Energy transport in a free Euler-Bernoulli beam in terms of Schr\"odinger's wave function

TL;DR

This paper establishes a direct correspondence between solutions of the Schrödinger equation and the classical Euler-Bernoulli beam equation, linking quantum wave functions with classical energy transport in beams.

Contribution

It demonstrates a one-to-one mapping between Schrödinger wave functions and beam dynamics, revealing that energy propagation in beams can be described using quantum mechanics concepts.

Findings

Energy density in the beam propagates like the probability density in Schrödinger's equation.

Solutions of the Schrödinger equation correspond to solutions of the Euler-Bernoulli beam equation.

The dynamics of a free Euler-Bernoulli beam can be equivalently described by the Schrödinger equation.

Abstract

The Schr\"odinger equation is not frequently used in the framework of the classical mechanics, though historically this equation was derived as a simplified equation, which is equivalent to the classical Germain-Lagrange dynamic plate equation. The question concerning the exact meaning of this equivalence is still discussed in modern literature. In this note, we consider the one-dimensional case, where the Germain-Lagrange equation reduces to the Euler-Bernoulli equation, which is used in the classical theory of a beam. We establish a one-to-one correspondence between the set of all solutions (i.e., wave functions $\psi$) of the 1D time-dependent Schr\"odinger equation for a free particle with arbitrary complex valued initial data and the set of ordered pairs of quantities (the beam strain and the particle velocity), which characterize solutions $u$ of the beam equation with arbitrary real valued initial data. Thus, the dynamics of a free infinite Euler-Bernoulli beam can be described by the Schr\"odinger equation for a free particle and vice versa. Finally, we show that for two corresponding solutions $u$ and $\psi$ the mechanical energy density calculated for $u$ propagates in the beam exactly in the same way as the probability density calculated for $\psi$.