Dirac Equations with Confining Potentials

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.

11 pages; RevTeX

References in corpus (4)

• Higher-order binding corrections to the Lamb shift of 2P states
• Foldy-Wouthuysen Transformation, Scalar Potentials and Gravity
• A simple Dirac wave function for a Coulomb potential with linear confinement
• Exact Solutions of the Dirac Equation for Modified Coulombic Potentials

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