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,[citation needed] which means "representation" in German.
Let Jx, Jy, Jz be generators of the Lie algebra of SU(2) and SO(3).
In all cases, the three operators satisfy the following commutation relations, where i is the purely imaginary number and the Planck constant ħ has been set equal to one.
The Casimir operator commutes with all generators of the Lie algebra.
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).
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.
Wigner gave the following expression:[1] The sum over s is over such values that the factorials are nonnegative, i.e.
In the often-used z-x-z convention of Euler angles, the factor
The realness of the d-matrix elements is one of the reasons that the z-y-z convention, used in this article, is usually preferred in quantum mechanical applications.
, which lead to: The complex conjugate of the D-matrix satisfies a number of differential properties that can be formulated concisely by introducing the following operators with
which have quantum mechanical meaning: they are space-fixed rigid rotor angular momentum operators.
Further, which have quantum mechanical meaning: they are body-fixed rigid rotor angular momentum operators.
satisfy anomalous commutation relations (have a minus sign on the right hand side).
act on the second (column) index of the D-matrix, and, because of the anomalous commutation relation the raising/lowering operators are defined with reversed signs, Finally, In other words, the rows and columns of the (complex conjugate) Wigner D-matrix span irreducible representations of the isomorphic Lie algebras generated by
form a set of orthogonal functions of the Euler angles
are matrix elements of a unitary transformation from one spherical basis
is represented by the relations:[3] The group characters for SU(2) only depend on the rotation angle β, being class functions, so, then, independent of the axes of rotation, and consequently satisfy simpler orthogonality relations, through the Haar measure of the group,[4] The completeness relation (worked out in the same reference, (3.95)) is whence, for
The set of Kronecker product matrices forms a reducible matrix representation of the groups SO(3) and SU(2).
, the D-matrix elements with second index equal to zero are proportional to spherical harmonics and associated Legendre polynomials, normalized to unity and with Condon and Shortley phase convention: This implies the following relationship for the d-matrix: A rotation of spherical harmonics
then is effectively a composition of two rotations, When both indices are set to zero, the Wigner D-matrix elements are given by ordinary Legendre polynomials: In the present convention of Euler angles,
This is one of the reasons that the z-y-z convention is used frequently in molecular physics.
From the time-reversal property of the Wigner D-matrix follows immediately There exists a more general relationship to the spin-weighted spherical harmonics: The absolute square of an element of the D-matrix, gives the probability that a system with spin
itself forms a real symmetric matrix, that depends only on the Euler angle
, is a scaled and shifted discrete Chebyshev polynomial, and the corresponding eigenvalue,
Using sign convention of Wigner, et al. the d-matrix elements
For j = 1/2 For j = 1 For j = 3/2 For j = 2[8] Wigner d-matrix elements with swapped lower indices are found with the relation: