Lecture 9 Non-degenerate & Degenerate Time Independent...

43
ECE 6451 - Dr. Alan Doolittle Georgia Tech Lecture 9 Non-degenerate & Degenerate Time Independent and Time Dependent Perturbation Theory: Reading: Notes and Brennan Chapter 4.1 & 4.2

Transcript of Lecture 9 Non-degenerate & Degenerate Time Independent...

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

Lecture 9

Non-degenerate & Degenerate Time Independent and Time Dependent Perturbation Theory:

Reading:

Notes and Brennan Chapter 4.1 & 4.2

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

Non-degenerate Time Independent Perturbation Theory

If the solution to an unperturbed system is known, including Eigenstates, Ψn(0) and Eigen

energies, En(0), ...

...then we seek to find the approximate solution for the same system under a slight perturbation (most commonly manifest as a change in the potential of the system).

To do this , we expand the Hamiltonian into modified form,

Where g is a dimensionless parameter meant to keep track of the degree of “smallness” (we will eventually set g=1, but for now, we keep it)

where H’ is the perturbation term in the Hamiltonian. As g→0, then H→Ho, Ψn→Ψn(0) and

En→En(0)

nnn EH Ψ=Ψ

pgHHH += 0

( ) nnnp EgHH Ψ=Ψ+0

)0()0()0(0 nnn EH Ψ=Ψ

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

Non-degenerate Time Independent Perturbation Theory

If the perturbation is small enough, it is reasonable to write the wavefunction as,

... and the energy as,

Thus, we can write the Schrodinger equation as,

or,

....)2(2)1()0( +Ψ+Ψ+Ψ=Ψ nnnn gg

....)2(2)1()0( +++= nnnn EggEEE

( )( ) ( )( )

( )( )( )

0)()(

0

4

3

)0()2()1()1()2()0()1()2(0

2

)0()1()1()0()0()1(0

1

)0()0()0(0

0

)2(2)1()0()2(2)1()0()2(2)1()0(0

=++

+

Ψ−Ψ−Ψ−Ψ+Ψ+

Ψ−Ψ−Ψ+Ψ+

Ψ−Ψ+

=Ψ−Ψ

+Ψ+Ψ+Ψ+++=+Ψ+Ψ+Ψ+

Ψ=Ψ

L

L

L

LLL

gg

EEEHHg

EEHHgEHg

EH

ggEggEEgggHHEH

nnnnnnnpn

nnnnnpn

nnn

nn

nnnnnnnnnp

nn

For any choice of g, each term in parenthesis must be equal to 0!

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

Non-degenerate Time Independent Perturbation Theory

Given,

1st order perturbation theory seeks values of En(1) and Ψn

(1) .

2nd order perturbation theory will seek values of En(2) and Ψn

(2) .

( )( )( )

0)()(

0

4

3

)0()2()1()1()2()0()1()2(0

2

)0()1()1()0()0()1(0

1

)0()0()0(0

0

=++

+

Ψ−Ψ−Ψ−Ψ+Ψ+

Ψ−Ψ−Ψ+Ψ+

Ψ−Ψ+

=Ψ−Ψ

L

L

L

gg

EEEHHg

EEHHgEHg

EH

nnnnnnnpn

nnnnnpn

nnn

nn

gEn(1)

En(0)

g2En(2)

gg=1

E

gYn(1)

Ψn(0)

g2Yn(2)

gg=1

Ψn

For any choice of g, each term in parenthesis must be equal to 0!

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

1st Order Perturbation Theory

Given the term that is 1st order in g,

and using the Fundamental Expansion Postulate for Ψn(1), using the basis vectors, Ψj

(0) ‘s

( ) 0)0()1()1()0()0()1(0 =Ψ−Ψ−Ψ+Ψ nnnnnpn EEHHg

)0()1()0()0()0()0(0

)0(0

)0(0

)0()1()0()0()0()0(0

)0()1()1()0()0()1(0

)0()1(

then,

since,But

nnj

jjnnpj

jjj

nn

nnj

jjnnpj

jj

nnnnnpn

jjjn

EaEHEa

EH

EaEHaH

EEHH

a

Ψ+

Ψ=Ψ+

Ψ

Ψ=Ψ

Ψ+

Ψ=Ψ+

Ψ

Ψ+Ψ=Ψ+Ψ

Ψ=Ψ

∑∑

∑∑

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1st Order Perturbation Theory

Cont’d,

Consider the case when m=n,

)0()0()1()0()0()0(0

)0(*)0()1()0()0(*)0(0

)0()0(

)0(*)0()1()0(*)0()0()0(*)0()0(*)0(0

*)0(

)0()1()0()0()0()0(0

or

j,m unless toorthogonal is fact that theUsing

space allover integrate and by multiply will we,many times done have weAs

nmnmnnpmmm

nmnmnnpmmm

jm

nmnj

jmjnnpmj

jmjj

m

nnj

jjnnpj

jjj

EaEHEa

dvEaEdvHEa

dvEdvaEdvHdvEa

EaEHEa

ΨΨ+=ΨΨ+

ΨΨ+=ΨΨ+

=ΨΨ

ΨΨ+

ΨΨ=ΨΨ+

ΨΨ

Ψ

Ψ+

Ψ=Ψ+

Ψ

∫∫

∫∫ ∑∫∫ ∑

∑∑

)0()0(

)0()0()1(

)0()0()1()0()0()0(0

)0()0()1()0()0()0(0

nn

npnn

nnnmmnpnmm

nmnmnnpmmm

HE

EaEHEa

EaEHEa

ΨΨ

ΨΨ=

ΨΨ+=ΨΨ+

ΨΨ+=ΨΨ+

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

1st Order Perturbation Theory

To find the coefficient, am’s consider the case where m≠n,

( )

( ) nmexcept n m, allfor validis which

0

)0()0(

)0()0(

)0()0()0()0(

)0()0()1()0()0()0(0

=−

ΨΨ=

+−=ΨΨ

ΨΨ+=ΨΨ+

mn

npmm

mmnnpm

nmnmnnpmmm

EEH

a

aEEH

EaEHEa

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1st Order Perturbation Theory

