Clebsch-Gordan coefficients

The Clebsch-Gordan coefficients are defined by

$\Psi _{JM}=\sum _{M=M_{1}+M_{2}}C_{M_{1}M_{2}}^{J}\Psi _{M_{1}M_{2}},$

where $J\equiv J_{1}+J_{2}$ and satisfies $(j_{1}j_{2}m_{1}m_{2}|j_{1}j_{2}m)=0$ for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(j_1j_2j;m_1m_2m)} (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)