3-j symbol

Summary

The Wigner 3-j symbol $\begin{pmatrix} j_1 & j_2 & j_3 \\ m_1 & m_2 & m_3 \end{pmatrix}$ is a number that describes how two irreducible representations of SU(2) combine into a third. It is the Clebsch–Gordan coefficient of the tensor product, rescaled and with a sign, so that it has simple symmetries under permutation of its columns. It vanishes unless $m_1 + m_2 + m_3 = 0$ and $j_1, j_2, j_3$ satisfy the triangle inequality. 3-j symbols appear in the coupling of angular momenta in quantum mechanics and in products of spherical harmonics.