Physics 452

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