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The American Journal of Physical Sciences and Applications

Periodicity of damped harmonic oscillator affected by magnetic field in timedependent noncommutative space

Published Jan 1, 2022• 0 citations
Publisher

Abstract

A model of two-dimensional damped harmonic oscillator affected by time dependent magnetic field in time dependent noncommutative space was studied in an earlier communication [Phys.Scr.96 (2021) 125224].In this article, the same model is considered to show that, for a suitable explicit choices of damping and magnetic field, it is possible to import a periodicity to the time evolution of the energy expectation value of the model system.

DOI
10.15864/ajps.2204

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A model of two-dimensional damped harmonic oscillator affected by time dependent magnetic field in time dependent noncommutative space was studied in an earlier communication [Phys.Scr.96 (2021) 125224].In this article, the same model is considered to show that, for a suitable explicit choices of damping and magnetic field, it is possible to import a periodicity to the time evolution of the energy expectation value of the model system.
This work is devoted to find the eigenstates of a two dimensional damped harmonic oscillator in the presence of an external magnetic field varying with respect to time with a time dependent spatial noncommutativity. It is observed that there are some specific choices of the damping factor, the time dependent frequency of the oscillator and the time dependent external magnetic field for which one can obtain interesting solutions of the time dependent noncommutative parameters. These solutions follow from the non-linear Ermakov-Pinney equation. We then obtain exact analytic forms for the phase which relates the eigenstates of the Hamiltonian with the eigenstates of the Lewis invariant. An important aspect of this study is to see the relationship between the damping and the applied magnetic field on the energy profile of the system. For this we compute the expectation value of the Hamiltonian. These are found to vary with time for different solutions of the Ermakov-Pinney equation corresponding to different choices of the damping factor, the time dependent frequency of the oscillator and the time dependent applied magnetic field. Finally, we compare our results with those in the absence of the magnetic field to determine the effect of the applied field on our system.
In this paper, decoherence of a damped anisotropic harmonic oscillator in the presence of a magnetic field is studied in the framework of the Lindblad theory of open quantum systems in noncommutative phase-space. General fundamental conditions that should follow our quantum mechanical diffusion coefficients appearing in the master equation are kindly derived. From the master equation, the expressions of density operator, the Wigner distribution function, the expectation and variance with respect to coordinates and momenta are obtained. Based on these quantities, the total energy of the system is evaluated and simulations show its dependency to phase-space structure and its improvement due to noncommutativity effects and the environmental temperature as well. In addition, we also evaluate the decoherence time scale and show that it increases with noncommutativity phase-space effects as compared to the commutative case. It turns out from simulations that this time scale is significantly improved under magnetic field effects.
A charged harmonic oscillator in a magnetic field, Landau problems, and an oscillator in a noncommutative space share the same mathematical structure in their Hamiltonians. We have considered a two-dimensional anisotropic harmonic oscillator with arbitrarily time-dependent parameters (effective mass and frequencies), placed in an arbitrarily time-dependent magnetic field. A class of quadratic invariant operators (in the sense of Lewis and Riesenfeld) have been constructed. The invariant operators (Î) have been reduced to a simplified representative form by a linear canonical transformation [the group Sp(4,R)]. An orthonormal basis of the Hilbert space consisting of the eigenvectors of Î is obtained. In order to obtain the solutions of the time-dependent Schrödinger equation corresponding to the system, both the geometric and dynamical phase-factors are constructed. A generalized Peres–Horodecki separability criterion (Simon’s criterion) for the ground state corresponding to our system has been demonstrated.
We consider several models of the damped oscillators in nonrelativistic quantum mechanics in a framework of a general approach to the dynamics of the time-dependent Schroedinger equation with variable quadratic Hamiltonians. The Green functions are explicitly found in terms of elementary functions and the corresponding gauge transformations are discussed. The factorization technique is applied to the case of a shifted harmonic oscillator. The time-evolution of the expectation values of the energy related operators is determined for two models of the quantum damped oscillators under consideration. The classical equations of motion for the damped oscillations are derived for the corresponding expectation values of the position operator.
We consider several models of the damped oscillators in nonrelativistic quantum mechanics in a framework of a general approach to the dynamics of the time-dependent Schroedinger equation with variable quadratic Hamiltonians. The Green functions are explicitly found in terms of elementary functions and the corresponding gauge transformations are discussed. The factorization technique is applied to the case of a shifted harmonic oscillator. The time-evolution of the expectation values of the energy related operators is determined for two models of the quantum damped oscillators under consideration. The classical equations of motion for the damped oscillations are derived for the corresponding expectation values of the position operator.