Proton-Size Resolution of the Hyperfine Puzzle in Hydrogen

TL;DR

The paper addresses the apparent divergence in the hydrogen hyperfine energy under a variational treatment, where E_hf,0 ∝ -1/R^3 would drive collapse as R→0. It incorporates finite-proton-size effects by modeling the proton magnetic density with a dipole form factor G_D(q^2) = 1/(1+q^2/Λ^2)^2 (Λ = 0.843 GeV), and computes the resulting E_hf(R) and total energy E_1(R) = E_0(R) + E_hf(R) with E_0(R) = 1/(2 m R^2) - α/R. The results show that the hyperfine contribution remains finite and the energy minimum stays at R = a_0, with a tiny, negligible shift (∼6×10^{-6} a_0), and that the outcome is robust to the details of ρ(r). This proton-size regularization resolves the hyperfine puzzle, confirming the Bohr-scale ground state as the variational minimum once proton structure is accounted for.

Abstract

Baym and Farrar (arXiv:2601.02300v1) have recently pointed out a puzzle in understanding the role of the hyperfine interaction in the ground state of a hydrogen atom. If one uses a variational wave function in which the Bohr radius, $a_0$ is replaced by a variational radius parameter, $R$, first-order perturbation theory can give a contribution to the energy proportional to $-1/R^3$. This raises the question of why the hyperfine interaction does not lead to collapse of hydrogen. I show that including the effects of the non-zero size of the proton leads to a resolution of the puzzle such that the variational procedure yields a value of $R$ that is indistinguishable from $a_0$.

Figures (1)

• Figure 1: $\epsilon_0(x)$ solid, $\epsilon_1(x)$ dashed. The curves overlap.