This is a study of proton transmission through planar channels of tungsten, where a proton beam is treated as an ensemble of noninteracting wave packets. For this system, the structural stability manifests in an appearance of caustic lines, and as an equivalence of self-interference produced waveforms with canonical diffraction patterns. We will show that coordination between particle self-interference is an additional manifestation of the structural stability existing only in ensembles. The main focus of the analysis was on the ability of the coordination to produce classical structures. We have found that the structures produced by the self-interference are organized in a very different manner. The coordination can enhance or suppress the quantum aspects of the dynamics. This behavior is explained by distributions of inflection, undulation, and singular points of the ensemble phase function, and their bifurcations. We have shown that the coordination has a topological origin which allows classical and quantum levels of reality to exist simultaneously. The classical behavior of the ensemble emerges out of the quantum dynamics without a need for reduction of the quantum to the classical laws of motion.
This is a study of proton transmission through planar channels of tungsten, where a proton beam is treated as an ensemble of noninteracting wave packets. For this system, the structural stability manifests in an appearance of caustic lines, and as an equivalence of self-interference produced waveforms with canonical diffraction patterns. We will show that coordination between particle self-interference is an additional manifestation of the structural stability existing only in ensembles. The main focus of the analysis was on the ability of the coordination to produce classical structures. We have found that the structures produced by the self-interference are organized in a very different manner. The coordination can enhance or suppress the quantum aspects of the dynamics. This behavior is explained by distributions of inflection, undulation, and singular points of the ensemble phase function, and their bifurcations. We have shown that the coordination has a topological origin which allows classical and quantum levels of reality to exist simultaneously. The classical behavior of the ensemble emerges out of the quantum dynamics without a need for reduction of the quantum to the classical laws of motion.
We are investigating the quantum dynamics of a well-collimated electron beam transmitting through planar channels of the Si crystal. Electron states were represented by wave packets while the electron beam is treated as an ensemble of noninteracting wave packets. We have investigated the relationship between classical caustic pattern and anharmonicity of the potential and analyzed how quantum dynamic depends on the wave packet impact parameter and beam's angular divergence. We found that the extrema of the electron trajectory period, considered as a function of the impact parameter, determine the shape of the caustic pattern. All wave packet probability densities have multiple maxima generated by a self-interference. Their sum, that represents probability density of an ensemble, was found to depends strongly on the beam angular divergence. For small divergence, most peaks of different wave packets are aligned causing wavelike behavior of the ensemble. For moderate angular divergence maxima of some wave packets, are aligned with minima of others, resulting in the emergence of the classical caustic pattern. We have shown and experimentally confirmed that the only indication that the observed caustic pattern is generated by the quantum dynamics is a slight systematic shift of the corresponding caustic maxima.
We have investigated the dynamics of a quantum particle in the optical lattice potential. Initially, the quantum particle was represented by a Gaussian wave packet, located in the center of the well. The corresponding Schrödinger equation was solved explicitly by the method of the Chebyshev global propagation. Obtained solutions were also used for the construction of the Wigner functions. We found a great number of local abrupt changes of the solution shape. To explain this behavior, we used the fact that structurally stable systems, which form the largest class of the low dimensional dynamical systems, can be modeled and classified according to the catastrophe theory. All important features of the exact solution were explained on the basis of the mathematical properties of the catastrophic model. Such an approach enabled us to extract relevant information out of numerical solutions without employing any kind of approximations. We have investigated the influence of the Wigner catastrophes on the details of the quantum-classical correspondence breakdown. The wave packet was found to expand rapidly, filling the whole classically available area of the phase space. It was found that its self-interference pattern saturates quickly. A region of the phase space emerges in which the Wigner function oscillations transform into the singularity driven fluctuations. Once this region covers the whole area of the phase space, a wave packet dynamics enters into the new regime where its Wigner function fluctuates around the ergodic average. It will be shown that all mentioned processes are caused by the proliferation of the catastrophes and their mutual interactions.
The behavior of classical and quantum wave beams in stationary media is shown to be ruled by a "Wave Potential" function encoded in Helmholtz-like equations, determined by the structure itself of the beam and taking, in the quantum case, the form of Bohm's "Quantum Potential", which is therefore not so much a "quantum" as a "wave" property. Exact, deterministic motion laws, mutually coupled by this term and describing wave-like features such as diffraction and interference, are obtained in terms of well defined Hamiltonian trajectories, and shown to reduce to the laws of usual geometrical optics and/or classical dynamics when this coupling term is neglected. As far as the quantum case is concerned, the approach proposed in the present paper, suggested by the direct extension of the treatment holding for classical waves, describes the motion of classical-looking, point-like particles, without resorting to the use of travelling wave-packets.
