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arXiv:1205.6239 · Symmetry Integrability and Geometry Methods and Applications

Harmonic Oscillator SUSY Partners and Evolution Loops

David J. Fernández C.

Published Jul 11, 2012• 3 citations
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Abstract

Supersymmetric quantum mechanics is a powerful tool for generating exactly solvable potentials departing from a given initial one.If applied to the harmonic oscillator, a family of Hamiltonians ruled by polynomial Heisenberg algebras is obtained.In this paper it will be shown that the SUSY partner Hamiltonians of the harmonic oscillator can produce evolution loops.The corresponding geometric phases will be as well studied.

DOI
10.3842/sigma.2012.041

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Supersymmetric quantum mechanics is a powerful tool for generating exactly solvable potentials departing from a given initial one.If applied to the harmonic oscillator, a family of Hamiltonians ruled by polynomial Heisenberg algebras is obtained.In this paper it will be shown that the SUSY partner Hamiltonians of the harmonic oscillator can produce evolution loops.The corresponding geometric phases will be as well studied.
The development of supersymmetric (SUSY) quantum mechanics has shown that some of the insights based on the algebraic properties of ladder operators related to the quantum mechanical harmonic oscillator (QMHO) carry over to the study of more general systems. At this level of generality, pairs of eigenfunctions of so-called partner Hamiltonians are transformed into each other, but the entire spectrum of any one of them cannot be deduced from this intertwining relationship in general – except in special cases. In this paper, we present a more general structure that provides all eigenvalues for a class of Hamiltonians that do not factor into a pair of operators satisfying canonical commutation relations. Instead of a pair of partner Hamiltonians, we consider two pairs that differ by an overall shift in their spectrum. This is called coupled supersymmetry. In that case, we also develop coherent states and present some uncertainty principles which generalise the Heisenberg uncertainty principle. Coupled SUSY is explicitly realised by an infinite family of differential operators which admit orthonormal bases of eigenfunctions of generalised harmonic oscillators.
We review the higher‐order supersymmetric quantum mechanics (H‐SUSY QM), which involves differential intertwining operators of order greater than one. The iterations of first‐order SUSY transformations are used to derive in a simple way the higher‐order case. The second order technique is addressed directly, and through this approach unexpected possibilities for designing spectra are uncovered. The formalism is applied to the harmonic oscillator: the corresponding H‐SUSY partner Hamiltonians are ruled by polynomial Heisenberg algebras which allow a straight construction of the coherent states.
Abstract We discussed transformations of supersymmetric quantum mechanics by considering a general six-parameter function, from which the superpotential and the supersymmetric partner potentials V − ( r ) and V + ( r ) were constructed. By specific choice of the parameters, V − ( r ) reproduced the radial harmonic oscillator, or its rationally extended version, with solutions containing X 1 -type Laguerre polynomials. The choice of parameters also determined which of the four possible SUSY transformations T i , i = 1, ...4 were employed. Within this formalism, the relation between the radial harmonic oscillator and its rational extension can be discussed in a straightforward way, furthermore, several aspects of SUSYQM and exactly solvable potentials can be demonstrated in a unified framework on a level suitable for basic quantum mechanics courses.
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.