Chapter 6. Time-Dependent Schrodinger Equation

(1)

Nonlinear Optics Lab

. .

Hanyang Univ.Hanyang Univ.

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 E_m to another energy E_n. A change from one orbit to another can occur if an external time-dependent force F_ext 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 (

F_ext t  V_ext 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)

(2)

. .

6.2 Time-Dependent Solutions

Time-independent Schrodinger equation ; H_a_n  E_n_n

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

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 t

i a V

E

a 



 







^

(6.2.2)

(6.2.3)

(3)

. .

mn n

m n

m

m      d 





 ⁽^r⁾ ^&



^, ^|



⁽^r⁾ ⁽^r⁾ ³^r

space all

* space

all

 

^ ^







n

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, V_mn(t) 



^*_m(r)V_ext(r,t)_n(r)d³r

: time-dependent Schrodinger equation

<Meaning of : probability amplitude>a_m 1

) , ( ) ,

( ³

*  



 ^r ^t ^r ^t ^d ^r _m ^a^m ^m _n ^aⁿ ⁿ ^d³^r

*



 



 



 



 



  

 

 

^ ^ ^ ^ ^ ^ ^



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)

(4)

. .

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⁽^k^^R^^^t⁾  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^{ t}ⁱ^ For a two-state system,

) r ( )

( )

r ( ) ( )

, r

(  ₁ ₁  ₂ ₂

 t a t a t

(6.2.8)  ia₁(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

 ia₁(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 )

(5)

. .

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   



^ⁱ^^t 

.) . ˆE

2 ( r 1 )

( ₂₁ ₀

21 t e e c c

V   



^ⁱ^^t  where, ^r¹² ^



^¹^*⁽^r⁾^r^²⁽^r⁾^d³^r

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  



^ⁱ^^t 



ⁱ^^t

) ( ) 2 (

) 1

( ₂₁ ₂ ₂₁ ₁₂^* ₁

2 t a e e a t

a

i 







^ⁱ^^t 



ⁱ^^t : Rabi frequency (field-atom interaction energy in freq. unit) (6.3.11)

(6)

. .

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

* 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

t c i

c t

c i









 _where,  



₂₁ ^{: detuning}





0 21

21 ( r ˆ) E

e





  

: Rabi frequency

(7)

. .

Solution) initial condition ; c₁(0) 

1, c₂(0)  0











 



 

 



 



 



 

 









2 / 2

2 / 1

sin 2 )

(

sin 2 cos 2

) (

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

1

t t

P

t t

P



(8)

. .

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

| a  a  a a  a a



where, ^r^ij ^



^^*ⁱ ⁽^r⁾^r^ ^j⁽^r⁾^d³^r

(9)

. .

. . r

r r

r  ₁₂a₁^*a₂  ₂₁a₂^*a₁  ₁₂a₁^*a₂ cc



For a case of linear polarization,



ˆE₀ is real.

0 r

, 0

V_ii  _ii 

)

|

| (|

) (

) 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^*a₂  a₁a₂^*)

)

|

| )(|

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)

. .

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

1  R

 

 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

zˆ

₂₁

(11)

. .

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 z  a



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)

. .

Classic



Quantum mechanics

m

e²



²^e² 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)

. .

6.5 Density Matrix and (Collisional) Relaxation

Two level system, time-dependent Schrodinger equation, )

r ( )

( )

r ( ) ( )

, r

(  ₁ ₁  ₂ ₂

 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

* 21 1

2 ) 1

(

2 ) 1

(

c c

t c i

c t

c i





(6.3.14)

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

(14)

. .

Define,



₁₂  c₁c^*₂

* 1 2 21  c c



2 1

* 1 1

11  c c | c |



2 2

* 2 2

22  c c | c |



* 1 2

1

* 2 2

*

* 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

   



  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)

. .

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

of relaxation such as collision.





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)

. .

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)

. .































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)

. .

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)

. .

<Special case> ₁  ₂  0,  

0



















) 2 (

A

) 2 (

A

21

* 12

22 21 22

21

* 12

22 21 11















i i





















) 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)

. .

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)