Transformation of the spinless Salpeter equation

TL;DR

This paper derives a Schrödinger-like equation for the relativistic spinless Salpeter system, analyzing bound states with free particle motion assumptions, and tests the model on hydrogen atom energy levels.

Contribution

It introduces a new relativistic invariant approach to the spinless Salpeter equation and provides multiple equivalent eigenvalue formulations.

Findings

Calculated hydrogen atom energy levels match NIST data

Obtained complex eigenmasses for the hydrogen atom

Developed a free particle hypothesis for bound states

Abstract

Spinless Salpeter equation for two bound particles is analyzed. We use the fact that in relativistic kinematics the spatial two particle relative momentum is relativistic invariant. Free particle hypothesis for the bound state is developed: comstituents move as free particles inside of the system. The Shr\"odinger-type wave equation is derived. Three equivalent forms of the eigenvalue equation are given. Relative motion of quarks in eigen states is described by the asymptotic solution in the form of the standing wave of $\cos(kx+a)$ for each spatial degree of freedom. To test the model the spin center-of-gravity energy levels for the hydrogen atom are calculated and compared with the NIST data. Complex eigenmasses for the $H$ atom are obtained.