The uncertainty principle in terms of isoperimetric inequalities

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 ${\sigma }_p\ge {\lambda }_1^{1\mathrm{/}2}\mathrm{\hslash }$, while ${\lambda }_1$ is the first Dirichlet eigenvalue of the Laplacian on $D$.