Dynamics of certain Euler-Bernoulli rods and rings from a minimal coupling quantum isomorphism

TL;DR

This paper explores a classical analogy between a minimally coupled quantum particle in 1D and a vibrating Euler-Bernoulli rod in 3D, revealing quantum-like phenomena such as uncertainty principles and quantized angular momentum emerging from classical rod dynamics.

Contribution

It demonstrates how classical Euler-Bernoulli rod dynamics can mimic quantum behaviors, including uncertainty relations and quantized angular momentum, through a minimal coupling framework.

Findings

Uncertainty principle governs transverse deformations of the rod.

Quantized orbital angular momentum emerges in ring configurations.

A twist quantum analogous to magnetic flux quantum appears in large wavenumber rings.

Abstract

In some parameter and solution regimes, a minimally coupled nonrelativistic quantum particle in 1d is isomorphic to a much heavier, vibrating, very thin Euler-Bernoulli rod in 3d, with ratio of bending modulus to linear density $(\hbar/2m)^2$. For $m=m_e$, this quantity is comparable to that of a microtubule. Axial forces and torques applied to the rod play the role of scalar and vector potentials, respectively, and rod inextensibility plays the role of normalization. We show how an uncertainty principle $\Delta x\Delta p_x\gtrsim\hbar$ governs transverse deformations propagating down the inextensible, force and torque-free rod, and how orbital angular momentum quantized in units of $\hbar$ or $\hbar/2$ (depending on calculation method) emerges when the force and torque-free inextensible rod is formed into a ring. For torqued rings with large wavenumbers, a ``twist quantum'' appears that is somewhat analogous to the magnetic flux quantum. These and other results are obtained from a purely classical treatment of the rod, i.e., without quantizing any classical fields.