Wigner D-matrix explained

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,

[J _{x,J} _{y]} = i J _{z,} [J _{z,J} _{x]} = i J _{y,} [J _{y,J} _{z]} = i J _{x,} where i is the purely imaginary number and the Planck constant has been set equal to one. The Casimir operatorJ ^{2} = J ^{2} _{x} + J ^{2} _{y} + J ^{2} _{z} 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, 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

l{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

D ^{j} _{m'm} (\alpha,\beta,\gamma) \equiv \langle jm' | l{R}(\alpha,\beta,\gamma)| jm \rangle = e ^{-im'\alpha} d ^{j} _{m'm} (\beta)e ^{-i} , whered ^{j} _{m'm} (\beta) = \langle jm' |e ^{- i\beta J _{y}} | jm \rangle = D ^{j} _{m'm} ( 0 ,\beta, 0 ) is an element of the orthogonal Wigner's (small) d-matrix.

That is, in this basis,

D ^{j} _{m'm} (\alpha, 0 , 0 ) = e ^{-im'\alpha} \delta _{m'm} is diagonal, like the γ matrix factor, but unlike the above β factor.

Wigner (small) d-matrix

Wigner gave the following expression:^[1]

d ^{j} _{m'm} (\beta) = [(j + m')!(j - m')!(j + m)!(j - m)!] ^{(1)/(2)} \sum ^{s _{max}} _{s = s _{min}} \left[ (( -1 ) ^{m' - m + s} \left(\cos (\beta)/(2) \right) ^{2j+m-m'-2s} \left(\sin (\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.

s _{min} = max ( 0 ,m - m') , s _{max} = 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

( -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.P ^{(a,b)} _{k(\cos\beta)} with nonnegative a and b. ^[2] Letk = min (j + m, j - m, j + m', j - m').

If

k = \begin{cases} j + m: & a = m' - m; λ= m' - m\\ j - m: & a = m - m'; λ= 0 \\ j + m': & a = m - m'; λ= 0 \\ j - m': & a = m' - m; λ= m' - m \\ \end{cases}

Then, with

b =2 j -2 k - a, the relation isd ^{j} _{m'm} (\beta) = ( -1 ) ^{λ} \binom{ 2 j - k}{k + a} ^{(1)/(2)} \binom{k + b}{b} ^{- (1)/(2)} \left(\sin (\beta)/(2) \right) ^{a} \left(\cos (\beta)/(2) \right) ^{b} P ^{(a,b)} _{k(\cos\beta),} where a,b \ge 0 .

It is also useful to consider the relations

a = |m' - m|, b = |m' + m|, λ = (m - m' - |m - m'|)/(2) , k = j - M , where M = max (|m|,|m'|) and N = min (|m|,|m'|) , which lead to:d ^{j} _{m'm} (\beta) = ( -1 ) ^{(m - m' - |m - m'|)/(2)} \left[ ((j + M)!(j - M)!)/((j + N)!(j - N)!) \right] ^{(1)/(2)} \left(\sin (\beta)/(2) \right) ^{|m-m'|} \left(\cos (\beta)/(2) \right) ^{|m+m'|} P ^{(|m - m'|,|m + m'|)} _{j - M} (\cos \beta).

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{align} \hat{ l{J}} _{1} & = i \left( \cos \alpha \cot \beta (\partial)/(\partial \alpha) + \sin \alpha {\partial \over \partial \beta} - {\cos \alpha \over \sin \beta} {\partial \over \partial \gamma} \right) \\ \hat{ l{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{ l{J}} _{3} & = - i {\partial \over \partial \alpha} \end{align} which have quantum mechanical meaning: they are space-fixed rigid rotor angular momentum operators.

Further,

\begin{align} \hat{ l{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{ l{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{ l{P}} _{3} & = - i {\partial\over \partial \gamma}, \\ \end{align} which have quantum mechanical meaning: they are body-fixed rigid rotor angular momentum operators.

