit plynomia 17,9) o degre m The
energy eigenfunctions can now be expressed as

EAN)
2 «eat )=0 u
à

PATRICE us
“The sar Hermite pyri a
m1 an
Ho 2 as
10 aso,
no eo

“The normalization condition lads 10

mr" tonos

MORE

(492)

118 Cuarta Machin

‘The normalized egenfuncions and th ground sat cigenfunción are given y

E.

(494)

“The wave functions y) an the probability density I, ofthe lowest four
states are alo ad in Figure 4.11

wo?

Mo E

De
2

s
EN

no

Figure 4.1 (a) Energy lee (E) and wave luna vo a
lowest four sites of te near hamon oscar
(0) probity dent IU ot te owes! our sas

Te may also he noted from the figure that este cigefunctions
even for even values of m and oud for odd values of. This is understandable
19 exp (372) I always even and 21,6) iv even or od according as I even
‘or dd. I may also o mota that the quantum ositos ean he found outside
‘the parabotiepototal hace since Y docs not vanish atthe clasica! tuning
points. In eer words, he pales can peneuate the busier 1 some exten.
‘This barrier penetration i an important feature of quantum mechanics.

Anotc interesting point tobe nod in connection with the nls the
nar of probability dien of classical and quant exiles. Classically
the probability of india the oseillator ata given pont is inverse proportional
10 its Velocity at that point. The ttl enray

ay ga
mo! + Ha? oe

ay Eros Prbans 119

"Therefore the clasica! probability

Ra (695)

“This probability is minimum at x = 0 (mean position) and maximum atthe
creme posos, In the quantum ease, for n= O, the probably e maximum
At =0 and as ie quantum number increase the maximum probity moves
towards the extreme positions. Figure 4.12 shows the probably density yi
and the classical probability distribution (owed line) for he same every.
‘Though the two distributions became more and more sitar for high quantum
numbers, the rapid achten of 19,7 I si a discrepan

Figure 4.12 The probably densiy yf orto state n 10 (ld eure)
“and form classical oneHlor Ih Sam tal enargy (ken Eur,

4.8. LINEAR HARMONIC OSCILLATOR: OPERATOR METHOD

"The operator method of solving for the energy cigemalucs fe
ill is based on the base commutation relation

REP 0)

whore and p ar the coondinate and momentum operators The Hamiltonian
(of a linear harmonic on

harmonie

sm

120 Cuarta Machin

(498)

> e»

Dr Ce

(4.100)

In the same way

on

um
we (4.103)
Tertre,
mm

= feat]

= M2 fo[ao]o[0.0]o]

= hon ion
in ie same way

té Ma hoe! us

‘To chain the energy levels of the osillor, we have 1 calculate the
marx of the Hamiltonian and dagonatize i However. if we work in the
energy representation in which the eigenvectors of 7 form the basis of the
Space, the mau of H would be diagonal

mitin) = Emin) = E, 4.106)

whore E, isthe energy cigalas ofthe mth tate, Next, we sal valle the
mata ofthe product aa

cremains try Foe Pts 11
casan = (la) dam
‘atic an abot ren
Cara) = m)
= (nlaln)"(mlala)
mazo aus
fron E(t and (th, 0
bbe ow met con

Now consider the eigenvalue equation Hin) = E, in). Operating From It by a
we get, an) = Bala). But from Ey. (1.104. al = Ha + hoa, and therefore,
wwe have

ta + hc) = Ea)
Hate

E, — hayate) wo)

“That i, iF In) isan cigonvoctor of M with an sigenvalus E, al) i also an
igenvcctr of H coresponding to an cigcnvaluc lowered by ha In a similar
vay, we can show that

Hat)

E, + hop) ann
“The operators a and. are spectively known as raising and lvering perte.
la follows dat the Le an) and al) ac al cigenkis of H corresponding
10 the eigenvalues (E, + No) and (E, - Ro) respectively.

Denoting the siound-state eigenket corresponding 10 the minimum
eigenvalue E by 10), we have

Hal) = (E — ha) any

Ehe ao
Ñ Ñ
5-4 Bey aus

Wien he cipenvalue equation 410) = E, 10) is operate from lft by a, we
ae edo

122 Cuarto Machin

Hat) = (E, + hoa) ano

“That is, of 10) is an cipenket of # with cigenvalve £, + hoo = (52) ha
Repcate operation hy al alos dhe cigemalus every time By ho Consequ

(rd)
wie ns 7 1

CRC

form à seis of igenkes corresponding to the eigenvalues E,
given by Eq. (4117).

“The form of these eigenkes can emily be obtained from Fa. (4.113)
Prom Eqs. (498) and (499), we get

à (mon on)

(ay

E + nh

aus
12,

qe m

inma]?

Substituting the value of a in Eq

3), we have

a de men
moc srt) y, m0 or Wo = MO ge
(mos ng) CO

Integrating, we get

ym loons

Repetel operation hy al from left gives

an

v

“This result is dental to the one we derived by the Schrodinger method.

According o Eg. (4.110), the operator a anniilats a quantum of energy
uo and ihereone it Te sometimes reed 10 as annihltien or destaction
‘operator, Similar the operator a creates a quantum of energy Nea Eq (4.111),
and therefore iti calle a creation operator. The creation and sition
operators al and a play important sols in the quantum theory of the
kcromagnei el,

ay Eros Probs. 123.

49 THE FREE PARTICLE
In tis section we shall look into the fee parle Schrdinger equation

ey (4122)

whore 2 ZE, Fora given valu of E, we have two solutions
YO) = Act and wi = Ae eus

“he free particle solutions are thus degenerate, The enerpy has 0 be real and
positive 0 make the cigenfunction finite atx = er and x =~. Th solution
2% cartesponds to the panicle moving in the positive x direction with +k
‘momentum and e “corresponds to the putcle moving in the negative
direction with “Ak momentum. However, the enerpy is same in ba the
cases, Thus, all wal postive values aro allowed cigenvalos

Box Normalization
In the vs sense it
asthe normalization integral diverges. One way to overcome the situation is
te sre the domain of he panic: under investigation to an atari large
length and 0 impose periodic boundary conditions atthe end-points. Because
ofthe perodie condition 4 i allowed to take only the discrete vales:

im

B n= 041.22, (ay

“The normalizar can now Be card out by integrating hetwsen the limits
LR and LR, which Teas o

Fe (4.135
4 ws
“The wave function is said 10 be box normalized. In three dimensions, he
normalization is done in an arbitrarily large but finite cubca box of length L.

vo

Delta Function Normalization

Normaliration of wave functions of he type in Big, (4.123) can lo he cac
‘ut using deta function. One of the representations of dela function also refer
10 Appendix ©) is

Rn
Base LF exp [ap 14.26)

Let ys) and yi. be two wave functions belonging to wave numbers and
The normalizaion integral in the infinite space i

124 Cuarto Macri

[rivas = be an

here dy ie Kronscker delt. A ho energy igenvaucs are continuous,
the Kronecker dela Bu, is replaco by dca function in he dit function
normalization. Equation 4.127) with det function normalization is weiten as

CE uns

Elvin
“The deta function normalized fre panicle wave function is then

wi) = Foe CE)

“Tho deta function normalisation is thas a uscfal technique for he normaliaion
of eigenfunction with continuous cipenvales,

HOKE EXAMPLES > __——

EXAMPLE 4.1. For an életon in a one-dimensional infinito potential well
af wih À, calcule) tb separation between the two lowest energy levels
(6) the frequency and wavelength of the Photon coresponding to à transition
encens tuo levis (ii in what segon ofthe elecnomagnetie spectrum
is this fequency/vavelensth?

