Download Creation of Colloidal Periodic Structure

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 : Fext (r, t )  Vext (r, t ) ,
and the system’s total Hamiltonian can be given by
 2
(6.1.1)
H  H a  Vext (r, t )  
  V (r )  Vext (r, t )
2m
The Schrodinger equation becomes

  2

  V (r )  Vext (r, t )  (r, t )  i

t
 2m

Nonlinear Optics Lab.
(6.1.2)
Hanyang Univ.
6.2 Time-Dependent Solutions
Time-independent Schrodinger equation ; H a  n  En  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 , t )   an  n ( r )
(6.2.1)
n
(6.1.2) =>
 an [ H a  Vext ]  n (r)   i
n
n
 an [ En  Vext ]  n (r)   i
n
n
an
 n (r )
t
(6.2.2)
an
 n (r )
t
(6.2.3)
Nonlinear Optics Lab.
Hanyang Univ.
  m (r ) &

,  m | n  
all space
*
3

(
r
)

(
r
)
d
 m n r   mn
all space
 ia m  Em am   an   *m (r )Vext  n (r )d 3 r
n
 iam  Em am  Vmn (t )an : time-dependent Schrodinger equation
n

*
3
where, Vmn (t )   m (r )Vext (r, t ) n (r ) d r (6.2.7)
<Meaning of am : probability amplitude>
  (r , t )(r , t )d r  1 
*
3
*


 3
a

a





m
m
n
n d r

m
 n

Probability that the
quantum system is
in its m-th orbit.
  am*  an   m |  n    am* an mn   | am |2  1
m
n
m
n
m
Nonlinear Optics Lab.
Hanyang Univ.
6.3 Two-State Quantum Systems and Sinusoidal External Forces
Time-dependent potential for the interaction between an EM field and an electron ;
Vext (r, R, t )  er  E(R, t ) : dipole approximation
For a monochromatic wave,
1
ˆ
E(R , t )  E 0 cos( k  R  t )  ˆE 0 ei ( kR  t )  c.c.
2
1
Put, R  0  ˆE 0 e i t )  c.c. (6.3.1)
2
For a two-state system,
(r, t )  a1 (t )1 (r)  a2 (t )2 (r)
(6.2.8)  ia1 (t )  E1a1 (t )  V11a1 (t )  V12a2 (t )
ia2 (t )  E2a2 (t )  V21a1 (t )  V22a2 (t )
0

ia1 (t )  E1a1 (t )  V12a2 (t )
ia2 (t )  E2a2 (t )  V21a1 (t )
Nonlinear Optics Lab.
Hanyang Univ.
(6.3.4)
Normalization condition ; | a1 (t ) |2  | a2 (t ) |2  1
(6.2.7), (6.3.1) =>
1
V12 (t )  er12  (ˆ E 0 e it  c.c.)
2
1
V21 (t )  er21  (ˆ E 0 e it  c.c.) where, r12   1* (r ) r  2 (r )d 3r
2
Define,
Set, E1  0
(6.3.4) =>
E2  E1
 21 
1
 i t
* i t


i
a
(
t
)


(

e


e )a2 (t )
1
12
21
E0
2
 21  e(r21  ˆ)

1
 i t
* i t

i
a
(
t
)


a

(

e


e )a1 (t )
2
21
2
21
12
E0
2
12  e(r12  ˆ)

(6.3.11)
: Rabi frequency (field-atom interaction energy in freq. unit)
Nonlinear Optics Lab.
Hanyang Univ.
i)   0 : E 0  0 ( radiation field=0)
 a1 (t )  a1 (0)  const.

a2 (t )  a2 (0) exp[ i 21t ]
ii)    21 ( nearly resonant radiation field)
Neglected by
rotating-wave approximation
1

 2 i t
*

i
c
(
t
)


(

e


)c2
1
12
21

2
 a1 (t )  c1 (t )
(6.3.11) => 

1
i t
* 2 i t
a
(
t
)

c
(
t
)
e


i
c
(
t
)

(



)
c

(



)c1
2
 2
2
21
2
21
12e
2

1 *


i
c
(
t
)


