Why bouncing droplets are a pretty good model of quantum mechanics

Abstract

In 2005, Couder, Protiere, Fort and Badouad showed that oil droplets bouncing on a vibrating tray of oil can display nonlocal interactions reminiscent of the particle-wave associations in quantum mechanics; in particular they can move, attract, repel and orbit each other. Subsequent experimental work by Couder, Fort, Protiere, Eddi, Sultan, Moukhtar, Rossi, Molacek, Bush and Sbitnev has established that bouncing drops exhibit single-slit and double-slit diffraction, tunnelling, quantised energy levels, Anderson localisation and the creation/annihilation of droplet/bubble pairs. In this paper we explain why. We show first that the surface waves guiding the droplets are Lorentz covariant with the characteristic speed c of the surface waves; second, that pairs of bouncing droplets experience an inverse-square force of attraction or repulsion according to their relative phase, and an analogue of the magnetic force; third, that bouncing droplets are governed by an analogue of Schroedinger's equation where Planck's constant is replaced by an appropriate constant of the motion; and fourth, that orbiting droplet pairs exhibit spin-half symmetry and align antisymmetrically as in the Pauli exclusion principle. Our analysis explains the similarities between bouncing-droplet experiments and the behaviour of quantum-mechanical particles. It also enables us to highlight some differences, and to predict some surprising phenomena that can be tested in feasible experiments.