Page 152 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

5.3 Application to orbital angular momentum 143
2l
I l ˆ I lÿ1 (5:43)
2l ‡ 1
Since I 0 is given by

1
I 0 ˆ dì ˆ 2
ÿ1
we can obtain I l by repeated application of equation (5.43) starting with I 0
(2l)(2l ÿ 2)(2l ÿ 4) ... 2 2 2l‡1 (l!) 2
I l ˆ I 0 ˆ
(2l ‡ 1)(2l ÿ 1)(2l ÿ 3) ... 3 (2l ‡ 1)!
where we have noted that
l
(2l)(2l ÿ 2) ... 2 ˆ 2 l!
(2l ‡ 1)(2l ÿ 1)(2l ÿ 3) ... 3

(2l ‡ 1)(2l)(2l ÿ 1)(2l ÿ 2)(2l ÿ 3) ... 3 3 2 3 1 (2l ‡ 1)!
ˆ ˆ
l
(2l)(2l ÿ 2) ... 2 2 l!
Substituting this result into equation (5.41), we ®nd that
r 
1 (2l ‡ 1)!
jA l jˆ
l
2 l! 2
iá
It is customary to let á equal zero in the phase factor e for È l,ÿl (è), so that
r 
1 (2l ‡ 1)! l
È l,ÿl (è) ˆ sin è (5:44)
2 l! 2
l
Combining equations (5.35), (5.40) and (5.44), we obtain the normalized
eigenfunction
r 
1 (2l ‡ 1)! l ÿilj
Y l,ÿl (è, j) ˆ sin è e (5:45)
2 l! 4ð
l

Spherical harmonics
The functions Y lm (è, j) are known as spherical harmonics and may be
^
obtained from Y l,ÿl (è, j) by repeated application of the raising operator L ‡
according to (5.38a). By this procedure, the spherical harmonics Y l,ÿl‡1 (è, j),
Y l,ÿl‡2 (è, j), ... , Y l,ÿ1 (è, j), Y l0 (è, j), Y l1 (è, j), ... , Y ll (è, j)may be
determined. Since the starting function Y l,ÿl (è, j) is normalized, each of the
spherical harmonics generated from equation (5.38a) will also be normalized.
We may readily derive a general expression for the spherical harmonic
^
Y lm (è, j) which results from the repeated application of L ‡ to Y l,ÿl (è, j). We
begin with equation (5.38a) with m set equal to ÿl
1
^
Y l,ÿl‡1 ˆ p  L ‡ Y l,ÿl (5:46)
2l "