Question regarding eigenvalues of angular momentum operator

[valy112]

Hello! First of all let me wish you a happy new year!

This is not a homework problem, but rather a curiosity of mine.
In Schwabl's Quantum Mechanics, one can find the proof of the fact that all eigenvalues of the angular momentum L_z are either integers or half-integers, raging from -l to l(l is just a notation they use). My question is, do all these eigenvalues have algebraic multiplicity 1, and if this is true, why?(I guess the fact, if true, should be proved using some subtle physical condition, since using math doesn't seem to work)

Also, does anyone know of a book where I could find information on unitary representation of SU(2)(or of SU(2) in general) and its application to Quantum Mechanics. Valentin

 

[BigJDubs]

. Yes, all eigenvalues of the angular momentum Lz have algebraic multiplicity 1. This is because the angular momentum operators in Quantum Mechanics are represented by matrices, and these matrices have only one nonzero element for each eigenvalue. Since the number of nonzero elements must equal the algebraic multiplicity, it follows that the algebraic multiplicity of each eigenvalue is 1. As for books on unitary representation of SU(2) and its application to Quantum Mechanics, I would suggest checking out the book "Group Theory and Its Applications in Physics" by V.V. Dodonov and V.I. Man'ko. It has a detailed discussion of SU(2) and its unitary representations, as well as their use in Quantum Mechanics.