Energy transport in a free Euler–Bernoulli beam in terms of Schrödinger’s wave function

Abstract

The Schrödinger 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 the 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 Schrödinger equation for a free particle with arbitrary complex initial values and the set of ordered pairs of quantities (the linear strain measure and the particle velocity), which characterize solutions $u$ of the beam equation with arbitrary real initial values. Thus, the dynamics of a free infinite Euler–Bernoulli beam can be described by the Schrödinger 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 $\psi$.