Talk:Wigner D-matrix

message: 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)
replies: text: Yes--P.wormer (talk) 10:01, 17 December 2014 (UTC)
thread_title: What's theta?

message: 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)
thread_title: sign error?

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

message: 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)
replies: text: 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)
thread_title:  z-y-z ? 

message: 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)
thread_title:  Some page refs 

message: 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^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)
replies: text: There we go again: somebody changed Jz to Jx, which is  wrong.. --P.wormer (talk) 07:33, 27 May 2024 (UTC)
thread_title:  Interchange alpha and gamma