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 C_(m_1m_2)^j, C_(m_1m_2m)^(j_1j_2j), (j_1j_2m_1m_2|j_1j_2jm), or <j_1j_2m_1m_2|j_1j_2jm>. 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


where J=J_1+J_2, and satisfy


for m_1+m_2!=m.

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 j_2=1, whereas ClebschGordan[{1, 0}, {1, 0}, {2, 0}] evaluates to the correct value sqrt(2/3).

The coefficients are subject to the restrictions that (j_1,j_2,j) be positive integers or half-integers, j_1+j_2+j is an integer, (m_1,m_2,m) are positive or negative integers or half integers,


and -|j_1|<=m_1<=|j_1|, -|j_2|<=m_2<=|j_2|, and -|j|<=m<=|j| (Abramowitz and Stegun 1972, p. 1006). In addition, by use of symmetry relations, coefficients may always be put in the standard form j_1<j_2<j and m>=0.

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


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

 (j_1j_2m_1m_2|j_1j_2jm)=(-1)^(m+j_1-j_2)sqrt(2j+1)(j_1 j_2 j; m_1 m_2 -m)
 V(j_1j_2j;m_1m_2m)=(-1)^(-j_1+j_2+j)(j_1 j_2 j_1; m_2 m_1 m_2).

They have the symmetry


and obey the orthogonality relationships


See also

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

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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 B_X 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.

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Clebsch-Gordan Coefficient

Cite this as:

Weisstein, Eric W. "Clebsch-Gordan Coefficient." From MathWorld--A Wolfram Web Resource.

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