Chem 120A
Perturbation Theory
03/12/07
Spring 2007
Lecture 22
READING:
(RS): Chapter 3
(AF): Chapter 6.16.7
Introduction to perturbation theory
The goal of the perturbation theory is to provide systematic approach for approximating solutions of complex
(hard) systems.
Suppose that we have an arbitrary Hamiltonian,
ˆ
H
, which we can express as the sum of another Hamiltonian
for which we know the exact solutions,
ˆ
H
0
, and change to the potential term,
V
,
ˆ
H
=
ˆ
H
0
+
V
.
(1)
The eigenvalues and eigenfunctions of
ˆ
H
0
are given by
ˆ
H
0
ψ
n
=
ε
n
ψ
n
, where we are using
ψ
n
to indicate a
state
,
vextendsingle
vextendsingle
ψ
n
)big
, rather than
ψ
(
x
)
which is an amplitude
. What we would like to know are the eigenfunctions and
eigenvalues of the perturbed Hamiltonian
ˆ
H
. Since
V
is small, we will seek solutions which are in the form
of a perturbative expansion. In order to keep track of the smallness of the perturbation
V
, it is convenient to
introduce a parameter
λ
, which can take on values between 0 and 1,
ˆ
H
(
λ
) =
ˆ
H
0
+
λ
V
.
(2)
Therefore, by adjusting the value of
λ
, we can adjust the Hamiltonian; when
λ
=
0 then
ˆ
H
=
ˆ
H
0
, while
when
λ
=
1, then we have the full system Hamiltonian. The new eigenvalue equation is given by
ˆ
H
(
λ
)
Ψ
n
(
λ
) =
E
n
(
λ
)
Ψ
n
(
λ
)
.
(3)
Since
V
must be small in order for perturbation theory to be valid, if we plot the eigenenergies of
ˆ
H
(
λ
)
vs.
λ
(Fig. 1), we should see that they are relatively constant, and that they do not cross. This is valid as long
as
V
<<
ε
n
. Now, since the
V
is small, we assume that we can write the eigenvalues and eigenfunction of
ˆ
H
(
λ
)
by using a Taylor series expansion in
λ
:
E
n
(
λ
)
=
ε
n
+
λε
′
n
+
λ
2
ε
′′
n
+
...
Ψ
n
(
λ
)
=
ψ
n
+
λψ
′
n
+
λ
2
ψ
′′
n
+
...
(4)
To first order, the correction to each energy,
ε
, due to the presence of the perturbation is
ε
′
, while
ψ
′
n
is the
first order correction to the eigenfunction. In order to determine what these corrections should be, we will
substitute Eqs. 4 into Eq. 3:
(
ˆ
H
0
+
λ
V
)(
ψ
n
+
λψ
′
n
+
λ
2
ψ
′′
n
+
...
) = (
ε
n
+
λε
′
n
+
λ
2
ε
′′
n
+
...
)(
ψ
n
+
λψ
′
n
+
λ
2
ψ
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 Spring '07
 Whaley
 Physical chemistry, pH, Schrodinger Equation, ψn, ﬁrst order correction, ψn ψn

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