Linkage between thermodynamic quantities and the uncertainty relation in harmonic oscillator model

Abstract

The Heisenberg uncertainty relation, ΔxΔpx ⩾ ℏ/2 (ℏ ≡ h/2π), is well known. The purpose of this paper is to study a linkage between macroscopic thermodynamic quantities and the uncertainty relation. To simplify matters, we consider one-dimensional harmonic oscillator model. The calculations are carried out by two steps. First, the thermal average of the uncertainty relation is obtained for canonical system in thermal equilibrium at temperature T, which is denoted by (Δx)T(Δpx)T. Second, the thermodynamic quantities are expressed analytically as a function of (Δx)T(Δpx)T within the harmonic oscillator model. Finally, a connection between the energy fluctuations and the uncertainty relation is clarified for the Einstein’s heat capacity model. The analysis is made on the basis of fluctuation theory. The uncertainty relation is valid not only in quantum state but also at finite temperature.