Emergence of wave–particle duality with interference pattern formation in a mathematically simulated double-slit configuration within the Schrödinger equation in a two-dimensional extension
This work extends the previous one-dimensional formulation of the double-slit problem to a two-dimensional framework within the Schrödinger equation, providing a more realistic representation of the physical configuration. The wavefunction is constructed through a spectral decomposition in terms of eigenstates, with a Gaussian distribution of initial energies. The temporal evolution of the system is explicitly analyzed, showing how interference patterns develop as a function of propagation time. In particular, the passage of electrons at different times is simulated, highlighting the progressive formation of fringes. A stochastic reconstruction of detection events is then introduced for the case of neutron beams, using physically realistic parameters, reproducing the emergence of the interference pattern from discrete particle-like impacts. The results show that both interference and particle-like behavior arise directly from the intrinsic mathematical structure of quantum mechanics, with no need for additional interpretative assumptions.
This work extends the previous one-dimensional formulation of the double-slit problem to a two-dimensional framework within the Schrödinger equation, providing a more realistic representation of the physical configuration. The wavefunction is constructed through a spectral decomposition in terms of eigenstates, with a Gaussian distribution of initial energies. The temporal evolution of the system is explicitly analyzed, showing how interference patterns develop as a function of propagation time. In particular, the passage of electrons at different times is simulated, highlighting the progressive formation of fringes. A stochastic reconstruction of detection events is then introduced for the case of neutron beams, using physically realistic parameters, reproducing the emergence of the interference pattern from discrete particle-like impacts. The results show that both interference and particle-like behavior arise directly from the intrinsic mathematical structure of quantum mechanics, with no need for additional interpretative assumptions.
In this work the author presents a fully explicit mathematical derivation of interference-like structures in a simulated double-slit configuration within the framework of the Schrödinger equation. The mathematical structure of the slits is directly incorporated into the initial wavefunction, avoiding simplified assumptions such as plane waves. The slit is modeled as a spatial constraint, and the resulting state is expanded over the eigenbasis of the system. The time evolution of the wavefunction naturally leads to the formation of structured spatial patterns. A multiscale potential based on the Livio function is introduced and treated through an explicit matrix formulation, allowing a direct comparison between the free case and a system with structured perturbations. In addition, a stochastic reconstruction of the probability density is performed by sampling individual detection events according to the Born rule, simulating the arrival of particles at a finite rate and showing the progressive emergence of the pattern over time. The results indicate that the interference fringes emerge as an intrinsic consequence of the spectral decomposition of a spatially structured quantum state. The close qualitative agreement between the simulated patterns and those observed in physical experiments therefore suggests that wave–particle-like behavior is intrinsically contained in the mathematical framework of quantum mechanics.
Quantum interference is traditionally interpreted as evidence of intrinsic wave–particle duality, invoking self-interference and wavefunction superposition. Here, we propose an alternative causal framework in which interference arises from interactions between a classical electron and a quantized electromagnetic field. In this picture, electrons follow definite, localized trajectories and produce discrete detection events, while the observed interference pattern emerges statistically from stochastic, nonlocal momentum transfers mediated by quantized field modes. Numerical simulations of the double-slit experiment show that dot-like interference fringes arise without invoking wavefunction collapse or intrinsic matter waves. Doubling the electron mass narrows the fringe spacing in agreement with de Broglie scaling, indicating that the effective wavelength emerges dynamically from particle–field coupling rather than representing an inherent property of the particle. The formulation is based on a rigorously defined Hamiltonian and derived equations describing electromagnetic coupling between particles and the slit environment. Although neutral particles such as neutrons or atoms carry no net charge monopoles, they possess electric or magnetic dipole and higher multipole moments that enable electromagnetic interactions with the slit apparatus, allowing discrete momentum transfer through the quantized field. Our simulations show that interference fringes arise from intermittent singleparticle detection events caused by stochastic momentum impulses from quantized electromagnetic modes. These results suggest a reinterpretation of wave–particle duality: the apparent wave-like behavior of matter may arise from interaction with the inherently wave-like quantized electromagnetic field—whose quanta are photons, the massless U(1)-gauge bosons—rather than from an intrinsic wave nature of the particles themselves. Our analysis is rigorous and is not speculative, because the proposed Hamiltonian for this system is physical. Within this nonlocal hidden-variable framework, the dot-like interference pattern observed in the double-slit experiment emerges from individual electrons traversing one slit at a time while receiving discrete momentum impulses from stochastic, spatially extended quantized electromagnetic fields.
