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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 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 Planck's constant $\hslash$ 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

$J^2|jm⟩=j(j+1)|jm⟩,J_z|jm⟩=m|jm⟩,$

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

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

The matrix with general element

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

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

Wigner (small) d-matrix

Wigner gave the following expression

$\begin{array}{lcl}{d}_{m^{\prime }m}^j(\beta )& =& [(j+m^{\prime })!(j-m^{\prime })!(j+m)!(j-m)!{]}^{1/2}\sum _s[\frac{(-1{)}^{m^{\prime }-m+s}}{(j+m-s)!s!(m^{\prime }-m+s)!(j-m^{\prime }-s)!}\\ & & \cdot {(\cos \frac{\beta }{2})}^{2j+m-m^{\prime }-2s}{(\sin \frac{\beta }{2})}^{m^{\prime }-m+2s}].\end{array}$

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 $(-1{)}^{m^{\prime }-m+s}$ 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

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

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

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

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

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

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

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

Further,

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

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

The operators satisfy the commutation relations

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

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,

$[{\mathcal{𝒫}}_i,{\mathcal{𝒥}}_j]=0,i,j=1,2,3,$

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,

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

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

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

and

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

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

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

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

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

Finally,

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

In other words, the rows and columns of the (complex conjugate) Wigner D-matrix span irreducible representations of the isomorphic Lie 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 $T$,

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

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 $(-1{)}^{2j-m^{\prime }-m}=(-1{)}^{m^{\prime }-m}$.

Orthogonality relations

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

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

This is a special case of the Schur orthogonality relations.

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

The set of Kronecker product matrices

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

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

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

The symbol $⟨jm{j}^{\prime }{m}^{\prime }|JM⟩$ 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:

$D_{m0}^{\ell }(\alpha ,\beta ,0)=\sqrt{\frac{4\pi }{2\ell +1}}{Y}_{\ell }^{m*}(\beta ,\alpha )=\sqrt{\frac{(\ell -m)!}{(\ell +m)!}}{P}_{\ell }^m(\cos \beta )e^{-im\alpha }$

This implies the following relationship for the d-matrix:

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

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:

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

Relation to Bessel functions

In the limit when $\ell \gg m,m^{\prime }$ we have $D_{m{m}^{\prime }}^{\ell }(\alpha ,\beta ,\gamma )\approx e^{-im\alpha -i{m}^{\prime }\gamma }{J}_{m-m^{\prime }}(\ell \beta )$ 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 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

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

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

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

See also

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

References

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External links

• PDG Table of Clebsch-Gordon Coefficients, Spherical Harmonics, and d-Functions