{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $j \\in \\set{0, \\frac12, 1, \\frac32, \\dots}$, and let $V_j$ be the [[irreducible representation|FX3NEFYFFN]] of [[SU(2)|E5P8AG6CK7]] of dimension $2j + 1$, with an invariant inner product.\nLet $J_x, J_y, J_z$ be the Hermitian operators on $V_j$ that represent the Lie algebra, with $[J_x, J_y] = i J_z$ and its cyclic permutations.\nLet $\\lvert j m \\rangle$, for $m = j, j - 1, \\dots, -j$, be the orthonormal basis of $V_j$ such that $J_z \\lvert j m \\rangle = m \\lvert j m \\rangle$ and $(J_x \\pm i J_y) \\lvert j m \\rangle = \\sqrt{j(j+1) - m(m \\pm 1)}\\, \\lvert j, m \\pm 1 \\rangle$.\nFor Euler angles $\\alpha, \\beta, \\gamma \\in \\R$, the *Wigner D-matrix* is the $(2j+1) \\times (2j+1)$ matrix with entries\n$$D^j_{m' m}(\\alpha, \\beta, \\gamma) = \\langle j m' \\rvert\\, e^{-i \\alpha J_z} e^{-i \\beta J_y} e^{-i \\gamma J_z} \\lvert j m \\rangle,$$\nwhere $m', m \\in \\set{j, j-1, \\dots, -j}$.\nThe *small d-matrix* is $d^j_{m' m}(\\beta) = D^j_{m' m}(0, \\beta, 0)$.\n\n## Remarks\n\nSince $J_z$ is diagonal, $D^j_{m' m}(\\alpha, \\beta, \\gamma) = e^{-i m' \\alpha}\\, d^j_{m' m}(\\beta)\\, e^{-i m \\gamma}$.\nThe entries of $d^j(\\beta)$ are real, and Wigner's formula is\n$$d^j_{m' m}(\\beta) = \\sum_{s} \\frac{(-1)^{m' - m + s} \\sqrt{(j+m')!\\,(j-m')!\\,(j+m)!\\,(j-m)!}}{(j+m-s)!\\, s!\\, (m'-m+s)!\\, (j-m'-s)!} \\left(\\cos\\tfrac{\\beta}{2}\\right)^{2j+m-m'-2s} \\left(\\sin\\tfrac{\\beta}{2}\\right)^{m'-m+2s},$$\nsummed over the integers $s$ for which all factorials have nonnegative arguments.\n\nExample: for $j = \\frac12$, $J_y = \\frac12 \\sigma_y$ with the Pauli matrix $\\sigma_y = \\begin{pmatrix} 0 \u0026 -i \\\\ i \u0026 0 \\end{pmatrix}$, and $\\sigma_y^2 = I$.\nSo $e^{-i \\beta J_y} = \\cos\\frac{\\beta}{2}\\, I - i \\sin\\frac{\\beta}{2}\\, \\sigma_y$, which gives $d^{1/2}_{\\frac12 \\frac12} = \\cos\\frac{\\beta}{2}$ and $d^{1/2}_{\\frac12, -\\frac12} = -\\sin\\frac{\\beta}{2}$.\nWigner's formula with $m' = \\frac12$, $m = -\\frac12$ has only the term $s = 0$, and it also gives $-\\sin\\frac{\\beta}{2}$.\n\nNon-example: the matrix $\\begin{pmatrix} \\cos\\frac{\\beta}{2} \u0026 \\sin\\frac{\\beta}{2} \\\\ \\sin\\frac{\\beta}{2} \u0026 \\cos\\frac{\\beta}{2} \\end{pmatrix}$ is not a D-matrix, since it is not unitary for $\\sin \\beta \\ne 0$, while every $D^j$ is unitary.\n\nFor integer $j = \\ell$, $d^\\ell_{00}(\\beta) = P_\\ell(\\cos\\beta)$, the [[Legendre polynomial|QFYGJVA996]].\nAlso $\\overline{D^\\ell_{m 0}(\\alpha, \\beta, \\gamma)} = \\sqrt{\\frac{4\\pi}{2\\ell+1}}\\, Y_\\ell^m(\\beta, \\alpha)$, where $Y_\\ell^m$ is a [[spherical harmonic|3F32Y21HP4]] with polar angle $\\beta$ and azimuth $\\alpha$.\nThe bra-ket notation $\\langle u \\rvert A \\lvert v \\rangle$ means the inner product of $u$ with $A v$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "QQPC9ASQHR",
    "kind": "definition",
    "msc": [
      "33C80",
      "22E70",
      "81R05"
    ],
    "name": "Wigner D-matrix",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Wigner_D-matrix"
      }
    ],
    "tag": "32SHBZX38H",
    "uses": [
      "E5P8AG6CK7",
      "FX3NEFYFFN",
      "9V6PRD1Z90"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/32SHBZX38H/",
  "links": [
    {
      "field": "defines",
      "target": "QQPC9ASQHR"
    },
    {
      "field": "uses",
      "target": "E5P8AG6CK7"
    },
    {
      "field": "uses",
      "target": "FX3NEFYFFN"
    },
    {
      "field": "uses",
      "target": "9V6PRD1Z90"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/32SHBZX38H.md",
  "msc": [
    "33C80",
    "22E70",
    "81R05"
  ],
  "name": "Wigner D-matrix",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/32S/32SHBZX38H.md",
  "status": null,
  "tag": "32SHBZX38H",
  "uri": "m2://32SHBZX38H",
  "url": "https://tamnd.github.io/math2/wiki/wigner-d-matrix-definition/",
  "writing": {
    "tag": "32SHBZX38H",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}