On the relation between momentum uncertainty and thermal wavelength

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