Ytdyklly

In finding the solutions to the wave equation for a spherically symmetric potential $V(r)$, we look for the eigenfunctions of $hat L^2$ and $hat L_z$ operators. However, what is the reasoning behind this? The time-independent Schrodinger equation:
$$hat H Psi = E Psi$$

This is an eigenvalue equation and any eigenfunction of $hat H$ this equation is obviously a solution for this equation. Now when there is a spherically symmetric potential $V(r)$, I understand that because of separation of variables we can look for solutions of the form:
$$Psi (vec r,theta, phi) = R(vec r) Y(theta, phi)$$

Now to find the angular part why do we look for eigenstates of $hat L^2$ and $hat L_z$ operators (I know that they commute)?

Well, to put it shortly, it's just because $L^2$ and $L_z$ are two observables that have no $r$ dependence. Since the kinetic terms of $H$ can be written as functions of $r$ and $L^2$, and the potential depends only on $r$, it is clear that the Hamiltonian commutes with the angular momenta. As such, it makes sense to write down $Y$ in terms of these two.

But if we'd like to go a little deeper, it is a bit related to representation theory. The fact that $L^2$ and $L_z$ form a complete description of vector rotations (they represent the Lie algebra $mathfrak{so}(3)$ of the Lie group $mathrm{SO}(3)$) and therefore are the most natural way to express the angular dependence of a rotation-invariant space in terms of orthonormalizable functions.

One objective of quantum mechanics is to fully labelled states of the system. In some cases energy is not enough: for instance in the hydrogen atom some energy levels occur more than once; the same happens in the 3d harmonic oscillator. In such cases, simply giving the energy is not enough to completely identify the state so we look for other ``quantum numbers'' to refine the identification.

Of course one would prefer to label states with constants of motion, since they are constant so what we call state $nell m$ at the start we can use the same name at the end. Hence the idea of using a complete set of commuting operators - the Hamiltonian and others that commute with it - so as to uniquely identify the quantum states with constants of motion.

Of course when the potential is spherical $V(r)$, what enters in the radial part of the Schrödinger equation is actually the effective potential
$$
V_{eff }=V(r)+frac{hbar^2}{2m}frac{ell(ell+1)}{r^2} tag{1}
$$
and so you see that the radial Schrödinger equation actually explicitly contains an angular momentum part so it is quite natural to look for functions $psi(r,theta,phi)$ which are also eigenfunctions $Y^ell_m(theta,phi)$ of $hat L^2$ as these will provide the correct extra factor in Eq.(1) when separating the variables. Since there are many functions $Y^ell_m(theta,phi)$ with the same $hbar^2ell(ell+1)$, one uses the eigenvalue $hbar m$ of $hat L_z$ to distinguish them.