6) We have 2a à LÀ 10% m. We aso know
Pee

Therefor,

ae sc]
ime” Exon ETES
1812 x 1073

= 11327 ev

6) We has
hem 1812107
27 x 100% the

Terre,

11x10"

(6) This frequency falls in the vacuum ulraviolt regio,

ay Eros Probs. 125.

EXAMPLE 4.2. Show thatthe energy and wave function ofa parick in a
square well of Finite dep Y, roces o the energy and wavs function of a
Square weil with gid walls Ih the Limit V, > =.

For a well of finite depth Eg. (4.21) gives

ka tan Ga)

me of tn à

Aso, fom as (419), and (416) we Rave

tan (ka)

When Y > =, an (da) > he, Then

wo Be
Th “a
= See

hic is Ea. (4.12,
‘Equation (118) pives the wave function for a parle in a square well
with finite depth, When Ya > =e > =, the wave funcion reduces 10

vor fu uen ur Le

hich isthe wave futon of a particle in a square well with righ walls

EXAMPLE 43 À hanmonicosilatr i inthe ground stat. i) Where isthe
probability density maximum? (i) What isthe value of maximum probably
Era

6

“The ground sito wave fonction is

of)" fe)

Probability density is

Pi

126 Cuarta Machen

tay Ce

2-0

‘Therefore, the probailiy des

Ey

EXAMPLE 44 A eV clctro go tapped inside the surface of a metal. I
‘he potential amer is 4.9 <V and the width ofthe arir 2A, callate the
probably ofits transmission

Ls de width ofthe harrier, the transmission ooeticent

ee]
2 x (RERO VE

a ET

CE

0085

EXAMPLE 45 Complet the steps involved in deriving Bq. (466) fom
Bq. (4.62).

Assuming the solution,
reset 10 x we Be

ap (my) and diferen oth des with

Semon inn wt LE anton in

Sutton ofthese vals in Eg. (462)

On solving, we get two values of my
Ht ky and ik by
‘Therefore, the final solution is
= Coxp Udy ih + D exp Ik + Mil, Dew <0
hic is, (4:60),

ay Eros Probs. 127.

EXAMPLE 46 Show th
Oscar in an arbitrary superpositions
10 the period ofthe oscillator,

‘The time-dependent wave Function of the incar harmonic osclltor in a
superposition ste is

the probability density of the nca harmonie
se a periodic with the period equal

here y,( the time independent wave funcion ofthe harmonie oselatoe
in ih mb state. Now. the probability density

10 Mr if Fis replaced by 1 + Caan,

BES al 5] sun 5

“|

ASE, E,)isan integral maple of That PO 1 period with period
Arie period ofthe linear harmonic oscillator.

EXAMPLE 47. For harmonic oscille wave functions, find the v
Ge av.
For Heemit polyuomials

Hy) = 29H06) + 208, 10)

Substituting the vales of Hayy Hy und H, ‚in
Functions (Eq. 493) and dropping

Er ef)

[env ay, satin,

1 of

as ofthe oxeilltor wave

fiom al tems, we get

(eet yy Evy tay, ya

Since y = (moja), inner product of this equation wih y, gives

128 Cuarta Machin

cord (PR) td eva) =

tan) (UE) our) + (BA) ou

nes

(av) = | fr
ano

o ikensı
EXAMPLE 48. Evaluate (+) (3), (Y) and (7) forthe stats of a harmonic
oscillate.

From Example 4.7, we have

Mat rom Mt by x and on hing ner product os song
un wih Y we Bet
(rote) «lapa d= 0

A)
Then

Sebringer equation for harmonie osilator is
wok

Substring his vals of Ze and using the result for (1). we get

wr

nr)

uno

> ns rte

One Omens Egy Eras Probie 129

side
on ae of pora ny
ce)
wo

=

Suhsttting the values of K and

cna teed ie

i i ini
A Stoe &

EXAMPLE 49 Show thatthe zero point energy of À of a linear harmonic
ciao is a manifestation of the uncertainty principle

‘The average positon and momentum of a clasica! harmonic octo
ound 10 the origin i zen. As per the Bhenfst's theorem this re must he
tne forthe quantum mechanical ease also. Hence

NN
DEE
For the total energy La

CE

and

Ap?) + IK). (here = mo)

up à Lea

Replacing (Ap)? with the hol of the relation.

ovat» À
wet

me
e a

+ Sean?

For the rpbhand sie be minimum, de differentia of (E) with respect vo
(0)? must be zero

1 na
Fr TREO oF nests
[Omen mw BS =

190 Guan Machin

EXAMPLE 410 A scam of panicles of mass and energy E moves wards
‘the patema sep W(x) =O for x <0 and VO = Y, for x> 0. the encrgy of
the particles E > Y Show that te sum of usos ofthe wansmited and reflected
Pareles is equal 1 the Mux of incident particles.

‘The Schrödinger equations for regions 1 and 2 (Figure 4.13) ac

Pa a ME tore < 0
Paja o
A en)
von
Re! toga
veo
| >.
Pare 413 Pot! mp
Te lin of he tv spin a
ne ed saeta (oro

Ve Dep is) ore > 0)

For comenience, the amplitud ofthe incent wave is taken a 1. The second
teem in yj, a wave travelling from sight to let iste reflected wave whereas
Ye isthe tansmited wave It may be noted that in region 2 we will not have
à save waveling from right Kf, The cominuiy conditions on y and is
eva a= 0 gies

Vee and KA)

Simplifying, we pet

an Bet a 2

2

‘The ox of pails for dhe incident wave (Probe 2.11) = Sat

‘The mapnitude of flux of parles for the reflected wave

‘The fh of paris forthe wansmitd wave

y Eerie Pres 181

Sum of ete and wanted fox = a]
[te AP a,
my qe

ich isthe incident or

EXAMPLE 411 A patch of mass m confined wo move ina poten Va) = 0
for D <<a and V(t) = = otherwise, The wave function of ti panico st ie

108 given by

(6) Normalize yx, 0) (i) Find YU 1) Gi) Is yix, a stationary state?
eis given that

Normaliz wir, 0) i

van

ce

192 Cuarta Machin

6) The time dependence of a star is given hy

OS

nee Ga, 2) in his cae i

Lg ct sg o)

Gi) is not à stationary state since y, 1) I à superposition sue

EXAMPLE 4.12 A partite of mass m and charge € moving in a on
dimensional Harmonie potential is subjected wo an elect field in the
direction, Find the energy levels and eigenfunctions

