Page 149 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

140 Angular momentum
^
The eigenvalue equation for L z is
" @
^ Y lm (è, j) ˆ m"Y lm (è, j) (5:33)
L z Y lm (è, j) ˆ
i @j
where equations (5.28b) and (5.31c) have been combined. Equation (5.33) may
be written in the form
dY lm (è, j)
ˆ im dj (è held constant)
Y lm (è, j)
the solution of which is
Y lm (è, j) ˆ È lm (è)e imj (5:34)
where È lm (è) is the `constant of integration' and is a function only of the
variable è. Thus, we have shown that Y lm (è, j) is the product of two functions,
one a function only of è, the other a function only of j
Y lm (è, j) ˆ È lm (è)Ö m (j) (5:35)
We have also shown that the function Ö m (j) involves only the parameter m
and not the parameter l.
The function Ö m (j) must be single-valued and continuous at all points in
^
^ 2
space in order for Y lm (è, j) to be an eigenfunction of L and L z .If Ö m (j) and
hence Y lm (è, j) are not single-valued and continuous at some point j 0 , then
the derivative of Y lm (è, j) with respect to j would produce a delta function at
the point j 0 and equation (5.33) would not be satis®ed. Accordingly, we
require that
Ö m (j) ˆ Ö m (j ‡ 2ð)
or
e imj ˆ e im(j‡2ð)
so that
2imð
e ˆ 1
This equation is valid only if m is an integer, positive or negative
m ˆ 0, 1, 2, ...
We showed in Section 5.2 that the parameter m for generalized angular
momentum can equal either an integer or a half-integer. However, in the case
of orbital angular momentum, the parameter m can only be an integer; the half-
integer values for m are not allowed. Since the permitted values of m are ÿl,
ÿl ‡ 1, ... , l ÿ 1, l, the parameter l can have only integer values in the case
of orbital angular momentum; half-integer values for l are also not allowed.

Ladder operators
The ladder operators for orbital angular momentum are