Wigner D-matrix

The Wigner D-matrix is a unitary matrix in an irreducible representation of the groups SU(2) and SO(3). It was introduced in 1927 by Eugene Wigner, and plays a fundamental role in the quantum mechanical theory of angular momentum. The complex conjugate of the D-matrix is an eigenfunction of the Hamiltonian of spherical and symmetric rigid rotors. The letter stands for ''Darstellung'', which means "representation" in German.

Definition of the Wigner D-matrix

Let 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,
: _x,J_y= i J_z,\quad _z,J_x= i J_y,\quad _y,J_z= i J_x,
where ''i'' is the purely imaginary number and Planck's constant has been set 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 .

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
:$J^2 , jm\rangle = j(j+1) , jm\rangle,\quad J_z , jm\rangle = m , jm\rangle,$
where ''j'' = 0, 1/2, 1, 3/2, 2, ... for SU(2), and ''j'' = 0, 1, 2, ... for SO(3). In both cases, .

A 3-dimensional rotation operator can be written as
:$\mathcal(\alpha,\beta,\gamma) = e^e^e^,$
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 2''j'' + 1 in this spherical basis with elements
:$D^j_(\alpha,\beta,\gamma) \equiv \langle jm' , \mathcal(\alpha,\beta,\gamma), jm \rangle =e^ d^j_(\beta)e^,$
where
:$d^j_(\beta)= \langle jm' , e^ , jm \rangle = D^j_(0,\beta,0)$
is an element of the orthogonal Wigner's (small) d-matrix.

That is, in this basis,
:$D^j_(\alpha,0,0) = e^ \delta_$
is diagonal, like the ''γ'' matrix factor, but unlike the above ''β'' factor.

Wigner (small) d-matrix

Wigner gave the following expression:

:d^j_(\beta) = j+m')!(j-m')!(j+m)!(j-m)! \sum_^ \left frac \right

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

''Note:'' The d-matrix elements defined here are real. In the often-used z-x-z convention of Euler angles, the factor $(-1)^$ in this formula is replaced by $(-1)^s i^,$ 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 $P^_k(\cos\beta)$ with nonnegative $a$ and $b.$ Let

:$k = \min(j+m, j-m, j+m', j-m').$

If

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

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

:$d^j_(\beta) = (-1)^ \binom^ \binom^ \left(\sin\frac\right)^a \left(\cos\frac\right)^b P^_k(\cos\beta),$

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 $(x, y, z) = (1, 2, 3),$
:$\begin \hat_1 &= i \left( \cos \alpha \cot \beta \frac + \sin \alpha - \right) \\ \hat_2 &= i \left( \sin \alpha \cot \beta - \cos \alpha - \right) \\ \hat_3 &= - i \end$
which have quantum mechanical meaning: they are space-fixed rigid rotor angular momentum operators.

Further,
:$\begin \hat_1 &= i \left( - \sin \gamma - \cot \beta \cos \gamma \right)\\ \hat_2 &= i \left( - - \cos \gamma + \cot \beta \sin \gamma \right) \\ \hat_3 &= - i , \\ \end$
which have quantum mechanical meaning: they are body-fixed rigid rotor angular momentum operators.

The operators satisfy the commutation relations
: \left mathcal_1, \mathcal_2\right= i \mathcal_3, \qquad \hbox\qquad \left mathcal_1, \mathcal_2\right= -i \mathcal_3,
and the corresponding relations with the indices permuted cyclically. The $\mathcal_i$ satisfy ''anomalous commutation relations'' (have a minus sign on the right hand side).

The two sets mutually commute,
:\left mathcal_i, \mathcal_j\right= 0,\quad i, j = 1, 2, 3,
and the total operators squared are equal,
:$\mathcal^2 \equiv \mathcal_1^2+ \mathcal_2^2 + \mathcal_3^2 = \mathcal^2 \equiv \mathcal_1^2+ \mathcal_2^2 + \mathcal_3^2.$

Their explicit form is,
:$\mathcal^2= \mathcal^2 =-\frac \left( \frac +\frac -2\cos\beta\frac \right)-\frac -\cot\beta\frac.$

