New Uncertainty Principle for a particle on a Torus Knot
arxiv(2024)
摘要
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.
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