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 ϰ−1 —, to the effect that the electrostatic pair interaction deviates significantly from Coulomb’s law only for distances much shorter than ϰ−1. Rigorous lower and upper bounds are constructed for the Schrödinger energy levels of the hydrogen atom, Eℓ,n(ϰ), for all ℓ∈{0,1,2,…} and n>ℓ. The energy levels E0,1(ϰ), E0,2(ϰ), and E1,2(ϰ) are also computed numerically and plotted versus ϰ−1. It is found that the BLTP theory predicts a nonrelativistic correction to the splitting of the Lyman-α 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-α line suggest that ϰ−1 must be smaller than ≈10−18m. 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 (×c2) 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.
| You currently do not have access to the full text article. |
|---|
