# 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?

• From what assumptions do you want to "derive" that? The Hamiltonians/Lagrangians of physics are always essentially guessed, both in the classical and in the quantum case. – ACuriousMind Feb 3 '17 at 17:35
• That said, renormalizable Lagrangians in QFT are highly constrained. QFT is still far from rigorous though. – TotallyRhombus Feb 3 '17 at 17:41
• So, we guess the Hamiltonian and then check if the predictions regarding the observables match the experiment. In this case, the guess that it might be the classical potential itself works well when tested experimentally. Am I correct? – Dvij Mankad Feb 3 '17 at 17:43
• There are more rigorous methods for Hydrogen, but they go beyond the Schroedinger equation: they take relativity into account and solve the Dirac equation for this atom producing vastly more accurate results. Improving on the Coulomb potential is a blind alley, in this context. – Cosmas Zachos Feb 3 '17 at 20:36

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.

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.

• I am aware of a book by A. Zee. Can you specify the exact title? – Dvij Mankad Feb 3 '17 at 18:14
• @Electrodynamist Done – tparker Feb 3 '17 at 18:17
• Isn't this the book by A. Zee? – Dvij Mankad Feb 3 '17 at 18:24
• @Electrodynamist "A." and "Tony" are two different abbreviations for his first name "Anthony." – tparker Feb 3 '17 at 19:35