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Journal of Chemical Theory and Computation

Computational Investigation of Wave Packet Scattering in the Complex Plane: Propagation on a Grid

Róbert E. Wyatt, Brad A. Rowland

Published Jan 29, 2009• 12 citations
Publisher

Abstract

The time-dependent scattering of a wave packet from a Gaussian barrier is investigated computationally in the complex z-plane. The initial wave packet and the potential energy are obtained through analytic continuation from functions specified on the real-axis. The wave packet is then propagated on the two-dimensional grid. For a low initial wave packet energy, the time evolution is followed by plotting the following functions: |ψ(z,t)|, real(ψ(z,t)), and the quantum momentum function (QMF), p(z,t). In the reflected packet, an important role is played by ripples (quasi-nodes) forming above the real axis. As these quasi-nodes move down across the real axis, they are 'detected' as 'interference oscillations' in the density. In contrast, the component of the packet below the real axis makes a significant contribution to the transmitted packet. Vector maps of the QMF show hyperbolic flow around quasi-nodes and counterclockwise circular flow around transient stagnation points, where the QMF vanishes. However, when the Pólya vector field (defined by P(z,t) = p*(z,t)) is plotted, circular counterclockwise flow is obtained near the quasi-nodes. The real and imaginary parts of the quantum action function S(z,t) are plotted and the vorticity, defined by the curl of the Pólya field, is used to pinpoint regions of nonanalyticity in the QMF.

DOI
10.1021/ct800248w

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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.

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The quantum trajectory method was recently developed to solve the hydrodynamic equations of motion in the Lagrangian, moving-with-the-fluid, picture. In this approach, trajectories are integrated for fluid elements (“particles”) moving under the influence of the combined force from the potential surface and the quantum potential. To accurately compute the quantum potential and the quantum force, it is necessary to obtain the derivatives of a function given only the values on the unstructured mesh defined by the particle locations. However, in some regions of space–time, the particle mesh shows compression and inflation associated with regions of large and small density, respectively. Inflation is especially severe near nodes in the wave function. In order to circumvent problems associated with highly nonuniform grids defined by the particle locations, adaptation of moving grids is introduced in this study. By changing the representation of the wave function in these local regions (which can be identified by diagnostic tools), propagation is possible to much longer times. These grid adaptation techniques are applied to the reflected portion of a wave packet scattering from an Eckart potential.

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Time-dependent wave packet resonant scattering for the double square barrier has been studied in terms of Bohm quantum trajectories. The high transmission probability for the wave packet with a resonant energy can be explained by the behavior of the quantum trajectories under the influence of the relatively slow formation of a node within the first barrier. This node splits the trajectories into reflected and transmitted components. During this stage, many particle trajectories pass through the double-barrier region and contribute to the transmitted part of the wave packet. Due to the transient nature of the nodes, trajectories in the reflected wave packet bunch together between the nodes for a finite period of time so that temporary structure (localization of particles and accompanying increase in the probability density) develops on small length scales. These calculations also show that the particles gain high momentum near the nodal points, and they reach a uniform momentum distribution after transmitting the barrier region. We have found that the presence of a node between the two barriers influences the different lifetimes of the quasi-bound states. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 206–213, 2001

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Note on the Transmission and Reflection of Wave Packets by Potential Barriers

L. A. MacColl

In previous studies, by the methods of wave mechanics, of one-dimensional motion of particles in cases in which there are intervals in which the value of the potential energy function $V(x)$ exceeds the value of the total energy $E$, attention has been confined to wave functions of the form $f(x, E)\mathrm{exp}(\ensuremath{-}\frac{2\ensuremath{\pi}\mathrm{iEt}}{h})$. In the present note wave packets are considered, instead of these trains of waves.The function $V(x)$ is taken as follows: ${V(x),=0 \mathrm{for} x<0 \mathrm{and}\mathrm{for} x>a,}{={V}_{0}>0 \mathrm{for} 0<x<a.}$ A wave function is set up which initially represents a wave packet moving toward the point $x=0$ from the left. The separation of the incident packet into a reflected packet and a transmitted packet is studied. It is found that the transmitted packet appears at the point $x=a$ at about the time at which the incident packet reaches the point $x=0$, so that there is no appreciable delay in the transmission of the packet through the barrier.

