Dirac Equations with Confining Potentials

TL;DR

This paper investigates the relativistic bound states of Dirac particles with combined Coulomb and linear confining potentials, revealing that certain energy levels are preserved due to an accidental cancellation mechanism specific to particular states.

Contribution

It demonstrates that the preservation of specific bound-state energies is due to an accidental cancellation mechanism, limited to individual states, and explores the effects on anti-particle states.

Findings

Certain bound-state energies retain Dirac--Coulomb values despite confining potentials.

The cancellation mechanism is accidental and state-specific, not a general symmetry.

The mechanism does not influence anti-particle (negative-energy) states.

Abstract

This paper is devoted to a study of relativistic eigenstates of Dirac particles which are simultaneously bound by a static Coulomb potential and added linear confining potentials. It has recently been shown that, despite the addition of radially symmetric, linear confining potentials, some specific bound-state energies surprisingly retain their exact Dirac--Coulomb values (in the sense of an "exact symmetry"). This observation raises pertinent questions as to the generality of the cancellation mechanism. A Foldy-Wouthuysen transformation is used to find the relevant nonrelativistic physical degrees of freedom, which include additional spin-orbit couplings induced by the linear confining potentials. The matrix elements of the effective operators obtained from the scalar, and time-like confining potentials mutually cancel for specific ratios of the prefactors of the effective operators, which must be tailored to the cancellation mechanism. The result of the Foldy-Wouthuysen transformation is used to explicitly show that the cancellation is accidental and restricted (for a given Hamiltonian) to only one reference state, rather than traceable to a more general relationship among the obtained effective low-energy operators. Furthermore, we show that the cancellation mechanism does not affect anti-particle (negative-energy) states.