Clebsch-Gordan coefficients

The Clebsch-Gordan coefficients are defined by

Ψ_{J M} = \sum_{M = M_{1} + M_{2}} C_{M_{1} M_{2}}^{J} Ψ_{M_{1} M_{2}},

where $J \equiv J_{1} + J_{2}$ and satisfies $(j_{1} j_{2} m_{1} m_{2} | j_{1} j_{2} m) = 0$ for $m_{1} + m_{2} \neq m$ . They are used to integrate products of three spherical harmonics (for example the addition of angular momenta). The Clebsch-Gordan coefficients are sometimes expressed using the related Racah V-coefficients, $V (j_{1} j_{2} j; m_{1} m_{2} m)$ (See also the Racah W-coefficients, sometimes simply called the Racah coefficients).

References

Robert E. Beck and Bernard Kolman "Racah's outer multiplicity formula", Computer Physics Communications 8 pp. 95-100 (1974)

Clebsch-Gordan coefficients

References

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