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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(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)
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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)
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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)
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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.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)
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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)
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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
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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
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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(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
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..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
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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
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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 !
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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
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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
Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.Univ.
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
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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>
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 -
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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
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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)
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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
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<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
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)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
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