Clebsch-Gordan coefficients are mathematical symbol used to integrate products of three spherical harmonics. Clebsch-Gordan coefficients
commonly arise in applications involving the addition of angular momentum in quantum
mechanics. If products of more than three spherical
harmonics are desired, then a generalization known as Wigner
6j-symbols or Wigner 9j-symbols
is used.

The Clebsch-Gordan coefficients are variously written as , , , or . The Clebsch-Gordan coefficients
are implemented in the Wolfram Language
as ClebschGordan[j1,
m1,
j2,
m2,
j,
m].

The Clebsch-Gordan coefficients are defined by

(1)

where ,
and satisfy

(2)

for .

Care is needed in interpreting analytic representations of Clebsch-Gordan coefficients since these coefficients are defined only on measure zero sets. As a result, "generic"
symbolic formulas may not hold it certain cases, if at all. For example, ClebschGordan[1,
0,
j2,
0,
2,
0]
evaluates to an expression that is "generically" correct but not correct
for the special case , whereas ClebschGordan[1, 0, 1, 0, 2, 0] evaluates to the correct value .

The coefficients are subject to the restrictions that be positive integers or half-integers, is an integer, are positive or negative integers or half integers,

(3)

(4)

(5)

and ,
,
and
(Abramowitz and Stegun 1972, p. 1006). In addition, by use of symmetry relations,
coefficients may always be put in the standard form and .

The Clebsch-Gordan coefficients are sometimes expressed using the related Racah
V-coefficients,