Abstract The double-slit experiment is a classic experiment that presents the clearest manifestation of wave-particle duality, one of the fundamental principles of quantum mechanics. The double-slit experiment still remains the phenomenon that demonstrates wave-particle duality at the most fundamental level. This phenomenon quantitatively and conceptually demonstrates that microscopic objects can behave like both waves and particles at the same time. This article provides an in-depth analysis of the physical nature of dualism, the mechanism of formation of the interference pattern, and the impact of the act of observation on the dynamics of quantum systems. The need for a new perspective on this topic is due to the fact that the current uncertainty in the interpretation of quantum reality has not yet been completely eliminated, and the boundaries between classical and quantum behavior in microscopic objects are still being explored. The results presented here restructure the deep mechanisms of the interference picture, more clearly frame the relationship between observation and wavefunction collapse, and provide a quantum. Since the article attempts to provide a complete answer to the mechanism of the formation of wave-particle duality, extensive space is given to sections such as historical development, theoretical models, modern research, etc. New approaches to this phenomenon have been developed using modern research and string theory. As a result, a new mathematical expression has been proposed that explains the reasons for the initial emergence of wave-particle duality, and is also important for string theory. The article explains the measurement problem with a more effective physical structure and ontological model.
Abstract Quantum interference is often seen as a manifestation of intrinsic wave–particle duality, invoking self-interference and wavefunction superposition. We propose an alternative causal framework where interference arises from interactions between a classical electron and a quantized electromagnetic field. Electrons follow definite, localized trajectories and yield discrete detection events. The wave-like interference pattern emerges statistically from stochastic, nonlocal momentum transfers mediated by quantized field modes. Simulations of double-slit experiments show that dot-like fringes arise without wavefunction collapse or intrinsic wave behavior. Doubling the electron mass narrows fringe spacing in line with de Broglie scaling, suggesting the effective wavelength results dynamically from field coupling, not as an inherent particle trait. This interpretation preserves classical realism while reproducing quantum interference as an emergent, ensemble-level phenomenon. It provides a deterministic account grounded in structured field interactions. In this view, wave–particle duality of electrons reflects the wave nature of photons—the massless U(1) gauge bosons mediating the field—rather than being a fundamental property of matter.
Unlike the usual interference experiment with two slits separated in space, a time double-slit experiment uses the equivalent of a single slit covered by a shutter that is opened and closed and then quickly opened and closed again. Such experiments have recently been performed, providing further confirmation of wave-particle duality. In either type of experiment, the wave function beyond the slit(s) initially has two peaks that then spread and interfere. This article derives the results of these experiments in a schematic, but analytic solution of the time-dependent Schrödinger equation with appropriate initial conditions. In the space double-slit case, in addition to the transverse interference pattern, the time evolution at a fixed position exhibits an oscillating transient behavior similar to that obtained in the time double-slit experiment. Finally, the schematic model also reproduces the observed progressive visibility of the interference pattern obtained as a movable mask is displaced in front of the double-slit.
Wave-particle duality remains one of the most striking and conceptually unresolved features of quantum mechanics. Here I present a dynamical mechanism for wave-particle duality based on spontaneous stochastic mass-energy interconversion at subatomic scales. By promoting particle mass to a fluctuating quantity consistent with Einstein’s mass-energy equivalence, I derive a Schrödinger equation with an additional stochastic kinetic-phase term. Applied to the canonical double-slit experiment, the framework shows that quantum interference arises from coherent, path-dependent phase accumulation driven by mass-energy fluctuations, while particle-like localization emerges naturally at detection. The formalism yields closed-form expressions for interference visibility, predicts a momentum- and mass-dependent decoherence rate, admits a path-integral interpretation, and enables direct experimental bounds using neutron, electron, and atom interferometry. The results provide a physically grounded account of wave-particle duality without modifying the axioms of quantum mechanics.