Equivalent quantum equations in a system inspired by bouncing droplets experiments

TL;DR

This paper demonstrates that a classical elastic medium system with a point-like concretion can produce equations equivalent to quantum mechanics, including guidance formulas, Schrödinger equation, and stationary states, bridging classical and quantum descriptions.

Contribution

It introduces a classical elastic medium model that reproduces quantum equations and phenomena, providing a new perspective on quantum-classical correspondence.

Findings

Derives a covariant guidance formula similar to de Broglie-Bohm theory.

Retrieves a Schrödinger-like equation for the system.

Shows energy and momentum expressions match quantum mechanical forms.

Abstract

In this paper we study a classical and theoretical system which consists of an elastic medium carrying transverse waves and one point-like high elastic medium density, called concretion. We compute the equation of motion for the concretion as well as the wave equation of this system. Afterwards we always consider the case where the concretion is not the wave source any longer. Then the concretion obeys a general and covariant guidance formula, which leads in low-velocity approximation to an equivalent de Broglie-Bohm guidance formula. The concretion moves then as if exists an equivalent quantum potential. A strictly equivalent free Schr\"odinger equation is retrieved, as well as the quantum stationary states in a linear or spherical cavity. We compute the energy (and momentum) of the concretion, naturally defined from the energy (and momentum) density of the vibrating elastic medium. Provided one condition about the amplitude of oscillation is fulfilled, it strikingly appears that the energy and momentum of the concretion not only are written in the same form as in quantum mechanics, but also encapsulate equivalent relativistic formulas.