Semi-relativistic wave-phase approximation for two-body spinless bound states in 1+1 dimensions

TL;DR

This paper develops an approximate two-body quantum equation for spinless particles in 1+1 dimensions that incorporates relativistic effects, providing a new method to analyze bound states with relativistic kinematics.

Contribution

It introduces a semi-relativistic wave-phase approximation for two-body spinless systems, derived from relativistic energy relations, and compares it to exact solutions for specific potentials.

Findings

The approximation captures relativistic effects in two-body bound states.

It yields a Bohr-Sommerfeld quantization condition.

Comparison shows good agreement with exact harmonic potential results.

Abstract

An approximate quantum-mechanical two-body equation for spinless particles incorporating relativistic kinematics is derived. The derivation is based on the relativistic energy-momentum relation $mc^{2}+\epsilon = \sqrt{m^{2}c^{4}+p^{2}c^{2}}+V$ for each single particle, where $mc^2$ is the particle rest mass energy, $p$ its linear momentum, $\epsilon$ its dynamical energy, and $V$ being the time-like vector interaction potential. The resulting two-body equation assumes rapid wave oscillations in a single, slowly varying potential well. A Bohr-Sommerfeld-type quantization condition is obtained. The approximation is compared to exact results for the harmonic potential.