Show that the Hamiltonian operator commutes with the angular momentum operator

Question

Show that the Hamiltonian operator $\hat{H}= (\hat{p}^2/2m)+\hat{V}$ commutes with all three components of $\vec{L}$, provided that $V$ depends only on $r$. (Thus $H$,$L^2$, $L_z$ are mutually compatible observables). ($\vec{L}$ is $\langle L_x, L_y, L_z\rangle$.)

Answer 1

Show that $L_x$ commutes with $(x^2+y^2+z^2)$. Then it follows that $L_x$ commutes with any function of $r^2$ and therefore with $V(r)$. Then show that $L_x$ commutes with $(p_x^2+p_y^2+p_z^2)$. So $L_x$ commutes with $H$. $L_y$ and $L_z$ follow by symmetry.