‘Additional potential energy due to the electric field € = ~eex. The
Schrödinger equation ofthe oscllto is:

her,
a
Inducing a new variable x, defined by

aaa

ue en) am

La Als
ei

CAE

$ 2 2
Since de = dez with hs pen fr the mode pn cry
CEA
eee
ee
wing pte

hey

an de
which ithe Schödinger equation of a single harmonic oscillator, The energy
Sgonvalcs at

ie
+ Lady = 1
aire

ay Egal Prins 133,

E

EXAMPLE 4.13. Consider a panicle of mass nin the one dimensional shot
range potential

Vo) = 450. V0
whore St) is Dirac dela function. Find the energy of the sys
“The Schrödinger equation for such a potential is

PY py 2m

ro Fast o
le. o
"cromatina sb uniy mr rl gen
tn A A tog m aly sl yore tb

orga

D) (Appendix ©) Hence, in

124 Cuarta Macs

(3)

Sehstiating the values of the et hand side trom Tg. ()

ay am,
OR

ko) veya 20

ame me

EXAMPLE 4.14. Consider the one dimensional problem of partiel of mass
min u potential V = = fr x <0; V = 0 for 0 < € a and V= Y, for x > a.
‘Oban the wave functions and show thatthe Bound sat encres (E < Y are

sven by
en PE E
D ME

‘The Schrodinger equation forthe different regions are:

2m,
ME «> a
FE

ae
‘The solution of these equations are:

YeAsinkes Books réa

waco’ pe Er

where A, 8, C and D are constant. Applying the boundary conditions = 0
Moya Data ne

sin O£réa

v

ives

ay Eros Prbans 135,

Dividing ono by the other

tan kaw À

Jrs)

EXAMPLE 4.15 Consider a steam of pales of mass m each moving in
the positive adición with kineUe energy towards the pote! bait

3e

Va) =0 for x0 and Vin

Find the faction ofthe panicles reed at x
‘The Seédinger equations fr the different reg

“The solution of the fst equation is
v

is the amplitude rflction eocficiet since e represents a ware
in the negative x direction. The solution of the sccond equation is

ae eg:

here
travel

vate, xo

here rs ts amplitude transmission cocficict i also oscillatory, since the
hei ofthe Bare isles han the kine energy’ ofthe partie, As the wave
funcion is cominaou tx

Lerma
As the derivative did is continuous at = 0,
kane

2

Solving the two equations y = 1/8 and hence one-ninth ofthe particles are
reflected at = 0.

196 Guan Machin

EXAMPLE 4.16 An clccion of mass pr is contained in a cube of side a
which is fail lrg Ii is in an clcromagnetc cl characterised hy the
Vector potential A= Bs the uni vector along y-axis determine the energy
levels and eigenfunctions,

‘The Hamiltonian operator of the electron having charge =e is

D CO +

here Py Pe Pate operator.

We can easily prove th Following commutation relations:
[.pJ=(pJ=0 and Up 40

Mene, by vine of Eg, (31063) p, and p, are constants, The Scbrólinger

‘equation is

ré BER 2 Bene

+ [Es , Rep, ni
me me

me
Re
PRE AA

Med MEN, le px
Pme me?” me

2m de 2 m:

“The forn ofthis equation i similar to hat ofthe Scbodingor equation for a
es he energy eigenvalues are

One Dimension Egg Een Probs. 137

crier «

ee
mé ÈS Be

‘The eigenfunctions ae piven by

ovale” a] mole ayer

ore

oh ch
EXAMPLE 4.17 À harmonic oscilor moves in a potential

La
vara
oat

valore is a constant, Find the energy cigovales,
‘Te Schrödiner equation i

Lego o) yea

mar

Changing the variable 0x, by defining x, = 4 ck

ona

1. A parle contin in a box must have a enain minimum eneray called
zero point energy. Comment

and 4 sas

‘Sketch graphs of wave Function y and of IP for the
fa pares tapped in a potential well of init dep,

3. Write the Schrödinger equation and the form ofthe wave function in the
différent regions of a square well with finite dep

4. Explsin symmerie and antisymmetric wave fonctions with examples
5. Exphin quantum mechanical tunnel,

6. In amic pencirton problem, why the exponentially increasing function

ea wher aris piven by

5

5

is abo an acceptable solution inside the barr

7. A particle having energy E is incident on a finite brrior of height
VAE < Va) Mustate the wave funcion in te different regions

8. Explain how barr tunneling accounts fr redocay by certain nu
9. What are Bloch functions? Stat and explain Bloch thsorem

10. In a one-dimensional crystal, the peiodiciy ofthe potential ed 10 the
concept of energy bands. Explain,

found outside te parabolic potential

LL Explai why the quantum oscila
bate

12. Sketch graph of y and Ip forthe fist 4 states ofthe one-dimensional
amont,

13, What is ere pont nergy of harmonic oscillator? Tow is it explain?

inal Energy Egon Potins 139

AAA

bin the eucry cigensalwos and eigenfunctions of a particle tapped in
the potential Va) = 0 for D € x Sa and Wo) = = otherwise

Show that the wavefunctions forthe different energy level ofthe parle
app in be square wel in Problem | are orogonal

3. Caleula the expectation values of poston (+) and of the momentum
0.) ofthe parle tapped inthe one-dimensional box of Problem 1

An electron in a one-dimensional infinite potential well (Scion 4.1)
008 fom the n = 4 to the n= 2 level. The frequency ofthe emite
Photon ie 3.43 x 10" Hz, Find the wid of the box

[Evaluate the probably of fading dhe wappcd pale of Problem 1
between = 0 and x = al, when itis in the mt sate

An alpha particle is app in a nucleus of radis LA % 10-8 m. What
is the probably that will escape from the nucleus. Js ener 8
2 MeV? The potential barrer a he surface ofthe auleus is 4 MeV.

‘The wave function ofa particle confined in a ox of length is

vine (2) un(e) exce

CCaleuate the probably o finding the particle int region 0 < x < a
ind (9) and (p) for ths nth stats of the liner harmonie site.

9. For them stat ofthe linear harmonic elle, cvaluat the uncertainty
product (As) (Ap)

10. A stream of panicles of mass m and energy E moves towards he potential
‘ep of Worked Example 4.10. Ihe energy of particles E< Y show tht
there isa Finke probability of finding the panics in the region > 0.
so, determine the flux of () incident paces (i) rte panics
(Gy the particles in region 2. Comment on the result,

11. A beam of 12 eV elcctrons is incident on a potential brie of height
3OSV and width DS am. Calcula the transmission cof

12. For the linear harmonie osito ia is ground sat, show tht the
probably of Finding the panico outside he classical mits e about
16 per am

13, An cron moves in a one-dimensional potential of width À and depth
12 eV. Find the number of Bound stats present.

14. A linear harmonic oscillator is inthe first excted state () where is its
probability density maximum? (i) what is the value of maximun
Probability densiy?

140 Cuarta Machin

15, Sketch the probability density y ofthe liner harmonie oscilar as à
function of x for m = 13. Compare dhe result with hat of the casical
oscar ofthe same total energy and discuss the limit » > =.

