Madelung Mechanics and Superoscillations

• URL: http://arxiv.org/abs/2405.06358v1
• Date: Fri, 10 May 2024 09:46:08 GMT
• Title: Madelung Mechanics and Superoscillations
• Authors: Mordecai Waegell,
• Abstract summary: In Madelung mechanics, the single-particle quantum state $Psi(vecx,t) = R(vecx,t) eiS(vecx,t)/hbar$ is interpreted as comprising an entire conserved fluid of classical point particles. The quantum potential can become negative and create a nonclassical boost in the kinetic energy.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In single-particle Madelung mechanics, the single-particle quantum state $\Psi(\vec{x},t) = R(\vec{x},t) e^{iS(\vec{x},t)/\hbar}$ is interpreted as comprising an entire conserved fluid of classical point particles, with local density $R(\vec{x},t)^2$ and local momentum $\vec{\nabla}S(\vec{x},t)$ (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 $Q(\vec{x},t) = -\frac{\hbar^2}{2m}\frac{\vec{\nabla}R(\vec{x},t)}{R(\vec{x},t)}$, 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 $\Psi$ exceeds its global band limit. Berry showed that for states of definite energy $E$, the regions of superoscillation are exactly the regions where $Q(\vec{x},t)<0$. For energy superposition states with band-limit $E_+$, the situation is slightly more complicated, and the bound is no longer $Q(\vec{x},t)<0$. 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 $Q(\vec{x},t)<0$ for general superpositions. Detailed examples are given which illustrate the role of the quantum potential and superoscillations in a range of scenarios.

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