New exact solution of Dirac-Coulomb equation with exact boundary condition

TL;DR

This paper presents a new exact solution to the Dirac-Coulomb equation by incorporating a realistic boundary condition considering the finite size of the atomic nucleus, resolving divergences and providing a novel energy level formula.

Contribution

The authors introduce an exact boundary condition based on the finite nuclear radius and derive a new solution to the Dirac-Coulomb equation that avoids divergences and virtual energies.

Findings

Wave functions are finite at the nucleus with the new boundary condition.

Derived energy levels differ from the traditional Dirac formula.

K value relates to the nuclear radius and is independent of energy.

Abstract

It usually writes the boundary condition of the wave equation in the Coulomb field as a rough form without considering the size of the atomic nucleus. The rough expression brings on that the solutions of the Klein-Gordon equation and the Dirac equation with the Coulomb potential are divergent at the origin of the coordinates, also the virtual energies, when the nuclear charges number Z > 137, meaning the original solutions do not satisfy the conditions for determining solution. Any divergences of the wave functions also imply that the probability density of the meson or the electron would rapidly increase when they are closing to the atomic nucleus. What it predicts is not a truth that the atom in ground state would rapidly collapse to the neutron-like. We consider that the atomic nucleus has definite radius and write the exact boundary condition for the hydrogen and hydrogen-like atom, then newly solve the radial Dirac-Coulomb equation and obtain a new exact solution without any mathematical and physical difficulties. Unexpectedly, the K value constructed by Dirac is naturally written in the barrier width or the equivalent radius of the atomic nucleus in solving the Dirac equation with the exact boundary condition, and it is independent of the quantum energy. Without any divergent wave function and the virtual energies, we obtain a new formula of the energy levels that is different from the Dirac formula of the energy levels in the Coulomb field.