Bounded solutions of neutral fermions with a screened Coulomb potential

TL;DR

This paper finds exact bound solutions for a relativistic fermion in a screened Coulomb potential with a background, revealing how the background influences bound states and addressing an uncertainty principle paradox.

Contribution

It introduces a mapping of the relativistic problem into a Sturm-Liouville form, leading to exact solutions and novel insights into the role of background potentials.

Findings

Number of bound states depends on the background potential

Eigenenergy behavior analyzed in detail

Resolution of an uncertainty principle paradox

Abstract

The intrinsically relativistic problem of a fermion subject to a pseudoscalar screened Coulomb plus a uniform background potential in two-dimensional space-time is mapped into a Sturm-Liouville. This mapping gives rise to an effective Morse-like potential and exact bounded solutions are found. It is shown that the uniform background potential determinates the number of bound-state solutions. The behaviour of the eigenenergies as well as of the upper and lower components of the Dirac spinor corresponding to bounded solutions is discussed in detail and some unusual results are revealed. An apparent paradox concerning the uncertainty principle is solved by recurring to the concepts of effective mass and effective Compton wavelength.