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
The Clebsch-Gordan coefficients are variously written as , , , or . The Clebsch-Gordan coefficients
are implemented in the Wolfram Language
The Clebsch-Gordan coefficients are defined by
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,
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,
(Abramowitz and Stegun 1972, p. 1006). In addition, by use of symmetry relations,
coefficients may always be put in the standard form and .