The behavior of classical and beams in stationary media is shown to be ruled by a Wave Potential function encoded in Helmholtz-like equations, determined by the structure itself of the beam and taking, in the case, the form of Bohm's Quantum Potential, which is therefore not so much a quantum as a wave property. Exact, deterministic motion laws, mutually coupled by this term and describing wave-like features such as diffraction and interference, are obtained in terms of well defined Hamiltonian trajectories, and shown to reduce to the laws of usual geometrical optics and/or classical dynamics when this coupling term is neglected. As far as the case is concerned, the approach proposed in the present paper, suggested by the direct extension of the treatment holding for classical waves, describes the motion of classical-looking, point-like particles, without resorting to the use of travelling wave-packets.
Interference is commonly regarded as the most direct manifestation of the superposition principle. This association is natural for the linear Schrödinger equation, where coherent alternatives combine at the level of probability amplitudes. However, the situation becomes less transparent when nonlinear couplings are present, or when the field is only partially coherent. In this work, we argue that a more robust organizing principle is provided by the local flow generated by phase variations. In this sense, phase-induced flow acts as a unifying mechanism for interference-like dynamics in nonlinear and partially coherent Schrödinger systems. The discussion is developed from a hydrodynamic, or Bohmian, perspective, understood here as a practical probing tool rather than as an additional ontology. Three representative situations are considered: interfering Bose--Einstein condensates described by the Gross--Pitaevskii equation, nonlinear Schrödinger dynamics obtained by modifying the quantum-potential contribution, and partially coherent Airy beams described through their cross-spectral density. Although these systems differ in physical origin and mathematical implementation, they share a common dynamical structure: density-related observables are shaped by velocity fields determined by phase, or ensemble-phase, information. From this viewpoint, interference-like traits, localization, self-acceleration and coherence loss can be interpreted in terms of the preservation, deformation or breaking of the symmetries displayed by the underlying flow. This provides a compact way of connecting interference, nonlinear dynamics, classicality, coherence loss, and structured-light propagation within a single trajectory-based framework.
Title:Interactions and Effective Forces Between Topological Defects in Emergent Schrödinger Dynamics Publication type:Preprint / Working Paper Description: This paper investigates the emergence of particle interactions in a classical wave system through topological defects. Building on previous work establishing an emergent Schrödinger equation in classical elastic media, we show how multiple particle-like excitations—identified as topological phase singularities—interact via phase-mediated mechanisms without introducing fundamental force laws. Key findings: Defect interactions arise automatically from the shared global wavefield, with no explicit interaction terms in the underlying wave equation. The interaction is long-ranged and geometry-driven, with like topological charges repelling and opposite charges attracting in a Coulomb-like manner. Inertia emerges from the distributed kinetic energy of the wavefield, yielding an effective mass that depends on the defect structure and system geometry. The dynamics exhibits deterministic scattering and bound-state behavior governed entirely by wave-mediated topology. Despite structural similarities to quantum mechanics, the framework remains fully classical and deterministic, with no superposition of trajectories, entanglement, or quantum statistics. Theoretical framework: The analysis is formulated within classical continuum mechanics, where an effective Schrödinger equation governs the envelope dynamics of elastic waves. Particle-like excitations correspond to phase singularities whose motion is constrained by a guidance equation. Multi-defect configurations are encoded within a single global wavefield whose phase topology captures all interactions. Significance: This work demonstrates how concepts typically regarded as fundamental—particles, mass, forces, and interactions—can emerge from continuous classical fields through topological constraints. The model provides a concrete and physically transparent example of emergence in physics, with implications for understanding: the relationship between wave and particle descriptions, geometric origins of effective forces, classical analogs of quantum-like behavior, and the role of topology in dynamical systems. Mathematical approach: The paper employs tools from: topological field theory (phase singularities, winding numbers), differential geometry (phase gradient and velocity fields), classical mechanics (effective mass, scattering dynamics), and continuum mechanics (elastic wave equations). Relation to existing work: This study completes a self-consistent framework connecting wave dynamics, particle motion, and interactions within classical physics. It relates to vortex dynamics in fluids, pilot-wave hydrodynamics, and emergent phenomena in condensed matter systems, while maintaining clear conceptual and physical distinctions from quantum mechanics.