ζ, μ À 1j
h
J
À
ð Þ
jζ, μi ¼ a μ
À
ð Þ
ζ, μ À 1
h
jζ, μ À 1i ¼ a μ
À
ð Þ ,
ð3:91Þ
where the second equality comes from that jζ, μ À 1i has been normalized; i.e.,
jjj ζ, μ À 1ijj ¼ 1. Meanwhile, taking adjoint of both sides of the first equation of
(3.77), we have
ζ, μj
h
J
þ
ð Þ
h
i { ¼ a μ
þ
ð Þ
h
i à ζ, μ þ 1j
h
:
ð3:92Þ
But, from (3.72) and the fact that J x and J y are Hermitian,
J
þ
ð Þ
h
i { ¼ J
À
ð Þ
:
ð3:93Þ
Using (3.93) and replacing μ in (3.92) with μ À 1, we get
ζ, μ À 1j
h
J
À
ð Þ
¼ a μÀ1
þ
ð Þ
h
i à ζ, μj
h
:
ð3:94Þ
Furthermore, multiplying jζ, μi on (3.94) from the right, we have
ζ, μ À 1j
h
J
À
ð Þ
jζ, μi ¼ a μÀ1
þ
ð Þ
h
i à ζ, μjζ, μi
h
¼ a μÀ1
þ
ð Þ
h
i Ã
,
ð3:95Þ
where again jζ, μi is assumed to be normalized. Comparing (3.91) and (3.95), we get
a μ
À
ð Þ
¼ a μÀ1
þ
ð Þ
h
i à :
ð3:96Þ
Taking an inner product regarding the first equation of (3.77) and its adjoint,
ζ, μj
h
J
À
ð Þ J
þ
ð Þ
jζ, μi ¼ a μ
þ
ð Þ
h
i Ã
a μ
þ
ð Þ
ζ, μ þ 1jζ, μ þ 1i ¼
h
a μ
þ
ð Þ
2 :
ð3:97Þ
Once again, the second equality of (3.97) results from the normalization of the
vector.
Using (3.83) as well as (3.71) and (3.86), (3.97) can be rewritten as
ζ, μj
h
J
2
À J z
2
À J z jζ, μi ¼ ζ, μj
h
j j þ 1
ð
ÞÀμ
2
À μjζ, μi
¼ hζ, μjζ, μi j À μ
ð
Þ j þ μ þ 1
ð
Þ¼ a μ
þ
ð Þ
2 :
ð3:98Þ
Thus, we get
76
3 Hydrogen-Like Atoms