Phys 852, Quantum mechanics II, Spring 2008
NonDegenerate TimeIndependent Perturbation Theory
1/14/2008
Prof. Michael G. Moore, Michigan State University
1
The central problem in timeindependent perturbation theory:
Let
H
0
be the unperturbed (a.k.a. ‘background’, ‘bare’) Hamiltonian whose eigenvalues and eigenvectors
are known. Let
E
(0)
n
be the
n
th
unperturbed energy eigenvalue, and

n
(0)
)
be the
n
th
unperturbed energy
eigenstate. They satisfy
H
0

n
(0)
)
=
E
(0)
n

n
(0)
)
(1)
and
(
n
(0)

n
(0)
)
= 1
.
(2)
Let
V
be a Hermitian operator which ‘perturbs’ the system, such that the full Hamiltonian is
H
=
H
0
+
V.
(3)
The standard approach is to instead solve
H
=
H
0
+
λV
(4)
, and use
λ
as for bookkeeping during the calculation, but set
λ
= 1 at the end of the calculation. However,
if there is a readily identifiable small parameter in the definition of the perturbation operator
V
, then use
it in the place of
λ
, and do not set it to 1. In these notes we will treat
λ
as a small parameter, and will
not set it equal to unity.
The goal is to find the eigenvalues and eigenvectors of the full Hamiltonian (4). Let
E
n
and

n
)
be the
n
th
eigenvalue and its corresponding eigenstate. They satisfy
H

n
)
=
E
n

n
)
(5)
and
(
n

n
)
= 1
.
(6)
Because
λ
is a small parameter, it is assumed that accurate results can be obtained by expanding
E
n
and

n
)
in powers of
λ
, and keeping only the leading term(s).
Formally expanding the perturbed quantities
gives
E
n
=
E
(0)
n
+
λE
(1)
n
+
λE
(2)
n
+
. . . ,
(7)
and

n
)
=

n
(0)
)
+
λ

n
(1)
)
+
λ
2

n
(2)
)
+
. . . ,
(8)
where
E
(
j
)
n
and

n
(
j
)
)
are yettobe determined expansion coefficients Inserting these expansions into the
eigenvalue equation (5) then gives
(
H
0
+
λV
)(

n
(0)
)
+
λ

n
(1)
)
+
λ
2

n
(2)
)
+
. . . .
) = (
E
(0)
n
+
λE
(1)
n
+
λ
2
E
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 Spring '10
 MichaelMoore
 mechanics, Eigenvalue, eigenvector and eigenspace, en, Singular perturbation, EMN, Vmk Vkn Vmn Vnn

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