In this document http://users.ece.gatech.edu/~alan/ECE6451/Lectures/StudentLectures/Reddy_3p1_AngMomNcentralForces.pdf pg.13 of 30, 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 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 Ψ = R(r) * Y_ml(θ,φ).."

(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 / angular parts of the wave function. Would you please explain that reasoning also?

In this PDF document (a lecture by Shivaly Reddy, page 13), 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 (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?