The bound-state solutions of the one-dimensional hydrogen atom

TL;DR

This paper investigates the bound-state solutions of a regularized one-dimensional hydrogen atom, clarifying the existence and behavior of even-parity states and their relation to odd-parity states as the regularization parameter approaches zero.

Contribution

It introduces a regularized model of the 1D hydrogen atom and analyzes the parity solutions, revealing how even-parity states behave and differ from the singular case.

Findings

Even-parity states, except the ground state, converge to the same form as odd-parity states.

Degeneracy occurs between even- and odd-parity solutions as the regularization cutoff approaches zero.

The regularized model differs from the singular Coulomb potential analysis regarding even-parity solutions.

Abstract

The one-dimensional hydrogen atom is an intriguing quantum mechanics problem that exhibits several properties which have been continually debated. In particular, there has been variance as to whether or not even-parity solutions exist, and specifically whether or not the ground state is an even-parity state with infinite negative energy. We study a "regularized" version of this system, where the potential is a constant in the vicinity of the origin, and we discuss the even- and odd-parity solutions for this regularized one-dimensional hydrogen atom. We show how the even-parity states, with the exception of the ground state, converge to the same functional form and become degenerate for $x > 0$ with the odd-parity solutions as the cutoff approaches zero. This differs with conclusions derived from analysis of the singular (i.e., without regularization) one-dimensional Coulomb potential, where even-parity solutions are absent from the spectrum.