Atomic Structure Theory

1.1 Orbital Angular Momentum - Spherical Harmonics 3
and we note that the states J±|λ, m〉 are also eigenstates of J 2 with eigenvalue
λ. Moreover, with the aid of (1.7), one can establish that J+|λ, m〉 and
J−|λ, m〉 are eigenstates of Jz with eigenvalues m ± 1, respectively:
JzJ+|λ, m〉 =(m +1)J+|λ, m〉, (1.12)
JzJ−|λ, m〉 =(m− 1) J−|λ, m〉. (1.13)
Since J+ raises the eigenvalue m by one unit, and J− lowers it by one unit,
these operators are referred to as raising and lowering operators, respectively.
Furthermore, since J 2 x + J 2 y is a positive definite hermitian operator, it follows
that
λ ≥ m 2 .
By repeated application of J− to eigenstates of Jz, one can obtain states of
arbitrarily small eigenvalue m, violating this bound, unless for some state
|λ, m1〉,
J−|λ, m1〉 =0.
Similarly, repeated application of J+ leads to arbitrarily large values of m,
unless for some state |λ, m2〉,
J+|λ, m2〉 =0.
Since m 2 is bounded, we infer the existence of the two states |λ, m1〉 and
|λ, m2〉. Starting from the state |λ, m1〉 and applying the operator J+ repeatedly,
one must eventually reach the state |λ, m2〉; otherwise, the value of m
would increase indefinitely. It follows that
m2 − m1 = k, (1.14)
where k ≥ 0 is the number of times that J+ must be applied to the state
|λ, m1〉 in order to reach the state |λ, m2〉. One finds from (1.8,1.9) that
leading to the identities
which can be rewritten
λ|λ, m1〉 =(m 2 1 − m1)|λ, m1〉,
λ|λ, m2〉 =(m 2 2 + m2)|λ, m2〉,
λ = m 2 1 − m1 = m 2 2 + m2, (1.15)
(m2 − m1 + 1)(m2 + m1) =0. (1.16)
Since the first term on the left of (1.16) is positive definite, it follows that
m1 = −m2. The upper bound m2 can be rewritten in terms of the integer k
in (1.14) as
m2 = k/2 =j.