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

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 $m{c}^2+ϵ=\sqrt{m^2{c}^4+p^2{c}^2}+V$ for each single particle, where $m{c}^2$ is the particle rest mass energy, $p$ its linear momentum, $ϵ$ 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.

Comments: 7 pages