The uncertainty principle in terms of isoperimetric inequalities

TL;DR

This paper establishes a new form of the uncertainty principle in quantum mechanics by linking the momentum uncertainty to the geometric properties of the particle's localization domain through isoperimetric inequalities.

Contribution

It introduces a novel bound on momentum uncertainty based on the first Dirichlet eigenvalue of the Laplacian, connecting quantum uncertainty with geometric analysis.

Findings

Momentum standard deviation is bounded below by the square root of the first Dirichlet eigenvalue times ħ.

The bound applies to particles localized in compact domains with boundaries in n-dimensional space.

The result links quantum uncertainty to geometric properties of the localization domain.

Abstract

Simultaneous measurements of position and momentum are considered in $n$ dimensions. We find, that for a particle whose position is strictly localized in a compact domain $D\subset \mathbb{R}^n$ (spatial uncertainty) with non-empty boundary, the standard deviation of its momentum is sharply bounded by $\sigma_p \geq \lambda_1^{1/2}\hbar$, while $\lambda_1$ is the first Dirichlet eigenvalue of the Laplacian on $D$.