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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

C^j_{m_1m_2}=(j_1j_2m_1m_2\vert j_1j_2jm)
\end{displaymath} (1)

and are defined by
\Psi_{JM}=\sum_{M=M_1+M_2} C^J_{M_1M_2}\Psi_{M_1M_2},
\end{displaymath} (2)

where $J\equiv J_1+J_2$. The Clebsch-Gordan coefficients are sometimes expressed using the related Racah V-Coefficient
\end{displaymath} (3)

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

(j_1j_2m_1m_2\vert j_1j_2m) = (-1)^{-j_1+j_2-m} \sqrt{2j+1}\pmatrix{j_1 & j_2 & j\cr m_1 & m_2 & -m\cr}
\end{displaymath} (4)

(j_1j_2m_1m_2\vert j_1j_2jm) = (-1)^{j+m}\sqrt{2j+1}V(j_1j_2j; m_1m_2\, -m)
\end{displaymath} (5)

V(j_1j_2j;m_1m_2m)=(-1)^{-j_1+j_2+j}\pmatrix{j_1 & j_2 & j_1\cr m_2 & m_1 & m_2}.
\end{displaymath} (6)

They have the symmetry
(j_1j_2m_1m_2\vert j_1j_2jm)=(-1)^{j_1+j_2-j}(j_2j_1m_2m_1\vert j_2j_1jm),
\end{displaymath} (7)

and obey the orthogonality relationships

\sum_{j,m}(j_1j_2m_1m_2\vert j_1j_2jm)(j_1j_2jm\vert j_1j_2m_1'm_2') = \delta_{m_1m_1'}\delta_{m_2m_2'}
\end{displaymath} (8)

\sum_{m_1, m_2}(j_1j_2m_1m_2\vert j_1j_2jm)(j_1j_2j'm'\vert j_1j_2m_1m_2) = \delta_{jj'}\delta_{mm'}.
\end{displaymath} (9)

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


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 $B_X$ 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.

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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