Two fermion relativistic bound states

TL;DR

This paper develops a covariant relativistic quantum framework for two interacting fermions, solves the eigenvalue problem numerically for Coulomb interaction, and explores spectral properties across different angular momenta and mass ratios.

Contribution

It introduces a covariant formulation and numerical solutions for relativistic two-fermion bound states with Coulomb interaction, extending understanding of their spectral characteristics.

Findings

Numerical solutions for J=0,1,2 states with Coulomb interaction.

Spectral curves as a function of mass ratio.

Extended analysis of ground states of positronium variants.

Abstract

We consider the relativistic quantum mechanics of a two interacting fermions system. We first present a covariant formulation of the kinematics of the problem and give a short outline of the classical results. We then quantize the system with a general interaction potential and deduce the explicit equations in a spherical basis. The case of the Coulomb interaction is studied in detail by numerical methods, solving the eigenvalue problem for J=0, J=1, J=2 and determining the spectral curves for a varying ratio of the mass of the interacting particles. Details of the computations, together with a perturbative approach in the mass ratio and an extended description of the ground states of the Para- and Orthopositronium are given in Appendix.