Wigner D-matrix#Relation to spherical harmonics and Legendre polynomials

Let {{math|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,

: $[J\_x,J\_y]\; =\; i\; J\_z,\backslash quad\; [J\_z,J\_x]\; =\; i\; J\_y,\backslash quad\; [J\_y,J\_z]\; =\; i\; J\_x,$

where i is the purely imaginary number and the Planck constant {{mvar|ħ}} 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 {{mvar|J_z}}.

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

: $J^2\; |jm\backslash rangle\; =\; j(j+1)\; |jm\backslash rangle,\backslash quad\; J\_z\; |jm\backslash rangle\; =\; m\; |jm\backslash rangle,$

where j = 0, 1/2, 1, 3/2, 2, ... for SU(2), and j = 0, 1, 2, ... for SO(3). In both cases, {{math|m {{=}} −j, −j + 1, ..., j}}.

A 3-dimensional rotation operator can be written as

: $\backslash mathcal\{R\}(\backslash alpha,\backslash beta,\backslash gamma)\; =\; e^\{-i\backslash alpha\; J\_z\}e^\{-i\backslash beta\; J\_y\}e^\{-i\backslash gamma\; J\_z\},$

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

: $D^j\_\{m\text{'}m\}(\backslash alpha,\backslash beta,\backslash gamma)\; \backslash equiv\; \backslash langle\; jm\text{'}\; |\; \backslash mathcal\{R\}(\backslash alpha,\backslash beta,\backslash gamma)|\; jm\; \backslash rangle\; =e^\{-im\text{'}\backslash alpha\; \}\; d^j\_\{m\text{'}m\}(\backslash beta)e^\{-i\; m\backslash gamma\},$

where

: $d^j\_\{m\text{'}m\}(\backslash beta)=\; \backslash langle\; jm\text{'}\; |e^\{-i\backslash beta\; J\_y\}\; |\; jm\; \backslash rangle\; =\; D^j\_\{m\text{'}m\}(0,\backslash beta,0)$

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

That is, in this basis,

: $D^j\_\{m\text{'}m\}(\backslash alpha,0,0)\; =\; e^\{-im\text{'}\backslash alpha\; \}\; \backslash delta\_\{m\text{'}m\}$

is diagonal, like the γ matrix factor, but unlike the above β factor.

Wigner gave the following expression:{{cite book |first=E. P. |last=Wigner |title=Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren |publisher=Vieweg Verlag |location=Braunschweig |year=1951 |orig-year=1931 |oclc=602430512}} Translated into English by {{cite book |translator-first=J.J. |translator-last=Griffin |title=Group Theory and its Application to the Quantum Mechanics of Atomic Spectra |publisher=Elsevier |isbn=978-1-4832-7576-5 |year=2013 |url={{GBurl|UITNCgAAQBAJ|pg=PR9}} |orig-year=1959 }}

: $d^j\_\{m\text{'}m\}(\backslash beta)\; =[(j+m\text{'})!(j-m\text{'})!(j+m)!(j-m)!]^\{\backslash frac\{1\}\{2\}\}\; \backslash sum\_\{s=s\_\{\backslash mathrm\{min\}\}\}^\{s\_\{\backslash mathrm\{max\}\}\}\; \backslash left[\backslash frac\{(-1)^\{m\text{'}-m+s\}\; \backslash left(\backslash cos\backslash frac\{\backslash beta\}\{2\}\backslash right)^\{2j+m-m\text{'}-2s\}\backslash left(\backslash sin\backslash frac\{\backslash beta\}\{2\}\backslash right)^\{m\text{'}-m+2s\}\}\{(j+m-s)!s!(m\text{'}-m+s)!(j-m\text{'}-s)!\}\; \backslash right].$

The sum over s is over such values that the factorials are nonnegative, i.e. $s\_\{\backslash mathrm\{min\}\}=\backslash mathrm\{max\}(0,m-m\text{'})$, $s\_\{\backslash mathrm\{max\}\}=\backslash mathrm\{min\}(j+m,j-m\text{'})$.