 21c2
1
where,   21   : detuning

2

E0
1
1
ˆ




(
e
r


)
ic2 (t )  ( 21   )c2   21c1   c2   c1
21
21

2
2

: Rabi frequency
trial solution,
Nonlinear Optics Lab.
Hanyang Univ.
Solution) initial condition ; c1 (0)  1, c2 (0)  0

t

t  it / 2

c
(
t
)

cos

i
sin

e
1


2

2 


t  it / 2
 

c2 (t )   i sin
e

2 
 

where,   (  2  2 )1/ 2 : Generalized
Rabi frequency
Probability ; P1 (t ) | a1 (t ) |2 , P2 (t ) | a2 (t ) |2
2
2

1    1  
 P1 (t )  1        cos t
2      2   


2
1


 
P
(
t
)

  [1  cos t ]
2

2

Nonlinear Optics Lab.
Hanyang Univ.
6.4 Quantum Mechanics and the Lorentz Model
- Lorentz (classical) model can’t give the oscillator stength, f
- Why the classical model offers good explanation for a wide variety of phenomena ?
Basic dynamic variable for an atomic electron : Displaceement, x in classical model,
Corresponding quantum displacement : expectation value,  r 
 r    * (r, t ) r  (r, t ) d 3 r
For the two-state atom,
 r   (a1*1*  a2* *2 ) r (a1 1  a2  2 ) d 3 r
| a1 |2 r11  | a2 |2 r22  a1*a2 r12  a2*a1r12

where, rij   *i ( r ) r  j (r ) d 3 r
Nonlinear Optics Lab.
Hanyang Univ.
Vii  0, rii  0
 r  r12a1*a2  r21a2*a1  r12a1*a2  c.c.
For a case of linear polarization, ˆE 0 is real.
d *
(a1 a2 )  i ( E2  E1 )a1*a2  iV21 (| a1 |2  | a2 |2 )
dt
2
d
  2 2 (a1*a2 )  i( E2  E1 ) 2 a1*a2  iV21 (| a1 |2  | a2 |2 )
dt
d
 i [V21 (| a1 |2  | a2 |2 )]
dt
Since r12 is real,  r  r12 (a1*a2  a1a2* )
(6.3.4)  
 d2
2e0
2
 2  0   r 
r12 (r21  E)(| a1 |2  | a2 |2 )

 dt

E2  E1
where,  0 

Nonlinear Optics Lab.
Hanyang Univ.
If we assume, | a2 |2  1 & | a1 |2  1
 d2

2e0
 2  0 2   r 
r12 (r21  E)

 dt

Suppose the E-field points in the z-direction, E  ẑE
 d2
2e0
2
 2  0   r 
r12 z21 E)

 dt

example) Let atomic state 1 and 2 be the 100 and 210 (1S and 2P)
1 (r)  R1, 0 (r)Y00 ( ,  )
 2 (r)  R2,1 (r)Y10 ( ,  )
z 21      *2 (r ) z 1 (r ) d 3r

2 
0
0 0
  r 3 R2*,1 (r ) R1,0 (r ) dr
r21
ẑ 21
*
Y
1
  ,0 ( , ) sin  cos Y0,0 ( , ) dd
Nonlinear Optics Lab.
Hanyang Univ.
Homework : Appendix 5.A !
Table 6.1, 6.2,

2
3 / 2  r   r / 2 a0 3  r / a0
e
r21  (2a0 )
(2a0 )  
re
dr  1.29a0
2a0 
3
0
2
 : Bohr radius
where, a0  40

0
.
53
A
me2
2

1 3
1
2
ẑ 21 
d

cos

sin

d


4 4 0 0
3
3 / 2
 z 21  r21zˆ21  0.745 a0
 d2
2e0
2e0
2


  2  0   r 
ẑ z12z 21E 
(z12 ) 2 ẑE


 dt

 d2
e
2
cf) 



x

ẑE in classical model
0 
 dt 2
m


Nonlinear Optics Lab.
Hanyang Univ.
Classic
2
e
m


Quantum mechanics
2e 2 0 2
z12

Oscillator Strength :
2
2
e
e
(3.7.5)

f
m
m
 f 
2m 0 2
z12

example) Hydrogen n=1 => n=2,
 , f  0.416 (Table 3.1)
  1216 A
2  3 108
2  9.110 
10
1216

