In [this PDF document (a lecture by Shivaly Reddy, page 13)](http://users.ece.gatech.edu/~alan/ECE6451/Lectures/StudentLectures/Reddy_3p1_AngMomNcentralForces.pdf), he says that  
>$L^2$ is independent of $r$; therefore it commutes with any function of $r$.

This seems related to a problem in [Schaum's Quantum Mechanics](http://www.amazon.com/Schaums-Outline-Quantum-Mechanics-Outlines/dp/0071623582) (Amazon link) dealing with a particle in a spherical potential well.  In the solution, after writing down the Hamiltonian, they say,  
>It is evident that $[H, L^2] = 0$;  hence, we write $\Psi = R(r) \cdot Y_{ml}(\theta,\varphi)$..."

 1.  Both sources seem to be saying that when the potential well is
     spherically symmetric, $[H, L^2] = 0$.  Why, exactly?

 2.  Schaum goes on to say that since $[H,L^2] = 0$, then we can separate
     out the radial and angular parts of the wave function.  Would you
     please explain that reasoning also?