For the case m=n, we have to consider, the normalization condition,

( )nmover performedonly is sum then,mfor 0a since :NOTE

becomes,

equation, original theThus,0a when allfor only true is which

11

, equal ,or form, theof termscross all sinceBut

1

1

m

)0()0()0(

)0()0()0(

)0()0(

)1()0(

*nn

*mm

*2*

mn)0(*)0()0(*)0(

)0()0(*

)0()0(

)1()0()1()0(

≠==

Ψ−

ΨΨ+Ψ=Ψ

Ψ+Ψ=Ψ

Ψ+Ψ=Ψ

====

=+++

ΨΨΨΨ

=

Ψ+Ψ

Ψ+Ψ

=Ψ+ΨΨ+Ψ

∑∫ ∫

∫ ∑∑

≠m

nm mn

npmnn

mm

mnn

nnn

mmmmm

mnnm

mmmn

mmmn

nnnn

EEH

g

agg

aaag

aaggaga

dvdv

dvagag

gg

δ

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

1st Order Perturbation Theory

In summary,

( ))0(

)0()0(

)0()0()1(

)1()0(

,

mnm mn

npmn

nnn

EEH

whereg

Ψ−

ΨΨ=Ψ

Ψ+Ψ=Ψ

∑≠

)0()0(

)0()0()1(

)1()0(

,

nn

npnn

nnn

HE

wheregEEE

ΨΨ

ΨΨ=

+=

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

1st Order Perturbation Theory

Things to consider:

1. To calculate the perturbed nth state wavefunction, all other unperturbed wavefunctions must be known.

2. Since the denominator is the difference in the energy of the unperturbed nth energy and all other unperturbed energies, only those energies close to the unperturbed nth energy significantly contribute to the 1st order correction to the wavefunction.

3. “g” can be set equal to 1 for convenience or rigidly determined by the normalization condition on Ψn.

( ))0(

)0()0(

)0()0()1(

)1()0(

,

mnm mn

npmn

nnn

EEH

whereg

Ψ−

ΨΨ=Ψ

Ψ+Ψ=Ψ

∑≠

)0()0(

)0()0()1(

)1()0(

,

nn

npnn

nnn

HE

wheregEEE

ΨΨ

ΨΨ=

+=

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

2nd Order Perturbation Theory

Given the term that is 2nd order in g,

and using the Fundamental Expansion Postulate for Ψn(2), using the basis vectors, Ψj

(0) ‘s,

( ) 0)0()2()1()1()2()0()1()2(0

2 =Ψ−Ψ−Ψ−Ψ+Ψ+ nnnnnnnpn EEEHHg

nmnnmnmjpmj

jmm

mjjmnn

nmnj

jmjnj

jmjn

jjjpm

jjjm

m

nnj

jjnj

jjnj

jjpj

jj

nnnnnnnpn

jjjn

EEaEbdvHaEb

dvEH

dvEdvaEdvbE

dvaHdvbH

EaEbEaHbH

EEEHH

b

δ

δ

)2()1()0()0(*)0()0(

)0(*)0()0(0

)0(0

)0(*)0()2()0(*)0()1()0(*)0()0(

)0(*)0()0(0

*)0(

*)0(

)0()2()0()1()0()0()0()0(0

)0()2()1()1()2()0()1()2(0

)0()2(

then,

and

since,But

space, allover gintegratin and by gmultiplyin Again,

0

++=ΨΨ+

=ΨΨΨ=Ψ

ΨΨ+

ΨΨ+

ΨΨ

=

ΨΨ+

ΨΨ

Ψ

Ψ+

Ψ+

Ψ=

Ψ+

Ψ

=Ψ−Ψ−Ψ−Ψ+Ψ

Ψ=Ψ

∫∑

∫∫ ∑∫ ∑

∫ ∑∫ ∑

∑∑∑∑

L

L

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

2nd Order Perturbation Theory

Cont’d,

To find the 2nd order energy correction, consider the case of m=n,

nmnnmnmjpmj

jmm

nmnnmnmjpmj

jmm

EEaEbHaEb

EEaEbdvHaEb

δ

δ

)2()1()0()0()0()0(

)2()1()0()0(*)0()0( or

++=ΨΨ+

++=ΨΨ+

∫∑

( )

( )∑

ΨΨ=

ΨΨ

ΨΨ=

ΨΨ=

ΨΨ−ΨΨ+ΨΨ=

−ΨΨ+ΨΨ=

=

−ΨΨ=

nj jn

jpnn

njjpm

mn

npmn

njjpmjn

npnnnpnnnj

jpmjn

n

nnnpnnnj

jpmjn

nnjpmj

jn

EE

HE

HEE

HE

a

HaE

HaHaHaE

E

EaHaHaE

EaHaE

)0()0(

2)0()0()2(

)0()0()0()0(

)0()0()2(

j

)0()0()2(

)0()0()0()0()0()0()2(

)1(

)1()0()0()0()0()2(

)1()0()0()2(

solution,order 1st thefrom for result theInserting

solution,order 1st thefrom for result theInserting

term,nm thesummation theofout pullingor

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2nd Order Perturbation Theory

( )

( ) ( )

( ) ( )

( )( )

( )( )

( )( )( )

