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.

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 thesethree operators are the components of a vector operator known as "angular momentum". Examplesare the angular momentum of an electronin an atom, electronic spin,and the angular momentumof a rigid rotor. In all cases the three operators satisfy the following commutation relations,:$[j\_x,j\_y]\; =\; i\; j\_z,quad\; [j\_z,j\_x]\; =\; i\; j\_y,quad\; [j\_y,j\_z]\; =\; i\; j\_x,$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 operatoris a convention), which commutes with $j^2$. That is, it can be shown that there is a complete set of kets with:$j^2\; |jm\; angle\; =\; j(j+1)\; |jm\; angle,quad\; j\_z\; |jm\; angle\; =\; m\; |jm\; angle,$where $j=0,\; frac\{1\}\{2\},\; 1,\; frac\{3\}\{2\},\; 2,\; dots$ and $m=-j,\; -j+1,\; ldots,\; j$. (For SO(3) the "quantum number" $j$ is integer.)

A rotation operator can be written as:where 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 $2j+1$ withgeneral element:The matrix with general element : is known as Wigner's (small) d-matrix.

Wigner (small) d-matrix

Wigner [E. P. Wigner, "Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren", Vieweg Verlag, Braunschweig (1931). Translated into English: J. J. Griffin, "Group Theory and its Application to the Quantum Mechanics of Atomic Spectra", Academic Press, New York (1959).] gave the following expression: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 $(-1)^\{m\text{'}-m+s\}$ in this formula is replaced by $(-1)^s,\; i^\{m-m\text{'}\}$, 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 with nonnegative $a,$ and $b,$. [L. C. Biedenharn and J. D. Louck, "Angular Momentum in Quantum Physics", Addison-Wesley, Reading, (1981).] Let :$k\; =\; min(j+m,,j-m,,j+m\text{'},,j-m\text{'}).$

:

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

:where $a,b\; ge\; 0.\; ,$

Properties of Wigner D-matrix

The complex conjugate of the D-matrix satisfies a number of differential propertiesthat can be formulated concisely by introducing the following operators with $(x,,\; y,,z)\; =\; (1,,2,,3)$,:which have quantum mechanical meaning: they are space-fixed rigid rotor angular momentum operators.

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

The operators satisfy the commutation relations:$left\; [mathcal\{J\}\_1,\; ,\; mathcal\{J\}\_2\; ight]\; =\; i\; mathcal\{J\}\_3,\; qquad\; hbox\{and\}qquadleft\; [mathcal\{P\}\_1,\; ,\; mathcal\{P\}\_2\; ight]\; =\; -i\; mathcal\{P\}\_3$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,:$left\; [mathcal\{P\}\_i,\; ,\; mathcal\{J\}\_j\; ight]\; =\; 0,quad\; i,,j\; =\; 1,,2,,3,$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,:

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

The operators $mathcal\{P\}\_i$ act on the second (column) index of the D-matrix:and because of the anomalous commutation relation the raising/lowering operatorsare defined with reversed signs,:

Finally,:

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 with the time reversal operator $T,$,:or: Here we used that $T,$ is anti-unitary (hence the complex conjugation after moving$T^dagger,$ from ket to bra), $T\; |\; jm\; angle\; =\; (-1)^\{j-m\}\; |\; j,-m\; angle$ and $(-1)^\{2j-m\text{'}-m\}\; =\; (-1)^\{m\text{'}-m\}$.

Orthogonality relations

The Wigner D-matrix elements form a complete setof orthogonal functions of the Euler angles $alpha$, and $gamma$::

This is a special case of the Schur orthogonality relations.

Relation with spherical harmonic functions

The D-matrix elements with second index equal to zero, are proportionalto spherical harmonics, normalized to unity and with Condon and Shortley phase convention, :In the present convention of Euler angles, $alpha$ is a longitudinal angle and is a colatitudinal angle (spherical polar anglesin 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:$left(\; Y\_\{ell\}^m\; ight)\; ^*\; =\; (-1)^m\; Y\_\{ell\}^\{-m\}.$There exists a more general relationship to the spin-weighted spherical harmonics::ell}m}(eta,alpha) e^{isgamma}.

Relation with Legendre polynomials

The Wigner small d-matrix elements with both indices set to zero are relatedto Legendre polynomials:

See also

* Clebsch-Gordan coefficients
* Eugene Paul Wigner

References

Cited references