Abstract
The Finite-Difference Time-Domain (FDTD) method is a well-known technique for the analysis of quantum devices. It solves a discretized Schrodinger 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 Schr?dinger 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.
Highlights
The 1D time-dependent linear Schrödinger equation, which is the basis of quantum mechanics [1,2], can be expressed as follows [3,4]: x, t t i 2 x, x2 t V x,t x, t (1)J·sec is Planck’s constant, V is the potential (J), x,t is a complex number, and i 1 The product of x,t with its complex conjugate, x,t x,t indicates the probability of a particle being at spatial location x at time t.It can be seen that the classic explicit two-level in time finite difference scheme, i.e., n 1 k t n 2m x2 x2 n (2)
We present a generalized Finite-Difference Time-Domain (FDTD) method with absorbing boundary condition for solving the one-dimensional (1D) time-dependent Schrödinger equation and obtain a more relaxed condition for stability
We develop a second-order absorbing boundary condition (ABC) which is obtained from analyzing the group velocity of the wavepacket at the boundaries [15]
Summary
The 1D time-dependent linear Schrödinger equation, which is the basis of quantum mechanics [1,2], can be expressed as follows [3,4]: x,. Sullivan [3] and Visscher [4] applied the finite-difference time-domain (FDTD) method, which is often employed in simulations of electromagnetic fields, to solve the above Schrödinger equation. The second author analyzed the stability of the FDTD-Q scheme using the discrete energy method and obtained a condition for determining the time step, t , so that the scheme is stable as follows [13]: m t x2. Schrödinger equation, so that a more relaxed condition for stability may be obtained
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