Madelung Mechanics and Superoscillations

In single-particle Madelung mechanics, the single-particle quantum state Ψ(x⃗,t) = R(x⃗,t) eiS(x⃗,t)/ is interpreted as comprising an entire conserved fluid of classical point particles, with local density R(x⃗,t)² and local momentum ∇⃗S(x⃗,t) (where R and S are real). The Schr\"{o}dinger equation gives rise to the continuity equation for the fluid, and the Hamilton-Jacobi equation for particles of the fluid, which includes a new density-dependent quantum potential energy term Q(x⃗,t) = -²/2m∇⃗R(x⃗,t)R(x⃗,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(x⃗,t)<0. For energy superposition states with band-limit E₊, the situation is slightly more complicated, and the bound is no longer Q(x⃗,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(x⃗,t)<0 for general superpositions. Detailed examples are given which illustrate the role of the quantum potential and superoscillations in a range of scenarios.