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 $P^\{(a,b)\}\_k(\backslash cos\backslash beta)$ with nonnegative $a$ and $b.${{cite book |first1=L. C. |last1=Biedenharn |first2=J. D. |last2=Louck |title=Angular Momentum in Quantum Physics |publisher=Addison-Wesley |location=Reading |year=1981 |isbn=0-201-13507-8 }} Let

: $k\; =\; \backslash min(j+m,\; j-m,\; j+m\text{'},\; j-m\text{'}).$

If

: $k\; =\; \backslash begin\{cases\}$

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{cases}

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

: $d^j\_\{m\text{'}m\}(\backslash beta)\; =\; (-1)^\{\backslash lambda\}\; \backslash binom\{2j-k\}\{k+a\}^\{\backslash frac\{1\}\{2\}\}\; \backslash binom\{k+b\}\{b\}^\{-\backslash frac\{1\}\{2\}\}\; \backslash left(\backslash sin\backslash frac\{\backslash beta\}\{2\}\backslash right)^a\; \backslash left(\backslash cos\backslash frac\{\backslash beta\}\{2\}\backslash right)^b\; P^\{(a,b)\}\_k(\backslash cos\backslash beta),$

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

It is also useful to consider the relations $a\; =\; |m\text{'}-m|,\; b\; =\; |m\text{'}+m|,\; \backslash lambda\; =\; \backslash frac\{m-m\text{'}-|m-m\text{'}|\}\{2\},\; k\; =\; j\; -\; M$, where $M\; =\; \backslash max(|m|,|m\text{'}|)$ and $N\; =\; \backslash min(|m|,|m\text{'}|)$, which lead to:

: $d^j\_\{m\text{'}m\}(\backslash beta)\; =(-1)^\{\backslash frac\{m-m\text{'}-|m-m\text{'}|\}\{2\}\}\backslash left[\; \backslash frac\{(j+M)!(j-M)!\}\{(j+N)!(j-N)!\}\backslash right]^\{\backslash frac\{1\}\{2\}\}\; \backslash left(\backslash sin\backslash frac\{\backslash beta\}\{2\}\backslash right)^$

The complex conjugate of the D-matrix satisfies a number of differential properties that can be formulated concisely by introducing the following operators with $(x,\; y,\; z)\; =\; (1,\; 2,\; 3),$

: $\backslash begin\{align\}$

\hat{\mathcal{J}}_1 &= i \left( \cos \alpha \cot \beta \frac{\partial}{\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{align}

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

Further,

: $\backslash begin\{align\}$

\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{align}

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

The operators satisfy the commutation relations

: $\backslash left[\backslash mathcal\{J\}\_1,\; \backslash mathcal\{J\}\_2\backslash right]\; =\; i\; \backslash mathcal\{J\}\_3,\; \backslash qquad\; \backslash hbox\{and\}\backslash qquad\; \backslash left[\backslash mathcal\{P\}\_1,\; \backslash mathcal\{P\}\_2\backslash right]\; =\; -i\; \backslash mathcal\{P\}\_3,$

and the corresponding relations with the indices permuted cyclically. The $\backslash mathcal\{P\}\_i$ satisfy anomalous commutation relations (have a minus sign on the right hand side).

The two sets mutually commute,

: $\backslash left[\backslash mathcal\{P\}\_i,\; \backslash mathcal\{J\}\_j\backslash right]\; =\; 0,\backslash quad\; i,\; j\; =\; 1,\; 2,\; 3,$

and the total operators squared are equal,

: $\backslash mathcal\{J\}^2\; \backslash equiv\; \backslash mathcal\{J\}\_1^2+\; \backslash mathcal\{J\}\_2^2\; +\; \backslash mathcal\{J\}\_3^2\; =\; \backslash mathcal\{P\}^2\; \backslash equiv\; \backslash mathcal\{P\}\_1^2+\; \backslash mathcal\{P\}\_2^2\; +\; \backslash mathcal\{P\}\_3^2.$

