Page 262 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

9.5 Degenerate perturbation theory 253
(1)
^
^ ^
(1)
^ ^
h÷ â j[A, H ]j÷ á iˆh÷ â jAH j÷ á iÿh÷ â jH (1) ^
Aj÷ á i
^ ^ (1) ^ (1)
ˆhA÷ â jH j÷ á iÿ ì á h÷ â jH j÷ á i
(0) ^ (1) (0)
ˆ (ì â ÿ ì á )hø jH jø i
nâ ná
^
ˆ (ì â ÿ ì á )H (1)
nâ,ná
ˆ 0
^
Since ì â 6ˆ ì á , the off-diagonal elements H (1) equal zero and the set of
nâ,ná
^
functions ø (0) ˆ ÷ á is the `correct' set. The parity operator Ð discussed in
ná
Section 3.8 can often be used in this context for selecting `correct' unperturbed
eigenfunctions.
First-order corrections to the eigenfunctions
To obtain the ®rst-order corrections ø (1) to the eigenfunctions ø ná , we multiply
ná
equation (9.62) by ø (0) for k 6ˆ n and integrate over all space
kâ
^
^
(0)
(0)
(0)
(0)
(0)
(1)
(0)
(1)
(1)
hø jH (0) ÿ E jø iˆÿhø jH jö i‡ E hø jö i
kâ n ná kâ ná ná kâ ná
^ (0)
Applying the hermitian property of H and noting that ø (0) is orthogonal to
kâ
(0)
all eigenfunctions belonging to the eigenvalue E ,we have
n
(0) (0) (0) (1) (0) ^ (1) (0)
(E ÿ E )hø jø iˆÿhø jH jö i (9:68)
k n kâ ná kâ ná
We next expand the ®rst-order correction ø (1) in terms of the complete set of
ná
unperturbed eigenfunctions
g j
X X
ø (1) ˆ a ná, jã ø (0) (9:69)
ná
jã
j(6ˆn) ãˆ1
where the terms with j ˆ n are omitted for the same reason that they are
omitted in equation (9.30). Substitution of equations (9.54) and (9.69) into
(9.68) gives
g j g n
X X X
(0) (0) (0) (0) ^ (1)
(E ÿ E ) a ná, jã hø jø iˆ ÿ c áã H
k n kâ jã kâ,nã
j(6ˆn) ãˆ1 ãˆ1
In view of the orthonormality relations, the summation on the left-hand side
may be simpli®ed as follows
g j g j
X X X X
(0) (0)
a ná, jã hø jø iˆ a ná, jã ä kj ä â㠈 a ná,kâ
kâ jã
j(6ˆn) ãˆ1 j(6ˆn) ãˆ1
Therefore, we have