Deducing Degeneracy in Spin from Commutation Relations

In summary, the conversation discusses the quantization of angular momentum-like operators and the difficulty in deducing the degeneracy of eigenstates from commutation relations. It is mentioned that the Wigner theory of unitary irreducible representations of the Poincare group can provide an explanation for this. The topic of rigged Hilbert spaces is also briefly mentioned, with suggestions for references. The conversation ends with a question about whether the dimensionality of eigenstates can be determined from mechanics alone.
  • #1
Manchot
473
4
In reviewing the derivation of the quantization of angular momentum-like operators from their commutation relations, I noticed that there is nothing a priori from which you can deduce the degeneracy of the eigenstates. While this is not a problem for angular momentum, in which other constraints come into play, it seems to me that it might be a problem for spin. Is it possible to deduce the dimensionality from the spin commutation relations alone? Or must one postulate it?

For example, I know from the commutation relations that for an electron, any non-trivial eigenstates of Sz have eigenvalues +1/2 or -1/2. But how do I know that there are only two degrees of freedom? That is, how do I know that there's no degeneracy in the m eigenvalues?

On a somewhat unrelated note, does anyone know of any good references on rigged Hilbert spaces? Thanks.
 
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  • #2
Manchot said:
For example, I know from the commutation relations that for an electron, any non-trivial eigenstates of Sz have eigenvalues +1/2 or -1/2. But how do I know that there are only two degrees of freedom? That is, how do I know that there's no degeneracy in the m eigenvalues?
This falls out directly from the Wigner theory of unitary irreducible representations of the
Poincare group. Weinberg vol-I covers this sort of thing is considerable detail.

On a somewhat unrelated note, does anyone know of any good references on rigged Hilbert spaces?
Depends on exactly what you want. Arno Bohm and colleagues have written heaps of
stuff on many aspects of that. Do a google search to find their website - I vaguely
remember it's in the University of Texas at Austin. Also google for "Gamow vectors"
which are related to rigged Hilbert spaces.

(If you can be more specific, I might be able to suggest something else.)
 
  • #3
strangerep said:
This falls out directly from the Wigner theory of unitary irreducible representations of the
Poincare group. Weinberg vol-I covers this sort of thing is considerable detail.
I'm at the lower graduate level, and haven't taken any field theory yet. So, am I to understand that finding the dimensionality is impossible from mechanics alone? From a pedagogical perspective, would I just be better off taking it as an axiom?

strangerep said:
(If you can be more specific, I might be able to suggest something else.)
I'm looking mostly for introductory material, but I'll start with what you gave me. Thanks.
 

1. What is the concept of degeneracy in spin?

Degeneracy in spin refers to the phenomenon where multiple quantum states have the same energy level. This means that these states cannot be distinguished based on their energy and are therefore considered degenerate.

2. How is degeneracy in spin related to the commutation relations?

The commutation relations, specifically the ones between angular momentum operators, can be used to determine the degeneracy of spin states. If the commutator between two operators is zero, it indicates that the corresponding quantum states are degenerate.

3. What are the implications of degeneracy in spin?

Degeneracy in spin has important implications in quantum mechanics. It can affect the stability of atoms and molecules, as well as the behavior of particles in magnetic fields. Understanding and predicting degeneracy is crucial for accurate calculations in quantum systems.

4. How can degeneracy be deduced from commutation relations?

To deduce degeneracy from commutation relations, one can use the ladder operator method. By finding the eigenvalues of the angular momentum operators, one can determine the degeneracy of a given spin state.

5. Is degeneracy in spin a common occurrence?

Yes, degeneracy in spin is a common phenomenon in quantum systems. It is especially prevalent in atoms with multiple electrons, where the spin of each electron can contribute to degeneracy. Furthermore, degeneracy can also occur in other types of particles with spin, such as protons and neutrons in atomic nuclei.

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