On the relation between momentum uncertainty and thermal wavelength

TL;DR

This paper derives an inequality linking momentum uncertainty and thermal wavelength for quantum particles in a Boltzmann state, extending previous work on the Heisenberg uncertainty bound.

Contribution

It introduces a new inequality connecting momentum uncertainty and thermal wavelength, based on a Boltzmann lower bound for the Heisenberg uncertainty product.

Findings

Established a lower bound for momentum uncertainty in terms of thermal wavelength.

Extended the Boltzmann lower bound to quantum particles in thermal states.

Provided a theoretical inequality relevant for quantum statistical mechanics.

Abstract

For quantum particles in a Boltzmann state, we derive an inequality between momentum uncertainty $\Delta p$ and thermal de Broglie wavelength $\lambda_{\rm th}$, expressed as $\Delta p \geq \sqrt{2\pi}\hbar/\lambda_{\rm th}$, as a corollary of the Boltzmann lower bound for the Heisenberg uncertainty product proposed in the previous work [EPL, 143, 20001 (2023)]