Uncertainty Relation and Minimum Wave Packet on Circle

TL;DR

This paper explores the uncertainty relation for a particle on a circle, defining position via Cartesian coordinates, deriving new URs, and analyzing minimum wave packets expressed by von Mises distributions, including extended UR series.

Contribution

It introduces a novel approach to UR on a circle using Cartesian variables, overcoming traditional angle-based difficulties, and constructs infinite series of total URs with extended variables.

Findings

Derived two new URs for Cartesian variables on a circle

Expressed minimum wave packets using von Mises distributions

Constructed infinite series of total URs for extended variables

Abstract

We discuss on the uncertainty relation (UR) for a closed one dimensional system (circle). In such a system, we cannot use the angle along the circle as a position variable. Otherwise we meet difficulties about the definition of the average position and the standard deviation (SD), and Hermitian property of angular momentum. From these reasons, we define the position variable as Cartesian variable $(X,Y)$ that have the periodic property for angle $\phi$. In the same way we define a SD by using that variables. Then we obtain two URs. We also discuss the minimum wave packet (MWP) on the circle. MWPs are expressed by von Mises distribution functions. Next we construct total URs by combining two URs for $X$ and $Y$. Furthermore, we extend the variables to $(X_n, Y_n)$ \; with $n= 1,2,3,\cdots$ and we have infinite series of total URs. We consider the meaning of such extended URs.