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International Journal of Quantum Chemistry

Supersymmetric treatment of a particle subjected to a ring-shaped potential

Gardo Blado

Published Jan 1, 1996• 35 citations
Publisher

Abstract

The ring-shaped Hartmann potential was introduced in quantum chemistry to describe ring-shaped molecules like benzene. In this article, fundamental concepts of supersymmetric quantum mechanics (SUSYQM) are discussed. The energy eigenvalues and (radial) eigenfunctions of the Hartmann potential are subsequently rederived using the techniques of SUSYQM. © 1996 John Wiley & Sons, Inc.

DOI
10.1002/(sici)1097-461x(1996)58:5<431::aid-qua1>3.0.co;2-z

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Journal of Southwest China Normal University•Jan 1, 2005•0 citations
The ring-shaped Hartmann potential was introduced in quantum chemistry to describe ring-shaped molecules like benzene. In this article, fundamental concepts of supersymmetric quantum mechanics (SUSYQM) are discussed. The energy eigenvalues and (radial) eigenfunctions of the Hartmann potential are subsequently rederived using the techniques of SUSYQM. © 1996 John Wiley & Sons, Inc.
The ring-shaped Hartmann potential V=ησ2ε0(2a0r−ηa02r2sin2θ)V = ησ^{2} ε_{0} \left( \frac{2 a_{0}}{r} - \frac{ηa_{0}^{2}}{r^{2} sin^{2} θ} \right)V=ησ2ε0​(r2a0​​−r2sin2θηa02​​) was introduced in quantum chemistry to describe ring-shaped molecules like benzene. In this article, fundamental concepts of supersymmetric quantum mechanics (SUSYQM) are discussed. The energy eigenvalues and (radial) eigenfunctions of the Hartmann potential are subsequently rederived using the techniques of SUSYQM.
The ring-shaped Hartmann potential $ V = ησ^{2} ε_{0} \left( \frac{2 a_{0}}{r} - \frac{ηa_{0}^{2}}{r^{2} sin^{2} θ} \right) $ was introduced in quantum chemistry to describe ring-shaped molecules like benzene. In this article, the supersymmetric features of the Hartmann potential are discussed. We first review the results of a previous paper in which we rederived the eigenvalues and radial eigenfunctions of the Hartmann potential using a formulation of one-dimensional supersymmetric quantum mechanics (SUSYQM) on the half-line [0,∞)\left[ 0, \infty \right)[0,∞). A reformulation of SUSYQM in the full line (−∞,∞)\left( -\infty, \infty \right)(−∞,∞) is subsequently developed. It is found that the second formulation makes a connection between states having the same quantum number LLL but different values of ησ2ησ^{2}ησ2 and quantum number NNN. This is in contrast to the first formulation, which relates states with identical values of the quantum number NNN and ησ2ησ^{2}ησ2 but different values of the quantum number LLL.
In this paper, the bound state solution of the modified Klein–Fock–Gordon equation is obtained for the Hulthén plus ring-shaped-like potential by using the developed scheme to overcome the centrifugal part. The energy eigenvalues and corresponding radial and azimuthal wave functions are defined for any [Formula: see text] angular momentum case on the conditions that scalar potential is whether equal and nonequal to vector potential, the bound state solutions of the Klein–Fock–Gordon equation of the Hulthén plus ring-shaped-like potential are obtained by Nikiforov–Uvarov (NU) and supersymmetric quantum mechanics (SUSY QM) methods. The equivalent expressions are obtained for the energy eigenvalues, and the expression of radial wave functions transformations to each other is revealed owing to both methods. The energy levels and the corresponding normalized eigenfunctions are represented in terms of the Jacobi polynomials for arbitrary [Formula: see text] states. A closed form of the normalization constant of the wave functions is also found. It is shown that the energy eigenvalues and eigenfunctions are sensitive to [Formula: see text] radial and [Formula: see text] orbital quantum numbers.