Can the impact point of a single electron be predictedfrom its launch point? Deterministic guidance,wave-source geometryand constraints from existing experiments
We ask whether the impact point of an individual electron in an interference experimentcan be predicted from its launch point, and what precision this would require. Particlesare described deterministically, guided by the ux of the wave as in the de BroglieBohmformulation, and wave sources are placed according to the base table of a directional-productmodel, xk,n = s ck + Cn with eight levels c = (0, 1, 1.5, 2.5, 3, 4, 5.5, 6). A simulation of thecontrolled electron double-slit experiment of Bach et al., with the electron wavelength writtenas the directional product (pc)2 = K(K + 2mec2), reproduces the interference pattern fromdeterministic, non-crossing paths (correlation 0.98), but shows that launch points spreadover one slit (54 nm) reach six fringes: predicting the fringe of a single electron requires itsposition in the slit plane to be known to about 5 nm. Propagating this requirement back toa nanotip source with the measured eective radius of 0.550.80 nm gives a required launchprecision of 0.48 pm for source-to-object distances of 0.11 mm at 600 eV, far below atomicdimensions. A review of single-electron experiments shows that sources already localise theemission point to below 1 nm without destroying coherence, and that single electrons fromsuch sources still arrive at dierent fringes; no experiment records the in-wave launch positionthat a deterministic prediction would need. A repeat-launch simulation shows why: xing thelaunch position by a measurement of 5 nm (1 nm) precision spreads the impacts over 4.3 (22)fringes and lowers the correlation with the interference pattern from 0.996 to 0.46 (−0.05).Within pilot-wave theory an individual prediction is therefore possible only for an ensembleout of quantum equilibrium, and the proposed test is a test of quantum non-equilibrium.We identify which existing data (per-pulse emission times and arrival positions) can testtiming-based prediction rules, and summarise a pre-registration protocol for a decisive test,in which the prediction rule is to be xed before any data are taken.
We ask whether the impact point of an individual electron in an interference experimentcan be predicted from its launch point, and what precision this would require. Particlesare described deterministically, guided by the ux of the wave as in the de BroglieBohmformulation, and wave sources are placed according to the base table of a directional-productmodel, xk,n = s ck + Cn with eight levels c = (0, 1, 1.5, 2.5, 3, 4, 5.5, 6). A simulation of thecontrolled electron double-slit experiment of Bach et al., with the electron wavelength writtenas the directional product (pc)2 = K(K + 2mec2), reproduces the interference pattern fromdeterministic, non-crossing paths (correlation 0.98), but shows that launch points spreadover one slit (54 nm) reach six fringes: predicting the fringe of a single electron requires itsposition in the slit plane to be known to about 5 nm. Propagating this requirement back toa nanotip source with the measured eective radius of 0.550.80 nm gives a required launchprecision of 0.48 pm for source-to-object distances of 0.11 mm at 600 eV, far below atomicdimensions. A review of single-electron experiments shows that sources already localise theemission point to below 1 nm without destroying coherence, and that single electrons fromsuch sources still arrive at dierent fringes; no experiment records the in-wave launch positionthat a deterministic prediction would need. A repeat-launch simulation shows why: xing thelaunch position by a measurement of 5 nm (1 nm) precision spreads the impacts over 4.3 (22)fringes and lowers the correlation with the interference pattern from 0.996 to 0.46 (−0.05).Within pilot-wave theory an individual prediction is therefore possible only for an ensembleout of quantum equilibrium, and the proposed test is a test of quantum non-equilibrium.We identify which existing data (per-pulse emission times and arrival positions) can testtiming-based prediction rules, and summarise a pre-registration protocol for a decisive test,in which the prediction rule is to be xed before any data are taken.
## AbstractA deterministic interpretation of the double-slit experiment is proposed, in which the electron is treated as a localized particle at all times. The observed interference pattern is attributed to a real physical energy wave in the electromagnetic field, generated by the electron's accelerated motion through the field. This energy wave reaches the slit barrier, where the slit geometry creates coherent waves that merge to form a constructive interference wave. This wave redistributes energy in the field, creating an Electron-EMF coupled pathway that guides the particle to the detector. The apparent randomness of individual impacts arises from incomplete knowledge of the initial conditions, not from intrinsic indeterminism. The measurement effect is modelled as a disturbance of the coherent energy waves, either preventing the formation of the coupled pathway or destroying it after formation. The model is extended to the quantum eraser experiment, where the restoration of coherence via a polarizer naturally explains the reappearance of interference. A further extension to the delayed-choice quantum eraser is proposed, introducing the Entangled Constructive Interference Wave (ECIW) as a physical mechanism for entanglement within the model. The mechanism is shown to apply across geometries beyond the double slit. Predictions include a correlation between landing position and flight time, offering a testable distinction from standard quantum mechanical interpretations. A preliminary, illustrative mathematical framework is provided, situating the model within the pilot-wave tradition. **Keywords: ** Double-slit experiment, pilot wave, electromagnetic field, quantum interpretation, determinism, hidden variables, measurement problem, energy wave, coupled pathway, quantum eraser, delayed-choice quantum eraser, entanglement.
