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Annalen der Physik

Time‐Dependent Free‐Particle Salpeter Equation: Numerical and Asymptotic Analysis in the Light of the Fundamental Solution

Amalia Torre, Ambra Lattanzi, Decio Levi B

Published Dec 19, 2016• 9 citations
Publisher

Abstract

Abstract A detailed study of the spinless (1+1)D free‐particle Salpeter equation is presented. It involves several aspects of the topic: from the analysis of the behavior of solutions of the equation, both numerically evaluated and asymptotically approximated for definite initial conditions, to the comparison with the behavior of the corresponding solutions of the Schrödinger equation in order to both highlight the differences and to possibly understand how the latter “flow” in the former. Interesting analogies with other fields of physics, in particular with optics, are suggested.

DOI
10.1002/andp.201600231

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Abstract A detailed study of the spinless (1+1)D free‐particle Salpeter equation is presented. It involves several aspects of the topic: from the analysis of the behavior of solutions of the equation, both numerically evaluated and asymptotically approximated for definite initial conditions, to the comparison with the behavior of the corresponding solutions of the Schrödinger equation in order to both highlight the differences and to possibly understand how the latter “flow” in the former. Interesting analogies with other fields of physics, in particular with optics, are suggested.
Abstract We investigate potential scattering and tunneling dynamics of a particle wavepacket evolving according to the relativistic Schrödinger equation (also known as the Salpeter equation). The tunneling properties of the Salpeter equation differ from those of the standard relativistic wave equations (such as the Klein–Gordon or Dirac equations). In particular, the tunneling solutions must be found by working in momentum space, given that the equation in configuration space contains a pseudo-differential operator. The resulting integral equations are derived and solved numerically for wavepackets scattering on model potential barriers. The solutions are characterized by the absence of Klein tunneling and an effect of the potential on the fraction of the transmitted wavepacket that propagates outside the light cone, a feature that has in the past been well-studied only for free propagation.
Quantum mechanics describes phenomena that are key to understanding processes that cannot be explained by classical physics. This paper analyzes the behavior of particles in the presence of a potential barrier through numerical solutions to both time-dependent and time-independent Schrödinger equations. These results enabled a detailed understanding of the energy structure and temporal dynamics of wave functions, including the phenomenon of quantum tunneling. The obtained data have a wide range of applications in technology: from semiconductor devices to quantum computing.
A system of two equations is found that has solutions which coincide with the solutions of the Klein‐Gordon equation in the rest frame. This system includes the Schrödinger equation for a free neutral spinless particle. Using the Schrödinger equation as an additional condition for solving the Klein‐Gordon equation in the rest frame leads to two Helmholtz equations. Helmholtz equations can be solved by specifying a particle model and boundary conditions. One of the Helmholtz equations leads to discreteness of the rest masses of relativistic particles.
Abstract The free propagation in time of a normalisable wave packet is the oldest problem of continuum quantum mechanics. Its motion from microscopic to macroscopic distance is the way in which most quantum systems are detected experimentally. Although much studied and analysed since 1927 and presented in many textbooks, here the problem is re‐appraised from the standpoint of semi‐classical mechanics. Particular aspects are the emergence of deterministic trajectories of particles emanating from a region of atomic dimension and the interpretation of the wave function as describing a single particle or an ensemble of identical particles. Of possible wave packets, that of Gaussian form is most studied due to the simple exact form of the time‐dependent solution in real and in momentum space. Furthermore, this form is important in laser optics. Here the equivalence of the time‐dependent Schrödinger equation to the paraxial equation for the propagation of light is demonstrated explicitly. This parallel helps to understand the relevance of trajectory concepts and the conditions necessary for the perception of quantum motion as classical. Key Points The paper addresses a problem of fundamental quantum mechanics, important in both physics and chemistry. Within physics, the subject is relevant to the disciplines of both quantum mechanics and classical optics. The matter presented is topical also for researchers in the history and philosophy of physics.
Chapter 7 is devoted to the free particle and the derivation of the time-dependent Schrödinger equation. The free particle might be thought to be the simplest quantum problem of all, but a detailed treatment of it, including both Born and Dirac normalization, serves to highlight some surprising features. Powerful tools are introduced that are of wider application, notably the Green function method and the Feynman propagator. This material paves the way for the study of the time-dependent Schrödinger equation, the introduction of the current density of particles and the distinction between pure states and mixtures; the equation is applied specifically to the study of coherent states. The chapter ends with an introduction to the propagator in the general case.
Optical vortices became a hot topic since almost two decades ago, when it was recognized that Laguerre-Gaussian laser modes carry orbital angular momentum [Allen et al. Phys Rev A 45, 8185 (1992)] related with a screw phase dislocation, and different from the spin angular momentum associated to circular polarization. In 1995, this dynamical quantity was transferred to matter in an optical micromanipulation system for the first time [H. He, et al., Phys. Rev. Lett. 75, 826 (1995)], and since then, a number of studies on angular momentum of light have unveiled different interesting aspects on the subject. However, there are still open questions, which have arisen together with the generation of novel light beams, such as vector vortices, for instance. In contrast with scalar vortices, with usual polarization states (linear, circular, elliptical), the orientation and magnitude of the electric field of vector vortices (solutions of the vector wave equation) is a function of space and time. In this work, we present an experimental study of the local angular momentum density of a Bessel vector vortex of first order by means of an optical trap. For this purpose, we used different probe particles in order to sense the local contribution to the optical angular momentum in each region of the beam. But optical fields are not the only wave fields that may exhibit phase dislocations or singularities. There are close analogies between light and sound fields that can be exploited in order to get a better understanding of common phenomena and study new aspects in both branches of physics. Here we also present the first experimental demonstration and theoretical analysis of acoustical vortices in free field, with similar properties to those of the optical vortices, including the angular momentum that can be transferred to matter. The corresponding analogies and differences with the optical case turn out to be very enlightening for the understanding of the phenomenon of angular momentum in wave fields.