Effects of a mixed vector-scalar kink-like potential for spinless particles in two-dimensional spacetime

TL;DR

This paper investigates the relativistic quantum behavior of spinless particles in two dimensions under mixed vector-scalar kink-like potentials, providing exact solutions and analyzing spectral properties and paradoxes related to the uncertainty principle.

Contribution

It introduces an exact solution approach for spinless particles in mixed potentials by mapping to the Rosen-Morse problem, revealing spectral characteristics and resolving a related paradox.

Findings

Exact bounded solutions for particles and antiparticles are obtained.

The spectrum behavior is analyzed in detail.

A paradox related to the uncertainty principle is addressed and resolved.

Abstract

The intrinsically relativistic problem of spinless particles subject to a general mixing of vector and scalar kink-like potentials ($\sim \mathrm{tanh} ,\gamma x$) is investigated. The problem is mapped into the exactly solvable Surm-Liouville problem with the Rosen-Morse potential and exact bounded solutions for particles and antiparticles are found. The behaviour of the spectrum is discussed in some detail. An apparent paradox concerning the uncertainty principle is solved by recurring to the concept of effective Compton wavelength.