Euler & Lagrange versus Heisenberg & Schroedinger: Dynamical Pictures in Classical and Quantum Mechanics

TL;DR

This paper explores the deep analogies between classical and quantum dynamical pictures, using the Koopman-von Neumann formalism to bridge concepts and demonstrate classical-quantum correspondences in dynamics.

Contribution

It introduces a framework linking classical fluid descriptions to quantum dynamical pictures, extending classical concepts like chaos to quantum mechanics.

Findings

Classical Eulerian and Lagrangian descriptions correspond to quantum Schrödinger and Heisenberg pictures.

The Koopman-von Neumann formalism enables a unified view of classical and quantum dynamics.

Classical evolution of particle ensembles under constant force is analytically described.

Abstract

Using quantum-classical analogies, we find that dynamical pictures of quantum mechanics have precise counterparts in classical mechanics. In particular, the Eulerian and Lagrangian descriptions of fluid dynamics in classical mechanics are the analogs of the Schroedinger and Heisenberg pictures in quantum mechanics, respectively. Similarities between classical and quantum dynamical pictures are explored within the framework of the Koopman-von Neumann formalism. These allow for a natural definition of various dynamical pictures in classical mechanics as well as the application of classical concepts to quantum dynamics. As an illustration, we use the interaction picture to find the classical evolution of an ensemble of particles of equal initial momenta and arbitrary configuration density under the action of a constant force in one dimension. As a second example, we discuss the extension of the ideas of sensitivity to initial conditions and chaos in classical mechanics to quantum mechanics.