Page 179 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

170 The hydrogen atom
d r
^
A l‡1 S l‡1,l ˆÿ r ‡ ‡ l S l‡1,l ˆ 0
dr 2
or
r
r dS l‡1,l ˆ l ÿ S l‡1,l
dr 2
Integration gives

r e
S l‡1,l ˆ [(2l ‡ 2)!] ÿ1=2 l ÿr=2 (6:49)
where the integration constant was evaluated using equations (6.25), (A.26),
and (A.28).
The eigenfunction S 21 from equation (6.49) is
1
S 21 ˆ p re ÿr=2

2 6
and equations (6.46) and (6.26b) for l ˆ 1give

1 d r 1
S 31 ˆ p  r ÿ ‡ 3 p  re ÿr=2
6 dr 2 2 6
1 ÿr=2
ˆ (4 ÿ r)re
12
r 
3 d r 1 ÿr=2
S 41 ˆ r ÿ ‡ 4 (4 ÿ r)re
40 dr 2 12
1 2 ÿr=2
ˆ p  (20 ÿ 10r ‡ r )re
8 30
.
. .
The functions S 31 , S 41 , ... are automatically normalized as speci®ed by
equation (6.25). The normalized eigenfunctions S nl (r) for l ˆ 2, 3, 4, ... with
n > (l ‡ 1) are obtained by the same procedure.
^
A general formula for S nl involves the repeated application of B k for
k ˆ l ‡ 2, l ‡ 3, ... , n ÿ 1, n to S l‡1,l in equation (6.49). The raising operator
must be applied (n ÿ l ÿ 1) times. The result is
ÿ1
S nl ˆ (b nl ) (b nÿ1,l ) ÿ1 ... (b l‡2,l ) ÿ1 ^ ^ ^
B n B nÿ1 ... B l‡2 S l‡1,l
1=2
(l ‡ 1)(2l ‡ 1)! d r
ˆ r ÿ ‡ n
n(n ‡ l)!(n ÿ l ÿ 1)!(2l ‡ 2)! dr 2

d r d r
l ÿr=2
3 r ÿ ‡ n ÿ 1 r ÿ ‡ l ‡ 2 r e (6:50)
dr 2 dr 2
Just as equation (6.46) can be used to go `up the ladder' to obtain S n,l from