Second-order stationary solutions for fermions in an external Coulomb field

TL;DR

This paper investigates second-order equations for fermions in Coulomb fields, revealing spectral similarities to Dirac equations and predicting an impermeable barrier affecting electron and positron behavior.

Contribution

It introduces a second-order spinor equation framework for fermions in Coulomb fields, showing spectral equivalence to Dirac solutions and identifying potential barriers impacting particle confinement.

Findings

Spectral spectrum matches Dirac equation for attractive Coulomb fields.

An impermeable barrier exists in repulsive Coulomb fields, depending on electron energy.

Barrier effects do not contradict experimental electron scattering results.

Abstract

We have studied self-conjugate second-order equations with spinor wavefunctions for fermions moving in an external Coulomb field. For stationary states, the equations are characterized by separated states with positive and negative energies, which render probabilistic interpretation possible. For the Coulomb field of attraction, the energy spectrum of the second-order equation coincides with the spectrum of the Dirac equation, while the probability densities of states are slightly different. For a Coulomb field of repulsion, there exists an impermeable potential barrier with radius depending on the classical electron radius and on the electron energy. The existence of the impermeable barrier does not contradict the results of experiment for determining the inner electron structure and does not affect (in the lowest order of perturbation theory) the Coulomb electron scattering cross section. The existence of the impermeable barrier can lead to positron confinement in supercritical nuclei with $Z \geq 170$ in case of realization of spontaneous emission of vacuum electron-positron pairs.