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

^{1}

^{2}

## Abstract

**:**

**[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.

## 1. Introduction

## 2. HUR by Feynman Periodic Paths

_{a}= x(0) = x(ħβ) = x

_{b}, with β the inverse of the thermal energy k

_{B}T (k

_{B}the Boltzmann constant) for a system in equilibrium at temperature T can be constructed by means of the Fourier series

_{m}=2πm/ħβ, m∈

**Z.**Under the condition of real paths, x

^{*}(τ) = x(τ), along the resulted relationship between the coefficients of periodic paths, x

^{*}

_{m}=x

_{−m}=x

_{m}, the series (3) can be rearranged as the expression

^{th}term x

_{0}being known as the Feynman centroid,

_{0}coordinate in terms of averaging of quantum periodic paths (orbits) for a given thermal energy k

_{B}T, stays as an elegant way of relating the classical with quantum nature of an observable (or experiment) without involving the fashioned Fisher information with the rate of entropy increase under Gaussian diffusion condition as a measure of measurement robustness [14].

_{0}) respecting the average of the observed coordinate (x

_{0}), by the Feynman integration rule founded in the ordinary quantum average (Eqution (6a))

_{0}) that appears in the final coordinate-momentum multiplied dispersions—being therefore incorporated in the HUR result—a feature not obviously revealed by earlier demonstrations.

## 3. Extended HUR and the Wave-Particle Quantum Status

- considering the condition (7) as an invariant of the measurement theory since it assures the connection between the average over quantum fluctuation of the coordinate and the observed averaged coordinate;
- specializing the quantum (average) relationship (8b) for the condition given by Equation (7);
- obtaining the average of the second order coordinate (8c);
- combining the steps i) and ii) is computing the coordinate dispersion Δx as given by Equation (2).

^{2}x

^{2}) that are most suited to represent the waves and particles, due to their obvious shapes, respectively. Such computations of averages are best performed employing the Fourier k-transformation as resulted from the de Broglie packet (6) equivalently rewritten successively as [25]:

#### 3.1. Observed Evolution

_{Obs}< 1, it appears that the general behavior of a quantum object is merely manifested as wave when observed, from which arises the efficacy of spectroscopic methods in assessing the quantum properties of matter.

#### 3.2. Free Evolution

**[0,1].**With Equations (17a) and (17c), the step iii) in above algorithm, one finds the coordinate dispersion

## 4. Discussion

_{Observed}, one can evaluate the appropriate particle-to-wave presence in a quantum complex for which experimental data are available: once knowing from a given measurement the quantities 〈 x

_{0}

^{2}〉

_{Exp}and 〈 x

^{2}〉

_{Exp}, with x

_{0}and x appropriately considered for each type of experiment (e.g., the statistical mean for classical records and the instantaneous values for quantum measurement of coordinate, respectively), one can employ Equations (15b) and (15c) to find the magnitude of the quantum fluctuation

_{0}whilst the scattered one departs from that incident with the amount Δλ = λ – λ

_{0}; such situation allows the immediate specialization of the quantum fluctuation magnitude (23) to its Compton form

_{Compton}of Equation (24) and consequently the increase of (P/W)

_{Compton}of Equation (16); this is in accordance with the fact that the scattered light on free electrons rises more and more its particle (photonic) behavior. On the other side, when the scattering is made on tight bonded electrons (e.g., electrons in atoms of a material), the Compton wavelength departure is negligible, Δλ→0, leaving from Equation (24) with the asymptotic higher quantum fluctuation magnitude n

_{Compton}→ ∞ that corresponds at its turn with (P/W)

_{Compton}= 0 in Equation (16). This matches with the fact that this case corresponds with complete wave manifestation of light that scatters bonded electrons, resembling the (classical) interpretation according which the scattered bounded electron by a wave entering in resonance with it while oscillating with the same frequency. Therefore, the reliability of the present (P/W)

_{Observed}formalism was paradigmatically illustrated, easily applied to other quantum experiments, while giving the numerical P/W estimations once having particular data at hand. Equally valuable is the free evolution (P/W)

_{Free}ratio of Equation (20) that may be employed for the wave-particle equivalency between the quantities (11) and (12)

_{Free}function (20) and of subsequent modified HUR may be explored also in modeling the various stages and parts of the Universe that cannot be directly observed, as well as when dealing with quantum hidden information in the sub-quantum or coherent states [27,28].

