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,
 |
(6)
|
or Wigner 3j-symbols. Connections among the three are
 |
(7)
|
 |
(8)
|
 |
(9)
|
They have the symmetry
 |
(10)
|
and obey the orthogonality relationships
 |
(11)
|
 |
(12)
|
in