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Revista Brasileira de Ensino de Física

Dissipative Dynamics and Uncertainty Measures of a Charged Oscillator in the Presence of the Aharonov-Bohm Effect

J. P. G. Nascimento, V. Aguiar, D. S. Cortez, I. Guedes

Published Jan 1, 2021• 2 citations
Publisher

Abstract

We analyze the effects of dissipation in a charged oscillator in the presence of the Aharonov-Bohm effect by using time-dependent mass (m(t)) Hamiltonians. We consider two different models for the dissipative Hamiltonian and analyze the uncertainties (Δr and Δp) and the quantum mechanical expectation value of energy (⟨E⟩) in terms of time (t), damping parameters and flux parameter (v). For the Caldirola-Kanai model, we observe that the flux parameter v decreases the energy dissipation in a quantum dot for a certain range of t.

DOI
10.1590/1806-9126-rbef-2020-0464

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We analyze the effects of dissipation in a charged oscillator in the presence of the Aharonov-Bohm effect by using time-dependent mass (m(t)) Hamiltonians. We consider two different models for the dissipative Hamiltonian and analyze the uncertainties (Δr and Δp) and the quantum mechanical expectation value of energy (⟨E⟩) in terms of time (t), damping parameters and flux parameter (v). For the Caldirola-Kanai model, we observe that the flux parameter v decreases the energy dissipation in a quantum dot for a certain range of t.
We begin by revisiting the so-called Caldirola - Kanai Hamiltonian and discuss its inherent ambiguity: does it represent a dissipative harmonic oscillator (HO) subject to a friction force, or does it describe an HO with a time-dependent mass (TDM)? Although classically both descriptions do coexist, in the quantum domain the solution of the Schrödinger equation (or Heisenberg equations of motion) with a TDM does not present inconsistencies, however, the dissipative Hamiltonian shows violation of the Heisenberg uncertainty principle. This violation is avoided by introducing a stochastic force in the equations of motion, which will take care of the fluctuations due to the environment. Once the distinction between the dissipative and amplifying Hamiltonian is made clear, we consider the problem of the quantum TDM HO subject to dissipation, showing that both phenomena may be merged and described by a single Hamiltonian, the amplifying - dissipative Hamiltonian . We obtain the solutions of the Heisenberg equations of motion for the canonical momentum and position; next, we specialize on the weak damping limit and analyse the effects of the amplifying - dissipative process on the mean values of the physical variables.
The amplitude of the Bohm-Aharonov oscillations of a particle moving around a ring threaded by a magnetic flux and coupled to different dissipative environments is studied. The decay of the oscillations on increasing the radius of the ring is shown to depend on the spatial features of the coupling. When the environment is modeled by the Caldeira-Leggett bath of oscillators, interference effects are suppressed beyond a finite length, even at zero temperature. A nontrivial renormalization of the Aharonov-Bohm oscillations is also found when the particle is coupled by the Coulomb potential to a dirty electron gas. A finite renormalization of the Aharonov-Bohm oscillations is obtained for other models of the environment.
A canonical description of a harmonic oscillator with energy dissipation is sought which combines the advantages of the Kanai-Caldirola Hamiltonian and a simple model of strangulation previously considered by the authors. The treatment is in the Heisenberg picture of quantum mechanics or alternatively in classical mechanics. A strangulation \ensuremath{\beta} is superimposed on the damping (or growth) \ensuremath{\gamma}, represented by an oscillator mass m(t)=m0{m}_{0}m0​exp(2\ensuremath{\gamma}t). In general, the expected dissipation of energy occurs only if \ensuremath{\beta}>\ensuremath{\Vert}\ensuremath{\gamma}\ensuremath{\Vert}. If \ensuremath{\beta}<\ensuremath{\Vert}\ensuremath{\gamma}\ensuremath{\Vert} the attempted strangulation induces a long-term growth in energy unless a special initial motion is chosen. A slight deviation from this initial motion results in a temporary decay in energy followed by growth. \ensuremath{\beta}<0 always induces a growth in energy.
The usual treatment of damping forces in quantum mechanics starts from the introduction of the explicitly time dependent Kanai Hamiltonian, which actually represents a variable mass particle, and the misinterpretation of this Hamiltonian as representing a particle of constant mass leads to certain physical difficulties. However the Hamiltonian can be modified so that it can be reinterpreted as describing a constant mass particle. Here we explicitly introduce the mass as a new dynamical variable, which allows us to write a linear, time independent Hamiltonian for the system, which can be solved by conventional methods. The damped harmonic oscillator and damped free particle are treated in detail, both for the Kanai Hamiltonian and for our case, and the solutions are compared. Our solution can be reduced to the Kanai one in appropriate circumstances, but in general it has a much greater versatility, as a result of which it can be more easily reinterpreted as describing a constant mass particle subject to a damping force, which reinterpretation is of course necessary if the method is to have practical applicability. We also show how such a reinterpretation can be carried out in detail by introducing a ’’dissipation variable’’, in terms of which one may avoid the concept of a variable mass altogether.