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ECE 3030 – Summer 2009 – Cornell University
Lecture 20
Time Independent Perturbation Theory - I
In this lecture you will learn:
• Time independent perturbation theory• Controlled expansions for changes in eigenenergies and eigenstates
ECE 3030 – Summer 2009 – Cornell University
Quantum SHO: A Recap
Consider a particle of charge q in a SHO potential:
The Hamiltonian is:x0
012 oE
1112 oE
2122 oE
3132 oE
1 1ˆ ˆ2 2o o o nH n n n n n E n
22 2 †ˆ 1 1 1ˆ ˆ ˆ ˆˆ
2 2 2 2o o o opH m x a a nm
The eigenstates are:
The wavefunctions are:
†
0
1 ˆ 0!
12 2!
nn
no
no
x x n x an
mx x xm xn
ECE 3030 – Summer 2009 – Cornell University
Perturbing the SHO with a DC Electric FieldNow suppose a small electric field is added to the potential:
22 2ˆ 1 ˆˆ ˆˆ ˆ
2 2 o x opH m x qE x H Om
+2 21
2 om x
xqE x
We know the final answer:
● All energies are shifted by a constant amount:
● All wavefunctions are also shifted in space by
2 22 2 2ˆ 1 1ˆ ˆ
2 2 2o o o opH m x x m xm
22 22
1 1 12 2 2
xnewn o o o n
o
qEE n m x E
m
newn n onewx x n x x
ˆ newnnew newH n E n
2x
oo
qExm
=ox
x x x
2212 o om x x
ECE 3030 – Summer 2009 – Cornell University
Perturbation Theory: The Basic Idea
Question: If we write the new energies and new wavefunctions as the old energies and old wavefunctions plus corrections, then can the corrections be expended as a power series in the perturbation strength?
2 31 2 3 .....new
n n n n x x xE E E E a E a E a E
A power series in the perturbation strength
2 31 2 3 ....new
n n n n x x xx x x x f x E f x E f x E
A power series in the perturbation strength
Question:How do we systematically find the coefficients a1 , a2 , a3, …… and how do we systematically find the functions f1(x) , f2(x) , f3(x) , ……. ???
ECE 3030 – Summer 2009 – Cornell University
Perturbation Theory: Statement of the Problem
Consider a problem for which the Hamiltonian is:
ˆo j j jH e E e
The eigenstates are:
ˆoH
1j jj
e e
The eigenstates form a complete orthonormal set:
j k jke e
Now suppose a small perturbation is added to the Hamiltonian:
ˆˆ ˆoH H O
O
Then new energy eigenstates and eigenvalues are given by:
ˆ new new newj j jH e E e
We write:newj j jE E E
newj j je e e
How do we find: jE
How do we find: je
ECE 3030 – Summer 2009 – Cornell University
Perturbation Theory: Change in Eigenstatesnewj j je e e 1j j
je e
j k jke e The change is (without loosing generality) orthogonal to the original state:
0j je e
1
j j m m jm
m j m m mm m j
e e e e e
e e e c e
We write:
Finding amounts to finding the coefficients je mc
ECE 3030 – Summer 2009 – Cornell University
Perturbation Theory: Smallness Parameter
We write the perturbation term as:
Here is a “smallness” parameter that will help us do a controlled perturbation expansion in powers of and we will set equal to unity when we are done with the calculations
ˆˆ ˆoH H O
O
1 2 32 3
...........
newj j j
j m mm j
j m m m mm j
e e e
e c e
e c c c e
A power series in the perturbation strength
A power series in the perturbation strength
Eigenenergies:
Eigenstates:
1 2 32 3 .....newj j j j j j jE E E E E E E
ECE 3030 – Summer 2009 – Cornell University
Perturbation Theory: EigenequationWe start from:
ˆ
ˆˆ
new new newj j j
new new newo j j j
H e E e
H O e E e
And we substitute the expressions: 1 2 32 3 .....new
j j j j j j jE E E E E E E
1 2 32 3 ...........newj j m m m m
m je e c c c e
To get:
1 2 32 3
1 2 3 1 2 32 3 2 3
ˆˆ ...........
..... ...........
o j m m m mm j
j j m m m mj j jm j
H O e c c c e
E E E E e c c c e
ECE 3030 – Summer 2009 – Cornell University
Perturbation Theory: First Order Eigenenergy Change
1 2 32 3
1 2 3 1 2 32 3 2 3
ˆˆ ...........
..... ...........
o j m m m mm j
j j m m m mj j jm j
H O e c c c e
E E E E e c c c e
STEP 1:Multiply by the bra from the left side and keep only the terms that are of zero-order in the parameter (i.e. that are independent of )
je
ˆj o j j j je H e E e e
STEP 2:Multiply by the bra from the left side and keep only the terms that are of first-order in the parameter
je
11 1
1
ˆ ˆ
ˆ
j j m j o m j j j m j mjm j m j
j jj
e O e c e H e E e e E c e e
E e O e
0 0
This means that the first order energy change of an eigenstate is just the mean value of the perturbation operator with respect to that eigenstate
Nothing new here
1 ˆ .....newj j j j jjE E E E e O e
ECE 3030 – Summer 2009 – Cornell University
Perturbation Theory: First Order Eigenstate Change
1 2 32 3
1 2 3 1 2 32 3 2 3
ˆˆ ...........