( )( )( )( ))0()0()0()0(

)0()0()0()0(

)0()0()0()0(

)0()0()0()0(

)0()0(

)0()0(

)0()0(

)0()0(

)0()0(

)0()0()0()0(

)0()0(

)0()0(

)0()0(

)0()0(

)0()0(

)0()0(

)0()0(

)1(

)0()0(

)0()0(

)1()0()0()0()0(

)2()1()0()0()0()0(

1

mnmn

npnnpm

nj mnjn

jpmnpjm

mn

npn

mn

npm

nj mn

jpmjn

npj

m

mn

nn

npnm

nj mn

jpmjm

mn

nm

nj mn

jpmjm

nmjpmnj

jmnm

nmnnmnmjpmnj

jmm

EEEEHH

EEEEHH

b

EE

HEE

H

EE

HEE

H

b

EE

Ha

EEHa

b

EEEa

EEHa

b

EaHaEEb

EEaEbHaEb

−−

ΨΨΨΨ−

−−

ΨΨΨΨ=

ΨΨ

ΨΨ

ΨΨ

ΨΨ

=

ΨΨ

ΨΨ

ΨΨ=

−−

ΨΨ=

−ΨΨ=−

++=ΨΨ+

δ

To find the coefficient, bm’s consider the case where m≠n,

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

2nd Order Perturbation TheoryWe also need to find the value of bm for the case where m=n. To do this, we again examine the normalization condition,

( )

( )( )( )

( )( )( )( ) ( )∑∑

∑∑

∫ ∫

∫ ∑∑∑∑

≠≠

ΨΨ−

−−

ΨΨΨΨ−

−−

ΨΨΨΨ=

Ψ≠

ΨΨ−=−=

ΨΨΨΨ

=

Ψ+Ψ+Ψ

Ψ+Ψ+Ψ

=Ψ+Ψ+ΨΨ+Ψ+Ψ

nj jn

npj

mnmn

npnnpm

nj mnjn

jpmnpjm

n

nj jn

npj

jjn

jnnj

jjj

jjjn

jjj

jjjn

nnnnnn

EE

H

EEEEHH

EEEEHH

b

EE

Hab

dvdv

dvbgagbgag

gggg

2)0()0(

2)0()0(

)0()0()0()0(

)0()0()0()0(

)0()0()0()0(

)0()0()0()0(

)2(m

2)0()0(

2)0()0(2

mn)0(*)0()0(*)0(

)0(2)0()0(

*

)0(2)0()0(

)2(2)1()0()2(2)1()0(

21

,for expression theinto thisinserting and nm withbfor result the to thisAdding

21

21

, equal or form, theof termscross all sinceBut

1

1

δ

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

2nd Order Perturbation TheoryCont’d:

( )( )( )

( )( )( )( ) ( )

( )( )

( )( )( )( )

( )( ) ( ))0(

2)0()0(

2)0()0(

)0()0()0()0(

)0()0()0()0(

)0()0()0()0(

)0()0()0()0(2

)0()0()0(

)0()0()0(

)0(2)0()0(

)2(2)1()0(

2)0()0(

2)0()0(

)0()0()0()0(

)0()0()0()0(

)0()0()0()0(

)0()0()0()0(

21

becomes,

equation, original theThus,

21

mm nj jn

npj

mnmn

npnnpm

nj mnjn

jpmnpj

mnm mn

npmnn

mm

mmm

mnn

nnnn

nj jn

npj

mnmn

npnnpm

nj mnjn

jpmnpjm

EE

H

EEEEHH

EEEEHH

g

EEH

g

bgaggg

EE

H

EEEEHH

EEEEHH

b

Ψ

ΨΨ−

−−

ΨΨΨΨ−

−−

ΨΨΨΨ

+Ψ−

ΨΨ+Ψ=Ψ

Ψ+Ψ+Ψ=Ψ

Ψ+Ψ+Ψ=Ψ

ΨΨ−

−−

ΨΨΨΨ−

−−

ΨΨΨΨ=

∑ ∑∑

∑∑

∑∑

≠≠

≠≠

L

L

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

2nd Order Perturbation TheoryIn Summary:

( )( )

( )( )( )( )

( )( ) ( ))0(

2)0()0(

2)0()0(

)0()0()0()0(

)0()0()0()0(

)0()0()0()0(

)0()0()0()0(2

)0()0()0(

)0()0()0(

21

mm nj jn

npj

mnmn

npnnpm

nj mnjn

jpmnpj

mnm mn

npmnn

EE

H

EEEEHH

EEEEHH

g

EEH

g

Ψ

ΨΨ−

−−

ΨΨΨΨ−

−−

ΨΨΨΨ

+Ψ−

ΨΨ+Ψ=Ψ

∑ ∑∑

≠≠

L

L

( )∑≠

ΨΨ+ΨΨ+=

nj jn

jpnnpnnn EE

HgHgEE )0()0(

2)0()0(2)0()0()0(

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

Perturbation TheoryConsider an important and illustrative example: Small electric field applied to an infinite potential barrier quantum well. What effect does this have on the ground state energy?

Ψ(0)n=1

E(0)n=1

Ψn=1

En=1E(0)

n=1

When the electric field is applied, the energy bands bend, resulting a redistribution of the electron in the well toward the right side. Since the energy on this side of the well is lower, the new energy of the ground state is expected to be smaller than the unperturbed ground state.

We previously solved this as an asymmetric solution (0<x<L) and had states that only depended on sin functions. For reasons that will become obvious, we redefine our limits as symmetric (- ½ a <x< + ½ a) which will require wave function solutions of the form:

-½a +½a0-½a +½a0Unperturbed Well Perturbed Well

( )2

222(0)

1m

)0()0(

21E

odd mfor cos2 and even mfor sin2

mam

axm

aaxm

a mm

ππ

==

=

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

Perturbation TheoryWe first attempt a 1st order perturbation solution of the form:

Ψ(0)n=1

E(0)n=1

Ψn=1

En=1E(0)

n=1

Note that since our ground state wave function solution is normalized, the denominator in the above equation is equal to 1.

In this case, the perturbation Hamiltonian, Hp= -qεοx (Electric field=Volts/meter times meters = Volts Energy by –q). Thus, for the ground state, n=1 (odd index).

