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Physical Review A

Quantum equations of motion for a dissipative system

R. J. Wysocki

Published Jan 7, 2000• 6 citations
Publisher

Abstract

We propose a method of determining quantum evolution equations for dissipative systems. Our approach is based on the observations that (1) the equations of quantum mechanics for a Hamiltonian system correspond to the classical equations describing the evolution of a special statistical ensemble of copies of the system, and (2) the quantum equations can be determined if the corresponding classical equations are known and the satisfaction of the uncertainty principle for particle position and momentum is postulated. Namely, if these statements are required to remain valid for a dissipative system, the quantum evolution equations for that system can be obtained.

DOI
10.1103/physreva.61.022104

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We propose a method of determining quantum evolution equations for dissipative systems. Our approach is based on the observations that (1) the equations of quantum mechanics for a Hamiltonian system correspond to the classical equations describing the evolution of a special statistical ensemble of copies of the system, and (2) the quantum equations can be determined if the corresponding classical equations are known and the satisfaction of the uncertainty principle for particle position and momentum is postulated. Namely, if these statements are required to remain valid for a dissipative system, the quantum evolution equations for that system can be obtained.
The translation from classical to quantum mechanics using the hamiltonian formalism for a non-dissipative systems is very well known. If in the classical picture there is dissipation, the equation of motion is given in terms of newtonian mechanics whose quantum equivalent is not obvious. Using results of previous works we show a way of obtaining a corresponding hamiltonian and discuss the wave function in physical space. We then proceed to analyze the quantum mechanical motion of gaussian wave packets with dissipation.
The Schrödinger free particle equation in 1+1 dimension describes second-order effects in ensembles of lattice random walks, in addition to its role in quantum mechanics, and its solutions represent the continuous limit of a property of ensembles of Brownian particles. In the present paper, the classical Schrödinger and Dirac equations have been derived from the Brownian motions of a particle, and it has been shown that the classical Schrödinger equation can be transformed into the usual Schrödinger quantum equation on applying the Heisenberg uncertainty principle between position and momentum, while the Dirac quantum equation follows from its classical counterpart on applying the Heisenberg uncertainty principle between energy and time, without applying any analytical continuation.
From a quantized many-body system a wave equation for the motion of a particle linearly coupled to a heat bath is derived. The effective Hamiltonian describing the motion of the single particle is explicitly time dependent, and for a quadratic potential, has a simple dependence on the initial position and momentum of the particle. For the case of dissipative harmonic motion, a time-dependent wave equation is derived and the ground-state wave function is determined. It is also shown that if the equations of motion for the many-body system is Galilean invariant, the reduced form of equation of motion for the single particle is not. However a generalized form of transformation for the position and momentum operators, to a coordinate system moving with constant velocity, is obtained, which reduces to the Galilean transformation when the coupling to the dissipative system is turned off.
This work proposes a conceptual and ontological reformulation of classical dynamics, thermodynamics, and quantum mechanics based on a single postulate: dissipation is a fundamental property of all physical processes. Traditional dynamics relies on the idealization of isolated, reversible systems, yet no real physical system satisfies such conditions. By introducing an intrinsic dissipative energy term into the equations of motion, we suggest that dissipation entails an irreversible loss of physical information, from which a classical uncertainty principle naturally follows. Stationary states are reinterpreted as zero‑dissipation configurations, lacking internal temporal evolution, thereby offering an ontological basis for quantization, non‑locality, and the Heisenberg uncertainty principle. Quantum mechanics can thus be viewed as the mathematical description of these stationary states and of the dissipative transitions connecting them. The framework is conceptually consistent with fluctuation‑dissipation theory, open‑system quantum dynamics, decoherence theory, and relaxation phenomena in NMR spectroscopy. Several experimental proposals are outlined to discriminate this ontological model from both classical mechanics and standard quantum mechanics.
Mechanics can be founded on a principle relating the uncertainty delta-q in the trajectory of an observable particle to its motion relative to the observer. From this principle, p.delta-q=const., p being the q-conjugated momentum, mechanical laws are derived and the meaning of the Lagrangian and Hamiltonian functions are discussed. The connection between the presented principle and Hamilton's Least Action Principle is examined. Wave mechanics and Schrodinger equation appear without additional assumptions by choosing the representation for delta-q in the case the motion is not trajectory describable. The Cramer-Rao inequality serves that purpose. For a particle hidden from direct observation, the position uncertainty determined by the enclosing boundaries leads to thermodynamics in a straightforward extension of the presented formalism. The introduction of uncertainty in classical mechanics formulation enables the translation of mechanical laws into the wide ranging conceptual framework of information theory. The boundaries between classical mechanics, thermodynamics and quantum mechanics are defined in terms of informational changes associated with the system evolution. As a direct application of the proposed formulation upper bounds for the rate of information transfer are derived.