Energy-level structure of the hydrogen atom confined by a penetrable cylindrical cavity

The bound-state energy spectrum and its evolution for a hydrogen atom located along the axis of a standard cylindrical confining cavity with either impenetrable or penetrable confining boundaries are obtained by solving the stationary Schrodinger equation using a finite differences approach. New results are first presented for a nuclear-centered position for the penetrable case as the barrier height and cavity size change. Special attention is then given to the energy-level dependence on the nuclear position along the cylinder axis, where proper identification for the evolution of states from the nuclear-centered position (centered states) up to the cylinder cap (cap states) is proposed, while the corresponding state evolution for intermediate nuclear positions (intermediate states) remains consistent with node conservation and symmetry. It is found that in general the energy levels evolve with an increasing value as the nuclear position is shifted from the central position up to a cylinder cap. As the barrier height (and cavity size) are reduced, a progressive extinction of bound states appears in the order cap states, intermediate states and centered states. Finally, a predominance of cavity-size over barrier-height effects on the energy level shift is found.