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,

${\displaystyle [J_{x},J_{y}]=iJ_{z},\quad [J_{z},J_{x}]=iJ_{y},\quad [J_{y},J_{z}]=iJ_{x},}$

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

${\displaystyle J^{2}=J_{x}^{2}+J_{y}^{2}+J_{z}^{2}}$

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

${\displaystyle J^{2}|jm\rangle =j(j+1)|jm\rangle ,\quad J_{z}|jm\rangle =m|jm\rangle ,}$

where j = 0, 1/2, 1, 3/2, 2, ... 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

${\displaystyle {\mathcal {R}}(\alpha ,\beta ,\gamma )=e^{-i\alpha J_{z}}e^{-i\beta J_{y}}e^{-i\gamma J_{z}},}$

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

${\displaystyle D_{m'm}^{j}(\alpha ,\beta ,\gamma )\equiv \langle jm'|{\mathcal {R}}(\alpha ,\beta ,\gamma )|jm\rangle =e^{-im'\alpha }d_{m'm}^{j}(\beta )e^{-im\gamma },}$

where

${\displaystyle d_{m'm}^{j}(\beta )=\langle jm'|e^{-i\beta J_{y}}|jm\rangle =D_{m'm}^{j}(0,\beta ,0)}$

is an element of the orthogonal Wigner's (small) d-matrix.

That is, in this basis,

${\displaystyle D_{m'm}^{j}(\alpha ,0,0)=e^{-im'\alpha }\delta _{m'm}}$

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

Wigner gave the following expression:[1]

${\displaystyle d_{m'm}^{j}(\beta )=[(j+m')!(j-m')!(j+m)!(j-m)!]^{\frac {1}{2}}\sum _{s=s_{\mathrm {min} }}^{s_{\mathrm {max} }}\left[{\frac {(-1)^{m'-m+s}\left(\cos {\frac {\beta }{2}}\right)^{2j+m-m'-2s}\left(\sin {\frac {\beta }{2}}\right)^{m'-m+2s}}{(j+m-s)!s!(m'-m+s)!(j-m'-s)!}}\right].}$

The sum over s is over such values that the factorials are nonnegative, i.e. ${\displaystyle s_{\mathrm {min} }=\mathrm {max} (0,m-m')}$, ${\displaystyle s_{\mathrm {max} }=\mathrm {min} (j+m,j-m')}$.

Note: The d-matrix elements defined here are real. In the often-used z-x-z convention of Euler angles, the factor ${\displaystyle (-1)^{m'-m+s}}$ in this formula is replaced by ${\displaystyle (-1)^{s}i^{m-m'},}$ 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 ${\displaystyle P_{k}^{(a,b)}(\cos \beta )}$ with nonnegative ${\displaystyle a}$ and ${\displaystyle b.}$[2] Let

${\displaystyle k=\min(j+m,j-m,j+m',j-m').}$

If

${\displaystyle k={\begin{cases}j+m:&a=m'-m;\quad \lambda =m'-m\\j-m:&a=m-m';\quad \lambda =0\\j+m':&a=m-m';\quad \lambda =0\\j-m':&a=m'-m;\quad \lambda =m'-m\\\end{cases}}}$

Then, with ${\displaystyle b=2j-2k-a,}$ the relation is

${\displaystyle d_{m'm}^{j}(\beta )=(-1)^{\lambda }{\binom {2j-k}{k+a}}^{\frac {1}{2}}{\binom {k+b}{b}}^{-{\frac {1}{2}}}\left(\sin {\frac {\beta }{2}}\right)^{a}\left(\cos {\frac {\beta }{2}}\right)^{b}P_{k}^{(a,b)}(\cos \beta ),}$

where ${\displaystyle a,b\geq 0.}$

The complex conjugate of the D-matrix satisfies a number of differential properties that can be formulated concisely by introducing the following operators with ${\displaystyle (x,y,z)=(1,2,3),}$

