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

Abstract The hydrogen atom perturbed by a constant 1-dimensional weak quadratic potential <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:msup> <mml:mi>z</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> 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.