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arXiv:quant-ph/9906082

Classical behaviour of many-body systems in Bohmian Quantum Mechanics

H. Geiger, Gustav M. Obermair, Ch. Helm

Published Jun 22, 1999• 0 citations
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Abstract

The classical behaviour of a macroscopic system consisting of a large number of microscopic systems is derived in the framework of the Bohmian interpretation of quantum mechanics. Under appropriate assumptions concerning the localization and factorization of the wavefunction it is shown explicitly that the center of mass motion of the system is determined by the classical equations of motion.

DOI
10.48550/arxiv.quant-ph/9906082

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The classical behaviour of a macroscopic system consisting of a large number of microscopic systems is derived in the framework of the Bohmian interpretation of quantum mechanics. Under appropriate assumptions concerning the localization and factorization of the wavefunction it is shown explicitly that the center of mass motion of the system is determined by the classical equations of motion.
The classical limit of quantum mechanics is investigated, by focusing on the study of the center of mass of a many-body system where each particle is described by quantum mechanics. We study how, in the limit when the number of particles diverges and under quite general assumptions, the center of mass of the system is not anymore described by quantum mechanics but by classical mechanics: the center of mass of the system becomes with a well-defined position and momentum, the state of the center of mass is fully determined by its position and by its momentum, and its dynamics is given by the classical law of dynamics. In order to get this result, three assumptions on the many-body system are necessary: the total mass of the system must be much larger than the mass of each particle composing the system; at most a finite number of particles has non-zero correlation in position with an infinite number of particles; finally, when the number of particles composing the system diverge, the variance in position of each particle converges to a finite value. These assumptions are commonly realized for a macroscopic solid and liquid systems. The results obtained in this paper show how the classical limit of quantum mechanics can be naturally achieved without the need to modify quantum mechanics, without the need to have an external environment and without the need of changing the interpretations of quantum mechanics.
An explicit model allowing a unified description of microscopic and macroscopic systems is exhibited. First, a modified quantum dynamics for the description of macroscopic objects is constructed and it is shown that it forbids the occurrence of linear superpositions of states localized in far-away spatial regions and induces an evolution agreeing with classical mechanics. This dynamics also allows a description of the evolution in terms of trajectories. To set up a unified description of all physical phenomena, a modification of the dynamics, with respect to the standard Hamiltonian one, is then postulated also for microscopic systems. It is shown that one can consistently deduce from it the previously considered dynamics for the center of mass of macroscopic systems. Choosing in an appropriate way the parameters of the so-obtained model one can show that both the standard quantum theory for microscopic objects and the classical behavior for macroscopic objects can all be derived in a consistent way. In the case of a macroscopic system one can obtain, by means of appropriate approximations, a description of the evolution in terms of a phase-space density distribution obeying a Fokker-Planck diffusion equation. The model also provides the basis for a conceptually appealing description of quantum measurement.
We discuss the conditions for the classicality of quantum states with a very large number of identical particles. By defining the center of mass from a large set of Bohmian particles, we show that it follows a classical trajectory when the distribution of the Bohmian particle positions in a single experiment is always equal to the marginal distribution of the quantum state in physical space. This result can also be interpreted as a single experiment generalization of the well-known Ehrenfest theorem. We also demonstrate that the classical trajectory of the center of mass is fully compatible with a quantum (conditional) wave function solution of a classical non-linear Schrödinger equation. Our work shows clear evidence for a quantum–classical inter-theory unification, and opens new possibilities for practical quantum computations with decoherence.
Bohmian mechanics is a deterministic theory of quantum mechanics that is based on a set of n velocity functions for n particles, where these functions depend on the wavefunction from the n-body time-dependent Schroedinger equation. It is well know that Bohmian mechanics is not applicable to stationary states, since the velocity field for stationary states is the zero function. Recently, an alternative to Bohmian mechanics has been formulated, based on a conservation of energy equation, where the velocity fields are not the zero function, but this formalism is only applicable to stationary states with real valued wavefunctions. In this paper, Bohmian mechanics is merged with the alternative to Bohmian mechanics. This is accomplished by introducing an interpretation of the Bohm quantum potential. The final formalism gives dynamic particles for all states, including stationary states. The final main working equation contains two kinetic energy terms and a term that contains a factor that can be interpreted as a pressure. The derivation is a simple n-body generalization of the recent generalization, or refinement, of the Madelung equations.
Bohmian mechanics is endowed with an ontological package that supposedly allows to solve the main interpretational problems of quantum mechanics. We are concerned in this work by the emergence of classicality from the quantum mechanical substrate. We will argue that although being superficially attractive, the de Broglie-Bohm interpretation does not shed new light on the quantum-to-classical transition. This is due to nature of the dynamical law of Bohmian mechanics by which the particles follow the streamlines of the probability flow. As a consequence, Bohmian trajectories can be highly non-classical even when the wavefunction propagates along classical trajectories, as happens in semiclassical systems. In order to account for classical dynamics, Bohmian mechanics needs non-spreading and non-interfering wave packets: this is achieved for practical purposes by having recourse to decoherence and dense measurements. However one then faces the usual fundamental problems associated with the meaning of reduced density matrices. Moreover the specific assets of the de Broglie-Bohm interpretation – in particular the existence of point-like particles pursuing well-defined trajectories – would play no role in accounting for the emergence of classical dynamics.