Eigenstates for a particle in a spherically symmetric potential Consider a particle in a spherically symmetric potential. Write down a complete set of commuting observables for the system, the relevant eingenvalue equations, and the corresponding eigenstates in a general form in the coordinate representation
My attempt: we can choose the complete set of commuting observables to be represented by the operators
 $\hat H$, $\hat L^2$, and $\hat L_z$.
We therefore require the eigenfunctions $\psi(r,\theta,\phi)$ to be simultaneous eigenfunctions of this C.S.C.O. Their eigenvalue equations would be given as follows:
$\hat H \psi (r)=E\psi (r)$
$\hat {L}^2 \psi (r)= \ell(\ell+1)\hbar^2 \psi (r)$
$\hat L_z \psi (r) = m\hbar\psi(r)$
Are the corresponding eigenstates given in the following form
$\psi(r)=R(r)Y^\ell_m(\theta,\phi)$ for fixed quantum numbers $\ell$  and $m$? 
Any insight or explanation regarding this question would be great thanks.
 A: You should write $\psi ( {\bf r})$ rather than $\psi ( r )$, that is the wave function is a function of the vector ${\bf r}$ rather than the distance $r$.
Because the problem is stated as being spherically symmetric, this means that the operator describing the Hamiltonian, $\hat H$, commutes with the operators $\hat L ^2$ and $\hat L_z$  (which also commute between themselves).
The eigenstates of $\hat L ^2$ and $\hat L_z$ are the spherical harmonics and are conventionally written as $Y_{l,m} (\theta,\phi)$.
It is a general observation that operators that commute with each other share common eigenstates.
Rather than repeating all the answers to other questions on this site, I give a link to one question with answers.
The only other comments to make are
(i) that the eigenstates of $\hat H$, can be written in the form $\psi (r)$, that is they depend upon the magnitude of the vector ${\bf r}$, $|{\bf r} | = r$.
(ii) the $\psi(r)$ will generally have some quantum number associated with them, say $n$, and should be written as $\psi_n (r)$.
Any eigenstate of the Hamiltonian is of the form $\psi ({\bf r}) = \psi_n(r) Y_{l,m} (\theta,\phi)$ and any allowed wave function is written as a linear combination of these, that is $$\sum_{n,l,m} a_{n,m,l} \psi_n(r) Y_{l,m} (\theta,\phi)$$
