Angular momentum quantum mechanics

[thenewbosco]

Consider a spin-less particle, mass M, confined on a sphere radius 1. It can move freely on the surface but is always at radius 1.
1. Write the Hamiltonian $$H=\frac{L_{op}^2}{2M}$$ in spherical polar coords.
2. Write the energy eigenvalues, specify degeneracy of each state. (not you can omit r part of wavefunction, concentrate on $$\theta$$ and $$\phi$$ dependence)

I have done part one. but i am not sure how to go about part two. I am thinking that it will be just the operator L^2 acting on a ket like |l m> ? then the eigenvals are l(l+1)hbar^2? i don't see where the degeneracy will come in...any help?

 

[dwintz02]

You have the right idea for the eigenvalues. For the degeneracies, think of how many states have the eigenvalue l(l+1)hbar^2. More directly, how many m's are possible for a given l?

 

[thenewbosco]

since m=-l...+l we have 2l+1, m values. how do i know which integers i will have for l in this case, would it be just 0 and 1? Since the hamiltonian has degeneracy, how can i find what else i need to specify a complete set of commuting observables?

 

[Marco_84]

since m=-l...+l we have 2l+1, m values. how do i know which integers i will have for l in this case, would it be just 0 and 1? Since the hamiltonian has degeneracy, how can i find what else i need to specify a complete set of commuting observables?

think what mean 2l+1: It is the dimension of SU(2) in function of the spin (weight) of the particle. In Our formal theories particles are just representations of groups, or maybe some tracks in experiments... well in any way you have a spinless particle...

bye
marco

 

[dwintz02]

You just need another operator that has eigenvalues that are functions of m that also commutes with the Hamiltonian.