${\displaystyle {\begin{aligned}{\hat {\mathcal {J}}}_{1}&=i\left(\cos \alpha \cot \beta {\frac {\partial }{\partial \alpha }}+\sin \alpha {\partial \over \partial \beta }-{\cos \alpha \over \sin \beta }{\partial \over \partial \gamma }\right)\\{\hat {\mathcal {J}}}_{2}&=i\left(\sin \alpha \cot \beta {\partial \over \partial \alpha }-\cos \alpha {\partial \over \partial \beta }-{\sin \alpha \over \sin \beta }{\partial \over \partial \gamma }\right)\\{\hat {\mathcal {J}}}_{3}&=-i{\partial \over \partial \alpha }\end{aligned}}}$

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

Further,

${\displaystyle {\begin{aligned}{\hat {\mathcal {P}}}_{1}&=i\left({\cos \gamma \over \sin \beta }{\partial \over \partial \alpha }-\sin \gamma {\partial \over \partial \beta }-\cot \beta \cos \gamma {\partial \over \partial \gamma }\right)\\{\hat {\mathcal {P}}}_{2}&=i\left(-{\sin \gamma \over \sin \beta }{\partial \over \partial \alpha }-\cos \gamma {\partial \over \partial \beta }+\cot \beta \sin \gamma {\partial \over \partial \gamma }\right)\\{\hat {\mathcal {P}}}_{3}&=-i{\partial \over \partial \gamma },\\\end{aligned}}}$

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

The operators satisfy the commutation relations

${\displaystyle \left[{\mathcal {J}}_{1},{\mathcal {J}}_{2}\right]=i{\mathcal {J}}_{3},\qquad {\hbox{and}}\qquad \left[{\mathcal {P}}_{1},{\mathcal {P}}_{2}\right]=-i{\mathcal {P}}_{3},}$

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

The two sets mutually commute,

${\displaystyle \left[{\mathcal {P}}_{i},{\mathcal {J}}_{j}\right]=0,\quad i,j=1,2,3,}$

and the total operators squared are equal,

${\displaystyle {\mathcal {J}}^{2}\equiv {\mathcal {J}}_{1}^{2}+{\mathcal {J}}_{2}^{2}+{\mathcal {J}}_{3}^{2}={\mathcal {P}}^{2}\equiv {\mathcal {P}}_{1}^{2}+{\mathcal {P}}_{2}^{2}+{\mathcal {P}}_{3}^{2}.}$

Their explicit form is,

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

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

${\displaystyle {\begin{aligned}{\mathcal {J}}_{3}D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}&=m'D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}\\({\mathcal {J}}_{1}\pm i{\mathcal {J}}_{2})D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}&={\sqrt {j(j+1)-m'(m'\pm 1)}}D_{m'\pm 1,m}^{j}(\alpha ,\beta ,\gamma )^{*}\end{aligned}}}$

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

${\displaystyle {\mathcal {P}}_{3}D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}=mD_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*},}$

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

${\displaystyle ({\mathcal {P}}_{1}\mp i{\mathcal {P}}_{2})D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}={\sqrt {j(j+1)-m(m\pm 1)}}D_{m',m\pm 1}^{j}(\alpha ,\beta ,\gamma )^{*}.}$

Finally,

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

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

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

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

or

${\displaystyle D_{m'm}^{j}(\alpha ,\beta ,\gamma )=(-1)^{m'-m}D_{-m',-m}^{j}(\alpha ,\beta ,\gamma )^{*}.}$

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

A further symmetry implies

${\displaystyle (-1)^{m'-m}D_{mm'}^{j}(\alpha ,\beta ,\gamma )=D_{m'm}^{j}(\gamma ,\beta ,\alpha )~.}$

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

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

This is a special case of the Schur orthogonality relations.

Crucially, by the Peter–Weyl theorem, they further form a complete set.

The fact that ${\displaystyle D_{mk}^{j}(\alpha ,\beta ,\gamma )}$ are matrix elements of a unitary transformation from one spherical basis ${\displaystyle |lm\rangle }$ to another ${\displaystyle {\mathcal {R}}(\alpha ,\beta ,\gamma )|lm\rangle }$ is represented by the relations:[3]

${\displaystyle \sum _{k}D_{m'k}^{j}(\alpha ,\beta ,\gamma )^{*}D_{mk}^{j}(\alpha ,\beta ,\gamma )=\delta _{m,m'},}$