16. Cateuate he energy levels and wave funcions of a parco of mass m
‘moving inthe on dimensional potential well defined by

je Den

VO mars toro

17. The srongest IR absorption hand of HCHO molecule occurs at 643 x
WH tHe IF he reduced mass of CHO ig 1185 x10 ™ kg, calcula
(0 the approximate zero-point encrpy (i) the force constant of the CO
bond,

18. For the nth state ofthe linear hannonie oscillator, what range of values
is allowed classically

19. An sictron is confined in th ground tte of a one dimensional harmonie
oscilar such that Ae = 10m, Assuming that (7)=(V), find the
‘ergy in eV quit excite ito is frst excited sas

20. An electron havia energy E = 1 eV is incident upon a retanpulr baie
‘of potential energy V, = 2 eV. How wide must the bari. 20 thi the
transition probably À 10

21. A particle of mass me confined to move in a potential Vix) = 0 for
05x50 and Vi) == otherwise, The wave function of the parle st
tine 1 = 0

renom (eut)

Gimp 0 Fly. oy e

22 Theor cent mr lo a
DT SEA ah ny ei
an ela heat

28 from Anm fa ag eat an

csotin-(ved)s

=aY o

Three-Dimensional
Energy Eigenvalue Problems

Inthe previous chaper we applied the hase de developed to certain important
one-dimensional potential problems. ln this chapter, we demonstrate how
effectively quantum mechanics explains mos ofthe important future of some
fF the thee dimensional problems

5.1 PARTICLE MOVING IN A SPHERICALLY SYMMETRIC
POTENTIAL

Ina spherically symmeri problem, the potential depends only on the distance

ofthe particle from a fied point, The timo independent Schrödinger equation

for such a system is

6»

Since spot spicy symmetric its convenient workin sprical
polar contlinates r, @g. (OS 7 < =, 0 4 OS 2 0 € 9 < 2. Expressing
1.5.1) i pola coordinates, we gt

12
Fr

(2%). 12 (sng), dr.
een

62)

142 Cuarta Machin

Separation ofthe Equation
xenon (52) can be separado dues ques by vins
Wr 8 6) = Ro) 0 (8) 86) 6a
Satin hs om ol Yin, (52) and maining by
Pinte
100
ve oi
29) + Steam Ile
6

“The letsand side of Eg. (5) à function of and 0 ad the sgh side is a
function of alone. This is posihe whom cach side is a Constant, ay m?. hon

EL OR
ra) ss
(a) sind ding 10)
Roa ar} © dO 4e, oe
Diving ot sides of a. (56) by sin" and seaanging, we go
12220, [ine 2) =
al" Band aol"? ao)" a

‘This i posible when both shes are qual 10 a constant, say A. Consoq
wwe get he Equation and the radial equation

wn

and

Le) eve .s

Pa

“Ts, the tree dimensional wave equation (5.2) is separated ito thre one«
dimensional equations (5.5, (5.7) and (5.8.

Solution of the equation
“The solution of Eq, (55) is srighorwanl and is given by

CO

‘Thee Dinero! Eve Eiger Preters 143

For 010 he single valsd, (4) = (9 + 28) Therefore
Acti = ein ar = enon

‘This is possible only if m= 0, 1,2, .. The quantum number mis called the
magnetic quantum number, The normalization condition gives

rare ut uo
Thea " "

Ua =

except for an arbitrary phase factor which can be taken as zero, The normalized
solutions then

om

Dee 69

Some of the normalized (9) ar given in Table 5.1. As sin (hn) and
cos (li at also solutions of Eg (35) the el form o the solutions are also
listed in the tbl,

“able 54. The Faso Nonmaized 0(9 unción

Im Comper jam Real foe
0 «4
Y
nn re

ou Lone
eur Fondo
ee uan

Solution of the @equation
To solve the @cquat
teren. we as

as

144 Guara Macs

We may alo write

In sms of Ba. 629 à
da lows
ee] (a Joa o (510)
ich act Lente equation. uation (510) hs poes =. For
Phys stable oon,

Ama. ll medal th

“Ths solution of Eq. (8.10) is the Legendre polynomial PAs) for m = 0 andthe
for m # 0. The normalized solution

000) = NP" sn
here A, i the normalization constant. The normalization condition is

6.12)

619

Where € = CIJ form > O and €

1 form 0 as por the established phase

Spherical Harmonics

‘Te solution ofthe angular par ofthe equation, calle the spherical armies

is independent of E and Vi) Combining Eg. (59) and (5.13) the normalizch

angular pat of the wave function is

ENTE
ae (m)

PAT EN (cost) emt 510)

Da mean. ss

“The spherical harmonies are muwallyorboponal and th fs fo of them are
given in Table 52

‘Thee Dinero! Er Eger Peters 145

‘Table 52. Tho Free Sorel Harmenioe

ea ome

Ya ("coco “E 3) a om oom 0» (18)
warf)" ra (2) “main

€
son (Bf neon
Jam. (2)

nn (E aomen

‘The presence of thy factor" makes the spherical harmonies complex in
general Ofen iti advamageows 10 work with real form, We have

hs fel form by taking a stale near combination of them, Similar
linc combinations are taken fr ether spherical harmonie ao. The ral
forms ofthe fire of thin are also listed in Table 52.

Radial Equation
“To solve the radial equation Eg. (5). the explicit form of the potential VO)
is needed. However, ths can be expressed a the form of a one-dimensional
equation by writing

my = #0 19

146 Quant Macs

“The radial equation now reduces 10
du ur

le vo- =o sn

ae der =) 2?

“This as the Form of one-dimensional Schridingsr sqution of à arte of
mass moving inthe direction of ria field of effective potent
hae ve

va = vos HE sn

‘The additional potential 1Q + 1) mr) is à repulsivo one and
coresponds o a free AU + 1) Mm). The eenuifogal force mre! can be
‚write in terms ofthe orbital angular momentum Las

1510)

“This form ofthe cotiza for andthe Force comesponding to additional
potential suggest hat 22 ean be taken as

TO (520)
“This showed puessrepadia the value of £ is put on fm tnorcical asis
Chapter $, As the quantum nunher Li associated withthe orbital angular
‘momentum, iis called the orbital angular mamentum quantum number. Often
We say thatthe orbital angular momentum is th though the exact vale is
Na+ DEA. EV) is Coulombic (22%), the additional potential is
mesigible at lage distances. However, this becomes the dominant tem at
lose distances. Figure $. gives a plot of Vaya a function ofr for a Coulomb
potential

Figure 5.1 The oc Ale Y vorne rfor à Color poeta

‘Thee Dinero! Eve Eiger Peters 147

52 SYSTEM OF TWO INTERACTING PARTICLES

‘So fr we have const ths motion of a pack npn cd.
there ae situations wherein we have tw inracting parties moving in à
three-tmensanal space. The wave equation of sucha system can be reduced
into two one-particle equations, one representing the wanslaional motion of
the coms of mass and the othr the rca motion of the two particles

Hamiltonian Operator

‘The position vectors and masses ofthe two particles are shown in Figure $2.
‘Te radius vector of the cone of mass

