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 $m_{1}+m_{2}\neq m$. 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. Giulio Racah "Theory of Complex Spectra. II", Physical Review 62 pp. 438-462 (1942)
2. Taro Shimpuku "General Theory and Numerical Tables of Clebsch-Gordan Coefficients", Progress of Theoretical Physics Supplement 13 pp. 1-135 (1960)
3. M. E. Rose "Elementary theory of angular momentum", John Wiley & Sons (1967) Appendix I
4. Robert E. Beck and Bernard Kolman "Racah's outer multiplicity formula", Computer Physics Communications 8 pp. 95-100 (1974)