On the Equivalence Between the Schrodinger Equation in Quantum Mechanics and the Euler-Bernoulli Equation in Elasticity Theory

TL;DR

The paper establishes a precise mathematical equivalence between the Schrödinger equation $i \dot{\psi} = -\Delta \psi$ and a pair of Euler–Bernoulli equations for two real fields, $\ddot{u} + \Delta^2 u = 0$ and $\ddot{v} + \Delta^2 v = 0$, via the decomposition $\psi = u + i v$ and dependent initial data derived from $\psi(0,\cdot) = \psi_0$. By differentiating in time and analyzing in both physical and Fourier domains, it shows that the EB system reproduces the Schrödinger dynamics, with explicit initial-data relations $u(0,x)=u_0(x)$, $\dot{u}(0,x) = -\Delta v_0(x)$, $v(0,x)=v_0(x)$, $\dot{v}(0,x)=\Delta u_0(x)$; the bridge is extended to generalized Hamiltonians $H$, curved-space Laplacians $\Delta_g$, and even $p$-adic operators $D^{\alpha}$. Core contributions include the explicit EB formulations with potential, spectral solutions $u(t) = \cos(Ht)u_0 + \sin(Ht)v_0$, and the parameter relation $\frac{\hbar^2}{4m^2} = \frac{FI}{\mu}$ linking elastic constants to quantum constants. The work also discusses implications for symplectic and quantum computing, a potential elastic-analogue for the two-slit experiment, and geometric/defect theories arising from this equivalence, broadening the conceptual toolkit for quantum-classical correspondences.

Abstract

In this note, we show that the Schrodinger equation in quantum mechanics is mathematically equivalent to the Euler-Bernoulli equation for vibrating beams and plates in elasticity theory, with dependent initial data. Remarks are made on potential applications of this equivalence for symplectic and quantum computing, the two-slit experiment using vibrating beams and plates, and the p-adic Euler-Bernoulli equation.