Nonlinear Optics Lab
.
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 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 ( )
, r ( V )
r
2 ( ext
2 (6.1.2)
Nonlinear Optics Lab
.
Hanyang Univ.6.2 Time-Dependent Solutions
Time-independent Schrodinger equation ; Han Enn
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)
Nonlinear Optics Lab
.
Hanyang Univ.mn n
m n
m
m d
(r) &
, |
(r) (r) 3rspace 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 a a d r
n
n n m
m m
3
*
m n m
m mn
n m
m n
n m
n
m a a a a
a* | *
| |2 1Probability that the quantum system is
in its m-th orbit.
(6.2.7)
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 ;
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
eit For a two-state system,) r ( )
( )
r ( ) ( )
, r
( 1 1 2 2
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)
Nonlinear Optics Lab
.
Hanyang Univ.Normalization condition ; | a1(t)|2 | a2(t) |21 (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 cc
V
it where, r12
1*(r)r2(r)d3rDefine,
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)
Nonlinear Optics Lab
.
Hanyang Univ.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,
i t
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
Nonlinear Optics Lab
.
Hanyang Univ.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)1/2 : Generalized Rabi frequencyProbability ; 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
Nonlinear Optics Lab
.
Hanyang Univ.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* a*2 *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 )
d3r
Nonlinear Optics Lab
.
Hanyang Univ.. . r
r r
r 12a1*a2 21a2*a1 12a1*a2 cc
For a case of linear polarization,
ˆE0 is real.0 r
, 0
Vii ii
)
|
|
| (|
) (
) (
) 4 . 3 . 6
(
a1*a2 i E2 E1 a1*a2 iV21 a1 2 a2 2 dtd
)
|
|
| (|
) (
)
( 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 dti 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
Nonlinear Optics Lab
.
Hanyang Univ.If we assume, | a2 |21 & |a1 |21 ) 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)Y
10( , ) 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
21Nonlinear Optics Lab
.
Hanyang Univ.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
edt
d
cf) 2 02 x zˆE
2
m e dt
d
in classical modelHomework : Appendix 5.A !
Nonlinear Optics Lab
.
Hanyang Univ.Classic
Quantum mechanicsm
e2
2e2 0 z122
(3.7.5)
Oscillator Strength : m f e m
e2 2
2
0 122m 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 8 31
f
Nonlinear Optics Lab
.
Hanyang Univ.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)
i t
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*a2 c.c., the combination variable a1*a2 and a1a*2 are more useful than either a1 or a2 alone.
Nonlinear Optics Lab
.
Hanyang Univ.Define,
12 c1c2** 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
* 2 1
12
)
2 ( 1
2 )
( 1
c c c i c i ci c
c c
c
) 2 ( 22 11
*
12
i i similarly,
) 2 (
22 1121
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
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>
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
t1
t
e i
t
level 1
to 2 from decay
: emission s
spontaneou -
Nonlinear Optics Lab
.
Hanyang Univ.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 1121
i iSimilarly,
) 2 (
1 )
( 22 11
* 12
12
i i Nonlinear Optics Lab
.
Hanyang Univ.
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) =>
( )
A
21 222
12 * 2111 1
11
i
) 2 (
) A
(
2 21 22 12 * 2122
i
emission s
Spontaneou and
effect collision
Inelastic 2)
Nonlinear Optics Lab
.
Hanyang Univ.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 1121
i
i
where, ( A )
2 1 1
21 2
1
: total relaxation rateNonlinear Optics Lab
.
Hanyang Univ.<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
A
212 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 )
(
wi v
Nonlinear Optics Lab
.
Hanyang Univ.) 0 ,
0 (&
real, is
that Assume
-
1 2
( )
) 2 2 (
)
(
21
12
21
22
11
12
22
11
i i i iw 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 matrixNonlinear Optics Lab