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arXiv:2509.04539

Wave Packet Sizes in Quantum Mechanical Scatterings: New Perspective

Kenzo Ishikawa, O. Jinnouchi

Published Sep 4, 2025• 0 citations
arXivHTMLPDF

Abstract

Particles exist with a probability of 1 and are represented by wave packets in quantum mechanics. These wave packets satisfy the Schrödinger equation and are classified as either bound states or continuum states with finite coherence lengths. They undergo transitions according to unambiguous absolute probabilities, the magnitudes of which depend on the sizes of the wave packets. Because these sizes are not universal constants but depend on the environment and experimental conditions, their accurate estimation is essential. The sizes of propagating wave packets are determined by their environmental coherence lengths, whereas those of bound states are determined by their wave functions. Here, we present a systematic analysis of wave-packet sizes.

DOI
10.48550/arxiv.2509.04539

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Particles exist with a probability of 1 and are represented by wave packets in quantum mechanics. These wave packets satisfy the Schrödinger equation and are classified as either bound states or continuum states with finite coherence lengths. They undergo transitions according to unambiguous absolute probabilities, the magnitudes of which depend on the sizes of the wave packets. Because these sizes are not universal constants but depend on the environment and experimental conditions, their accurate estimation is essential. The sizes of propagating wave packets are determined by their environmental coherence lengths, whereas those of bound states are determined by their wave functions. Here, we present a systematic analysis of wave-packet sizes.
Coherence lengths of one particle states described by quantum wave functions are studied. We show that one particle states in various situations are not described by simple plane waves but are described by wave packets that are superpositions of plane waves. Wave packet is an approximate eigenstate of the free Hamiltonian and has a finite spatial size which we call the coherence length. The coherence lengths in the coordinate space and in the momentum space are studied in the present paper. We investigate several mechanisms of forming wave packets, stabilities of wave packets, and transformations of wave packets. 1 1
The coherence lengths of one-particle states described using quantum wave functions are studied. We show that one particle states in various situations are not described using simple plane waves but using wave packets that are superpositions of plane waves. A wave packet is an approximate eigenstate of the free Hamiltonian and has a finite spatial size that we call the coherence length. The coherence lengths in the coordinate space and momentum space are studied in this paper. We investigate several mechanisms of forming wave packets, stabilities of wave packets, and transformations of wave packets.
The wave-particle duality introduced by de Broglie and developed mathematically by Schrödinger remains one of the deepest conceptual achievements of modern physics. Yet Schrödinger's original physical intuition contained a problem that has never entirely disappeared: if a particle is represented by a wave, how can an extended and generally dispersive wave structure correspond to the localized events observed in experiments? Schrödinger initially hoped that a particle could be understood as a localized wave packet formed by the coherent superposition of many elementary waves. The program was attractive because localization could then arise from interference rather than being imposed as an independent particle property. However, generic wave packets disperse under free evolution, while stable bound states of quantum systems are not generally localized packets moving along classical trajectories. The subsequent Born interpretation provided an extraordinarily successful connection between the wave function and experimental statistics, identifying ∣Ψ∣2|\Psi|^2 with a probability density. Nevertheless, this interpretation leaves open a distinct physical question: how does a spatially extended complex amplitude structure give rise to localized, discrete detection events? This paper revisits that question from the historical development of wave mechanics to the operational structure of early scattering experiments. Particular attention is given to the experiments of Davisson and Kunsman, Ramsauer's measurements of slow-electron scattering, and Dymond's measurements of the velocity and angular distribution of electrons scattered in helium. Although these experiments differ in physical arrangement and measured quantities, they share a common operational structure: an extended incident quantum state undergoes propagation and interaction, after which the experiment registers discrete events within a finite spatial, angular, or energy acceptance region. The central proposal of this paper is therefore not a new derivation of Born's rule, but a reformulation of the problem that precedes such a derivation. Instead of asking only why a quantum state has a probability distribution, one may ask how a propagating wave-amplitude structure is dynamically converted into a localized event rate. In symbolic form, Ψ⟶D⟶Γ,\Psi \quad \longrightarrow \quad \mathcal{D} \quad \longrightarrow \quad \Gamma, where Ψ\Psi denotes the propagating amplitude structure, D\mathcal{D} represents the interaction and detection dynamics, and Γ\Gamma denotes the resulting localized event rate. Born's rule then appears as the empirical relation Γ(r)∝∣Ψ(r)∣2,\Gamma(\mathbf r)\propto |\Psi(\mathbf r)|^2, whose dynamical origin remains to be established. The paper concludes by reconsidering Schrödinger's original wave intuition rather than simply declaring it defeated. The failure of the persistent localized wave packet may indicate not that wave ontology was fundamentally misguided, but that propagation and localization are physically distinct stages of quantum dynamics. This distinction provides a conceptual starting point for a subsequent series of investigations into the dynamical origin of quantum probability.
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The quantum mechanics is most easily started from the experimentally observed discrete energy states of bound particles. The free particle may take on any energy whatever, which makes the quantum description of a free particle more awkward conceptually and mathematically than the quantum description of a bound particle. The construction of moving quantum spatial probabilities will require a superposition of stationary state wave functions of different energies. Any coherent superposition of energy eigen states for a particle represents a possible quantum state of that particle. The most important property of the spatial probability function for superposed energy states is that it changes with time. A moving probability that is limited spatially is called a packet and the corresponding quantum-mechanical description is called a packet state. The free-particle packet states examples satisfy the position-momentum uncertainty relation. Probably the most important application of the energy time uncertainty relation concerns the lifetimes of the excited states of atoms and nuclei.