Page 337 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

328 Appendix G
l ‡ 1, F 0 ˆ a 0 r l
1
l ‡ 2, F 1 ˆ a 0 r l (1 ÿ r
2l ‡ 2
1 1
l ‡ 3, F 2 ˆ a 0 r l 1 ÿ r ‡ r 2
l ‡ 1 (2l ‡ 3)(2l ‡ 2)
. . . . . .

Since l is an integer with values 0, 1, 2, ... , the parameter ë takes on integer values n,
n ˆ 1, 2, 3, .. . , so that n ˆ l ‡ 1, l ‡ 2, ... When the quantum number n equals 1,
the value of l is 1; when n ˆ 2, we have l ˆ 0, 1; when n ˆ 3, we have l ˆ 0, 1, 2;
etc.
The energy E of the hydrogen-like atom is related to ë by equation (6.21). If we
solve this equation for E and set ë equal to n, we obtain
2
ìZ e9 4
E n ˆÿ 2 2 , n ˆ 1, 2, 3, .. .
2" n
in agreement with equation (6.48).
To identify the polynomial solutions for F(r), we make the substitution
l
F(r) ˆ r u(r) (G:50)
in the differential equation (G.43) and set ë equal to n to obtain
ru0 ‡ [2(l ‡ 1) ÿ r]u9 ‡ (n ÿ l ÿ 1)u ˆ 0 (G:51)
Since n and l are integers, equation (G.51) is identical to the associated Laguerre
differential equation (F.16) with k ˆ n ‡ l and j ˆ 2l ‡ 1. Thus, the solutions u(r)
are proportional to the associated Laguerre polynomials L 2l‡1 (r), whose properties are
n‡l
discussed in Appendix F
u(r) ˆ cL 2l‡1 (r) (G:52)
n‡l
Combining equations (G.42), (G.50), and (G.52), we obtain
l ÿr=2 2l‡1
S nl (r) ˆ c nl r e L (r) (G:53)
n‡l
where c nl are the normalizing constants. Equation (G.53) agrees with equation (6.53).