Nonlinear Optics Lab. Hanyang Univ. Chapter 6. Time-Dependent Schrodinger Equation 6.1 Introduction...

Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.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(F extext tVt

),r(V)r(2

),r(VHH ext2

ext tVm

ta (6.1.1)

The Schrodinger equation becomes

tittV

m

),r(),r(V)r(

2 ext2 (6.1.2)


6.2 Time-Dependent Solutions

Time-independent Schrodinger equation ; nnna EH

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

Following Dirac, the exact time-dependent wave function can be expressed by

a sum of ;s'n

)r(),r( n

nnat (6.2.1)

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

nn

nan t

aiVHa

)r()r(][ ext nn

nn

nnn t

aiVEa

(6.2.2)

(6.2.3)


mnnmnmm d r)r()r(|,&)r( 3

space all

*

space all

n

nmnmmm dVaaEai r)r()r( 3ext

*

n

nmnmmm atVaEai )(

where, r)r(),r()r()( 3ext

* dtVtV nmmn

: time-dependent Schrodinger equation

<Meaning of : probability amplitude>ma

1),(),( 3* rdtrtr rdaan

nnm

mm3

*

m n m

mmnnmm n

nmnm aaaaa 1||| 2**

Probability that thequantum system is

in its m-th orbit.

(6.2.7)


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.E2

1)Rkcos(E),E(R )Rk(

00 tiett

0R

),R(Er),Rr,(ext tetV : dipole approximation

For a monochromatic wave,

Put, c.c.E2

1 )0 tie

For a two-state system,

)r()()r()(),r( 2211 tatat

(6.2.8) )()()()( 212111111 taVtaVtaEtai

)()()()( 222121222 taVtaVtaEtai 0

)()()( 212111 taVtaEtai

)()()( 121222 taVtaEtai

(6.3.1)

(6.3.4)


Normalization condition ; 1|)(||)(| 2

22

1 tata

(6.2.7), (6.3.1) =>

.).Eˆ(2

1r)( 01212 cceetV ti

.).Eˆ(2

1r)( 02121 cceetV ti where, r)r(r)r(r 3

2*112 d

Define,

12

21

EE

0

2121

E)ˆr( e

0

1212

E)ˆr( e

Set, 01 E (6.3.4) =>

)()(2

1)( 2

*21121 taeetai titi

)()(2

1)( 1

*12212212 taeeatai titi

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


0E:0 0 i) ( radiation field=0)

]exp[)0()(

const.)0()(

2122

11

tiata

ata

21 ii) ( nearly resonant radiation field)

