On Heisenberg Uncertainty Relationship, Its Extension, and the Quantum Issue of Wave-Particle Duality

Abstract

Within the path integral Feynman formulation of quantum mechanics, the fundamental Heisenberg Uncertainty Relationship (HUR) is analyzed in terms of the quantum fluctuation influence on coordinate and momentum estimations. While introducing specific particle and wave representations, as well as their ratio, in quantifying the wave-to-particle quantum information, the basic HUR is recovered in a close analytical manner for a large range of observable particle-wave Copenhagen duality, although with the dominant wave manifestation, while registering its progressive modification with the factor 1-n2, in terms of magnitude n∈[0,1]. of the quantum fluctuation, for the free quantum evolution around the exact wave-particle equivalence. The practical implications of the present particle-to-wave ratio as well as of the free-evolution quantum picture are discussed for experimental implementation, broken symmetry and the electronic localization function.

Spectral Inverse Quantum (Spectral-IQ) Method for Modeling Mesoporous Systems: Application on Silica Films by FTIR

“…The wave-particle issue was in the “heart” of quantum mechanics, even in its very principles, Heisenberg one in particular; see [28] and references therein. Currently assumed as a complementarity reality, it was just recently quantified with the aim of the path integrals’ quantum fluctuation factor ( n ) through considering the quantum averages for the Gaussian wave packet to the harmonic one for the particle and wave representations, respectively.…”

“…Currently assumed as a complementarity reality, it was just recently quantified with the aim of the path integrals’ quantum fluctuation factor ( n ) through considering the quantum averages for the Gaussian wave packet to the harmonic one for the particle and wave representations, respectively. The results were finite and apart of consistently explaining the atomic (and thus the matter) stability through particle-wave equivalency at the quantum level; they permit also a general formulation of the particle-to-wave ratio content for an observed event [28]:…”

“…Unfortunately, so far, the computational methods available for extracting from experimental spectra such information are missing and in favor of meso-to-macro analysis. Instead, this work makes the advancement of combining the observed data from FTIR spectra with the recent original method of modeling the wave/particle dual information by use of the spectroscopic assignment of the inverse quantum fluctuation factors [28]. This way, the present method fills the quantum-to-meso gap by the so-called spectral-inverse quantum (Spectral-IQ) algorithm, lending itself to being generalized and adapted for a wide type of spectra and gas-solid or sol-gel physicochemical interactions.…”

Valence atom with bohmian quantum potential: the golden ratio approach

“…Atomic stability and periodicity remain major issues in the structural theories of matter; fortunately, they both have been largely solved by wave-particle (W/P) complementarily quantum behavior; phenomenologically, such relationship can be expressed as “ WAVE ⊗ PARTICLE = constant ”, while it may be quantized (by Planck’s constant h ) in the light of Heisenberg principle as [36]…”

Review of Feynman’s Path Integral in Quantum Statistics: from the Molecular Schrödinger Equation to Kleinert’s Variational Perturbation Theory

“…Kleinert's variational perturbation (KP) theory [34] for the centroid density [32,70,[84][85][86][87][88][89]92] of Feynman path integrals [2,[32][33][34][60][61][62][63][64][65][66][67][68][69][70][71][72][73][74][75] provides a complete theoretical foundation for developing non-stochastic/non-sampling methods to systematically incorporate internuclear quantum-statistical effects in condensed phase systems. Similar to the complementary interplay between the rapidly growing quantum Monte Carlo simulations [146][147][148][149] and the well-established ab initio or density-functional theories (DFT) for electronic structure calculations [4,5,[25][26][27]29], non-sampling/non-stochastic pathintegral methods can complement the conventional Fourier or discretized path-integral Monte-Carlo (PIMC) [131,136,[139][140]…”