-½a +½a0-½a +½a0Unperturbed Well Perturbed Well

)0()0(

)0()0()1()1()0( ,

nn

npnnnnn

HEwheregEEE

ΨΨ

ΨΨ=+=

( ) ( )

function) odd an is (integrand 0

1cos)(2 )(

)1(1

2

2

22

2

)0(1

*)0(1

)0(1

)0(1

)1(1

=

=

−=Ψ−Ψ=ΨΨ=

=

+

+

−===== ∫∫

n

a

a o

a

a nonnpnn

E

dxa

xnxqa

dxxqHE πεε

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

Perturbation TheorySince our 1st order perturbation correction to the ground state energy resulted in a zero correction factor that deviated from our physical understanding of the system (i.e. we expect a lower ground state energy), we now must consider the 2nd order correction.

Ψ(0)n=1

E(0)n=1

Ψn=1

En=1E(0)

n=1

Note that since our 1st order correction was equal to 0, the middle term in the above equation is equal to 0.

We will examine odd and even indexes in the summation separately.

-½a +½a0-½a +½a0Unperturbed Well Perturbed Well

( )∑≠

ΨΨ+ΨΨ+=

nj jn

jpnnpnnn EE

HgHgEE )0()0(

2)0()0(2)0()0()0(

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

Perturbation TheoryConsider the Odd indexes in the summation:

( )

( )( )

( ) ( ) ( )( )

( ) ( ) ( ) 0cos21cos2

odd, jfor Consider

2

2

*

)0()0(1

)0(2

2

*)0(1

)0()0(1

)0()0(1

2)0()0(1)2(

)0()0(1

2)0()0(1)2(

=

=

=Ψ−Ψ

Ψ−Ψ=Ψ−Ψ

Ψ−Ψ=

ΨΨ=

+

=

+

==

≠ =

=

≠ =

=

dxaxj

axq

axn

axq

dxxqxq

EE

xqE

EE

HE

o

a

ajon

jo

a

anjon

nj jn

jonn

nj jn

jpnn

πεπε

εε

ε

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

Perturbation TheoryConsider the Even indexes in the summation:

( )

( )( )

( ) ( ) ( )( )

( ) ( ) ( )

( ) ( )( )

( )( )

+

+

+

−=Ψ−Ψ

=

=

=Ψ−Ψ

Ψ−Ψ=Ψ−Ψ

Ψ−Ψ=

ΨΨ=

=

+

=

+

==

≠ =

=

≠ =

=

21sin

121sin

12

0sin21cos2

even, jfor Consider

22)0()0(

1

2

2

*

)0()0(1

)0(2

2

*)0(1

)0()0(1

)0()0(1

2)0()0(1)2(

)0()0(1

2)0()0(1)2(

ππ

ππ

εε

πεπε

εε

ε

jjaj

ja

aqxq

dxaxj

axq

axn

axq

dxxqxq

EE

xqE

EE

HE

ojon

o

a

ajon

jo

a

anjon

nj jn

jonn

nj jn

jpnn

So the Even indexes in the summation contribute non-zero values!

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

Perturbation TheoryThus, the correction term for Even indexes is:

( )

( )( )

( )( )

( )∑

=

+

+

+

=

ΨΨ=

=

=

odd is j2

222

2

222

222

)2(

odd is j)0()0(

1

2)0()0(1)2(

221

21sin

121sin

12

maj

man

jjaj

ja

aq

E

EE

HE

o

n

jn

jpnn

hh ππ

ππ

ππ

ε

Technically, this is an infinite summation. However, since the denominator increases proportionally to j2, the relative weight of higher order j terms rapidly decreases and the solution converges after just a few terms.

Note: Since the denominator is always negative and the numerator is always positive, the 2nd order correction is always negative resulting in a lower energy than the unperturbed ground state as expected. I.e.,

( )( ) ( )Number Negative0 2)0(

11

)0()0(1

2)0()0(12)0(

1)0(1

)0(11

ggEE

EE

HgHgEE

nn

nj jn

jpnnpnnn

++=

ΨΨ+ΨΨ+=

==

≠ =

=

==== ∑

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

Degenerate Perturbation TheoryThus far, it was assumed that the energy values of each state are never the same. If this is not the case, the correction terms can “blow up” to infinity.

( )( )

( )( )( )( )

( )( ) ( ))0(

2)0()0(

2)0()0(

)0()0()0()0(

)0()0()0()0(

)0()0()0()0(

)0()0()0()0(2

)0()0()0(

)0()0()0(

21

mm nj jn

npj

mnmn

npnnpm

nj mnjn

jpmnpj

mnm mn

npmnn

EE

H

EEEEHH

EEEEHH

g

EEH

g

Ψ

ΨΨ−

−−

ΨΨΨΨ−

−−

ΨΨΨΨ

+Ψ−

ΨΨ+Ψ=Ψ

∑ ∑∑

≠≠

L

L

( )∑≠

ΨΨ+ΨΨ+=

nj jn

jpnnpnnn EE

HgHgEE )0()0(

2)0()0(2)0()0()0(

How do we handle the case where the energy of multiple states is degenerate.

Ψ1(0)

Ψ2(0)

Ψ3(0)

Ψ4(0)

E1(0)=E2

(0)=E3(0)=E4

(0)

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

( )( )

( )( )( )( )

( )( ) ( ))0(

2)0()0(

2)0()0(

)0()0()0()0(

)0()0()0()0(

)0()0()0()0(

)0()0()0()0(2

)0()0()0(

)0()0()0(

21

mm nj jn

npj

mnmn

npnnpm

nj mnjn

jpmnpj

mnm mn

npmnn

EE

H

EEEEHH

EEEEHH

g

EEH

g

Ψ

ΨΨ−

−−

ΨΨΨΨ−

−−

ΨΨΨΨ

+Ψ−

ΨΨ+Ψ=Ψ

∑ ∑∑

≠≠

L

L

( )∑≠

ΨΨ+ΨΨ+=

nj jn

jpnnpnnn EE

HgHgEE )0()0(

2)0()0(2)0()0()0(

Degenerate Perturbation TheoryThe solution to this problem is to transform the degenerate basis vectors into another set which results in zero numerator terms.

