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arXiv:quant-ph/0206013

The interplay of supersymmetry and PT symmetry in quantum mechanics: a case study for the Scarf II potential

GÉZA LÉVAI, Miloslav Znojil

Published Jun 3, 2002• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.quant-ph/0206013

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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.
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.
We consider PT-symmetric quantum mechanical potentials, having an underlying supersymmetry (SUSY). The eigenspectra of both the broken and unbroken phases of PT-symmetry are analytically obtained through SUSY and shape invariance (SI). The SI is characterized by real and imaginary parametric shifting for unbroken and broken phases of PT, respectively. The use of SUSY and PT is shown to yield several complex potential systems, originating from Coulomb, Poschl-Teller, and other solvable problems. In the PT-symmetric phase, spontaneous breaking of SUSY is observed in some parametric domains, which leads to non-trivial shape invariances to obtain the energy spectra. A conserved non-local correlation in PT-symmetric problems is shown to explain observed transmission and reflection behavior in optical systems, possessing PT-symmetry. Interestingly, the stationary states with real energy emerge in the PT-symmetric phase. The correlation also plays a crucial role in the broken PT case, having the complex conjugate energy eigenvalues and the corresponding states related.
We study the interrelation of https://www.w3.org/1998/Math/MathML" display="inline"> P T symmetry, supersymmetry and Lie symmetries in the description of one-dimensional quantum mechanical potentials. Compared to their Hermitian counterparts, the complex https://www.w3.org/1998/Math/MathML" display="inline"> P T potentials usually possess a second set of normalizable solutions, and the two sets can be distinguished by the quasi-parity quantum number q = ±1. We demonstrate that this doubling mechanism leads to larger potential algebras and also to the doubling of the SUSY partners of these potentials.
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.
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.
The PT\cal PTPT-symmetric complexified Scarf II potential $V(x)= - V_1 \sech^{2}x + {\rm i} V_2 \sech x \tanh x$, V1>0V_1>0V1​>0 , V2≠0V_{2}\neq 0V2​=0 is revisited to study the interplay among its coupling parameters. The existence of an isolated real and positive energy level that has been recently identified as a spectral singularity or zero-width resonance is here demonstrated through the behaviour of the corresponding wavefunctions and some property of the associated pseudo-norms is pointed out. We also construct four different rationally-extended supersymmetric partners to V(x)V(x)V(x), which are PT\cal PTPT-symmetric or complex non-PT\cal PTPT-symmetric according to the coupling parameters range. A detailed study of one of these partners reveals that SUSY preserves the V(x)V(x)V(x) spectral singularity existence.