By Ángel S. Sanz
Trajectory-based formalisms are an intuitively attractive approach of describing quantum tactics simply because they permit using "classical" suggestions. starting at an introductory point compatible for college kids, this two-volume monograph provides (1) the basics and (2) the purposes of the trajectory description of simple quantum procedures. this primary quantity is focussed at the classical and quantum history 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 structures and to optics also are proposed and discussed.
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Additional info for A Trajectory Description of Quantum Processes. I. Fundamentals: A Bohmian Perspective
Quasiperiodic trajectories will give rise to regular islands and chaotic trajectories to randomly distributed points on the PSOS. The local stability of periodic motions is an important issue. 40), the evolution of trajectories starting at conditions which slightly deviate from it in small amounts δx are studied. 39) and then expanding to linear order in δx yields δ x˙ = ∂x F(x)δx. 44) where M(x0 , t) is the fundamental matrix, which obeys the evolution equation ˙ 0 , t) = ∂x0 F[ M(x t (x0 )]M(x0 , t)δx0 .
126) where p is the pressure. This equation describes the evolution of a non-viscous flow and is obtained by setting the shear (η) and bulk (κ) viscosity coefficients equal to zero. On the contrary, for a viscous flow the hydrodynamics is ruled by the Navier– Stokes equation, ρfl ∂ η + v · ∇ v = η∇ 2 v + + κ ∇(∇ · v) − ∇ p(r, t). 127) As it will be seen in Chap. 6, these two equations also play a fundamental role in the understanding of the dynamics of quantum systems when they are described or interpreted as quantum fluids .
The motion with frequency ω− is called the symmetric mode, η− = x1 + x2 , because both particles oscillate in the same direction (in phase). On the other hand, ω+ gives rise to the so-called antisymmetric mode, η+ = x1 −x2 , because of their oscillation in opposite directions (out of phase). In terms of these modes, the total Hamiltonian reads as p12 p2 1 1 + (κ + κ12 )x12 + 2 + (κ + κ12 )x22 − κ12 x1 x2 2m 2 2m 2 p¯ 2 p¯ 2 1 2 2 1 2 2 = + + ω+ η + + − + ω− η− . 104) That is, one passes from two coupled oscillators (H1 + H2 + V12 ) to two uncoupled effective oscillators (H+ + H− , ) although the phase coherence is preserved implicitly: the coherence remains present in the collective (in phase or out of phase) motion.