Simple Example: Changing to an equivalent basis set in real space. A convenient set of equivalent basis vectors can be selected for any given problem.

Ψ1(0)

Ψ2(0)

Ψ3(0)

Ψ1(0)

Ψ2(0)αΨ3

(0)α

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

h ..., 3, 2, 1,m any value be can m e wher)0(, =Ψ mn

Degenerate Perturbation TheoryLets assume that “h” various unperturbed states result in the same (degenerate) energies. Any specific one of these states, say the mth state can be written as:

n still represents the state of the electron while m represents which of the h degenerate states are being considered.

Since each of the above unperturbed states is an Eigenfunction of Ho, with h degenerate Eigen energies, En

(0), any linear combination of these degenerate basis states is also an Eigenfunction of Ho (see Brennan Chapter 1). Thus, we can construct a new set of basis sets, labeled α, that are linear combinations of the degenerate Eigenstates.

h ..., 3, 2, 1,m where 1

)0(,

)0( =Ψ=Ψ ∑=

h

mmnmn bαα

The task is simply to find the appropriate values of the coefficients, bmα. To do this, we

simply find the set of bmα’s that force the numerator,

to equal zero when the denominator is also zero, thus removing the singularity.

( )( ) ( )

( ))0(

)0()0(

)0(*)0(

)1(

)0()0()0(

)0()0()1(

m

nm mn

npm

n

mnm mn

npmn

EE

dxH

EEH

Ψ−

ΨΨ=Ψ

Ψ−

ΨΨ=Ψ

∑∫

α

α

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

Ψ

ΨΨ

ΨΨ

+=

==

==

==

)0(

)0(1

)0(,

)0(2,2

)0(1,1

1

1

1111

111111

n

hn

hmhn

mn

mn

nnn

hhh

nh

p

HH

HH

HHHHHH

H

M

M

LLLLL

MOM

MOM

M

MOM

M

LLL

Degenerate Perturbation TheoryThe problem reduces to a diagonalization of the sub-matrix of index h by defining new basis sets such that,

Ψ

ΨΨ

ΨΨ

+=

==

==

==

+

+

)0(

)0(1

)0(,

)0(2,2

)0(1,1

1

1)1(

11

1)1(111

000

00000

n

hn

hmhn

mn

mn

nnn

h

hh

nh

p

HH

HH

HHHH

H

M

M

LLLLL

MOM

MO

ML

MMOM

M

LL

α

α

α

∑ Ψ=Ψi

iic

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

Time Dependent Perturbation TheoryIn all previous cases, nothing observable happens because the states are assumed static –unchanging. Most useful systems require transitions between states. For example, optical absorption and electron-hole pair recombination require a change from one state to another. This inherently requires TIME DEPENDENT Perturbation Theory.

Ψ(0)n=1

E(0)n=1

Ψn=1

En=1E(0)

n=1 E(0)n=1

Ψn=1

En=1-½a +½a0

-½a +½a0-½a +½a0

Given wave functions are sluggish in nature (waves do not change instantly when the perturbation changes – consider a water wave as an analogy), an instantaneous change in the perturbation results is a “more” gradual time change in the wave function and thus the distribution of particles. Note, the expectation value of the total potential energy is assumed to change instantly as the perturbation energy changes instantly but the expectation value of the kinetic energy changes “gradually”.

Time t<0 Time t=0 Time t>>0

Note: terms like “gradually” and “sluggish” are somewhat misleading in that these changes can often happen in fractions of a nanosecond. However, compared to the instantaneous perturbation these changes are considered “slower”.

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

Time Dependent Perturbation TheoryThe starting point for time dependent perturbation theory is the time dependent Schrodinger equation (see lecture 6).

( )

( ) ( )∑

∑=

=

=

=

∂∂

−=+

∂∂

−=

nnn

nn

nnno

aa

ta

EH

1 toreduces , ions,eigenfunct individual theoflity orthonorma given which

condition, ionnormalizat thesubject to )(:is onperturbati thebefore solutiondependent timefor the solution general theThus,

,eΨ Since

begins, (t) H'before solution thegConsiderinti

(t)gH'H

(t), H'part,dependent timeSMALLa andpart , Ht,independen a time into thisBreakingor

tiH

*(0)n

(0)(0)

(0)n

(0)

)0()0()0(

tEi(0)n

(0)n

o

o

(0)n

φφφ

φφ

φφ

φ

φφ

φφ

h

h

h

Ψ= h

Etiezyxtzyx ),,(),,,(φ

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

Time Dependent Perturbation TheoryFor the time dependent solution,

( )

∑∑ ∫

∫ ∑∫ ∑

∑∑

∑∑∑∑

∑∑∑

=∂∂

=∂∂

∂∂

=

∂∂

−=

∂∂

∂∂

−=+

∂∂

−=+

=

∂∂

−=+

nnm

nnm

nn

nn

nn

nn

nn

nn

nn

nn

nn

nn

nn

nn

txtxtatadvtxtxtata

dvtxtxtadvtxtxta

tx

txtatxta

txtatxtatxtatxta

or

txtatxtatxta

txtatx

),((t)H'),()(i-g)(t

or ),((t)H'),()(i-g)(t

),(),()(t

),((t)H'),()(ig-

space, allover integrate and ,),( function, wavespecifica by multiply webefore, many times have weAs

),()(ti

),((t)gH')(

:leaving cancel terms4th and1st thesolution, dunperturbe thefromBut

),(t

)(i

),()(ti

),((t)gH')(),(H)(

),()(ti

),()((t)gH'),()(H

yeilds,r EquationSchrodinge theinto expression thisof onSubstituti

),()(),( ions, wavefunctdunperturbe ofset basis complete theof ncombinatiolinear a as functions wave theexpress We

ti(t)gH'H

(0)n

(0)m

(0)n

(0)m

(0)n

(0)m

(0)n

(0)m

(0)m

(0)n

(0)n

(0)n

(0)n

(0)n

(0)no

(0)n

(0)n

(0)no

(0)n

o

φφφφ

φφφφ

φ

φφ

φφφφ

φφφ

φφ

φφ

hh

h

h

hh

h

h

NOTE: The time evolution of the wave function depends on the time evolution of the weighing coefficients (to be determined) and the known time evolution of the basis unperturbed states (see previous slide)

The time dependent perturbation can be described as a time evolution of the coefficients of the basis set wave vectors.

