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Journal of Physics A Mathematical and Theoretical

Exact analytical description of quantum transients in one-dimensional scattering

Sergio Cordero, Gastón García‐Calderón

Published Jun 27, 2011• 4 citations
Publisher

Abstract

An exact analytical solution of the transient behaviour in one-dimensional time-dependent scattering for initial states with a sharp wavefront is derived. This work extends to the full line previous work on the subject for the transmitted and internal interaction regions. We exemplify our findings by considering single and quadruple barrier systems using cutoff plane wave initial states.

DOI
10.1088/1751-8113/44/30/305302

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An exact analytical solution of the transient behaviour in one-dimensional time-dependent scattering for initial states with a sharp wavefront is derived. This work extends to the full line previous work on the subject for the transmitted and internal interaction regions. We exemplify our findings by considering single and quadruple barrier systems using cutoff plane wave initial states.
An exact analytic solution of the time-dependent Schrödinger equation with cutoff wave initial conditions, that goes into the stationary solution at asymptotically long times, is used to investigate the transient behavior in the time evolution of the probability density along the internal and transmitted regions of a potential barrier. The approach involves the complex poles and residues (resonant states) of the outgoing Green's function of the problem. The solution exhibits absence of propagation along the classically forbidden internal region of the potential, and along the transmitted region it splits into two propagating structures. One of them is a transient effect describing the earliest response of the system to the initial condition whereas the other structure evolves towards the stationary solution.
The time dependence, from the transient to the stationary regime, of a quantum state initially prepared as a cutoff plane wave impinging on a one-dimensional square barrier is analyzed in detail using a decomposition based on a discrete set of critical points in the complex momentum plane. The motion and shape of the main wave front, the forerunners, and asymptotic (large time) features of the wave are discussed by means of the exact solution and in terms of velocities and characteristic times associated with the partial contributions of the critical points. \textcopyright{} 1996 The American Physical Society.
Exact analytical solutions of the time-dependent Schrödinger equation with the initial condition of an incident cutoff wave are used to investigate the traversal time for tunneling. The probability density starts from a vanishing value along the tunneling and transmitted regions of the potential. At the barrier width it exhibits, at early times, a distribution of traversal times that typically has a peak τpτ_pτp​ and a width ΔτΔτΔτ. Numerical results for other tunneling times, as the phase-delay time, fall within ΔτΔτΔτ. The Büttiker traversal time is the closest to τpτ_pτp​. Our results resemble calculations based on Feynman paths if its noisy behaviour is ignored.
In this paper we reexamine, and analyze solutions to, the recently derived time-independent wave packet-Schrödinger (TIWS) and time-independent wave packet-Lippmann–Schwinger (TIWLS) equations. These equations are so named because they are inhomogeneous, with the inhomogeneity being the initial L2 wave packet from an underlying time-dependent treatment of the dynamics. We explicitly show that a particular solution of the homogeneous Schrödinger equation can be constructed out of two particular solutions of the inhomogeneous TIW equation satisfying causal and anticausal boundary conditions. The structure of this solution of the homogeneous equation is shown to depend sensitively on the nature of the initial wave packet inhomogeneity, but, as we demonstrate, correct scattering information can be obtained even when the initial wave packet is nonzero only in the target region. It thus becomes possible to carry out quantum scattering calculations in which one need not propagate the wave packet from the noninteracting to the interacting region. The method is illustrated by calculations for two 1-D scattering problems, namely the transmission of an electron through a single barrier and through a double barrier. The latter is especially challenging because of the occurrence of long-lived resonances with the electron trapped inside the double barrier. In addition, we show that the method can still make use of absorbing potentials to decrease grid size, and we show how the formalism can be used to treat inelastic and reactive scattering, and radial scattering variables, thereby constituting a general approach to time-independent wave packet quantum scattering.
Exact analytical solutions of the time-dependent Schrodinger equation with the initial condition of an incident cutoff wave are used to investigate the traversal time for tunneling. The probability density starts from a vanishing value along the tunneling and transmitted regions of the potential. At the barrier width it exhibits, at early times, a distribution of traversal times that typically has a peakp and a width ��. Numerical results for other tunneling times, as the phase-delay time, fall within ��. The Buttiker traversal time is the closest top. Our results resemble calculations based on Feynman paths if its noisy behaviour is ignored.
In two previous studies, the time-dependent scattering of a wave packet from a Gaussian barrier was investigated computationally in the complex z-plane. One of these involved the 'direct' propagation of the wave packet in the complex space, and the other used numerical analytic continuation techniques to generate the dynamics in the complex plane from the wave function computed on the real-axis. In the current study, the dynamics of exact quantum trajectories are analyzed for the same barrier scattering problem. Thousands of quantum trajectories were launched from positions near the center of the initial wave packet. These trajectories were computed by integrating equations-of-motion involving the quantum momentum function, which was obtained from the time-dependent wave function and its derivative. In order to analyze the dynamics, many trajectories were plotted on space-time diagrams. Particular emphasis was placed upon trajectories undergoing reflection in the barrier region. Some groups of strongly correlated trajectories form long-lived highly organized patterns, including helical wrappings around a series of stagnation filaments. These curves alternate with quasi-nodes where the amplitude of the wave function reaches low values. In addition, other trajectories for short times follow hyperbolic paths as they propagate near vorticity tubes surrounding these quasi-nodes.