Wigner D-matrix: Difference between revisions
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The '''Wigner D-matrix''' (also known as the Wigner rotation matrix) is a square matrix, of dimension <math>2j+1</math>, given by | 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> ) | ||
:<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 (Eqs. 4.11 and 4.13 of <ref name="rose"> </ref>) | ||
:<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')!]^{1/2} | ||
\ | \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'- | &&\times \left(\cos\frac{\beta}{2}\right)^{2j+m-m'-2\chi}\left(-\sin\frac{\beta}{2}\right)^{m'-m+2\chi} | ||
\end{array} </math> | \end{array} </math> | ||
This represents a rotation of <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 === | === 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> | ||
==References== | |||
<references/> | |||
'''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== | ==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] | ||
[[Category: Mathematics]] | [[Category: Mathematics]] | ||
[[category: Quantum mechanics]] | |||
Latest revision as of 12:09, 26 October 2010
The Wigner D-matrix (also known as the Wigner rotation matrix)[1] is a square matrix, of dimension , given by (Eq. 4.12 of [2] )
where and are Euler angles, and where , known as Wigner's reduced d-matrix, is given by (Eqs. 4.11 and 4.13 of [2])
![{\displaystyle {\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 _{\chi }{\frac {(-1)^{\chi }}{(j-m'-\chi )!(j+m-\chi )!(\chi +m'-m)!\chi !}}\\&&\times \left(\cos {\frac {\beta }{2}}\right)^{2j+m-m'-2\chi }\left(-\sin {\frac {\beta }{2}}\right)^{m'-m+2\chi }\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b328493f438280cb71675325910dfd5c2f233826)
The sum over is restricted to those values that do not lead to negative factorials. This function represents a rotation of about the (initial frame) axis.
Relation with spherical harmonic functions[edit]
The D-matrix elements with second index equal to zero, are proportional to spherical harmonics (normalized to unity)
References[edit]
- ↑ Eugene Paul Wigner "Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren", Vieweg Verlag, Braunschweig (1931)
- ↑ 2.0 2.1 M. E. Rose "Elementary theory of angular momentum", John Wiley & Sons (1967) ISBN 0486684806
Related reading
- 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)
- Holger Dachsel "Fast and accurate determination of the Wigner rotation matrices in the fast multipole method", Journal of Chemical Physics 124 144115 (2006)