New Uncertainty Principle for a particle on a Torus Knot

TL;DR

This paper derives new quantum uncertainty relations for a particle constrained on a torus knot, revealing how local geometric properties influence uncertainty measures beyond topological features.

Contribution

It introduces generalized uncertainty relations for particles on torus knots, explicitly involving geometric parameters and showing local geometry's role over topology.

Findings

Derived generalized uncertainty relations involving torus and knot parameters

Identified restrictions on wave functions due to knot geometry

In the thin torus limit, results reduce to particle on a circle

Abstract

The present work deals with quantum Uncertainty Relations (UR) subjected to the Standard Deviations (SD) of the relevant dynamical variables for a particle constrained to move on a torus knot. It is important to note that these variables have to obey the two distinct periodicities of the knotted paths embedded on the torus. We compute generalized forms of the SDs and the subsequent URs (following the Kennard-Robertson formalism). These quantities explicitly involve the torus parameters and the knot parameters where restrictions on the latter have to be taken into account. These induce restrictions on the possible form of wave functions that are used to calculate the SDs and URs and in our simple example, two distinct SDs and URs are possible. In a certain limit (thin torus limit), our results will reduce to the results for a particle moving in a circle. An interesting fact emerges that in the case of the SDs and URs, the local geometry of the knots plays the decisive role and not their topological properties.