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From Prepared Electron Flux to Localized Detection Events: Schrödinger Flow, Boundary Transport, and the Structure of Born Statistics

Kaisheng Li, Longji Li

Published Sep 22, 2026• 0 citations
Publisher

Abstract

The Born rule provides the empirically successful statistical connection between the Schrödinger wave function and the outcomes of quantum experiments. Yet two logically distinct questions are often compressed into this rule: what the wave function describes during propagation, and why repeated localized particle events exhibit the statistics associated with its squared amplitude. In an earlier analysis, we proposed a deliberately restricted interpretation of the single-particle Schrödinger wave function as a complex-amplitude representation of the spatial--directional structure of quantum propagation. The present paper advances that proposal by one dynamical step. We begin not from an abstract probability distribution, but from the operational structure of electron-scattering experiments. Irrespective of the detailed electron-generation mechanism, a completed preparation procedure produces an actual electron flux crossing a source-exit surface. We therefore introduce a \textbf{preparation boundary} $\Sigma_0$, and define $N_0$ as the actual number of electrons crossing $\Sigma_0$ during a specified time interval. A second surface, $\Sigma_D$, represents an idealized detection boundary. For a Schrödinger state $\Psi(\mathbf r,t)$, the exact continuity structure defines\[\rho_\Psi=|\Psi|^2,\qquad\mathbf j_\Psi=\frac{\hbar}{m}\operatorname{Im}\left(\Psi^*\nabla\Psi\right),\]and, wherever $\rho_\Psi\neq0$,\[\mathbf v_\Psi=\frac{\mathbf j_\Psi}{\rho_\Psi}.\] We investigate the trajectory hypothesis that a localized electron configuration crossing $\Sigma_0$ at $(\mathbf X_0,t_0)$ evolves according to\[\frac{d\mathbf X}{dt}=\mathbf v_\Psi(\mathbf X,t).\]This generates a boundary-to-boundary flow map\[\Phi_\Psi:(\mathbf X_0,t_0)\longrightarrow(\mathbf X_D,t_D).\] An ensemble of actual source crossings therefore defines a physical source measure $\mu_0$, while the corresponding distribution of detector crossings is the push-forward\[\mu_D=(\Phi_\Psi)_*\mu_0.\] The Born problem can then be reformulated sharply: under what physical conditions does the source-generated measure $\mu_0$, transported by the Schrödinger flow, produce the standard Born distribution at $\Sigma_D$? The trajectory law and the equivariance of $|\Psi|^2$ are closely related to established de Broglie--Bohm theory, and the connection between quantum flux and asymptotic surface crossings is rigorously developed in the flux-across-surfaces literature. We do not claim these mathematical structures as new. The contribution of the present work is instead to reconstruct the quantum-statistical problem from the complete experimental chain\[\begin{aligned}\text{preparation}&\rightarrow\text{source crossing}\rightarrow\text{Schrödinger flow}\\&\rightarrow\text{trajectory}\rightarrow\text{detection crossing}\rightarrow\text{statistics},\end{aligned}\]and thereby isolate a distinct \textbf{source-measure problem}. If Born statistics are to be dynamically accounted for rather than imposed at the outset, the remaining question is not only how $|\Psi|^2$ is transported, but why the microscopic preparation process produces the particular source ensemble whose boundary transport yields Born statistics. The present paper does not claim to solve that final problem. It formulates it as a concrete boundary-transport problem that can be addressed analytically, numerically, and ultimately through explicit models of electron preparation.

DOI
10.5281/zenodo.22886037

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The physical meaning of Schrödinger’s wave function remains afoundational interpretive question in quantum mechanics. This paperseparates two questions that are often introduced together: what thewave function describes during propagation, and why experimentaloutcomes obey Born statistics. Rather than taking∣Ψ∣2|\Psi|^2∣Ψ∣2 as the starting definition of Ψ\PsiΨ, we begin from theoperational structure of early electron-scattering and transmissionexperiments surrounding the emergence of wave mechanics and Born’s1926 collision theory. Davisson and Kunsman measured angle-dependent collected current,Ramsauer measured energy-dependent electron transmission throughgases, and Dymond later resolved the energy and angular redistributionof electrons after collisions. In each case, the apparatus returnedphysical signals---charge, current, or flux---from which reproducibleoutcome distributions were reconstructed. From this common structure we propose a deliberately limitedinterpretation: for a single nonrelativistic particle, Ψ\PsiΨ may beread first as a complex-amplitude representation of thespatial--directional organization of quantum propagation. Thisproposal is anchored not only in the experimental sequence but also inthe mathematics of the wave function. Its Fourier representationresolves propagation into wave-vector components, while theSchrödinger continuity relation defines a conserved density--currentpair; in polar formΨ=Rexp⁡(iS/ℏ), \Psi = R\exp(iS/\hbar),Ψ=Rexp(iS/ℏ),the associated current is proportional toR2∇S. R^2\nabla S.R2∇S. These facts show that Ψ\PsiΨ contains both spatial weighting anddirectional phase structure before any additional ontological claim ismade. We do not infer a material wave medium, deterministic particletrajectories, hidden variables, or an alternative dynamics. Nor do wederive the Born rule. Instead, we isolate a separate mapping problem:how does a structured propagation state, through a specific detectorinteraction, become the observed statistics of localized or collectedparticle events? In scattering, any acceptable account must recover the standard fluxand cross-section relations. The contribution of the present paper istherefore interpretive and programmatic: it identifies a constrainedphysical reading of Ψ\PsiΨ and states precisely what remains to beexplained.
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