By Ángel S. Sanz
Trajectory-based formalisms are an intuitively attractive means of describing quantum approaches simply because they enable using "classical" options. starting at an introductory point appropriate for college kids, this two-volume monograph offers (1) the basics and (2) the purposes of the trajectory description of simple quantum strategies. this primary quantity is focussed at the classical and quantum heritage essential to comprehend the basics of Bohmian mechanics, which are thought of the most subject of this paintings. Extensions of the formalism to the fields of open quantum platforms and to optics also are proposed and discussed.
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Extra resources for A Trajectory Description of Quantum Processes. I. Fundamentals: A Bohmian Perspective
It can be noticed that the system dynamics is independent of the particular choice of the initial ensemble distribution, ρ0 . 58). Within this formulation, classical statistical mechanics can then be interpreted as a field theory. Actually, if u and ρcl are assigned to the velocity field and density of an incompressible and non-rotational fluid respectively, the formalism can be tackled from a hydrodynamic viewpoint. 12a), an Euler equation is reached where the velocity field does not depend on the fluid density, 30 1 From Trajectories to Ensembles in Classical Mechanics ∂u ∇V + (u · ∇)u = − .
2). Nonetheless, in order to compare later on with quantum and Bohmian mechanics, consider the problem of an ensemble of identical, non-interacting free classical particles with mass m in one dimension. Moreover, it will also be assumed that their velocity is the same (v) and constant. This is equivalent to say that their energy is the same, since V = 0. Essentially, this ensemble is described by a microcanonical distribution function. 76) can be expressed in one dimension as 32 1 From Trajectories to Ensembles in Classical Mechanics ∂ρcl ∂ρcl = −v .
In general, they appear in dynamical systems described by nonlinear differential equations. More specifically, according to the Poincaré–Bendixon theorem , a system will display a chaotic dynamics if it is described by at least a set of three autonomous (time-independent) coupled equations or two if the system is nonautonomous (the third equation is supplied by the time-dependence). Hence, in the case of an autonomous Hamiltonian system, it is necessary, at least, two degrees of freedom with a nonseparable potential function.