Any Rigorous Approach to Hydrogen Atom? According to the book by David Griffiths on Quantum Mechanics, while solving the Schrodinger equation for the electron of Hydrogen atom, the Potential Function appearing in the Schrodinger equation is expressed as 
$$V=-\dfrac{e^2}{4\pi\epsilon_0r}.$$
I think that this is inappropriate as we don't know apriori that the Potential function appearing in Schrodinger equation will be determined by the Coulomb's classical formula. Is there any more rigorous method using which we can determine the Potential function appearing in the Schrodinger equation of the electron in Hydrogen atom within non-relativistic Quantum Mechanics? 
 A: In a comment you noted the relevance of fitting experiments, I'll expand on that a little. We expect the quantum-mechanical potential to be "the same as" the classical one (albeit promoted to an operator of course), otherwise the quantum model can't recover the classical model, nor its theoretical motivation or pre-quantum empirical successes.
For example, not only does the empirically successful $r^{-2}$ force follow from Gauss's law, but it's also one of only two central-force choices that leads to stable elliptical orbits classically. (The other choice has some applicability to covalent bonds.) Obviously electrons aren't "really" orbiting on the quantum view, but a quantum equivalent of Bertrand's theorem constrains the potential for electrons that stay close to the nucleus indefinitely without spiraling in or away.
There is also some post-quantum evidence justifying an $r^{-1}$ potential. For example, an $r^{-n}$ potential's value of $n$ can be computed from the eigenenergies using the virial theorem.
A: The effective nonrelativistic, "classical" potential can be derived from Quantum Electrodynamics by Fourier transforming the renormalized photon propagator, as explained in Tony Zee's book "Quantum Field Theory in a Nutshell," section I.5.
