Madelung mechanics and superoscillations

Abstract

In single-particle Madelung mechanics, the single-particle quantum state Ψ(x→,t)=R(x→,t)eiS(x→,t)/ℏ is interpreted as comprising an entire conserved fluid of classical point particles, with local density R(x→,t)2 and local momentum ∇→S(x→,t) (where R and S are real). The Schrödinger equation gives rise to the continuity equation for the fluid, and the Hamilton–Jacobi equation for particles of the fluid, which includes an additional density-dependent quantum potential energy term Q(x→,t)=−ℏ22m∇→R(x→,t)R(x→,t) , which is all that makes the fluid behavior nonclassical. In particular, the quantum potential can become negative and create a nonclassical boost in the kinetic energy. This boost is related to superoscillations in the wavefunction, where the local frequency of Ψ exceeds its global band limit. Berry showed that for states of definite energy E, the regions of superoscillation are exactly the regions where Q(x→,t)<0 . For energy superposition states with band-limit E+ , the situation is slightly more complicated, and the bound is no longer Q(x→,t)<0 . However, the fluid model provides a definite local energy for each fluid particle which allows us to define a local band limit for superoscillation, and with this definition, all regions of superoscillation are again regions where Q(x→,t)<0 for general superpositions. An alternative interpretation of these quantities involving a reduced quantum potential is reviewed and advanced, and a parallel discussion of superoscillation in this picture is given. Detailed examples are given which illustrate the role of the quantum potential and superoscillations in a range of scenarios.