PECES

wi vn
Figure 52. System a wo racing sarees.
‘The relative positon vector i given by
or 22)
From Ex. ($21) and (52), we bave
gene, gem Me 23)
“The momenta of the two pas can he writen as
Pps oni = mi + ues By 62
where
pee 125

is called the reduced mass of the panicles. Assuming the pol to be
dependent only on the distance hetwson ths two particles, the Hamnitonian of
the sytem is

148 Guri Macs

(529

Edd 2
anto sn
here Af = m, + mpg = MR and p,= jr. Replacing the dynamical variables
by the corresponding operators and writing

vi ve. 6.28)
u

Tin Ss en
room Ble) com

n for Relative Motion
Fquaion (830) can he separated mo two equations by writing

VER DR) vn 1530
With this form of y CR.) E (530) reduces 10

Pl .
PACE iy MO HEY (533)
For the validity of Eq ($32), each side must be ual 0 a constant, say E,
“That

Ex >
and

(E ED ae) 1530

AS Eg. (533) isthe same as à fc paricle equation of mass Mit describes
‘he transiional mation ofthe system in space. Equation (5-34) isthe same as
the Schrödinger equation ofa particle of mass moving in afield of potential

‘Thee Dinero! Eve Eiger Peters 149

Vr) and represents ih relative motion of he two pales, The energy for the
relate motion is Ey ~ E, = E, In the coordinate system in which Ih centre
‘of muss i attest, £, = O'and then isthe total energy ofthe system. Thus
the Seluodinger equation for relative motion is

Evi pins Vie pie = Ente) >

EN]
In the following ssctons we shall consider Iwo important systems of two
panico, the gid ator and hydiogen atom

53 RIGID ROTATOR

A rid roto consists of emo masses m, and m, separate by a fixed distance
+. Consder th rotation ofthe system about an axis passing trough the centre
SF mass and porpendiolar 1 th plane containing the two masses. For free
rotation, the potenti Y) = 0. AS rs fixed, the wave funcion will depend
‘nly on the angles @ and 6, In spleical polar coordinates, tbe Sebringer
‘ition fr relative motion redaes 10

ad 1 #
Eat gd (ied) ran man 536
Ala ae) 1_ Pe | wee
ads EOI » m)
Wong
ze es
5
sod
CUP 6%
Eguasion (5.3) reduces tothe following wo equations:
ED ato sn
and
(eee) a) 0000 Gan
sind da” 40. sin’, à
er 2 pr he momen of nia of the sor and m sa constant

“Those equations ars the same as Eqs. (8.8) and (5.7). Hen, the rigid rotor
‘wave funcion are the sphrical hannoaics Y,, (@ @- From th solution, it
follows tat À = AU + 1). From Eq. (535) the energy eigenvalues ate

150 Quart Machen

fea

a
“This consittes a sc of quand energy levels with (8 DAL separation
between any two consccwivo levels, (is the quantum number ofthe lower
stats). Since (21+ 1) values of are possible for a givn val Of, cach ta
ds 21+ te degenerae

E 12, sa

54 HYDROGEN ATOM

‘Theory of hydrogen atom i of fundamental importance as it provides the basis
for the theory of many eecron systems. Als, this I be only atom for which
ext solution of the Schrdinger equation is possible, For discussion we shal
consider hydrogen lke atom which consists 0 à nucous of charge Ze and an
cleetron of charge -e separated by a distance 1. The potential is Coulombic and
is given by

vin = ZE son

“The timesindependent Schrödinger equation for relative motion is given by

e

1540

Radial Equation

Expressing Ey. (54) in spherical polar oordiats (7 9 and separating the
variables as in Section (5.1) by writing

V6 0.0 = Ro 90% sas
ve ui etn wt
a eus

"The solution of the angular parti the spherical harmonies YO)
To sole Bg, (5:46) since avaible and à consta À defined by

we Ta

E sn
ne 2)

AS E is negative for hound stats, p and À ar ral qua. I ts ofthe
mew variable, Eg, (546) becomes

648

‘Thee Omarion! Eres Eger Protons 151

Solo ote radia gaben. sap ota cam De mi
fas When pos mE) wdc
fe
we
1 ton ae A= 097 and 2, Qu al hse wo soon. ny is
esti nee °F O Te ast slo 6 (88)

Rip) =e? Rp) 1540)

Substitution of Eq. (549) in Eq. (548) gives the differential equation satsied
by ip) as

R=0

er ar
later 6

When p = 0, we ast
N+ )RO)=0 or ROO, 140 ss)

Therefore if we ty a power seis sation for Tap) lt must not contain a
constant wem. Hence

r= Sa" es
With is vale o Fp, Ba. ($0) reluce 0
ace Dades Presi Pongo
59

Equation ($53 is vali for al values ofp only i te coefficient of each power
of p vanishes separately. Equating the coefiient of pe 1 zero, we have

(070)

Therefor,
ert or cra ss»

We = (+ 1), the Fit term in FU) would he ap! which tends infinity

à p 0. Hence c= isthe only acceptable val. Setag the coefficient of

PE in By, (853) to zero, we obtain

teksto’

Hanae a

1555

182 Cuarta Macri

"This recursion lation allows us to determine the enfin dy ty y in
term ot a, which is quite array. For large values of e gt fom EQ 5.55)

Im the expansion

Menge as =>», the series for Fi) behaves lke plo? and
Ro)

This value of Rip) isnot acceptable and therefore the series must break ff
after a certain value off, say For hi to happen a, must be sto, Then,
fiom Ua. (5.55)

Pape

tana W012. 650)
Energy Eigenvalues
Defining a new quantum number by
x me
A ME

Squaring and simpiying
ube
Gee, RR

LR SSD

Since n° and 1 are integers including zero.

nahn (539)
[Asm 214 1, the highest posible valu of Ys = 1. Thus
1201.2, (1) 65)

‘The aew quantum number a is called the principal quant number whieh
detrmines the energy. Tor hydrogen Z = 1 and the reduced mass yr = m, the
mas of electro,

The eaersy E, isthe same as the one obtained by Bohr on the basis of
quantum ideas. The major diflrene is the occurence of the concept of
Stationary stats and the quantization of energy as à consequence ofthe solution
of the Sehringer equation

‘Thee Derio) Er Eiger Probens 188

Radial Wave Functions
‘The above sesion in energy makes te series for Fp) into a polynomial
Writing

Tp) = po) sw

Equation (5:50) reduce to

eu
ofa

ra BA

up
+@-I-Dup=0 60
7 w)

“The asociated Laoerr polynomial of order and degree (= p), denoted as
1 Gp}. saisies the equation

2

a ”
ren noo (50)

Equations (5.61) and (5.62) ane denial if Lp) i akon as £3}

Ry = Ne MPR) e

“The normalization integral

Ter
allows the determination ofthe constant Hence
ferran) paper
mie wlan) ap:
Using the ortogonal promise of anche Laguero polonia

plo oy
In Dr

ss»

59)

