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

• URL: http://arxiv.org/abs/2010.13946v1
• Date: Mon, 26 Oct 2020 23:16:49 GMT
• Title: The bound-state solutions of the one-dimensional hydrogen atom
• Authors: Rufus Boyack and Frank Marsiglio
• Abstract summary: We show how the even-parity states 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.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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