Energy transport in the Schrödinger plate
Serge N. Gavrilov, Anton M. Krivtsov, Ekaterina V. Shishkina
TL;DR
The paper defines the Schrödinger plate as an infinite two-dimensional micro-polar Cosserat plate on a specially designed elastic foundation, establishing a direct link between its free dynamics and the time-dependent Schrödinger equation for a particle in a potential $V$, where the foundation encodes the spatial dependence of $V$. By deriving the governing equations via the direct Cosserat approach and applying a Kirchhoff constraint, the authors show how the Germain–Lagrange plate equation can be reformulated as the Schrödinger equation in this mechanical setting, and they introduce a complex wave function $\psi$ that satisfies $\mathcal S_+\psi=0$, providing an energetic interpretation that ties $\operatorname{Im}\psi$ to kinetic energy density and, under certain conditions, $\operatorname{Re}\psi$ to potential energy. In the zero-potential case ($V=0$), the work identifies specific moduli relations that yield a strain-energy form $2W=c_1(\Delta w)^2$, enabling the mechanical energy density to propagate exactly like the quantum probability density, though this stresses a non-positive-definite energy and shows the Schrödinger plate is not a Kirchhoff–Love plate (the equivalent Poisson ratio would be $\nu=1$, forbidden for 3D isotropic materials). For nonzero $V$, the authors introduce a modified foundation energy $\Pi^M$ so that a globally conserved energy density $\mathcal E^M$ yields $p=2\lambda\mathcal E^M$, preserving the energy–probability correspondence via a conserved energy flux. These results furnish a mechanical analogue of quantum dynamics and open avenues for analyzing energy transport in Cosserat plate systems with quantum-like behavior.
Abstract
In this paper, we introduce "the Schrödinger 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 Schrödinger 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 Schrödinger plate cannot be imagined as a thin elastic body composed of an isotropic linear material. The main property of the Schrödinger 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 Schrödinger equation.