Page 183 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

174 The hydrogen atom

1
2
e [L
c 2 nl r 2l‡1 ÿr 2l‡1 (r)] dr ˆ 1
n‡l
0
The value of the integral is given by equation (F.25) with á ˆ n ‡ l and
j ˆ 2l ‡ 1, so that
2n[(n ‡ l)!] 3
c 2 ˆ 1
nl
(n ÿ l ÿ 1)!
and S nl (r) in equation (6.53) becomes
1=2
(n ÿ l ÿ 1)! l ÿr=2 2l‡1
S nl (r) ˆÿ r e L n‡l (r) (6:54)
2n[(n ‡ l)!] 3
Taking the negative square root maintains the sign of S nl (r).
j
Equations (6.39) and (F.22), with S nl (r) and L (r) related by (6.54), are
k
identical. From equations (F.23) and (F.24), we ®nd
s 

1 (n ÿ l)(n ‡ l ‡ 1)
2 1
S nl (r)S n 1,l (r)r dr ˆÿ
2
0 n(n ‡ 1)

1
2
S nl (r)S n9,l (r)r dr ˆ 0, n9 6ˆ n, n 1
0
The normalized radial functions R nl (r) may be expressed in terms of the
associated Laguerre polynomials by combining equations (6.22), (6.23), and
(6.54)
s 
4(n ÿ l ÿ 1)!Z 3 2Zr l
R nl (r) ˆÿ e ÿ Zr=na 0 L 2l‡1 (2Zr=na ì ) (6:55)
n‡l
3 3
4
n [(n ‡ l)!] a ì na ì
Solution for positive energies
There are also solutions to the radial differential equation (6.17) for positive
values of the energy E, which correspond to the ionization of the hydrogen-like
atom. In the limit r !1, equations (6.17) and (6.18) for positive E become
2
d R(r) 2ìE
‡ R(r) ˆ 0
dr 2 " 2
for which the solution is
R(r) ˆ ce i(2ìE) 1=2 r="
where c is a constant of integration. This solution has oscillatory behavior at
in®nity and leads to an acceptable, well-behaved eigenfunction of equation
(6.17) for all positive eigenvalues E. Thus, the radial equation (6.17) has a
continuous range of positive eigenvalues as well as the discrete set (equation
(6.48)) of negative eigenvalues. The corresponding eigenfunctions represent