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arXiv:1407.4250 · Communications in Computational Physics

A Model of a Quantum Particle in a Quantum Environment: A Numerical Study

Raffaele Carlone, Rodolfo Figari, Claudia Negulescu

Published Jan 1, 2015• 7 citations
arXivHTMLPDF

Abstract

We define and investigate, via numerical analysis, a one dimensional toy-model of a cloud chamber. An energetic quantum particle, whose initial state is a superposition of two identical wave packets with opposite average momentum, interacts during its evolution and exchanges (small amounts of) energy with an array of localized spins. Triggered by the interaction with the environment, the initial superposition state turns into an incoherent sum of two states describing the following situation: or the particle is going to the left and a large number of spins on the left side changed their states, or the same is happening on the right side. This evolution is reminiscent of what happens in a cloud chamber where a quantum particle, emitted as a spherical wave by a radioactive source, marks its passage inside a supersaturated vapour-chamber in the form of a sequence of small liquid bubbles arranging themselves around a possible classical trajectory of the particle.

DOI
10.4208/cicp.270814.311214a

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We define and investigate, via numerical analysis, a one dimensional toy-model of a cloud chamber. An energetic quantum particle, whose initial state is a superposition of two identical wave packets with opposite average momentum, interacts during its evolution and exchanges (small amounts of) energy with an array of localized spins. Triggered by the interaction with the environment, the initial superposition state turns into an incoherent sum of two states describing the following situation: or the particle is going to the left and a large number of spins on the left side changed their states, or the same is happening on the right side. This evolution is reminiscent of what happens in a cloud chamber where a quantum particle, emitted as a spherical wave by a radioactive source, marks its passage inside a supersaturated vapour-chamber in the form of a sequence of small liquid bubbles arranging themselves around a possible classical trajectory of the particle.
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.
The behaviour of localized quantum wave packet has lately become an object of much interest to forge a link between classical and quantum mechanics. Wave packet is often taken as the best approximation of quantum theory to a classical particle. If the wave packet is regarded as a point, with zero amplitude everywhere except in a very small region, then its motion can be interpreted as the motion of a single particle. The advent of short-pulsed lasers made it possible to produce and detect superposition of electron states for many physical systems. Such superposition leads to the formation of localized electron wave packets. The localized wave packet solution disperses or spreads in time, however, at a later time, it reforms and appears as its initial state. Revivals in quantum systems were first studied in the Jaynes–Cumming model, describing a two-level atom interacting with a resonant monochromatic field.
We consider a simple one-dimensional quantum system consisting of a heavy and a light particle interacting via a point interaction. The initial state is chosen to be a product state, with the heavy particle described by a coherent superposition of two spatially separated wave packets with opposite momentum and the light particle localized in the region between the two wave packets. We characterize the asymptotic dynamics of the system in the limit of small mass ratio, with an explicit control of the error. We derive the corresponding reduced density matrix for the heavy particle and explicitly compute the (partial) decoherence effect for the heavy particle induced by the presence of the light one for a particular set up of the parameters.
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.
Two-particle scattering probabilities in tunneling scenarios with exchange interaction are analyzed with quasi-particle wave packets. Two initial one-particle wave packets (with opposite central momentums) are spatially localized at each side of a barrier. After impinging upon a tunneling barrier, each wave packet splits into transmitted and reflected components. When the initial two-particle anti-symmetrical state is defined as a Slater determinant of any type of (normalizable) one-particle wave packet, it is shown that the probability of detecting two (identically injected) electrons at the same side of the barrier is different from zero in very common (single or double barrier) scenarios. In some particular scenarios, the transmitted and reflected components become orthogonal and the mentioned probabilities reproduce those values associated to distinguishable particles. These unexpected non-zero probabilities are still present when non-separable Coulomb interaction or non-symmetrical potentials are considered. On the other hand, for initial wave packets close to Hamiltonian eigenstates, the usual zero two-particle probability for electrons at the same side of the barrier found in the literature is recovered. The generalization to many-particle scattering probabilities with quasi-particle wave packets for low and high phase-space density are also analyzed. The far-reaching consequences of these non-zero probabilities in the accurate evaluation of quantum noise in mesoscopic systems are briefly indicated.