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American Journal of Physics

Numerical and analytical study of the bound states of the −α/x2 potential

Thanh Nguyen, F. Marsiglio

Published Aug 19, 2020• 8 citations
Publisher

Abstract

The quantum mechanical bound states of the −α/x2 potential are truly anomalous. We revisit this problem by adopting a slightly modified version of this potential, one that adopts a cutoff in the potential arbitrarily close to the origin. The resulting solutions are completely well-defined and “normal,” and provide us with additional insight into the solutions of the “bare” −α/x2 potential. We present results here as a case study in undergraduate research—two independent methodologies are used: one analytical (albeit with very unfamiliar non-elementary functions) and one numerical (with a very straightforward methodology). These play complementary roles in arriving at solutions and achieving insights into this problem.

DOI
10.1119/10.0001533

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The quantum mechanical bound states of the −α/x2 potential are truly anomalous. We revisit this problem by adopting a slightly modified version of this potential, one that adopts a cutoff in the potential arbitrarily close to the origin. The resulting solutions are completely well-defined and “normal,” and provide us with additional insight into the solutions of the “bare” −α/x2 potential. We present results here as a case study in undergraduate research—two independent methodologies are used: one analytical (albeit with very unfamiliar non-elementary functions) and one numerical (with a very straightforward methodology). These play complementary roles in arriving at solutions and achieving insights into this problem.
Numerical and Analytical Study of the Bound States of the −α/x2-α/x^2−α/x2 Potential
The quantum mechanical bound states of the −α/x2-α/x^2−α/x2 potential are truly anomalous. We revisit this problem by adopting a slightly modified version of this potential, one that adopts a cutoff in the potential arbitrarily close to the origin. The resulting solutions are completely well-defined and ``normal.'' We present results here as a case study in undergraduate research --- two independent methodologies are used: one analytical (with very unfamiliar non-elementary functions) and one numerical (with very straightforward methodology). These play complementary roles in arriving at solutions and achieving insights in this problem.
In quantum mechanics a localized attractive potential typically supports a (possibly infinite) set of bound states, characterized by a discrete spectrum of allowed energies, together with a continuum of scattering states, characterized (in one dimension) by an energy-dependent phase shift. The 1∕x2 potential on 0<x<∞ confounds all of our intuitions and expectations. Resolving its paradoxes requires sophisticated theoretical machinery: regularization, renormalization, anomalous symmetry-breaking, and self-adjoint extensions. Our goal is to introduce the essential ideas at a level accessible to advanced undergraduates.
The procedure commonly used in textbooks for determining the eigenvalues and eigenstates for a particle in an attractive Coulomb potential is not symmetric in the way the boundary conditions at r = 0 and r → ∞ are considered. We highlight this fact by solving a model for the Coulomb potential with a cutoff (representing the finite extent of the nucleus). In the limit that the cutoff is reduced to zero, we recover the standard result, albeit in a non-standard way. This example is used to emphasize that a more consistent approach to solving the Coulomb problem in quantum mechanics requires an examination of the non-standard solution. The end result is, of course, the same.
We have investigated S-wave bound states composed of three identical bosons interacting via regulated delta function potentials in non-relativistic quantum mechanics. For low-energy systems, these short-range potentials serve as an approximation to the underlying physics, leading to an effective field theory. A method for perturbatively expanding the three-body bound-state equation in inverse powers of the cutoff is developed. This allows us to extract some analytical results concerning the behavior of the system. Further results are obtained by solving the leading order equations numerically to 11 or 12 digits of accuracy. The limit-cycle behavior of the required three-body contact interaction is computed, and the cutoff-independence of bound-state energies is shown. By studying the relationship between the two- and three-body binding energies, we obtain a high accuracy numerical calculation of Efimov's universal function. Equations for the first order corrections, necessary for the study of cutoff dependence, are derived. However, a numerical solution of these equations is not attempted.
In this study, we focus on investigating the exact relativistic bound-state spectra for supersymmetric, PT -supersymmetric and non -Hermitian versions of the q -deformed parameter Hulthén potential. The Hamiltonian hierarchy mechanism, namely the factorization method , is adopted within the framework of SUSYQM. This algebraic approach is used in solving the Klein-Gordon equation with the potential cases. The results obtained analytically by executing the straightforward calculations are in consistent forms for certain values of q . Achieving the results may have a particular interest for such applications. That is, they can be involved in determining the quantum structural properties of molecules for ro-vibrational states, and optical spectra characteristics of semiconductor devices with regard to the lattice dynamics. They are also employed to construct the broken or unbroken case of the supersymmetric particle model concerning the interaction between the elementary particles.