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  "body": "\n## Summary\n\nThe Wigner D-matrix $D^j(\\alpha, \\beta, \\gamma)$ is the matrix of a rotation, given by its Euler angles, in the irreducible representation of spin $j$ of [[SU(2)|E5P8AG6CK7]].\nIts entries are functions of the three angles, and they factor as $e^{-i m' \\alpha} d^j_{m' m}(\\beta) e^{-i m \\gamma}$.\nThe entries for all $j$ form an orthogonal basis of the square-integrable functions on $\\mathrm{SU}(2)$.\nThose with integer $j$ form such a basis for the rotation group $\\mathrm{SO}(3)$.\nThey contain the [[spherical harmonics|3F32Y21HP4]] and the [[Legendre polynomials|QFYGJVA996]] as special cases, and they are the standard tool for rotations in quantum mechanics.\n\n## Explanations\n\n### L1 human\n\nA rotation of space acts on functions on the sphere, and on states of a quantum system with angular momentum $j$.\nIn a fixed basis of $2j + 1$ states this action is a square matrix, and this matrix is the Wigner D-matrix.\nFor spin $j = 1/2$ it is a $2 \\times 2$ unitary matrix; a rotation by $\\beta$ about the $y$-axis gives the rows $(\\cos \\frac{\\beta}{2}, -\\sin \\frac{\\beta}{2})$ and $(\\sin \\frac{\\beta}{2}, \\cos \\frac{\\beta}{2})$.\nThe half angles show that a full turn by $2\\pi$ gives $-1$ and not $1$, a famous feature of spin.\n",
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