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Physical Review A

Stability of particle trajectories and generalized coherent states

Kiyotaka Kakazu, Shuichi Matsumoto

Published Nov 1, 1990• 4 citations
Publisher

Abstract

We show that particle trajectories in the Wilson cloud chamber are straight and stable, using perturbation theory and generalized coherent states. The model consists of an incoming particle, many molecules, and the immediate environment of each molecule acting as a detector. Each such detector has many harmonic oscillators whose states are totally described by generalized coherent states, because of which particle trajectories are almost stable. That is, if a molecule is ionized through its interaction with the particle, the ionized excited state does not return to the lower states. Many molecules can be ionized when the line joining them is practically parallel to the direction of motion of the particle. However, when the molecules do not lie on a straight line, all of them cannot be ionized. The particle trajectory that is made up of the ionized molecules, therefore, is straight.

DOI
10.1103/physreva.42.5093

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We show that particle trajectories in the Wilson cloud chamber are straight and stable, using perturbation theory and generalized coherent states. The model consists of an incoming particle, many molecules, and the immediate environment of each molecule acting as a detector. Each such detector has many harmonic oscillators whose states are totally described by generalized coherent states, because of which particle trajectories are almost stable. That is, if a molecule is ionized through its interaction with the particle, the ionized excited state does not return to the lower states. Many molecules can be ionized when the line joining them is practically parallel to the direction of motion of the particle. However, when the molecules do not lie on a straight line, all of them cannot be ionized. The particle trajectory that is made up of the ionized molecules, therefore, is straight.
Particle trajectory in the Wilson cloud chamber is explained by means of the perturbation theory and generalized coherent states. The model consists of an incoming particle, molecules and detectors (the immediate environment of the molecules). The particle and the molecules are treated as typical quantum systems, while each detector has many harmonic oscillators whose state are totally described by generalized coherent states. If a molecule is ionized through interaction with a particle, the ionized excited state is stable, that is, the molecule does not return to the ground state. The stability of excited states of the molecule is a result of properties of the generalized coherent states of the detector interacting with the molecule.
The aim of this paper is to discuss in a pedagogical way the problem of the emergence of classical behaviour in certain physical systems which, in principle, are correctly described by quantum mechanics. It is stressed that the limit ℏ → 0 is not sufficient and the crucial role played by the environment must be taken into account. In particular, recalled is the old problem raised by Mott in 1929 ( Proc. R. Soc. Lond. A 126 79) concerning the straight tracks observed in a cloud chamber, produced by an α-particle emitted by a source in the form of a spherical wave. The conceptual relevance of the problem for a clearer understanding of the classical limit is discussed in a historical perspective. Moreover, a simple mathematical model is proposed, where the result of Mott is obtained in a rigorous mathematical way.
In the coherent state of the harmonic oscillator, the probability density is that of the ground state subjected to an oscillation along a classical trajectory. Senitzky and others pointed out that there are states of the harmonic oscillator corresponding to an identical oscillatory displacement of the probability density of any energy eigenstate. These generalizations of the coherent state are rarely discussed, yet they furnish an interesting set of quantum states of light that combine features of number states and coherent states. Here, we give an elementary account of the quantum optics of generalized coherent states.
The Mott problem concerns a long-standing paradox in quantum mechanics: although alpha-particle wavefunctions emitted from radioactive sources are spherically symmetric, cloud chambers consistently record straight-line tracks. Standard explanations rely on decoherence, probabilistic branching, or implicit classical assumptions, yet none fully resolve how directional trajectories emerge from an initially isotropic quantum state. This paper presents a geometric interpretation of the Mott problem within the Dimensional Evolution (DE) framework, in which wavefunction collapse is treated as a resonance- and identity-mediated geometric process rather than a purely probabilistic event. Measurement is modeled as a structured interaction between system and detector identity wells, producing directional collapse funnels that anchor sequential localization events. Trajectory formation is described as a process of coherent tether crystallization, whereby successive measurement interactions align collapse curvature along a stable resonant direction. Straight tracks arise not from classical particle paths, but from the geometric reinforcement of collapse orientation through detector–system coupling. Inverse-domain resonance matching replaces environment-driven decoherence as the primary mechanism governing localization. The DE approach resolves the Mott paradox without invoking hidden variables, many-worlds branching, or ad hoc classical transitions, and provides a unified geometric explanation for trajectory emergence in quantum measurement systems. Implications for quantum foundations, measurement theory, and experimental testability are discussed.