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

Area of Science:

• Quantum mechanics
• Classical mechanics
• Materials science

Background:

• A quantum particle in one dimension can exhibit behaviors analogous to a vibrating Euler-Bernoulli rod in three dimensions.
• The ratio of bending modulus to linear density for an electron mass (m_e) is comparable to that of a microtubule.

Purpose of the Study:

• To explore the classical mechanics analogy for quantum phenomena.
• To demonstrate quantum principles using a classical mechanical model of a rod.

Main Methods:

• Classical treatment of a vibrating Euler-Bernoulli rod.
• Analysis of rod deformations under axial forces and torques.
• Investigation of rod behavior in ring configurations.

Main Results:

• The uncertainty principle (ΔxΔp_x ≳ ℏ) governs transverse rod deformations.
• Quantized orbital angular momentum emerges in ring configurations.
• A "twist quantum" analogous to the magnetic flux quantum appears in torqued rings.

Conclusions:

• Classical mechanics of a vibrating rod can model quantum mechanical behaviors.
• This analogy provides a novel perspective on quantum phenomena without explicit quantization.
• The model offers insights into the behavior of systems like microtubules.