On the Schrödinger spectrum of a hydrogen atom with electrostatic Bopp-Landé-Thomas-Podolsky interaction between electron and proton

Abstract

The Schrödinger spectrum of a hydrogen atom, modelled as a two-body system consisting of a point electron and a point proton, interacting with a modification of Coulomb's law proposed, in the 1940s, by Bopp, Landé--Thomas, and Podolsky (BLTP). The BLTP theory hypothesizes the existence of an electromagnetic length scale of nature --- the Bopp length ${\varkappa }^{-1}$ ---, to the effect that the electrostatic pair interaction deviates significantly from Coulomb's law only for distances much shorter than ${\varkappa }^{-1}$. Rigorous lower and upper bounds are constructed for the Schrödinger energy levels of the hydrogen atom, $E_{\ell ,n}(\varkappa )$, for all $\ell \in \{0,1,2,...\}$ and $n>\ell$. The energy levels $E_{0,1}(\varkappa )$, $E_{0,2}(\varkappa )$, and $E_{1,2}(\varkappa )$ are also computed numerically and plotted versus ${\varkappa }^{-1}$. It is found that the BLTP theory predicts a non-relativistic correction to the splitting of the Lyman-$\alpha$ line in addition to its well-known relativistic fine-structure splitting. Under the assumption, that this splitting doesn't go away in a relativistic calculation, it is argued that present-day precision measurements of the Lyman-$\alpha$ line suggest that ${\varkappa }^{-1}$ must be smaller than $\approx 10^{-18}\mathrm{m}$. Finite proton size effects are found not to modify this conclusion. As a consequence, the electrostatic field energy of an elementary point charge, although finite in BLTP electrodynamics, is much larger than the empirical rest energy of an electron. If, as assumed in all `renormalized theories' of the electron, the empirical rest mass of a physical electron is the sum of its bare rest mass plus its electrostatic field energy ($/c^2$), then in BLTP electrodynamics the electron has to be assigned a negative bare rest mass.

revised version, 23 pages, 9 figures. To appear in Int. J. Modern Phys. A