Wigner D-matrix

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 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,

$[J_x,J_y]=i{J}_z,[J_z,J_x]=i{J}_y,[J_y,J_z]=i{J}_x,$

where i is the purely imaginary number and the Planck constant ħ has been set equal to one. The Casimir operator

$J^2=J_x^2+J_y^2+J_z^2$

commutes with all generators of the Lie algebra. Hence, it may be diagonalized together with J_z.

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

$J^2|jm⟩=j(j+1)|jm⟩,J_z|jm⟩=m|jm⟩,$

where j = 0, 1/2, 1, 3/2, 2, ... for SU(2), and j = 0, 1, 2, ... for SO(3). In both cases, m = −j, −j + 1, ..., j.

A 3-dimensional rotation operator can be written as

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

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 unitary square matrix of dimension 2j + 1 in this spherical basis with elements

$D_{m^{\prime }m}^j(\alpha ,\beta ,\gamma )\equiv ⟨j{m}^{\prime }|ℛ(\alpha ,\beta ,\gamma )|jm⟩=e^{-i{m}^{\prime }\alpha }{d}_{m^{\prime }m}^j(\beta )e^{-im\gamma },$

where

$d_{m^{\prime }m}^j(\beta )=⟨j{m}^{\prime }|e^{-i\beta J_y}|jm⟩=D_{m^{\prime }m}^j(0,\beta ,0)$

is an element of the orthogonal Wigner's (small) d-matrix (sometimes referred to as the reduced Wigner D-matrix).

That is, in this basis,

$D_{m^{\prime }m}^j(\alpha ,0,0)=e^{-i{m}^{\prime }\alpha }{\delta }_{m^{\prime }m}$

is diagonal, like the γ matrix factor, but unlike the above β factor.

Wigner (small) d-matrix

Wigner gave the following expression:

$d_{m^{\prime }m}^j(\beta )=[(j+m^{\prime })!(j-m^{\prime })!(j+m)!(j-m)!{]}^{\frac{1}{2}}\sum _{s=s_{\mathrm{m}\mathrm{i}\mathrm{n}}}^{s_{\mathrm{m}\mathrm{a}\mathrm{x}}}\left[\frac{(-1{)}^{m^{\prime }-m+s}{(\cos \frac{\beta }{2})}^{2j+m-m^{\prime }-2s}{(\sin \frac{\beta }{2})}^{m^{\prime }-m+2s}}{(j+m-s)!s!(m^{\prime }-m+s)!(j-m^{\prime }-s)!}\right].$

The sum over s is over such values that the factorials are nonnegative, i.e. $s_{\mathrm{m}\mathrm{i}\mathrm{n}}=\mathrm{m}\mathrm{a}\mathrm{x}(0,m-m^{\prime })$, $s_{\mathrm{m}\mathrm{a}\mathrm{x}}=\mathrm{m}\mathrm{i}\mathrm{n}(j+m,j-m^{\prime })$.

Note: The d-matrix elements defined here are real. In the often-used z-x-z convention of Euler angles, the factor $(-1{)}^{m^{\prime }-m+s}$ in this formula is replaced by $(-1{)}^s{i}^{m-m^{\prime }},$ 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 $P_k^{(a,b)}(\cos \beta )$ with nonnegative $a$ and $b.$ Let

$k=\min (j+m,j-m,j+m^{\prime },j-m^{\prime }).$

If

$k=\{\begin{array}{cc}j+m:& a=m^{\prime }-m;\lambda =m^{\prime }-m\\ j-m:& a=m-m^{\prime };\lambda =0\\ j+m^{\prime }:& a=m-m^{\prime };\lambda =0\\ j-m^{\prime }:& a=m^{\prime }-m;\lambda =m^{\prime }-m\end{array}$

