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Journal of Physics A Mathematical and General

Fluid-dynamical interpretation of quantum damped oscillators

Mohammed A. Z. Habeeb

Published Dec 1, 1987• 1 citation
Publisher

Abstract

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.

DOI
10.1088/0305-4470/20/17/025

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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 a quantum version of the Arnold transformation of classical mechanics, all quantum dynamical systems whose classical equations of motion are non-homogeneous linear second-order ordinary differential equations (LSODE), including systems with friction linear in velocity such as the damped harmonic oscillator, can be related to the quantum free-particle dynamical system. This implies that symmetries and simple computations in the free particle can be exported to the LSODE-system. The quantum Arnold transformation is given explicitly for the damped harmonic oscillator, and an algebraic connection between the Caldirola-Kanai model for the damped harmonic oscillator and the Bateman system will be sketched out.
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.
We construct a Madelung fluid model with time variable parameters as a dissipative quantum fluid and linearize it in terms of Schrödinger equation with time-dependent parameters. It allows us to find exact solutions of the nonlinear Madelung system in terms of solutions of the Schrödinger equation and the corresponding classical linear ordinary differential equation with variable frequency and damping. For the complex velocity field, the Madelung system takes the form of a nonlinear complex Schrödinger–Burgers equation, for which we obtain exact solutions using complex Cole–Hopf transformation. In particular, we give exact results for nonlinear Madelung systems related with Caldirola–Kanai-type dissipative harmonic oscillator. Collapse of the wave function in dissipative models and possible implications for the quantum cosmology are discussed.
Quantum dissipation is studied within two model oscillators, the Caldirola-Kanai (CK) oscillator as an open system with one degree of freedom and the Bateman-Feshbach-Tikochinsky (BFT) oscillator as a closed system with two degrees of freedom. Though these oscillators describe the same classical damped motion, the CK oscillator retains the quantum coherence, whereas the damped subsystem of the BFT oscillator exhibits both quantum decoherence and classical correlation. Furthermore the amplified subsystem of the BFT oscillator shows the same degree of quantum decohernce and classical correlation.