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Chapter 6. Time-Dependent Schrodinger Equation

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Chapter 6. Time-Dependent Schrodinger Equation

6.1 Introduction

Energy can be imparted or taken from a quantum system only if the system can jump from one energy Em to another energy En. A change from one orbit to another can occur if an external time-dependent force Fext acts on the quantum system.

We can associate this force with a new potential energy : , and the system’s total Hamiltonian can be given by

) , r ( )

, r (

Fext t  Vext t )

, r ( V )

r 2 (

) , r ( V H

H ext 2 V ext t

t m

a      

  (6.1.1)

The Schrodinger equation becomes

i t t

t

m V

 



 

    (r) V (r, ) (r, ) 

2 ext

2 (6.1.2)

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6.2 Time-Dependent Solutions

Time-independent Schrodinger equation ; HanEnn

n: Complete & Orthonormal => Any function can be expressed by the n's

*

Following Dirac, the exact time-dependent wave function can be expressed by a sum of ;n's

) r ( )

, r

(

n

n

an

t (6.2.1)

(6.1.2) => [ ext] (r) n(r)

n

n n

n

a

n t

i a V

H

a

 

) r ( )

r ( ]

[ ext n

n

n n

n

n

n t

i a V

E

a

 

(6.2.2)

(6.2.3)

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mn n

m n

m

m      d

(r) &

, |

(r) (r) 3r

space all

* space

all

 

n

n m

n m

m

m E a a V d

a

i * (r) ext (r) 3r

n

n mn

m m

m E a V t a

a

i ( )

where, Vmn(t) 

*m(r)Vext(r,t)n(r)d3r

: time-dependent Schrodinger equation

<Meaning of : probability amplitude>am 1

) , ( ) ,

( 3

*  

r t r t d r m am m n an n d3r

*



 

 



 

 

  

 

 

m n m

m mn

n m

m n

n m

n

m a a a a

a* | *

| |2 1

Probability that the quantum system is

in its m-th orbit.

(6.2.7)

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6.3 Two-State Quantum Systems and Sinusoidal External Forces

Time-dependent potential for the interaction between an EM field and an electron ;

c.c.

ˆE 2 ) 1 R

k cos(

ˆE )

,

E(R t

0  

t

0ei(kRt)  0

R 

) , R ( E r )

, R r,

ext( t e t

V    : dipole approximation

For a monochromatic wave,

Put, ˆE c.c.

2

1 )

0

e ti For a two-state system,

) r ( )

( )

r ( ) ( )

, r

(  1122

t a t a t

(6.2.8)  ia1(t)  E1a1(t)V11a1(t) V12a2(t) ) ( )

( )

( )

( 2 2 21 1 22 2

2 t E a t V a t V a t

a

i   

0

ia1(t)  E1a1(t)V12a2(t) ) ( )

( )

( 2 2 21 1

2 t E a t V a t

a

i  

(6.3.1 )

(6.3.4 )

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Normalization condition ; | a1(t)|2  | a2(t)|21 (6.2.7), (6.3.1) =>

.) . ˆE

2 ( r 1 )

( 12 0

12 t e e c c

V   

it

.) . ˆE

2 ( r 1 )

( 21 0

21 t e e c c

V   

itwhere, r12

1*(r)r2(r)d3r

Define,

1 2

21

E E

0 21

21

) E r ˆ

(

 e

0 12

12

) E r ˆ

(

 e

Set, E1  0(6.3.4) =>

) ( ) 2 (

) 1

( 12 21* 2

1 t e e a t

a

i  

it

it

) ( ) 2 (

) 1

( 21 2 21 12* 1

2 t a e e a t

a

i 

it

it : Rabi frequency (field-atom interaction energy in freq. unit) (6.3.11)

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0

E :

0 0

i) ( radiation field=0)



] exp[

) 0 ( )

(

const.

