For quantum particles in a Boltzmann state, we derive an inequality between momentum uncertainty $Δp$ and thermal de Broglie wavelength $λ_{\rm th}$, expressed as $Δp \geq \sqrt{2π}\hbar/λ_{\rm th}$, as a corollary of the Boltzmann lower bound for the Heisenberg uncertainty product proposed in the previous work [EPL, 143, 20001 (2023)]
For quantum particles in a Boltzmann state, we derive an inequality between momentum uncertainty Δp and thermal de Broglie wavelength λth, expressed as Δp≥2πℏ/λth, as a corollary of the Boltzmann lower bound for the Heisenberg uncertainty product proposed in the previous work [EPL, 143, 20001 (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.
The idea to base the uncertainty relation for photons on the electromagnetic energy distribution in space enabled us to derive a sharp inequality that expresses the uncertainty relation [Phys. Rev. Lett. 108, 140401 (2012)]. An alternative version of the uncertainty relation derived in this paper is closer in spirit to the original Heisenberg relation because it employs the analog of the position operator for the photon---the center of the energy operator. The noncommutativity of the components of the center of the energy operator results in the increase of the bound $3\ensuremath{\hbar}/2$ in the standard Heisenberg uncertainty relation in three dimensions. This difference diminishes with the increase of the photon energy. In the infinite-momentum frame, the lower bound in the Heisenberg uncertainty relations for photons is the same as in nonrelativistic quantum mechanics. A similar uncertainty relation is also derived for coherent photon beams. This relation has direct experimental consequences since it gives a precise relationship between the spectral composition of the laser beam and the minimal focal volume.
Standard-deviation and entropic formulations of uncertainty principle capture the spread of the probability distribution but say little about the probability itself contained in a small region. We introduce the confidence uncertainty Δcx(θx) as the minimal Lebesgue measure of the support set in which the particle is found with probability at least θx, and the companion interval confidence uncertainty ΔIx(θx) which restricts the support to a single interval. We prove two complementary uncertainty inequalities. (i) For θx+θp≤1 both confidence uncertainties can be made arbitrarily small simultaneously, so that no nontrivial product bound holds; in particular, position and momentum can be jointly localised with probability at least~50%. (ii) For θx+θp>1 a lower bound holds: combining Lenard's projection inequality with the Donoho--Stark operator-norm bound we obtain ΔcxΔcp≥2πℏ(θxθp−(1−θx)(1−θp))2, and for the interval version we obtain the sharp implicit Landau--Pollak bound ΔIxΔIp≥4ℏλ0−1((θxθp−(1−θx)(1−θp))2), where λ0(c) is the largest prolate-spheroidal eigenvalue. We support the analytical bounds with numerical evaluation of λ0(c), provide closed-form small-c and large-c asymptotics, compute the optimal Slepian-superposition states that saturate the interval bound, and compare the resulting product against the variance Heisenberg--Kennard, the Białynicki-Birula--Mycielski entropic, and the Donoho--Stark concentration bounds. The unified picture provides a complete phase diagram on (θx,θp)∈[0,1]2.
Recently, S\'anchez-Moreno et al. [New J. Phys. 8, 330 (2006)] have shown that the three-dimensional Heisenberg uncertainty relation ${\ensuremath{\sigma}}_{r}{\ensuremath{\sigma}}_{p}\ensuremath{\ge}3\ensuremath{\hbar}/2$ might be significantly sharpened if the relevant quantum state describes the particle in a central potential. I extend that result to the case of states which are not the eigenstates of the square of the angular momentum operator. I derive a lower bound for ${\ensuremath{\sigma}}_{r}{\ensuremath{\sigma}}_{p}$ which involves the mean value and the variance of the square of the angular momentum operator.