## Saturday, 23 July 2011

### 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)