Elucidating Fermi's golden rule via bound-to-bound transitions in a confined hydrogen atom

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.