The operators $\mathcal_i$ act on the first (row) index of the D-matrix,
:$\begin \mathcal_3 D^j_(\alpha,\beta,\gamma)^* &=m' D^j_(\alpha,\beta,\gamma)^* \\ (\mathcal_1 \pm i \mathcal_2) D^j_(\alpha,\beta,\gamma)^* &= \sqrt D^j_(\alpha,\beta,\gamma)^* \end$

The operators $\mathcal_i$ act on the second (column) index of the D-matrix,
:$\mathcal_3 D^j_(\alpha,\beta,\gamma)^* = m D^j_(\alpha,\beta,\gamma)^* ,$
and, because of the anomalous commutation relation the raising/lowering operators are defined with reversed signs,
:$(\mathcal_1 \mp i \mathcal_2) D^j_(\alpha,\beta,\gamma)^* = \sqrt D^j_(\alpha,\beta,\gamma)^* .$

Finally,
:$\mathcal^2 D^j_(\alpha,\beta,\gamma)^* =\mathcal^2 D^j_(\alpha,\beta,\gamma)^* = j(j+1) D^j_(\alpha,\beta,\gamma)^*.$

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

An important property of the Wigner D-matrix follows from the commutation of
$\mathcal(\alpha,\beta,\gamma)$ with the time reversal operator
,
:$\langle jm' , \mathcal(\alpha,\beta,\gamma), jm \rangle = \langle jm' , T^ \mathcal(\alpha,\beta,\gamma) T, jm \rangle =(-1)^ \langle j,-m' , \mathcal(\alpha,\beta,\gamma), j,-m \rangle^*,$
or
:$D^j_(\alpha,\beta,\gamma) = (-1)^ D^j_(\alpha,\beta,\gamma)^*.$
Here, we used that $T$ is anti-unitary (hence the complex conjugation after moving $T^\dagger$ from ket to bra), $T , jm \rangle = (-1)^ , j,-m \rangle$ and $(-1)^ = (-1)^$.

A further symmetry implies
:$(-1)^D^_(\alpha,\beta,\gamma)=D^_(\gamma,\beta,\alpha)~.$

Orthogonality relations

The Wigner D-matrix elements $D^j_(\alpha,\beta,\gamma)$ form a set of orthogonal functions of the Euler angles $\alpha, \beta,$ and $\gamma$:

:$\int_0^ d\alpha \int_0^\pi d\beta \sin \beta \int_0^ d\gamma \,\, D^_(\alpha,\beta,\gamma)^\ast D^j_(\alpha, \beta, \gamma) = \frac \delta_\delta_\delta_.$

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 $D^j_(\alpha,\beta,\gamma)$ are matrix elements of a unitary transformation from one spherical basis $, lm \rangle$ to another $\mathcal(\alpha,\beta,\gamma) , lm \rangle$ is represented by the relations:
:$\sum_k D^j_(\alpha, \beta, \gamma)^* D^j_(\alpha, \beta, \gamma) = \delta_,$
:$\sum_k D^j_(\alpha, \beta, \gamma)^* D^j_(\alpha, \beta, \gamma) = \delta_.$

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

:$\chi^j (\beta)\equiv \sum_m D^j_(\beta)=\sum_m d^j_(\beta) = \frac,$

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

:$\frac \int _0^ d\beta \sin^2 \left (\frac \right ) \chi^j (\beta) \chi^(\beta)= \delta_.$

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

:$\sum_j \chi^j (\beta) \chi^j (\beta')= \delta (\beta -\beta'),$

whence, for $\beta' =0,$

:$\sum_j \chi^j (\beta) (2j+1)= \delta (\beta ).$

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

The set of Kronecker product matrices
:$\mathbf^j(\alpha,\beta,\gamma)\otimes \mathbf^(\alpha,\beta,\gamma)$
forms a reducible matrix representation of the groups SO(3) and SU(2). Reduction into irreducible components is by the following equation:
:$D^j_(\alpha,\beta,\gamma) D^_(\alpha,\beta,\gamma) = \sum_^ \langle j m j' m' , J \left(m + m'\right) \rangle \langle j k j' k' , J \left(k + k'\right) \rangle D^J_(\alpha,\beta,\gamma)$
The symbol $\langle j_1 m_1 j_2 m_2 , j_3 m_3 \rangle$ 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:
:$D^_(\alpha,\beta,\gamma) = \sqrt Y_^ (\beta, \alpha ) = \sqrt \, P_\ell^m ( \cos ) \, e^.$
This implies the following relationship for the d-matrix:
:$d^_(\beta) = \sqrt \, P_\ell^m ( \cos ).$