10
f 
 0.417
34
1.054 10
31
Nonlinear Optics Lab.
Hanyang Univ.
6.5 Density Matrix and (Collisional) Relaxation
Two level system, time-dependent Schrodinger equation,
(r, t )  a1 (t )1 (r)  a2 (t )2 (r)
 a1 (t )  c1 (t )

i t
a
(
t
)

c
(
t
)
e
2
 2
(6.3.2)
(6.3.12)
1 *


i
c
(
t
)


 21c2
 1
2

(6.3.14)
1
ic2 (t )   c2   c1
2

*
*
Via (6.4.3),  r  r12a1*a2  c.c., the combination variable a1 a2 and a1a2
are more useful than either a1 or a2 alone.
Nonlinear Optics Lab.
Hanyang Univ.
*
Define, 12  c1c2
 21  c2c1*
11  c1c1* | c1 |2
 22  c2c2* | c2 |2
1
1
 12  c1c2*  c1c2*  i (  *c2 )c2*  c1 (ic2  i c1 )*
2
2
 i12  i
similarly,
*
2
(  22  11 )

 21  i 21  i (  22  11 )
2
i
* 11, 22 : level' s occupation probabilit y
11   ( 12   *  21 )
2
(population )
i
* 12 , 21 : complex amplitude
 22  ( 12   *  21 )
2
of the electron' s displaceme nt,  r 
Nonlinear Optics Lab.
Hanyang Univ.
The equations are not yet in their most useful form, since they do not reflect the existence
of relaxation such as collision.
<Relaxation Processes>
elastic collision : oscillatio n phase change
- collision 
 inelastic collision : decay to other levels
- spontaneou s emission : decay from 2 to 1 level
1) Elastic collision effect
* 11,  22  const.  only change the 12 ,  21.
* if the radiation field is steady.    const.
* Collision occurs at t 1 , and  21 (t ) |t t1  0
(6.3.14) =>
 21 (t )  
 (  22  11 )
2
(1  e i ( t t1 ) )
Nonlinear Optics Lab.
Hanyang Univ.
Average value
  21 (t )  
 (  22  11 )
2
t
 dt e
 ( t t1 ) / 
1
(1  e
i ( t t1 )
)

 (  22  11 )
2
1
  i /
This result can also be reached by a simple modification of the original equation of motion ;

i

1

  21  (  i)  21  i (  22  11 )

2
Similarly,
1
*
12  (  i) 12  i (  22  11 )

2
Nonlinear Optics Lab.
Hanyang Univ.
2) Inelastic collision effect and Spontaneou s emission
i) 11 ,  22







(  22 ) col  2  22
(  22 )spon  A 21  22
( 11 ) col  1  22
2
2
A21
1
( 11 )spon   A 21 22
1
i
2
i
 22  (2  A 21 )  22  ( 12   *  21 )
2
*
(6.5.2) => 11  1 11  A 21  22  ( 12    21 )
Nonlinear Optics Lab.
Hanyang Univ.
ii) 12 ,  21
By definition (6.5.1) [ 12  c1c2* ,  21  c2 c1* ],
each effect on 11 and  22 contribute s to 12 or  21 evenly.
 average of the two effects
(6.5.2) =>
12  (   i) 12  i
 21  (   i)  21  i
where,

*
2

2
(  22  11 )
(  22  11 )
1
1
 (1  2  A 21 ) : total relaxation rate
 2
Nonlinear Optics Lab.
Hanyang Univ.
<Special case> 1  2  0,   0
i
 
*


A


(



 21 )
21 22
12
 11
2

i
  22  A 21 22  ( 12   *  21 )
2

 11   22  0  11  22  1
No dynamic information !

*
(  22  11 )
 12   12  i
2

  21   21  i  (  22  11 ) ,   1  1 A
21
2

 2
So, we can pay attention solely to the differences, 22  11, 12  21
v  i(  21  12 )

 w   22  11
Nonlinear Optics Lab.
Hanyang Univ.
- Assume that  is real, (& 1  2  0,   0)







   i (  21  12 )  i  21  i (  22  11 )  12  i (  22  11 ) 
2
2


  v  w
w   22  11   A21 22 
i
i
 ( 12   21 )  A21 22   ( 12   21 )
2
2
  A21( 22  1  11)  v   A21(1  w)  v
(Chapter 8 : Bloch equation)
The notation used for
 's
 11

  21
12 
 22 
: density matrix
Nonlinear Optics Lab.
Hanyang Univ.
Nonlinear Optics Lab.
Hanyang Univ.