Mott problem

Abstract: We analyze a one dimensional quantum system consisting of a test particle interacting with two harmonic oscillators placed at the positions $a_1$, $a_2$, with $a_1 >0$, $|a_2|>a_1$, in the two possible situations: $a_2>0$ and $a_2 0$ at time $t > |a_2| v_0^{-1}$ when $a_2 0$). We prove that $\mathcal{P}_{n_1 n_2}^- (t)$ is negligible with respect to $\mathcal{P}_{n_1 n_2}^+ (t)$, up to second order in time dependent perturbation theory. The system we consider is a simplified, one dimensional version of the original model of a cloud chamber introduced by Mott in \cite{m}, where the result was argued using euristic arguments in the framework of the time independent perturbation theory for the stationary Schr\"{o}dinger equation. The method of the proof is entirely elementary and it is essentially based on a stationary phase argument. We also remark that all the computations refer to the Schr\"{o}dinger equation for the three-particle system, with no reference to the wave packet collapse postulate.

Abstract: The hypothesis (Sparenberg et al. in EPJ Web Conf 58:01016, [1]. https://doi.org/10.1051/epjconf/20135801016 ) that the particular linear tracks appearing in the measurement of a spherically-emitting radioactive source in a cloud chamber are determined by the (random) positions of atoms or molecules inside the chamber is further explored in the framework of a recently established one-dimensional model (Carlone et al. Comm Comput Phys 18:247, [2]. https://doi.org/10.4208/cicp.270814.311214a ). In this model, meshes of localized spins 1/2 play the role of the cloud-chamber atoms and the spherical wave is replaced by a linear superposition of two wave packets moving from the origin to the left and to the right, evolving deterministically according to the Schrodinger equation. We first revisit these results using a time-dependent approach, where the wave packets impinge on a symmetric two-sided detector. We discuss the evolution of the wave function in the configuration space and stress the interest of a non-symmetric detector in a quantum-measurement perspective. Next we use a time-independent approach to study the scattering of a plane wave on a single-sided detector. Preliminary results are obtained, analytically for the single-spin case and numerically for up to 8 spins. They show that the spin-excitation probabilities are sometimes very sensitive to the parameters of the model, which corroborates the idea that the measurement result could be determined by the atom positions. The possible origin of decoherence and entropy increase in future models is finally discussed.

Abstract: In the α decay of a nucleus, the tracks left in the medium by the α particle are linear, even though its initial wave function is spherically symmetric Understanding this quantum phenomenon has been called “the Mott problem”, ever since Mott’s fundamental paper on the subject (Mott in Proc R Soc London Ser A 126:79 1929) Here we study a one dimensional version of the Mott problem The particle emitted in the decay is represented as a superposition of waves, one traveling to the left, the other to the right The atoms with which the particle interacts are modeled as two level systems The wave equation obeyed by the particle is taken to be the massless Dirac equation For a certain space-time structure for the particle-atom interaction, it is possible to derive an explicit space-time solution for the entire system, for an arbitrary number of atoms In the one dimensional solution, the coherent superposition of right and left-moving wave packets leaves behind tracks of excited atoms The Mott problem on the nature of the tracks left behind is addressed using the reduced density matrix, defined by taking the trace over all particle degrees of freedom It is found that the reduced density matrix is the incoherent sum of two terms, one involving excited atoms only on the right; the other involving excited atoms only on the left, implying that tracks will show excited atoms on one side or the other In one dimension, tracks which involve excited atoms exclusively on one side or the other are the analog of straight tracks in three dimensions

Abstract: We analyze a one dimensional quantum system consisting of a test particle interacting with two harmonic oscillators placed at the positions a1 and a2, with a1>0 and ∣a2∣>a1, in the two possible situations: a2>0 and a2 0 at time t>∣a2∣v0−1 when a2 0). We prove that Pn1n2−(t) is negligible with respect to Pn1n2+(t) up to second order in time dependent perturbation theory. The system we consider is a simplified, one dimensional version of the original model of a cloud chamber introduced by Mott [“The wave mechanics of α-ray tracks,” Proc. R. Soc. London, Ser. A 126, 79 (1929)], ...