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  "body": "\n## Summary\n\nThe Wigner 3-j symbol $\\begin{pmatrix} j_1 \u0026 j_2 \u0026 j_3 \\\\ m_1 \u0026 m_2 \u0026 m_3 \\end{pmatrix}$ is a number that describes how two irreducible representations of [[SU(2)|E5P8AG6CK7]] combine into a third.\nIt 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.\nIt vanishes unless $m_1 + m_2 + m_3 = 0$ and $j_1, j_2, j_3$ satisfy the triangle inequality.\n3-j symbols appear in the coupling of angular momenta in quantum mechanics and in products of [[spherical harmonics|3F32Y21HP4]].\n\n## Explanations\n\n### L1 human\n\nTwo spinning particles with spins $j_1$ and $j_2$ together can have total spin $j_3$ only for a few values of $j_3$.\nThe 3-j symbols are the coefficients that express a state of total spin in terms of the states of the two particles.\nThey are arranged in a table with three columns, one per spin, and the bottom row lists the projections on an axis.\nFor two spins $\\frac12$ that combine to spin $0$, the symbols are $\\pm 1/\\sqrt{2}$, which gives the singlet state $(\\uparrow\\downarrow - \\downarrow\\uparrow)/\\sqrt{2}$.\n",
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