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Journal of Physics A Mathematical and General

Analysis of inverse-square potentials using supersymmetric quantum mechanics

Asim Gangopadhyaya, Prasanta Kumar Panigrahi, U. Sukhatme

Published Jun 21, 1994• 20 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/27/12/032

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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.
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 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.
In this paper, we show that the exact energy eigenvalues and eigenfunctions of the Schrödinger equation for charged particles moving in a certain class of noncentral potentials can be easily calculated analytically in a simple and elegant manner by using Supersymmetric method (SUSYQM). We discuss the Poschl-Teller plus Scarf non-central potential systems. Then, by operating the lowering operator we get the ground state wave function, and the excited state wave functions are obtained by operating raising operator repeatedly. The energy eigenvalue is expressed in the closed form obtained using the shape invariant properties. The results are in exact agreement with other methods.Keyword: supersymmetry, non-central potentials, poschl teller plus scarf.
In this paper, we show that the exact energy eigenvalues and eigenfunctions of the Schrödinger equation for charged particles moving in a certain class of noncentral potentials can be easily calculated analytically in a simple and elegant manner by using Supersymmetric method (SUSYQM). We discuss the Poschl-Teller plus Scarf non-central potential systems. Then, by operating the lowering operator we get the ground state wave function, and the excited state wave functions are obtained by operating raising operator repeatedly. The energy eigenvalue is expressed in the closed form obtained using the shape invariant properties. The results are in exact agreement with other methods.Keyword: supersymmetry, non-central potentials, poschl teller plus scarf.
Abstract One-dimensional potentials defined by V ( S ) ( x ) = S ( S + 1 ) ℏ 2 π 2 / [ 2 ma 2 sin 2 ( π x / a ) ] (for integer S ) arise in the repeated supersymmetrization of the infinite square well, here defined over the region (0, a ). We review the derivation of this hierarchy of potentials and then use the methods of supersymmetric quantum mechanics, as well as more familiar textbook techniques, to derive compact closed-form expressions for the normalised solutions, ψ n ( S ) ( x ) , for all V ( S ) ( x ) in terms of well-known special functions in a pedagogically accessible manner. We also note how the solutions can be obtained as a special case of a family of shape-invariant potentials, the trigonometric Pöschl–Teller potentials, which can be used to confirm our results. We then suggest additional avenues for research questions related to, and pedagogical applications of, these solutions, including the behaviour of the corresponding momentum-space wave functions ϕ n ( S ) ( p ) for large ∣ p ∣ and general questions about the supersymmetric hierarchies of potentials which include an infinite barrier.
Starting from the orthonormal eigenfunctions which are the solutions of the Schrödinger equation for a given potential it is shown that sets of functions orthogonal to the eigenstates inside a region of varying radius R may be constructed. These sets of functions may in turn be used to construct a new set of functions which satisfy bound-state boundary conditions and are themselves solutions of the Schrödinger equation in a new potential. The bound-state spectrum of the new potential is related to the spectrum of the original potential in a definite manner. The relationship of this construction of a new potential to the inverse scattering theory approach based on the Gelfand–Levitan equation and the potentials constructed using supersymmetric quantum mechanics (SUSYQM) is explored. The connection of this approach to other approaches based on the implementation of orthogonality and their relation to Pauli exclusion principle is examined.