Bound states of bosons and fermions in a mixed vector-scalar coupling with unequal shapes for the potentials

TL;DR

This paper explores bound states of bosons and fermions in relativistic quantum equations with specific vector and scalar potentials, revealing differences in localization and the importance of the effective Compton wavelength.

Contribution

It introduces a general condition for Klein-Gordon and Dirac equations with hyperbolic potentials, providing solutions and analyzing physical properties like localization and eigenvalues.

Findings

Bosons are better localized than fermions with the same mass.

Eigenvalues and eigenfunctions are explicitly derived.

The effective Compton wavelength is a key physical quantity.

Abstract

The Klein-Gordon and the Dirac equations with vector and scalar potentials are investigated under a more general condition, $V_{v}+V_{s}= \mathrm{constant}$. These intrinsically relativistic and isospectral problems are solved in a case of squared hyperbolic potential functions and bound states for either particles or antiparticles are found. The eigenvalues and eigenfuntions are discussed in some detail and the effective Compton wavelength is revealed to be an important physical quantity. It is revealed that a boson is better localized than a fermion when they have the same mass and are subjected to the same potentials.