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Journal of Physics A Mathematical and Theoretical

Quantum dynamics of parasupersymmetric and shape-invariant coupled systems

A N F Aleixo, A. B. Balantekin

Published Jul 3, 2007• 3 citations
Publisher

Abstract

A class of bound-state problems which represents the coupling of a three-level atom with a two-dimensional system involving two shape-invariant potentials was introduced in a previous paper. In this paper, we considered second-order parasupersymmetric quantum-mechanical models and two possible kinds for the coupling Hamiltonian (linear and nonlinear in the potential ladder operators). In the present paper, using an algebraic formulation for shape-invariant potential systems, we study the quantum dynamics of these coupled systems and obtain the temporal behaviour of some dynamical variables related with the atom and the coupling potentials. An application is given for a couple of shape-invariant potentials (Poschl–Teller + self-similar potentials).

DOI
10.1088/1751-8113/40/29/016

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A class of bound-state problems which represents the coupling of a three-level atom with a two-dimensional system involving two shape-invariant potentials was introduced in a previous paper. In this paper, we considered second-order parasupersymmetric quantum-mechanical models and two possible kinds for the coupling Hamiltonian (linear and nonlinear in the potential ladder operators). In the present paper, using an algebraic formulation for shape-invariant potential systems, we study the quantum dynamics of these coupled systems and obtain the temporal behaviour of some dynamical variables related with the atom and the coupling potentials. An application is given for a couple of shape-invariant potentials (Poschl–Teller + self-similar potentials).
The quantum dynamics of a two-level system coupled to a shape-invariant potential is investigated. Shape-invariant potentials share an integrability condition called shape invariance which identifies an underlying algebraic structure, in general infinite dimensional, that transforms the potential parameters such as strength, range and diffuseness. We determine the time-evolution operator, the density operator of the system, and obtain the temporal behaviour of various dynamical variables by considering either pure or mixed initial states of the system, constructed with the generalized coherent state of the shape-invariant coupling potential. We consider specific examples of shape-invariant coupling potentials (harmonic oscillator, Pöschl–Teller and self-similar potentials). The results obtained for all dynamical variables exhibit rapid oscillations which periodically collapse and regenerate in different ways depending on the coupling potential nature.
Generalized coherent states for shape invariant potentials are constructed using an algebraic approach based on supersymmetric quantum mechanics. We show this generalized formalism is able to: a) supply the essential requirements necessary to establish a connection between classical and quantum formulations of a given system (continuity of labeling, resolution of unity, temporal stability, and action identity); b) reproduce results already known for shape-invariant systems, like harmonic oscillator, double anharmonic, Poschl-Teller and self-similar potentials and; c) point to a formalism that provides an unified description of the different kind of coherent states for quantum systems.
Generalized coherent states for shape invariant potentials are constructed using an algebraic approach based on supersymmetric quantum mechanics. We show this generalized formalism is able to: a) supply the essential requirements necessary to establish a connection between classical and quantum formulations of a given system (continuity of labeling, resolution of unity, temporal stability, and action identity); b) reproduce results already known for shape-invariant systems, like harmonic oscillator, double anharmonic, Poschl-Teller and self-similar potentials and; c) point to a formalism that provides an unified description of the different kind of coherent states for quantum systems.