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

Hefei National Research Center for Physical Sciences at the Microscale, University of Science and Technology of China - Hefei, Anhui 230026, China

^(a) E-mail: wy2010@ustc.edu.cn (corresponding author)

Received: 20 May 2023
Accepted: 11 July 2023

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 and . Our new bound is expressed in terms of the ratio between 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.