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Chemical Physics Letters

Computational investigation of wave packet barrier scattering in the complex plane: Examination of the complex quantum potential utilizing numerical analytic continuation techniques

Brad A. Rowland, Róbert E. Wyatt

Published Jul 3, 2008• 11 citations
Publisher
DOI
10.1016/j.cplett.2008.06.085

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Computational investigation of wave packet scattering in the complex plane: Numerical analytic continuation techniques

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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

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Complex quantum trajectories for barrier scattering

Brad A. Rowland

We have directed much attention towards developing quantum trajectory methods which can accurately predict the transmission probabilities for a variety of quantum mechanical barrier scattering processes. One promising method involves solving the complex quantum Hamilton-Jacobi equation with the Derivative Propagation Method (DPM). We present this method, termed complex valued DPM (CVDPM(n)). CVDPM(n) has been successfully employed in the Lagrangian frame to accurately compute transmission probabilities on 'thick' one dimensional Eckart and Gaussian potential surfaces. CVDPM(n) is able to reproduce accurate results with a much lower order of approximation than is required by real valued quantum trajectory methods, from initial wave packet energies ranging from the tunneling case (E[subscript o]=0) to high energy cases (twice the barrier height). We successfully extended CVDPM(n) to two-dimensional problems (one translational degree of freedom representing an Eckart or Gaussian barrier coupled to a vibrational degree of freedom) in the Lagrangian framework with great success. CVDPM helps to explain why barrier scattering from "thick" barriers is a much more well posed problem than barrier scattering from "thin" barriers. Though results in these two cases are in very good agreement with grid methods, the search for an appropriate set of initial conditions (termed an 'isochrone) from which to launch the trajectories leads to a time-consuming search problem that is reminiscent of the rootsearching problem from semi-classical dynamics. In order to circumvent the isochrone problem, we present CVDPM(n) equations of motion which are derived and implemented in the arbitrary Lagrangian-Eulerian frame for a metastable potential as well as the Eckart and Gaussian surfaces. In this way, the isochrone problem can be circumvented but at the cost of introducing other computational difficulties. In order to understand why CVDPM may give better transmission probabilities than real valued counterparts, much attention we have been studying and applying numerical analytic continuation techniques to visualize complex-extended wave packets as well as the complex-extended quantum potential. Numerical analytic continuation techniques have also been used to analytically continue a discrete real-valued potential into the complex plane for CVDPM with very promising results.

•Dec 1, 2007•0 citations
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
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.
We have directed much attention towards developing quantum trajectory methods which can accurately predict the transmission probabilities for a variety of quantum mechanical barrier scattering processes. One promising method involves solving the complex quantum Hamilton-Jacobi equation with the Derivative Propagation Method (DPM). We present this method, termed complex valued DPM (CVDPM(n)). CVDPM(n) has been successfully employed in the Lagrangian frame to accurately compute transmission probabilities on 'thick' one dimensional Eckart and Gaussian potential surfaces. CVDPM(n) is able to reproduce accurate results with a much lower order of approximation than is required by real valued quantum trajectory methods, from initial wave packet energies ranging from the tunneling case (E[subscript o]=0) to high energy cases (twice the barrier height). We successfully extended CVDPM(n) to two-dimensional problems (one translational degree of freedom representing an Eckart or Gaussian barrier coupled to a vibrational degree of freedom) in the Lagrangian framework with great success. CVDPM helps to explain why barrier scattering from "thick" barriers is a much more well posed problem than barrier scattering from "thin" barriers. Though results in these two cases are in very good agreement with grid methods, the search for an appropriate set of initial conditions (termed an 'isochrone) from which to launch the trajectories leads to a time-consuming search problem that is reminiscent of the rootsearching problem from semi-classical dynamics. In order to circumvent the isochrone problem, we present CVDPM(n) equations of motion which are derived and implemented in the arbitrary Lagrangian-Eulerian frame for a metastable potential as well as the Eckart and Gaussian surfaces. In this way, the isochrone problem can be circumvented but at the cost of introducing other computational difficulties. In order to understand why CVDPM may give better transmission probabilities than real valued counterparts, much attention we have been studying and applying numerical analytic continuation techniques to visualize complex-extended wave packets as well as the complex-extended quantum potential. Numerical analytic continuation techniques have also been used to analytically continue a discrete real-valued potential into the complex plane for CVDPM with very promising results.