Clebsch-Gordan coefficients

The Clebsch-Gordan coefficients are defined by

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 \Psi_{JM}= \sum_{M=M_1 + M_2} C_{M_1 M_2}^J \Psi_{M_1 M_2},}

where 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 J \equiv J_1 + J_2} and satisfies 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 (j_1j_2m_1m_2|j_1j_2m)=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, $V(j_{1}j_{2}j;m_{1}m_{2}m)$ (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)