Editing 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 =
 e^{-im'\alpha } d^j_{m'm}(\beta)e^{-i m\gamma} </math>

where <math>d^j_{m'm}(\beta)</math>, known as ''Wigner's (small) d-matrix'', is given by

:<math>\begin{array}{lcl}
d^j_{m'm}(\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} 
</math>
==References==
#E. P. Wigner, ''Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren'', Vieweg Verlag, Braunschweig (1931).
[[Category: Mathematics]]
@@ Line 1: / Line 1: @@
-The '''Wigner D-matrix''' (also known as the Wigner rotation matrix)<ref>Eugene Paul Wigner "Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren", Vieweg Verlag, Braunschweig (1931)</ref> is a square matrix, of dimension <math>2j+1</math>, given by (Eq. 4.12 of <ref name="rose">M. E. Rose "Elementary theory of angular momentum", John Wiley & Sons (1967) ISBN 0486684806</ref> )
+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 =
   e^{-im'\alpha } d^j_{m'm}(\beta)e^{-i m\gamma} </math>
-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 (small) d-matrix'', is given by
-where <math>d^j_{m'm}(\beta)</math>, known as Wigner's reduced  d-matrix, is given by (Eqs. 4.11 and 4.13 of  <ref name="rose"> </ref>)
 :<math>\begin{array}{lcl}
-d^j_{m'm}(\beta) &=& D^j_{m'm}(0,\beta,0) \\
+d^j_{m'm}(\beta) &=& \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)!]^{1/2}
-&=& [(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)!} \\
-\sum_\chi \frac{(-1)^{\chi}}{(j-m'-\chi)!(j+m-\chi)!(\chi+m'-m)!\chi!} \\
+&&\times \left(\cos\frac{\beta}{2}\right)^{2j+m-m'-2s}\left(\sin\frac{\beta}{2}\right)^{m'-m+2s}
-&&\times \left(\cos\frac{\beta}{2}\right)^{2j+m-m'-2\chi}\left(-\sin\frac{\beta}{2}\right)^{m'-m+2\chi}
+\end{array}
-\end{array} </math>
+</math>
-The sum over <math>\chi</math> is restricted to those values that do not lead to negative factorials.
-This function represents a rotation of <math>\beta</math> about the (initial frame) <math>Y</math> axis.
-=== Relation with spherical harmonic functions ===
-The D-matrix elements with second index equal to zero, are proportional
-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>
 ==References==
-<references/>
+#E. P. Wigner, ''Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren'', Vieweg Verlag, Braunschweig (1931).
-'''Related reading'''
-*[http://dx.doi.org/10.1016/S0166-1280(97)00185-1 Miguel A. Blanco, M. Flórez and M. Bermejo "Evaluation of the rotation matrices in the basis of real spherical harmonics", Journal of Molecular Structure: THEOCHEM '''419''' pp. 19-27 (1997)]
-*[http://dx.doi.org/10.1063/1.2194548 Holger Dachsel "Fast and accurate determination of the Wigner rotation matrices in the fast multipole method", Journal of Chemical Physics '''124''' 144115 (2006)]
-==External links==
-*[http://en.wikipedia.org/wiki/Wigner_D-matrix Wigner D-matrix page on Wikipedia]
 [[Category: Mathematics]]
-[[category: Quantum mechanics]]