) 0 ( )

(

21 2

2

1 1

t i

a t

a

a t

a

21

ii) ( nearly resonant radiation field) trial solution,



 ti

e t c t

a

t c t

a

)

( )

(

) ( )

(

2 2

1

1 (6.3.11) =>





1 2

* 12 21

2 21

2

2

* 21 2

12 1

) 2 (

) 1 (

) (

) 2 (

) 1 (

c e

c t

c i

c e

t c i

t i t

i

Neglected by

rotating-wave approximation





 

1 2

1 21 2

21 2

2

* 21 1

2 1 2

) 1 (

) (

2 ) 1

(

c c

c c

t c i

c t

c i

where,  

21: detuning

0 21

21 ( r ˆ) E

e

  

: Rabi frequency

(7)

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Solution) initial condition ; c1(0) 

1, c2(0)  0







 

 

 



 

 

 

 

2 / 2

2 / 1

sin 2 )

(

sin 2 cos 2

) (

t i

t i

t e i

t c

t e t i

t c

where, (

2 2: Generalized )1/2 Rabi frequency

Probability ; P1(t) | a1(t)|2, P2(t) | a2(t)|2





 

 

 

 

 

 





 

 

 

] cos 1

2 [ ) 1 (

2 cos 1 1

2 ) 1 (

2 2

2 2

1

t t

P

t t

P

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6.4 Quantum Mechanics and the Lorentz Model

- Lorentz (classical) model can’t give the oscillator stength,

- Why the classical model offers good explanation for a wide variety of phenomena ? f Basic dynamic variable for an atomic electron : Displaceement, in classical model, Corresponding quantum displacement : expectation value,

x

 r



 r *(r,t)r (r,t)d3r

For the two-state atom,



 r (a1* 1* a2* *2)r(a1 1 a2 2)d3r

12 1

* 2 12

2

* 1 22

2 2 11

2

1 | r | | r r r

| aaa aa a

where, rij

*i (r)r j(r)d3r

(9)

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. . r

r r

r  12a1*a221a2*a112a1*a2cc

For a case of linear polarization,

ˆE0 is real.

0 r

, 0

Viiii

)

|

|

| (|

) (

) (

) 4 . 3 . 6

( a1*a2 i E2 E1 a1*a2 iV21 a1 2 a2 2 dt

d     

 

)

|

|

| (|

) (

)

( 1* 2 2 1 2 1* 2 21 1 2 2 2

2

2 2 a a i E E a a iV a a

dt

d     

 

)]

|

|

| (|

[V21 a1 2 a2 2 dt

i d

 

Since r12 is real,  r  r12(a1*a2a1a2*)

)

|

|

| )(|

E r

( 2 r

r 0 12 21 1 2 2 2

2 2 0

2

a e a

dt

d    

 

 

 

 

where,

1 2

0

E E

(10)

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If we assume, | a2 |21 & | a1 |2

1 ) E r

( 2 r

r 0 12 21

2 2 0

2   

 

 

e

dt d

Suppose the E-field points in the z-direction, E  zˆE )

E 2 r

r 0 12 21

2 2 0

2

e z dt

d

 

 

 

example) Let atomic state 1 and 2 be the 100 and 210 (1S and 2P)

) , ( (r)Y r)

( 1,0 00

1R

 

 2(r)  R2,1(r)Y10(

,

) r

) r ( )

r

( 1 3

* 2

21 z d

z



 

d d Y

Y d

R

R



0

2

0 0

0 , 0

* 0 , 1 0

, 1

* 1 , 2

3 (r) (r) r ( , )sin cos ( , )

r

r21

21

(11)

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Table 6.1, 6.2,

0 0

/ 3 r 2 / r 0

2 / 3 0 2

/ 3 0

21 r r 1.29

2 ) r

2 3 ( ) 2 2

(

r e 0 e 0 d a

a a

a  a a

 

where, 4 2 0.53A

2 0

0    

a



me : Bohr radius

 

 

2

0 0

2

21 3

sin 1 4 cos

3 4

zˆ 1 d d

0 21

21

21 ˆ 0.745

z  r za

E zˆ ) z 2 (

E z z 2 zˆ

r 0 12 21 0 12 2

2 2 0

2

e

e

dt

d   

 

 

cf) 2 02 x zˆE

2

m e dt

d  

 

 

in classical model

Homework : Appendix 5.A !

