A numerical experiment that simulates and observes the de Broglie relation lambda=h/p inside a single-source double-slit classical wave model. An electron is modeled as a spatially localized wave built from an equal-amplitude sum of odd harmonics (N>1). The central result is NOT the value of h (which is a near-tautological self-consistency, since lambda0=h/p is the input and the resonance lattice is dense ~1e-19 m): it is (i) that a highly localized odd-harmonic classical wave can survive two-slit interference as a sharp single peak only under an inter-harmonic alignment condition, and (ii) that the observed central fringe spacing DeltaX = lambda' D/W is set by the fundamental wavelength and is independent of the harmonic count N (N controls only the localization sharpness ~1/(N+1)). Scope is stated explicitly: this is neither a first-principles derivation of p*lambda=h nor a derivation of a Delta_x*Delta_p ~ h uncertainty relation. Each electron is given independent Monte Carlo draws of source position (cos^2, +/-lambda0/2) and initial wavelength (cos^2, +/-1%), with interference, alignment and exclusion recomputed exactly per trial. Bilingual (Japanese + English) drafts with source programs.
A numerical experiment that simulates and observes the de Broglie relation lambda=h/p inside a single-source double-slit classical wave model. An electron is modeled as a spatially localized wave built from an equal-amplitude sum of odd harmonics (N>1). The central result is NOT the value of h (which is a near-tautological self-consistency, since lambda0=h/p is the input and the resonance lattice is dense ~1e-19 m): it is (i) that a highly localized odd-harmonic classical wave can survive two-slit interference as a sharp single peak only under an inter-harmonic alignment condition, and (ii) that the observed central fringe spacing DeltaX = lambda' D/W is set by the fundamental wavelength and is independent of the harmonic count N (N controls only the localization sharpness ~1/(N+1)). Scope is stated explicitly: this is neither a first-principles derivation of plambda=h nor a derivation of a Delta_xDelta_p ~ h uncertainty relation. Each electron is given independent Monte Carlo draws of source position (cos^2, +/-lambda0/2) and initial wavelength (cos^2, +/-1%), with interference, alignment and exclusion recomputed exactly per trial. Bilingual (Japanese + English) drafts with source programs.
Thought experiment (model calculation), not a measurement. Extends the position-readout push-forward (companion paper, isSupplementTo) to a localized source: localization is represented by a constant-amplitude odd-harmonic sum (isolated peak wave S_N). Although S_N is a sum of odd harmonics, it localizes in the far field WITHOUT changing shape only when the source-slit diagonal r_k=sqrt(L^2+(W/2)^2) is an integer/half-integer multiple of the fundamental wavelength: sqrt(L^2+(W/2)^2)=(m/2) lambda0. Alignment is a band of width ~1/(2N) (narrower for larger N); a ~3% change in lambda0 decides interference vs scattering. The central-alignment tolerance is W/L-independent; larger W gives greater off-axis fragility. An aligned localized source inherits the cos^2 fringe-shift readout under positional fluctuation but its peaks fall below cos^2 off-axis (extra scattering loss). The single wavelength N=1 has none of this fragility and reproduces the companion paper to machine precision. Net result is negative: localization adds an alignment constraint and off-axis fragility; shape preservation is conditional and fragile, with N=1 the robust special case. Distinguished from spatial coherence / van Cittert-Zernike visibility loss.
The double-slit experiment is commonly interpreted as evidence that a single electron must be described by a spatially extended wavefunction whose path amplitudes interfere. Here, we present an alternative geometric formulation within the Bivector Standard Model, in which the observed far-field interference pattern is reproduced while the phase is attributed to an internal bivector clock carried by the electron. In this approach, each electron remains localized and produces a single detection event, while the familiar fringe pattern emerges statistically from the accumulation of many impacts. Interference arises from the comparison of clock phases associated with geometrically distinct paths, rather than from the superposition of spatial waves. The resulting probability distribution recovers the standard two-slit interference factor, the single-slit diffraction envelope, and the usual fringe-spacing relation in the Fraunhofer regime. The de Broglie wavelength emerges as the spatial manifestation of this transported phase. This formulation provides a geometric account of the origin of phase in single-electron interference, consistent with standard results while offering a distinct physical interpretation.
