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European Journal of Physics

Introducing supersymmetric quantum mechanics via point canonical transformations

Gabriel González

Published Mar 11, 2020• 0 citations
Publisher

Abstract

Abstract We propose an alternative way to introduce supersymmetric quantum mechanics to physics students using a point canonical coordinate transformation. We provide two point canonical transformations that allow us to obtain new supersymmetric partner Hamiltonians with iso-spectral and non-iso-spectral energy eigenvalues starting from a known solvable Hamiltonian. We illustrate this new approach by applying it to the harmonic oscillator potential.

DOI
10.1088/1361-6404/ab7ebe

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Abstract We propose an alternative way to introduce supersymmetric quantum mechanics to physics students using a point canonical coordinate transformation. We provide two point canonical transformations that allow us to obtain new supersymmetric partner Hamiltonians with iso-spectral and non-iso-spectral energy eigenvalues starting from a known solvable Hamiltonian. We illustrate this new approach by applying it to the harmonic oscillator potential.
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.
In this work we present an introduction to Supersymmetry in the context of 1-dimensional Quantum Mechanics. For that purpose we develop the concept of hamiltonians factorization using the simple harmonic oscillator as an example, we introduce the supersymmetric oscilator and, next, we generalize these concepts to introduce the fundamentals of Supersymmetric Quantum Mechanics. We also discuss useful tools to solve problems in Quantum Mechanics which are intrinsecally related to Supersymmetry as hierarchy of hamiltonians and shape invariance. We present two approximation methods which will be specially useful: the well known Variational Method and the Logarithmic Perturbation Theory, the latter being closely related to the concepts of superpotentials and hierarchy of hamiltonians. Finally, we present problems related to superpotentials which are monomials in even powers of the x coordinate multiplied by the sign function epsilon(x), which seems to be a new class of problems in Supersymmetric Quantum Mechanics.
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.
This undergraduate thesis contains one dimensional introductory treatment of Supersymmetric Quantum Mechanics (SUSY QM). The factorisation method is used to obtain Partner Hamiltonians, and the notions of partner potentials and superpotential are introduced. It demonstrated how energy spectra and eigenfunctions of partner potentials can be correlated for the cases of unbroken and broken supersymmetry. Generalisations of these cases attributed to Sets of Hamiltonians and shape invariant potentials are introduced. The devel oped formalisation is used to obtain energy eigenfunctions and spectra of partner potentials of some well-known quantum systems, namely the Infinite Square Well and the Free Particle. In these systems, the partner potentials are trigonometric and hyperbolic functions, respectively. A comparison of these two physically distinct but mathematically related partner potentials is provided. A new philosophical interpretation of SUSY QM is introduced, with an ontological discussion of supersymmetric notions and SUSY QM. The current observation problem of SUSY is discussed, and on that basis, two opposing arguments are presented. The method of Falsification by Karl Popper is briefly introduced, and an argument is produced regarding its application to SUSY.
Factorization of quantum mechanical Hamiltonians has been a useful technique for some time. This procedure has been given an elegant description by supersymmetric quantum mechanics, and the subject has become well developed. We demonstrate that the existence of raising and lowering operators for the harmonic oscillator (and many other potentials) can be extended to their supersymmetric partners. The double supersymmetry (or a factorization chain) is used to obtain non-singular isospectral potentials, and the explicit expressions for the ladder operators, wavefunctions and probability densities are provided. This application avoids the technical complexities of the most general approaches, and requires relatively modest methods from supersymmetric quantum mechanics.