Then, with $b=2j-2k-a,$ the relation is

$d_{m^{\prime }m}^j(\beta )=(-1{)}^{\lambda }{(\genfrac{}{}{0ex}{}{2j-k}{k+a})}^{\frac{1}{2}}{(\genfrac{}{}{0ex}{}{k+b}{b})}^{-\frac{1}{2}}{(\sin \frac{\beta }{2})}^a{(\cos \frac{\beta }{2})}^b{P}_k^{(a,b)}(\cos \beta ),$

where $a,b\ge 0.$

It is also useful to consider the relations $a=|m^{\prime }-m|,b=|m^{\prime }+m|,\lambda =\frac{m-m^{\prime }-|m-m^{\prime }|}{2},k=j-M$, where $M=\max (|m|,|m^{\prime }|)$ and $N=\min (|m|,|m^{\prime }|)$, which lead to:

$d_{m^{\prime }m}^j(\beta )=(-1{)}^{\frac{m-m^{\prime }-|m-m^{\prime }|}{2}}{\left[\frac{(j+M)!(j-M)!}{(j+N)!(j-N)!}\right]}^{\frac{1}{2}}{(\sin \frac{\beta }{2})}^{|m-m^{\prime }|}{(\cos \frac{\beta }{2})}^{|m+m^{\prime }|}{P}_{j-M}^{(|m-m^{\prime }|,|m+m^{\prime }|)}(\cos \beta ).$

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

$\begin{array}{cc}{\widehat{𝒥}}_1& =i(\cos \alpha \cot \beta \frac{\partial }{\partial \alpha }+\sin \alpha \frac{\partial }{\partial \beta }-\frac{\cos \alpha }{\sin \beta }\frac{\partial }{\partial \gamma })\\ {\widehat{𝒥}}_2& =i(\sin \alpha \cot \beta \frac{\partial }{\partial \alpha }-\cos \alpha \frac{\partial }{\partial \beta }-\frac{\sin \alpha }{\sin \beta }\frac{\partial }{\partial \gamma })\\ {\widehat{𝒥}}_3& =-i\frac{\partial }{\partial \alpha }\end{array}$

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

Further,

$\begin{array}{cc}{\widehat{𝒫}}_1& =i(\frac{\cos \gamma }{\sin \beta }\frac{\partial }{\partial \alpha }-\sin \gamma \frac{\partial }{\partial \beta }-\cot \beta \cos \gamma \frac{\partial }{\partial \gamma })\\ {\widehat{𝒫}}_2& =i(-\frac{\sin \gamma }{\sin \beta }\frac{\partial }{\partial \alpha }-\cos \gamma \frac{\partial }{\partial \beta }+\cot \beta \sin \gamma \frac{\partial }{\partial \gamma })\\ {\widehat{𝒫}}_3& =-i\frac{\partial }{\partial \gamma },\\ \end{array}$

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

The operators satisfy the commutation relations

$[{𝒥}_1,{𝒥}_2]=i{𝒥}_3,\text{and}[{𝒫}_1,{𝒫}_2]=-i{𝒫}_3,$

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

The two sets mutually commute,

$[{𝒫}_i,{𝒥}_j]=0,i,j=1,2,3,$

and the total operators squared are equal,

${𝒥}^2\equiv {𝒥}_1^2+{𝒥}_2^2+{𝒥}_3^2={𝒫}^2\equiv {𝒫}_1^2+{𝒫}_2^2+{𝒫}_3^2.$

Their explicit form is,

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

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

$\begin{array}{cc}{𝒥}_3{D}_{m^{\prime }m}^j(\alpha ,\beta ,\gamma {)}^{*}& =m^{\prime }{D}_{m^{\prime }m}^j(\alpha ,\beta ,\gamma {)}^{*}\\ ({𝒥}_1\pm i{𝒥}_2)D_{m^{\prime }m}^j(\alpha ,\beta ,\gamma {)}^{*}& =\sqrt{j(j+1)-m^{\prime }(m^{\prime }\pm 1)}{D}_{m^{\prime }\pm 1,m}^j(\alpha ,\beta ,\gamma {)}^{*}\end{array}$

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

${𝒫}_3{D}_{m^{\prime }m}^j(\alpha ,\beta ,\gamma {)}^{*}=m{D}_{m^{\prime }m}^j(\alpha ,\beta ,\gamma {)}^{*},$

