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 1d is isomorphic to a much heavier, vibrating, very thin Euler-Bernoulli rod in 3d, with ratio of bending modulus to linear density $(\hslash /2m{)}^2$. 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 $\Delta x\Delta p_x\gtrsim \hslash$ governs transverse deformations propagating down the inextensible, force and torque-free rod, and how orbital angular momentum quantized in units of $\hslash$ or $\hslash /2$ (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.

arXiv comment7 pages, 1 table, 1 figure, accepted for publication in Phys. Rev. E

References in corpus (10)

• Topological Boundary Modes in Isostatic Lattices
• Modeling semiflexible polymer networks
• Topological modes bound to dislocations in mechanical metamaterials
• Edge waves in plates with resonators: An elastic analogue of the quantum valley Hall effect
• Dynamic buckling of elastic rings in a soap film
• Topological boundary modes in jammed matter
• Dynamic buckling of an inextensible elastic ring: Linear and nonlinear analyses
• The deformation of an elastic rod with a clamp sliding along a smooth and curved profile
• Vibrations and fractional vibrations of rods, plates and Fresnel pseudo-processes
• The quantum character of buckling instabilities in thin rods

Cited by in corpus (2)

• Energy transport in a free Euler-Bernoulli beam in terms of Schrödinger's wave function
• Geometric Localization of Waves on Thin Elastic Structures