Page 144 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

5.2 Generalized angular momentum 135
^
^
J ^ 2 because J ^ 2 and J ‡ commute. However, if the operator J z acts on the
^
function J ‡ jëmi,wehave
^ ^ ^ ^ ^ ^ ^
J z J ‡ jëmiˆ J ‡ J z jëmi‡ "J ‡ jëmiˆ m"J ‡ jëmi‡ "J ‡ jëmi
^
ˆ (m ‡ 1)"J ‡ jëmi (5:20)
^
where equations (5.19a) and (5.16b) were used. Thus, the function J ‡ jëmi is
^
an eigenfunction of J z with eigenvalue (m ‡ 1)". Writing equation (5.16b) as
^
J z jë, m ‡ 1iˆ (m ‡ 1)"jë, m ‡ 1i
^
we see from equation (5.20) that J ‡ jëmi is proportional to jë, m ‡ 1i
^
J ‡ jëmiˆ c ‡ jë, m ‡ 1i (5:21)
^
where c ‡ is the proportionality constant. The operator J ‡ is, therefore, a
raising operator, which alters the eigenfunction jëmi for the eigenvalue m" to
the eigenfunction for (m ‡ 1)".
The proportionality constant c ‡ in equation (5.21) may be evaluated by
squaring both sides of equation (5.21) to give
^ ^ 2
hëmjJ ÿ J ‡ jëmiˆjc ‡ j hë, m ‡ 1jë, m ‡ 1i
^
^
since the bra hëmjJ ÿ is the adjoint of the ket J ‡ jëmi. Using equations (5.16)
and (5.19g) and the normality of the eigenfunctions, we have
^
^
^
2
2
2
2
jc ‡ j ˆhëmjJ ÿ J ÿ "J z jëmiˆ (ë ÿ m ÿ m)" 2
z
and equation (5.21) becomes
p 
^
J ‡ jëmiˆ ë ÿ m(m ‡ 1) "jë, m ‡ 1i (5:22)
In equation (5.22) we have arbitrarily taken c ‡ to be real and positive.
^
^
^ 2
We next let the operators J and J z act on the function J ÿ jëmi to give
^ ^ ^ ^ 2 2 ^
2
J J ÿ jëmiˆ J ÿ J jëmiˆ ë" J ÿ jëmi
^ ^ ^ ^ ^ ^
J z J ÿ jëmiˆ J ÿ J z jëmiÿ "J ÿ jëmiˆ (m ÿ 1)"J ÿ jëmi
where we have used equations (5.16), (5.19b), and (5.19d). The function
^
^
^ 2
2
J ÿ jëmi is a simultaneous eigenfunction of J and J z with eigenvalues ë" and
^
(m ÿ 1)", respectively. Accordingly, the function J ÿ jëmi is proportional to
jë, m ÿ 1i
^
J ÿ jëmiˆ c ÿ jë, m ÿ 1i (5:23)
^
where c ÿ is the proportionality constant. The operator J ÿ changes the eigen-
function jëmi to the eigenfunction jë, m ÿ 1i for a lower value of the eigen-
^
value of J z and is, therefore, a lowering operator.
To evaluate the proportionality constant c ÿ in equation (5.23), we square
^
both sides of (5.23) and note that the bra hëmjJ ‡ is the adjoint of the ket
^
J ÿ jëmi, giving