Am I correct in assuming that the Wigner-Eckert theorem only holds for spherical harmonics? Is there an analogous theorem for cylindrical harmonics?
The most general statement of Wigner-Eckhart theorem can be found in group theory books. It gives the relation between matrix elements of certain operators. The operators and the states from which the matrix elements are computed should belong to irreducible representations of some group. The cylindrical harmonics probably form irreducible representations of the group of rotations about fixed axis. If there are operators forming irreducible representations of the same group, there probably is Wigner-Eckhart theorem for the matrix elements. Unfortunately Google didn't return a good illustration of the above.
No. As smallphi says, there is a more general, group-theoretical version of the Wigner-Eckert theorem. I don't think so, but I am far from sure. Representations of the rotattion group SO(3) lead to the spherical harmonics. smallphi presented a good idea, look at SO(2) as a subgroup of SO(3), but I think it's slightly too restrictive. When this is done, the spherical harmonics reduce to functions of the form exp(i m phi), and this is just what's needed for representations of SO(2). [Edit] Going from rotations in three dimensions to rotations in two dimensions means going from spherical harmonics to exp(i m phi). Going from translations and rotations in three dimensions to translations and rotations in two dimensions means going from spherical Bessel functions to Bessel functions. I don't know of an application of the Wigner-Eckert theorem to spherical Bessel functions. [End Edit] Cylindrical harmonics (Bessel functions of the first kind?) relate to representations of the the group generated by rotations and translations in two dimensions, and, because of the translations, this group is not compact. I think the Wigner-Eckert theorem applies only to (certain) compact groups. The general Wigner-Eckert theorem is stated in, for example, Group Theory in Physics by Tung, and in the more advanced and more rigorous Theory of Group Representations and Applications by Barut and Raczka. I suspect the multi-volume work by Cornwell also covers it.
Tinkham covers it as well as a host of other group therory and QM texts. I would be willing to bet that you could reduce to 2-d in some limit, i.e. if your harmonic was a function of theta only, not of phi or for the special case of [itex] L_{z} = 0 [/itex].
Thanks for the input guys. Initially, I tried something like that which was suggested... I've run into a few problems when attempting to apply this theorem to a real world proglem. I'll look at a little more. If you guys consider Tung elementary, you must be much more experienced than I am. Quantum Field Theory was the last physics class I ever took, and we only skimmed Tung a little.
Yea I'll echo what George said and state that I think this theorem is only valid for the spherical harmonic case. Going through the steps in my mind I seem to be missing some commutators that I'd want. Hmm! Non trivial question. You might have more luck in the math forum.
I didn't say that Tung is elementary, I said that Barut and Raczka is more advanced and more rigourous. Tung isn't rigourous, but it is quite sophisticated. It's tough to learn group theory and representation theory from the bits seen in physics courses that use, but that don't specialize in, group theory and representattion theory. This is part of a bigger problem in today's physics - there is so more useful stuff than there is time for learning the stuff. One has to make choices.
OK, these questions will reveal my ignorance, but I'll ask it any way. After doing some research I think I have found an old paper which addresses this issue. So my related question is: am I dealing with O(2,1). I thought I was dealing with the orthogonal group O(2). What does the 1 mean, and how is that relevant to a physical example?