Elucidating Fermi's Golden Rule via bound-to-bound transitions in a confined hydrogen atom

TL;DR

This paper introduces a method to calculate bound-to-continuum cross-sections in confined quantum systems, demonstrating convergence with analytical solutions and revealing how confinement affects photoionization behavior.

Contribution

It presents a novel approach for computing bound-to-continuum transitions in confined atoms, clarifying Fermi's Golden Rule implementation and analyzing confinement effects on photoionization.

Findings

Convergence between the new method and analytical solutions for large confinement volumes.

Identification of scaling laws for photoionization cross-sections across different atomic states.

Discussion of how confinement radius influences physical behavior and transition dynamics.

Abstract

We demonstrate an effective method for calculating bound-to-continuum cross-sections by examining transitions to bound states above the ionization energy that result from placing the system of interest within an infinite spherical well. Using photoionization of the hydrogen atom as an example, we demonstrate convergence between this approach for a large volume of confinement and an exact analytical alternate approach that uses energy-normalized continuum wavefunctions, which helps to elucidate the implementation of Fermi's Golden Rule. As the radius of confinement varies, the resulting changes in physical behavior of the system are presented and discussed. The photoionization cross-sections from a variety of atomic states with principal quantum number $n$ are seen to obey particular scaling laws.