Comment on the uncertainty relation with periodic boundary conditions

TL;DR

This paper examines the uncertainty relation for a free particle in a periodic boundary condition setting, proposing a formulation analogous to Bloch waves and a simplified Robertson-type relation for practical use.

Contribution

It introduces a formulation of the uncertainty relation for particles with periodic boundary conditions and proposes a practical Robertson-type relation minimizing boundary effects.

Findings

Uncertainty relation adapted for periodic boundary conditions

Construction of free particle states similar to Bloch waves

A simplified Robertson-type relation for practical applications

Abstract

The Kennard-type uncertainty relation $\Delta x\Delta p >\frac{\hbar}{2}$ is formulated for a free particle with given momentum < \hat{p}>$ inside a box with periodic boundary conditions in the large box limit. Our construction of a free particle state is analogous to that of the Bloch wave in a periodic potential. A simple Robertson-type relation, which minimizes the effect of the box boundary and may be useful in some practical applications, is also presented.