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arXiv:hep-th/9305082

An Alternate Approach to Transition Potentials

Asim Gangopadhyaya, Prasanta Kumar Panigrahi, U. Sukhatme

Published May 18, 1993• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.hep-th/9305082

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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.
We analyse transition potentials, i.e. potentials exhibiting limiting inverse-square behaviour V(r) to r approximately 0 alpha r -2 in non-relativistic quantum mechanics using the techniques of supersymmetry. For the range - 1/4 <or= alpha < 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' wavefunctions as the conventional approach and thus provides a simple justification for the usual practice in the literature. Two examples (the Poschl-Teller II potential and a two-anyon system on the plane) have been worked out for illustrative purposes.
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 a recent short note (Bergeron et al 2010 Europhys. Lett. 92 60003), we have presented the good properties of a new family of semi-classical states for Pöschl–Teller potentials. These states are built from a supersymmetric quantum mechanics (SUSYQM) approach and the parameters of these 'coherent' states are points in the classical phase space. In this paper, we develop all the mathematical aspects that have been left out of the previous paper (proof of the resolution of unity, detailed calculations of the quantized version of classical observables and mathematical study of the resulting operators: problems of domains, self-adjointness or self-adjoint extensions). Some additional questions such as asymptotic behavior are also studied. Moreover, the framework is extended to a larger class of Pöschl–Teller potentials. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Coherent states: mathematical and physical aspects'.
We review SUSY in non-relativistic quantum mechanics as a powerful tool for studying one dimensional potentials and discuss some applications. The Dirac operatorial factorization method for the harmonic oscillator in QM is generalized to a broader scope. A Schr¨odinger second order operator can be factorized into two first-order ones by solving a Riccati equation and determining the corresponding superpotential. We apply the technique to square well, P¨oschl-Teller and finite barrier potentials, analyzing the resulting energy spectra and the transmission and reflection coefficients. Both discrete and continuous energy spectrum cases are discussed.
This work prolongs recent investigations by Bergeron et al [see 2012 J. Phys. A: Math. Theo. 45 244028] on new SUSYQM coherent states for Pöschl-Teller potentials. It mainly addresses explicit computations of eigenfunctions and spectrum associated to the higher order hierarchic supersymmetric Hamiltonian. Analysis of relevant properties and normal and anti-normal forms is performed and discussed. Coherent states of the hierarchic first order differential operator Am,ν,βA_{m,ν,β}Am,ν,β​ of the Pöschl-Teller Hamiltonian Hν,β(m)H_{ν,β}^{(m)}Hν,β(m)​ and their characteristics are studied.
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.