The operators satisfy the commutation relations

\left[ l{J} _{1,} l{J} _{2\right]} = i l{J} _{3,} \hbox{and} \left[ l{P} _{1,} l{P} _{2\right]} = - i l{P} _{3,} and the corresponding relations with the indices permuted cyclically. The l{P} _{i} satisfy anomalous commutation relations (have a minus sign on the right hand side). The two sets mutually commute,\left[ l{P} _{i,} l{J} _{j\right]} = 0 , i, j = 1 , 2 , 3 , and the total operators squared are equal,l{J} ^{2} \equiv l{J} ^{2+} _{1} l{J} ^{2} _{2} + l{J} ^{2} _{3} = l{P} ^{2} \equiv l{P} ^{2+} _{1} l{P} ^{2} _{2} + l{P} ^{2 .} _{3}

Their explicit form is,

l{J} ^{2=} l{P} ^{2} =- (1)/(\sin ^{2\beta}) \left( (\partial ^{2})/(+ \partial \alpha ^{2}) (\partial ^{2})/(-2 \cos\beta \partial \gamma ^{2}) (\partial ^{2})/(\right) - \partial\alpha\partial \gamma) (\partial ^{2})/(- \cot\beta \partial \beta ^{2}) (\partial)/(\partial \beta) .

The operators

l{J} _{i} act on the first (row) index of the D-matrix,\begin{align} l{J} _{3} D ^{j} _{m'm} (\alpha,\beta,\gamma) ^{*} & = m' D ^{j} _{m'm} (\alpha,\beta,\gamma) ^{*} \\ ( l{J} _{1} \pm i l{J} _{2)} D ^{j} _{m'm} (\alpha,\beta,\gamma) ^{*} & = \sqrt{j(j +1 ) - m'(m'\pm 1 )} D ^{j} _{m'\pm 1 , m} (\alpha,\beta,\gamma) ^{*} \end{align}

The operators

l{P} _{i} act on the second (column) index of the D-matrix,l{P} _{3} D ^{j} _{m'm} (\alpha,\beta,\gamma) ^{*} = m D ^{j} _{m'm} (\alpha,\beta,\gamma) ^{*} , and, because of the anomalous commutation relation the raising/lowering operators are defined with reversed signs,( l{P} _{1} \mp i l{P} _{2)} D ^{j} _{m'm} (\alpha,\beta,\gamma) ^{*} = \sqrt{j(j +1 ) - m(m\pm 1 )} D ^{j} _{m', m\pm 1} (\alpha,\beta,\gamma) ^{*} .

Finally,

l{J} ^{2} D ^{j} _{m'm} (\alpha,\beta,\gamma) ^{*} = l{P} ^{2} D ^{j} _{m'm} (\alpha,\beta,\gamma) ^{*} = j(j +1 ) D ^{j} _{m'm} (\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

\{ l{J} _{i\}} and \{ - l{P} _{i\}} .

An important property of the Wigner D-matrix follows from the commutation of

l{R}(\alpha,\beta,\gamma) with the time reversal operator,\langle jm' | l{R}(\alpha,\beta,\gamma)| jm \rangle = \langle jm' | T l{R}(\alpha,\beta,\gamma) T| jm \rangle = ( -1 ) ^{m'-m} \langle j, - m' | l{R}(\alpha,\beta,\gamma)| j, - m \rangle ^{*,} orD ^{j} _{m'm} (\alpha,\beta,\gamma) = ( -1 ) ^{m'-m} D ^{j} _{- m', - m} (\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} .

A further symmetry implies

( -1 ) ^{m'-m} D ^{j} _{mm'} (\alpha,\beta,\gamma) = D ^{j} _{m'm} (\gamma,\beta,\alpha)~.