${\displaystyle \sum _{k}D_{km'}^{j}(\alpha ,\beta ,\gamma )^{*}D_{km}^{j}(\alpha ,\beta ,\gamma )=\delta _{m,m'}.}$

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

${\displaystyle \chi ^{j}(\beta )\equiv \sum _{m}D_{mm}^{j}(\beta )=\sum _{m}d_{mm}^{j}(\beta )={\frac {\sin \left({\frac {(2j+1)\beta }{2}}\right)}{\sin \left({\frac {\beta }{2}}\right)}},}$

and consequently satisfy simpler orthogonality relations, through the Haar measure of the group,[4]

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

The completeness relation (worked out in the same reference, (3.95)) is

${\displaystyle \sum _{j}\chi ^{j}(\beta )\chi ^{j}(\beta ')=\delta (\beta -\beta '),}$

whence, for ${\displaystyle \beta '=0,}$

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

The set of Kronecker product matrices

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

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

${\displaystyle D_{mk}^{j}(\alpha ,\beta ,\gamma )D_{m'k'}^{j'}(\alpha ,\beta ,\gamma )=\sum _{J=|j-j'|}^{j+j'}\langle jmj'm'|J\left(m+m'\right)\rangle \langle jkj'k'|J\left(k+k'\right)\rangle D_{\left(m+m'\right)\left(k+k'\right)}^{J}(\alpha ,\beta ,\gamma )}$

The symbol ${\displaystyle \langle j_{1}m_{1}j_{2}m_{2}|j_{3}m_{3}\rangle }$ is a Clebsch–Gordan coefficient.

For integer values of ${\displaystyle l}$, 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:

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

This implies the following relationship for the d-matrix:

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

A rotation of spherical harmonics ${\displaystyle \langle \theta ,\phi |\ell m'\rangle }$ then is effectively a composition of two rotations,

${\displaystyle \sum _{m'=-\ell }^{\ell }Y_{\ell }^{m'}(\theta ,\phi )~D_{m'~m}^{\ell }(\alpha ,\beta ,\gamma ).}$

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

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

In the present convention of Euler angles, ${\displaystyle \alpha }$ is a longitudinal angle and ${\displaystyle \beta }$ 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

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

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

${\displaystyle D_{ms}^{\ell }(\alpha ,\beta ,-\gamma )=(-1)^{s}{\sqrt {\frac {4\pi }{2{\ell }+1}}}{}_{s}Y_{\ell }^{m}(\beta ,\alpha )e^{is\gamma }.}$[5]

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

${\displaystyle F_{mm'}(\beta )=|D_{mm'}^{j}(\alpha ,\beta ,\gamma )|^{2},}$

gives the probability that a system with spin ${\displaystyle j}$ prepared in a state with spin projection ${\displaystyle m}$ along some direction will be measured to have a spin projection ${\displaystyle m'}$ along a second direction at an angle ${\displaystyle \beta }$ to the first direction. The set of quantities ${\displaystyle F_{mm'}}$ itself forms a real symmetric matrix, that depends only on the Euler angle ${\displaystyle \beta }$, as indicated.

Remarkably, the eigenvalue problem for the ${\displaystyle F}$ matrix can be solved completely:[6][7]

${\displaystyle \sum _{m'=-j}^{j}F_{mm'}(\beta )f_{\ell }^{j}(m')=P_{\ell }(\cos \beta )f_{\ell }^{j}(m)\qquad (\ell =0,1,\ldots ,2j).}$

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

In the limit when ${\displaystyle \ell \gg m,m^{\prime }}$ we have

${\displaystyle D_{mm'}^{\ell }(\alpha ,\beta ,\gamma )\approx e^{-im\alpha -im'\gamma }J_{m-m'}(\ell \beta )}$

where ${\displaystyle J_{m-m'}(\ell \beta )}$ is the Bessel function and ${\displaystyle \ell \beta }$ is finite.

