# Hydrogen in a cavity

## TL;DR

This paper investigates the quantum behavior of a hydrogen atom confined within a spherical cavity, emphasizing the importance of boundary conditions on its energy spectrum using advanced numerical methods.

## Contribution

It derives the differential equation for a hydrogen atom in a cavity and applies the Gaussian Expansion Method for accurate numerical solutions, highlighting boundary condition effects.

## Key findings

- Boundary conditions significantly affect the energy spectrum.
- The Gaussian Expansion Method provides efficient numerical results.
- Correct boundary implementation is crucial for small cavities.

## Abstract

The system of a proton and an electron in an inert and impenetrable spherical cavity is studied by solving Schr\"{o}dinger equation with the correct boundary conditions. The differential equation of a hydrogen atom in a cavity is derived. The numerical results are obtained with the help a power and efficient few-body method, Gaussian Expansion Method. The results show that the correct implantation of the boundary condition is crucial for the energy spectrum of hydrogen in a small cavity.