Energy transport in a free Euler-Bernoulli beam in terms of Schrödinger's wave function
Serge N. Gavrilov, Anton M. Krivtsov, Ekaterina V. Shishkina
TL;DR
The paper establishes a one-to-one correspondence between solutions of the 1D free-particle Schrödinger equation and the Euler-Bernoulli beam equation by relating beam strain γ and particle velocity v to a Schrödinger wave function ψ. It shows that an initial-value problem for a real beam can be reformulated as a pair of conjugate Schrödinger IVPs, with real data mapping to conjugate complex initial data, and, for real data, the beam displacement emerges from a single Schrödinger evolution via u = 2 Re Ψ_+. An energy-transport identity is derived, linking the mechanical energy density 𝔼 to the quantum probability density ρ through ρ = λ|ψ|^2 = 2λ𝔼, and the corresponding fluxes are shown to obey a balance equation. The work provides a constructive framework to compute beam dynamics from Schrödinger solutions and suggests generalizations to nonzero potentials and higher dimensions, potentially via Cosserat-type continua; this offers a conceptually clear bridge between classical energy dynamics and quantum probability flow in a one-dimensional setting.
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 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 $ψ$) of the 1D time-dependent Schrödinger 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 Schrödinger equation for a free particle and vice versa. Finally, we show that for two corresponding solutions $u$ and $ψ$ the mechanical energy density calculated for $u$ propagates in the beam exactly in the same way as the probability density calculated for $ψ$.