Wigner D-matrix

The Wigner D-matrix is a square matrix, of dimension $2j+1$, given by

$D_{m'm}^{j}(\alpha ,\beta ,\gamma ):=\langle jm'|{\mathcal {R}}(\alpha ,\beta ,\gamma )|jm\rangle =e^{-im'\alpha }d_{m'm}^{j}(\beta )e^{-im\gamma }$

where $\alpha ,\;\beta ,$ and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \gamma \;} are Euler angles, and where $d_{m'm}^{j}(\beta )$, known as Wigner's reduced d-matrix, is given by

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{array}{lcl}d_{m'm}^{j}(\beta )&=&\langle jm'|e^{-i\beta j_{y}}|jm\rangle \\&=&[(j+m')!(j-m')!(j+m)!(j-m)!]^{1/2}\sum _{s}{\frac {(-1)^{m'-m+s}}{(j+m-s)!s!(m'-m+s)!(j-m'-s)!}}\\&&\times \left(\cos {\frac {\beta }{2}}\right)^{2j+m-m'-2s}\left(\sin {\frac {\beta }{2}}\right)^{m'-m+2s}\end{array}}}

Relation with spherical harmonic functions

The D-matrix elements with second index equal to zero, are proportional to spherical harmonics (normalized to unity)

$D_{m0}^{\ell }(\alpha ,\beta ,\gamma )^{*}={\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell }^{m}(\beta ,\alpha )$

References

1. E. P. Wigner, Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren, Vieweg Verlag, Braunschweig (1931).