Page 174 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

6.3 The radial equation 165
^
H9 l S ël ˆÿëS ël (6:35)
^
where H9 l is de®ned by
d 2 d r l(l ‡ 1)
^
H9 l ˆ r ‡ 2 ÿ ÿ (6:36)
dr 2 dr 4 r
^
By means of integration by parts, we can readily show that this operator H9 l is
hermitian for a weighting function w(r) equal to r, thereby implying the
orthogonal relationships

1
S ël (r)S ë9l (r)r dr ˆ 0 for ë 6ˆ ë9 (6:37)
0
In order to complete the characterization of integrals of S ël (r), we need to
consider the case where ë ˆ ë9 for w(r) ˆ r. Recall that the functions S ël (r)
2
are normalized for w(r) ˆ r as expressed in equation (6.25). The same result
does not apply for w(r) ˆ r. We begin by expressing the desired integral in a
slightly different form

1 1
2 1 2 2
[S ël (r)] r dr ˆ [S ël (r)] d(r )
2
0 0
Integration of the right-hand side by parts gives
1
1 1 d
2 1 2 2 2
[S ël (r)] r dr ˆ 2 r [S ël (r)] ÿ r S ël S ël dr
0 0 0 dr
If S ël (r) is well-behaved, the integrated term vanishes. From equation (6.26a)
we may write
d r
^
r ˆÿA ë ÿ ‡ ë ÿ 1
dr 2
so that
d
^
r S ël ˆÿA ë S ël ÿ rS ël ‡ (ë ÿ 1)S ël
1
dr 2
1
ˆÿa ël S ëÿ1,l ÿ rS ël ‡ (ë ÿ 1)S ël
2
where equation (6.32) has been introduced. The integral then takes the form

1 1 1
2 2
2 1 [S ël ] r dr
[S ël (r)] r dr ˆ a ël S ël S ëÿ1,l r dr ‡
2
0 0 0

1
2
ÿ (ë ÿ 1) [S ël ] r dr
0
Since the ®rst integral on the right-hand side vanishes according to equation
(6.37) and the second integral equals unity according to (6.25), the result is