Wigner D-matrix

The Wigner D-matrix is a unitary matrix in an irreducible representation of the groups SU(2) and SO(3). It was introduced in 1927 by Eugene Wigner, and plays a fundamental role in the quantum mechanical theory of angular momentum. The complex conjugate of the D-matrix is an eigenfunction of the Hamiltonian of spherical and symmetric rigid rotors. The letter D stands for Darstellung, which means "representation" in German. Let Jx, Jy, Jz be generators of the Lie algebra of SU(2) and SO(3). In quantum mechanics, these three operators are the components of a vector operator known as angular momentum. Examples are the angular momentum of an electron in an atom, electronic spin, and the angular momentum of a rigid rotor. In all cases, the three operators satisfy the following commutation relations, where i is the purely imaginary number and Planck's constant ħ has been set equal to one. The Casimir operator commutes with all generators of the Lie algebra. Hence, it may be diagonalized together with Jz. This defines the spherical basis used here. That is, there is a complete set of kets (i.e. orthonormal basis of joint eigenvectors labelled by quantum numbers that define the eigenvalues) with where j = 0, 1/2, 1, 3/2, 2, ... for SU(2), and j = 0, 1, 2, ... for SO(3). In both cases, m = −j, −j + 1, ..., j. A 3-dimensional rotation operator can be written as where α, β, γ are Euler angles (characterized by the keywords: z-y-z convention, right-handed frame, right-hand screw rule, active interpretation). The Wigner D-matrix is a unitary square matrix of dimension 2j + 1 in this spherical basis with elements where is an element of the orthogonal Wigner's (small) d-matrix. That is, in this basis, is diagonal, like the γ matrix factor, but unlike the above β factor. Wigner gave the following expression: The sum over s is over such values that the factorials are nonnegative, i.e. , . Note: The d-matrix elements defined here are real. In the often-used z-x-z convention of Euler angles, the factor in this formula is replaced by causing half of the functions to be purely imaginary.