Relativistic Coulomb problem for Z larger than 137

TL;DR

This paper introduces a new relativistic Hermitian theory for Coulomb problems with charges exceeding 137, resolving issues like vacuum instability and providing accurate energy spectra and wavefunctions.

Contribution

It presents a novel relativistic Hermitian framework for high-Z Coulomb problems that remains stable and consistent with known physics.

Findings

Vacuum stability for all Z values.

Accurate relativistic bound state energies and wavefunctions.

Elimination of positive-negative energy state transitions.

Abstract

We propose a relativistic one-parameter Hermitian theory for the Coulomb problem with an electric charge greater than 137. In the non-relativistic limit, the theory becomes identical to the Schr\"odinger-Coulomb problem for all Z. Moreover, it agrees with the Dirac-Coulomb problem to order (\alph Z)^2, where \alpha is the fine structure constant. The vacuum in the theory is stable and does not suffer from the "charged vacuum" problem for all Z. Moreover, transition between positive and negative energy states could be eliminated. The relativistic bound states energy spectrum and corresponding spinor wavefunctions are obtained.