(12)

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Classic

Quantum mechanics

m

e2

2e2 0 z122

(3.7.5)

Oscillator Strength : m f e m

e2 2

 2 0 122

m z

f

example) Hydrogen n=1 => n=2,

1216A, f  0.416 (Table 3.1)

417 . 10 0

054 . 1

10 1216

10 3

10 2 1

. 9 2

34

10 31 8

  

 

 

f

(13)

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6.5 Density Matrix and (Collisional) Relaxation

Two level system, time-dependent Schrodinger equation, )

r ( )

( )

r ( ) ( )

, r

(  1122

t a t a t (6.3.2)



 ti

e t c t

a

t c t

a

)

( )

(

) ( )

(

2 2

1

1 (6.3.12)





 

1 2

2

2

* 21 1

2 ) 1

(

2 ) 1

(

c c

t c i

c t

c i

(6.3.14)

Via (6.4.3),  r  r12a1*a2c.c., the combination variable a1*a2 and aa1 2* are more useful than either a1 or a2 alone.

(14)

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Define,

12c1c*2

* 1 2 21c c

2 1

* 1 1

11c c| c |

2 2

* 2 2

22c c| c |

* 1 2

1

* 2 2

*

* 2 1

* 2 1

12 )

2 ( 1

2 )

( 1 c c c i c i c

i c

c c

c

 

        

   

) 2 ( 22 11

*

12

  

 

i i

similarly,

) 2 ( 22 11

21

21

   

  i i  )

2 ( 21

* 12

11

  

  i

)

2 ( 21

* 12

22

  

i

y probabilit occupation

s level' :

,

* 11 22

) population (

amplitude complex

: ,

* 12 21

 r nt, displaceme s

electron' the

of

(15)

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The equations are not yet in their most useful form, since they do not reflect the existence

of relaxation such as collision.

<Relaxation Processes>



levels other

decay to :

collision inelastic

change phase

n oscillatio :

collision elastic

collision -

0

| ) ( and

, at t occurs Collision

*

const.

steady.

is field radiation

the if

*

. ,

the change only

const.

,

*

effect collision

Elastic 1)

21 1

1

21 12

22 11

t

t t

(6.3.14) => (1 )

2

) ) (

( 22 11 ( )

21 1

t t

e i

t

 

  

level 1

to 2 from decay

: emission s

spontaneou

-

(16)

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Average value

 

/ 1 2

) ) (

1 2 (

) ) (

( 22 11 1 ( )/ ( ) 22 11

21 1 1

e i e

dt t

t

t t i t

t

 

 

 



This result can also be reached by a simple modification of the original equation of motion ;

i

) 2 (

1 )

( 21 22 11

21

   

 

   

  i i

Similarly,

) 2 (

1 )

( 22 11

* 12

12

   

  

i i

(17)

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



22 21 spon

11

22 1 col

11

22 21 spon

22

22 2 col

22 22 11

A )

(

) (

A )

(

) (

, i)

1 2

A21

2

1

(6.5.2) => ( )

A21 22 2 12 * 21

11 1

11

    

    i

) 2 (

) A

( 2 21 22 12 * 21

22

   

     i

emission s

Spontaneou and

effect collision

Inelastic 2)

(18)

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effects two

the of average

evenly.

or to

s contribute and

on effect each

], ,

[ (6.5.1) definition

By

, ii)

21 12

22 11

* 1 2 21

* 2 1 12

21 12

c c c

c

(6.5.2) => ( )

) 2

( 22 11

* 12

12

    

   i i  ) 2 (

)

( 21 22 11

21

    

   i i

where, ( A )

2 1 1

21 2

1   

: total relaxation rate

(19)

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<Special case> 1  2  0,  

0





) 2 (

A

) 2 (

A

21

* 12

22 21 22

21

* 12

22 21 11





i i





) 2 (

) 2 (

11 22

21 21

11 22

* 12

12

 



 



i i

A21

2 1 ,  1 

 

1

0 11 22

22

11     

 

No dynamic information !

So, we can pay attention solely to the differences,

22

11,

12

21



11 22

12

21 )

(

w

i v

(20)

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0, 0) (&

real, is

that Assume

-

1  2   



 

     

 ( )

) 2 2 (

)

(

21

12



21

 

22

11



12

 

22

11

i   i i i

w v

) 2 (

)

2 ( 12 21 21 22 12 21

22 21 11

22

        

       

i

i A A

w  

v w

A v

A

 

   

21( 22 1 11) 21(1 )

(Chapter 8 : Bloch equation)

The notation used for

's



 

 

22 21

12 11

 

: density matrix

(21)

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