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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

Since its inception, the Heisenberg Uncertainty Relationship (HUR) [

as independently proved by Robertson and Schrodinger [

it was eventually criticized as being no more than the experimental realization of the operatorial (non)commutation relation [

In this context, the actual quest is to present a clear yet effective discussion on how HUR becomes valid without involving any operatorial commutation constraints, through explicitly including the quantum fluctuation, while providing the complementary wave-particle analytical description in which the extended-HUR (E-HUR) is not only possible but necessary.

The background of the present approach is the Feynman path integral formulation of quantum mechanics [

Yet, for being adequate for the measurement conditions the periodic paths have to be considered, _{a}_{b}_{B}_{B}

in terms of the so called Matsubara frequencies _{m}^{*}(τ) = ^{*}_{m}_{−m}_{m}

with the 0^{th} term _{0} being known as the Feynman centroid,

It represents more than the “zero-oscillating” mode of motion but the thermally averaged path over entire quantum sample [

Being, thus, appropriately interpreted as the average of the observed coordinate at given equilibrium temperature _{0} coordinate in terms of averaging of quantum periodic paths (orbits) for a given thermal energy _{B}

Instead, here, the philosophy is to introduce appropriately the quantum fluctuation information _{0}) respecting the average of the observed coordinate (_{0}), by the Feynman integration rule founded in the ordinary quantum average (

for the normalized Gaussian wave-function (

recovering the de Broglie wave-packet [

It is obvious that the

The next test is about the validity of the

Then, through combining the expression

with the prescription (

that, when plugged in the basic

featuring it in a direct relationship with the quantum fluctuation width.

In the same manner, the evaluations for the integrals of the first and second orders of kinetic moment unfold as

while when plugging them in

Worth noting is that from the coordinate and momentum dispersions,

However, when multiplying the expressions (

this way resembling in an elegant manner the previous result of statistical complementary observables of position and momentum [

For the sake of experimental precision it is worth noting that the error in coordinate localization is given _{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.

Yet, another important idea was raised, namely that the coordinate and momentum dispersions, although in reciprocal relationship with quantum fluctuation,

We like to identify the general quantum fluctuation conditions in which the HUR is valid and when it is eventually extended. We already note that, whereas the momentum dispersion computation is fixed by relations (

considering the condition (

specializing the quantum (average) relationship (

obtaining the average of the second order coordinate (

combining the steps i) and ii) is computing the coordinate dispersion Δ

The present algorithm may be naturally supplemented with the analysis of the wave-particle duality. This is accomplished by means of considering further averages over the quantum fluctuations for the mathematical objects exp(−^{2}^{2}) that are most suited to represent the waves and

With the rule (

and

It is worth observing that the practical rule (

Next, the ratio of

giving the working tool in estimating the particle-to-wave content for a quantum object by considering various coordinate average information; this will be achieved by (v) making the

since they nevertheless emerge from quantum average operations (measurements).

Now we are ready for presenting the two possible scenarios for quantum evolutions along the associate HUR realization and the wave-particle behavior.

For the case of observed quantum evolution, the averaged observed position is considered in relation with the quantum fluctuation by the general relationship

implying that the average of the second order of Feynman centroid looks like

When (

Not surprisingly, when further combining relations (

showing that the wave-particle duality is indeed a reality that can be manifested in various particle-wave (complementary) proportions—yet never reaching the perfect equivalence (the ratio approaching unity). Moreover, because (P/W)_{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.

Moving to the treatment of the

since the quantum object, although existing, is not observed (see the spontaneous broken symmetry mechanism in the Discussion Section 4 below).

