The Klein-Gordon equation with relativistic mass: a relativistic Schr\"odinger equation

TL;DR

This paper introduces a modified Klein-Gordon equation using relativistic mass, which overcomes previous limitations by ensuring positive probability density and aligning with the Schrödinger equation in the non-relativistic limit, thus acting as a relativistic Schrödinger equation.

Contribution

It proposes replacing rest mass with relativistic mass in the Klein-Gordon equation to retain relativistic features while ensuring positive probability density and reducing to Schrödinger equation non-relativistically.

Findings

Probability density becomes positive definite.

Equation reduces to Schrödinger in non-relativistic limit.

Loss of Lorentz invariance due to relativistic mass substitution.

Abstract

The Klein-Gordon equation describes the wave-like behavior of spinless particles since it is Lorentz invariant. While it seemed initially ripe for explaining the electronic structure of the hydrogen atom, the lack of a unconditional positive probability density really limited its applications. Yet, it is intimately connected with fermions. Any solution to the Dirac equation is automatically a solution to the Klein-Gordon equation. What is even more surprising, the Klein-Gordon equation for a free particle turns into the Schr\"odinger equation in the non-relativistic limit. In this work we show that these problems disappear when we use the relativistic mass instead of the rest mass. While the Klein-Gordon equation losses its Lorentz invariance because of this transformation, it gains most of the features present the Schr\"odinger equation, including the unconditional positivity of probability density, while keeping most of its relativistic characteristics intact, including the matter-wave dispersion relation. What is even more surprising, the non-relativistic, quasi-static limit of the Klein-Gordon equation with relativistic mass is simply the Schr\"odinger equation under all possible conditions. So, it can be argued that this Klein-Gordon equation is a sort of relativistic Schr\"odinger equation.