Wigner D-matrix

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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.

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 Planck's constant has been put equal to one. The operator

is a Casimir operator of SU(2) (or SO(3) as the case may be). It may be diagonalized together with (the choice of this operator is a convention), which commutes with . That is, it can be shown that there is a complete set of kets with

where j = 0, 1/2, 1, 3/2, 2,... and m = -j, -j + 1,..., j. For SO(3) the quantum number j is integer.

A 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 square matrix of dimension 2j + 1 with general element

The matrix with general element

is known as Wigner's (small) d-matrix.

Wigner (small) d-matrix

Wigner[1] gave the following expression

The sum over s is over such values that the factorials are nonnegative.

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 and .[2] Let

Then, with , the relation is


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.


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,


In other words, the rows and columns of the (complex conjugate) Wigner D-matrix span irreducible representations of the isomorphic Lie algebra's generated by and .

An important property of the Wigner D-matrix follows from the commutation of with the time reversal operator ,


Here we used that is anti-unitary (hence the complex conjugation after moving from ket to bra), and .

Orthogonality relations

The Wigner D-matrix elements form a complete set of orthogonal functions of the Euler angles , and :

This is a special case of the Schur orthogonality relations.

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 , 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:

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, is a longitudinal angle and 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:

Relation to Bessel functions

In the limit when we have where is the Bessel function and 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 [3]

Wigner d-matrix elements with swapped lower indices are found with the relation:


See also


  1. {{#invoke:citation/CS1|citation |CitationClass=book }} Translated into English by {{#invoke:citation/CS1|citation |CitationClass=book }}
  2. {{#invoke:citation/CS1|citation |CitationClass=book }}
  3. {{#invoke:Citation/CS1|citation |CitationClass=journal }}

External links