We consider a semiclassical approximation, first derived by Heller and coworkers, for the time evolution of an originally gaussian wave packet in terms of complex trajectories.We also derive additional approximations replacing the complex trajectories by real ones. These yield three different semiclassical formulae involving different real trajectories. One of these formulae is Heller's thawed gaussian approximation. The other approximations are non-gaussian and may involve several trajectories determined by mixed initial-final conditions. These different formulae are tested for the cases of scattering by a hard wall, scattering by an attractive gaussian potential, and bound motion in a quartic oscillator. The formula with complex trajectories gives good results in all cases. The non-gaussian approximations with real trajectories work well in some cases, whereas the thawed gaussian works only in very simple situations.

We present an exhaustive numerical analysis of violations of local realism by families of multipartite quantum states. As an indicator of nonclassicality we employ the probability of violation for randomly sampled observables. Surprisingly, it rapidly increases with the number of parties or settings and even for relatively small values local realism is violated for almost all observables. We have observed this effect to be typical in the sense that it emerged for all investigated states including some with randomly drawn coefficients. We also present the probability of violation as a witness of genuine multipartite entanglement.Comment: 8 pages, 2 figures, 3 tables, journal versio

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In quantum theory we refer to the probability of finding a particle between positions x and x + dx at the instant t, although we have no capacity of predicting exactly when the detection occurs. In this work, first we present an extended non-relativistic quantum formalism where space and time play equivalent roles. It leads to the probability of finding a particle between x and x + dx during [t,t + dt]. Then, we find a Schrödinger-like equation for a "mirror" wave function φ(t, x) associated with the probability of measuring the system between t and t + dt, given that detection occurs at x. In this framework, it is shown that energy measurements of a stationary state display a non-zero dispersion, and that energy-time uncertainty arises from first principles. We show that a central result on arrival time, obtained through approaches that resort to ad hoc assumptions, is a natural, built-in part of the formalism presented here. In Schrödinger quantum mechanics (QM) there is a clear asymmetry between time and space. Time is a continuous parameter that can be chosen with arbitrary precision and used to label the solution of the wave equation. In contrast, the position of a particle is seen as an operator, and therefore its value under a measurement is inherently probabilistic. It is common to hear that this asymmetry is due to the non-relativistic character of the Schrödinger equation (SE). Although partially correct, this argument is largely insufficient to justify all the disparity between space and time in the formalism of QM.A clear illustration is as follows. In a position measurement, ψ(x, t) = x|ψ(t) gives the probability amplitude of finding the particle within [x, x + dx], given that the time of detection is t. Would it not be equally reasonable, even in the non-relativistic domain, to ask about the probability of measuring the particle between x and x + dx, and t and t + dt? In this broader scenario, inquiring about the state of a particle at a given time t (as we often do), should make as much sense as asking about the state of that particle in a given position x (which we never do). In addition, if symmetry is to hold at this level, then there should exist a "mirror" wave function φ(t, x) = t|φ(x) , where x is a continuous parameter and t is the eigenvalue of an observable. If the location of particle becomes a physical reality only when a measurement is made, then it is a tenable position to expect that time should emerge in the same way. To earnestly consider these issues is the main goal of this manuscript.Time has been addressed in different contexts in QM . Common to several of these works is the attempt to remain within the borders of the standard theory. However, the solution to the arrival-time problem is considered by several authors to lay outside the framework of QM. It concerns the arrival of a particle in a spatially localized apparatus, where a time operator may be defined so that the relation [T ,Ĥ] = i is satisfied, and * corresponding author: eduardodias@df.ufpe.br † parisio@df.ufpe.br ...

The injection of a fluid into another of larger viscosity in a Hele-Shaw cell usually results in the formation of highly branched patterns. Despite the richness of these structures, in many practical situations such convoluted shapes are quite undesirable. In this Brief Report, we propose an efficient and easily reproducible way to restrain these instabilities based on a simple piecewise-constant pumping protocol. It results in a reduction in the size of the viscous fingers by one order of magnitude.

The semiclassical propagation of Gaussian wave packets by complex classical trajectories involves multiple contributing and noncontributing solutions interspersed by phase space caustics. Although the phase space caustics do not generally lie exactly on the relevant trajectories, they might strongly affect the semiclassical evolution depending on their proximity to them. In this paper, we derive a third-order regular semiclassical approximation which correctly accounts for the caustics and which is finite everywhere. We test the regular formula for the potential V (x) = 1/x 2 , where the complex classical trajectories and phase space caustics can be computed analytically. We make a detailed analysis of the structure of the complex functions involved in the saddle point approximations and show how the changes in the steepest descent integration contour control both the contributing and noncontributing trajectories and the type of Airy function that appears in the regular approximation.

In this work we investigate the probability of violation of local realism under random measurements in parallel with the strength of these violations as described by resistance to white noise admixture. We address multisetting Bell scenarios involving up to 7 qubits. As a result, in the first part of this manuscript we report statistical distributions of a quantity reciprocal to the critical visibility for various multipartite quantum states subjected to random measurements. The statistical relevance of different classes of multipartite tight Bell inequalities violated with random measurements is investigated. We also introduce the concept of typicality of quantum correlations for pure states as the probability to generate a nonlocal behaviour with both random state and measurement. Although this typicality is slightly above 5.3% for the CHSH scenario, for a modest increase in the number of involved qubits it quickly surpasses 99.99%.

The question of how Bell nonlocality behaves in bipartite systems of higher dimensions is addressed. By employing the probability of violation of local realism under random measurements as the figure of merit, we investigate the nonlocality of entangled qudits with dimensions ranging from d = 2 to d = 7. We proceed in two complementary directions. First, we study the specific Bell scenario defined by the Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequality. Second, we consider the nonlocality of the same states under a more general perspective, by directly addressing the space of joint probabilities (computing the frequencies of behaviours outside the local polytope). In both approaches we find that the nonlocality decreases as the dimension d grows, but in quite distinct ways. While the drop in the probability of violation is exponential in the CGLMP scenario, it presents, at most, a linear decay in the space of behaviours. Furthermore, in both cases the states that produce maximal numeric violations in the CGLMP inequality present low probabilities of violation in comparison with maximally entangled states, so, no anomaly is observed. Finally, the nonlocality of states with non-maximal Schmidt rank is investigated.