Orthogonality relations

The Wigner D-matrix elements

D ^{j} _{mk} (\alpha,\beta,\gamma) form a set of orthogonal functions of the Euler angles \alpha, \beta, and \gamma :\int ^{2 \pi} _{0} d\alpha \int ^{\pi} _{0} d\beta \sin \beta \int ^{2 \pi} _{0} d\gamma D ^{j'} _{m'k'} (\alpha,\beta,\gamma) ^{\ast} D ^{j} _{mk} (\alpha, \beta, \gamma) = (8 \pi ^{2})/(2 j +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

D ^{j} _{mk} (\alpha,\beta,\gamma) are matrix elements of a unitary transformation from one spherical basis | lm \rangle to another l{R}(\alpha,\beta,\gamma) | lm \rangle is represented by the relations:^[3] \sum _{k} D ^{j} _{m'k} (\alpha, \beta, \gamma) ^{*} D ^{j} _{mk} (\alpha, \beta, \gamma) = \delta _{m,m'} , \sum _{k} D ^{j} _{k m'} (\alpha, \beta, \gamma) ^{*} D ^{j} _{km} (\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,

\chi ^{j} (\beta)\equiv \sum _{m} D ^{j} _{mm} (\beta) = \sum _{m} d ^{j} _{mm} (\beta) = (\sin\left ( (( 2 j +1 )\beta)/(2) \right ))/(\sin \left ( (\beta)/(2) \right )) ,

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

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

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) ( 2 j +1 ) = \delta (\beta ).

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

The set of Kronecker product matrices

D ^{j(\alpha,\beta,\gamma) ⊗} 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 ^{j} _{m k} (\alpha,\beta,\gamma) D ^{j'} _{m' k'} (\alpha,\beta,\gamma) = \sum ^{j + j'} _{J = |j - j'|} \langle j m j' m' | J \left(m + m'\right) \rangle \langle j k j' k' | J \left(k + k'\right) \rangle D ^{J} _{\left(m + m'\right) \left(k + k'\right)} (\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 proportionalto spherical harmonics and associated Legendre polynomials, normalized to unity and with Condon and Shortley phase convention:D ^{\ell} _{m 0} (\alpha,\beta,\gamma) = \sqrt{ (4 \pi)/(2 \ell +1) } Y_^ (\beta, \alpha) = \sqrt \, P_\ell^m (\cos) \, e^.This implies the following relationship for the d-matrix:d ^{\ell} _{m 0} (\beta) = \sqrt{ ((\ell - m)!)/((\ell + m)!) } \, 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} _{m' =- \ell} Y ^{m'} _{\ell} (\theta, \phi ) ~ D ^{\ell} _{m' ~m} (\alpha,\beta,\gamma).

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

D ^{\ell} _{0 , 0} (\alpha,\beta,\gamma) = d ^{\ell} _{0 , 0} (\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 anglesin the physical definition of such angles). This is one of the reasons that the z-y-zconvention is used frequently in molecular physics.From the time-reversal property of the Wigner D-matrix follows immediately\left( Y ^{m} _{\ell} \right) ^{*} = ( -1 ) ^{m} Y ^{- m} _{\ell} . There exists a more general relationship to the spin-weighted spherical harmonics:D ^{\ell} _{m s} (\alpha,\beta, - \gamma) = ( -1 ) ^{s} \sqrt (4 \pi)/(2 {\ell) +1 } {} _{sY} ^{m(\beta,\alpha)} ^{\ell} e ^{is\gamma} . ^[5]

Connection with transition probability under rotations

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

F _{mm'} (\beta) = | D ^{j} _{mm'} (\alpha,\beta,\gamma) | ^{2,} gives the probability that a system with spin j prepared in a state with spin projection m alongsome 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 _{mm'} itself forms a real symmetric matrix, thatdepends only on the Euler angle \beta , as indicated.

Remarkably, the eigenvalue problem for the

F matrix can be solved completely:^{[6] [7]} \sum ^{j} _{m' =- j} F _{mm'} (\beta) f ^{j} _{\ell} (m') = P _{\ell} (\cos\beta) f ^{j} _{\ell} (m) (\ell = 0 , 1 , \ldots, 2 j).

Here, the eigenvector,

f ^{j} _{\ell} (m) , is a scaled and shifted discrete Chebyshev polynomial, and the corresponding eigenvalue, P _{\ell} (\cos\beta) , is the Legendre polynomial.