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

Time Dependent Perturbation TheoryImportant Observations: Since the weighting coefficients are known before the perturbation, knowing the time derivative (rate of change) of the coefficients implies full knowledge of the weighting coefficients after the perturbation occurs.

∑∑ ∫ =∂∂

=∂∂

nnm

nnm txtxtatadvtxtxtata ),((t)H'),()(i-g)(

tor ),((t)H'),()(i-g)(

t(0)n

(0)m

(0)n

(0)m φφφφ

hh

Each individual state is “connected” through the perturbation Hamiltonian. I.e. state “m”is connected to every other state (all n states) via the perturbation Hamiltonian. If the Hamiltonian does not allow a transition from one state to another (i.e. the matrix element is zero) then the weighting coefficient for that state remains unchanged (i.e. static in time).

The change in each individual coefficient in time depends on couplings between ALLother available states!

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

Time Dependent Perturbation TheoryConsider an expanded form of the previous equation:

MMMM

LLh

MMMM

LLh

LLh

h

=

++++=∂∂

=

++++=∂∂

++++=∂∂

=∂∂

==========

==========

==========

),((t)H'),()(),((t)H'),()(),((t)H'),()()(t

),((t)H'),()(),((t)H'),()(),((t)H'),()()(t

),((t)H'),()(),((t)H'),()(),((t)H'),()()(t

),((t)H'),()(i-g)(t

(0)jn

(0)m

(0)1n

(0)m1

(0)0n

(0)jm0

(0)jn

(0)1m

(0)1n

(0)1m1

(0)0n

(0)1m01

(0)jn

(0)0m

(0)1n

(0)0m1

(0)0n

(0)0m00

(0)n

(0)m

txtxtatxtxtatxtxtatagi

txtxtatxtxtatxtxtatagi

txtxtatxtxtatxtxtatagi

txtxtata

jjnjnnjm

jnnnm

jnnnm

nnm

φφφφφφ

φφφφφφ

φφφφφφ

φφ

Considering the vastness of this system of equations a simplification similar to what we did for the unperturbed system is in order. Since the wave function evolution in time is entirely determined by the coefficients, we will expand the coefficients in orders of g.

L

L

+++=

+++=

''2')0(

2

22)0(

jjjj

jjjj

aggaaaor

dgad

gdgda

gaa

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

Time Dependent Perturbation TheoryUsing this expansion, the previous system of equations becomes:

For our purposes we will only consider 1st order time dependent perturbation theory (THANK GOD!). For this we will consider only the linear terms in g. Again, for any choice of g, the terms on the left hand side must equate to the terms of equal magnitude on the right hand side. This is done on the following page...

( ) ( ) ( )( ) LLL

LLLLh

),((t)H'),(

),((t)H'),(),((t)H'),(t

(0)n

(0)jm

''2')0(

(0)1n

(0)jm

''1

2'1

)0(1

(0)0n

(0)jm

''0

2'0

)0(0

''2')0(

txtxaggaag

txtxaggaagtxtxaggaagaggaai

jjjj

jjj

==

====

++++

++++++++=+++∂∂

φφ

φφφφ

( ) ( ) ( )( ) LLL

LLLLh

),((t)H'),(

),((t)H'),(),((t)H'),(t

(0)n

(0)0m

''2')0(

(0)1n

(0)0m

''1

2'1

)0(1

(0)0n

(0)0m

''0

2'0

)0(0

''0

2'0

)0(0

txtxaggaag

txtxaggaagtxtxaggaagaggaai

jjjj ==

====

++++

++++++++=+++∂∂

φφ

φφφφ

( ) ( ) ( )( ) LLL

LLLLh

),((t)H'),(

),((t)H'),(),((t)H'),(t

(0)n

(0)1m

''2')0(

(0)1n

(0)1m

''1

2'1

)0(1

(0)0n

(0)1m

''0

2'0

)0(0

''1

2'1

)0(1

txtxaggaag

txtxaggaagtxtxaggaagaggaai

jjjj ==

====

++++

++++++++=+++∂∂

φφ

φφφφ

MM

MM

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

Time Dependent Perturbation TheoryConsidering equality of the linear terms,

The importance of this equation is that it is an approximate 1st order solution that only depends on the stationary unperturbed coefficients and the unperturbed basis wave functions. Thus, it depends on the initial condition (unperturbed coefficients and wave functions) of the system.

Each equation in the above system of equations represents the growth or decay of an individual Eigenfunction’s magnitude and thus represents the growth or decay of the probability of the state being occupied or empty.

All states are connected via the perturbation Hamiltonian, H’(t). Thus, if a particular state, say m=3, grows (or decays) in amplitude, it must come at the expense of some other state, for example, m=11 decaying (or growing) in amplitude. The rate of growth or decay is set by the 3rd and 11th equations. NOTE: this is just a simple example and in practice all states are coupled so growth and decay involves transitions from all states. In some cases, the perturbation Hamiltonian will prevent certain transitions resulting in a inability to transition between states – called forbidden transitions.

