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arXiv:2308.07613 · Europhysics Letters (EPL)

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

Yao Wang

Published Jul 1, 2023• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1209/0295-5075/ace64f

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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.
For quantum particles in a Boltzmann state, we derive an inequality between momentum uncertainty ΔpΔpΔp and thermal de Broglie wavelength λthλ_{\rm th}λth​, expressed as Δp≥2πℏ/λthΔp \geq \sqrt{2π}\hbar/λ_{\rm th}Δp≥2π​ℏ/λth​, as a corollary of the Boltzmann lower bound for the Heisenberg uncertainty product proposed in the previous work [EPL, 143, 20001 (2023)]
Uncertainty relations represent a foundational principle in quantum mechanics, imposing inherent limits on the precision with which \textit{mechanically} conjugate variables such as position and momentum can be simultaneously determined. This work establishes analogous relations for \textit{thermodynamically} conjugate variables -- specifically, a classical intensive parameter θθθ and its corresponding extensive quantum operator O^\hat{O}O^ -- in equilibrium states. We develop a framework to derive a rigorous thermodynamic uncertainty relation for such pairs, where the uncertainty of the classical parameter θθθ is quantified by its quantum Fisher information Fθ\mathcal{F}_θFθ​. The framework is based on an exact integral representation that relates Fθ\mathcal{F}_θFθ​ to the autocorrelation function of operator O^\hat{O}O^. From this representation, we derive a tight upper bound for the quantum Fisher information, which yields a thermodynamic uncertainty relation: Δθ ΔO‾≥kBTΔθ\,\overline{ΔO} \ge k_\text{B}TΔθΔO≥kB​T with ΔO‾≡∂θ⟨O^⟩ Δθ\overline{ΔO}\equiv\partial_θ\langle\hat{O}\rangle\,ΔθΔO≡∂θ​⟨O^⟩Δθ and TTT is the system temperature. The result establishes a fundamental precision limit for quantum sensing and metrology in thermal systems, directly connecting it to the thermodynamic properties of linear response and fluctuations.
According to Heisenberg’s uncertainty principle [1], both, position and momentum of a particle, cannot be known with ultimate accuracy; the product: Δ x . Δ p > h . Whereas in the case of Laser-cooled atoms, we know that velocity of the particle is close to zero; and its position x is so perfectly known, that an atom is said to be held in Laser forceps [2]. These experiments show that both position and momentum of atoms are accurately knowable simultaneously! Similarly, when momentum of an atom is zero, the wavelength of its de Broglie wave is very very long; and according to quantum-mechanical interpretation, the atom is likely to be detected anywhere within its de Broglie wavelength, with higher probability at the peaks of the wave. Whereas the experiments with Laser-cooled atoms show that the atom is confined precisely within the Laser tweezers. Based on this observation a new experiment is proposed here in which, instead of quantum mechanical waves, which have wavelength h / m v , corresponding to phase velocity of the wave, we can carefully observe the waves corresponding to group-velocity, of the wavelength h v / m c^2. And we can expect that a particle may deterministically tunnel from one peak of the group-wave to next peak of the group wave; and can be detected at deterministically predictable points.
Boltzmann’s statistical interpretation of the second law of thermodynamics indicates that systems evolve from less probable to more probable states. In an isolated system, entropy reaches its maximum at equilibrium, corresponding to the greatest number of accessible microstates. The quantum uncertainty principle, a fundamental feature of quantum mechanics, reflects the intrinsic probabilistic nature of physical systems. This study proposes that the equilibrium state defined by maximal entropy and microstate accessibility also represents a condition of maximal intrinsic uncertainty. We examine the connection between Boltzmann’s entropy framework and the quantum mechanical uncertainty relation between position and momentum, focusing on localized harmonic oscillator systems. Properties of the uncertainty product are analyzed in detail. Both the uncertainty principle and the second law of thermodynamics are rooted in probabilistic foundations. Our findings suggest that the uncertainty principle provides a conceptual perspective on thermodynamic behavior.
Within the framework of thermo-field-dynamics, we define the reduced probability densities, and investigate the sum of information-theoretic entropies in quantum mechanics associated with the measurements of the position and momentum of a particle surrounded by the thermal environment. It is found that this quantity cannot be made arbitrarily small but has a universal lower bound dependent on the temperature. We develop the finite-temperature variational technique for the reduced system, and show that this relation, termed the thermal information-entropic uncertainty relation, is exactly saturated by the thermal coherent state. We also show that the Heisenberg uncertainty relation at finite temperature can be derived from this relation.