Macroscopic Oil Droplets Mimicking Quantum Behaviour: How Far Can We Push an Analogy?

Abstract

We describe a series of experimental analogies between fluid mechanics and quantum mechanics recently discovered by a team of physicists. These analogies arise in droplet systems guided by a surface (or pilot) wave. We argue that these experimental facts put ancient theoretical work by Madelung on the analogy between fluid and quantum mechanics into new light. After re-deriving Madelung’s result starting from two basic fluid mechanical equations (the Navier–Stokes equation and the continuity equation), we discuss the relation with the de Broglie–Bohm theory. This allows to make a direct link with the droplet experiments. It is argued that the fluid mechanical interpretation of quantum mechanics, if it can be extended to the general N-particle case, would have a considerable advantage over the Bohm interpretation: it could rid Bohm’s theory of its non-local character.

Acknowledgements

We would like to thank, for instructive discussions, Chérif Hamzaoui, Jean-Pierre Blanchet, and Yvon Gauthier; we are also grateful to two unnamed referees of this journal for particularly stimulating questions and remarks, which certainly helped to improve the article.

Notes

[1] In somewhat more detail, the wave field itself results from the superposition of the waves generated by the periodic impacts of the droplet on the film. It thus contains a memory of the past trajectory of the particle—a mild form of non-locality. A related type of non-locality in the system stems from the fact that the detailed characteristics of the wave field depend on the parameters of the whole experimental set-up, including the precise geometry of the bath. (These are mild forms of non-locality because per se they obviously do not invoke faster-than-light forces—which amount to strong, pathological non-locality.)

[2] In the semi-classical Bohr–Sommerfeld approximation, these discrete energy levels also lead to discrete radiuses on which the electron can move, just as in the droplets case.

[3] In the Bohr–Sommerfeld approximation, cf. previous note.

[4] To that end it would not be necessary to derive the numerical value of ; it would suffice to show which combination of fluid-mechanical constants formally takes the role of .

[5] Another argument that is sometimes advanced is that in Bohm's theory the ψ-field acts on the particle, but not the particle on the field (in the sense that the field is the same for different trajectories); which is again highly unusual.

[6] In the hydrodynamic framework, the ‘hidden variables’ of Bell's theorem are the (initial) positions of the streamlines, just as in Bohm's framework.