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

$({𝒫}_1\mp i{𝒫}_2)D_{m^{\prime }m}^j(\alpha ,\beta ,\gamma {)}^{*}=\sqrt{j(j+1)-m(m\pm 1)}{D}_{m^{\prime },m\pm 1}^j(\alpha ,\beta ,\gamma {)}^{*}.$

Finally,

${𝒥}^2{D}_{m^{\prime }m}^j(\alpha ,\beta ,\gamma {)}^{*}={𝒫}^2{D}_{m^{\prime }m}^j(\alpha ,\beta ,\gamma {)}^{*}=j(j+1)D_{m^{\prime }m}^j(\alpha ,\beta ,\gamma {)}^{*}.$

In other words, the rows and columns of the (complex conjugate) Wigner D-matrix span irreducible representations of the isomorphic Lie algebras generated by $\{{𝒥}_i\}$ and $\{-{𝒫}_i\}$.

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

$⟨j{m}^{\prime }|ℛ(\alpha ,\beta ,\gamma )|jm⟩=⟨j{m}^{\prime }|T^{†}ℛ(\alpha ,\beta ,\gamma )T|jm⟩=(-1{)}^{m^{\prime }-m}⟨j,-m^{\prime }|ℛ(\alpha ,\beta ,\gamma )|j,-m{⟩}^{*},$

or

$D_{m^{\prime }m}^j(\alpha ,\beta ,\gamma )=(-1{)}^{m^{\prime }-m}{D}_{-m^{\prime },-m}^j(\alpha ,\beta ,\gamma {)}^{*}.$

Here, we used that $T$ is anti-unitary (hence the complex conjugation after moving $T^{†}$ from ket to bra), $T|jm⟩=(-1{)}^{j-m}|j,-m⟩$ and $(-1{)}^{2j-m^{\prime }-m}=(-1{)}^{m^{\prime }-m}$.

A further symmetry implies

$(-1{)}^{m^{\prime }-m}{D}_{m{m}^{\prime }}^j(\alpha ,\beta ,\gamma )=D_{m^{\prime }m}^j(\gamma ,\beta ,\alpha ).$

Orthogonality relations

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

$\underset{0}{\overset{2\pi }{\int }}d\alpha \underset{0}{\overset{\pi }{\int }}d\beta \sin \beta \underset{0}{\overset{2\pi }{\int }}d\gamma D_{m^{\prime }{k}^{\prime }}^{j^{\prime }}(\alpha ,\beta ,\gamma {)}^{\ast }{D}_{mk}^j(\alpha ,\beta ,\gamma )=\frac{8{\pi }^2}{2j+1}{\delta }_{m^{\prime }m}{\delta }_{k^{\prime }k}{\delta }_{j^{\prime }j}.$

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 $D_{mk}^j(\alpha ,\beta ,\gamma )$ are matrix elements of a unitary transformation from one spherical basis $|lm⟩$ to another $ℛ(\alpha ,\beta ,\gamma )|lm⟩$ is represented by the relations:

$\sum _k{D}_{m^{\prime }k}^j(\alpha ,\beta ,\gamma {)}^{*}{D}_{mk}^j(\alpha ,\beta ,\gamma )={\delta }_{m,m^{\prime }},$

$\sum _k{D}_{k{m}^{\prime }}^j(\alpha ,\beta ,\gamma {)}^{*}{D}_{km}^j(\alpha ,\beta ,\gamma )={\delta }_{m,m^{\prime }}.$

The group characters for SU(2) only depend on the rotation angle β, being class functions, so, then, independent of the axes of rotation,

${\chi }^j(\beta )\equiv \sum _m{D}_{mm}^j(\beta )=\sum _m{d}_{mm}^j(\beta )=\frac{\sin \left(\frac{(2j+1)\beta }{2}\right)}{\sin \left(\frac{\beta }{2}\right)},$

and consequently satisfy simpler orthogonality relations, through the Haar measure of the group,

$\frac{1}{\pi }\underset{0}{\overset{2\pi }{\int }}d\beta {\sin }^2\left(\frac{\beta }{2}\right){\chi }^j(\beta ){\chi }^{j^{\prime }}(\beta )={\delta }_{j^{\prime }j}.$

The completeness relation is (cf. Eq. (3.95) in ref., or Eq. (4.10.7) in ref.)