Physical Review•May 15, 1932•349 citations
The time-dependent scattering of a wave packet from a Gaussian barrier is investigated computationally in the complex z-plane. The initial wave packet and the potential energy are obtained through analytic continuation from functions specified on the real-axis. The wave packet is then propagated on the two-dimensional grid. For a low initial wave packet energy, the time evolution is followed by plotting the following functions: |ψ(z,t)|, real(ψ(z,t)), and the quantum momentum function (QMF), p(z,t). In the reflected packet, an important role is played by ripples (quasi-nodes) forming above the real axis. As these quasi-nodes move down across the real axis, they are 'detected' as 'interference oscillations' in the density. In contrast, the component of the packet below the real axis makes a significant contribution to the transmitted packet. Vector maps of the QMF show hyperbolic flow around quasi-nodes and counterclockwise circular flow around transient stagnation points, where the QMF vanishes. However, when the Pólya vector field (defined by P(z,t) = p*(z,t)) is plotted, circular counterclockwise flow is obtained near the quasi-nodes. The real and imaginary parts of the quantum action function S(z,t) are plotted and the vorticity, defined by the curl of the Pólya field, is used to pinpoint regions of nonanalyticity in the QMF.
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.
Abstract In this study, we utilize Numerical Analytic Continuation (NAC) techniques to compute the complex‐extension of wave packets computed on the real‐axis. Various methods for doing this are reviewed, and the accuracy and stability of one particular method (the Cauchy Method) are examined using a test function. We focus on generating numerically accurate maps of the complex‐extended time‐dependent wave packet for the Gaussian and Eckart barrier scattering problems. We also examine quantum momentum function vector fields and the associated Pólya vector fields for this problem. In a companion study, we present a novel method for solving the complex‐extended Schrödinger equation for points in the complex plane for wave packet scattering from a Gaussian barrier. Several fascinating features of the solution (such as quantized vortices) were presented, but we could only check the veracity of those results on the real‐axis. The method utilized in this study provides an additional check of these results. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2011
The quantum trajectory method was recently developed to solve the hydrodynamic equations of motion in the Lagrangian, moving-with-the-fluid, picture. In this approach, trajectories are integrated for fluid elements (“particles”) moving under the influence of the combined force from the potential surface and the quantum potential. To accurately compute the quantum potential and the quantum force, it is necessary to obtain the derivatives of a function given only the values on the unstructured mesh defined by the particle locations. However, in some regions of space–time, the particle mesh shows compression and inflation associated with regions of large and small density, respectively. Inflation is especially severe near nodes in the wave function. In order to circumvent problems associated with highly nonuniform grids defined by the particle locations, adaptation of moving grids is introduced in this study. By changing the representation of the wave function in these local regions (which can be identified by diagnostic tools), propagation is possible to much longer times. These grid adaptation techniques are applied to the reflected portion of a wave packet scattering from an Eckart potential.
Time-dependent wave packet resonant scattering for the double square barrier has been studied in terms of Bohm quantum trajectories. The high transmission probability for the wave packet with a resonant energy can be explained by the behavior of the quantum trajectories under the influence of the relatively slow formation of a node within the first barrier. This node splits the trajectories into reflected and transmitted components. During this stage, many particle trajectories pass through the double-barrier region and contribute to the transmitted part of the wave packet. Due to the transient nature of the nodes, trajectories in the reflected wave packet bunch together between the nodes for a finite period of time so that temporary structure (localization of particles and accompanying increase in the probability density) develops on small length scales. These calculations also show that the particles gain high momentum near the nodal points, and they reach a uniform momentum distribution after transmitting the barrier region. We have found that the presence of a node between the two barriers influences the different lifetimes of the quasi-bound states. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 206–213, 2001
The wave-packet simulation (WPS) method for calculating the time a tunneling particle spends inside a one-dimensional potential barrier is reexamined using the Feynman path-integral technique. Following earlier work by Sokolovski and Baskin [Phys. Rev. A 36, 4604 (1987)], the tunneling (or traversal) time tpackT{\mathit{t}}_{\mathrm{pack}}^{\mathit{T}}tpackT​ is defined as a matrix element of a classical nonlocal functional between two states that represent the initial and transmitted wave packets. These states do not lie on the same orbit in Hilbert space; as a result, tpackT{\mathit{t}}_{\mathrm{pack}}^{\mathit{T}}tpackT​ is complex-valued. It is shown that RetpackT{\mathit{t}}_{\mathrm{pack}}^{\mathit{T}}tpackT​ reduces to the standard WPS result, tphaseT{\mathit{t}}_{\mathrm{phase}}^{\mathit{T}}tphaseT​, for conditions similar to those employed in the conventional WPS analysis. Similarly, ImtpackT{\mathit{t}}_{\mathrm{pack}}^{\mathit{T}}tpackT​ is shown to contain information about the energy dependence of the transmission probability. Under semiclassical conditions, ImtpackT{\mathit{t}}_{\mathrm{pack}}^{\mathit{T}}tpackT​ reduces to the well-known Wentzel-Kramers-Brillouin expression for the tunneling time. It is shown there are different definitions for the traversal time of a classical moving object, whose size is comparable to the width of the region of interest. In the quantum case, these different definitions correspond to different ways of analyzing the WPS experiment. The path-integral approach demonstrates that the tunneling-time problem is one of understanding the physical significance of complex-valued off-orbit matrix elements of an operator or functional. The physical content of complex-valued tunneling times is discussed. It is emphasized that the use of complex tunneling times includes real-time approaches as a special case. Nevertheless, there is a limitation in the description of tunneling experiments using tunneling times, whether real or complex. The path-integral approach does not supply a universal traversal time, analogous to a classical time, that can be used in quantum situations. It is demonstrated that the often expressed hope of finding a well-defined and universal real tunneling time is erroneous.
In previous studies, by the methods of wave mechanics, of one-dimensional motion of particles in cases in which there are intervals in which the value of the potential energy function V(x)V(x)V(x) exceeds the value of the total energy EEE, attention has been confined to wave functions of the form $f(x, E)\mathrm{exp}(\ensuremath{-}\frac{2\ensuremath{\pi}\mathrm{iEt}}{h})$. In the present note wave packets are considered, instead of these trains of waves.The function V(x)V(x)V(x) is taken as follows: V(x),=0forx<0andforx>a,=V0>0for0<x<a.{V(x),=0 \mathrm{for} x<0 \mathrm{and}\mathrm{for} x>a,}{={V}_{0}>0 \mathrm{for} 0<x<a.}V(x),=0forx<0andforx>a,=V0​>0for0<x<a. A wave function is set up which initially represents a wave packet moving toward the point x=0x=0x=0 from the left. The separation of the incident packet into a reflected packet and a transmitted packet is studied. It is found that the transmitted packet appears at the point x=ax=ax=a at about the time at which the incident packet reaches the point x=0x=0x=0, so that there is no appreciable delay in the transmission of the packet through the barrier.