Page 250 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

9.3 Non-degenerate perturbation theory 241

This equation has the form
X k
f (å) ˆ b k å ˆ 0
k
where

k
1 @ f
b k ˆ
k! @å k åˆ0
Since f (å) is identically zero, the coef®cients b k all vanish. Thus, the coef®-
k
cients of ë on the left-hand side of equation (9.21) are equal to the coef®cients
0
k
of ë on the right-hand side. The coef®cients of ë give equation (9.18) for the
unperturbed system. The coef®cients of ë yield
^
(1)
(0)
^
(H (0) ÿ E )ø (1) ˆÿ(H (1) ÿ E )ø (0) (9:22)
n
n
n
n
2
while the coef®cients of ë give
^
^
^
(2)
(0)
(1)
(H (0) ÿ E )ø (2) ‡ (H (1) ÿ E )ø (1) ˆÿ(H (2) ÿ E )ø (0) (9:23)
n n n n n n
and so forth.
First-order corrections
To ®nd the ®rst-order correction E (1) to the eigenvalue E n , we multiply
n
equation (9.22) by the complex conjugate of ø (0) and integrate over all space to
n
obtain
^
^
(1)
(0)
(0)
(0)
(1)
(1)
(0)
(0)
(0)
hø jH jø iÿ E hø jø iˆÿhø jH jø i‡ E (1)
n n n n n n n n
^ (0)
where we have noted that ø (0) is normalized. Since H is hermitian, the ®rst
n
integral on the left-hand side takes the form
(0) ^ (0) (1) ^ (0) (0) (1) (0) (0) (1)
hø jH jø iˆhH ø jø iˆ E hø jø i
n n n n n n n
and therefore cancels the second integral on the left-hand side. The ®rst-order
^ (1)
correction E (1) is, then, the expectation value of the perturbation H in the
n
unperturbed state
^
^
(0)
(1)
(0)
E (1) ˆhø jH jø i H (1) (9:24)
n n n nn
The ®rst-order correction ø (1) to the eigenfunction is obtained by multi-
n
plying equation (9.22) by the complex conjugate of ø (0) for k 6ˆ n and
k
integrating over all space to give
(0) ^ (0) (1) (0) (0) (1) (0) ^ (1) (0) (1) (0) (0)
hø jH jø iÿ E hø jø iˆÿhø jH jø i‡ E hø jø i
k n n k n k n n k n
(9:25)
Noting that the unperturbed eigenfunctions are orthogonal
(0) (0)
hø jø iˆ ä kn (9:26)
k n