**M**is always zero, <

**M**>= 0, since +

**M**and −

**M**occur with the same probability [30]. In the case of condensation (for instance Bose-Einstein), the order parameter 〈ψ〉 that is obtained from averaging the bosonic fields on the canonical ensemble gives zero result in free (untouched) evolution, 〈ψ〉 = 0, due to the inner annihilation nature of the bosonic field 〈ψ〉, beside the total Hamiltonian is global gauge invariant under the transformation ψ(x) →e

^{iθ}ψ(x), ∀ θ ∈ℜ that corresponds with the conservation of the total number of particles in the system [31]. However, either case is resolved within experiments by simple observation (e.g., the ferromagnets and the superfluid

^{4}He appear under natural conditions without special experimental conditions) through the so–called “Goldstone excitations” (spin waves and the phonons for ferromagnets and superfluids, respectively) that eventually turns (brakes) the microscopic (free evolution) Hamiltonian symmetry into the macroscopic (observed or directional evolution) symmetry. This mechanism of broken symmetry fits with the present free-to-observed quantum evolution picture since, when revealed, it involves a countless number of zero-energy (yet orthogonal) ground states, leading with the rising of the locally (Goldstone) excited state from one of the ground states that gradually changes over the space from the zero energy and infinity wavelength to some finite non-zero energy and long wavelength; such behavior parallels the turning of the condition of Equation (17a) into that of Equation (15a), where the exact Heisenberg principle is obeyed—however in different Particle/Wave ratios (depending on the phenomenon and experiment), see the above discussion and the Figure 1.

_{P/W}index (26) for explaining—for instance—the molecular aromaticity [35] in terms of geometry of bonding and the amount of quantum fluctuation present, are in progress and will be in the future communicated.

_{U}is accompanied by the time operator t

_{U}= − iħ∂

_{μ}with the ∂

_{μ}= d/d

_{μ}(ɛ) relating the integrable measure μ(ɛ) as depending of the energetic spectra (ɛ) on the associate generalized Hilbert space [37]. On the other side, quantitatively, the time-energy HUR faces with the practical problem in evaluating the general yield of the Hamiltonian variance

_{0}) of the de Broglie wave-function (6b) as related with the averaged potential over the quantum fluctuations 〈V(x)〉

_{a}

_{2(}

_{x}

_{0)}; a self-consistent equation is this way expected, while the final time-energy HUR may further depend on the ground or excited (Wigner) states considered, i.e., within the inverse of the thermal energy limits β → ∞ or β → 0, respectively. Nevertheless, this remains a challenging subject that will be also approached in the near future.

## 5. Conclusion

**[0,1].**

## Acknowledgements

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**Figure 1.**The chart of Heisenberg Uncertainty Relationship (HUR) appearance for observed and free quantum evolutions covering the complete scale of the particle to wave ratios as computed from the Equations (16) and (20), respectively; the points Ω and α correspond to wave-particle precise equivalence and to the special extended-HURs of Equations (21) and (22), respectively.

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**MDPI and ACS Style**

Putz, M.V. On Heisenberg Uncertainty Relationship, Its Extension, and the Quantum Issue of Wave-Particle Duality. *Int. J. Mol. Sci.* **2010**, *11*, 4124-4139.
https://doi.org/10.3390/ijms11104124

**AMA Style**

Putz MV. On Heisenberg Uncertainty Relationship, Its Extension, and the Quantum Issue of Wave-Particle Duality. *International Journal of Molecular Sciences*. 2010; 11(10):4124-4139.
https://doi.org/10.3390/ijms11104124

**Chicago/Turabian Style**

Putz, Mihai V. 2010. "On Heisenberg Uncertainty Relationship, Its Extension, and the Quantum Issue of Wave-Particle Duality" *International Journal of Molecular Sciences* 11, no. 10: 4124-4139.
https://doi.org/10.3390/ijms11104124