Is the 2-D Wigner-Eckert Theorem applicable to cylindrical harmonics?

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In summary: The 1 means that the spherical harmonics are not orthogonal. The relevance to a physical example is that the Wigner-Eckert theorem is not always valid for orthogonal groups.
  • #1
Surrealist
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Am I correct in assuming that the Wigner-Eckert theorem only holds for spherical harmonics? Is there an analogous theorem for cylindrical harmonics?
 
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  • #2
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.
 
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  • #3
Surrealist said:
Am I correct in assuming that the Wigner-Eckert theorem only holds for spherical harmonics?

No. As smallphi says, there is a more general, group-theoretical version of the Wigner-Eckert theorem.

Is there an analogous theorem for cylindrical harmonics?

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.
 
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  • #4
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].
 
  • #5
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.
 
  • #6
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.
 
  • #7
Surrealist said:
If you guys consider Tung elementary, you must be much more experienced than I am.

I didn't say that Tung is elementary, I said that Barut and Raczka is more advanced and more rigourous. :biggrin: Tung isn't rigourous, but it is quite sophisticated.

Quantum Field Theory was the last physics class I ever took, and we only skimmed Tung a little.

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. :mad:
 
  • #8
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?
 

1. What is the 2-D Wigner-Eckert Theorem?

The 2-D Wigner-Eckert Theorem is a mathematical theorem that relates to the representation theory of Lie groups. It states that for any two-dimensional irreducible representation of a Lie group, there exists a unique set of basis operators that can be used to describe the representation.

2. How is the 2-D Wigner-Eckert Theorem used in physics?

The 2-D Wigner-Eckert Theorem is used in quantum mechanics to describe the symmetries of a system. It allows for the simplification of calculations and provides a framework for understanding the behavior of particles in various physical systems.

3. What is the significance of the 2-D Wigner-Eckert Theorem?

The 2-D Wigner-Eckert Theorem is significant because it provides a fundamental understanding of the symmetries in physical systems. It also has important applications in quantum information theory, quantum computing, and other areas of physics.

4. Can the 2-D Wigner-Eckert Theorem be generalized to higher dimensions?

Yes, the 2-D Wigner-Eckert Theorem can be generalized to higher dimensions. However, the higher-dimensional versions are more complex and require more advanced mathematical techniques to understand and apply.

5. Are there any limitations to the 2-D Wigner-Eckert Theorem?

The 2-D Wigner-Eckert Theorem has limitations in that it only applies to two-dimensional representations of Lie groups. It also assumes that the group is compact and connected. These limitations can be overcome by using more advanced mathematical techniques, but they should be considered when applying the theorem in a real-world context.

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