$\sum _j{\chi }^j(\beta ){\chi }^j({\beta }^{\prime })=\delta (\beta -{\beta }^{\prime }),$

whence, for ${\beta }^{\prime }=0,$

$\sum _j{\chi }^j(\beta )(2j+1)=\delta (\beta ).$

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

The set of Kronecker product matrices

${𝐃}^j(\alpha ,\beta ,\gamma )\otimes {𝐃}^{j^{\prime }}(\alpha ,\beta ,\gamma )$

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

$D_{mk}^j(\alpha ,\beta ,\gamma )D_{m^{\prime }{k}^{\prime }}^{j^{\prime }}(\alpha ,\beta ,\gamma )=\sum _{J=|j-j^{\prime }|}^{j+j^{\prime }}⟨jm{j}^{\prime }{m}^{\prime }|J(m+m^{\prime })⟩⟨jk{j}^{\prime }{k}^{\prime }|J(k+k^{\prime })⟩D_{(m+m^{\prime })(k+k^{\prime })}^J(\alpha ,\beta ,\gamma )$

The symbol $⟨j_1{m}_1{j}_2{m}_2|j_3{m}_3⟩$ 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:

$D_{m0}^{\ell }(\alpha ,\beta ,\gamma )=\sqrt{\frac{4\pi }{2\ell +1}}{Y}_{\ell }^{m*}(\beta ,\alpha )=\sqrt{\frac{(\ell -m)!}{(\ell +m)!}}{P}_{\ell }^m(\cos \beta )e^{-im\alpha }.$

This implies the following relationship for the d-matrix:

$d_{m0}^{\ell }(\beta )=\sqrt{\frac{(\ell -m)!}{(\ell +m)!}}{P}_{\ell }^m(\cos \beta ).$

A rotation of spherical harmonics $⟨\theta ,\varphi |\ell m^{\prime }⟩$ then is effectively a composition of two rotations,

$\sum _{m^{\prime }=-\ell }^{\ell }{Y}_{\ell }^{m^{\prime }}(\theta ,\varphi )D_{m^{\prime }m}^{\ell }(\alpha ,\beta ,\gamma ).$

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

$D_{0,0}^{\ell }(\alpha ,\beta ,\gamma )=d_{0,0}^{\ell }(\beta )=P_{\ell }(\cos \beta ).$

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:

$D_{ms}^{\ell }(\alpha ,\beta ,-\gamma )=(-1{)}^s\sqrt{\frac{4\pi }{2\ell +1}}{}_s{Y}_{\ell }^m(\beta ,\alpha )e^{is\gamma }.$

Connection with transition probability under rotations

The absolute square of an element of the D-matrix,

$F_{m{m}^{\prime }}(\beta )=|D_{m{m}^{\prime }}^j(\alpha ,\beta ,\gamma ){|}^2,$

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 $m^{\prime }$ along a second direction at an angle $\beta$ to the first direction. The set of quantities $F_{m{m}^{\prime }}$ 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:

$\sum _{m^{\prime }=-j}^j{F}_{m{m}^{\prime }}(\beta )f_{\ell }^j(m^{\prime })=P_{\ell }(\cos \beta )f_{\ell }^j(m)(\ell =0,1,\dots ,2j).$

Here, the eigenvector, $f_{\ell }^j(m)$, is a scaled and shifted discrete Chebyshev polynomial, and the corresponding eigenvalue, $P_{\ell }(\cos \beta )$, is the Legendre polynomial.

Relation to Bessel functions

In the limit when $\ell \gg m,m^{\prime }$, one obtains

$D_{m{m}^{\prime }}^{\ell }(\alpha ,\beta ,\gamma )\approx e^{-im\alpha -i{m}^{\prime }\gamma }{J}_{m-m^{\prime }}(\ell \beta )$

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 $d_{m^{\prime }m}^j(\theta )$ for j = 1/2, 1, 3/2, and 2 are given below.

