## Clebsch-Gordan Coefficient

A mathematical symbol used to integrate products of three Spherical Harmonics. Clebsch-Gordan coefficients commonly arise in applications involving the addition of angular momentum in quantum mechanics. If products of more than three Spherical Harmonics are desired, then a generalization known as Wigner 6j-Symbol or Wigner 9j-Symbol is used. The Clebsch-Gordan coefficients are written

 (1)

and are defined by
 (2)

where . The Clebsch-Gordan coefficients are sometimes expressed using the related Racah V-Coefficient
 (3)

or Wigner 3j-Symbol. Connections among the three are

 (4)

 (5)

 (6)

They have the symmetry
 (7)

and obey the orthogonality relationships

 (8)

 (9)

See also Racah V-Coefficient, Racah W-Coefficient, Wigner 3j-Symbol, Wigner 6j-Symbol, Wigner 9j-Symbol

References

Abramowitz, M. and Stegun, C. A. (Eds.). Vector-Addition Coefficients.'' §27.9 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 1006-1010, 1972.

Cohen-Tannoudji, C.; Diu, B.; and Laloë, F. Clebsch-Gordan Coefficients.'' Complement in Quantum Mechanics, Vol. 2. New York: Wiley, pp. 1035-1047, 1977.

Condon, E. U. and Shortley, G. §3.6-3.14 in The Theory of Atomic Spectra. Cambridge, England: Cambridge University Press, pp. 56-78, 1951.

Fano, U. and Fano, L. Basic Physics of Atoms and Molecules. New York: Wiley, p. 240, 1959.

Messiah, A. Clebsch-Gordan (C.-G.) Coefficients and 3j' Symbols.'' Appendix C.I in Quantum Mechanics, Vol. 2. Amsterdam, Netherlands: North-Holland, pp. 1054-1060, 1962.

Shore, B. W. and Menzel, D. H. Coupling and Clebsch-Gordan Coefficients.'' §6.2 in Principles of Atomic Spectra. New York: Wiley, pp. 268-276, 1968.

Sobel'man, I. I. `Angular Momenta.'' Ch. 4 in Atomic Spectra and Radiative Transitions, 2nd ed. Berlin: Springer-Verlag, 1992.