Quantum particles from classical statistics

Abstract

Quantum particles and classical particles are described in a common setting of classical statistical physics. The property of a particle being "classical" or "quantum" ceases to be a basic conceptual difference. The dynamics differs, however, between quantum and classical particles. We describe position, motion and correlations of a quantum particle in terms of observables in a classical statistical ensemble. On the other side, we also construct explicitly the quantum formalism with wave function and Hamiltonian for classical particles. For a suitable time evolution of the classical probabilities and a suitable choice of observables all features of a quantum particle in a potential can be derived from classical statistics, including interference and tunneling. Besides conceptual advances, the treatment of classical and quantum particles in a common formalism could lead to interesting cross-fertilization between classical statistics and quantum physics.

Quantum Particles from Classical Probabilities in Phase Space

“…[8], this sign is largely determined by analyticity properties. The classical wave function plays for the description of a single classical particle the same role as the quantum wave function for a quantum particle.…”

“…(10) in the limit of infinitely sharp probability distributions. For any nonzero width of w(x, p), however, the distribution will broaden in the course of the time evolution [8,9], as well known from the dispersion of quantum particles. With the use of the Liouville operator (9) our discussion of the classical wave function shares some important features with the Hilbert space formulation of classical mechanics by Koopman and von Neumann [10].…”

“…[8] that it is possible, in principle, to formulate a classical statistical ensemble for a quantum particle such that the position and momentum observables can be associated to standard classical observables which take a fixed value in every state of some microscopic ensemble. However, for the description of a quantum particle by a probability distribution in phase space w(x, p) it is not clear a priori if the position and momentum observables should be directly associated with x and p. We will actually see that this is not the case.…”

“…In the present paper we include in our statistical formulation a new type of "statistical observables" [8] that measure statistical properties of the probability distribution as its roughness in position or momentum. The expectation value of such a statistical observable is computable for any given w(x, p).…”

“…II we recall the probabilistic formulation for a single classical particle in terms of a wave function [8]. The concept of the classical wave function allows for the introduction of statistical observables which can be computed from the probability density for a classical particle.…”