For j = 1/2

$\begin{array}{cc}{d}_{\frac{1}{2},\frac{1}{2}}^{\frac{1}{2}}& =\cos \frac{\theta }{2}\\ d_{\frac{1}{2},-\frac{1}{2}}^{\frac{1}{2}}& =-\sin \frac{\theta }{2}\end{array}$

For j = 1

$\begin{array}{cc}{d}_{1,1}^1& =\frac{1}{2}(1+\cos \theta )\\ d_{1,0}^1& =-\frac{1}{\sqrt{2}}\sin \theta \\ d_{1,-1}^1& =\frac{1}{2}(1-\cos \theta )\\ d_{0,0}^1& =\cos \theta \end{array}$

For j = 3/2

$\begin{array}{cc}{d}_{\frac{3}{2},\frac{3}{2}}^{\frac{3}{2}}& =\frac{1}{2}(1+\cos \theta )\cos \frac{\theta }{2}\\ d_{\frac{3}{2},\frac{1}{2}}^{\frac{3}{2}}& =-\frac{\sqrt{3}}{2}(1+\cos \theta )\sin \frac{\theta }{2}\\ d_{\frac{3}{2},-\frac{1}{2}}^{\frac{3}{2}}& =\frac{\sqrt{3}}{2}(1-\cos \theta )\cos \frac{\theta }{2}\\ d_{\frac{3}{2},-\frac{3}{2}}^{\frac{3}{2}}& =-\frac{1}{2}(1-\cos \theta )\sin \frac{\theta }{2}\\ d_{\frac{1}{2},\frac{1}{2}}^{\frac{3}{2}}& =\frac{1}{2}(3\cos \theta -1)\cos \frac{\theta }{2}\\ d_{\frac{1}{2},-\frac{1}{2}}^{\frac{3}{2}}& =-\frac{1}{2}(3\cos \theta +1)\sin \frac{\theta }{2}\end{array}$

For j = 2

$\begin{array}{cc}{d}_{2,2}^2& =\frac{1}{4}{(1+\cos \theta )}^2\\ d_{2,1}^2& =-\frac{1}{2}\sin \theta (1+\cos \theta )\\ d_{2,0}^2& =\sqrt{\frac{3}{8}}{\sin }^2\theta \\ d_{2,-1}^2& =-\frac{1}{2}\sin \theta (1-\cos \theta )\\ d_{2,-2}^2& =\frac{1}{4}{(1-\cos \theta )}^2\\ d_{1,1}^2& =\frac{1}{2}(2{\cos }^2\theta +\cos \theta -1)\\ d_{1,0}^2& =-\sqrt{\frac{3}{8}}\sin 2\theta \\ d_{1,-1}^2& =\frac{1}{2}(-2{\cos }^2\theta +\cos \theta +1)\\ d_{0,0}^2& =\frac{1}{2}(3{\cos }^2\theta -1)\end{array}$

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

$d_{m^{\prime },m}^j=(-1{)}^{m-m^{\prime }}{d}_{m,m^{\prime }}^j=d_{-m,-m^{\prime }}^j.$

Symmetries and special cases

$\begin{array}{cc}{d}_{m^{\prime },m}^j(\pi )& =(-1{)}^{j-m}{\delta }_{m^{\prime },-m}\\ d_{m^{\prime },m}^j(\pi -\beta )& =(-1{)}^{j+m^{\prime }}{d}_{m^{\prime },-m}^j(\beta )\\ d_{m^{\prime },m}^j(\pi +\beta )& =(-1{)}^{j-m}{d}_{m^{\prime },-m}^j(\beta )\\ d_{m^{\prime },m}^j(2\pi +\beta )& =(-1{)}^{2j}{d}_{m^{\prime },m}^j(\beta )\\ d_{m^{\prime },m}^j(-\beta )& =d_{m,m^{\prime }}^j(\beta )=(-1{)}^{m^{\prime }-m}{d}_{m^{\prime },m}^j(\beta )\end{array}$

See also

• Clebsch–Gordan coefficients
• Tensor operator
• Symmetries in quantum mechanics