# Electrostatic potential of a proton

I have been working on a quantum mechanical problem regarding the ground state of the Hydrogen atom. It appears that the best way to solve the underlying problem is to modify the electrostatic potential that is generated by the proton and felt by the electron. The modification is only required for very small distances r. While the modified potential is not precisely defined, it should have the following general structure:

$V(r) = A - Br^2 + Cr^4 - Dr^6 + \dots \quad \text{for} \quad r < R$

$V(r) = 1/r \quad \text{for} \quad r > R$

Here $R$ is some unspecified distance, much smaller than the Bohr radius ($10^{-10} \mathrm{m}$). I suppose $R$ may well related to the proton size. For very small separation between proton and electron the potential diminishes approximately like a Cosine function, a Bessel function [i.e. $\sin(x)/x$], or a Gaussian. For distances $r$ larger than $R$ the usual Coulomb potential of a point charge can be used. The transition between the two regions ar $r=R$ is smooth.

My question to the experts is: Is there any evidence or confirmation, theoretical or experimental, for the result described above?

• Can you clarify what it is you're trying to solve? The Hydrogen atom has a well-known potential and doesn't require modification; the principal term is the coulomb potential, but there are also fine and hyperfine corrections arising from relativity. – chase Feb 11 '14 at 23:12
• The charge distribution of the proton can be derived from the form-factors or parton distribution functions. However, these quantities depend on the interaction energy and are measured at much high energies than you are interested in. There are good theoretical reasons motivating the usual projections to $Q^2 = 0$, and I think the shift of Hydrogen-atom quantities has been verified to at least a modest degree. You could approach this from either the nuclear or atomic side, and I'd start by looking in advanced texts from either discipline. – dmckee Feb 11 '14 at 23:49
• @chase - Sure! The problem I have been working on is the computation of the higher moments of the kinetic energy operator T. It turns out that these moments appear undefined, since T^n acting on Psi = exp(-r) is not self-adjoint. I found out, that this is because of the non-analytic behaviour [cusp] of Psi around r=0. By adjusting Psi slightly, self-adjointness was obtained and I obtained the moments uniquely. The adjustment of Psi around r=0 can be attributed to (or interpreted as) an adjustment of the electrostatic potential for small r. – M. Wind Feb 12 '14 at 0:15