Solution
= Solution
With $L=xp_y-yp_x$, the canonical commutators give
$$
[L,x]=i\hbar y,
\quad
[L,y]=-i\hbar x,
\quad
[L,p_x]=i\hbar p_y,
\quad
[L,p_y]=-i\hbar p_x.
$$
It follows that
$$
[L,x^2+y^2]=0,
\qquad
[L,p_x^2+p_y^2]=0.
$$
Therefore rotational symmetry of the isotropic oscillator gives
$$
\boxed{[L,H]=0}.
$$