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arXiv:2105.10784 · Physical review. E

Exact transparent boundary condition for the multidimensional Schrödinger equation in hyperrectangular computational domain

Ruslan M. Feshchenko, A. V. Popov

Published Aug 17, 2021• 7 citations

Abstract

In this paper an exact transparent boundary condition for the multidimensional Schrödinger equation in a hyperrectangular computational domain is proposed. It is derived as a generalization of exact transparent boundary conditions for two-dimensional (2D) and 3D equations reported before. An exact fully discrete (i.e., derived directly from the finite-difference scheme used) 1D transparent boundary condition is also proposed. Several numerical experiments using an improved unconditionally stable numerical implementation in the 3D space demonstrate propagation of Gaussian wave packets in free space and penetration of a particle through a 3D spherically asymmetrical barrier. The application of the multidimensional transparent boundary condition to the dynamics of the 2D system of two noninteracting particles is considered. The proposed boundary condition is simple, robust, and can be useful in the field of computational quantum mechanics, when an exact solution of the multidimensional Schrödinger equation (including multiparticle problems) is required.

DOI: 10.1103/physreve.104.025306

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In this paper an exact transparent boundary condition for the multidimensional Schrödinger equation in a hyperrectangular computational domain is proposed. It is derived as a generalization of exact transparent boundary conditions for two-dimensional (2D) and 3D equations reported before. An exact fully discrete (i.e., derived directly from the finite-difference scheme used) 1D transparent boundary condition is also proposed. Several numerical experiments using an improved unconditionally stable numerical implementation in the 3D space demonstrate propagation of Gaussian wave packets in free space and penetration of a particle through a 3D spherically asymmetrical barrier. The application of the multidimensional transparent boundary condition to the dynamics of the 2D system of two noninteracting particles is considered. The proposed boundary condition is simple, robust, and can be useful in the field of computational quantum mechanics, when an exact solution of the multidimensional Schrödinger equation (including multiparticle problems) is required.

We construct exact transparent boundary conditions (TBCs) for a time-continuous, spatially discrete Schrödinger equation that models a one-dimensional quantum lattice. Using a recently developed exact solution for the discrete system, we derive the Dirichlet-to-Neumann maps analytically via Laplace transforms. This yields a convolution-type boundary condition governed by Bessel functions. We rigorously demonstrate the consistency of this discrete formulation with its continuous counterpart in the continuum limit. Additionally, we present an efficient time-discretization scheme based on the trapezoidal rule for practical implementation. Numerical experiments using a Crank-Nicolson solver verify that our proposed TBCs eliminate spurious backscattering entirely and preserve reflectionless propagation of a Gaussian wave packet exiting the computational domain

The open-system boundary conditions for the one-dimensional Schr\"odinger equation are derived by dividing the unbounded domain into a finite system and two semi-infinite reservoirs. The resulting boundary conditions on the system are non-Markovian, as they contain a convolution over the history of the system. Thus, time-irreversibility arises in a pure-state problem. The propagator which appears in the boundary condition is derived for a simple discrete model. The correctness of the boundary conditions is verified and the usefulness of the discrete model is demonstrated by a numerical calculation of the time-evolution of a wave packet.

Many systems of interest at the forefront of technological development, for example, in quantum computation, consist of weakly interacting elements that obey quantum mechanics. A challenge for modern theoretical physics is to develop a systematic methodology for approaching these “open” quantum systems. In particular, applied mathematicians have formulated the method of transparent boundary conditions (TBCs) to model interacting quantum systems of infinite extent. Here, we consider a particular application of TBCs to the escape of a particle from a one-dimensional infinite well when one of the bounding barriers is extinguished. We analytically obtain the exact time-dependent wave function by a Green's function method and then show that a numerical solution based on the TBC method is in excellent agreement. This work was motivated by an experiment carried out at the University of Texas on escape from a gravitational wedge billiard. Physicists have used billiards to understand and explore both classical and quantum chaos. Although the original experiment was carried out in the classical regime, current work is probing lower temperatures, where a quantum mechanical formulation is required.

A method for the solution of the time-dependent Schrödinger equation in two dimensions is presented which is based on the hydrodynamic analogy to quantum mechanics. The continuum introduced in that analogy is approximated by a finite number of particles, whose trajectories are computed. The procedure is applied here to the dynamics of a Gaussian wave packet on a two-dimensional quadratic potential surface containing a saddle point. Comparison with the known analytical solution indicates that the method is capable of accurate results with substantially less computer time required than in methods previously employed. This particular check problem was chosen because of its close relationship to the wave-packet dynamics of chemical reaction rates.