trial solution,

tietcta

tcta)()(

)()(

22

11 (6.3.11) =>

12*

12212212

2*21

2121

)(2

1)()(

)(2

1)(

cectci

cetci

ti

ti

Neglected by rotating-wave approximation

121212212

2*211

2

1

2

1)()(

2

1)(

cccctci

ctci

where, 21 : detuning

0

2121 )ˆr(E

e

: Rabi frequency


Solution) initial condition ; 0)0(,1)0( 21 cc

2/2

2/1

2sin)(

2sin

2cos)(

ti

ti

et

itc

et

it

tc

where, 2/122 )( : Generalized Rabi frequency

Probability ; 222

211 |)(|)(,|)(|)( tatPtatP

]cos1[2

1)(

cos2

11

2

1)(

2

2

22

1

ttP

ttP


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(r),r(r 3* dtt

For the two-state atom,

r)(r)(r 32211

*2

*2

*1

*1 daaaa

121*2122

*122

2211

21 rrr||r|| aaaaaa

where, r)r(r)r(r 3* djiij


..rrrr 2*1121

*2212

*112 ccaaaaaa

For a case of linear polarization, real.isE0

0r,0V iiii

)|||(|)()()4.3.6( 22

21212

*1122

*1 aaiVaaEEiaa

dt

d

)|||(|)()( 22

21212

*1

2122

*12

22 aaiVaaEEiaadt

d

)]|||(|[ 22

2121 aaV

dt

di

Since real,isr12 )(rr *212

*112 aaaa

)|||)(|Er(r2

r 22

212112

0202

2

aae

dt

d

where,

120

EE


If we assume, 1||&1|| 2

12

2 aa

)Er(r2

r 211202

02

2

e

dt

d

Suppose the E-field points in the z-direction, EzE

)Er2

r 211202

02

2

ze

dt

d

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

),((r)Yr)( 000,11 R ),((r)Yr)( 101,22 R

r)r()r( 31

*221 dzz

ddYYdRR

0

2

0 0

0,0*0,10,1

*1,2

3 ),(cossin),(r)r()r(r

21r21z


Table 6.1, 6.2,

0

0

/r32/r

0

2/30

2/3021 29.1rr

2

r)2(

3

2)2(r 00 adee

aaa aa

where, A53.042

2

00

me

a : Bohr radius

2

0 0

221 3

1sincos

4

3

4

1z dd

0212121 745.0ˆz azr

Ez)z(2

Ezzz2

r 212

02112

0202

2

ee

dt

d

cf) Ezx202

2

m

e

dt

d

in classical model

Homework : Appendix 5.A !


Classic Quantum mechanics

m

e2

212

022

ze

(3.7.5)

Oscillator Strength :

fm

e

m

e 22

212

02z

mf

example) Hydrogen n=1 => n=2, 3.1)(Table416.0,A1216 f

417.010054.1

1012161032

101.92

34

10

831

f


6.5 Density Matrix and (Collisional) Relaxation

Two level system, time-dependent Schrodinger equation,

)r()()r()(),r( 2211 tatat (6.3.2)

tietcta

tcta)()(

)()(

22

11(6.3.12)

122

2*211

2

1)(

2

1)(

cctci

ctci

(6.3.14)

Via (6.4.3), ..rr 2*112 ccaa , the combination variable

*212

*1 and aaaa

are more useful than either alone.or 21 aa


Define,

*2112 cc*1221 cc

21

*1111 || ccc

22

*2222 || ccc

*121

*22

**21

*2112 )

2

1()

2

1( cicicccicccc

)(2 1122

*

12 ii

similarly,

)(2 11222121 ii

)(2 21

*1211

i

)(2 21

*1222

i

yprobabilitoccupationslevel':,* 2211 )population(

amplitudecomplex :,* 2112 r nt,displaceme selectron' theof


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

of relaxation such as collision.

<Relaxation Processes>

levelsother decay to :collision inelastic

change phasen oscillatio:collision elasticcollision -

0|)( and,at t occursCollision *

const.steady. is fieldradiation theif*

., thechangeonly const., *

effectcollision Elastic 1)

1211

21122211

ttt

(6.3.14) => )1(2

)()( )(1122

211ttiet

level 1 to2 fromdecay :emission sspontaneou -


Average value

/

1

2

)()1(

2

)()( 1122)(/)(

11122

2111

ieedtt

tttitt

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

i

)(2

)1

( 11222121

ii

Similarly,

)(2

)1

( 1122

*

1212

ii


2221spon11

221col11

2221spon22

222col22

2211

A)(

)(

A)(

)(

, i)

1

2

A21

2

1

(6.5.2) => )(2

A 21*

12222111111 i

)(2

)A( 21*

122221222 i

emission sSpontaneou andeffect collision Inelastic 2)


effects two theof average

evenly.or toscontribute and on effect each

],,[ (6.5.1) definitionBy

, ii)

21122211

*1221

*2112

2112

cccc

(6.5.2) => )(2

)( 1122

*

1212 ii

)(2

)( 11222121 ii

where, )A(2

112121

: total relaxation rate


<Special case> 0,021

)(2

A

)(2

A

21*

12222122

21*

12222111

i

i

)(2

)(2

11222121

1122

*

1212

i

i

21A2

11,

10 22112211

No dynamic information !

So, we can pay attention solely to the differences, 21121122 ,

1122

1221 )(

w

iv


)0,0 (& real, is that Assume - 21

)(

2)(

2)( 1122121122211221 iiii

wv

)(2

)(2 21122221211222211122

iA

iAw

vwAvA )1()1( 21112221

(Chapter 8 : Bloch equation)

The notation used for s'

2221

1211

: density matrix

Nonlinear Optics Lab. Hanyang Univ. Chapter 6. Time-Dependent Schrodinger Equation 6.1 Introduction...

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Transcript of Nonlinear Optics Lab. Hanyang Univ. Chapter 6. Time-Dependent Schrodinger Equation 6.1 Introduction...