Atomic Structure Theory

12 1 Angular Momentum
1.3 Clebsch-Gordan Coefficients
One common problem encountered in atomic physics calculations is finding
eigenstates of the sum of two angular momenta in terms of products of the
individual angular momentum eigenstates. For example, as mentioned in section
(1.2.1), the products Yl,m(θ, φ)χµ are eigenstates of L2 and Lz, aswellas
S2 and Sz. The question addressed in this section is how to combine product
states such as these to find eigenstates of J 2 and Jz, whereJ = L + S.
Generally, let us suppose that we have two commuting angular momentum
vectors J 1 and J 2. Let|j1,m1〉 be an eigenstate of J 2 1 and J1z with eigenvalues
(in units of ¯h) j1(j1 +1) andm1, respectively. Similarly, let |j2,m2〉
be an eigenstate of J 2 2 and J2z with eigenvalues j2(j2 +1) andm2. We set
J = J1 + J2 and attempt to construct eigenstates of J 2 and Jz as linear
combinations of the product states |j1,m1〉|j2,m2〉:
|j, m〉 =
C(j1,j2,j; m1,m2,m)|j1,m1〉|j2,m2〉 . (1.57)
m1,m2
The expansion coefficients C(j1,j2,j; m1,m2,m), called Clebsch-Gordan coefficients,
are discussed in many standard quantum mechanics textbooks
[for example, 34, chap. 10]. One sometimes encounters notation, such as
〈j1,m1,j2,m2|j, m〉 for the Clebsch-Gordan coefficient C(j1,j2,j; m1,m2,m).
Since Jz = J1z + J2z, it follows from (1.57) that
m|j, m〉 =
(m1 + m2)C(j1,j2,j; m1,m2,m)|j1,m1〉|j2,m2〉 . (1.58)
m1,m2
Since the states |j1,m1〉|j2,m2〉 are linearly independent, one concludes from
(1.58) that
(m1 + m2 − m)C(j1,j2,j; m1,m2,m)=0. (1.59)
It follows that the only nonvanishing Clebsch-Gordan coefficients are those
for which m1 + m2 = m. The sum in (1.57) can be expressed, therefore, as
asumoverm2only, the value of m1 being determined by m1 = m − m2.
Consequently, we rewrite (1.57) as
|j, m〉 =
C(j1,j2,j; m − m2,m2,m)|j1,m− m2〉|j2,m2〉 . (1.60)
m2
If we demand that all of the states in (1.60) be normalized, then it follows
from the relation
〈j ′ ,m ′ |j, m〉 = δj ′ jδm ′ m ,
that
m ′ 2 ,m2
C(j1,j2,j ′ ; m ′ − m ′ 2,m ′ 2,m ′ )C(j1,j2,j; m − m2,m2,m)×
〈j1,m ′ − m ′ 2|j1,m− m2〉〈j2,m ′ 2|j2,m2〉 = δj ′ jδm ′ m.