Page 335 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

326 Appendix G

ø lm (è, j) ˆ c lm P jmj (cos è)e imj (G:40)
l
where c lm are the normalization constants. A comparison of equation (G.40) with
(5.59) shows that the functions ø lm (è, j) are the spherical harmonics Y lm (è, j).

Radial equation for the hydrogen-like atom
The radial differential equation for the hydrogen-like atom is given by equation (6.24)
as
2
d S 2 dS 1 ë l(l ‡ 1)
‡ ‡ÿ ‡ ÿ S ˆ 0 (G:41)
dr 2 r dr 4 r r 2
where l is a positive integer. If a power series solution is applied directly to equation
(G.41), the resulting recursion relation involves a k‡2 , a k‡1 , and a k . Since such a three-
term recursion relation is dif®cult to handle, we ®rst examine the asymptotic behavior
of S(r). For large values of r, the terms in r ÿ1 and r ÿ2 become negligible and
equation (G.41) reduces to
2
d S S
ˆ
dr 2 4
or
S ˆ ce r=2
where c is the integration constant. Since r, as de®ned in equation (6.22), is always
real and positive for E < 0, the function e r=2 is not well-behaved, but e ÿr=2 is.
Therefore, we let S(r) take the form
S(r) ˆ F(r)e ÿr=2 (G:42)
Substitution of equation (G.42) into (G.41) yields
2
r F 0 ‡ r(2 ÿ r)F9 ‡ [(ë ÿ 1)r ÿ l(l ‡ 1)]F ˆ 0 (G:43)
2 r=2
where we have multiplied through by r e . To solve this differential equation by the
series solution method, we substitute equations (G.2), (G.3), and (G.4) for F, F9, and
F 0 to obtain
1 1
X k‡s X k‡s‡1
a k [(k ‡ s)(k ‡ s ‡ 1) ÿ l(l ‡ 1)]r ‡ a k (ë ÿ 1 ÿ k ÿ s)r ˆ 0
kˆ0 kˆ0
(G:44)
s
The indicial equation is given by the coef®cient of r as
a 0 [s(s ‡ 1) ÿ l(l ‡ 1)] ˆ 0 (G:45)
with solutions s ˆ l and s ˆÿ(l ‡ 1). For the case s ˆÿ(l ‡ 1), we have
F(r) ˆ a 0 r ÿ(l‡1) ‡ a 1 r ÿl ‡ a 2 r ÿl‡1 ‡ (G:46)
4
which diverges at the origin. Thus, the case s ˆ l is the only acceptable solution.
Omitting the vanishing ®rst term in the ®rst summation on the left-hand side of
(G.44) and replacing k by k ‡ 1 in that summation, we have

4 The reason for rejecting the solution s ˆÿ(l ‡ 1) is actually more complicated for states with l ˆ 0.
I. N. Levine (1991) Quantum Chemistry, 4th edition (Prentice-Hall, Englewood Cliffs, NJ), p. 124,
summarizes the arguments with references to more detailed discussions. The complications here strengthen
the reasons for preferring the ladder operator technique used in the main text.