Wigner D-matrix

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

$D_{m^{\prime }m}^j(\alpha ,\beta ,\gamma ):=⟨j{m}^{\prime }|\mathcal{R}(\alpha ,\beta ,\gamma )|jm⟩=e^{-i{m}^{\prime }\alpha }{d}_{m^{\prime }m}^j(\beta )e^{-im\gamma }$

where $\alpha ,\beta ,$ and $\gamma$ are Euler angles, and where $d_{m^{\prime }m}^j(\beta )$, known as Wigner's reduced d-matrix, is given by

$\begin{array}{ccc}{d}_{m^{\prime }m}^j(\beta )& =& ⟨jm\text{'}\left|e^{-i\beta j_y}\right|jm⟩\\ & =& [(j+m\text{'})!(j-m\text{'})!(j+m)!(j-m)!{]}^{1/2}{\sum }_s\frac{(-1{)}^{m^{\prime }-m+s}}{(j+m-s)!s!(m^{\prime }-m+s)!(j-m^{\prime }-s)!}\\ & & \times {(\cos \frac{\beta }{2})}^{2j+m-m^{\prime }-2s}{(\sin \frac{\beta }{2})}^{m^{\prime }-m+2s}\end{array}$

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).