Using sign convention of Wigner, et al. the d-matrix elements ${\displaystyle d_{m'm}^{j}(\theta )}$ for j = 1/2, 1, 3/2, and 2 are given below.

for j = 1/2

${\displaystyle {\begin{aligned}d_{{\frac {1}{2}},{\frac {1}{2}}}^{\frac {1}{2}}&=\cos {\frac {\theta }{2}}\\[6pt]d_{{\frac {1}{2}},-{\frac {1}{2}}}^{\frac {1}{2}}&=-\sin {\frac {\theta }{2}}\end{aligned}}}$

for j = 1

${\displaystyle {\begin{aligned}d_{1,1}^{1}&={\frac {1}{2}}(1+\cos \theta )\\[6pt]d_{1,0}^{1}&=-{\frac {1}{\sqrt {2}}}\sin \theta \\[6pt]d_{1,-1}^{1}&={\frac {1}{2}}(1-\cos \theta )\\[6pt]d_{0,0}^{1}&=\cos \theta \end{aligned}}}$

for j = 3/2

${\displaystyle {\begin{aligned}d_{{\frac {3}{2}},{\frac {3}{2}}}^{\frac {3}{2}}&={\frac {1}{2}}(1+\cos \theta )\cos {\frac {\theta }{2}}\\[6pt]d_{{\frac {3}{2}},{\frac {1}{2}}}^{\frac {3}{2}}&=-{\frac {\sqrt {3}}{2}}(1+\cos \theta )\sin {\frac {\theta }{2}}\\[6pt]d_{{\frac {3}{2}},-{\frac {1}{2}}}^{\frac {3}{2}}&={\frac {\sqrt {3}}{2}}(1-\cos \theta )\cos {\frac {\theta }{2}}\\[6pt]d_{{\frac {3}{2}},-{\frac {3}{2}}}^{\frac {3}{2}}&=-{\frac {1}{2}}(1-\cos \theta )\sin {\frac {\theta }{2}}\\[6pt]d_{{\frac {1}{2}},{\frac {1}{2}}}^{\frac {3}{2}}&={\frac {1}{2}}(3\cos \theta -1)\cos {\frac {\theta }{2}}\\[6pt]d_{{\frac {1}{2}},-{\frac {1}{2}}}^{\frac {3}{2}}&=-{\frac {1}{2}}(3\cos \theta +1)\sin {\frac {\theta }{2}}\end{aligned}}}$

for j = 2[8]

${\displaystyle {\begin{aligned}d_{2,2}^{2}&={\frac {1}{4}}\left(1+\cos \theta \right)^{2}\\[6pt]d_{2,1}^{2}&=-{\frac {1}{2}}\sin \theta \left(1+\cos \theta \right)\\[6pt]d_{2,0}^{2}&={\sqrt {\frac {3}{8}}}\sin ^{2}\theta \\[6pt]d_{2,-1}^{2}&=-{\frac {1}{2}}\sin \theta \left(1-\cos \theta \right)\\[6pt]d_{2,-2}^{2}&={\frac {1}{4}}\left(1-\cos \theta \right)^{2}\\[6pt]d_{1,1}^{2}&={\frac {1}{2}}\left(2\cos ^{2}\theta +\cos \theta -1\right)\\[6pt]d_{1,0}^{2}&=-{\sqrt {\frac {3}{8}}}\sin 2\theta \\[6pt]d_{1,-1}^{2}&={\frac {1}{2}}\left(-2\cos ^{2}\theta +\cos \theta +1\right)\\[6pt]d_{0,0}^{2}&={\frac {1}{2}}\left(3\cos ^{2}\theta -1\right)\end{aligned}}}$

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

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

${\displaystyle {\begin{aligned}d_{m',m}^{j}(\pi )&=(-1)^{j-m}\delta _{m',-m}\\[6pt]d_{m',m}^{j}(\pi -\beta )&=(-1)^{j+m'}d_{m',-m}^{j}(\beta )\\[6pt]d_{m',m}^{j}(\pi +\beta )&=(-1)^{j-m}d_{m',-m}^{j}(\beta )\\[6pt]d_{m',m}^{j}(2\pi +\beta )&=(-1)^{2j}d_{m',m}^{j}(\beta )\\[6pt]d_{m',m}^{j}(-\beta )&=d_{m,m'}^{j}(\beta )=(-1)^{m'-m}d_{m',m}^{j}(\beta )\end{aligned}}}$

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

• Amsler, C.; et al. (Particle Data Group) (2008). "PDG Table of Clebsch-Gordan Coefficients, Spherical Harmonics, and d-Functions" (PDF). Physics Letters B667.