Page 331 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

322 Appendix G
2 3
2 ë(2 ÿ ë) 2 ë(2 ÿ ë)(4 ÿ ë)
2 4 6
u 1 ˆ a 0 1 ÿ ëî ÿ î ÿ î ÿ (G:16a)
4! 6!

2
3
2(1 ÿ ë) 3 2 (1 ÿ ë)(3 ÿ ë) 5 2 (1 ÿ ë)(3 ÿ ë)(5 ÿ ë) 7
u 2 ˆ a 1 î ‡ î ‡ î ‡ î ‡
3! 5! 7!
(G:16b)
The solution u 1 is an even function of the variable î and u 2 is an odd function of î.
Accordingly, u 1 and u 2 are independent solutions. For the case s ˆ 1, we again obtain
the solution u 2 .
The ratio of consecutive terms in either series solution u 1 or u 2 is given by the
recursion formula with s ˆ 0as
a k‡2 î k‡2 2(k ÿ ë)
ˆ î 2
a k î k (k ‡ 2)(k ‡ 1)
In the limit as k !1, this ratio approaches zero
a k‡2 î k‡2 2
2
lim ! lim î ! 0
k!1 a k î k k!1 k
so that the series u 1 and u 2 converge for all ®nite values of î. To see what happens to
u 1 and u 2 as î ! 1, we consider the Taylor series expansion of e î 2
1 î 2n î 4 î 6
X
2
e î 2 ˆ ˆ 1 ‡ î ‡ ‡ ‡
n! 2! 3!
nˆ0
The coef®cient a n is given by a n ˆ 1=(n=2)! for n even and a n ˆ 0 for n odd, so that

n
a n‡2 î n‡2 2 ! 1 2î 2
2
2 î as n !1
a n î n ˆ n ‡ 2 î ˆ n n
! ‡ 1
2 2
Thus, u 1 and u 2 behave like e î 2 as î ! 1. For large jîj, the function ö(î) behaves
like
2
î
2
2
2
ö(î) ˆ u(î)e ÿî =2 e e ÿî =2 ˆ e î =2 !1 as î ! 1
which is not satisfactory behavior for a wave function.
In order to obtain well-behaved solutions for the differential equation (G.8), we need
to terminate the in®nite power series u 1 and u 2 in (G.16) to a ®nite polynomial. If we
let ë equal an integer n (n ˆ 0, 1, 2, 3, .. .), then we obtain well-behaved solutions
ö(î)
2
n ˆ 0, ö 0 ˆ a 0 e ÿî =2 , a 1 ˆ 0
2
n ˆ 1, ö 1 ˆ a 1 îe ÿî =2 , a 0 ˆ 0
2
2
n ˆ 2, ö 2 ˆ a 0 (1 ÿ 2î )e ÿî =2 , a 1 ˆ 0
2
2 2
n ˆ 3, ö 3 ˆ a 1 î(1 ÿ î )e ÿî =2 , a 0 ˆ 0
3
2
2
4 4
n ˆ 4, ö 4 ˆ a 0 (1 ÿ 4î ‡ î )e ÿî =2 , a 1 ˆ 0
3
2
4 2
4
4
n ˆ 5, ö 5 ˆ a 1 î(1 ÿ î ‡ î )e ÿî =2 , a 0 ˆ 0
3 15
. . . . . .