The Dirac and Klein-Gordon equations with equal scalar and vector potentials

摘要

We study the three-dimensional Dirac and Klein-Gordon equations with scalar and vector potentials of equal amplitudes as an attempt to present a critique for the interpretation of this class of problems which has recently been accumulating interest. We consider a large class of these problems in which the potentials are noncentral (angular-dependent) such that the equations separate completely in spherical coordinates. The relativistic energy spectra are obtained and compared to well-established results. Consequently, we prove (by examples) that solutions of such problems do not constitute the correct relativistic extension of the given potentials despite the fact that the non-relativistic limit is correct. The Coulomb, Oscillator and Hartmann potentials are considered. This shows that although the nonrelativistic limit is well-defined and unique, the relativistic extension is not. Additionally, we investigate the Klein-Gordon equation with uneven mix of potentials leading to the correct relativistic extension. We consider the case of spherically symmetric exponential-type potentials resulting in the s-wave Klein-Gordon-Morse problem.