Page 330 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

Series solutions of differential equations 321

We ®rst investigate the asymptotic behavior of ö(î). For large values of î, the
2
constant 2ë ‡ 1 may be neglected in comparison with î and equation (G.8) becomes
2
ö0 ˆ î ö
The approximate solutions of this differential equation are
2
ö ˆ ce î =2
because we have
ö0 ˆ (î 1)ö î ö for large î
2
2
2
The function e î =2 is not a satisfactory solution because it becomes in®nite as
2
î ! 1, but the function e ÿî =2 is well-behaved. This asymptotic behavior of ö(î)
suggests that a satisfactory solution of equation (G.8) has the form
2
ö(î) ˆ u(î)e ÿî =2 (G:9)
where u(î) is a function to be determined.
Substitution of equation (G.9) into (G.8) gives
u0 ÿ 2îu9 ‡ 2ëu ˆ 0 (G:10)
We solve this differential equation by the series solution method. Applying equations
(G.2), (G.3), and (G.4), we obtain
1 1
X k‡sÿ2 X k‡s
a k (k ‡ s)(k ‡ s ÿ 1)î ‡ a k [ÿ2(k ‡ s) ‡ 2ë]î ˆ 0 (G:11)
kˆ0 kˆ0
The coef®cient of î sÿ2 gives the indicial equation
a 0 s(s ÿ 1) ˆ 0 (G:12)
with two solutions, s ˆ 0 and s ˆ 1. The coef®cient of î sÿ1 gives
a 1 (s ‡ 1)s ˆ 0 (G:13)
For the case s ˆ 0, the coef®cient a 1 has an arbitrary value; for s ˆ 1, we have a 1 ˆ 0.
If we omit the ®rst two terms (they vanish according to equations (G.12) and
(G.13)) in the ®rst summation on the left-hand side of (G.11) and replace the dummy
index k by k ‡ 2 in that summation, we obtain
1
X k‡s
fa k‡2 (k ‡ s ‡ 2)(k ‡ s ‡ 1) ‡ a k [ÿ2(k ‡ s) ‡ 2ë]gî ˆ 0 (G:14)
kˆ0
Setting the coef®cient of each power of î equal to zero gives the recursion formula
2(k ‡ s ÿ ë)
a k‡2 ˆ a k (G:15)
(k ‡ s ‡ 2)(k ‡ s ‡ 1)
For the case s ˆ 0, the constants a 0 and a 1 are arbitrary and we have the following
two sets of expansion constants
a 0 a 1
2(1 ÿ ë)
a 2 ˆÿëa 0 a 3 ˆ a 1
3!
2
2
2(2 ÿ ë) 2 ë(2 ÿ ë) 2(3 ÿ ë) 2 (1 ÿ ë)(3 ÿ ë)
a 4 ˆ a 2 ˆÿ a 0 a 5 ˆ a 3 ˆ a 1
.
.
4 3 4! 5 4 5!
3
3
2(4 ÿ ë) 2 ë(2 ÿ ë)(4 ÿ ë) 2(5 ÿ ë) 2 (1 ÿ ë)(3 ÿ ë)(5 ÿ ë)
a 6 ˆ a 4 ˆÿ a 0 a 7 ˆ a 5 ˆ a 1
.
.
6 5 6! 7 6 7!
. .
. . . .
Thus, the two solutions of the second-order differential equation (G.10) are