Physics 452

Quantum mechanics II

Winter 2012

Karine Chesnel

Phys 452

Homework

Friday Mar 9: assignment # 14

Pb 9.1, 9.2, 9.5, 9.6

Tuesday Mar 13: assignment # 15

Pb 9.7, 9.18, 9.19

Two- level systemsTime- dependent perturbation

Phys 452

aE

bE ' 'ij i jH H

In case ofoff-diagonalperturbation

' ' 0aa bbH H

0' i taab b

dc iH e c t

dt

0' i tbba a

dc iH e c t

dt

/ /( ) a biE t iE ta a b bt c t e c t e

Two- level systemsTime- dependent perturbation

Phys 452

aE

bE

0 'H H H

Perturbed state:

/ /( ) a biE t iE ta a b bt c t e c t e

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

Goal: solve for ca and cb

Initial conditions: (0) (0) 1a ac t c

(0) (0) 0b bc t c

0

10' i tb

ba a

dc iH e c t

dt

0

10' i ta

ab b

dc iH e c t

dt

First 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 ' '' " '

tti t i t

a ab bac t H t e H t e dt dt

(2) (1)b bc t c t

Two- level systemsTime- dependent perturbation

Phys 452

aE

bE

(0) 1ac t (0) 0bc t Initial conditions

Application: sinusoidal perturbation

' cosH V t

0 '(1) '

0

' 't

i tb ab

ic t H t e dt

First order

Probability of transition:

2 20

22

0

sin / 2ab

a b

tVP t

Two- level systemsTime- dependent perturbation

Phys 452

aE

bE

Sinusoidal perturbation ' cosH V t

Probability of transition:

2 20

22

0

sin / 2ab

a b

tVP t

t

P(t)for a given frequency

Two- level systemsTime- dependent perturbation

Phys 452

aE

bE

Sinusoidal perturbation ' cosH V t

Probability of transition:

2 20

22

0

sin / 2ab

a b

tVP

P()for a given time t

0

Two- level systemsTime- dependent perturbation

Phys 452

aE

bE

Pb 9.7/2' i tH Ve

Probability of transition: 2 2

2 2

sin / 2raba b

r

tVP t

P()

'0

22

0

1/

2r abV

Rabi flopping frequency

Solving exactly for

Time- dependent perturbationPhys 452

Pb 9.1

'H eE t z

Electric field perturbation

n=1

n=2 'ij i jH eE t z

,mi nlm nl lR r Y

Simulation of atomic dipole transitionwww.falstad.com/qmatomrad/

Time- dependent perturbationPhys 452

Pb 9.2Apply this theory

for time-independent perturbation

0' i taab b

dc iH e c t

dt

0' i tbba a

dc iH e c t

dt

2

2...bd c

dt Second order

differentialequation

2 21a bc t c t

Transition between two states still possible

Time- dependent perturbationPhys 452

Pb 9.6Apply this theory

for time-independent perturbation

0' i taab b

dc iH e c t

dt

0' i tbba a

dc iH e c t

dt

Use the perturbation theoryAnd integrate at second order

0 '(1) '

0

' 't

i tb ab

ic t H t e dt

0 0'' "(2) ' '

2 00

11 ' '' " '

tti t i t

a ab bac t H t e H t e dt dt

Compare with the exact answer of problem 9.2

Time- dependent perturbationPhys 452

Pb 9.5Two levels system

with different initial conditions

0' i taab b

dc iH e c t

dt

0' i tbba a

dc iH e c t

dt

0ac a

Initial conditions

0bc b