Wigner D-matrix: Difference between revisions

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(Minor changes on definition of Wigner reduced matrix)
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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==

Latest revision as of 13:09, 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])

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]

  1. Eugene Paul Wigner "Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren", Vieweg Verlag, Braunschweig (1931)
  2. 2.0 2.1 M. E. Rose "Elementary theory of angular momentum", John Wiley & Sons (1967) ISBN 0486684806

Related reading

External links[edit]