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The '''Wigner D-matrix''' | The '''Wigner D-matrix''' is a square matrix, of dimension <math>2j+1</math>, given by | ||
:<math> D^j_{m'm}(\alpha,\beta,\gamma) := \langle jm' | \mathcal{R}(\alpha,\beta,\gamma)| jm \rangle = | :<math> D^j_{m'm}(\alpha,\beta,\gamma) := \langle jm' | \mathcal{R}(\alpha,\beta,\gamma)| jm \rangle = | ||
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where <math>\alpha, \; \beta, </math> and <math>\gamma\;</math> are [[Euler angles]], and | where <math>\alpha, \; \beta, </math> and <math>\gamma\;</math> are [[Euler angles]], and | ||
where <math>d^j_{m'm}(\beta)</math>, known as Wigner's reduced d-matrix, is given by | where <math>d^j_{m'm}(\beta)</math>, known as Wigner's reduced d-matrix, is given by | ||
:<math>\begin{array}{lcl} | :<math>\begin{array}{lcl} | ||
d^j_{m'm}(\beta) &=& D^j_{m'm}(0,\beta,0) \\ | d^j_{m'm}(\beta) &=& D^j_{m'm}(0,\beta,0) \\ | ||
&=& \langle jm' |e^{-i\beta j_y} | jm \rangle\\ | &=& \langle jm' |e^{-i\beta j_y} | jm \rangle\\ | ||
&=& [(j+m)!(j-m)!(j+m | &=& [(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'- | &&\times \left(\cos\frac{\beta}{2}\right)^{2j+m-m'-2s}\left(\sin\frac{\beta}{2}\right)^{m'-m+2s} | ||
\end{array} | \end{array} | ||
</math> | |||
=== Relation with spherical harmonic functions === | === Relation with spherical harmonic functions === | ||
The D-matrix elements with second index equal to zero, are proportional | The D-matrix elements with second index equal to zero, are proportional | ||
to [[spherical harmonics]] (normalized to unity) | to [[spherical harmonics]] (normalized to unity) | ||
:<math>D^{\ell}_{m 0}(\alpha,\beta,\gamma)^* = \sqrt{\frac{4\pi}{2\ell+1}} Y_{\ell}^m (\beta, \alpha )</math> | :<math>D^{\ell}_{m 0}(\alpha,\beta,\gamma)^* = \sqrt{\frac{4\pi}{2\ell+1}} Y_{\ell}^m (\beta, \alpha )</math> | ||
==External links== | ==External links== | ||
*[http://en.wikipedia.org/wiki/Wigner_D-matrix Wigner D-matrix page on Wikipedia] | *[http://en.wikipedia.org/wiki/Wigner_D-matrix Wigner D-matrix page on Wikipedia] | ||
==References== | |||
#E. P. Wigner, ''Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren'', Vieweg Verlag, Braunschweig (1931). | |||
[[Category: Mathematics]] | [[Category: Mathematics]] | ||