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} | ||
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:<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== | ||
<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== | ||
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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2j+1} , given by (Eq. 4.12 of [2] )
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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} }
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha, \; \beta, } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma\;} are Euler angles, and where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d^j_{m'm}(\beta)} , known as Wigner's reduced d-matrix, is given by (Eqs. 4.11 and 4.13 of [2])
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{lcl} d^j_{m'm}(\beta) &=& D^j_{m'm}(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} }
The sum over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} is restricted to those values that do not lead to negative factorials. This function represents a rotation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta} about the (initial frame) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y} 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)
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D^{\ell}_{m 0}(\alpha,\beta,\gamma)^* = \sqrt{\frac{4\pi}{2\ell+1}} Y_{\ell}^m (\beta, \alpha )}
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)