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Discusi´ on Sobre La Disipaci´ on En Mec´ anica Cu´ antica: Modelo De Cardirola-Kanai Discussion On The Dissipation In Quantum Mechanics: Kanai Model

O. A. Rold, Herbert Vinck-Posada

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

In this work we present the Cardirola-Kanai model for the damped-quantum harmonic oscillator. We obtain the propagator with path-integral techniques. The evolution for a squeezed state and its variance is found (a straightforward approach to a coherent-squeezed state is possible). The results show that if we take a damped oscillator with zero initial velocity its evolution has similar uncertainty relations to those of a coherent state in the quantum case.

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In this work we present the Cardirola-Kanai model for the damped-quantum harmonic oscillator. We obtain the propagator with path-integral techniques. The evolution for a squeezed state and its variance is found (a straightforward approach to a coherent-squeezed state is possible). The results show that if we take a damped oscillator with zero initial velocity its evolution has similar uncertainty relations to those of a coherent state in the quantum case.
We study the time evolution of SU(1,1) coherent states driven by a damped harmonic oscillator as described by the Kanai-Caldirola Hamiltonian. The coherence of these states is preserved and their time evolution is described by nonlinear ``classical'' equations. The trajectories of all SU(1,1) coherent states in phase space asymptotically approach a common point as t\ensuremath{\rightarrow}\ensuremath{\infty}. In all cases, this approach is accompanied by a damping of the associated energy expectation value to zero.
The model of a quantum damped oscillator based on the Caldirola-Kanai equation is investigated by exploiting its coherent state basis. It is shown how these coherent states facilitate the derivation of fluid-dynamical equations corresponding to this model in the classical limit. From the conservative form of these equations it is concluded that the system represented by the Caldirola-Kanai equation cannot be considered as the quantum analogue of the classical damped oscillator. This is contrasted with the case of a harmonic oscillator interacting with a heat bath.
Using the Caldirola-Kanai Hamiltonian for the quantum dissipative system, the author expresses the propagator as the modified Feynman path integral in the configuration space, which can then be evaluated for the damped harmonic oscillator by Montroll's method. The propagator of the damped harmonic oscillator can also be calculated beyond and at caustics with the help of Horvathy-Feynman formula. The new results are confirmed by investigating the classical paths joining two fixed end-point positions, Finally the author obtains the time-dependent wavefunctions from the propagator of the dynamical system.