Clebsch-Gordan coefficients

The Clebsch-Gordan coefficients are defined by

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Psi _{JM}=\sum _{M=M_{1}+M_{2}}C_{M_{1}M_{2}}^{J}\Psi _{M_{1}M_{2}},}

where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle J\equiv J_{1}+J_{2}} and satisfies Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (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

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