Clebsch-Gordan coefficients

The Clebsch-Gordan coefficients are defined by

${\mathrm{\Psi }}_{JM}={\sum }_{M=M_1+M_2}{C}_{M_1{M}_2}^J{\mathrm{\Psi }}_{M_1{M}_2},$

where $J\equiv J_1+J_2$ and satisfies $(j_1{j}_2{m}_1{m}_2|j_1{j}_{2}m)=0$ for $m_1+m_2\ne m$. 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)