Time Dependent Perturbation Theory

The presentation is about how to evaluate the probability of finding the system in any particular state at any later time when the simple Hamiltonian was added by time dependent perturbation. So now the wave function will have perturbation-induced time dependence.

1. Time-Dependent
Perturbation Theory
Prepared by: James Salveo L. Olarve
Graduate Student


January 28, 2010

2. Introduction
The presentation is about how to evaluate the
probability of finding the system in any particular state at
any later time when the simple Hamiltonian was added
by time dependent perturbation. So now the wave
function will have perturbation-induced time
dependence.
The intended reader of this presentation were physics
students. The author already assumed that the reader
knows dirac braket notation.
This presentation was made to facilitate learning in
quantum mechanics.

3. ()tVHH +=
o
()tV
Time-Dependent Perturb Hamiltonian
We look at a Hamiltonian
is some time-dependent perturbation in which we assumed
to be small compared to the time-independent part.
here
nEnH
n
=
o
unperturbed eigenvalue equation
Note: We label here not since for a time-dependent
Hamiltonian, the energy will not be conserved. Therefore energy
corrections are futile to solve
n
E
o
n
E
Now, even for V=0, the wave functions have the usual time dependence
() nec
tiE
n
n
t
n/-
S=y
sc
n
n
'Swith as constant

4. On introducing the acquire time dependence
Time-Dependent Perturb Hamiltonian
()tV sc
n'
() () netc
tiE
n
n
t
n/-
S=y
This time dependence can be determined by Schödinger’s Equation
with ()tVHH +=
o
() ()( ) ()åå
--
+=
¶
¶
n
tiE
n
n
tiE
n netctVHnetc
t
i
nn


// o
() () () ()( )ååå ·+=·
ú
û
ù
ê
ë
é
¶
¶
+
¶
¶
---
n
tiE
n
tiE
n
n
n
n
tiE
ntVHetcne
t
tctc
t
ei
nnn o

///
() () () ()( )ååå ·+=·
ú
û
ù
ê
ë
é
÷
ø
ö
ç
è
æ-
+
---
n
n
tiE
n
tiEn
n
n
n
n
tiE
ntVEetcne
iE
tctc
dt
d
ei
nnn 


///
() () () () ()åååå
----
·
=-+
n
tiE
n
n
n
tiE
n
n
n
tiE
n
n
tiE
n ntVetcnEetcnEetcnetci
nnnn 

////

5. Time-Dependent Perturb Hamiltonian
Now, () () () netctVnetci
tiE
n
n
tiE
n
n
nn


// --
·
åå =
Taking the inner product with the bra
and introducing
/tiE
m
em

nm
mn
EE-
=w
()
( )
() ()
( )
netctVmnmetci
tnEmEi
nm
n
n
tEEi
n
n



/
/
-
åå =
-
·
() () ()ntVmetcci
ti
n
n
m
mn
w
å=
·

() ()
mn
ti
n
n
m Vetcci
mn
w
å=
·


6. Time-Dependent Perturb Hamiltonian
This is a matrix differential equation for the sc
n
'
and solving this set of coupled equations will give us the , and
hence the probability of finding the system in any particular state at any
later time.

7. If the system is in initial state at t = 0, the probability amplitude
for it being in state at time t is to leading order in the perturbation
Time-Dependent Perturb Hamiltonian
i
f
The probability that the system is in fact in state at time t is therefore f
Obviously, this is only going to be a good approximation if it predicts
that the probability of transition is small—otherwise we need to go to
higher order, using the Interaction Representation

8. Reference:
Retrieved from
http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/Time_Ind_PT.htm
, January 19 2010, Michael Fowler.
Introduction to Quantum Mechanics. David
J. Griffiths. 1994