The hydrogen atom perturbed by a 1-dimensional Simple Harmonic Oscillator (1d-SHO) potential

TL;DR

This paper analyzes the hydrogen atom under a weak 1D quadratic potential using first-order perturbation theory, providing insights relevant to quadratic Zeeman effects and van der Waals interactions.

Contribution

It introduces a first-order perturbation solution for the hydrogen atom with a 1D quadratic potential using the coupled basis eigenstates.

Findings

Derived the perturbed energy levels of the hydrogen atom

Applicable to weak quadratic Zeeman and van der Waals effects

Provides a framework for further perturbative studies

Abstract

The hydrogen atom perturbed by a constant 1-dimensional weak quadratic potential $\lambda z^2$ is solved at first-order perturbation theory using the eigenstates of the total angular momentum operator - the coupled basis. Physical applications of this result could be found, for example, in the study of a quadratic Zeeman effect weaker than fine-structure effects, or in a perturbation caused by instantaneous generalised van der Waals interactions.