Lecture 9 Non-degenerate & Degenerate Time Independent...
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
∑∑
∑∑
∑
ECE 6451 - Dr. Alan DoolittleGeorgia Tech
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
ECE 6451 - Dr. Alan DoolittleGeorgia Tech
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
ECE 6451 - Dr. Alan DoolittleGeorgia Tech
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
hπ
ππ
==
=Ψ
=Ψ
=
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
=Ψ
+=
==
==
==
)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
=Ψ
+=
==
==
==
+
+
)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
tω
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
tω
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
tω
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.