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) =<phi,theta| R(-alpha,-beta,-gamma)|j,m>

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}=<j,m'| exp(i alpha Lz) exp(i beta Ly) e^(i gamma Lz) |j,m> =exp( i alpha m'+i gamma m) * d_m',m(beta)

where d_m',m(beta)=<j,m'|exp(i beta Ly)|j,m>

(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)=<j,m'|exp(-i beta Ly)|j,m>)

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:

• 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, doi:10.1145/1321261.1321303

-- Jheald (talk) 07:46, 24 July 2011 (UTC)

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 × 3 rotation matrices and Hilbert space operators, etc. Somebody interchanged α and γ 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:

${\hat {\mathcal {J}}}_{3}\,D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}\equiv -i\;{\partial \over \partial \alpha }D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}=-i\;{\partial \over \partial \alpha }\,e^{im'\gamma }d_{m'm}^{j}(\beta )e^{im\alpha }=m'\,D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{*}$

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)