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Journal of Physics A Mathematical and General

Supersymmetric quantum merchanics, orthogonal states and the Pauli exclusion principle

C V Sukumar, D.M. Brink

Published May 11, 2004• 5 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/37/21/014

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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.
The harmonic oscillator is a quantum mechanical system that represents one of the most basic potentials. In order to understand the behavior of a particle within this system, the time-independent Schrödinger equation was solved; in other words, its eigenfunctions and eigenvalues were found. The first goal of this study was to construct a family of single parameter potentials and corresponding eigenfunctions with a spectrum similar to that of the harmonic oscillator. This task was achieved by means of supersymmetric quantum mechanics, which utilizes an intertwining operator that relates a known Hamiltonian with another whose potential is to be built. Secondly, a generalization of the technique was used to work with the time-dependent Schrödinger equation to construct new potentials and corresponding solutions.
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.
In the standard formulation of quantum mechanics, one starts by proposing a potential function that models the physical system. The potential is then inserted into the Schrödinger equation, which is solved for the wavefunction, bound states energy spectrum, and/or scattering phase shift. In this work, however, we propose an alternative formulation in which the potential function does not appear. The aim is to obtain a set of analytically realizable systems, which is larger than in the standard formulation and may or may not be associated with any given or previously known potential functions. We start with the wavefunction, which is written as a bounded infinite sum of elements of a complete basis with polynomial coefficients that are orthogonal on an appropriate domain in the energy space. Using the asymptotic properties of these polynomials, we obtain the scattering phase shift, bound states, and resonances. This formulation enables one to handle not only the well-known quantum systems but also previously untreated ones. Illustrative examples are given for two- and three-parameter systems.
The quantum inverse scattering method is applied to the nonlinear Schrodinger model of fermions with attractive interactions. Both scattering and bound-state operators are constructed. Among state operators, there are generators of infinitely many conserved quantities and the creators of the eigenstates of these quantities. The commutation relations and the eigenvalues of physical interest are calculated. The quantum Gelfand-Levitan equations are established. The Fourier transform of the connected part of the 4-point Green function and the two body S matrix are calculated explicitly.