# Quantum Mechanics of Particle on a torus knot: Curvature and Torsion Effects

## TL;DR

This paper investigates the quantum behavior of a particle constrained to a torus knot, explicitly incorporating curvature and torsion effects to reveal how the knot's topology influences energy spectra and eigenfunctions.

## Contribution

It introduces a novel approach using Geometry Induced Potential to account for curvature and torsion in quantum dynamics on a torus knot, highlighting topological effects.

## Key findings

- Curvature and torsion significantly affect energy eigenvalues.
- The knottedness influences quantum states and spectra.
- Identified a potential topological invariant related to the knot.

## Abstract

Constraints play an important role in dynamical systems. However, the subtle effect of constraints in quantum mechanics is not very well studied. In the present work we concentrate on the quantum dynamics of a point particle moving on a non-trivial torus knot. We explicitly take into account the role of curvature and torsion, generated by the constraints that keep the particle on the knot. We exploit the "Geometry Induced Potential (GIP) approach" to construct the Schrodinger equation for the dynamical system, obtaining thereby new results in terms of particle energy eigenvalues and eigenfunctions. We compare our results with existing literature that completely ignored the contributions of curvature and torsion. In particular, we explicitly show how the "knottedness" of the path influences the results. In the process we have revealed a (possibly un-noticed) "topological invariant".