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arXiv:0902.0141

Classical mechanics as the many-particle limit of quantum mechanics

Gabriele Carcassi

Published Feb 2, 2009• 1 citation
arXivHTMLPDF

Abstract

We derive the classical limit of quantum mechanics by describing the center of mass of a system constituted by a large number of particles. We will show that in that limit the commutator between the position and velocity of the center of mass is infinitesimal, which allows both to be known with great precision. We then show how the infinitesimal commutator allows for the definition of functions of position and velocity, and how the commutator reduces to a Poisson bracket.

DOI
10.48550/arxiv.0902.0141

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We derive the classical limit of quantum mechanics by describing the center of mass of a system constituted by a large number of particles. We will show that in that limit the commutator between the position and velocity of the center of mass is infinitesimal, which allows both to be known with great precision. We then show how the infinitesimal commutator allows for the definition of functions of position and velocity, and how the commutator reduces to a Poisson bracket.
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.
The framework of entropic dynamics (ED) allows one to derive quantum mechanics as an application of entropic inference. In this work we derive the classical limit of quantum mechanics in the context of ED. Our goal is to find conditions so that the center of mass (CM) of a system of N particles behaves as a classical particle. What is of interest is that ~ remains finite at all steps in the calculation and that the classical motion is obtained as the result of a central limit theorem. More explicitly we show that if the system is sufficiently large, and if the CM is initially uncorrelated with other degrees of freedom, then the CM follows a smooth trajectory and obeys the classical Hamilton-Jacobi with a vanishing quantum potential.
The framework of entropic dynamics (ED) allows one to derive quantum mechanics as an application of entropic inference. In this work we derive the classical limit of quantum mechanics in the context of ED. Our goal is to find conditions so that the center of mass (CM) of a system of N particles behaves as a classical particle. What is of interest is that ~ remains finite at all steps in the calculation and that the classical motion is obtained as the result of a central limit theorem. More explicitly we show that if the system is sufficiently large, and if the CM is initially uncorrelated with other degrees of freedom, then the CM follows a smooth trajectory and obeys the classical Hamilton-Jacobi with a vanishing quantum potential.
The framework of entropic dynamics (ED) allows one to derive quantum mechanics as an application of entropic inference. In this work we derive the classical limit of quantum mechanics in the context of ED. Our goal is to find conditions so that the center of mass (CM) of a system of N particles behaves as a classical particle. What is of interest is that ~ remains finite at all steps in the calculation and that the classical motion is obtained as the result of a central limit theorem. More explicitly we show that if the system is sufficiently large, and if the CM is initially uncorrelated with other degrees of freedom, then the CM follows a smooth trajectory and obeys the classical Hamilton-Jacobi with a vanishing quantum potential.
The framework of entropic dynamics (ED) allows one to derive quantum mechanics as an application of entropic inference. In this work we derive the classical limit of quantum mechanics in the context of ED. Our goal is to find conditions so that the center of mass (CM) of a system of N particles behaves as a classical particle. What is of interest is that ħ remains finite at all steps in the calculation and that the classical motion is obtained as the result of a central limit theorem. More explicitly we show that if the system is sufficiently large, and if the CM is initially uncorrelated with other degrees of freedom, then the CM follows a smooth trajectory and obeys the classical Hamilton-Jacobi with a vanishing quantum potential.