Their explicit form is,

: $\backslash mathcal\{J\}^2=\; \backslash mathcal\{P\}^2\; =-\backslash frac\{1\}\{\backslash sin^2\backslash beta\}\; \backslash left(\; \backslash frac\{\backslash partial^2\}\{\backslash partial\; \backslash alpha^2\}\; +\backslash frac\{\backslash partial^2\}\{\backslash partial\; \backslash gamma^2\}\; -2\backslash cos\backslash beta\backslash frac\{\backslash partial^2\}\{\backslash partial\backslash alpha\backslash partial\; \backslash gamma\}\; \backslash right)-\backslash frac\{\backslash partial^2\}\{\backslash partial\; \backslash beta^2\}\; -\backslash cot\backslash beta\backslash frac\{\backslash partial\}\{\backslash partial\; \backslash beta\}.$

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

: $\backslash begin\{align\}$

\mathcal{J}_3 D^j_{m'm}(\alpha,\beta,\gamma)^* &=m' D^j_{m'm}(\alpha,\beta,\gamma)^* \\

(\mathcal{J}_1 \pm i \mathcal{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 $\backslash mathcal\{P\}\_i$ act on the second (column) index of the D-matrix,

: $\backslash mathcal\{P\}\_3\; D^j\_\{m\text{'}m\}(\backslash alpha,\backslash beta,\backslash gamma)^*\; =\; m\; D^j\_\{m\text{'}m\}(\backslash alpha,\backslash beta,\backslash gamma)^*\; ,$

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

: $(\backslash mathcal\{P\}\_1\; \backslash mp\; i\; \backslash mathcal\{P\}\_2)\; D^j\_\{m\text{'}m\}(\backslash alpha,\backslash beta,\backslash gamma)^*\; =\; \backslash sqrt\{j(j+1)-m(m\backslash pm\; 1)\}\; D^j\_\{m\text{'},\; m\backslash pm1\}(\backslash alpha,\backslash beta,\backslash gamma)^*\; .$

Finally,

: $\backslash mathcal\{J\}^2\; D^j\_\{m\text{'}m\}(\backslash alpha,\backslash beta,\backslash gamma)^*\; =\backslash mathcal\{P\}^2\; D^j\_\{m\text{'}m\}(\backslash alpha,\backslash beta,\backslash gamma)^*\; =\; j(j+1)\; D^j\_\{m\text{'}m\}(\backslash alpha,\backslash beta,\backslash gamma)^*.$

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

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

$\backslash mathcal\{R\}(\backslash alpha,\backslash beta,\backslash gamma)$ with the time reversal operator

{{mvar|T}},

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

or

: $D^j\_\{m\text{'}m\}(\backslash alpha,\backslash beta,\backslash gamma)\; =\; (-1)^\{m\text{'}-m\}\; D^j\_\{-m\text{'},-m\}(\backslash alpha,\backslash beta,\backslash gamma)^*.$

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

A further symmetry implies

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

The Wigner D-matrix elements $D^j\_\{mk\}(\backslash alpha,\backslash beta,\backslash gamma)$ form a set of orthogonal functions of the Euler angles $\backslash alpha,\; \backslash beta,$ and $\backslash gamma$:{{Cite journal|last=Van de Wiele|first=Jacques|url=https://hal.science/in2p3-00019832 |title=Rotations et moments angulaires en mécanique quantique|date=2001|journal=Annales de Physique|volume=26|pages=1-169}}