154 Quart Macri

here
sr
ne

a (566)
“The negative sign is selected to make Ry postive. As ys approximately equal
to the scan mass, ay = 2, Me Bohr rads. Some of the radial wave
functions ae given in Table $3 I may be noted that atthe origin the wave
funcions Ry Ray Ru ate finite whereas Ry Ky Ry ae 200

able 53 The Fini Facial Wave Functans ofa Hydrogen ae Alam

Ejea

Wave Functions of Hydrogen-like Atom
‘The complete wave function for hydrogensike atom is given by

Wan 0.0.9) = RADY. © sn
ver
REL D3 OLA Dee D me HA cay

‘The explicit form of the wave function for some of the tte ate

Gas

‘Thee Omarion Er Eiger Pobers 185

au (5.68)

Yon
Von sw
1 (zy on
Y gr 50
tna" (E) sna so“

In may be note that be expressions forthe U = 1 stato contain de facto
“The f= 2 sates will have the factor © and so on. The presence of the factor
‘mas the wave Function zero at = O excep Fr the Sat

Radial Probability Density

For the state specified by the wave funtion y, the probability of finding the
een in à volume clement de is

Iva Par = RE 11,2 sin Ode dO de

The probability of finding the electron in a thin spherical shell bounded by
radi and (rd) is then

arm 8 deff sin 04020

Since the spherical harmonies are normalized 10 unity
Pig) dr = IR Pde

“Ths radia probability density Ps defined as he probability of

electron of the hydrogen atom St a distance 7 from the nues. Thea

Ps = PP 570)

Plots of radial prohailty density for some of the stats ae lust in
Figure 53,

For the ground tt, a maximum probably density Py exit ata radia
positon given by

Meg o (oro = ré

a )

‘The maximum oceurs a a distance equal 10 the Bohr rads Tom the
origin. Though the radial probability density is maximum at the Bohr
radius, aspherical distribution of the ground state probability cannot be
‘overlooked,

156 Guara Machen

Figure 53. The ral pobabity density P,{ for ne sogen Ike atom

55 HYDROGENIC ORBITALS

‘The three quantum numbers 2 mr specify de hydrogen atom wave fonctions
and describe the motion ofthe electron, The wave function ole electron
in he hydrogen atom rod to a he roger orbital When 1 = 0, Y,
2.3... the cketon is defined as ans pdf, ete. electron, This notation
is derived from an oM description of spectal series: sharp series, principal
series, dlls series and fondamental seres, An electron with /=0 I an se
schen andthe corresponding ware function is called an sorhial, The wae
Functions coresponding 1 1 2 1 ar th prota and those for 1= 2 and
‘orbitals and so on. Ths the symbol 3d is shortened notation or th electron
save function having à =3, = 2 and = 2, 1,0,

“The wave fonction y, 8 6) can e writen a the product oF the uo
Canis A) and Y D. Tor a gen va fm can a he as
LA D (2 eu Doch, and the radial part RAP Is ib same
for al ihe (2+ 1) ave functions Hence, the ware functions ate represented
en by the angular part Y, 9) nly. Thus at having w= 2, 1 1 have
r= 1, 0,1 andthe sates are sometimes dnote as 2p. 27, and 2p y. We
have already discussed in Section 5.1 how real frm of spherical harmonics

‘Thee Derio! Er Eiger Pobers 187

ar formed, The linces combination of 2p, and 2p gives one combination
withthe factor cos and the fer withthe factor sin 9 (Tale 5.2). The one
With the os 6 factors denoted as 2p, andthe oras 2p, The 2p, is denotd
a 2p, Similar notons are used for higher sates also,

Represetion of oebitals ase usual done in two ways: in he Fit
method, graphs of 7,8.) and in the second method Contour suraces of
constant probability density ae drawn. Polar representations of the angular
par of s,p and d orbitals are ils in Figure 5.4. These plots represent
Surfaces in hr dimensions, ih distance rom the origin 10 a pot om the graph
wil he proportional tothe squire of the angulr par, PCA O) ofthe ori

‘The sorbital wave fonctions depend oaly on r and therefore they are
spherically symmeuie. Each p-orbil bas two lobes. For p, p, and p the lobes
are along x, yand 2 ases with y, az and xy as nodal planes. Four of the ive
‘emia have four lobes and evo nodal planes cach Ths th one orbital)
has two Jobe along the gris and a rng (charge disibution) in he Sy plane.

©

ay int aay ais mat

Se Cho: bo

RE

+ +

Figure $4 Polar representations Y! yatogen s,p and d
“al. Tho danes ef the cano om ine orgie ropemens|
10 te oma ofthe angler pan ol the amis otal

158 Quart Machin

55 THE FREE PARTICLE

A free particle has three à
Sotrodingercquation is

ccs of frecdom and it time-indspend

Vins EM) om

Plane Wave Solution
In cartesian coordinates, writing
win XAO 67

[Equation (5.1) can be separated imo the res equations

LE rs emo
CELINA 3 mE,
er E o
Dre) : em»
ono
tes
Base
Kanes
Een om
Th aon of ES
Mn = Comp 9 ey

XC) will be fit forall weal values of 4. Similar solutions for y and +
vorinaes are possible, Combining the thee solution, we pet

veo) = A ex kn (5.16)
she À 6 the mormalization constant In he usal sont, he cigenfuncions
are not normale In Such stations oc can rson o what i know as or
normalization by vesicng the domain of ye o an rbitriy lange tu iio
fabs o side L cerred atthe origin, The box normalized plane wave solution
ls given by

wn

ep Uk) sm

Sphorical Wave Solution

Laplacian V2ye) in Eg. (571) is expressed in (7 8 9) coordinates and the
variables are separated by writing (e, a in Eq (55). The radial equation,
Ea. (58) with V = 0 takes ie form.

The Derio) Erray Suomi Pobens 159

1 (200) [ane 10),
Hr) [e so
BE, ou pair sm
7
nen 3.7) vice
ERICA LLO Paper ”
wlan sw
by wing
kp) = Zp" san
auton (530) can be writen as
een, 1, anus ou
ap. Late wT ap »
which is Boss caen ls gene soon is
TONI + Lor
Then
“ ©
Ro Fini) + le
# e
= ann es»

het À and B ae consta, jp) and np) ar he sherical Bessel fonctions
and spherical Neumann funcions respectively and ar defined by

or (El anto
s(t 5

no (ZY Auto ss
lit expression for de stew 7 md ae

6560

ss

6560,

560
se rn
2 6

min Da
. Dar sn

For small and large valut ofp, the hsbasiou of fr) and n/d) are given y

in

140 2 mo a Lolo to) sm)

ta Mo ole h) emo
138.10 fo)

mo a ED, nip) 2 Leo) om
Cri CRE

“The solution ofthe fee parci equation comesponding 1 à dein encres
de oil angular momentum [104 DJ can be

valo 8 9 = Ml) + Bp, OCB 9) sw
“The most general solution corresponding o a definite enery is

win @o~ Sun manınan 6m
ic a an an Be
pace do es ee oe

vir a à = Fara on 00

Ihe partite moves only in he segon Ar > 0, he solution whic is not regular
as the origin is also as important as the regular one and we have 19 use
Eq, (59). nthe theory of scaring we Will aye occasion fo us is impor
result

