.

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 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,

\( [J_x,J_y] = i J_z,\quad [J_z,J_x] = i J_y,\quad [J_y,J_z] = i J_x, \)

where i is the purely imaginary number and Planck's constant \hbar has been put equal to one. The operator

\( J^2 = J_x^2 + J_y^2 + J_z^2 \)

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

\( J^2 |jm\rangle = j(j+1) |jm\rangle,\quad J_z |jm\rangle = m |jm\rangle, \)

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

\( \mathcal{R}(\alpha,\beta,\gamma) = e^{-i\alpha J_z}e^{-i\beta J_y}e^{-i\gamma J_z}, \)

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

\( D^j_{m'm}(\alpha,\beta,\gamma) \equiv \langle jm' | \mathcal{R}(\alpha,\beta,\gamma)| jm \rangle = e^{-im'\alpha } d^j_{m'm}(\beta)e^{-i m\gamma}. \)

The matrix with general element

\( d^j_{m'm}(\beta)= \langle jm' |e^{-i\beta J_y} | jm \rangle \)

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

Wigner (small) d-matrix

Wigner[1] gave the following expression

\( \begin{array}{lcl} d^j_{m'm}(\beta) &=& [(j+m')!(j-m')!(j+m)!(j-m)!]^{1/2} \sum\limits_s \left[\frac{(-1)^{m'-m+s}}{(j+m-s)!s!(m'-m+s)!(j-m'-s)!} \right.\\ &&\left. \cdot \left(\cos\frac{\beta}{2}\right)^{2j+m-m'-2s}\left(\sin\frac{\beta}{2}\right)^{m'-m+2s} \right]. \end{array} \)

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 \( (-1)^{m'-m+s}\) in this formula is replaced by \( (-1)^s\, i^{m-m'}\) , 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 \( P^{(a,b)}_k(\cos\beta) with nonnegative a\, and b\,.[2] Let

\( k = \min(j+m,\,j-m,\,j+m',\,j-m'). \)

\( \hbox{If}\quad k = \begin{cases} j+m: &\quad a=m'-m;\quad \lambda=m'-m\\ j-m: &\quad a=m-m';\quad \lambda= 0 \\ j+m': &\quad a=m-m';\quad \lambda= 0 \\ j-m': &\quad a=m'-m;\quad \lambda=m'-m \\ \end{cases} \)

Then, with \( b=2j-2k-a\,, \) the relation is

\( d^j_{m'm}(\beta) = (-1)^{\lambda} \binom{2j-k}{k+a}^{1/2} \binom{k+b}{b}^{-1/2} \left(\sin\frac{\beta}{2}\right)^a \left(\cos\frac{\beta}{2}\right)^b P^{(a,b)}_k(\cos\beta), \)

where \( a,b \ge 0. \,\)

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 \((x,\, y,\,z) = (1,\,2,\,3),\)

\( \begin{array}{lcl} \hat{\mathcal{J}}_1 &=& i \left( \cos \alpha \cot \beta \, {\partial \over \partial \alpha} \, + \sin \alpha \, {\partial \over \partial \beta} \, - {\cos \alpha \over \sin \beta} \, {\partial \over \partial \gamma} \, \right) \\ \hat{\mathcal{J}}_2 &=& i \left( \sin \alpha \cot \beta \, {\partial \over \partial \alpha} \, - \cos \alpha \; {\partial \over \partial \beta } \, - {\sin \alpha \over \sin \beta} \, {\partial \over \partial \gamma } \, \right) \\ \hat{\mathcal{J}}_3 &=& - i \; {\partial \over \partial \alpha} , \end{array} \)

which have quantum mechanical meaning: they are space-fixed rigid rotor angular momentum operators.

Further,

\( \begin{array}{lcl} \hat{\mathcal{P}}_1 &=& \, i \left( {\cos \gamma \over \sin \beta} {\partial \over \partial \alpha } - \sin \gamma {\partial \over \partial \beta } - \cot \beta \cos \gamma {\partial \over \partial \gamma} \right) \\ \hat{\mathcal{P}}_2 &=& \, i \left( - {\sin \gamma \over \sin \beta} {\partial \over \partial \alpha} - \cos \gamma {\partial \over \partial \beta} + \cot \beta \sin \gamma {\partial \over \partial \gamma} \right) \\ \hat{\mathcal{P}}_3 &=& - i {\partial\over \partial \gamma}, \\ \end{array} \)

which have quantum mechanical meaning: they are body-fixed rigid rotor angular momentum operators.

