Relativistic Coulomb Problem for Z Larger Than 137

Abstract

We propose a relativistic one-parameter Hermitian theory for the Coulomb problem with an electric charge greater than 137. In the non-relativistic limit, the theory becomes identical to the Schrödinger-Coulomb problem for all Z. Moreover, it agrees with the Dirac-Coulomb problem to order, where α is the fine structure constant. The vacuum in the theory is stable and does not suffer from the "charged vacuum" problem for all Z. Moreover, transition between positive and negative energy states could be eliminated. The relativistic bound states energy spectrum and corresponding spinor wavefunctions are obtained.

Klein-Gordon Equation with Coulomb Potential in the Presence of a Minimal Length

“…Another alternative approach was discussed in Ref. [14] to deal with the strong attractive Couomb potential in the context of Dirac equation to overcome the dependence of the problem on the numbre Z.…”

“…It is important to mention that solution (14) reduces to an hypergeometric function in the particular case β = β ′ as follow:…”

Solution of Effective-Mass Dirac Equation with Scalar-Vector and Pseudoscalar Terms for Generalized Hulthén Potential

“…In the present work, we extend the search including the solutions of the Dirac equation having a pseudoscalar interaction term, as in Refs. [33][34][35][36], for the q-parameter…”

“…In the present work, we extend the search including the solutions of the Dirac equation having a pseudoscalar interaction term, as in Refs. [33][34][35][36], for the q-parameter Hulthén potential within the position-dependent mass (PDM) formalism. This formalism gives an opportunity such as writing the analytical results for the case where the mass is constant.…”

The Dirac–Coulomb problem: a mathematical revisit

“…Using the Dirac equation, which accounts for the relativistic effects, it was shown that the wavefunction of the electron for atoms with 137 Z  is oscillatory and unbounded. The explanation of this mistery is a subject of interest to the physics community and one of the authors recently made a contribution in this regard [3].…”