5.7 THREE-DIMENSIONAL SQUARE-WELL POTENTIAL

‘The tree dimensional square well of finite dep is ilstated
and is defined by

Figure 55

‘The Dinara Energy Eiger Prtione_ 1

[4 derca

wo ra

1590
were a isthe adios of the sphere having spherical symmetry

wo)

Figure 55 Throe-dmersiona square plant

rom Eg. (5.5), the radial equation fr a state with deis angular momentum
is given hy

MED Drew (592

1
4 a
? ro

(0%), (pena

Fala 7

“The solos of tes equations for cues with foe angular momentum

A Gi too much involved since one has 1 deal wit peril Bese a
ane fonts. L w conser eth impor case of te = O) wich

issue for win th grand se of stot eme oF

mere To solve Eqs (592 and 393) it u we

ra 69%)

694)

19

(590)

162 Cuarta Macri

"The respective solutions of these equations arc

a

Asin Gy) + 8 cos (hr) sm
and
Min) = Comp hy + Dow yr) 698)

here AB, C and D are consta. As ul) must tend to zero as > D, the
constant B has to be zoo. The solution exp (7) is wot Finite as = = This
makes D = 0. Henee the acceptable solutions are

MV = A sin ty) Derca 1599)
und

wine Cepek ra 5.100)
Applying the continuity conditions on ur) and dur at = a, we have

A sin ha) = Comp he) sion
and

Ale cos (ka) = RC exp Cho) (5.1013)
Dividing ono by the other, we have

yc kya) = hy (5.102)
Equation (5102) is similar to Eq, (123) ofthe one-dimensional square well
“Therefor, te prapbial solution of Section 4.2 is applicable I follows that no
solution exists unless Ve > (84) and tere will be a bound state if

(5.103)

5.8 THE DEUTERON

‘An interesting application ofthe ttee-dimensionalsquae-well potential isto
the round state of deutron mucus. Deuteen isthe Small mucous in which
a proton and a ncuron ars held together by the muckar potential. Stuy oF tis
‘wo particle system helps to understand the nature of nuclear force, aueear
size, ete. The Binding encrpy of deuteon El = 2 226 MeV and therefore k can
ba calculated from Eg (593). Equation (5.108) ca thon be used 19d

CR,
wie)

15100

Since he bindin energy lis small compared to the depth ofthe potential Y»
to get an approximate range-depth relation, we cn set

vet
wm

-0 os

Thee Derio) Er Eiger Pobers 163

and thereone

(5.106)

ete m = m, = m, the mass of the nucion.
"The range of the nuclear potential is approximately 14 x 10m, Hence

sn

which juss our asumption V, >> EL. Further insight can he obtained fom
the wave fonction in de region 7 > a, The form ofthe function exp (ck)
suggests that LA, may be taken as a measure ofthe spatial extent of the
Acuteron, Tha ithe probably oF Finding the partite is maximum in the
region Der < TA,

¿e
Ela (5:08)

From Cas. (5.106) and (5.108), we have

—
mm, 2%
rae sn

means thatthe range of the wave function ofthe deuteron (ik) is considerably
greater than the range ofthe potential (a). That is, the deuten is a manch
Bound structure with the nucicos sending considerable me outside the range
‘ofthe potential

ALES —————

EXAMPLE $1 A parie of mass m moves in a tree dimensional box of
side a, c.f he potential is zero inside and infinity outsid the box, ind
the energy eigenvalues and eigenfunctions.

AS the potential infinity the wave function outside the box must he
vo. Inside the box, the Scbriager equation is given by

‘The equation can he separated into thre equations by writing
yee». ANA

164 Guar Machin

Suhsiuing this vals of y and simplifying

2x09)
PAO e no

20, Barone

and

here E = E, + E, + E, Use ofthe houndary condition Xt) = at
t= and the normalration condon pives

Then

voy

EXAMPLE 82 In Example $1, ithe box ita cubical ons of side a drive
expression for energy cipemalues and eigenfunctions. What is the 2r0-piat
scr of the syst? What is the degeneracy of the fast and second excited
“The energy eigenvalues and eigenfunction are
gan, EE (ah er)

wn

Ten

ro-point energy Ey, = ER
Zero poi acres Ey = A
‘The the independent having quanto numbers (1. 1,2.1,2, D, 1,1)
for nn; m have the enersy

‘Three Dinero! Errgy Eiger Probes 165

ww

Epa or

which th fist excited sate and stes old degenerate, The energy ofthe
second exc sae is

Ein" Ea En ©

Is ao à tre dol degencras

EXAMPLE £3 A rigid rotator is constraiod to rotate about a Fed avis
Find ou ts normalized sipenfunctons and cigonvalcs.

As the as of rotation is always along a fixed direction, the rotator moves
articular plane. If his plan taken a Ihe y plnc, @is always 90° and
the wave function y is a fonction of 6 only. The SeluOdinger equation now
reduces to

“Te solution ofthis equation is
YOO) = A ep Gimp), m = 0, 41.22.
“The energy eigenvalues ae given by

Em
5

Nomalied eigenfunetions are

Deren

vo- exp tine, m

EXAMPLE SA Calculate the energy difference hetwecn the stationary states
1= 1 and = 2 ofthe rigid molecule. Use Bobr frequency rule to estimate
the requeney of radiation involved during transition betwcom these wo stats
Suggest a method for determining the bond length of hydrogen molecu

The energy of a rigid rotor iy given by

£

swe 2
Me, ana

here

Moment of inertia 2

A

ere m i the mas of hydrogen atom and rs the bond eng of hydrogen
molecule. Substiating this value of 7

a

EXAMPLES Solve the time-independent Schrdinger equation for a thse
dimensional harmonic oscillator whose patel encrgy ls

vo La ekg? + ke)
The tory we devloped for cr harmonic nie can ey be

extended t the eats of tree dimensional our. The Seridingr equation
for the system is

Loria rre
‘isin nh pt e en rg wa on
ws) son 20

ue Shigeo ow sp i etn he un

exo

alte)

here 6, + 6, +, E the total energy ofthe system, and

Tiros Dimension Era gente Probleme 167

Using se of sa amo ir cm 4)
Pelee ees
= (ne Dae,

f= fed

fnetons ae given by

Yan = MEN

„om Goer Ae rel]
ere N the maman cota and
ee.

se of Eu, (192) gives

sn

”

EXAMPLE 56 For the ground state of the hydrogen atom, evaluate the
expeciión vals ofthe radius vector of the sico,
Wave function of the ground state

vos +

Ge [iar de = aie aff seve

‘Te integration over the angular coordinates gives Ar. Using the relation in
Appendix A, the rintgral can be evaluated

Oe Far *

‘The expectation valve of rin the ground state of hydrogen atom is 34/2

EXAMPLE 57 Show thatthe de 2peigenfunctions of hydrogen sto are
rhogonal cach ar

We have the lee 2reigenfuntions (Eq, S6Re and 5.680)

re cs

Yon

168 Cuarta Macnee

and
ves = ar sinne

whore can re constants, The dependent par of Y, Va, 8 € 2

“The corespondin 6 integral becomes

The diner of [vio vou sen [4 4920

Tus ong fia

llores the ee 2p sigenfoncsions of hydrogen atom are orbogonal 10 ach
other

EXAMPLE 58 Prove that the Is, 2p and Arial of a hydrogen Hike atom
show à single maximum inthe radial probability curves. Obtain the values at
which these maxima occu.

rail probability dens

P= PRP, where

Rs Conan) op [-)

Ra ems rep (2)

Rue toma (2)

Lt be maximum shen dP für = 0. Therefore

co coms (rev (2

at

Beno sives

a
In the same way

mm [2

In pene Fg = E

Note The rol = 7 sage le sia o rom shen ropero
da e bee ae mabe.