Relation to Bessel functions

In the limit when

\ell \gg m, m ^{\prime} we haveD ^{\ell} _{mm'} (\alpha,\beta,\gamma) ≈ e ^{-im\alpha-im'\gamma} J _{m-m'} (\ell\beta) where J _{m-m'} (\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} _{m'm} (\theta) for j = 1/2, 1, 3/2, and 2 are given below.

For j = 1/2

\begin{align} d ^{(1)/(2)} _{(1)/(, (1)/(2) 2)} & = \cos (\theta)/(2) \\[ 6 pt] d ^{(1)/(2)} _{(1)/(, - (1)/(2) 2)} & = - \sin (\theta)/(2) \end{align}

For j = 1

\begin{align} d ^{1} _{1 , 1} & = (1)/(2) ( 1+ \cos \theta) \\[ 6 pt] d ^{1} _{1 , 0} & =- (1)/(\sqrt{ 2) } \sin \theta \\[6pt]d_^ &= \frac (1-\cos \theta) \\[6pt]d_^ &= \cos \theta\end

For j = 3/2

\begin{align} d ^{(3)/(2)} _{(3)/(, (3)/(2) 2)} & = (1)/(2) ( 1+ \cos \theta) \cos (\theta)/(2) \\[ 6 pt] d ^{(3)/(2)} _{(3)/(, (1)/(2) 2)} & =- (\sqrt{ 3)/() } (1+\cos \theta) \sin \frac \\[6pt]d_^ &= \frac (1-\cos \theta) \cos \frac \\[6pt]d_^ &= -\frac (1-\cos \theta) \sin \frac \\[6pt]d_^ &= \frac (3\cos \theta - 1) \cos \frac \\[6pt]d_^ &= -\frac (3\cos \theta + 1) \sin \frac\end

For j = 2^[8]

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

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

d ^{j} _{m', m} = ( -1 ) ^{m-m'} d ^{j} _{m, m'} = d ^{j.} _{- m, - m'}

Symmetries and special cases

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

See also

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

External links

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

Notes and References

1. Book: Wigner, E. P. . Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren . Vieweg Verlag . Braunschweig . 1951 . 1931 . 602430512. Translated into English by Book: J.J. . Griffin . [{{GBurl|UITNCgAAQBAJ|pg=PR9}} Group Theory and its Application to the Quantum Mechanics of Atomic Spectra ]. Elsevier . 978-1-4832-7576-5 . 2013 . 1959 .
2. Book: L. C. . Biedenharn . J. D. . Louck . Angular Momentum in Quantum Physics . Addison-Wesley . Reading . 1981 . 0-201-13507-8 .
3. Book: Rose, Morris Edgar. [{{GBurl|3lSiev-MnLQC|pg=PR7}} Elementary theory of angular momentum]. 1995. Dover. 0-486-68480-6. 1957. 31374243.
4. Schwinger . J. . On Angular Momentum . Harvard University, Nuclear Development Associates . NYO-3071, TRN: US200506%%295 . January 26, 1952 . 10.2172/4389568.
5. M. . Shiraishi . Appendix A: Spin-Weighted Spherical Harmonic Function . https://link.springer.com/content/pdf/bbm:978-4-431-54180-6/1.pdf . Probing the Early Universe with the CMB Scalar, Vector and Tensor Bispectrum . 2013 . 978-4-431-54180-6 . Nagoya University . PhD . 153–4.
6. A. . Meckler . Majorana formula . Physical Review . 1958 . 111 . 6 . 1447 . 10.1103/PhysRev.111.1447.
7. N.D. . Mermin . G.M. . Schwarz . Joint distributions and local realism in the higher-spin Einstein-Podolsky-Rosen experiment . Foundations of Physics . 1982 . 12 . 2 . 101 . 10.1007/BF00736844 . 121648820.
8. 10.1002/cmr.a.10061 . Edén, M.. Computer simulations in solid-state NMR. I. Spin dynamics theory. Concepts in Magnetic Resonance Part A. 17A. 1. 117–154. 2003.