Rotational invariance and degeneracy (quantum mechanics)

[LikingPhysics]

Homework Statement

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

Homework Equations

The professor told us to recall that

$J: \vec{L}=(L_x,L_y,L_z)$

$L_z|l,m\rangle=m|l,m\rangle$

$L_\pm=L_x\pm iL_y$

$L_\pm|l,m\rangle= \hbar\sqrt{l(l+1)-m(m\pm1)} |l,m\pm 1\rangle$

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.