# The hydrogen atom as relativistic bound system

## TL;DR

This paper models the hydrogen atom as a relativistic bound system using a complex-mass scheme, deriving a wave equation and calculating energy levels that incorporate proton structure and relativistic effects.

## Contribution

It introduces a novel relativistic wave equation for two interacting spinless particles with a position-dependent mass, accounting for proton structure in the hydrogen atom.

## Key findings

- Eigenmasses of the hydrogen atom are obtained as complex values.
- Calculated energy levels agree with known relativistic equations and experimental data.
- The approach provides a new perspective on relativistic bound systems.

## Abstract

The hydrogen atom as relativistic bound-state system of a proton and an electron in the complex-mass scheme is investigated. Interaction of a proton and an electron in the atom is described by the Lorentz-scalar Coulomb potential; the proton structure is taken into account. The concept of position dependent particle mass is developed. Relativistic wave equation for two interacting spinless particles is derived; asymptotic method is used to solve the equation. % Asymptotic solution of the equation for the system in the form of %standing wave and eigenmasses of the $H$ atom are obtained. Complex eigenmasses for the $H$ atom are obtained. The spin center-of-gravity energy levels for the $H$ atom are calculated and compared with ones obtained from solution of some known relativistic wave equations % the Shr\"odinger, Klein-Gordon and tabulated NIST data.