Quantum central force problem and variable separations

Question

When dealing with a Hamiltonian of the type:

$$H=\frac{p^2}{2m}+V(r)$$

I have a big problem in understanding why we factor the total eigenfunction of the Hamiltonian into a radial and an angular term.

I think I understand that since $H, L^2$ and $L_z$ are three operators which, in this context, commute, it is convenient to look for the simultaneous eigenfunctions of the three operators. Therefore, when we write the TISE

$$H\psi_n(r,\theta,\phi)=E\psi_n(r,\theta,\phi)$$

the $\psi_n(r,\theta,\phi)$ refers to the simultaneous eigenfunctions of the three operators.

The main problem is why we write

$\psi_n(r,\theta,\phi)=f(R)F(\theta,\phi)$.

Is this related to the fact that the Hamiltonian can be separated into a radial term $H_r$ and an angular term $H_{\theta}$ and so we apply the standard resolution for a differential equation of this type, namely: we introduce a "test" function which consists of the product of single function each of one depends only on one variable: $T(x,y)=X(x)Y(y)$ and then we divide the whole differential equations by $T(x,y)$ and we obtain two differents ordinary differential equation?

For example in this discussion Quantum Central Force Problem and Angular Momentum in the first answer in the "Factorization of eigenvalues" paragraph, I read

its obvious, this equation does not depend anymore on θ or ϕ and therefore the solution must simplify as: $\psi_{nlm}(r,\theta,\phi)=u_{nl}(r)Y_{lm}(\theta,\phi)$

I ask you: why the solution must simplify in that way? Is that answer related to mine?

If possible I would like an explanation that uses only time-independent Schrodinger equations and is not too difficult because I am on my first quantum mechanics course and am not very knowledgeable.

Answer 1

One common way of solving a partial differential equation is by separation of variables. With a bit of experience it’s clear that, for potentials of the type $V(r)$, the angular part will not depend on $V$, and only the radial part will depend on $V$.

Separation of variables then allows you to transform the PDE into a set of ODEs linked by the separation constants.

It is not the only way to proceed but its advantages are numerous since you get out of that a complete set of functions, so that any solution can be written as a sum of products of separated functions. Also it simplifies the search for solution since you only need to “solve again” the radial part : the radial ODE is different for every potential but the angular part remains the same for all $V(r)$ so you only need to do that angular part once.