From quantum to classical world: emergence of trajectories in a quantum system.

This note deals with models of quantum systems where the emergence of a classical behavior can be concretely analyzed. We first briefly review some well known difficulties arising in the classical limit of quantum mechanics according to the Copenhagen interpretation. Then we discuss the seminal contribution by Mott (1929) on the tracks observed in a cloud chamber, where the problem can be approached in a particularly transparent way. Finally, we propose a model Hamiltonian, with interaction described by spin dependent point interactions, where Mott’s analysis can be rephrased and the result can be rigorously formulated

Emergence of classical trajectories in quantum systems: the cloud chamber problem in the analysis of Mott (1929)

We analyze the paper "The wave mechanics of $\alpha$-ray tracks" (Mott, 1929), published in 1929 by N.F. Mott. In particular, we discuss the theoretical context in which the paper appeared and give a detailed account of the approach used by the author and the main result attained. Moreover, we comment on the relevance of the work not only as far as foundations of Quantum Mechanics are concerned but also as the earliest pioneering contribution in decoherence theory

EUROMOD: the European Union tax-benefit microsimulation model

This paper aims to provide an introduction to the current state of the art of EUROMOD, the European Union tax-benefit microsimulation model. It explains the original motivations for building a multi-country EU-wide model and summarises its current organisation. It provides an overview of EUROMOD components, covering its policy scope, the input data, the validation process and some technical aspects such as the tax-benefit programming language and the user interface. The paper also reviews some recent applications of EUROMOD and, finally, considers future developments

Joint excitation probability for two harmonic oscillators in dimension one and the Mott problem

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 zero the harmonic oscillators are in their ground state and the test particle is in a superposition state of two wave packets centered in the origin with opposite mean momentum. %$\pm M v_0$. Under suitable assumptions on the physical parameters of the model, we consider the time evolution of the wave function and we compute the probability $\mathcal{P}^{-}_{n_1 n_2} (t)$ (resp. $\mathcal{P}^{+}_{n_1 n_2} (t)$) that both oscillators are in the excited states labelled by $n_1$, $n_2 >0$ 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.Comment: 26 page

Effective equation for a system of mechanical oscillators in an acoustic field

We consider a one dimensional evolution problem modeling the dynamics of an acoustic field coupled with a set of mechanical oscillators. We analyze solutions of the system of ordinary and partial differential equations with time-dependent boundary conditions describing the evolution in the limit of a continuous distribution of oscillators.Comment: Improved Theorem 2. Updated introduction and references. Added 1 figure. 11 page

Two-dimensional Time-dependent Point Interactions

We study the time-evolution of a quantum particle subjected to time-dependent zero-range forces in two dimensions. After establishing a conceivable ansatz for the solution to the Schr\"{o}dinger equation, we prove that the wave packet time-evolution is completely specified by the solutions of a system of Volterra-type equations -- the {\it charge equations} -- involving the coefficients of the singular part of the wave function, thus extending to the two-dimensional case known results in one and three dimensions.Comment: 17 pages, AMS-LaTex; presentation of the model changed, small changes to Lemma 2.1 and Proposition 2.

GINI DP 2: Are European Social Safety Nets Tight Enough? Coverage and adequacy of minimum income schemes in 14 EU countries

This paper explores and compares the effectiveness of Minimum Income (MI) schemes in protecting persons of working age from poverty in the European Union. Using the European microsimulation model EUROMOD we estimate indicators of coverage and adequacy of MI schemes in 14 EU countries. In terms of coverage, we find that in several countries a significant number of individuals are ineligible for MI even when they fall below a poverty line set at 40 per cent of median income. With respect to adequacy, we show that in certain countries a large fraction of those entitled to MI remain at very low levels of income even when MI benefit is added. Overall, our findings suggest that the clustering of MI schemes in Europe may be more complex than previous literature has hitherto allowed for.