The de Broglie wavelength λ= h/mv of simple particles (photon, electron, neutron) atoms (helium, neon) and molecules (fullerene, fluorous porphyrin) deduced from the de Broglie (dB) relation and those measured by double slits interferometry are compared.The de Broglie relation is verified in very large domains of velocity v and mass m of the particles, respectively 8 and 13 orders of magnitude.In this paper one analyses the interference spectra of photons and electrons and give the Intensity Relations IR(II-DI) between the interference intensity (II) by two slits and the diffracted intensity (DI) by one slit.The double slits interference in the de Broglie-Bohm interpretation is an interference of particle and pilot waves, internal and external waves each passing by a different slit.The velocities of these waves are discussed.From the universal de Broglie relation and the relation vp = vw between the particle and pilot waves one shows that only in the domain of the maximum of the light spectrum (λm~ 550 µm) the dB relation leads to the Plank-Einstein (PE) relation hν=mc 2 .A measure of the mass of the photon m* = 4 10 -36 Kg is deduced.The de Broglie relation for different wave lengths shows that the PE relation must be modified, the velocity of light is not constant as shown by various experiments: Global Positioning Systems GPS, red shift in astronomy etc.This has been discussed by Marmet in years 2000.The same conclusion was given by Sato in 2010 for electromagnetic waves.The trajectories of electrons observed by Jönsson by double slits interferometry and then calculated by Gondran according to the de Broglie-Bohm interpretation are then analyzed in detail.The correlations observed between the properties of these trajectories at short and long distances from the slits permit to give the general equation of the trajectories.The origin of these properties is discussed in the framework model of De Broglie and Bohm, assuming that at any point the intensity of the two waves (particle and pilot wave) passing by the 2 slits are deceasing functions of the distance to the slits.The changes of direction of the trajectories are explained.
This paper presents a minimal arc-geometric reconstruction of single-electron double-slit interference within the conceptual framework of Arc Theory. The central aim is not to reject the empirical success of standard quantum mechanics, but to show that a nonstandard geometric ontology can be expressed in a traditional mathematical-physical form, with explicit definitions, axioms, propositions, derivations, and stated limitations. Starting from the proposed arc-geometric constraint R = DA, where R denotes the curvature radius of the external zero-weight ring and DA denotes the intrinsic arc-radius of the electronic arc-helix, the paper derives the de Broglie relation lambdaA = h/p through phase closure and action quantization. The double-slit condition d sin(theta) = m lambdaA is then reconstructed as a consequence of arc-branch phase closure under boundary-induced splitting. The paper further formulates the double-slit process in terms of four structural operators: the arc-branching operator SAlpha, the inter-branch coherence kernel SigmaAlphaBeta, the arc-difference field DeltaAlphaBeta, and the measurement-localization operator LambdaD. In this formulation, the Born density is interpreted not as a primitive probabilistic substance, but as the observable projection density of an underlying high-dimensional arc trajectory onto a lower-dimensional detection interface. In compact form, the proposed interpretation may be written as: rhoArc(x) = |A(x)|^2 where A(x) denotes the projected arc amplitude and rhoArc(x) denotes the observable arc-projection density. A central part of the work is the quantitative formulation of topological friction. The proposed self-decoherence rate GammaSelf is expressed as a functional of curvature mismatch, torsion mismatch, and deviation from the R = DA constraint: GammaSelf = Integral over GammaA of [ gammaKappa * (DeltaKappaA)^2 + gammaTau * (DeltaTauA)^2 + gammaR * ((R - DA)/R)^2 ] dmuA This gives a precise geometric meaning to internal self-decoherence and separates it from ordinary external environmental decoherence. The paper also demystifies observational collapse by representing measurement as environment-induced arc localization. In the proposed formalism, the disappearance of interference is described by the suppression of off-diagonal coherence terms: rhoAlphaBeta -> muAlphaBeta * rhoAlphaBeta with muAlphaBeta -> 0 corresponding to complete which-path distinguishability or environmental trace formation. The scope of the paper is deliberately limited. It does not claim to derive the numerical values of Planck's reduced constant, electron mass, electric charge, magnetic moment, or the electron g-factor. Rather, it provides a mathematically disciplined reconstruction of the kinematics of single-electron double-slit interference and proposes a possible geometric foundation beneath the standard probabilistic formalism. The paper is intended as a preprint-level contribution to the foundations of quantum theory, mathematical physics, and alternative geometric models of microscopic phenomena.
In a recent paper, titled “Explanation for the observed wave-particle-duality of Light” (Tank, H.K. http://vixra.org/abs/1402.0153, 2014) it was explained that: “Since at very high frequencies, narrowband filtering and generation of purely monochromatic light of one Hertz bandwidth is not yet technically possible, there has been quite a wide bandwidth of waves involved, in the double slit experiments and photoelectric experiments performed so far. And since this wide ‘band’ of waves coherently add only at discrete points in space and time, we get ‘particles’ at the detectors.” Now, to further confirm the validity of this explanation, a new kind of experiment is proposed here, in which protons and electrons are accelerated at appropriate velocities vp ve , such that their de Broglie wavelengths are equal; i.e. ( h / mp vp )= ( h / me ve ) and let them pass through closely spaced slits so that they can interfere with each other. Similarly, in another experiment, their wave-functions: A exp i ( kp X - ωp t ) = A exp i ( ke X - ωe t ) , and let them pass through closely spaced slits, and see how the two matter-waves interfere.