Electron double-slit experiments, the essence of quantum mechanical “wave/particle duality”, have been extensively and continuously studied because their unrealistic interpretations have attracted many researchers. Here, we discuss recent sophisticated V-shaped double-slit experiments to elucidate electron-path information, i.e., which slit electrons pass through, which path they pursue, and where they arrive and interfere on the image plane. In a transmission electron microscope, an optically zero-propagation distance condition was realized, where the double-slit position was imaged just on the detector plane. Interference fringes composed of dot images were controlled by using two electron biprisms. Using a V-shaped double slit, we confirmed that it is possible to observe interference fringes only when the path information of individual electrons is not available. Furthermore, we studied how the interference fringes would shift their detected positions when electron wave phases were modulated before illumination to the double slit. Phase shifts were generated by tilting electron beams. We confirmed that each single electron as a wave changed its phase even when it traveled alone inside the microscope.
Electron double-slit experiments, the essence of quantum mechanical “wave/particle duality,” have been extensively and continuously studied because their unrealistic interpretations have attracted many researchers. Here, we discuss recent sophisticated V-shaped double-slit experiments to elucidate electron-path information, i.e., which slit electrons pass through, which path they pursue, and where they arrive and interfere on the image plane. In a transmission electron microscope, an optically zero-propagation distance condition was realized, where the double-slit position was imaged just on the detector plane. Interference fringes composed of dot images were controlled by using two electron biprisms. Using a V-shaped double slit, we confirmed that it is possible to observe interference fringes only when the path information of individual electrons is not available. Furthermore, we studied how the interference fringes would shift their detected positions when electron wave phases were modulated before illumination to the double slit. Phase shifts were generated by tilting electron beams. We confirmed that each single electron as a wave changed its phase even when it traveled alone inside the microscope.
Abstract One of the greatest mysteries that has puzzled and seemingly hoodwinked our very understanding of nature is the single electron double slit experiment – one that Feynman termed the central mystery of quantum mechanics. At the heart of this mystery is how can an electron that is a physical particle exhibit a wave-like property? We’ve proposed a non-counterintuitive way to explain this phenomenon called the randomly-jumping electron hypothesis, in which the electron randomly “jumps” in discrete succession to specific points only and not others which is what causes the electron to indeed be a classical particle only but is able to produce a wave-like interference pattern. To provide supporting evidence for our hypothesis, we’ve developed a computer program to simulate the single electron double slit experiment by treating the electron as a classical particle while still exhibiting wave-like fringes in agreement with real-world observed data of the single electron double slit experiment performed. Our interpretation does not set out to “solve” the mystery of why an electron can produce a wave-like interference pattern once and for all, however our hypothesis is a monism ontological interpretation that does address some of the crucial debates that have remained unsolved for 100 years since the formulation of quantum mechanics.
We propose a trajectory density formulation of the double-slit interference pattern that reproduces observed fringe intensities without wave superposition. The intensity at screen position x is the weighted density of particle trajectories arriving there, suppressed exponentially by the deflection cost in energy space: I(x) = V_{Trit}(KE) \times \sum_n w_n \cdot \delta\!\left(x - \frac{n\lambda L}{d}\right) \cdot e^{-k|n|} where k = dθ_eff/λ_dB is the coherence parameter set by the effective illumination semiangle θ_eff, and V_Trit(KE) = 2γβ/(1+γ²β²) is a new energy-dependent visibility factor from the Trit framework [1–3]. The parameter k connects to the effective source temperature through the emission physics: higher source temperature → larger emission area → larger θ_eff → larger k → fewer visible fringes. We test the formula against five independent experiments across five orders of magnitude in particle energy and three particle types. For thermionic and Schottky electron sources the formula outperforms standard QM by a factor of 2–6× in R². For cold FEG sources and C60 molecules the formula correctly reduces to QM (k → 0). The C60 result leads to a clean new distinction: V_Trit(KE) applies to elementary particles, where the energy space geometry directly governs wave-particle balance; for composite particles V_Trit = 1 and interference is from the centre-of-mass de Broglie wave. This difference is falsifiable: varying electron energy should reveal a visibility peak at KE = 211.66 keV; varying C60 velocity should show no such peak.
The double-slit experiment is usually taken as evidence that a particle is also a wave. We show by simulation that a definite particle, moving through a physical medium and guided by its own wave, produces the same interference pattern, with no constant fitted to the pattern. Space is modeled as a compressible particle medium with a short-range stiffness. A particle set in motion disturbs it once. The disturbance disperses, and the particle carries with it the part of the wave whose group velocity matches its own speed, so the wavelength at the particle is fixed by its speed alone, λ ∝ 1/v, with no oscillator and no continued pushing. That wave passes through two slits and interferes as any wave does. The particle passes through one slit, keeps its speed, and takes its direction from the flow of the wave around it. It lands where the wave is bright. Hundreds of particles, sent one at a time, reproduce the wave's own pattern: at three speeds and two slit spacings, up to nine bright lanes with empty dark bands between them, lane spacing that scales with the wavelength of the wave it carries, the same pattern for a stream of particles as for one, and no lanes when one slit is closed. The rule that steers the particle has no strength, no frequency, and no free constant, and it is the same rule at every scale because each particle's wave is sized to itself. The result depends on one condition: that the particle's launch point lie well beyond the near-field distance of the slit pair, which laboratory sources satisfy. We record the rules that were tried and failed and the numerical pitfalls, and we state what remains open: the silent coast is a postulate, the mass dependence of the wavelength is untested, and the work is two-dimensional.