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

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.

Figure 1

Propagation of a sum of Gaussian wave packets in free space. The curves show the difference $V$ [see (59)] between the numerical and exact solutions for a number of values of parameters $\tau$ and $h$. The apparent maximum is caused by the discretization errors. The top figure corresponds to the FD scheme with the fully discrete TBC, while the bottom one corresponds to the same FD scheme with the discretized Baskakov-Popov TBC

Figure 2

Evolution of wave function in a semispherical potential well with $r_0=70$. Distributions of ${log}_{10}{\left|\psi \right|}^2$ in $(x,z)$ plane at $y=0$ are plotted at four different times: (a) 320 arb. units, (b) 640 arb. units, (c) 1280 arb. units, (d) 1920 arb. units.