The relation with quantum fluctuation may be nevertheless gained by the average of the second order of the Feynman centroid–considered under the form

Note that

Next, through recalling the referential

The result (

with the immediate consequence in adjusting the basic HUR as

On the other hand, within conditions fixed by

Through characterizing the numerical results of

It is clear that whereas the omega case of

It is very instructive to present in a unitary manner the observed and free quantum evolution cases in the chart of

Note that the possibility a quantum object is manifested

However, the wave-particle

Yet, having the analytical expressions for both observed and free quantum evolutions may considerably refine our understanding of macro- and micro-universe. For instance, with various (P/W)_{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 〈 _{0}^{2}〉_{Exp}^{2}〉_{Exp}_{0} and

that when replaced into

It is worth giving a working example for emphasizing the reliability of the present approach and to choose for this aim the fundamental Compton quantum experiment. In this case, the incoming photonic beam carries the wavelength λ_{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 (

Now we can interpret the various experimental situations encountered, employing the output of _{Compton}_{Compton}_{Compton}_{Compton}_{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

with an important role in assessing the stability of matter, from atom to molecule. As an example, the justification of the Hydrogen stability was successfully proved through setting the ratio P/W = 1 in the omega point of function (_{Free} function (

On the other side, one would wish to further discuss the free quantum ^{iθ}^{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

For advanced molecular physical chemistry, it is worth pointing out that the particle/wave ratio (P/W) of

tells us that, in accordance with the recent interpretation of ELF as error in electronic localization [_{P/W} index (

Finally, for spectroscopic analysis, one could ask upon the corresponding time-energy uncertainty relationship [_{U}_{U}_{μ}_{μ}_{μ}

since containing the non-specified external potential dependency:

Yet, the present periodic path approach may be eventually employed to assess the problem through reconsidering the width _{0}) of the de Broglie wave-function (_{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,

It is widely recognized that despite the huge success of quantum mechanics, since forecasting the experimental observations, its basic conceptual consequences, namely the

The present endeavor made such a step towards providing a unified answer on these fundamental quantum problems by the aid of the Feynman periodic path methodology adapted to compute the coordinate and momentum standard deviations in terms of the quantum fluctuation and the averages of the observed coordinate (the Feynman centroid).

The approach successfully resembles the basic Heisenberg uncertainty relationship (HUR) by showing the reciprocal quantum fluctuation contributions in coordinate and momentum dispersions, yet without employing any operatorial identity or commutation rule. However, the present HUR proof emphasizes the correct role the quantum fluctuation rather than the Planck constant has in uncertainty, it being directly related with coordinate and inversely correlated with momentum uncertainties in measurements.

Moreover, the wave-particle quantum issue was adequately unfolded as well by assessing two types of quantum fluctuation contributions to the first and second orders of coordinate averages. This way, it was found that the wave-particle complex covers two continuously connected realities: one observed and the other of free evolution, yet each of them being analytically characterized by a specific P(article)/W(ave) ratio function.

We found that while the observed reality is fully covered by the standard HUR albeit with an undulatory predominant manifestation of the quantum objects, P/W ∈ [0, 0.952], the free evolution corresponds with isolated (not measured) quantum systems/states with a symmetrical appearance between the particle and wave dominant manifestations around their perfect equivalency, P/W ∈ [0.952, 1.048], however, with the price of altering HUR realization with the factor

Overall, the present work offers strong analytical arguments in favor of Copenhagen interpretation (consecrated either by the Bohr’s complementarity or by the de Broglie pilot-wave/double-solution pictures) [

On the other side, the ever residual particle manifestation in whatever system that accompany the wave character of quantum observed evolutions, further allows characterization of the chemical bond by the covalent-ionic mixture as an important molecular specialization of the wave-particle quantum physical paradigm; moreover, the particle-to-wave ratio may provide a working electronic localization function to be further used in understanding bonding properties in direct relation with molecular data assay through the recorded information and computed quantum fluctuation magnitude: see

However, through the Heisenberg uncertainty it is hopefully better integrated in the quantum “measurement dogma” herewith, the numerical predictions of the wave-particle character for both experimental and theoretical approaches are advanced within the reunited {observed ∪ free} evolutions of the quantum objects, by means of the associate P/W functions depending only on the quantum fluctuation magnitude factor rather than on other statistical information.

The author kindly thanks Emeritus Hagen Kleinert, the last disciple of Feynman, for the enlightening discussions on path integrals in general, and on the periodic paths in particular, during mutual visits from 2008–2010 at the Free University of Berlin and West University of Timisoara. This work was supported by CNCSIS-UEFISCSU, project number PN II-RU TE16/2010.

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