Wigner D-matrix

Summary

The 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). Its entries are functions of the three angles, and they factor as $e^{-i m' \alpha} d^j_{m' m}(\beta) e^{-i m \gamma}$. The entries for all $j$ form an orthogonal basis of the square-integrable functions on $\mathrm{SU}(2)$. Those with integer $j$ form such a basis for the rotation group $\mathrm{SO}(3)$. They contain the spherical harmonics and the Legendre polynomials as special cases, and they are the standard tool for rotations in quantum mechanics.