Rotational invariance and degeneracy (quantum mechanics)

[LikingPhysics]

Homework Statement

Show that if a Hamiltonian [itex]H[/itex] is invariant under all rotations, then the eigenstates of [itex]H[/itex] are also eigenstates of [itex]L^{2}[/itex] and they have a degeneracy of [itex]2l+1[/itex].

Homework Equations

The professor told us to recall that

[itex]J: \vec{L}=(L_x,L_y,L_z) [/itex]

[itex] L_z|l,m\rangle=m|l,m\rangle [/itex]

[itex] L_\pm=L_x\pm iL_y [/itex]

[itex] L_\pm|l,m\rangle= \hbar\sqrt{l(l+1)-m(m\pm1)} |l,m\pm 1\rangle[/itex]

The Attempt at a Solution

I have been reading as much materials as I can, but I still have no clue at all on how to solve it at this moment. Can anyone help? Thank you so much!

 

[LikingPhysics]

I solved it!

Thank goodness! I have solved it now.

One can do the calculations in either generic Hilbert space for general rotations or in 3D real Hilbert space.