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

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 $\psi$.