Wigner D-matrix

The Wigner D-matrix is a matrix in an irreducible representation of the groups SU(2) and SO(3). The complex conjugate of the elements of the D-matrix with integral indices are eigenfunctions of the Hamiltonian of spherical and symmetric rigid rotors. The matrix was introduced by E. Wigner in 1927

Definition 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,

$[j_{x},j_{y}]=ij_{z},\quad [j_{z},j_{x}]=ij_{y},\quad [j_{y},j_{z}]=ij_{x},$

where i is the purely imaginary number and Planck's constant $\hbar$ 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 $j=0,{\tfrac {1}{2}},1,{\tfrac {3}{2}},2,\dots$ and $m=-j,-j+1,\ldots ,j$. (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 $\alpha ,\;\beta ,$ and $\gamma \;$ 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

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

The matrix with general element

$d_{m'm}^{j}(\beta )=\langle jm'|e^{-i\beta j_{y}}|jm\rangle$

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

Wigner (small) d-matrix

Wigner gave the following expression

${\begin{array}{lcl}d_{m'm}^{j}(\beta )&=&[(j+m')!(j-m')!(j+m)!(j-m)!]^{1/2}\sum _{s}{\frac {(-1)^{m'-m+s}}{(j+m-s)!s!(m'-m+s)!(j-m'-s)!}}\\&&\times \left(\cos {\frac {\beta }{2}}\right)^{2j+m-m'-2s}\left(\sin {\frac {\beta }{2}}\right)^{m'-m+2s}\end{array}}$

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 $(-1)^{m'-m+s}$ 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 $P_{k}^{(a,b)}(\cos \beta )$ with nonnegative $a\,$ and $b\,$. 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

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

where $a,b\geq 0.\,$

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

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

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,

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

and

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

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

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

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

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

Finally,

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

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

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

or

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

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

Relation with spherical harmonic functions

The D-matrix elements with second index equal to zero, are proportional to spherical harmonics, normalized to unity and with Condon and Shortley phase convention,

$D_{m0}^{\ell }(\alpha ,\beta ,\gamma )^{*}={\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell }^{m}(\beta ,\alpha ).$

The left-hand side contains γ in the expression exp[−i⋅0⋅γ] and hence does not depend on γ. The angle γ does not appear in the right-hand side either, so that the expression is valid for any γ. 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

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

References

Cited references