Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/9511040

A Supersymmetric Treatment of a Particle Subjected to a Ring-shaped Potential

Gardo Blado

Published Nov 28, 1995• 0 citations
arXivHTMLPDF

Abstract

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, 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.

DOI
10.48550/arxiv.quant-ph/9511040

Related papers

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

Gardo Blado

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.

International Journal of Quantum Chemistry•Jan 1, 1996•35 citations

Supersymmetry and the Hartmann Potential of Theoretical Chemistry

Gardo Blado

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 $\left[ 0, \infty \right)$. 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 $L$ but different values of $ησ^{2}$ and quantum number $N$. This is in contrast to the first formulation, which relates states with identical values of the quantum number $N$ and $ησ^{2}$ but different values of the quantum number $L$.

arXiv:quant-ph/9602005•Feb 8, 1996•0 citations

Solving the Schrödinger Equation with Hartmann Potential by Factorization Method and Supersymmetry

Hossein Rahbar, Jafar Sadeghi

In this paper ,we study Schrödinger equation with Hartman potential, and discussed the fundamental concepts of supersymmetric quantum mechanics (SUSYQM) , and factorization method in radial and angular part separately .The energy eigenvalues and (radial) eigenfunctions of the Hartmann potential are subsequently rederived using the techniques of SUSYQM

Theoretical Physics•Dec 30, 2016•3 citations

Bound state solution of the Klein–Fock–Gordon equation with the Hulthén plus a ring-shaped-like potential within SUSY quantum mechanics

A. I. Ahmadov, Sh. M. Nagiyev, Mafkura Gojayeva +2 more

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.

arXiv:1812.11179•International Journal of Modern Physics A•Nov 30, 2018•31 citations

Exact Solutions of Bound States of the Schrdinger Equation with the Ring-Shaped Hartmann Potential

Zidong Chen

The Schrdinger equation with the ring-shaped Hartmann potential is solved by using the usual method of variable separation in spherical polar coordinates. The exact energy equation is obtained by then solved angle equation and radial equation. The normalized angle wavefunction and radial wavefunction are presented. The general calculation formulas for radial matrix elements and radial average values of the Hartmann potential are obtained.

Journal of Southwest China Normal University•Jan 1, 2005•0 citations

An Alternate Approach to Transition Potentials

Asim Gangopadhyaya, Prasanta Kumar Panigrahi, U. Sukhatme

We analyze transition potentials $(V(r) \stackrel{r\sim 0}{\rightarrow} {αr^{-2}})$ in non-relativistic quantum mechanics using the techniques of supersymmetry. For the range $-1/4 < α< 3/4$, the eigenvalue problem becomes ill-defined (since it is not possible to choose a unique eigenfunction based on square integrability and boundary conditions). It is shown that supersymmetric quantum mechanics (SUSYQM) provides a natural prescription for a unique determination of the spectrum. Interestingly, our SUSYQM based approach picks out the same "less singular" wave functions as the conventional approach, and thus provides a simple justification for the usual practice in the literature. Two examples (the Pöschl-Teller II potential and a two anyon system on the plane) have been worked out for illustrative purposes.

arXiv:hep-th/9305082•May 18, 1993•0 citations
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.
We analyze transition potentials (V(r)→r∼0αr−2)(V(r) \stackrel{r\sim 0}{\rightarrow} {αr^{-2}})(V(r)→r∼0αr−2) in non-relativistic quantum mechanics using the techniques of supersymmetry. For the range −1/4<α<3/4-1/4 < α< 3/4−1/4<α<3/4, the eigenvalue problem becomes ill-defined (since it is not possible to choose a unique eigenfunction based on square integrability and boundary conditions). It is shown that supersymmetric quantum mechanics (SUSYQM) provides a natural prescription for a unique determination of the spectrum. Interestingly, our SUSYQM based approach picks out the same "less singular" wave functions as the conventional approach, and thus provides a simple justification for the usual practice in the literature. Two examples (the Pöschl-Teller II potential and a two anyon system on the plane) have been worked out for illustrative purposes.