---
tag: YJRJVZ998D
kind: concept
name: 3-j symbol
msc: [33C80, 81R05, "22E70"]
sources:
  - url: https://en.wikipedia.org/wiki/3-j_symbol
same_as:
  wikidata: Q2570568
  wikipedia: 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)|E5P8AG6CK7]] 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|3F32Y21HP4]].

## Explanations

### L1 human

Two spinning particles with spins $j_1$ and $j_2$ together can have total spin $j_3$ only for a few values of $j_3$.
The 3-j symbols are the coefficients that express a state of total spin in terms of the states of the two particles.
They are arranged in a table with three columns, one per spin, and the bottom row lists the projections on an axis.
For two spins $\frac12$ that combine to spin $0$, the symbols are $\pm 1/\sqrt{2}$, which gives the singlet state $(\uparrow\downarrow - \downarrow\uparrow)/\sqrt{2}$.