Wigner D-matrix

The Wigner D-matrix is a matrix in an irreducible representation of the groups SU(2) and SO(3). The complex conjugate of the D-matrix is an eigenfunction of the Hamiltonian of spherical and symmetric rigid rotors. The matrix was introduced in 1927 by Eugene Wigner. D stands for darstellung, which means "representation" in German.

Let J_x, J_y, J_z 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 ${\displaystyle \hbar }$ has been put equal to one. The Casimir operator

commutes with all generators of the Lie algebra. Hence it may be diagonalized together with ${\displaystyle J_{z}}$. That is, it can be shown that there is a complete set of kets with

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