The bound-state solutions of the one-dimensional pseudoharmonic oscillator

• URL: http://arxiv.org/abs/2111.12833v1
• Date: Wed, 24 Nov 2021 23:03:10 GMT
• Title: The bound-state solutions of the one-dimensional pseudoharmonic oscillator
• Authors: Rufus Boyack, Asadullah Bhuiyan, Aneca Su, Frank Marsiglio
• Abstract summary: We study the bound states of a quantum mechanical system governed by a constant $alpha$. For attractive potentials within the range $-1/4leqalpha0$, there is an even-parity ground state with increasingly negative energy. We show how the regularized excited states approach their unregularized counterparts.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the bound states of a quantum mechanical system consisting of a simple harmonic oscillator with an inverse square interaction, whose interaction strength is governed by a constant $\alpha$. The singular form of this potential has doubly-degenerate bound states for $-1/4\leq\alpha<0$ and $\alpha>0$; since the potential is symmetric, these consist of even and odd-parity states. In addition we consider a regularized form of this potential with a constant cutoff near the origin. For this regularized potential, there are also even and odd-parity eigenfunctions for $\alpha\geq-1/4$. For attractive potentials within the range $-1/4\leq\alpha<0$, there is an even-parity ground state with increasingly negative energy and a probability density that approaches a Dirac delta function as the cutoff parameter becomes zero. These properties are analogous to a similar ground state present in the regularized one-dimensional hydrogen atom. We solve this problem both analytically and numerically, and show how the regularized excited states approach their unregularized counterparts.

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