A Generalized FDTD Method with Absorbing Boundary Condition for Solving a Time-Dependent Linear Schrodinger Equation

Abstract

The Finite-Difference Time-Domain (FDTD) method is a well-known technique for the analysis of quantum devices. It solves a discretized Schrödinger equation in an iterative process. However, the method provides only a second-order accurate numerical solution and requires that the spatial grid size and time step should satisfy a very restricted condition in order to prevent the numerical solution from diverging. In this article, we present a generalized FDTD method with absorbing boundary condition for solving the one-dimensional (1D) time-dependent Schrödinger equation and obtain a more relaxed condition for stability. The generalized FDTD scheme is tested by simulating a particle moving in free space and then hitting an energy potential. Numerical results coincide with those obtained based on the theoretical analysis.

Analyzing Quantum Time‐Dependent Singular Potential Systems in One Dimension

“…Several methods for numerical treatment of time-dependent Schrödinger equations are known. If we enumerate some of them, they are the finite difference time domain (FDTD) method [26][27][28][29][30][31], the discretization method that takes advantage of the asymptotic behavior correspondence (ABC) [32,33], and the discrete local discontinuous Galerkin method [34]. In particular, the FDTD method has been widely applied to obtain numerical solutions of mechanical problems of dynamical systems including Maxwell-Schrödinger equations for electromagnetic fields [30,31].…”

Simulation of Time-Dependent Schrödinger Equation in the Position and Momentum Domains

“…The position-domain approximation of the kinetic energy operator is derived using Lagrange polynomials and consistent with results from [4]. In the position domain the approximation to the kinetic energy operator is fourth-order accurate.…”

“…Given an initial state 0 y at 0 t = , the four-step predictor-corrector requires the creation of n j n y y j t − = − ∆ for 1, , 4 j =  in order to compute the first predictor-corrector time-step. A simple backwards Euler method, outlined in [4], [7]- [12], is used to generate the wave function at these early time-steps. Each of these early states for 1, , 4 j =  is re-normalized after their creation to ensure minimum initial error.…”

Nonuniform and Higher-order FDTD Methods for the Schrödinger Equation

“…Therefore, it is essential to develop efficient and accurate tools capable of solving the time-dependent Schrödinger equation. The Finite-Difference Time-Domain (FDTD) method is a well-known technique that discretizes the wave function on a space-time structured grid [1][2][3][4][5][6][7][8][9]. Depending on the discretization of the temporal derivative, the resulting scheme is either implicit or explicit.…”

“…In this work, the focus is on the nonuniform and higher-order spatial discretization, which is applied for one possible -but widely used -temporal discretization scheme. Still, it is applicable to other temporal discretizations [1,7,12,13]. The methods are validated using analytical solutions to a dynamical problem and their accuracy and efficiency are demonstrated.…”