[Paper Review] Dualit\'e onde-corpuscule form\'ee par une masselotte oscillante dans un milieu \'elastique : \'etude th\'eorique et similitudes quantiques

Christian Borghesi|arXiv (Cornell University)|Sep 29, 2016

Fluid Dynamics and Heat Transfer3 citations

TL;DR

This paper proposes a macroscopic wave-particle system where a bead oscillating in an elastic medium governed by the Klein-Gordon equation exhibits dual wave-particle behavior. Using a unified mathematical formalism, it derives the bead's effective velocity and shows that the wave function ψ it generates obeys an equation analogous to the free Schrödinger equation under specific conditions, with particle and wave characteristics proportionally linked in linear and spherical cavities.

ABSTRACT

We introduce a dual wave-particle macroscopic system, where a bead oscillator oscillates in an elastic medium which obeys the Klein-Gordon equation. This theoretical system is mostly inspired by bouncing droplets experiments and bead sliding on a vibrating string experiments. This system is studied using a common and simple mathematical formalism. We compute the motion equation of the bead as well as the wave equation of the system. We introduce the effective velocity of the bead with respect to the elastic medium and the wave $\psi$, created by the bead, which modulates the natural wave of the medium. Provided some conditions, $\psi$ obeys an equation analogous to the free Schrodinger equation. In the case of linear and spherical cavities, the particle-like characteristics of the bead, expressed with its effective velocity, are proportional to the corresponding wave-like characteristics of the system. This paper is a translation of Dualite onde-corpuscule formee par une masselotte oscillante dans un milieu elastique : etude theorique et similitudes quantiques.

Motivation & Objective

• To model a macroscopic system exhibiting wave-particle duality inspired by bouncing droplet and vibrating string experiments.
• To derive the motion equation of a bead oscillating in an elastic medium governed by the Klein-Gordon equation.
• To investigate the emergence of a wave function ψ that modulates the medium and obeys a Schrödinger-like equation under specific conditions.
• To establish a quantitative proportionality between the particle-like effective velocity of the bead and the wave-like characteristics of ψ in confined geometries.

Proposed method

• Formulating the dynamics of a bead oscillating in an elastic medium using the Klein-Gordon equation as the underlying wave equation.
• Defining the effective velocity of the bead relative to the elastic medium and the wave ψ it generates.
• Deriving the wave equation for ψ and showing its formal similarity to the free Schrödinger equation under specific physical conditions.
• Analyzing the system in linear and spherical cavities to explore the relationship between particle and wave characteristics.
• Applying a unified mathematical formalism to describe both the particle motion and wave propagation in a consistent framework.

Experimental results

Research questions

• RQ1 Can a macroscopic bead oscillating in an elastic medium exhibit behavior analogous to quantum wave-particle duality?
• RQ2 Under what conditions does the wave ψ generated by the bead obey an equation similar to the free Schrödinger equation?
• RQ3 How are the particle-like effective velocity of the bead and the wave-like properties of ψ related in confined geometries?
• RQ4 What mathematical formalism enables a unified description of both wave and particle dynamics in this system?

Key findings

• The wave ψ generated by the oscillating bead satisfies an equation formally analogous to the free Schrödinger equation under specific physical conditions.
• The effective velocity of the bead is proportional to the wave-like characteristics of ψ in linear and spherical cavities.
• The system demonstrates a direct quantitative link between particle-like motion and wave-like behavior in confined elastic media.
• The mathematical formalism successfully unifies the description of particle motion and wave propagation in the system.