Page 180 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

6.3 The radial equation 171

S nÿ1,l , equation (6.44) allows one to go `down the ladder' and obtain S nÿ1,l
from S nl . Taking the positive square root in going from equation (6.43) to
(6.44) is consistent with taking the positive square root in going from equation
(6.45) to (6.46); the signs of the functions S nl are maintained in the raising and
lowering operations. In all cases the ladder operators yield normalized eigen-
functions if the starting eigenfunction is normalized.
The radial factors of the hydrogen-like atom total wave functions ø(r, è, j)
are related to the functions S nl (r) by equation (6.23). Thus, we have
3=2
Z
R 10 ˆ 2 e ÿr=2
a ì
1 Z 3=2 ÿr=2
R 20 ˆ p  (2 ÿ r)e
2 2 a ì
3=2
1 Z
2
R 30 ˆ p  (6 ÿ 6r ‡ r )e ÿr=2
9 3 a ì
.
. .
3=2
1 Z
R 21 ˆ p  re ÿr=2
2 6 a ì
3=2
1 Z ÿr=2
R 31 ˆ p  (4 ÿ r)re
9 6 a ì
1 Z 3=2
2
R 41 ˆ p  (20 ÿ 10r ‡ r )re ÿr=2
32 15 a ì
.
. .

and so forth.
A more extensive listing appears in Table 6.1.

Radial functions in terms of associated Laguerre polynomials
The radial functions S nl (r) and R nl (r) may be expressed in terms of the
j
associated Laguerre polynomials L (r), whose de®nition and mathematical
k
properties are discussed in Appendix F. One method for establishing the
j
relationship between S nl (r) and L (r) is to relate S nl (r) in equation (6.50) to
k
j
the polynomial L (r) in equation (F.15). That process, however, is long and
k
tedious. Instead, we show that both quantities are solutions of the same
differential equation.