In Fourier analysis and quantum mechanics, the Heisenberg uncertainty principle stands as a fundamental result. Its classical version provides a product lower bound for the variances of position and frequency. More recently, for vector-valued functions, Dang, Deng and Qian introduced an improved inequality featuring an extra covariance term derived from the current density. The present work performs a one-dimensional numerical comparison of the two lower bounds using four test functions: a pure Gaussian, an asymmetric sum of two Gaussians, a linear chirp, and a vector-valued signal with spatially separated orthogonal components. For the Gaussian, the covariance term vanishes up to machine precision, making the two bounds identical. For the asymmetric sum, the refined bound exceeds the classical one by about three percent. The chirp signal yields a dramatically larger refined bound, approximately sixteen times above the classical one. For the vector-valued case with each component a real Gaussian, the refined bound is about eight times higher. These numerical outcomes confirm the validity of the refined inequality and highlight the sensitivity of the covariance term to phase structure and vectorial coupling.
In Fourier analysis and quantum mechanics, the Heisenberg uncertainty principle stands as a fundamental result. Its classical version provides a product lower bound for the variances of position and frequency. More recently, for vector-valued functions, Dang, Deng and Qian introduced an improved inequality featuring an extra covariance term derived from the current density. The present work performs a one-dimensional numerical comparison of the two lower bounds using four test functions: a pure Gaussian, an asymmetric sum of two Gaussians, a linear chirp, and a vector-valued signal with spatially separated orthogonal components. For the Gaussian, the covariance term vanishes up to machine precision, making the two bounds identical. For the asymmetric sum, the refined bound exceeds the classical one by about three percent. The chirp signal yields a dramatically larger refined bound, approximately sixteen times above the classical one. For the vector-valued case with each component a real Gaussian, the refined bound is about eight times higher. These numerical outcomes confirm the validity of the refined inequality and highlight the sensitivity of the covariance term to phase structure and vectorial coupling.
A sharper uncertainty inequality which exhibits a lower bound larger than that in the classical N -dimensional Heisenberg’s uncertainty principle is obtained, and extended from N -dimensional Fourier transform domain to two N -dimensional fractional Fourier transform domains. The conditions that reach the equality relation of the uncertainty inequalities are deduced. Example and simulation are performed to illustrate that the newly derived uncertainty principles are truly sharper than the existing ones in the literature. The new proposals’ applications in time-frequency analysis and optical system analysis are also given.
In this paper we revisit the quantitative formulation of the Heisenberg uncertainty principle. The primary version of this principle establishes the impossibility of refined simultaneous measurement of position x and momentum u for a (1-dimensional) quantum particle in terms of variances: <||x||^2> <||u||^2> ≥ 1/4. Since this inequality applies provided each variance exists, some authors proposed entropic versions of this principle as an alternative (employing Shannon's or Rényi's entropies). As another alternative, we consider moments-based formulations and show that inequalities involving moments of orders other than 2 can be found. Our procedure is based on the Rényi entropic versions of the Heisenberg relation together with the search for the maximal entropy un- der statistical moments' constraints ( <||x||^a> and <||u||^b> ). Our result improves a relation proposed very recently by Dehesa et al. [1] where the same approach was used but starting with the Shannon version of the entropic uncertainty relation. Furthermore, we show that when a = b, the best bound we can find with our approach coincides with that of Ref. [1] and, in addition, for a = b = 2 the variance-based Heisenberg relation is recovered. Finally, we illustrate our results in the cases of d-dimensional hydrogenic systems.
ABSTRACT. The well-known second moment Heisenberg-Weyl inequality (or uncertainty relation) states: Assume that f: R → C is a complex valued function of a random real variable x such that f ∈ L2 (R), where R = (−∞, ∞). Then the product of the second moment of the random real x for |f | 2 and the second moment of the random real ξ for ∣ f ∣ 2 is at least E 2 R,|f | 4π, where f is the Fourier transform of f, f (ξ) = ∫ R e−2iπξxf (x) dx and f (x) = ∫ R e2iπξx ˆ f (ξ) dξ, and E 2 R,|f | = ∫ R |f (x)|2 dx. This uncertainty relation is wellknown in classical quantum mechanics. In 2004, the author generalized the afore-mentioned result to the higher order moments for L2 (R) functions f. In this paper, a refined form of the generalized Heisenberg-Weyl type inequality is established.
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.
Heisenberg’s uncertainty principle establishes a fundamental restriction on the simultaneous resolution of certain pairs of quantum observables. In its familiar position–momentum form, the standard deviations of position and momentum satisfy a lower bound determined by the reduced Planck constant. This physical restriction, however, is distinct from the computational problem of resolving an uncertainty state from incomplete, discrete, noisy, or progressively acquired observational data. This paper explores the latter problem through the lens of a Calculus of Uncertainty, or CoU. The CoU framework considered here begins with partial data and applies a recursive reconstruction procedure consisting of rectangular approximation, trapezoidal approximation, midpoint approximation, numerical refinement, bounded logistic formulation, validation, and recursive state resolution. Applied to the uncertainty principle, this procedure does not attempt to recover simultaneously exact values of incompatible observables, nor does it modify the quantum-mechanical lower bound. Instead, it reconstructs estimates of the statistical moments of position and momentum distributions from partial observations, derives corresponding estimates of their standard deviations, evaluates those estimates relative to the uncertainty relation, and recursively determines whether the resulting state is sufficiently supported to be accepted. A distinction is therefore maintained between quantum uncertainty, which is a property of the admissible statistical structure of a quantum state, and reconstruction uncertainty, which arises from partial sampling, numerical approximation, truncation, measurement limitations, and incomplete contextual information. The proposed formulation treats the uncertainty relation as a constraint within the CoU context and validation structure. A state may be accepted only when its numerical reconstruction has converged to a specified tolerance and its estimated uncertainty product remains consistent with the quantum-mechanical lower bound. The resulting approach provides a tolerance-aware, constraint-consistent method for representing how an inferred state develops as additional evidence is acquired, approximations are refined, and prior states are recursively revised or retained. A worked minimum-uncertainty Gaussian reconstruction demonstrates that quadrature may converge accurately on a truncated domain while the reconstructed uncertainty product remains inadmissible until sufficient distributional mass is captured. **Keywords:** Calculus of Uncertainty, Heisenberg uncertainty principle, quantum uncertainty, partial data, numerical reconstruction, midpoint rule, logistic state formulation, recursive state resolution
A sharper uncertainty inequality which exhibits a lower bound tighter than that in the classical N-dimensional Heisenberg's uncertainty principle is obtained. The condition that reaches an equality relation of the uncertainty inequality is deduced. Example and simulations are carried out to illustrate that the newly derived uncertainty principle is truly sharper than the classical one. The new proposal's applications in time-frequency analysis, optical signal processing and physical optics propagation are also given.