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Journal of Physics A Mathematical and Theoretical

An extended phase-space SUSY quantum mechanics

Gagik Ter-Kazarian

Published Jan 6, 2009• 3 citations
Publisher

Abstract

In the present paper, we will concern ourselves with the extended phase-space quantum mechanics of particles which have both bosonic and fermionic degrees of freedom, i.e., the quantum field theory in (0 + 1) dimensions in q -(position) and p -(momentum) spaces, exhibiting supersymmetry. We present ( N = 2) realization of extended supersymmetry algebra and discuss the vacuum energy and topology of super-potentials. Shape invariance of exactly solvable extended SUSY potentials allows us to obtain analytic expressions for the entire energy spectrum of an extended Hamiltonian with, for example, Scarf potential without ever referring to an underlying differential equation.

DOI
10.1088/1751-8113/42/5/055302

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In the present paper, we will concern ourselves with the extended phase-space quantum mechanics of particles which have both bosonic and fermionic degrees of freedom, i.e., the quantum field theory in (0 + 1) dimensions in q -(position) and p -(momentum) spaces, exhibiting supersymmetry. We present ( N = 2) realization of extended supersymmetry algebra and discuss the vacuum energy and topology of super-potentials. Shape invariance of exactly solvable extended SUSY potentials allows us to obtain analytic expressions for the entire energy spectrum of an extended Hamiltonian with, for example, Scarf potential without ever referring to an underlying differential equation.
Supersymmetry (SUSY) postulates that for every fermion there is boson of equal mass (i.e. energy). In quantum mechanics this comes about because for every quantum Hamiltonian there is a partner Hamiltonian that has the same energy spectrum above the ground state of the original system. In particle physics, one “sector” is populated by bosons and the other sector by fermions and SUSY predicts that the lowest lying fermion state is energetically degenerate with the first excited boson state. Evidence for SUSY has proven to be elusive and it is now believed that SUSY is a broken symmetry. In January 2009 at a conference dedicated to Bob Wyatt, we began to look at SUSY as a way to develop new computational methods and approaches. Up until now, SUSY has been more of a mathematical technique that has been used more or less as a way to obtain stationary solutions to the Schrödinger equation for a variety of T&F Cat # K11224, Chapter 4, Page 54, 15-9-2010 one-dimensional potential systems. In this chapter, we will discuss some of the work we have been doing in developing “SUSY” inspired methods for performing quantum many-body calculations and quantum scattering calculations. We begin with a brief overview of the SUSY theory and some of its elementary results. We then discuss how we have used the approach to develop both analytical and numerical solutions of the stationary Schrödinger equation. Finally, we conclude by discussing our recent extension of SUSY to higher dimensions and for scattering theory.
We review supersymmetry (SUSY) in nonrelativistic quantum mechanics emphasizing algebraic aspects.We discuss the Hamiltonian subgroup implementing supersymmetry as well as the corresponding algebra of the SUSY generators.In the SUSYQM framework, the two distinct partner potentials are connected by a superpotential satisfying a Riccati differential equation.A full Hamiltonian operator in an extended Hilbert space is defined in order to render the supersymmetry manifest.As a result, the eigenfunctions of the original potentials are connected by generalized ladder operators.We provide an explicit realization of the abstract supersymmetry group for SUSYQM depending on one real and two Grassmann parameters.
Motivated by the duality of normalizable states and the presence of the quasi-parity quantum number q = ± 1 in 𝒫𝒯 symmetric (non-Hermitian) quantum mechanical potential models, the relation of 𝒫𝒯 symmetry and supersymmetry (SUSY) is studied. As an illustrative example, the 𝒫𝒯 invariant version of the Scarf II potential is presented, and it is shown that the ‘bosonic’ Hamiltonian has two different ‘fermionic’ SUSY partner Hamiltonians (potentials) generated from the ground-state solutions with q = 1 and q = −1. It is shown that the ‘fermionic’ potentials cease to be 𝒫𝒯 invariant when the 𝒫𝒯 symmetry of the ‘bosonic’ potential is spontaneously broken. A modified 𝒫𝒯 symmetry inspired SUSY construction is also discussed, in which the SUSY charge operators contain the antilinear operator 𝒯. It is shown that in this scheme the ‘fermionic’ Hamiltonians are just the complex conjugates of the original ‘fermionic’ Hamiltonians, and thus possess the same energy eigenvalues.
Motivated by the duality of normalizable states and the presence of the quasi-parity quantum number q=+/-1 in PT symmetric (non-Hermitian) quantum mechanical potential models, the relation of PT symmetry and supersymmetry (SUSY) is studied. As an illustrative example the PT invariant version of the Scarf II potential is presented, and it is shown that the "bosonic" Hamiltonian has two different "fermionic" SUSY partner Hamiltonians (potentials) generated from the ground-state solutions with q=1 and q=-1. It is shown that the "fermionic" potentials cease to be PT invariant when the PT symmetry of the "bosonic" potential is spontaneously broken. A modified PT symmetry inspired SUSY construction is also discussed, in which the SUSY charge operators contain the antilinear operator T. It is shown that in this scheme the "fermionic" Hamitonians are just the complex conjugate of the original "fermionic" Hamiltonians, and thus possess the same energy eigenvalues.
With a view to increasing the scope of applications of supersymmetric quantum mechanics (SUSY QM), we formulate the same in a complex phase space. Within this framework, the concept of shape invariance is reinvestigated and an insight into the eigenvalue spectra of non-Hermitian Hamiltonians is sought. The results are applied to a variety of potentials. We claim that the shape invariance for these potentials in the complex phase space can be retrieved in terms of the prescriptions already proposed in the conventional SUSY QM, in that the transformation of potential parameters takes the form of a reflection in the parameter space. Interestingly, some of these features turn out to be the generalization of the concept of quasi-parity used recently in the context of SUSY QM of PT-symmetric potentials. Further, a correspondence of the present approach with other complex formulations of SUSY QM and also in two real dimensions is demonstrated.
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.