Page 258 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

9.5 Degenerate perturbation theory 249

this section we modify the perturbation method to allow for degenerate
eigenvalues. In view of the complexity of this new procedure, we consider only
the ®rst-order perturbation corrections to the eigenvalues and eigenfunctions.
The eigenvalues and eigenfunctions for the unperturbed system are given by
equation (9.18), but now the eigenvalue E (0) is g n -fold degenerate. Accord-
n
(0)
ingly, there are g n eigenfunctions with the same eigenvalue E . For greater
n
clarity, we change the notation here and denote the eigenfunctions correspond-
(0)
ing to E (0) as ø , á ˆ 1, 2, ... , g n . Equation (9.18) is then replaced by the
n ná
equivalent expression
(0)
(0)
(0)
H ø (0) ˆ E ø , á ˆ 1, 2, ... , g n (9:53)
ná
ná
n
Each of the eigenfunctions ø (0) is orthogonal to all the other unperturbed
ná
eigenfunctions ø (0) for k 6ˆ n, but is not necessarily orthogonal to the other
ká
(0)
eigenfunctions for E . Any linear combination ö ná of the members of the set
n
ø (0)
ná
g n
X (0)
ö ná ˆ c áâ ø , á ˆ 1, 2, ... , g n (9:54)
nâ
âˆ1
(0)
is also a solution of equation (9.53) with the same eigenvalue E .As
n
discussed in Section 3.3, the members of the set ø (0) may be constructed to be
ná
orthonormal and we assume that this construction has been carried out, so that
(0)
(0)
hø jø iˆ ä áâ , á, ⠈ 1, 2, ... , g n (9:55)
nâ ná
By suitable choices for the coef®cients c áâ in equation (9.54), the functions
ö ná may also be constructed as an orthonormal set
hö nâ jö ná iˆ ä áâ , á, ⠈ 1, 2, ... , g n (9:56)
Substitution of equation (9.54) into (9.56) and application of (9.55) give
g n
X

c c á㠈 ä áâ , á, ⠈ 1, 2, ... , g n (9:57)
âã
ãˆ1
È
The Schrodinger equation for the perturbed system is
^
Hø ná ˆ E ná ø ná , á ˆ 1, 2, ... , g n (9:58)
^
where the Hamiltonian operator H is given by equation (9.16), E ná are the
eigenvalues for the perturbed system, and ø ná are the corresponding eigen-
functions. While the unperturbed eigenvalue E (0) is g n -fold degenerate, the
n
^
perturbation H9 in the Hamiltonian operator often splits the eigenvalue E (0)
n
into g n different values. For this reason, the perturbed eigenvalues E ná require
the additional index á. The perturbation expansions of E ná and ø ná in powers
of ë are