Why is it that in separable Hamiltonian problems the total eigenfunction is equal to the product of the individual eigenfunctions, but the individual Hamiltonians must commute?
In mathematics, when the method of separating variables is used, for example for some PDEs, it is assumed that the total solution is product of functions of the individual variables.
But why, in quantum mechanics, must the individual Hamiltonians commute with each other in order to use this method of resolution?
I refer for example to the case of the hydrogen atom: after the change of variables, one can write the total Hamiltonian as the sum of two Hamiltonians that commute with each other and the eigenfunction of $H$ is the product of the individual eigenfunctions.
In this case (relative coordinates hamiltonian of a 3D hydrogen atom in spherical coordinates) my book only states: "Since a central hamiltonian commutes with $L^2$ and $L_z$, we can write the solutions of TISE as: $\psi(r,\theta,\phi)=F(\theta,\phi)R(r)."$