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Physics 452
Quantum mechanics IIWinter 2012
Karine Chesnel
Homework
Phys 452
Friday Mar 9: assignment # 149.1, 9.2, 9.5, 9.6
Time-dependent Schrödinger equation
Phys 452
dH idt
How to describe dynamical phenomena ?
1E
2E Quantum jumps…Lifetime of excited states…Oscillation between states…
Simulation of atomic dipole transitionwww.falstad.com/qmatomrad/
Quiz 21
Phys 452
In case of time-independent Schrödinger equation, the solution for the wave function
does not depend on time
A. True
B. False
“Static” potentialPhys 452
dH idt
Eigenstates: n n nH E
General solution: /niE tn n
n
c e
Probability to measure the energy En: 2
nc
Potential V constant in time
Dynamical systemsPhys 452
dH idt
V depends on time
General solution: /niE tn n
n
c t e
Probability to measure the energy En
and to find the particle in state : 2nc t
n
Dynamical systemssolved by
perturbation theory
Phys 452
Time-dependent part
0 ' ( )H H H t
/niE tn n
n
c t e
0 1 2( ) ( ), ( ), ( ),...n n n nc t c t c t c tApproximation
Two- level systemsPhys 452
aE
bE
0a a aH E
0b b bH E
Solution for unperturbed state:
/ /( ) a biE t iE ta a b bt c e c e
Two- level systemsTime- dependent perturbation
Phys 452
aE
bE
Perturbed state:
/ /( ) a biE t iE ta a b bt c t e c t e
0 'H H H
Perturbation theory:
Apply: dH idt
Two- level systemsTime- dependent perturbation
Phys 452
aE
bE
Perturbed state:
/ /( ) a biE t iE ta a b bt c t e c t e
0 ' dH H idt
( ) /' ' b ai E E taa aa b ab
dc i c H c H edt
( ) /' ' b ai E E tbb bb a ba
dc i c H c H edt
Two- level systemsTime- dependent perturbation
Phys 452
' 'ij i jH H
aE
bE
Perturbationmatrix
0' i taab b
dc i H e c tdt
0' i tbba a
dc i H e c tdt
In case ofoff-diagonalperturbation
with 0 /b aE E
Two- level systemsTime- dependent perturbation
Phys 452
aE
bE
Goal: solve for ca and cb
0
10' i tb
ba adc i H e c tdt
0
10' i ta
ab bdc i H e c tdt
At first order:
Approximation:
0 1 2( ) ( ), ( ), ( ),...a a a ac t c t c t c t 0 1 2( ) ( ), ( ), ( ),...b b b bc t c t c t c t
Two- level systemsTime- dependent perturbation
Phys 452
aE
bE
Zeroth order =Initial conditions:
(0) (0) 1a ac t c
(0) (0) 0b bc t c
First order: (1) (0) 1a ac t c t
0 '(1) '
0
' 't
i tb ab
ic t H t e dt
Two- level systemsTime- dependent perturbation
Phys 452
aE
bE
0
21' i tb
ba adc i H e c tdt
0
21' i ta
ab bdc i H e c tdt
Second order:
Two- level systemsTime- dependent perturbation
Phys 452
aE
bE
Zero order: (0) 1ac t (0) 0bc t
First order: (1) 1ac t 0 '(1) '
0
' 't
i tb ab
ic t H t e dt
Second order:
0 0'' "(2) ' '
2 00
11 ' '' " 't
ti t i ta ab bac t H t e H t e dt dt
(2) (1)b bc t c t
Two- level systemsPhys 452
Pb 9.1 Hydrogen atom and electrical field
Calculate ' 'ij i jH H
' ( )H eE t z
Express , , , and100 200 210 211 21 1
Non-zero terms Selection rules forElectric dipolar transition
Two- level systemsPhys 452
Pb 9.2 Application to time-independent perturbation
0' i taab b
dc i H e c tdt
0' i tbba a
dc i H e c tdt
with
0 /b aE E
(0) 1(0) 0a
b
cc