Quantum models of classical mechanics: maximum entropy packets

TL;DR

This paper develops quantum maximum entropy packets to model classical mechanics, showing they generalize Gaussian wave packets and better approximate classical trajectories when variances are large.

Contribution

It introduces a quantum maximum entropy framework for classical mechanics, extending the partition-function method and providing a new way to approximate classical states.

Findings

ME packets generalize Gaussian wave packets

Large variances improve classical approximation

A new method for calculating averages in quantum states

Abstract

In a previous paper, a project of constructing quantum models of classical properties has been started. The present paper concludes the project by turning to classical mechanics. The quantum states that maximize entropy for given averages and variances of coordinates and momenta are called ME packets. They generalize the Gaussian wave packets. A non-trivial extension of the partition-function method of probability calculus to quantum mechanics is given. Non-commutativity of quantum variables limits its usefulness. Still, the general form of the state operators of ME packets is obtained with its help. The diagonal representation of the operators is found. A general way of calculating averages that can replace the partition function method is described. Classical mechanics is reinterpreted as a statistical theory. Classical trajectories are replaced by classical ME packets. Quantum states approximate classical ones if the product of the coordinate and momentum variances is much larger than Planck constant. Thus, ME packets with large variances follow their classical counterparts better than Gaussian wave packets.