( ) ( ) ( ) ( ) LLh ),((t)H'),(),((t)H'),(),((t)H'),(

t(0)n

(0)jm

)0((0)1n

(0)jm

)0(1

(0)0n

(0)jm

)0(0

' txtxatxtxatxtxaai jjj ====== +++=

∂∂

− φφφφφφ

( ) ( ) ( ) ( ) LLh ),((t)H'),(),((t)H'),(),((t)H'),(

t(0)n

(0)0m

)0((0)1n

(0)0m

)0(1

(0)0n

(0)0m

)0(0

'0 txtxatxtxatxtxaa

i jj ====== +++=∂∂

− φφφφφφ

( ) ( ) ( ) ( ) LLh ),((t)H'),(),((t)H'),(),((t)H'),(

t(0)n

(0)1m

)0((0)1n

(0)1m

)0(1

(0)0n

(0)1m

)0(0

'1 txtxatxtxatxtxaa

i jj ====== +++=∂∂

− φφφφφφ

MM

MM

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

Time Dependent Perturbation TheoryExample 1: Consider the case when the initial state of a system is known to be in onlythe “jth” state (all zero coefficients except aj=1) and the perturbation is turned on at time, t=0 (i.e. H’(t)=H’u(t) where H’ is a constant and u(t) is a step function.

The systems of equations ...

( ) ( ) ),((t)H'),(1t

(0)n

(0)0m

)0('0 txtxaa

i jj ====∂∂

− φφh

( ) ( ) ),((t)H'),(1t

(0)n

(0)1m

)0('1 txtxaa

i jj ====∂∂

− φφh

M

( ) ( )( )

( ) 0t

and jmbut 2, 1, 0,mfor )(H')(1t

'(0)n

(0)m

')0()0(

≈∂∂

≠=ΨΨ=∂∂

−−

= j

tEEi

jm aexxai

jm

Lh

h

( )

( ) ( ) ( ) ( ) ( )h

hh

h

)0()0(

mjti(0)

j(0)m

tEi

(0)j

tEi(0)m

tEi(0)n

(0)n

where,eΨH'ΨexΨH'exΨ

then,exΨ sincefact that theused have wewhere

mj

(0)j(0)

m

(0)n

jm EExx −==

=

ω

φ

ω

t=0 t

H’

H’(t)=H’u(t)

... reduces to equations of the form:

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

Time Dependent Perturbation Theory

( ) jmbut 2, 1, 0,mfor )(H')(1t

(0)n

(0)m

' ≠=ΨΨ

=∂∂

= Lh

tijm

mjexxia ω

t=0 t

H’

H’(t)=H’u(t)

Thus, this general equation,

can be directly integrated to result in:

0 general, inbut ion wavefunct totalthe

of ionnormalizat throughfound be can 0,for t

and, 0 thenaa 0for t Since

'

'

'(0)jj

>

==≤

j

j

j

a

a

a

( ) jmbut 2, 1, 0,mfor u(t) K )(H')(11 (0)n

(0)m

' ≠=

+ΨΨ

−= = L

h

tij

mjm

mjexxta ω

ω

The integration constant can be evaluated by restricting a’m(t)=0 at t=0 resulting in,

( ) ( ) jmbut 2, 1, 0,mfor u(t)1 - )(H')(1 (0)n

(0)m

' ≠=

ΨΨ−= = L

h mj

ti

jm

mjexxtaω

ω

Finally, since the perturbation is defined as small, the jth coefficient is simply,

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

Time Dependent Perturbation Theory

Important Observations:

1) The matrix element in the above expression ( <...> term) is known as the transition matrix element and describes which transitions are allowed and disallowed and how strongly the mth and jth states are coupled.

2) If the transition matrix element integrand is odd (a very common occurrence) the integral is 0 meaning that a transition between the mth and jth state is forbidden.

3) In general, this transition matrix element is responsible for a variety of “selection rules” in atomic, nuclear and semiconductor bulk/quantum well optical spectra (emission or absorption resulting from electron/hole transitions between states).

4) Even though a specific transition is forbidden from say the jth state to the mth state, the mth state may still eventually become populated by indirect (and thus slower) transitions of the form j k m, etc...

( ) ( ) jmbut 2, 1, 0,mfor u(t)1 - )(H')(1 (0)n

(0)m

' ≠=

ΨΨ−= = L

h mj

ti

jm

mjexxtaω

ω

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

Time Dependent Perturbation Theory

( )

( ) ( )

( ) ( ) jmbut 2, 1, 0,mfor ee)(A(x))(t

jmbut 2, 1, 0,mfor )(u(t)eeA(x))(t

jmbut 2, 1, 0,mfor )((t)H')(1t

tiωtiω(0)n

(0)m

'

(0)n

tiωtiω(0)m

'

(0)n

(0)m

'

oo

oo

≠=+ΨΨ

=∂∂

≠=Ψ+Ψ

=∂∂

≠=ΨΨ

=∂∂

−=

=−

=

Lh

Lh

Lh

tijm

tijm

tijm

mj

mj

mj

exxia

exxia

exxia

ω

ω

ω

t=0 t

Example 2: Turning on a “Harmonic Perturbation” at time t=0. This case is important as many excitations such as electromagnetic radiation, ac electric and magnetic fields all can be periodic in nature. This problem proceeds identical to the previous example except:

can be directly integrated to result in:

0for t 0 and 0for t 0 '' >≈≤= jj aa

Where the integration constant was again evaluated by restricting a’m(t)=0 at t=0.

Finally, since the perturbation is defined as small, the jth coefficient is simply,

( )u(t)eeA(x)(t)H' tiωtiω oo −+=

( )( )( )

( )( )( )

( ) jmbut 2, 1, 0,mfor u(t)ω

1-ω

1-)(A(x))(

o

o

tω(0)n

(0)m'

oo

≠=

−−

+

ΨΨ−=

−+=

Lh mj

i

mj

ij

m

mjmj eexxta

ωω

ωω

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

Time Dependent Perturbation Theory

Important Observations:

1) Two resonances exist in the above coupling equation ωmj=+/-ωo where the coupling between the mth and jth state is extremely strong. These resonances occur when the mth

state is exactly +/-ħωο away from the initially occupied jth state.2) States not on resonance still can be coupled to the initially occupied jth state, but with a

lesser coupling strength than those directly on resonance.3) While this equation predicts a strong coupling at resonance, a’m(t) ∞, our restriction

of a small perturbation (i.e. a’m(t) small) indicates that 1st order time dependent perturbation theory is insufficient to accurately describe this case and thus, it is expected that a’m(t) will NOT actually ∞ and that higher order approximations will be needed to accurately describe this condition.

