Talk:Wigner D-matrix

What's theta?
In the section "List of d-matrix elements" functions of theta are listed. What's theta? I don't see any reference to theta in the article. Is theta=beta? — Preceding unsigned comment added by 66.188.89.180 (talk) 03:21, 17 December 2014 (UTC)
 * Yes--P.wormer (talk) 10:01, 17 December 2014 (UTC)

sign error?
I get the following:

Euler rotations = i) rotation about z by alpha ii) rotation about new y by beta iii) rotation about new z by gamma

giving the rotation operator

R(alpha,beta,gamma)=exp(-i gamma Lz) exp (-i beta Ly) exp (-i alpha Lz)

To rotate a function, e.g. psi^l_m(phi,theta) by the Euler angles, we have

R(alpha, beta, gamma) psi^l_m(phi,theta) = psi(R(alpha, beta, gamma):phi,theta) =

where R(alpha,beta,gamma):phi,theta indicates a rotation of phi,theta on the unit sphere.

which means we want to expand

R(-alpha,-beta,-gamma)|j,m>=sum_m' D_m',m |j,m'>

D_{l,m',m}= =exp( i alpha m'+i gamma m) * d_m',m(beta)

where d_m',m(beta)=

(BTW, I worked out d_m',m(beta) from first principles and get the same expression as in the article, which is odd, because the article is calculating the expression d_m',m(beta)=)

I checked this numerically with a short computer program to rotate spherical harmonics and it seems to work.

It may be that this is just a different sign convention, or that I'm using a different definition, or I've misread the article or I've just made a mistake. (As usual)

christianjb — Preceding unsigned comment added by 109.255.41.76 (talk) 04:29, 4 November 2011 (UTC)

acknowlegement
I thank Joseph.romano for finding and correcting an error--P.wormer 13:23, 29 December 2006 (UTC)

z-y-z ?
In the 'Definition Wigner D-matrix' section, is "z-y-z convention" correct or should it be "x-y-z convention"? If "z-y-z convention" is incorrect, then the definition of the rotation operator above it is also probably incorrect. TBond (talk) 02:48, 15 May 2011 (UTC)
 * z-y-z is correct (or, as our page on Euler angles calls it, Z-Y'-Z '' ). The motivation being that the effect of a rotation around the Z axis in the current co-ordinate system is particularly simple to compute, such a rotation does not mix different spherical harmonics, it only change their phase.  Jheald (talk) 07:25, 24 July 2011 (UTC)

Some page refs
Lodging some page refs here, for future convenience:


 * E.P. Wigner (1959), Group Theory and its Application to the Quantum Mechanics of Atomic Spectra. pp. 153-156, 167-168
 * M. Hamermesh (1962), Group Theory and Its Application to Physical Problems pp. 334-337.
 * A. Messiah (1961), Quantum Mechanics, vol 2, pp. 1070-1075. (where the matrices are called "Rotation matrices", R(J)MM')
 * Baylis (1999), Electrodynamics: a modern geometric approach, ch. 13, (but key pages not on Google Books)
 * M. E. Rose (1957), Elementary Theory of Angular Momentum pp. 48-57.  (Agrees with all conventions here.) Cuzkatzimhut (talk) 14:25, 1 February 2018 (UTC)

A couple of recent papers: -- Jheald (talk) 07:46, 24 July 2011 (UTC)
 * J. Pagaran, S. Fritzschea, G. Gaigalas (2006), Maple procedures for the coupling of angular momenta. IX. Wigner D-functions and rotation matrices, Computer Physics Communications 174, 616–630
 * Ian G. Lisle, S.-L. Tracy Huang (2007), Algorithms for spherical harmonic lighting,

Interchange alpha and gamma
The choice of Euler angles is extremely subtle, has to do with active and passive (intrinsic and extrinsic) conventions, homomorphism or anti-homomorphism between 3 &times; 3 rotation matrices and Hilbert space operators, etc. Somebody interchanged &alpha; and &gamma; in the definition of the D-matrix, probably because of the order in the Greek alphabet. I don't want to enter an edit war, so I won't change anything, but let me just point out that after the change the relation in the article:

\hat{\mathcal{J}}_3\,D^j_{m'm}(\alpha,\beta,\gamma)^* \equiv - i \; {\partial \over \partial \alpha} D^j_{m'm}(\alpha,\beta,\gamma)^* =- i  \; {\partial \over \partial \alpha}\,e^{im'\gamma } d^j_{m'm}(\beta)e^{i m\alpha} =   m' \,  D^j_{m'm}(\alpha,\beta,\gamma)^* $$ is  wrong. .  --P.wormer (talk) 14:48, 17 March 2016 (UTC)

I agree with P.wormer and corrected the equations accordingly. gerritg (talk) 14:24, 14 April 2016 (UTC)


 * There we go again: somebody changed Jz to Jx, which is  wrong. . --P.wormer (talk) 07:33, 27 May 2024 (UTC)
 * A couple of days ago Kkumarsshasshank changed Jx back to Jz, which is correct. --P.wormer (talk) 09:05, 3 July 2024 (UTC)