Energy transport in the Schr\"odinger plate

TL;DR

This paper introduces the Schr"odinger plate, a novel elastic medium whose energy transport mimics quantum probability propagation, linking classical elasticity with quantum mechanics through a specialized elastic foundation.

Contribution

It develops a new elastic model called the Schr"odinger plate, connecting classical plate theory with the Schr"odinger equation for the first time.

Findings

Energy propagates exactly as quantum probability density.

The plate's properties are governed by a modified strain energy.

The model links classical elasticity to quantum dynamics.

Abstract

In this paper, we introduce "the Schr\"odinger plate." This is an infinite two-dimensional linear micro-polar elastic medium, with out-of-plane degrees of freedom, lying on a linear elastic foundation of a special kind. Any free motion of the plate can be corresponded to a solution of the two-dimensional Schr\"odinger equation for a single particle in the external potential field $V$. The specific dependence of the potential $V$ on the position is taken into account in the properties of the plate elastic foundation. The governing equations of the plate are derived as equations of the two-dimensional constraint Cosserat continuum using the direct approach. The plate dynamics can be described by the classical Germain-Lagrange equation for a plate, but the strain energy is different from the one used in the classical Kirchhoff-Love plate theory. Namely, the Schr\"odinger plate cannot be imagined as a thin elastic body composed of an isotropic linear material. The main property of the Schr\"odinger plate is as follows: the mechanical energy propagates in the plate exactly in the same way as the probability density propagates according to the corresponding Schr\"odinger equation.