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  "body": "\n## Definition\n\nFor $j \\in \\set{0, \\frac12, 1, \\dots}$, let $V_j$ be the [[irreducible representation|FX3NEFYFFN]] of [[SU(2)|E5P8AG6CK7]] of dimension $2j+1$, with the standard [[orthonormal basis|0S9GGTR0EX]] $\\lvert j m \\rangle$, $m = j, j-1, \\dots, -j$, of the Condon–Shortley convention.\nLet $j_1, j_2, J$ be such that $V_J$ occurs in the [[tensor product|0BF2DE2SYK]] $V_{j_1} \\otimes V_{j_2}$, that is, $\\abs{j_1 - j_2} \\le J \\le j_1 + j_2$ and $j_1 + j_2 + J \\in \\Z$.\nLet $\\lvert J M \\rangle$ be the basis of a copy of $V_J$ in $V_{j_1} \\otimes V_{j_2}$.\nThe *Clebsch–Gordan coefficients* $\\langle j_1 m_1, j_2 m_2 \\mid J M \\rangle$ are its coordinates:\n$$\\lvert J M \\rangle = \\sum_{m_1, m_2} \\langle j_1 m_1, j_2 m_2 \\mid J M \\rangle\\, \\lvert j_1 m_1 \\rangle \\otimes \\lvert j_2 m_2 \\rangle,$$\nwhere the copy is fixed by the convention that all coefficients are real and $\\langle j_1 j_1, j_2 (J - j_1) \\mid J J \\rangle \u003e 0$.\nThe *3-j symbol* is\n$$\\begin{pmatrix} j_1 \u0026 j_2 \u0026 j_3 \\\\ m_1 \u0026 m_2 \u0026 m_3 \\end{pmatrix} = \\frac{(-1)^{j_1 - j_2 - m_3}}{\\sqrt{2 j_3 + 1}}\\, \\langle j_1 m_1, j_2 m_2 \\mid j_3, -m_3 \\rangle.$$\nIt is set to $0$ when $j_1, j_2, j_3$ fail the conditions above or some $\\abs{m_i} \u003e j_i$.\n\n## Remarks\n\nThe symbol is $0$ unless $m_1 + m_2 + m_3 = 0$, since $J_z$ is additive on tensor products.\nIt is unchanged by an even permutation of its columns and is multiplied by $(-1)^{j_1 + j_2 + j_3}$ by an odd permutation.\nThe orthogonality relation reads $\\sum_{m_1, m_2} (2 j_3 + 1) \\begin{pmatrix} j_1 \u0026 j_2 \u0026 j_3 \\\\ m_1 \u0026 m_2 \u0026 m_3 \\end{pmatrix} \\begin{pmatrix} j_1 \u0026 j_2 \u0026 j_3' \\\\ m_1 \u0026 m_2 \u0026 m_3' \\end{pmatrix} = \\delta_{j_3 j_3'} \\delta_{m_3 m_3'}$.\n\nExample: for $j_1 = j_2 = \\frac12$ and $J = 0$, the invariant vector is $\\lvert 0 0 \\rangle = \\frac{1}{\\sqrt2}\\left(\\lvert \\tfrac12 \\tfrac12 \\rangle \\otimes \\lvert \\tfrac12, -\\tfrac12 \\rangle - \\lvert \\tfrac12, -\\tfrac12 \\rangle \\otimes \\lvert \\tfrac12 \\tfrac12 \\rangle\\right)$.\nSo $\\begin{pmatrix} \\frac12 \u0026 \\frac12 \u0026 0 \\\\ \\frac12 \u0026 -\\frac12 \u0026 0 \\end{pmatrix} = \\frac{(-1)^{0}}{1} \\cdot \\frac{1}{\\sqrt2} = \\frac{1}{\\sqrt2}$.\nThis agrees with the general value $\\begin{pmatrix} j \u0026 j \u0026 0 \\\\ m \u0026 -m \u0026 0 \\end{pmatrix} = \\frac{(-1)^{j-m}}{\\sqrt{2j+1}}$.\n\nNon-example: $\\begin{pmatrix} 1 \u0026 1 \u0026 1 \\\\ 1 \u0026 1 \u0026 0 \\end{pmatrix} = 0$, because $m_1 + m_2 + m_3 = 2 \\ne 0$.\n\nPhysicists write the coefficients $\\langle j_1 m_1, j_2 m_2 \\mid J M \\rangle$ also as $C^{J M}_{j_1 m_1 j_2 m_2}$.\n",
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