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?