PARTICLE TRAJECTORIES AND THE PERCEPTION OF CLASSICAL MOTION IN THE FREE PROPAGATION OF WAVE PACKETS.

The free propagation in time of a normalisable wave packet is the oldest problem of continuum quantum mechanics. Its motion from microscopic to macroscopic distance is the way in which most quantum systems are detected experimentally. Although much studied and analysed since 1927 and presented in many text books, here the problem is re-appraised from the standpoint of semi-classical mechanics. Particular aspects are the emergence of deterministic trajectories of particles emanating from a region of atomic dimensions and the interpretation of the wave function as describing a single particle or an ensemble of identical particles. Of possible wave packets, that of gaussian form is most studied due to the simple exact form of the time-dependent solution in real and in momentum space. Furthermore, this form is important in laser optics. Here the equivalence of the time-dependent Schroedinger equation to the paraxial equation for the propagation of light is demonstrated explicitly. This parallel helps to understand the relevance of trajectory concepts and the conditions necessary for the perception of motion as classical.

Relativistically invariant Bohmian trajectories of photons

Bohmian mechanics is a nonlocal hidden-variable interpretation of quantum theory which predicts that particles follow deterministic trajectories in spacetime. Historically, the study of Bohmian trajectories has been restricted to nonrelativistic regimes due to the widely held belief that the theory is incompatible with special relativity. Here we derive expressions for the relativistic velocity and spacetime trajectories of photons in a Michelson-Sagnac-type interferometer. The trajectories satisfy quantum-mechanical continuity, the relativistic velocity addition rule. Our new velocity equation can be operationally defined in terms of weak measurements of momentum and energy. We finally propose a modified Alcubierre metric which could give rise to these trajectories within the paradigm of general relativity.

Experimental Comparison of Bohm-like Theories with Different Primary Ontologies

The de Broglie-Bohm theory is a hidden-variable interpretation of quantum mechanics which involves particles moving through space along deterministic trajectories. This theory singles out position as the primary ontological variable. Mathematically, it is possible to construct a similar theory where particles are moving through momentum-space, and momentum is singled out as the primary ontological variable. In this paper, we construct the putative particle trajectories for a two-slit experiment in both the position and momentum-space theories by simulating particle dynamics with coherent light. Using a method for constructing trajectories in the primary and non-primary spaces, we compare the phase-space dynamics offered by the two theories and show that they do not agree. This contradictory behaviour underscores the difficulty of selecting one picture of reality from the infinite number of possibilities offered by Bohm-like theories.

Stochastic Generation and Deformation of Toroidal Oscillations in Neuron Model

The stochastic Hindmarsh–Rose neuron model in the parametric regions of complex nonlinear dynamics with quiescent and torus bursting regimes is studied. We show that in the zone where the deterministic system exhibits the quiescence, noise can lead to the generation of toroidal dynamical structure with the transition to the bursting regime. The studies of dispersion of random trajectories as well as power spectral density and interspike intervals distribution confirm these changes in system dynamics. Moreover, the stochastic emergence of torus is followed by the noise-induced chaotization. We show that the generation of the torus bursting oscillations is associated with the specificity of the arrangement of deterministic trajectories in the vicinity of the equilibrium as well as its stochastic sensitivity. A probabilistic mechanism of the stochastic generation of torus is investigated on the basis of the analysis of the stochastic sensitivity and geometry of confidence regions. We also show that in the parametric range of the torus bursting, noise can lead to the increase of number of bursts per time interval and to the reduction of spiking and quiescent phase duration.

Stochastic dynamics of snow avalanche occurrence by superposition of Poisson processes

We study the dynamics of systems with deterministic trajectories randomly forced by instantaneous discontinuous jumps occurring according to two different compound Poisson processes. One process, with constant frequency, causes instantaneous positive random increments, whereas the second process has a state-dependent frequency and describes negative jumps that force the system to restart from zero (renewal jumps). We obtain the probability distributions of the state variable and the magnitude and intertimes of the jumps to zero. This modelling framework is used to describe snow-depth dynamics on mountain hillsides, where the positive jumps represent snowfall events, whereas the jumps to zero describe avalanches. The probability distributions of snow depth, together with the statistics of avalanche magnitude and occurrence, are used to explain the correlation between avalanche occurrence and snowfall as a function of hydrologic, terrain slope and aspect parameters. This information is synthesized into a ‘prediction entropy’ function that gives the level of confidence of avalanche occurrence prediction in relation to terrain properties.