Page 145 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

136 Angular momentum
^
^
^ ^
^
2
2
2
2
jc ÿ j ˆhëmjJ ‡ J ÿ jëmiˆhëmjJ ÿ J ‡ "J z jëmiˆ (ë ÿ m ‡ m)" 2
z
where equation (5.19f) was also used. Equation (5.23) then becomes
p 
^
J ÿ jëmiˆ ë ÿ m(m ÿ 1) "jë, m ÿ 1i (5:24)
where we have taken c ÿ to be real and positive. This choice is consistent with
the selection above of c ‡ as real and positive.
Determination of the eigenvalues
We now apply the raising and lowering operators to ®nd the eigenvalues of J ^ 2
^
and J z . Equation (5.17) tells us that for a given value of ë, the parameter m has
a maximum and a minimum value, the maximum value being positive and the
minimum value being negative. For the special case in which ë equals zero, the
parameter m must, of course, be zero as well.
We select arbitrary values for ë,say î, and for m,say ç, where 0 < ç < î
2
^
so that (5.17) is satis®ed. Application of the raising operator J ‡ to the
^
corresponding ket jîçi gives the ket jî, ç ‡ 1i. Successive applications of J ‡
give jî, ç ‡ 2i, jî, ç ‡ 3i, etc. After k such applications, we obtain the ket
2
jîji, where j ˆ ç ‡ k and j < î. The value of j is such that an additional
^
2
application of J ‡ produces the ket jî, j ‡ 1i with (j ‡ 1) . î (that is to say, it
2
produces a ket jëmi with m . ë), which is not possible. Accordingly, the
^
sequence must terminate by the condition J ‡ jîjiˆ 0. From equation (5.22),
this condition is given by
p 
^
J ‡ jîjiˆ î ÿ j(j ‡ 1) "jî, j ‡ 1iˆ 0
which is valid only if the coef®cient of jî, j ‡ 1i vanishes, so that we have
î ˆ j(j ‡ 1).
^
We now apply the lowering operator J ÿ to the ket jîji successively to
construct the series of kets jî, j ÿ 1i, jî, j ÿ 2i, etc. After a total of n
^
applications of J ÿ , we obtain the ket jîj9i, where j9 ˆ j ÿ n is the minimum
value of m allowed by equation (5.17). Therefore, this lowering sequence must
terminate by the condition
p 
^
J ÿ jîj9iˆ î ÿ j9(j9 ÿ 1) "jî, j9 ÿ 1iˆ 0
where equation (5.24) has been introduced. This condition is valid only if the
coef®cient of jî, j9 ÿ 1i vanishes, giving î ˆ j9(j9 ÿ 1).
The parameter î has two conditions imposed upon it
î ˆ j(j ‡ 1)
î ˆ j9(j9 ÿ 1)

giving the relation