Uncertainty Principle for Ensembles

Abstract

Uncertainty principles for the average fluctuations of momentum and position per particle in an ensemble are derived. For bosons, these results are similar to the usual uncertainty principle, with slight corrections for the correlated motion of the particles. For fermions, a large correction for the Pauli exclusion principle is necessary, which causes $〈p^2〉〈{\mathcal{r}}^2〉$ to be proportional to the two-thirds power of the number of particles in the ensemble.