Page 171 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

162 The hydrogen atom

ized. Since the radial part of the volume element in spherical coordinates is
2
r dr, the normalization criterion is

1
2 2
[R El (r)] r dr ˆ 1 (6:19)
0
Through an explicit integration by parts, we can show that

1 1
^ 2 ^ 2
R El (r)[H l R E9l (r)]r dr ˆ R E9l (r)[H l R El (r)]r dr
0 0
^
Thus, the operator H l is hermitian and the radial functions R El (r) constitute an
orthonormal set with a weighting function w(r) equal to r 2

1
2
R El (r)R E9l (r)r dr ˆ ä EE9 (6:20)
0
where ä EE9 is the Kronecker delta and equation (6.19) has been included.
We next make the following conventional change of variables
ìZe9 2
ë ˆ (6:21)
"(ÿ2ìE) 1=2
2
2(ÿ2ìE) 1=2 r 2ìZe9 r 2Zr
r ˆ ˆ ˆ (6:22)
" ë" 2 ëa ì
2
2
where a ì ˆ " =ìe9 . We also make the substitution
3=2
2Z
R El (r) ˆ S ël (r) (6:23)
ëa ì
Equations (6.17) and (6.18) now take the form
!
d 2 d r 2
r 2 ‡ 2r ‡ ër ÿ S ël ˆ l(l ‡ 1)S ël (6:24)
dr 2 dr 4
where the ®rst term has been expanded and the entire expression has been
2
multiplied by r .
To be a suitable wave function, S ël (r) must be well-behaved, i.e., it must be
continuous, single-valued, and quadratically integrable. Thus, rS ël vanishes
when r !1 because S ël must vanish suf®ciently fast. Since S ël is ®nite
everywhere, rS ël also vanishes at r ˆ 0. Substitution of equations (6.22) and
(6.23) into (6.19) shows that S ël (r) is normalized with a weighting function
w(r) equal to r 2

1
2 2
[S ël (r)] r dr ˆ 1 (6:25)
0
Equation (6.24) may be solved by the Frobenius or series solution method as
presented in Appendix G. However, in this chapter we employ the newer
procedure using ladder operators.