By Ángel S. Sanz
Trajectory-based formalisms are an intuitively attractive method of describing quantum techniques simply because they permit using "classical" techniques. starting at an introductory point appropriate for college students, this two-volume monograph provides (1) the basics and (2) the functions of the trajectory description of simple quantum strategies. this primary quantity is focussed at the classical and quantum historical past 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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Additional info for A Trajectory Description of Quantum Processes. I. Fundamentals: A Bohmian Perspective
Thus, a periodic orbit of period n on the Poincaré map can only appear (or disappear, if it already existed) whenever Tr (Mn ) = 2. 48) This is called a bifurcation. , those with smaller periods and the simplest topology in general), since periodic orbits of higher periods usually originate from them. 3 From Regular to Chaotic Dynamics 19 is satisfied, where m is an integer such that the cosine is modulo π. , 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].
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 . 45) Within this context, this matrix is known as the stability matrix. If the trajectory x is periodic with period T, M(x0 , T ) is also known as the monodromy matrix. The eigenvalues (λi ) and eigenvectors of this matrix determine the local behavior of neighboring trajectories, since they describe the deformation of a neighborhood δx for a finite time t.
However, one is often interested in obtaining information for a particular range of values of such parameters, which implies a parametric analysis of evolution of the phase-space structure. The Kolmogorov–Arnold–Moser (KAM) theorem [23, 47] gives a detailed account on the destruction of individual tori in phase space under perturbations. 3 A suitable starting point consists in defining the main families of periodic orbits according to Weinstein’s theorem , which in the vicinity of an equilibrium point of the potential guarantees the existence of as many periodic orbits as system degrees of freedom.
A Trajectory Description of Quantum Processes. I. Fundamentals: A Bohmian Perspective by Ángel S. Sanz