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The American Journal of Physical Sciences and Applications

A study of an Exotic Quantum Damped

Published Jan 1, 2021• 0 citations
Publisher

Abstract

We have studied a two-dimensional damped harmonic oscillator in time dependent noncommutative space in an earlier communication [Int.J. Theor.Phys.59, 3852 (2020)].In this paper we have reconsidered the same model and extended the discussion by introducing a generalised solution of Ermakov-Pinney (EP) equation.We proceed to estimate the expectation value of energy in this state.

DOI
10.15864/ajps.1402

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We have studied a two-dimensional damped harmonic oscillator in time dependent noncommutative space in an earlier communication [Int.J. Theor.Phys.59, 3852 (2020)].In this paper we have reconsidered the same model and extended the discussion by introducing a generalised solution of Ermakov-Pinney (EP) equation.We proceed to estimate the expectation value of energy in this state.
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.
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.
The system of a two-dimensional damped harmonic oscillator is revisited in the extended phase space. It is an old problem that has already been addressed by many authors that we present here with some fresh points of view and carry on a whole discussion. We show that the system is singular. The classical Hamiltonian is proportional to the first-class constraint. We pursue with the Dirac’s canonical quantization procedure by fixing the gauge and provide a reduced phase space description of the system. As a result, the quantum system is simply modeled by the original quantum Hamiltonian.