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Journal of Mathematical Physics

Inverse generators in the algebraic description of quantum confined systems: Properties and connections with nonlinear quantum deformation process and phase operators

A N F Aleixo, A. B. Balantekin

Published Dec 1, 2017• 2 citations
Publisher

Abstract

We introduce, via their action on the eigenstate basis, the inverse or the one-sided inverse of the algebraic generators for a class of important potentials used to model quantum confined systems in several fields of physics. We obtain the complete algebraic formulation of the systems with the inclusion of the inverse generators. We establish the relations of this complete algebraic approach with the nonlinear quantum deformation approach and the generalization of the phase operators’ definition for these systems. We outline extensions for all the results obtained in this study for the case of the two-parameter quantum deformed systems which preserves the algebraic nature of the undeformed systems.

DOI
10.1063/1.4999035

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We introduce, via their action on the eigenstate basis, the inverse or the one-sided inverse of the algebraic generators for a class of important potentials used to model quantum confined systems in several fields of physics. We obtain the complete algebraic formulation of the systems with the inclusion of the inverse generators. We establish the relations of this complete algebraic approach with the nonlinear quantum deformation approach and the generalization of the phase operators’ definition for these systems. We outline extensions for all the results obtained in this study for the case of the two-parameter quantum deformed systems which preserves the algebraic nature of the undeformed systems.
Using an expanded algebraic formalism with the inclusion of inverse operators, we construct raised and decreased coherent states for a set of exactly solvable quantum confined systems. We assume in this procedure both the ladder-operator and the displacement-operator methods, showing the equivalence between the two approaches. For each coherent state defined, we present its expansion in the Hilbert eigenstate space Hes, eigenvalue equation, overcompleteness relation, as well as other intrinsic properties. Whenever possible, we present an interpretation based on nonlinear deformation models for these new forms of coherent states. We evaluate the relevance of the new coherent states in quantum entanglement and squeezing by taking, as an example, the case of a coupled system.
We consider quantum confined systems represented by supersymmetric quantum mechanics where the Hamiltonians are products of two operators. We introduce inverses of these operators as well as pair operators, which are products of the original operators and their inverses. We show that the eigenstates of these pair operators are purely squeezed states, and using them we construct raised and decreased extensions of the squeezed states. We apply this procedure both to the ladder-operator and to the displacement-operator methods, showing the equivalence between those the two approaches. Finally, we show how the raised and decreased squeezed states can be dynamically produced by coupling the Hamiltonian with a two-level atom.
In this paper, within a parasupersymmetric and quantum deformed formalism, we introduce a class of bound-state problems which represents the coupling of a three-level atom with a two-dimensional potential system. We consider second-order parasupersymmetric quantum-mechanical models and a nonlinear deformed algebraic formulation for shape-invariant potential systems to study the quantum dynamics of physical observable, such as atomic level occupation, level transition probabilities, entropy and entanglement, in terms of time and deformation intensity. An application is given for a couple of shape-invariant potentials widely used to model quantum confined systems in several fields of physics, assuming a simple exponential form for the nonlinear deformation function.
Generalized coherent states for primary shape invariant potential systems quantum deformed by different models are constructed using an algebraic approach based on supersymmetric quantum mechanics. We show that this generalized formalism is able (a) to supply the essential requirements necessary to establish a connection between classical and quantum formulations of a given system, (b) to reproduce, as particular cases, results already known for shape-invariant systems (such as standard harmonic oscillator and Pöschl–Teller potentials as well as quantum deformed harmonic oscillator models) and (c) point to a formalism that provides a unified description of the different kind of coherent states for quantum systems, deformed or not deformed.
A nonlinear deformed algebraic theory for quantum confined systems is constructed using a supersymmetric and shape-invariant approach. In the framework of this model, we present a mathematical Barut–Girardello procedure which leads to nonlinear deformed coherent states by the successive application of a deformed raising operator on the ground state of these systems. As an application, we use a simple exponential form for the nonlinearity function to study in detail the mathematical and physical properties of such states and their evolution in time.
Deformation of the Heisenberg-Weyl algebra W s of creation-annihilation operators is studied and the problem of eigenvalues of the Hamiltonian for s deformed oscillators with interaction is solved within this algebra. At first, types of deformation are found for which solutions could be presented analytically and a simple q−deformation is considered by the means of the perturbation theory. Cases of reducing Hamiltonians which do not preserve the total particle number to that studied here are indicated. Investigations of new types of symmetries in different areas of mathematical physics by using the Inverse Scattering Method (ISM) and nonlinear differential equations, led to the appearance of notions ”Quantum Group”, ”Quantum Algebra” [1]. Nowadays, the deformed quantum statistics, parastatistics and corresponding algebras of creation-annihilation operators are widely studied. The common Heisenberg-Weyl algebra W s consists of generators ai, a + , ni, i =1 ,...,s ,