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Physical review. B, Condensed matter

Non-Markovian open-system boundary conditions for the time-dependent Schrödinger equation

James R. Hellums, William R. Frensley

Published Jan 15, 1994• 64 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.49.2904

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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.
We study the problem of the boundary conditions in the numerical simulation of closed and open quantum systems, described by a Schrödinger equation. On one hand, we show that a closed quantum system is defined by local boundary conditions. On the other hand, we argue that, because of the uncertainty principle, no local boundary condition can be defined for open quantum systems. For this reason plane waves or wave packet trains cannot be simulated on a finite numerical lattice with the usual procedures. We suggest a method that avoids these difficulties by using only a small numerical lattice and maintains the correspondence with the physical picture, in which the incident and scattered waves may be infinitely extended.
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
This project looks into the time evolution of a wave function within a two-dimensional quantum well. We start by solving the time-dependent Schrödinger equation for stationary states in a quantum well. Next, we express any wave function as a linear combination of stationary states, allowing us to understand their time evolution. Two methods are presented: one relies on decomposing the wave function into a basis of stationary states and the other on discretisation of the time-dependent Schrödinger equation, incorporating three-point formulas for derivatives. These approaches necessitate confronting intricate boundary conditions and require maintaining energy conservation for numerical accuracy. We further demonstrate the methods using a wave packet, revealing fundamental phenomena in quantum physics. Our results demonstrate the utility of these methods in understanding quantum systems, despite the challenges in determining stationary states for a given potential. This study enhances our comprehension of the dynamics of quantum states in constrained systems, essential for fields like quantum computing and nanotechnology.
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Analytical solutions to the time-dependent Shrödinger equation in one dimension are developed for time-independent potentials, one consisting of an infinite wall and a repulsive delta function. An exact solution is obtained by means of a convolution of time-independent solutions spanning the given Hilbert space with appropriately chosen spectral functions. Square-integrability and the boundary conditions are satisfied. The probability for the particle to be found inside the potential well is calculated and shown to exhibit non-exponential decay decreasing at large times as t−3t^{-3}t−3. The result is generalized for all square-integrable solutions to this problem.