Page 182 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

6.3 The radial equation 173

Table 6.1. (cont.)

(Z=a ì ) 3=2 2 3 4 ÿr=2
R 61 ˆ p  (840 ÿ 840r ‡ 252r ÿ 28r ‡ r )r e
432 210
(Z=a ì ) 3=2 2 3 2 ÿr=2
R 62 ˆ p  (336 ÿ 168r ‡ 24r ÿ r )r e
864 105
(Z=a ì ) 3=2 2 3 ÿr=2
R 63 ˆ p  (72 ÿ 18r ‡ r )r e
2592 35
(Z=a ì ) 3=2
4 ÿr=2
R 64 ˆ p  (10 ÿ r)r e
12 960 7
(Z=a ì ) 3=2
5 ÿr=2
R 65 ˆ p  r e
12 960 77

We observe that the solutions S nl (r) of the differential equation (6.24)
l ÿr=2
contain the factor r e . Therefore, we de®ne the function F nl (r)by
l ÿr=2
S nl (r) ˆ F nl (r)r e
and substitute this expression into equation (6.24) with ë ˆ n to obtain
2
d F nl dF nl
r ‡ (2l ‡ 2 ÿ r) ‡ (n ÿ l ÿ 1)F nl ˆ 0 (6:51)
dr 2 dr
where we have also divided the equation by the common factor r.
The differential equation satis®ed by the associated Laguerre polynomials is
given by equation (F.16) as
2
d L j dL j
j
r k ‡ (j ‡ 1 ÿ r) k ‡ (k ÿ j)L ˆ 0
dr 2 dr k
If we let k ˆ n ‡ l and j ˆ 2l ‡ 1, then this equation takes the form
2 2l‡1
d L dL 2l‡1
r n‡l ‡ (2l ‡ 2 ÿ r) n‡l ‡ (n ÿ l ÿ 1)L 2l‡1 ˆ 0 (6:52)
dr 2 dr n‡l
We have already found that the set of functions S nl (r) contains all the
solutions to (6.24). Therefore, a comparison of equations (6.51) and (6.52)
shows that F nl is proportional to L 2l‡1 . Thus, the function S nl (r) is related to
n‡1
the polynomial L 2l‡1 (r)by
n‡l
l ÿr=2 2l‡1
S nl (r) ˆ c nl r e L (r) (6:53)
n‡l
The proportionality constants c nl in equation (6.53) are determined by the
normalization condition (6.25). When equation (6.53) is substituted into (6.25),
we have