4) Even though a specific transition is forbidden from say the jth state to the mth state, the mth state may still eventually become populated by indirect (and thus slower) transitions of the form j k m, etc...

( )( )( )

( )( )( )

( ) jmbut 2, 1, 0,mfor u(t)ω

1-ω

1-)(A(x))(

o

o

tω(0)n

(0)m'

oo

≠=

−−

+

ΨΨ−=

−+=

Lh mj

i

mj

ij

m

mjmj eexxta

ωω

ωω

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

Time Dependent Perturbation TheoryFermi’s Golden Rule: Since the probability of state m being occupied is found by (Ψm)*(Ψm) and due to the normalization of the basis wave functions this reduces to (a’

m(t))*(a’m(t)). Thus, we need to

consider the value of (a’m(t))*(a’

m(t)). To simplify this procedure, we note that only one term in the square braces need be considered for the two cases near resonance.

( )( ) ( )( )( )

( )

( )( ) ( )( )( )

( ) o(0)j

(0)m2

omj

omj2

2

2(0)jn

(0)m'

m*'

m

o(0)j

(0)m2

omj

omj2

2

2(0)jn

(0)m'

m*'

m

ωEfor E u(t)ωω

tωω21sin(x)ΨA(x)(x)Ψ

4tata

ωEfor E u(t)ωω

tωω21sin(x)ΨA(x)(x)Ψ

4tata

hh

hh

−≈

+

+

+≈

=

=

and

The reason we make this simplification is that in deriving Fermi’s Golden Rule, the functional form of the above equations will be convenient for integration. Before we proceed lets examine these equations.

( )( )( )

( )( )( )

( ) jmbut 2, 1, 0,mfor u(t)ω

1-ω

1-)(A(x))(

o

o

tω(0)n

(0)m'

oo

≠=

−−

+

ΨΨ−=

−+=

Lh mj

i

mj

ij

m

mjmj eexxta

ωω

ωω

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

Log{

(a’ m

(t))*

(a’ m

(t))}

o(0)j

(0)m ωEE h−≈

t≤0

t>0

t>>0

o(0)j

(0)m ωEE h+≈(0)

jE

The probability of all states including “far off-resonant” states becoming occupied increases with time.

( )( ) ( )( )( )

( )

( )( ) ( )( )( )

( ) o(0)j

(0)m2

omj

omj2

2

2(0)jn

(0)m'

m*'

m

o(0)j

(0)m2

omj

omj2

2

2(0)jn

(0)m'

m*'

m

ωEfor E u(t)ωω

tωω21sin(x)ΨA(x)(x)Ψ

4tata

ωEfor E u(t)ωω

tωω21sin(x)ΨA(x)(x)Ψ

4tata

hh

hh

−≈

+

+

+≈

=

=

and

Time Dependent Perturbation Theory

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

Instead of discrete state to state transitions, it is often useful to consider a discrete state to “band of states” transition. Examples of this are donor states to conduction band transitions, acceptor states to valence band transitions or simply defect/impurity states to either conduction/valence band transitions.These bands can be described by their density per unit energy (see lecture 7) or since E=ħω, the density (number) per frequency ω centered around the transition frequency = ρ(ωmj).

Assuming that this density of states, ρ(ωmj), does not change quickly with ωmj, we can find the new probability density of state m by integrating the previous expression over ωmj. For example,

Time Dependent Perturbation Theory

( )( ) ( )( )( )

( )( ) o

(0)j

(0)mmjmj2

omj

omj2

2

2(0)jn

(0)m'

m*'

m ωEfor E ωω

21

tωω21sin(x)ΨA(x)(x)Ψ

tata hh

+≈

≈ ∫

=ωωρ d

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

Making some assumptions about this function:1) The transition matrix element is a slowly varying function of ωmj.2) The density of states, ρ(ωmj), also does not change quickly with ωmj.3) Both of the above two conditions can be achieved by noting that since the [sin2...]

function sharpens in in ωmj with increasing time, we can always wait long enough in time to make this part of the integrand the most rapidly varying portion of the integrand in ωmj.

...

Time Dependent Perturbation Theory

( )( ) ( )( )( )

( )( ) o

(0)j

(0)mmjmj2

omj

omj2

2

2(0)jn

(0)m'

m*'

m ωEfor E ωω

21

tωω21sin(x)ΨA(x)(x)Ψ

tata hh

+≈

≈ ∫

=ωωρ d

ECE 6451 - Dr. Alan DoolittleGeorgia Tech

...with these assumptions, the above expression becomes:

Time Dependent Perturbation Theory

( )( ) ( )( ) ( )( )

( )

( )( ) ( )( ) ( )[ ]

,expression above theof derivative timethemerely is states of banda tostate discretea from change of rate theHence

ωEfor E 2(x)ΨA(x)(x)Ψ

tata

ωEfor E ωω

21

tωω21sin(x)ΨA(x)(x)Ψ

tata

o(0)j

(0)mmj

2

2

(0)jn

(0)m'

m*'

m

o(0)j

(0)mmj2

omj

omj2

mj2

2(0)jn

(0)m'

m*'

m

hh

hh

+≈

+≈

=

=

t

d

πωρ

ωωρ

Fermi’s Golden Rule:

( ) ( )o)0(2(0)

jn(0)m

2'mmj (x)ΨA(x)(x)Ψ2taW ωρπ

hh

+=≈= =→ jEEdtd

Where, since E=ħω, we have replaced ρ(ω) by the equivalent energy density of states, ρ(ω) ħρ(E).One Final Note: Since Fermi’s Golden Rule was derived from 1st order time dependent perturbation theory, it is only valid for “short” times for which the initial state occupancy does not significantly change. If you wait long enough after a perturbation this will eventually not be the case and higher order theories will be required.