# Exact solution of the Schr\"{o}dinger equation for an hydrogen atom at the interface between the vacuum and a topologically insulating surface

## TL;DR

This paper provides an exact analytical solution to the Schrödinger equation for a hydrogen atom at a topologically insulating surface, revealing how electromagnetic boundary effects influence atomic energy levels.

## Contribution

It presents the first exact solution of the hydrogen atom's Schrödinger equation at a topological insulator interface, including energy shifts due to axionic electromagnetic effects.

## Key findings

- Exact solutions for energy levels and wave functions obtained.
- Perturbation theory results agree with exact solutions for small θ.
- High θ-values used to highlight topological effects.

## Abstract

When an hydrogen atom is brought near to the interface between $\theta$-media, the quantum-mechanical motion of the electron will be affected by the electromagnetic interaction between the atomic charges and the $\theta$-interface, which is described by an axionic extension of Maxwell electrodynamics in the presence of a boundary. In this paper we investigate the atom-surface interaction effects upon the energy levels and wave functions of an hydrogen atom placed at the interface between a $\theta$-medium and the vacuum. In the approximation considered, the Schr\"{o}dinger equation can be exactly solved by separation of variables in terms of hypergeometic functions for the angular part and hydrogenic functions for the radial part. In order to make such effects apparent we deal with unrealistic high values of the $\theta$-parameter. We also compute the energy shifts using perturbation theory for a particular small value of $\theta$ and we demonstrate that they are in a very good agreement with the ones obtained from the exact solution.