Wigner D-matrix: Difference between revisions
Carl McBride (talk | contribs) m (Corrected typo.) |
Carl McBride (talk | contribs) (Added a reference and corrected equation) |
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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 (Ref. 2 Eq. 4.11 and 4.13) | ||
:<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>\beta</math> about the (inital frame) <math>Y</math> axis. | This represents a rotation of <math>\beta</math> about the (inital frame) <math>Y</math> axis. | ||
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==References== | ==References== | ||
#Eugene Paul Wigner "Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren", Vieweg Verlag, Braunschweig (1931). | #Eugene Paul Wigner "Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren", Vieweg Verlag, Braunschweig (1931). | ||
#M. E. Rose "Elementary theory of angular momentum", John Wiley & Sons (1967) | |||
#[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/10.1016/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)] | #[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)] | ||
[[Category: Mathematics]] | [[Category: Mathematics]] | ||
Revision as of 16:50, 18 June 2008
The Wigner D-matrix (also known as the Wigner rotation matrix) 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
- 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 (Ref. 2 Eq. 4.11 and 4.13)
- 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} }
This 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 (inital 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
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 )}
External links
References
- Eugene Paul Wigner "Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren", Vieweg Verlag, Braunschweig (1931).
- M. E. Rose "Elementary theory of angular momentum", John Wiley & Sons (1967)
- 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)