Wigner 3j-Symbol

The Wigner $3j$ symbols are written

$$\pmatrix{j_1 & j_2 & j\cr m_1 & m_2 & m\cr}$$ (1)

and are sometimes expressed using the related Clebsch-Gordan Coefficients

$$C^j_{m_1m_2}=(j_1j_2m_1m_2\vert j_1j_2jm)$$ (2)

(Condon and Shortley 1951, pp. 74-75; Wigner 1959, p. 206), or Racah V-Coefficient

$$V(j_1j_2j;m_1m_2m).$$ (3)

Connections among the three are

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

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

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

The Wigner $3j$-symbols have the symmetries

$$\left(\begin{array}{ccc}j_1 & j_2 & j\\ m_1 & m_2 & m\end{array}\right) = \left(\begin{array}{ccc}j_2 & j & j_1\\ m_2 & m & m_1\end{array}\right)$$

$$= \left(\begin{array}{ccc}j & j_1 & j_2\\ m & m_1 & m_2\end{array}... ...+j_2+j}\left(\begin{array}{ccc}j_2 & j_1 & j\\ m_2 & m_1 & m\end{array}\right)$$

$$= (-1)^{j_1+j_2+j}\left(\begin{array}{ccc}j_1 & j & j_2\\ m_1 & m & m_2\end{array}\right)$$

$$= (-1)^{j_1+j_2+j}\left(\begin{array}{ccc}j & j_2 & j_1\\ m & m_2 & m_1\end{array}\right)$$

$$= (-1)^{j_1+j_2+j}\left(\begin{array}{ccc}j_1 & j_2 & j\\ -m_1 & -m_2 & -m\end{array}\right).$$ (7)

The symbols obey the orthogonality relations

$$\sum_{j,m} (2j+1)\pmatrix{j_1 & j_2 & j\cr m_1 & m_2 & m\cr}... ...j_2 & j\cr m_1' & m_2' & m} = \delta_{m_1m_1'}\delta_{m_2m_2'}$$ (8)

$$\sum_{m_1,m_2} \pmatrix{j_1 & j_2 & j\cr m_1 & m_2 & m\cr}\p... ...{j_1 & j_2 & j'\cr m_1 & m_2 & m'} = \delta_{jj'}\delta_{mm'},$$ (9)

where $\delta_{ij}$ is the Kronecker Delta.

General formulas are very complicated, but some specific cases are

$\pmatrix{j_1 & j_2 & j_1+j_2\cr m_1 & m_2 & -m_1-m_2\cr} = (-1)^{j_1-j_2+m_1+m_2}$
$\times\left[{{(2j_1)!(2j_2)!\over (2j_1+2j_2+1)!(j_1+m_1)!}{(j_1+j_2+m_1+m_2)!(j_1+j_2-m_1-m_2)!\over (j_1-m_1)!(j_2+m_2)!(j_2-m_2)!}}\right]^{1/2}\quad$	(10)

$\pmatrix{j_1 & j_2 & j\cr j_1 & -j_1- & m\cr}=(-1)^{-j_1+j_2+m}$
$\times\left[{{(2j_1)!(-j_1+j_2+j)!\over (j_1+j_2+j+1)!(j_1-j_2+j)!}{(j_1+j_2+m)!(j-m)!\over (j_1+j_2-j)!(-j_1+j_2-m)!(j+m)!}}\right]^{1/2}\quad$	(11)

$$\left(\begin{array}{ccc} j_1 & j_2 & j\\ 0 & 0 & 0 \end{ar... ...f\ } J=2g\\ 0\\ \quad {\rm if\ } J=2g+1, \end{array}\right.$$ (12)

for $J\equiv j_1+j_2+j$.

For Spherical Harmonics $Y_{lm}(\theta,\phi)$,

$Y_{l_1m_1}(\theta,\phi)Y_{l_2m_2}(\theta,\phi)$
$=\sum_{l,m}\sqrt{(2l_1+1)(2l_2+1)(2l+1)\over 4\pi}\pmatrix{l_1 & l_2 & l\cr m_1 & m_2 & m\cr} Y_{lm}^*(\theta,\psi)\pmatrix{l_1 & l_2 & l\cr 0 & 0 & 0\cr}.\quad$	(13)

For values of $l_3$ obeying the Triangle Condition $\Delta(l_1l_2l_3)$,

$\int Y_{l_1m_1}(\theta,\phi)Y_{l_2m_2}(\theta,\phi)Y_{l_3m_3}(\theta,\phi)\sin\theta\,d\theta\,d\phi$
$= \sqrt{(2l_1+1)(2l_2+1)(2l_3+1)\over 4\pi} \pmatrix{l_1 & l_2 & l_3\cr 0 & 0 & 0\cr}\pmatrix{l_1 & l_2 & l_3\cr m_1 & m_2 & m_3\cr}\quad$	(14)

and

$${1\over 2}\int P_{l_1}(\cos\theta)P_{l_2}(\cos\theta)P_{l_3}... ...\theta\,d\theta = \pmatrix{l_1 & l_2 & l_3\cr 0 & 0 & 0\cr}^2.$$ (15)

See also Clebsch-Gordan Coefficient, Racah V-Coefficient, Racah W-Coefficient, 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.

Condon, E. U. and Shortley, G. The Theory of Atomic Spectra. Cambridge, England: Cambridge University Press, 1951.

de Shalit, A. and Talmi, I. Nuclear Shell Theory. New York: Academic Press, 1963.

Gordy, W. and Cook, R. L. Microwave Molecular Spectra, 3rd ed. New York: Wiley, pp. 804-811, 1984.

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.

Rotenberg, M.; Bivens, R.; Metropolis, N.; and Wooten, J. K. The $3j$ and $6j$ Symbols. Cambridge, MA: MIT Press, 1959.

Shore, B. W. and Menzel, D. H. Principles of Atomic Spectra. New York: Wiley, pp. 275-276, 1968.

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

Wigner, E. P. Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, expanded and improved ed. New York: Academic Press, 1959.