Page 146 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

5.2 Generalized angular momentum 137
j(j ‡ 1) ˆ j9(j9 ÿ 1)

The solution to this quadratic equation gives j9 ˆÿj. The other solution,
j9 ˆ j ‡ 1, is not physically meaningful because j9 must be less than j.We
have shown, therefore, that the parameter m ranges from ÿj to j
ÿj < m < j

If we combine the conclusion that j9 ˆÿj with the relation j9 ˆ j ÿ n, we see
that j ˆ n=2, where n ˆ 0, 1, 2, ... Thus, the allowed values of j are the
1 3 5
integers 0, 1, 2, ... (if n is even) and the half-integers , , , ... (if n is odd)
2 2 2
and the allowed values of m are ÿj, ÿj ‡ 1, ... , j ÿ 1, j.
We began this analysis with an arbitrary value for ë, namely ë ˆ î, and an
arbitrary value for m, namely m ˆ ç. We showed that, in order to satisfy
requirement (5.17), the parameter î must satisfy î ˆ j(j ‡ 1), where j is
restricted to integral or half-integral values. Since the value î was chosen
arbitrarily, we conclude that the only allowed values for ë are

ë ˆ j(j ‡ 1) (5:25)

The parameter ç is related to j by j ˆ ç ‡ k, where k is the number of
^
successive applications of J ‡ until jîçi is transformed into jîji. Since k must
be a positive integer, the parameter ç must be restricted to integral or half-
integral values. However, the value ç was chosen arbitrarily, leading to the
conclusion that the only allowed values of m are m ˆÿj, ÿj ‡ 1, ... , j ÿ 1,
j. Thus, we have found all of the allowed values for ë and for m and, therefore,
^
^ 2
all of the eigenvalues of J and J z .
In view of equation (5.25), we now denote the eigenkets jëmi by jjmi.
Equations (5.16) may now be written as
2
^
2
3
1
J jjmiˆ j(j ‡ 1)" jjmi, j ˆ 0, ,1, ,2, ... (5:26a)
2 2
^
J z jjmiˆ m"jjmi, m ˆÿj, ÿj ‡ 1, ... , j ÿ 1, j (5:26b)
^ 2
Each eigenvalue of J is (2j ‡ 1)-fold degenerate, because there are (2j ‡ 1)
values of m for a given value of j. Equations (5.22) and (5.24) become
p 
^
J ‡ jjmiˆ j(j ‡ 1) ÿ m(m ‡ 1) "jj, m ‡ 1i
p 
ˆ (j ÿ m)(j ‡ m ‡ 1) "jj, m ‡ 1i (5:27a)
^ p 
J ÿ jjmiˆ j(j ‡ 1) ÿ m(m ÿ 1) "jj, m ÿ 1i
p 
ˆ (j ‡ m)(j ÿ m ‡ 1) "jj, m ÿ 1i (5:27b)