Page 143 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

134 Angular momentum

since the eigenfunctions jëmi are normalized. Using equation (5.14) we may
also write
^ 2 ^ 2 ^ 2 ^ 2
hJ iˆhJ i‡hJ i‡hJ i
z
y
x
^
^
^ 2
^ 2
Since J x and J y are hermitian, the expectation values of J and J are real
x y
and positive, so that
^ 2 ^ 2
hJ i > hJ i
z
from which it follows that
2
ë > m > 0 (5:17)
Ladder operators
We have already introduced the use of ladder operators in Chapter 4 to ®nd the
eigenvalues for the harmonic oscillator. We employ the same technique here to
^ 2 ^ ^ ^
obtain the eigenvalues of J and J z . The requisite ladder operators J ‡ and J ÿ
are de®ned by the relations
^ ^ ^
J ‡ J x ‡ iJ y (5:18a)
^ ^ ^
J ÿ J x ÿ iJ y (5:18b)
^
^
Neither J ‡ nor J ÿ is hermitian. Application of equation (3.33) shows that they
are adjoints of each other. Using the de®nitions (5.18) and (5.14) and the
commutation relations (5.13) and (5.15), we can readily prove the following
relationships
^ ^ ^
[J z , J ‡ ] ˆ "J ‡ (5:19a)
^ ^ ^
[J z , J ÿ ] ˆÿ"J ÿ (5:19b)
^
^
2
[J , J ‡ ] ˆ 0 (5:19c)
^
^
2
[J , J ÿ ] ˆ 0 (5:19d)
^ ^ ^ (5:19e)
[J ‡ , J ÿ ] ˆ 2"J z
^ ^ ^ 2 ^ 2 ^
J ‡ J ÿ ˆ J ÿ J ‡ "J z (5:19f)
z
^ ^ ^ 2 ^ 2 ^
J ÿ J ‡ ˆ J ÿ J ÿ "J z (5:19g)
z
^
If we let the operator J ^ 2 act on the function J ‡ jëmi and observe that,
^
^ 2
according to equation (5.19c), J and J ‡ commute, we obtain
^ ^ ^ ^ 2 2 ^
2
J J ‡ jëmiˆ J ‡ J jëmiˆ ë" J ‡ jëmi
^
where (5.16a) was also used. We note that J ‡ jëmi is an eigenfunction of J ^ 2
^
2
with eigenvalue ë" . Thus, the operator J ‡ has no effect on the eigenvalues of