By Ángel S. Sanz
Trajectory-based formalisms are an intuitively beautiful manner of describing quantum procedures simply because they permit using "classical" recommendations. starting at an introductory point appropriate for college kids, this two-volume monograph offers (1) the basics and (2) the functions of the trajectory description of easy quantum approaches. 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 resources for A Trajectory Description of Quantum Processes. I. Fundamentals: A Bohmian Perspective
Q = q(q0 , p0 , t), p = p(q0 , p0 , t). 61) Given a Hamiltonian, the corresponding trajectories can never cross in phase space due to the uniqueness of the classical equations of motion, as seen in Sect. 1. This classical non-crossing rule is comparable to the one arising in Bohmian mechanics (see Sect. 1), although the latter is more restrictive in the sense that it directly applies on the configuration space defined by q. The Liouville equation can be seen as the infinitesimal generator of the group of time translations induced in the phase space of the probability densities.
As in Sect. 106) by means of the Fourier method. 106), the latter becomes 4 This example has been chosen here for its simplicity and without loss of generality. Thus, the results could also be readily extended to nonhomogeneous, three-dimensional media. 109) ξ˜ (α, t) = ξ˜ (α, 0)eiαvt . 111) ξ˜ (α, 0)e−iα(x−vt) dα = ξ(x − vt). 107) is finally reached, 1 ξ(x, t) = √ 2π Consider that the perturbation or pulse has also initially a Gaussian shape with width σ , ξ(x, 0) = ξ0 e−x 2 /2σ 2 . , the Gaussian propagates with positive velocity along the string.
64) its coefficient, and f 0 (q0 , p0 ) any classical initial distribution function. The formal comparison between the solutions of the quantum and classical Liouville equations may lead to think that the superposition principle also holds classically. Some superpositions of eigendistributions for different eigenvalues of the Liouville operator describing the classical harmonic oscillator can be found in . In those cases where one is only interested in the distribution of some of the degrees of freedom, reduced distribution functions can be easily obtained by integrating over all those phase coordinates which are not of interest.
A Trajectory Description of Quantum Processes. I. Fundamentals: A Bohmian Perspective by Ángel S. Sanz