Page 333 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

324 Appendix G
e 2ðim ˆ 1
so that m is an integer, m ˆ 0, 1, 2, .. .
To solve the differential equation (G.22), we introduce a change of variable
ì ˆ cos è (G:25)
The function È(è) then becomes a new function F(ì) of the variable ì, È(è) ˆ F(ì),
so that
dÈ dF dì dF 2 1=2 dF
ˆ ˆÿsin è ˆÿ(1 ÿ ì ) (G:26)
dè dì dè dì dì
Substitution of equations (G.25) and (G.26) into (G.22) gives
!
2
d 2 dF m
(1 ÿ ì ) ‡ ë ÿ F ˆ 0
dì dì 1 ÿ ì 2
or
!
m 2
2
(1 ÿ ì )F 0 ÿ 2ìF9 ‡ ë ÿ F ˆ 0 (G:27)
1 ÿ ì 2
A power series solution of equation (G.27) yields a recursion formula relating a k‡4 ,
a k‡2 , and a k , which is too complicated to be practical. Accordingly, we make the
further de®nition
2 jmj=2
F(ì) ˆ (1 ÿ ì ) G(ì) (G:28)
from which it follows that
2 ÿ1
2 jmj=2
F9 ˆ (1 ÿ ì ) [G9 ÿjmjì(1 ÿ ì ) G] (G:29)
2 ÿ1
2 ÿ1
2 jmj=2
F 0 ˆ (1 ÿ ì ) [G0 ÿ 2jmjì(1 ÿ ì ) G9 ÿjmj(1 ÿ ì ) G
2 ÿ2
2
‡jmj(jmjÿ 2)ì (1 ÿ ì ) G] (G:30)
2 jmj=2
Substitution of (G.28), (G.29), and (G.30) into (G.27) with division by (1 ÿ ì )
gives
2
(1 ÿ ì )G0 ÿ 2(jmj‡ 1)ìG9 ‡ [ë ÿjmj(jmj‡ 1)]G ˆ 0 (G:31)
To solve this differential equation, we substitute equations (G.2), (G.3), and (G.4)
for G, G9, and G0 to obtain
1 1
X k‡sÿ2 X k‡s
a k (k ‡ s)(k ‡ s ÿ 1)ì ‡ a k [ë ÿ (k ‡ s ‡jmj)(k ‡ s ‡jmj‡ 1)]î
kˆ0 kˆ0
ˆ 0 (G:32)
Equating the coef®cient of ì sÿ2 to zero, we obtain the indicial equation
a 0 s(s ÿ 1) ˆ 0 (G:33)
with solutions s ˆ 0 and s ˆ 1. Equating the coef®cient of ì sÿ1 to zero gives
a 1 s(s ‡ 1) ˆ 0 (G:34)
For the case s ˆ 0, the coef®cient a 1 has an arbitrary value, while for s ˆ 1, the
coef®cient a 1 must vanish.
If we replace the dummy index k by k ‡ 2 in the ®rst summation on the left-hand
side of equation (G.32), that equation becomes