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International Journal of Quantum Chemistry

Reconstructing interference fringes in slit experiments by complex quantum trajectories

Ciann‐Dong Yang, Kuan‐Chang Su

Published Jul 10, 2012• 14 citations
Publisher

Abstract

Abstract There are external and internal representations for a quantum state Ψ. External representation is commonly adopted in the standard quantum mechanics by exploiting probability density function Ψ*Ψ to explain the observed interference fringes in slit experiments. On the other hand, in quantum Hamilton mechanics, the quantum state Ψ has a dynamical representation that reveals the internal mechanism underlying the externally observed interference fringes. The internal representation of Ψ is described by a set of Hamilton equations of motion, by which quantum trajectories of a particle moving in Ψ can be solved. In this article, millions of complex quantum trajectory connecting slits to a screen are solved from the Hamilton equations, and the statistical distribution of their arrivals on the screen is shown to reproduce the observed interference fringes. This appears to be the first quantitative verification of the equivalence between the trajectory‐based statistics and the wavefunction‐based statistics on slit experiments. © 2012 Wiley Periodicals, Inc.

DOI
10.1002/qua.24269

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Abstract There are external and internal representations for a quantum state Ψ. External representation is commonly adopted in the standard quantum mechanics by exploiting probability density function Ψ*Ψ to explain the observed interference fringes in slit experiments. On the other hand, in quantum Hamilton mechanics, the quantum state Ψ has a dynamical representation that reveals the internal mechanism underlying the externally observed interference fringes. The internal representation of Ψ is described by a set of Hamilton equations of motion, by which quantum trajectories of a particle moving in Ψ can be solved. In this article, millions of complex quantum trajectory connecting slits to a screen are solved from the Hamilton equations, and the statistical distribution of their arrivals on the screen is shown to reproduce the observed interference fringes. This appears to be the first quantitative verification of the equivalence between the trajectory‐based statistics and the wavefunction‐based statistics on slit experiments. © 2012 Wiley Periodicals, Inc.
Single particle interference experiments have documented the existence of trajectory of particles ’ during the accumula-tion of an interference pattern. The expression for probability of particle’s arrivals to the screen was derived previously [22] by approximating the trajectories with straight lines, which start at various points of the slits, and using wave function of the transverse motion in the momentum representation. We use this expression to determine particle distribution at various distances along the lines parallel to a multiple slit grating. The agreement with usual probability density given by the mod-ulus square of the wave function in coordinate representation is very good in the far field. In the near field the agreement is poor. This result supports the view that the trajectories of the particles do exist and that they are straight lines in the far fields, but further study on the form of the trajectories of particles ’ in the near field is necessary. 164 Concepts of Physics, Vol. II (2005) On the (non)existence of quantum particles’...
We demonstrate how the interference effects for a quantum particle in the double-slit experiment can be described by classical probabilities. We investigate a classical field theory for a complex scalar field with probabilistic initial conditions. A central element are conserved charges leading to the concept of particles. These are statistical observables which describe properties of the probability distribution for field configurations. The conserved charges define subsystems for particle excitations of a vacuum state. The classical probability distribution for the one-particle subsystem can be expressed in terms of a complex wave function. The Liouville equation for the classical probability distribution implies that the time evolution of this wave function obeys the Schrödinger equation for a quantum particle in a potential. The potential arises from space-dependent external fields in the otherwise relativistic classical field theory. It can be chosen arbitrarily, realizing the typical quantum effects of interference, tunneling or discrete energy spectra.
Abstract We show that the quantum central‐force problems can be modeled and solved exactly by quantum Hamilton–Jacobi formulation, from which the quantum operators L^\hat{L}L^ z, L^\hat{L}L^ 2, and H^\hat{H}H^ can be derived without using the quantization principle p → (ℏ\hbarℏ /i)∂/∂x. Quantum conservation laws expressed by the Poisson bracket show that the eigenvalues of these quantum operators are just equal to the constants of motion along the eigen‐trajectories defined in a complex domain. The shell structure observed in bound systems, such as the hydrogen atom, is found to stem from the structure of the quantum potential, by which the quantum forces acting on the electron can be uniquely determined, the stability of atomic configuration can be justified, and the quantum trajectories of the electron can be obtained by integrating the related quantum Lagrange equations. On solving the quantum equations of motion, the solution of the Schrödinger equation serves as the first integration of the second‐order quantum Lagrange equations. The stable equilibrium points of the derived first‐order nonlinear quantum dynamics are shown to be identical to the positions with maximum probability predicted by standard quantum mechanics. The internal mechanism of how the quantum dynamics evolve continuously to classical dynamics and of how the quantum conservation laws transit continuously to the classical conservation laws as n → ∞ are analyzed in detail. The construction of the quantum scattering trajectory by searching for an unbound solution for the Schrödinger equation is investigated. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006
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.
This article examines the double-slit experiment from a configurational perspective and compares it with the standard quantum-mechanical, Bohmian, and Many-Worlds interpretations. Rather than focusing on microscopic trajectories or wavefunction collapse, the work argues that the central question posed by the experiment is not “how” individual particles behave, but “why” a stable interference pattern emerges with overwhelming statistical certainty. By framing the experiment in terms of constrained configuration spaces and typicality, the article shows that interference can be understood as an inevitable macroscopic outcome, not a mysterious dynamical process.
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.