Classical and quantum space splitting: the one-dimensional hydrogen atom

Abstract

The one-dimensional (1D) hydrogen atom with potential energy V(q) = −e2/|q|, with e the electron charge and q its position coordinate, has been a source of discussion and controversy for more than 55 years. A number of incorrect claims have been made about its spectrum; for example, that its ground state has infinite binding energy, that bound states associated with a continuum of negative energy values exist, or that anomalous non-Balmer energy levels are present in the system. Given such claims and the ongoing controversy, we have re-analysed the 1D hydrogen atom, first from a classical and then from a quantum perspective both in the coordinate and in the momentum representations. This work exhibits that certain classical properties of the system may serve to clarify the properties of the quantum problem. Using the Dirichlet boundary condition, we show that the singularity of the potential prevents any relation between the right and left sides of the origin. Hence we prove that the attractive potential V(q) acts in that case as an impenetrable barrier splitting the coordinate space into two independent regions. We show that such splitting appears both in the classical and in the quantum descriptions of the system. The analysis presented in this paper may serve as a pedagogical tool for the comparison between classical and quantum problems, as well as an illustrative example of a problem involving a singular potential that can be approached both from its position and momentum representations.