A rotation of spherical harmonics $\langle \theta, \phi, \ell m'\rangle$ then is effectively a composition of two rotations,
:$\sum^\ell_ Y_^ (\theta, \phi ) ~ D^_(\alpha,\beta,\gamma).$

When both indices are set to zero, the Wigner D-matrix elements are given by ordinary Legendre polynomials:
:$D^_(\alpha,\beta,\gamma) = d^_(\beta) = P_(\cos\beta).$

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
:$\left( Y_^m \right) ^* = (-1)^m Y_^.$
There exists a more general relationship to the spin-weighted spherical harmonics:
:$D^_(\alpha,\beta,-\gamma) =(-1)^s \sqrt\frac _sY_^m(\beta,\alpha) e^.$

Connection with transition probability under rotations

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

:$F_(\beta) = , D^j_(\alpha,\beta,\gamma) , ^2,$

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

Remarkably, the eigenvalue problem for the $F$ matrix can be solved completely:

:$\sum_^j F_(\beta) f^j_(m') = P_(\cos\beta) f^j_(m) \qquad (\ell = 0, 1, \ldots, 2j).$

Here, the eigenvector, $f^j_(m)$, is a scaled and shifted discrete Chebyshev polynomial, and the corresponding eigenvalue, $P_(\cos\beta)$, is the Legendre polynomial.

Relation to Bessel functions

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

:$D^\ell_(\alpha,\beta,\gamma) \approx e^J_(\ell\beta)$

where $J_(\ell\beta)$ 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 $d^j_(\theta)$
for ''j'' = 1/2, 1, 3/2, and 2 are given below.

for ''j'' = 1/2

:\begin
d_^ &= \cos \frac \\ ptd_^ &= -\sin \frac
\end

for ''j'' = 1

:\begin
d_^ &= \frac (1+\cos \theta) \\ ptd_^ &= -\frac \sin \theta \\ ptd_^ &= \frac (1-\cos \theta) \\ ptd_^ &= \cos \theta
\end

for ''j'' = 3/2

:\begin
d_^ &= \frac (1+\cos \theta) \cos \frac \\ ptd_^ &= -\frac (1+\cos \theta) \sin \frac \\ ptd_^ &= \frac (1-\cos \theta) \cos \frac \\ ptd_^ &= -\frac (1-\cos \theta) \sin \frac \\ ptd_^ &= \frac (3\cos \theta - 1) \cos \frac \\ ptd_^ &= -\frac (3\cos \theta + 1) \sin \frac
\end

for ''j'' = 2

:\begin
d_^ &= \frac\left(1 +\cos \theta\right)^2 \\ ptd_^ &= \frac\sin \theta \left(1 + \cos \theta\right) \\ ptd_^ &= \sqrt\sin^2 \theta \\ ptd_^ &= -\frac\sin \theta \left(1 - \cos \theta\right) \\ ptd_^ &= \frac\left(1 -\cos \theta\right)^2 \\ ptd_^ &= \frac\left(2\cos^2\theta + \cos \theta-1 \right) \\ ptd_^ &= -\sqrt \sin 2 \theta \\ ptd_^ &= \frac\left(- 2\cos^2\theta + \cos \theta +1 \right) \\ ptd_^ &= \frac \left(3 \cos^2 \theta - 1\right)
\end

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

:$d_^j = (-1)^d_^j = d_^j.$

Symmetries and special cases

:\begin
d_^(\pi) &= (-1)^ \delta_ \\ ptd_^(\pi-\beta) &= (-1)^ d_^(\beta)\\ ptd_^(\pi+\beta) &= (-1)^ d_^(\beta)\\ ptd_^(2\pi+\beta) &= (-1)^ d_^(\beta)\\ ptd_^(-\beta) &= d_^(\beta) = (-1)^ d_^(\beta)
\end

See also

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

References

External links

* {{cite web , first=C. , last=Amsler , author2=''et al.'' (Particle Data Group) , title=PDG Table of Clebsch-Gordan Coefficients, Spherical Harmonics, and d-Functions , date=2008 , work=Physics Letters B667 , url=http://pdg.lbl.gov/2008/reviews/clebrpp.pdf

Representation theory of Lie groups
Matrices
Special hypergeometric functions
Rotational symmetry