On the classical Schr\"odinger equation

TL;DR

This paper explores the classical Schrödinger equation, deriving it from classical mechanics and Bohmian mechanics, and analyzes its properties and numerical behavior, highlighting non-classical features and the quantum-to-classical transition.

Contribution

It provides a theoretical derivation of the classical Schrödinger equation from both classical and quantum perspectives, and investigates its properties through numerical simulations.

Findings

Classical Schrödinger equation derived from classical mechanics with non-crossing trajectories.

Quantum-to-classical transition leads to a classical Schrödinger equation for large particle systems.

Numerical simulations illustrate differences between classical and quantum Schrödinger equations.

Abstract

In this paper, the classical Schr\"odinger equation, which allows the study of classical dynamics in terms of wave functions, is analyzed theoretically and numerically. First, departing from classical (Newtonian) mechanics, and assuming an additional single-valued condition for the Hamilton's principal function, the classical Schr\"odinger equation is obtained. This additional assumption implies inherent non-classical features on the description of the dynamics obtained from the classical Schr\"odinger equation: the trajectories do not cross in the configuration space. Second, departing from Bohmian mechanics and invoking the quantum-to-classical transition, the classical Schr\"odinger equation is obtained in a natural way for the center of mass of a quantum system with a large number of identical particles. This quantum development imposes the condition of dealing with a narrow wave packet, which implicitly avoids the non-classical features mentioned above. We illustrate all the above points with numerical simulations of the classical and quantum Schr\"odinger equations for different systems.