Page 176 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

6.3 The radial equation 167
1=2
^ ë ÿ 1
A ë S ël ˆ (ë ‡ l)(ë ÿ l ÿ 1) S ëÿ1,l (6:44)
ë
where we have arbitrarily taken the positive square root.
The numerical constant b ël , de®ned in equation (6.34), may be determined
by an analogous procedure, beginning with the square of both sides of equation
(6.34) and using equations (6.27), (6.26a), (6.34), (6.30), and (6.39). We obtain

ë ë
2
b ˆ [ë(ë ÿ 1) ÿ l(l ‡ 1)] ˆ (ë ‡ l)(ë ÿ l ÿ 1) (6:45)
ël
ë ÿ 1 ë ÿ 1
so that equation (6.34) becomes
1=2
ë
^
B ë S ëÿ1,l ˆ (ë ‡ l)(ë ÿ l ÿ 1) S ë,l (6:46)
ë ÿ 1
Taking the positive square root here will turn out to be consistent with the
choice in equation (6.44).

Quantization of the energy
The parameter ë is positive, since otherwise the radial variable r, which is
inversely proportional to ë, would be negative. Furthermore, the parameter ë
cannot be zero if the transformations in equations (6.21), (6.22), and (6.23) are
to remain valid. To ®nd further restrictions on ë we must consider separately
the cases where l ˆ 0 and where l > 1.
For l ˆ 0, equation (6.44) takes the form
^
A ë S ë0 ˆ (ë ÿ 1)S ëÿ1,0 (6:47)
Suppose we begin with a suitably large value of ë,say î, and continually apply
the lowering operator to both sides of equation (6.47) with ë ˆ î
^ ^
A îÿ1 A î S î0 ˆ (î ÿ 1)(î ÿ 2)S îÿ2,0
^ ^ ^
A îÿ2 A îÿ1 A î S î0 ˆ (î ÿ 1)(î ÿ 2)(î ÿ 3)S îÿ3,0
.
. .
Eventually this procedure produces an eigenfunction S îÿk,0 , k being a positive
integer, such that 0 ,(î ÿ k) < 1. The next step in the sequence would give a
function S îÿkÿ1,0 or S ë0 with ë ˆ (î ÿ k ÿ 1) < 0, which is not allowed. Thus,
the sequence must terminate with the condition
^
A îÿk S îÿk,0 ˆ (î ÿ k ÿ 1)S îÿkÿ1,0 ˆ 0
which can only occur if (î ÿ k) ˆ 1. Thus, î must be an integer and the
minimum value of ë for l ˆ 0is ë ˆ 1.
2
For the situations in which l > 1, we note that the quantities a in equation
ël