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.

Definition of the Wigner D-matrix

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 the Planck 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, . 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 (small) d-matrix

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. 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. The d-matrix elements are related to Jacobi polynomials with nonnegative a and b. Let If Then, with b=2j-2k-a, the relation is where a,b \ge 0. It is also useful to consider the relations, where and , which lead to:

Properties of the Wigner D-matrix

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. The operators satisfy the commutation relations and the corresponding relations with the indices permuted cyclically. The satisfy anomalous commutation relations (have a minus sign on the right hand side). The two sets mutually commute, and the total operators squared are equal, Their explicit form is, The operators act on the first (row) index of the D-matrix, The operators 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 and. An important property of the Wigner D-matrix follows from the commutation of with the time reversal operator T, or Here, we used that T is anti-unitary (hence the complex conjugation after moving T^\dagger from ket to bra), and. A further symmetry implies

Orthogonality relations

The Wigner D-matrix elements form a set of orthogonal functions of the Euler angles and \gamma: This is a special case of the Schur orthogonality relations. Crucially, by the Peter–Weyl theorem, they further form a complete set. The fact that are matrix elements of a unitary transformation from one spherical basis to another is represented by the relations: 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, The completeness relation (worked out in the same reference, (3.95)) is whence, for \beta' =0,

Kronecker product of Wigner D-matrices, Clebsch–Gordan series

The set of Kronecker product matrices forms a reducible matrix representation of the groups SO(3) and SU(2). Reduction into irreducible components is by the following equation: The symbol is a Clebsch–Gordan coefficient.

Relation to spherical harmonics and Legendre polynomials

For integer values of l, 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, \alpha is a longitudinal angle and \beta is a colatitudinal angle (spherical polar angles in the physical definition of such 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:

Connection with transition probability under rotations

The absolute square of an element of the D-matrix, gives the probability that a system with spin j prepared in a state with spin projection m along some direction will be measured to have a spin projection m' along a second direction at an angle \beta to the first direction. The set of quantities F_{mm'} itself forms a real symmetric matrix, that depends only on the Euler angle \beta, as indicated. Remarkably, the eigenvalue problem for the F matrix can be solved completely: Here, the eigenvector,, is a scaled and shifted discrete Chebyshev polynomial, and the corresponding eigenvalue, , is the Legendre polynomial.

Relation to Bessel functions

In the limit when we have where is the Bessel function and \ell\beta is finite.

List of d-matrix elements

Using sign convention of Wigner, et al. the d-matrix elements for j = 1/2, 1, 3/2, and 2 are given below. For j = 1/2 For j = 1 For j = 3/2 For j = 2 Wigner d-matrix elements with swapped lower indices are found with the relation:

Symmetries and special cases