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

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

(1)

and are defined by

$\begin{displaymath} \Psi_{JM}=\sum_{M=M_1+M_2} C^J_{M_1M_2}\Psi_{M_1M_2}, \end{displaymath}$

(2)

where $J\equiv J_1+J_2$ . The Clebsch-Gordan coefficients are sometimes expressed using the related Racah V-Coefficient

$\begin{displaymath} V(j_1j_2j;m_1m_2m) \end{displaymath}$

(3)

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

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

(4)

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

(5)

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

(6)

They have the symmetry

$\begin{displaymath} (j_1j_2m_1m_2\vert j_1j_2jm)=(-1)^{j_1+j_2-j}(j_2j_1m_2m_1\vert j_2j_1jm), \end{displaymath}$

(7)

and obey the orthogonality relationships

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

(8)

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

(9)

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.