Wigner D-matrix - AbsoluteAstronomy.com

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 rotor

Rigid rotor

The rigid rotor is a mechanical model that is used to explain rotating systems.An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space three angles are required. A special rigid rotor is the linear rotor which isa 2-dimensional object, requiring...

s. The matrix was introduced in 1927 by Eugene Wigner.

Definition Wigner D-matrix

Let

be generators of the Lie algebra

Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

of SU(2) and SO(3). In quantum mechanics

Quantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

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

Spin (physics)

In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...

,and the angular momentum
of a rigid rotor

Rigid rotor

. In all cases the three operators satisfy the following commutation relations,

where i is the purely imaginary number

Imaginary number

An imaginary number is any number whose square is a real number less than zero. When any real number is squared, the result is never negative, but the square of an imaginary number is always negative...

and Planck's constant

has been put equal to one. The operator

is a Casimir operator

Casimir invariant

In mathematics, a Casimir invariant or Casimir operator is a distinguished element of the centre of the universal enveloping algebra of a Lie algebra...

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

and

. (For SO(3) the quantum number

is integer.)

A rotation operator can be written as

where

and

are Euler angles

Euler angles

The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body. To describe such an orientation in 3-dimensional Euclidean space three parameters are required...

(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

with
general element

The matrix with general element

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

Wigner (small) d-matrix

Wigner 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

Jacobi polynomials

In mathematics, Jacobi polynomials are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight ^\alpha ^\beta on the interval [-1, 1]...

with nonnegative

and

. Let

Then, with

, the relation is

where

Properties of 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

Rigid rotor

angular momentum operators.

Further,

which have quantum mechanical meaning: they are body-fixed rigid rotor

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,

and

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 algebra's

Lie algebra

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

Schur orthogonality relations

In mathematics, the Schur orthogonality relations express a central fact about representations of finite groups.They admit a generalization to the case of compact groups in general, and in particular compact Lie groups, such as therotation group SO....

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

Spherical harmonics

In mathematics, spherical harmonics are the angular portion of a set of solutions to Laplace's equation. Represented in a system of spherical coordinates, Laplace's spherical harmonics Y_\ell^m are a specific set of spherical harmonics that forms an orthogonal system, first introduced by Pierre...

and associated Legendre polynomials

Associated Legendre polynomials

In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation\,y -2xy' + \left\,y = 0,\,or equivalently...

, 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

Spin-weighted spherical harmonics

Spin-weighted spherical harmonics are generalizations of the standard spherical harmonics and—like the usual spherical harmonics—are functions on the sphere. Unlike ordinary spherical harmonics, the spin-weighted harmonics are U gauge fields rather than scalar fields: mathematically, they take...

Relation to Bessel Function

In the limit when

we have

where

is the Bessel function

Bessel function

In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...

and