# A Fully Relativistic Treatment of Confined Hydrogen-Like Atoms

## TL;DR

This paper presents a relativistic calculation of the energy levels of a hydrogen-like atom confined within a spherical barrier using the Dirac equation, revealing how confinement affects energy states.

## Contribution

It introduces a fully relativistic approach to model confined hydrogen-like atoms with a penetrable spherical barrier, extending understanding of energy state transfer as confinement diminishes.

## Key findings

- Derived binding energies for various barrier heights and radii.
- Showed energy states transition to free atom states without quantum number n.
- Demonstrated the relativistic treatment's consistency with known free atom energies.

## Abstract

The Dirac equation is used to provide a relativistic calculation of the binding energy of a hydrogen-like atom confined within a penetrable spherical barrier. We take the potential to be Coulombic within the barrier and constant outside the barrier. Binding energies are derived for the ground state of hydrogen for various barrier heights and confining radii. In addition, it is shown that without the introduction of the principle quantum number $n$, all energy states of the confined relativistic hydrogen atom, determined by a single quantum number $k$, transfer into the known energy states of the free relativistic hydrogen atom as the radius of confinement becomes large.