Clebsch–Gordan coefficients

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 Mathematica 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)