Quantum Trajectories: The Dynamics of Knowledge

Richard Lombardini, Bill Poirier

In recent decades, the de Broglie-Bohm or ‘pilot wave’ interpretation of quantum mechanics–especially its concept of the ‘quantum trajectory’–has inspired a plethora of computational methods in molecular and chemical physics. One particularly promising subclass of such methods are the ‘interacting quantum trajectory’ (IQT) methods, for which an ensemble of trajectories is propagated independently of–and without any recourse to–an underlying wavefunction. In this work, an IQT version of the Pauli equation is developed–enabling spin to be incorporated into IQT theories for the first time, to the authors' knowledge. The new trajectory-based dynamical equations are then used to solve two iconic quantum problems involving spin: the ‘quantum spin flipper’, and the Stern-Gerlach experiment.

Sophya V. Garashchuk

The de Broglie-Bohm formulation of the Schrodinger equation implies conservation of the wave function probability density associated with each quantum trajectory in closed systems. This conservation property greatly simplifies numerical implementations of the quantum trajectory dynamics and increases its accuracy. The reconstruction of a wave function, however, becomes expensive or inaccurate as it requires fitting or interpolation procedures. In this paper we present a method of computing wave packet correlation functions and wave function projections, which typically contain all the desired information about dynamics, without the full knowledge of the wave function by making quadratic expansions of the wave function phase and amplitude near each trajectory similar to expansions used in semiclassical methods. Computation of the quantities of interest in this procedure is linear with respect to the number of trajectories. The introduced approximations are consistent with approximate quantum potential dynamics method. The projection technique is applied to model chemical systems and to the H+H(2) exchange reaction in three dimensions.

Maciej Błaszak, Ziemowit Domański

Michail A. Zak

The paper extends the mathematical formalism of quantum physics to include economics. The novelty of the approach is based upon a human-factor-based behavioral model of an intelligent agent. The model is quantum inspired: it is represented by a modified Madelung equation in which the gradient of quantum potential is replaced by a specially chosen information force. It consists of motor dynamics simulating actual behavior of the agent, and mental dynamics representing evolution of the corresponding knowledge base and incorporating this knowledge in the form of information flows into the motor dynamics. Due to feedback from mental dynamics, the motor dynamics attains quantum-like properties: its trajectory splits into a family of different trajectories, and each of those trajectories can be chosen with the probability prescribed by the mental dynamics. All of these departures actually extend and complement the classical methods making them especially successful in analysis of communications of agents represented by new mathematical formalism.