On the lower bound of the Heisenberg uncertainty product in the Boltzmann states

TL;DR

This paper derives a refined lower bound on the Heisenberg uncertainty product for particles in Boltzmann states, linking it to thermal de Broglie wavelength and enhancing understanding of thermodynamic measurement limits.

Contribution

It introduces a new lower bound on the uncertainty product in Boltzmann states, improving the characterization of thermodynamic precision over the traditional Heisenberg limit.

Findings

New bound expressed in terms of ratio between Δx and thermal de Broglie wavelength

Application to Brownian oscillator system demonstrating increased precision

Comparison showing the new bound is more accurate for thermodynamic limits

Abstract

The uncertainty principle lies at the heart of quantum mechanics, as it describes the fundamental trade-off between the precision of position and momentum measurements. In this work, we study the quantum particle in the Boltzmann states and derive a refined lower bound on the product of {\Delta}x and {\Delta}p. Our new bound is expressed in terms of the ratio between {\Delta}x and the thermal de Broglie wavelength, and provides a valuable tool for characterizing thermodynamic precision. We apply our results to the Brownian oscillator system, where we compare our new bound with the well-known Heisenberg uncertainty principle. Our analysis shows that our new bound offers a more precise measure of the thermodynamic limits of precision.