The operators satisfy the commutation relations

\( \left[\mathcal{J}_1, \, \mathcal{J}_2\right] = i \mathcal{J}_3, \qquad \hbox{and}\qquad \left[\mathcal{P}_1, \, \mathcal{P}_2\right] = -i \mathcal{P}_3 \)

and the corresponding relations with the indices permuted cyclically. The \( \mathcal{P}_i \) satisfy anomalous commutation relations (have a minus sign on the right hand side).

The two sets mutually commute,

\( \left[\mathcal{P}_i, \, \mathcal{J}_j\right] = 0,\quad i,\,j = 1,\,2,\,3, \)

and the total operators squared are equal,

\( \mathcal{J}^2 \equiv \mathcal{J}_1^2+ \mathcal{J}_2^2 + \mathcal{J}_3^2 = \mathcal{P}^2 \equiv \mathcal{P}_1^2+ \mathcal{P}_2^2 + \mathcal{P}_3^2 . \)

Their explicit form is,

\( \mathcal{J}^2= \mathcal{P}^2 = -\frac{1}{\sin^2\beta} \left( \frac{\partial^2}{\partial \alpha^2} +\frac{\partial^2}{\partial \gamma^2} -2\cos\beta\frac{\partial^2}{\partial\alpha\partial \gamma} \right) -\frac{\partial^2}{\partial \beta^2} -\cot\beta\frac{\partial}{\partial \beta}. \)

The operators \mathcal{J}_i act on the first (row) index of the D-matrix,

\( \mathcal{J}_3 \, D^j_{m'm}(\alpha,\beta,\gamma)^* = m' \, D^j_{m'm}(\alpha,\beta,\gamma)^* ,\)

and

\( (\mathcal{J}_1 \pm i \mathcal{J}_2)\, D^j_{m'm}(\alpha,\beta,\gamma)^* = \sqrt{j(j+1)-m'(m'\pm 1)} \, D^j_{m'\pm 1, m}(\alpha,\beta,\gamma)^* . \)

The operators \( \mathcal{P}_i act on the second (column) index of the D-matrix

\( \mathcal{P}_3 \, D^j_{m'm}(\alpha,\beta,\gamma)^* = m \, D^j_{m'm}(\alpha,\beta,\gamma)^* , \)

and because of the anomalous commutation relation the raising/lowering operators are defined with reversed signs,

\( (\mathcal{P}_1 \mp i \mathcal{P}_2)\, D^j_{m'm}(\alpha,\beta,\gamma)^* = \sqrt{j(j+1)-m(m\pm 1)} \, D^j_{m', m\pm1}(\alpha,\beta,\gamma)^* . \)

Finally,

\( \mathcal{J}^2\, D^j_{m'm}(\alpha,\beta,\gamma)^* = \mathcal{P}^2\, D^j_{m'm}(\alpha,\beta,\gamma)^* = j(j+1) D^j_{m'm}(\alpha,\beta,\gamma)^*. \)

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 \( \{\mathcal{J}_i\} \) and \( \{-\mathcal{P}_i\}.\)

An important property of the Wigner D-matrix follows from the commutation of \( \mathcal{R}(\alpha,\beta,\gamma) \) with the time reversal operator \( T\,,\)

\(\langle jm' | \mathcal{R}(\alpha,\beta,\gamma)| jm \rangle = \langle jm' | T^{\,\dagger} \mathcal{R}(\alpha,\beta,\gamma) T| jm \rangle = (-1)^{m'-m} \langle j,-m' | \mathcal{R}(\alpha,\beta,\gamma)| j,-m \rangle^*, \)

\( D^j_{m'm}(\alpha,\beta,\gamma) = (-1)^{m'-m} D^j_{-m',-m}(\alpha,\beta,\gamma)^*. \)

Here we used that \( T\, is anti-unitary (hence the complex conjugation after moving \( T^\dagger\, from ket to bra), T | jm \rangle = (-1)^{j-m} | j,-m \rangle and \( (-1)^{2j-m'-m} = (-1)^{m'-m}.\)

Orthogonality relations

The Wigner D-matrix elements \( D^j_{mk}(\alpha,\beta,\gamma)\) form a complete set of orthogonal functions of the Euler angles \(\alpha, \beta, \) and \( \gamma:\)

\( \int_0^{2\pi} d\alpha \int_0^\pi \sin \beta d\beta \int_0^{2\pi} d\gamma \,\, D^{j'}_{m'k'}(\alpha,\beta,\gamma)^\ast D^j_{mk}(\alpha,\beta,\gamma) = \frac{8\pi^2}{2j+1} \delta_{m'm}\delta_{k'k}\delta_{j'j}. \)

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

\( \mathbf{D}^j(\alpha,\beta,\gamma)\otimes \mathbf{D}^{j'}(\alpha,\beta,\gamma) \)

forms a reducible matrix representation of the groups SO(3) and SU(2). Reduction into irreducible components is by the following equation:

