Clebsch-Gordan Coefficient

Clebsch-Gordan coefficients are 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-symbols or Wigner 9j-symbols is used.

The Clebsch-Gordan coefficients are variously written as , , , or . The Clebsch-Gordan coefficients are implemented in the Wolfram Language as ClebschGordan[j1, m1, j2, m2, j, m].

The Clebsch-Gordan coefficients are defined by

(1)

where , and satisfy

(2)

for .

Care is needed in interpreting analytic representations of Clebsch-Gordan coefficients since these coefficients are defined only on measure zero sets. As a result, "generic" symbolic formulas may not hold it certain cases, if at all. For example, ClebschGordan[1, 0, j2, 0, 2, 0] evaluates to an expression that is "generically" correct but not correct for the special case , whereas ClebschGordan[1, 0, 1, 0, 2, 0] evaluates to the correct value .

The coefficients are subject to the restrictions that  be positive integers or half-integers,  is an integer,  are positive or negative integers or half integers,

			(3)
			(4)
			(5)

and , , and  (Abramowitz and Stegun 1972, p. 1006). In addition, by use of symmetry relations, coefficients may always be put in the standard form  and .

The Clebsch-Gordan coefficients are sometimes expressed using the related Racah V-coefficients,

(6)

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

(7)

(8)

(9)

They have the symmetry

(10)

and obey the orthogonality relationships

(11)

(12)

REFERENCES:

Abramowitz, M. and Stegun, I. 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.

Rose, M. E. Elementary Theory of Angular Momentum. New York: Dover, 1995.

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.