Wigner D-matrix

The Wigner D-matrix (also known as the Wigner rotation 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 $\gamma \;$ are Euler angles, and where $d_{m'm}^{j}(\beta )$ , known as Wigner's reduced d-matrix, is given by

{\begin{array}{lcl}d_{m'm}^{j}(\beta )&=&D_{m'm}^{j}(0,\beta ,0)\\&=&\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}}

This represents a rotation of $\beta$ about the (inital frame) $Y$ axis.

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 )

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