By Ángel S. Sanz
Trajectory-based formalisms are an intuitively attractive approach of describing quantum techniques simply because they enable using "classical" strategies. 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 tactics. this primary quantity is focussed at the classical and quantum historical past essential to comprehend the basics of Bohmian mechanics, that 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 info for A Trajectory Description of Quantum Processes. I. Fundamentals: A Bohmian Perspective
76) can be expressed in one dimension as 32 1 From Trajectories to Ensembles in Classical Mechanics ∂ρcl ∂ρcl = −v . 80) This is the Euler equation corresponding to an incompressible flow. 80), it is possible to switch from ρcl to its Fourier transform or spectral decomposition, 1 ρcl (x, t) = √ 2π ρ˜cl (α, t)e−iαx dα. 82) ρ˜cl (α, t) = ρ˜cl (α, 0)eiαvt . 81) and rearranging terms, it is found that ρcl (x, t) = ρcl (x − vt). 84) Physically, this result means that the density distribution function describing an incompressible, free fluid does not spread, but preserves its initial shape along time: it is dispersionless.
Physically, this means that it is not necessary to describe in a very detailed way the deterministic evolution of the adsorbates, but the effect induced by the surface on their positions (or velocities) allows us to accounted for them as Gaussian stochastic variables (see Appendix B). As a consequence, particles exhibit a Brownian motion. 51)—processes where particles do not undergo a Brownian motion—are generically called anomalous diffusion processes. 51) is replaced by |x(t) − x(0)|2 ∼ Dα t α .
Do not give rise to new ones), but may change their stability. In two-dimensional Hamiltonian systems there are only five types of bifurcations [57–59]. For example, the period-doubling bifurcation (n = 2, m = 1, Tr(M1 ) = − 2) changes the stability of the period-1 motion. In such a case, locating the most important periodic orbits of period 1 and following the evolution of the monodromy matrix with energy will allow us to obtain valuable information about how the phase-space structure changes through this type of bifurcation.