Wigner D-matrix - Revision history

Carl McBride: Changed references to Cite format

2010-10-26T11:09:04Z

Changed references to Cite format

← Older revision		Revision as of 13:09, 26 October 2010
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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 (~~Ref. 2~~ Eq. 4.12)		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 (~~Ref. 2 Eq~~. 4.11 and 4.13)		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==
	~~#Eugene Paul Wigner "Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren", Vieweg Verlag, Braunschweig (1931).~~		<references/>
	~~#M. E. Rose "Elementary theory of angular momentum", John Wiley & Sons (1967) ISBN 0486684806~~		'''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/10.1016/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)]

	==External links==		==External links==

Eva.noya: Minor changes on definition of Wigner reduced matrix

2009-05-22T15:34:43Z

Minor changes on definition of Wigner reduced matrix

← Older revision		Revision as of 17:34, 22 May 2009
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	&&\times \left(\cos\frac{\beta}{2}\right)^{2j+m-m'-2\chi}\left(-\sin\frac{\beta}{2}\right)^{m'-m+2\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} </math>		\end{array} </math>
	This represents a rotation of <math>\beta</math> about the (~~inital~~ frame) <math>Y</math> axis.		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

Nice and Tidy: /* References */ Added an ISBN for Rose.

2009-02-24T15:54:44Z

References: Added an ISBN for Rose.

← Older revision		Revision as of 17:54, 24 February 2009
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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)		#M. E. Rose "Elementary theory of angular momentum", John Wiley & Sons (1967) ISBN 0486684806
	#[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)]

	==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]]		[[category: Quantum mechanics]]

Nice and Tidy: Placed external links at foot of page.

2008-12-11T18:34:58Z

Placed external links at foot of page.

← Older revision		Revision as of 20:34, 11 December 2008
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	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==~~
	*[http://en.wikipedia.org/wiki/Wigner_D-matrix Wigner D-matrix page on Wikipedia]
	==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).
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	#[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)]
			==External links==
			*[http://en.wikipedia.org/wiki/Wigner_D-matrix Wigner D-matrix page on Wikipedia]
	[[Category: Mathematics]]		[[Category: Mathematics]]
	[[category: Quantum mechanics]]		[[category: Quantum mechanics]]

Carl McBride: Added Quantum mechanics category

2008-07-21T11:17:53Z

Added Quantum mechanics category

← Older revision		Revision as of 13:17, 21 July 2008
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	#[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]]
			[[category: Quantum mechanics]]

Carl McBride: Added equation number

2008-06-18T15:52:55Z

Added equation number

← Older revision		Revision as of 17:52, 18 June 2008
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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) is a square matrix, of dimension <math>2j+1</math>, given by (Ref. 2 Eq. 4.12)

	:<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 =

Carl McBride: Added a reference and corrected equation

2008-06-18T15:50:22Z

Added a reference and corrected equation

← Older revision		Revision as of 17:50, 18 June 2008
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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)!]^{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} </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]]

Carl McBride: Corrected typo.

2008-06-17T15:21:26Z

Corrected typo.

← Older revision		Revision as of 17:21, 17 June 2008
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	&&\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'-2s}\left(\sin\frac{\beta}{2}\right)^{m'-m+2s}
	\end{array} </math>		\end{array} </math>
	This represents a rotation of <math>\~~theta~~</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.
	=== 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
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	*[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==		==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).
	#[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]]

Carl McBride: /* References */ Added a reference

2008-06-17T14:29:15Z

References: Added a reference

← Older revision		Revision as of 16:29, 17 June 2008
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	*[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==		==References==
	#~~E. P.~~ 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).
			#[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]]

Carl McBride: /* References */ Added a reference

2008-06-17T14:19:08Z

References: Added a reference

← Older revision		Revision as of 16:19, 17 June 2008
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	==References==		==References==
	#E. P. Wigner, ''Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren'', Vieweg Verlag, Braunschweig (1931).		#E. P. Wigner, ''Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren'', Vieweg Verlag, Braunschweig (1931).
			#[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]]