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

Abstract

In some parameter and solution regimes, a minimally coupled nonrelativistic quantum particle in one dimension is isomorphic to a much heavier, vibrating, very thin Euler-Bernoulli rod in three dimensions with ratio of bending modulus to linear density (ℏ/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 ΔxΔp_{x}≳ℏ governs transverse deformations propagating down the inextensible, force and torque-free rod, and how orbital angular momentum quantized in units of ℏ or ℏ/2 (depending on calculation method) emerges when the force and torque-free inextensible rod is formed into a ring. For torqued rings with large wave numbers, 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.