Two-body bound states through Yukawa forces and perspectives on hydrogen and deuterium

Abstract

We present and discuss numerical solutions to a two-body quantum bound state problem closely related to that of the hydrogen atom and the deuterium nucleus. The forces binding the particles of our system are Yukawa forces, which fall off with distance faster than the Coulomb force and arise for non-relativistic particles whose interactions are mediated by massive scalar or vector particles; the Coulomb force arises in the limit of massless mediators such as photons. We use the solutions to explain several features of deuterium. We also present heuristic estimates of ground state energies and energy level degeneracy breaking. Studying this Yukawa two-body system provides insights into hydrogen, deuterium, and two-body bound states, generally. We describe our undergraduate-accessible procedure for numerically solving non-relativistic two-body bound state problems with mathematica in the appendix.