The Classical and Quantum Mechanics of a Particle on a Knot

TL;DR

This paper investigates the classical and quantum behavior of a particle constrained to move along a knot on a torus, providing exact solutions and perturbative corrections, and comparing it to the particle on a circle.

Contribution

It offers a novel analysis of a particle on a knot, deriving classical solutions, exact quantum spectra in the thin torus limit, and perturbative corrections for finite thickness.

Findings

Classical equations of motion are solved in closed form.

Exact quantum energy spectrum obtained via Mathieu equation in the thin torus limit.

Finite-thickness corrections are incorporated perturbatively.

Abstract

A free particle is constrained to move on a knot obtained by winding around a putative torus. The classical equations of motion for this system are solved in a closed form. The exact energy eigenspectrum, in the thin torus limit, is obtained by mapping the time-independent Schrodinger equation to the Mathieu equation. In the general case, the eigenvalue problem is described by the Hill equation. Finite-thickness corrections are incorporated perturbatively by truncating the Hill equation. Comparisons and contrasts between this problem and the well-studied problem of a particle on a circle (planar rigid rotor) are performed throughout.