\( D^j_{m k}(\alpha,\beta,\gamma) D^{j'}_{m' k'}(\alpha,\beta,\gamma) = \sum_{J=|j-j'|}^{j+j'} \sum_{M=-J}^J \sum_{K=-J}^J \langle j m j' m' | J M \rangle \langle j k j' k' | J K \rangle D^J_{M K}(\alpha,\beta,\gamma) \)

The symbol \( \langle j m j' m' | J M \rangle \) 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:

\( D^{\ell}_{m 0}(\alpha,\beta,0) = \sqrt{\frac{4\pi}{2\ell+1}} Y_{\ell}^{m*} (\beta, \alpha ) = \sqrt{\frac{(\ell-m)!}{(\ell+m)!}} \, P_\ell^m ( \cos{\beta} ) \, e^{-i m \alpha } \)

This implies the following relationship for the d-matrix:

\( d^{\ell}_{m 0}(\beta) = \sqrt{\frac{(\ell-m)!}{(\ell+m)!}} \, P_\ell^m ( \cos{\beta} ) \)

When both indices are set to zero, the Wigner D-matrix elements are given by ordinary Legendre polynomials:

\( D^{\ell}_{0,0}(\alpha,\beta,\gamma) = d^{\ell}_{0,0}(\beta) = P_{\ell}(\cos\beta). \)

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

\( \left( Y_{\ell}^m \right) ^* = (-1)^m Y_{\ell}^{-m}. \)

There exists a more general relationship to the spin-weighted spherical harmonics:

\( D^{\ell}_{-m s}(\alpha,\beta,-\gamma) =(-1)^m \sqrt\frac{4\pi}{2{\ell}+1} {}_sY_{{\ell}m}(\beta,\alpha) e^{is\gamma}. \)

Relation to Bessel functions

In the limit when \( \ell \gg m, m^\prime \) we have \( D^\ell_{mm^\prime}(\alpha,\beta,\gamma) \approx e^{-im\alpha-im^\prime\gamma}J_{m-m^\prime}(\ell\beta) \) where \( J_{m-m^\prime}(\ell\beta) is the Bessel function and \ell\beta\) 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

\( d_{1/2,1/2}^{1/2} = \cos (\theta/2)\)
\( d_{1/2,-1/2}^{1/2} = -\sin (\theta/2)\)

for j = 1

\( d_{1,1}^{1} = \frac{1+\cos \theta}{2}\)
\( d_{1,0}^{1} = \frac{-\sin \theta}{\sqrt{2}}\)
\( d_{1,-1}^{1} = \frac{1-\cos \theta}{2}\)
\( d_{0,0}^{1} = \cos \theta\)

for j = 3/2

\( d_{3/2,3/2}^{3/2} = \frac{1+\cos \theta}{2} \cos \frac{\theta}{2}\)
\( d_{3/2,1/2}^{3/2} = -\sqrt{3} \frac{1+\cos \theta}{2} \sin \frac{\theta}{2}\)
\( d_{3/2,-1/2}^{3/2} = \sqrt{3} \frac{1-\cos \theta}{2} \cos \frac{\theta}{2}\)
\( d_{3/2,-3/2}^{3/2} = - \frac{1-\cos \theta}{2} \sin \frac{\theta}{2}\)
\( d_{1/2,1/2}^{3/2} = \frac{3\cos \theta - 1}{2} \cos \frac{\theta}{2}\)
\( d_{1/2,-1/2}^{3/2} = - \frac{3\cos \theta + 1}{2} \sin \frac{\theta}{2}\)

for j = 2 [3]

\( d_{2,2}^{2} = \frac{1}{4}\left(1 +\cos \theta\right)^2\)
\( d_{2,1}^{2} = -\frac{1}{2}\sin \theta \left(1 + \cos \theta\right)\)
\( d_{2,0}^{2} = \sqrt{\frac{3}{8}}\sin^2 \theta\)
\( d_{2,-1}^{2} = -\frac{1}{2}\sin \theta \left(1 - \cos \theta\right)\)
\( d_{2,-2}^{2} = \frac{1}{4}\left(1 -\cos \theta\right)^2\)
\( d_{1,1}^{2} = \frac{1}{2}\left(2\cos^2\theta + \cos \theta-1 \right)\)
\( d_{1,0}^{2} = -\sqrt{\frac{3}{8}} \sin 2 \theta\)
\( d_{1,-1}^{2} = \frac{1}{2}\left(- 2\cos^2\theta + \cos \theta +1 \right)\)
\( d_{0,0}^{2} = \frac{1}{2} \left(3 \cos^2 \theta - 1\right)\)

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

\( d_{m', m}^j = (-1)^{m-m'}d_{m, m'}^j = d_{-m,-m'}^j.\)