: $\backslash int\_0^\{2\backslash pi\}\; d\backslash alpha\; \backslash int\_0^\backslash pi\; d\backslash beta\; \backslash sin\; \backslash beta\; \backslash int\_0^\{2\backslash pi\}\; d\backslash gamma\; \backslash ,\backslash ,\; D^\{j\text{'}\}\_\{m\text{'}k\text{'}\}(\backslash alpha,\backslash beta,\backslash gamma)^\backslash ast\; D^j\_\{mk\}(\backslash alpha,\; \backslash beta,\; \backslash gamma)\; =\; \backslash frac\{8\backslash pi^2\}\{2j+1\}\; \backslash delta\_\{m\text{'}m\}\backslash delta\_\{k\text{'}k\}\backslash delta\_\{j\text{'}j\}.$

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 $D^j\_\{mk\}(\backslash alpha,\backslash beta,\backslash gamma)$ are matrix elements of a unitary transformation from one spherical basis $|\; lm\; \backslash rangle$ to another $\backslash mathcal\{R\}(\backslash alpha,\backslash beta,\backslash gamma)\; |\; lm\; \backslash rangle$ is represented by the relations:{{Cite book|last=Rose|first=Morris Edgar|url={{GBurl|3lSiev-MnLQC|pg=PR7}} |title=Elementary theory of angular momentum|date=1995|publisher=Dover|isbn=0-486-68480-6|orig-year=1957|oclc=31374243}}

: $\backslash sum\_k\; D^j\_\{m\text{'}k\}(\backslash alpha,\; \backslash beta,\; \backslash gamma)^*\; D^j\_\{mk\}(\backslash alpha,\; \backslash beta,\; \backslash gamma)\; =\; \backslash delta\_\{m,m\text{'}\},$

: $\backslash sum\_k\; D^j\_\{k\; m\text{'}\}(\backslash alpha,\; \backslash beta,\; \backslash gamma)^*\; D^j\_\{km\}(\backslash alpha,\; \backslash beta,\; \backslash gamma)\; =\; \backslash delta\_\{m,m\text{'}\}.$

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

: $\backslash chi^j\; (\backslash beta)\backslash equiv\; \backslash sum\_m\; D^j\_\{mm\}(\backslash beta)=\backslash sum\_m\; d^j\_\{mm\}(\backslash beta)\; =\; \backslash frac\{\backslash sin\backslash left\; (\backslash frac\{(2j+1)\backslash beta\}\{2\}\; \backslash right\; )\}\{\backslash sin\; \backslash left\; (\backslash frac\{\backslash beta\}\{2\}\; \backslash right\; )\},$

and consequently satisfy simpler orthogonality relations, through the Haar measure of the group,{{cite tech report |last=Schwinger |first=J. |url=https://www.osti.gov/biblio/4389568-angular-momentum |title=On Angular Momentum |institution=Harvard University, Nuclear Development Associates |id=NYO-3071, TRN: US200506%%295 |date=January 26, 1952 |doi=10.2172/4389568}}

: $\backslash frac\{1\}\{\backslash pi\}\; \backslash int\; \_0^\{2\backslash pi\}\; d\backslash beta\; \backslash sin^2\; \backslash left\; (\backslash frac\{\backslash beta\}\{2\}\; \backslash right\; )\; \backslash chi^j\; (\backslash beta)\; \backslash chi^\{j\text{'}\}(\backslash beta)=\; \backslash delta\_\{j\text{'}j\}.$

The completeness relation is (cf. Eq. (3.95) in ref. )

: $\backslash sum\_j\; \backslash chi^j\; (\backslash beta)\; \backslash chi^j\; (\backslash beta\text{'})=\; \backslash delta\; (\backslash beta\; -\backslash beta\text{'}),$

whence, for $\backslash beta\text{'}\; =0,$

: $\backslash sum\_j\; \backslash chi^j\; (\backslash beta)\; (2j+1)=\; \backslash delta\; (\backslash beta\; ).$

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:

: $$

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 $\backslash langle\; j\_1\; m\_1\; j\_2\; m\_2\; |\; j\_3\; m\_3\; \backslash rangle$ is a Clebsch–Gordan coefficient.