..... ...........
o j m m m mm j
j j m m m mj j jm j
H O e c c c e
E E E E e c c c e
STEP 3:Multiply by the bra (where p ≠ j) from the left side and keep only the terms that are of first-order in the parameter
pe
11 1
1 1
1
ˆ ˆ
ˆ
ˆ
p j m p o m p j j m p mjm j m j
p j p p j p
p jp
j p
e O e c e H e E e e E c e e
e O e c E E c
e O ec
E E
0
This means the first order change in the eigenstate is:
1ˆ
..... ......m jnewj j m m j m
m j m j j m
e O ee e c e e e
E E
ECE 3030 – Summer 2009 – Cornell University
Perturbation Theory: Second Order Eigenenergy Change
1 2 32 3
1 2 3 1 2 32 3 2 3
ˆˆ ...........
..... ...........
o j m m m mm j
j j m m m mj j jm j
H O e c c c e
E E E E e c c c e
STEP 4:Multiply by the bra from the left side and keep only the terms that are of second-order in the parameter
je
1 22
2 1 1 22 2
22 1
ˆ
ˆˆ
m j m m j mm j m j
j j m j m j m j mj jm j m j
j mm j mj
m j m j j m
c e O e c e e
E e e E c e e E c e e
e O eE c e O e
E E
0
0 0
1ˆ
m jm
j m
e O ec
E E
This means that up to second order the energy change of an eigenstate is:
2
1 2ˆ
ˆ..... ......j mnew
j j j j jj jm j j m
e O eE E E E E e O e
E E
ECE 3030 – Summer 2009 – Cornell University
Perturbed SHO: Exact Solution
+2 21
2 om x
xqE x =ox
x x x
2212 o om x x
22
12
xnewj j
o
qEE E
m
2
22
1 ........2!
j jnewj j o j o o
x xx x x x x x
x x
22 2ˆ 1 ˆˆ ˆˆ ˆ
2 2 o x opH m x qE x H Om
ECE 3030 – Summer 2009 – Cornell University
Perturbed SHO: Exact Solution for the Ground State
+2 21
2 om x
xqE x =ox
x x x
2212 o om x x
20 0 2
12
xnew
o
qEE E
m
00 0 0
00
0 1 2
........
........
......2
0 0 1 ......2
newo o
o
o x
o
xnew
o o
xx x x x x
xx
x xx
m qEx xm
qEm
ECE 3030 – Summer 2009 – Cornell University
Perturbation Theory for the SHO: Ground State
1
0 0
ˆ 00 0 ..... 0 ......x
mnew m j m m
m qE xc m m
E E
only m=1 will give a non-zero contribution
11
0 1
ˆ1 00 0 1 ..... 0 1 ......
ˆ1 0 0 1 ......
ˆ1 0 0 1 ......
0 1 ......2
xmnew
x
o
xo
x
o o
qE xc
E EqE x
xqE
qEm
Matches with the exact solution to first order in the electric field
ECE 3030 – Summer 2009 – Cornell University
Perturbation Theory for the SHO: Ground State Energy
21 2
0 0 00 00 0
ˆ0ˆ..... 0 0 ......x mnew
xm m
qE x eE E E E E qE x
E E
2
0 00 1
2
0
2
0 2
ˆ0 1ˆ0 0 ......
2 ......
1 ........2
xnewx
xo
o
x
o
qE xE E qE x
E E
qEm
E
qEE
m
0
Matches with the exact solution to second order in the electric field
only m=1 will give a non-zero contribution
ECE 3030 – Summer 2009 – Cornell University
Perturbation Theory Example
Finite Potential Well
1
2
3
The Finite Potential Well with a Bump
V=UV=U
x
V=0
02L 2L
A potential bump
ECE 3030 – Summer 2009 – Cornell University
Perturbation Theory Example: Stark Effect
Finite Potential Well
1
2
3
Finite Potential Well in a DC Electric Field
Shifts in the eigenenergies are called Stark shifts
V=U – qExx
x
V=0
02L 2L
V=U – qExx
V= – qExx
21 2 ˆ
ˆ..... ......j x mnew
j j j j x jj jm j j m
e qE x eE E E E E e qE x e
E E
Stark shift
Johannes Stark(1874-1957)Nobel Prize
ECE 3030 – Summer 2009 – Cornell University
Stark Effect in Telecom: Electro-Absorption Modulators
Modulated Laser Light
ECE 3030 – Summer 2009 – Cornell University
Stark Effect in Telecom: Electro-Absorption Modulators
Light absorption can be switched on or off by applying a DC electric field to a semiconductor quantum well
ECE 3030 – Summer 2009 – Cornell University
