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 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}]=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 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 $J_{z}$ . 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,... for SU(2), and j = 0, 1, 2, ... for SO(3). In both cases, m = -j, -j + 1,..., j.

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 elements

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

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

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

Wigner (small) d-matrix

Wigner^[1] gave the following expression

{\begin{array}{lcl}d_{m'm}^{j}(\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 $P_{k}^{(a,b)}(\cos \beta )$ 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_{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 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_{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 ^{*},

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

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 $\gamma$ :

\int _{0}^{2\pi }d\alpha \int _{0}^{\pi }\sin \beta d\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.

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:^[3]

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

The symbol $\langle jmj'm'|JM\rangle$ 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_{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 $D_{mm^{\prime }}^{\ell }(\alpha ,\beta ,\gamma )\approx e^{-im\alpha -im^{\prime }\gamma }J_{m-m^{\prime }}(\ell \beta )$ 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 ^[4]

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

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}

References

↑ Wigner, E. P. (1931). Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren. Braunschweig: Vieweg Verlag. Translated into English by Griffin, J. J. (1959). Group Theory and its Application to the Quantum Mechanics of Atomic Spectra. New York: Academic Press.
↑ Biedenharn, L. C.; Louck, J. D. (1981). Angular Momentum in Quantum Physics. Reading: Addison-Wesley. ISBN 0-201-13507-8.
↑ Rose, M. E. Elementary Theory of Angular Momentum. New York, JOHN WILEY & SONS, 1957.
↑ Edén, M. (2003). "Computer simulations in solid-state NMR. I. Spin dynamics theory". Concepts Magn. Reson. 17A (1): 117–154. doi:10.1002/cmr.a.10061.

External links

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

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