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^{\ell}_{m 0}(\alpha,\beta,\gamma) = \sqrt{\frac{4\pi}{2\ell+1}} Y_{\ell}^{m*} (\beta, \alpha ) = \sqrt{\frac{(\ell-m)!}{(\ell+m)!}} \, P_\ell^m ( \cos{\beta} ) \, e^{-i m \alpha }.

This implies the following relationship for the d-matrix:

: $$

d^{\ell}_{m 0}(\beta) = \sqrt{\frac{(\ell-m)!}{(\ell+m)!}} \, P_\ell^m ( \cos{\beta} ).

A rotation of spherical harmonics $\backslash langle\; \backslash theta,\; \backslash phi|\; \backslash ell\; m\text{'}\backslash rangle$ then is effectively a composition of two rotations,

: $$

\sum^\ell_{m'=-\ell} Y_{\ell}^ {m'} (\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, $\backslash alpha$ is

a longitudinal angle and $\backslash beta$ 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^{\ell}_{m s}(\alpha,\beta,-\gamma) =(-1)^s \sqrt\frac{4\pi}{2{\ell}+1} {}_sY_{\ell}^m(\beta,\alpha) e^{is\gamma}.

{{cite thesis |first=M. |last=Shiraishi |chapter=Appendix A: Spin-Weighted Spherical Harmonic Function |chapter-url=https://link.springer.com/content/pdf/bbm:978-4-431-54180-6/1.pdf |title=Probing the Early Universe with the CMB Scalar, Vector and Tensor Bispectrum |date=2013 |isbn=978-4-431-54180-6 |publisher=Nagoya University |type=PhD |url= |pages=153–4}}

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 $m$ along

some direction will be measured to have a spin projection $m\text{'}$ along a second direction at an angle $\backslash beta$ to the first direction. The set of quantities $F\_\{mm\text{'}\}$ itself forms a real symmetric matrix, that

depends only on the Euler angle $\backslash beta$, as indicated.

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

{{cite journal

| first = A. |last=Meckler

| title = Majorana formula

| journal = Physical Review

| year = 1958

| volume = 111

| issue=6

| page = 1447 |doi=10.1103/PhysRev.111.1447

}}

{{cite journal

| first1 = N.D. |last1=Mermin |first2 = G.M. |last2=Schwarz

| title = Joint distributions and local realism in the higher-spin Einstein-Podolsky-Rosen experiment

| journal = Foundations of Physics

| year = 1982

| volume = 12 |issue=2

| page = 101 |doi=10.1007/BF00736844

|s2cid=121648820

}}

: $$

\sum_{m' = -j}^j F_{mm'}(\beta) f^j_{\ell}(m') = P_{\ell}(\cos\beta) f^j_{\ell}(m) \qquad (\ell = 0, 1, \ldots, 2j).

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

In the limit when $\backslash ell\; \backslash gg\; m,\; m^\backslash prime$, one obtains

: $D^\backslash ell\_\{mm\text{'}\}(\backslash alpha,\backslash beta,\backslash gamma)\; \backslash approx\; e^\{-im\backslash alpha-im\text{'}\backslash gamma\}J\_\{m-m\text{'}\}(\backslash ell\backslash beta)$

where $J\_\{m-m\text{'}\}(\backslash ell\backslash beta)$ is the Bessel function and $\backslash ell\backslash beta$ is finite.

Using sign convention of Wigner, et al. the d-matrix elements $d^j\_\{m\text{'}m\}(\backslash theta)$

for j = 1/2, 1, 3/2, and 2 are given below.