‘Thee Dinero! Errgy Eiger Pobors 109

EXAMPLE 59 IF the
wold Be ls ground ta

Helium atom has two elctons and Z
wave function of hydrogenlike atom are

role pushen in liu is ignora, what
€ energy and wave funcion?

2. The ground-state energy and

Zn = 13622 ev
2

and

no

Win the imercletoni repulsion is glas the eneray ofthe system the
sum of he enrgis of the uo clcetrns and the wave funtion is the product
ofthe two funcion. Then

Energy E 2 1362 13.62

Wane nin ve vn = LE] o 402]
tera dis ves of Sam ad? pea

EXAMPLE 5.10 Evaluate the most probable distance of the cleo ofthe
Iydropen atom in its 2p state, What isthe radial probability at tht distance?

The radial probability density 20) = 7 ARE, whore

)

arts)

mel Se

Therefor,

For Py, to be maximum

rr = day, Ths most probable distance i four times the Bohr rains,

EXAMPLE 5.1 A posiuon and an clecron form a sborlived atom called
positronium betore the two annihilate to produce gamma rays. Calculate in
"Seton volt, the ground. sto energy of poston

170 Cuarta Machin

"The positon Nas à charge 4e and mass equal to the clecron mas. The
sassy inthe encray expression of hydrogen atom (Eg. 55) 1 he reduced
‘mass which for postronium atom is

mm

2

et, m, isthe electron mass. Hence the energy of the poshronium stom is
Bull the energy of hydrogen atom:

ne 23
ar
“Te ground tate energy I hen
288 ey a6 8ev

EXAMPLE 5.12 A mesie atom is formed by a muon of mass 207 tines the
electron mas, charge e and the hydrogen muccus. Calcula () the energy
levels ofthe mesic stom (1) raus of the mesic atom and (i) wavelength of
the 2p > Ls transition,

(6) The sytem i similar to that oF hydrogen atom, Hence th energy levels
are given by
we

CES

12,3,

her Js the proon-mun reducod mass which i 186 m,
(i) Rais ofthe mesic atom wil also be Similar 1 tht of Bohr ads,

1.0119)
La. e898 10 N mic? (Lg 1:16)
hue
AS 1 1

Bossi Naco HERO aro

22882210 m 280201070 m

32 fa

CE

EME x 10° Nic 2} 086x0.1x10 (LS x 10 PO
20010

= 19033 0

Tiros Dinero! Eng gene Palome_171

De (5625 10" 19 8210 ms)
EE amraxıo

065278 x 10" m 04653 am

EXAMPLE S13 A paniclof mass in a spherically symmetric atractivo
potential of aus a Find the minimum depth ofthe potential needed to have
no bound states of ero angular momentum.

‘The spherically symmetric ancre Potential is defined by

vo

as

When the parle is baving zero angular momentum, the radial equation is
sven by Eqs. (595) and (590),

O<r<a

For solution refor seton 57. For one hound stats the depth Y, is gin by
Ba (5.103)

EXAMPLE $14 A rigid routor having moment of nr 1, rotates fesly
in de ay plan. IQ isthe angle between the axis and he rar as ind
(ye energy cgenvalues and eigenfunctions (i) the angular speed i) ya)
fort > OH YO) = À eos? @

The eneryy eigenvalues and eipenfuntions (refer Example 53) ae

=e
El

spine), m= 0,

2.

vo-a pto somo

172 Cuarto Machin

von

‘The fs tem correspond tom =D. In the second term, one fxm corresponds
tom = 2 and the other one to m = 2,

9) The angular spood $ is given by

iy yo

A

ll imftos)

EXAMPLE SS A panicle of mass m is confined t he inter 08 a all
spheial cavity of ads A, with impenetrable walls. Find the pressure exeted
où the walls of the avity by the panicle in ts ground state

The taal wave equation with Vi) = 0 is
LL (ait) [int ta)

Pa a) Tr

For the ground state, = 0, Writing

a

the rial equation reduce to [eter Eq. (SID

Rn=

ame,

ER :
Er Bear: ree
ALES

‘The solution oF this equation is:

2 A sin Br à Boos br. A and B are constants

‘is finite a = 0, That i a r= 0, x= Ar 0, This leads to B= 0, Hence
2= A sin kr
ar Ry gives

= Asin KR,

“The condition that R

Tiros Diners Enea gerade Probar 173

AS A cannot he ero,
Akane or be, nel

ones the solution i

Normaliation gives

&

withthe condition that

LL ay, am). de
Be

E
‘The parce i in its ground state, Hence n = 1 and

an.
CE
‘The pressure exer on the walls:
ae A
Po aan Sma

EXAMPLE $46 Attimo

0, the wave Function for hydrogen atom is

(2% as + Pao + VE Pa + a)

her the subscripts ate values of dhe quantum numbers, I m

(69) What is the expectation value forthe energy of the system?
16) What i th probably of Finding the system with T= I,m = 17

16) The expectation value oF the emery of the system
(e)= (ve)

ROM + o + a O
o + + + VM)

174 Cuarto Machin

Eo + Va + Van + SH)
RE + Es go + VIEW, + BE)

(46, +66)

6) The required probaly is given by

2 2
AS

EE QUESTIONS Jam

1. Wat re spherical harmonies? A the mutually orthogonal?
2 Are the rig rotator energy levels degenerne? Explain

3. Assuming he oxygen molsculoto he rad, give a method of dstemining
its bond length

. List he quantum numbers required to specly the sate of the electron in
the hydrogen atom. What ate thei allowed values?

. What is angular momentum quantum number? Why’ si called 30?

What is magnetic quantum number? Explain ts significance

What ae atomic orbitals? Explain some ofthe uses of atomic orbitals

Y What ate p and d-orials? Give polar representations of their angular
pan

9. For what stats of the hydrogen atom, ihe ecto
distihution sphcrially symmetric

10. Name the quantum system for which tb spacing between adjacent energy
levels) increases as energy increas, (i) nereases as energy decrease
and (i) remains constant a6 nergy increases,

11. What psiuoaium? Why isits energy cigenvalues ae half of the energy
Sigenvalucs of the hydrogen stom?

12. A negative muon can take the place of electron around a les, Muonic
ons have bydeogeclke spectra with energy eigenvalues higher by a
factor of about 20027, Explain.

13. Ualike the finite square-wel potential, the Coulomb potential gives an
Infinite number of bound states. Why?

probability density

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