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, modeled as a two-body system consisting of a point electron and a point proton, changes when the usual Coulomb interaction between point particles is replaced with an interaction which results from a modification of Maxwell’s law of the electromagnetic vacuum. Empirical spectral data thereby impose bounds on the theoretical parameters involved in such modified vacuum laws. In the present paper the vacuum law proposed, in the 1940s, by Bopp, Landé–Thomas, and Podolsky (BLTP) is scrutinized in such a manner. The BLTP theory hypothesizes the existence of an electromagnetic length scale of nature — the Bopp length [Formula: see text] —, to the effect that the electrostatic pair interaction deviates significantly from Coulomb’s law only for distances much shorter than [Formula: see text]. Rigorous lower and upper bounds are constructed for the Schrödinger energy levels of the hydrogen atom, [Formula: see text], for all [Formula: see text] and [Formula: see text]. The energy levels [Formula: see text], [Formula: see text], and [Formula: see text] are also computed numerically and plotted versus [Formula: see text]. It is found that the BLTP theory predicts a nonrelativistic correction to the splitting of the Lyman-[Formula: see text] line in addition to its well-known relativistic fine-structure splitting. Under the assumption that this splitting does not go away in a relativistic calculation, it is argued that present-day precision measurements of the Lyman-[Formula: see text] line suggest that [Formula: see text] must be smaller than [Formula: see text]. 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 mass ([Formula: see text]) 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 and its electrostatic field energy, then in BLTP electrodynamics the electron has to be assigned a negative bare rest mass.