For j = 1/2

: $\backslash begin\{align\}$

d_{\frac{1}{2},\frac{1}{2}}^{\frac{1}{2}} &= \cos \frac{\theta}{2} \\[6pt]

d_{\frac{1}{2},-\frac{1}{2}}^{\frac{1}{2}} &= -\sin \frac{\theta}{2}

\end{align}

For j = 1

: $\backslash begin\{align\}$

d_{1,1}^{1} &= \frac{1}{2} (1+\cos \theta) \\[6pt]

d_{1,0}^{1} &= -\frac{1}{\sqrt{2}} \sin \theta \\[6pt]

d_{1,-1}^{1} &= \frac{1}{2} (1-\cos \theta) \\[6pt]

d_{0,0}^{1} &= \cos \theta

\end{align}

For j = 3/2

: $\backslash begin\{align\}$

d_{\frac{3}{2}, \frac{3}{2}}^{\frac{3}{2}} &= \frac{1}{2} (1+\cos \theta) \cos \frac{\theta}{2} \\[6pt]

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

d_{\frac{3}{2},-\frac{1}{2}}^{\frac{3}{2}} &= \frac{\sqrt{3}}{2} (1-\cos \theta) \cos \frac{\theta}{2} \\[6pt]

d_{\frac{3}{2},-\frac{3}{2}}^{\frac{3}{2}} &= -\frac{1}{2} (1-\cos \theta) \sin \frac{\theta}{2} \\[6pt]

d_{\frac{1}{2}, \frac{1}{2}}^{\frac{3}{2}} &= \frac{1}{2} (3\cos \theta - 1) \cos \frac{\theta}{2} \\[6pt]

d_{\frac{1}{2},-\frac{1}{2}}^{\frac{3}{2}} &= -\frac{1}{2} (3\cos \theta + 1) \sin \frac{\theta}{2}

\end{align}

For j = 2{{cite journal | doi = 10.1002/cmr.a.10061 | author = Edén, M.

| title = Computer simulations in solid-state NMR. I. Spin dynamics theory| journal = Concepts in Magnetic Resonance Part A| volume=17A| issue=1| pages=117–154| year=2003}}

: $\backslash begin\{align\}$

d_{2,2}^{2} &= \frac{1}{4}\left(1 +\cos \theta\right)^2 \\[6pt]

d_{2,1}^{2} &= -\frac{1}{2}\sin \theta \left(1 + \cos \theta\right) \\[6pt]

d_{2,0}^{2} &= \sqrt{\frac{3}{8}}\sin^2 \theta \\[6pt]

d_{2,-1}^{2} &= -\frac{1}{2}\sin \theta \left(1 - \cos \theta\right) \\[6pt]

d_{2,-2}^{2} &= \frac{1}{4}\left(1 -\cos \theta\right)^2 \\[6pt]

d_{1,1}^{2} &= \frac{1}{2}\left(2\cos^2\theta + \cos \theta-1 \right) \\[6pt]

d_{1,0}^{2} &= -\sqrt{\frac{3}{8}} \sin 2 \theta \\[6pt]

d_{1,-1}^{2} &= \frac{1}{2}\left(- 2\cos^2\theta + \cos \theta +1 \right) \\[6pt]

d_{0,0}^{2} &= \frac{1}{2} \left(3 \cos^2 \theta - 1\right)

\end{align}

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

: $d\_\{m\text{'},\; m\}^j\; =\; (-1)^\{m-m\text{'}\}d\_\{m,\; m\text{'}\}^j\; =\; d\_\{-m,-m\text{'}\}^j.$

: $\backslash begin\{align\}$

d_{m',m}^{j}(\pi) &= (-1)^{j-m} \delta_{m',-m} \\[6pt]

d_{m',m}^{j}(\pi-\beta) &= (-1)^{j+m'} d_{m',-m}^{j}(\beta)\\[6pt]

d_{m',m}^{j}(\pi+\beta) &= (-1)^{j-m} d_{m',-m}^{j}(\beta)\\[6pt]

d_{m',m}^{j}(2\pi+\beta) &= (-1)^{2j} d_{m',m}^{j}(\beta)\\[6pt]

d_{m',m}^{j}(-\beta) &= d_{m,m'}^{j}(\beta) = (-1)^{m'-m} d_{m',m}^{j}(\beta)

\end{align}

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

{{reflist}}

• {{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}}

Category:Representation theory of Lie groups

Category:Matrices (mathematics)

Category:Special hypergeometric functions

Category:Rotational symmetry