Pauli equation in spaces of constant curvature and extended Nikiforov-Uvarov method

TL;DR

This paper investigates the application of the extended Nikiforov-Uvarov method to the non-relativistic Dirac equation with Coulomb potential in curved spaces, revealing limitations in its effectiveness for such quantum problems.

Contribution

It demonstrates the limited applicability of the extended Nikiforov-Uvarov method to curved space quantum equations, highlighting issues with polynomial solutions and spectrum calculations.

Findings

Energy spectrum matches Schrödinger equation results, minus geometric potential.

Necessary polynomial solutions conditions are not met, questioning method reliability.

Extended Nikiforov-Uvarov method has limited usefulness in curved space quantum mechanics.

Abstract

We apply the extended Nikiforov-Uvarov method to the non-relativistic limit of the Dirac equation with a Coulomb potential in spaces of constant curvature. In this case, the radial equation reduces to the Heun equation, and the extended Nikiforov-Uvarov method easily yields a quantization condition which leads to necessary condition under which the resulting Heun equation can have polynomial solutions. The energy spectrum implied by the quantization condition is virtually identical to the spectrum of a spinless particle obtained using the Schr\"{o}dinger equation, except for the absence of the ``geometric potential", confirming the non-commutativity of the naive non-relativistic limit with the ``squaring" of the Dirac equation, first discovered on curved surfaces. However, the necessary conditions for the existence of polynomial solutions cannot be met, and this fact undermines the reliability of the results obtained. This circumstance forces us to conclude that the extended Nikiforov-Uvarov method has limited, if any, value when considering similar problems in quantum mechanics.