Relativistic correction to the Bohr radius and electron distance expectation value via Dirac Equation

Abstract

In this article and beginning with the Dirac solution to the Hydrogen atom in its ground state, the exact results corresponding to the expectation value of the distance of the electron to the proton and the maximum probability distance are found. For hydrogen-like atoms and in contrast to the non relativistic Bohr radius and expectation values, for $Z=\frac{\sqrt3}{2}\frac{1}{\alpha}$ (close to 118), the expectation distance to the nucleus is zero thus putting an end to the periodic table for Z = 118 nicely matching the last element discovered (Oganesson).