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Physics 6210/Spring 2007/Lecture 32
Lecture 32
Relevant sections in text:
§
5.2
Degenerate Perturbation Theory (cont.)
Degenerate perturbation theory leads to the following conclusions (see the text for
details of the derivation). To compute the ﬁrstorder corrections to the energies and eigen
vectors when there is degeneracy in the unperturbed eigenvalue one proceeds as follows.
Step 1
Consider the restriction,
˜
V
of the perturbation
V
to the
d
dimensional degenerate
subspace
D
.
˜
V
is deﬁned as follows. The action of
V
on a vector from
D
is some other
vector in the Hilbert space. Take the component of this vector along
D
,
i.e.,
project this
vector back into
D
. This process deﬁnes a Hermitian linear mapping
˜
V
from
D
to itself.
In practice, the most convenient way to compute
˜
V
is to pick a basis for
D
. Compute the
matrix elements of
V
in this basis. One now has a
d
×
d
Hermitian matrix representing
˜
V
.
Step 2
Find the eigenvectors and eigenvalues of
˜
V
. Again, it is most convenient to do this
using the matrix representation of
˜
V
.
Step 3
The ﬁrstorder energy corrections to
E
(0)
n
are the eigenvalues of
˜
V
. The zeroth order
limit of the true eigenvectors that correspond to the true eigenvalues are the eigenvectors
of
˜
V
that correspond to the ﬁrstorder eigenvalue corrections.
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 Spring '07
 M
 Physics, mechanics

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