High-Accurate Transparent Boundary Conditions for Time-Dependent Quantum Wave Packet Method

Complex absorbing potential is usually required in a time-dependent wave packet method to accomplish the calculation in a truncated region.Usually it works effectively but becomes inefficient when the wave function involves translational energy of broad range,particularly involving ultra-low energy.In this work,a new transparent boundary condition(TBC)is proposed for the time-dependent wave packet method.It in principle is of spectral accuracy when typical discrete variable representations are applied.The prominent merit of the new TBC is that its accuracy is insensitive to the translational energy distribution of the wave function,in contrast with the complex absorbing potential.Application of the new TBC is given to one-dimensional particle wave packet scatterings from a barrier with a potential well,which supports resonances states.

A Review of Transparent and Artificial Boundary Conditions Techniques for Linear and Nonlinear Schrodinger Equations

In this review article we discuss different techniques to solve numerically the time-dependent Schrodinger equation on unbounded domains.We present in detail the most recent approaches and describe briefly alternative ideas pointing out the relations between these works.We conclude with several numerical examples from different application areas to compare the presented techniques.We mainly focus on the one-dimensional problem but also touch upon the situation in two space dimensions and the cubic nonlinear case.

Study and Design of Spiral Bent Waveguide Configuration

A new version of the scalar transverse electric(TE) wave equation in the bent waveguide is introduced. Then, TE polarized field in curved single-mode waveguides is analyzed by using the finite-difference beam propagation method(FD-BPM). The bending loss in bent waveguides is gotten for the optical fields obtained from BPM and comparisons are made among losses of the waveguides with various curvature radiuses, refractive index differences and cross sections. Based on the results, the design of spiral bent waveguide configuration is proposed as follows: refractive index difference being of 0.007, both width and thickness of waveguides being of 6 μm, the curvature radius in the spiral centre being of 4 mm, and the bending loss coefficient of the designed spiral bent waveguide being of 0.302 3 dB/cm.

A Node-Based Smoothed Finite Element Method with Linear Gradient Fields for Elastic Obstacle Scattering Problems

In this paper,a node-based smoothed finite element method(NS-FEM)with linear gradient fields(NS-FEM-L)is presented to solve elastic wave scattering by a rigid obstacle.By using Helmholtz decomposition,the problem is transformed into a boundary value problem with coupled boundary conditions.In numerical analysis,the perfectly matched layer(PML)and transparent boundary condition(TBC)are introduced to truncate the unbounded domain.Then,a linear gradient is constructed in a node-based smoothing domain(N-SD)by using a complete order of polynomial.The unknown coefficients of the smoothed linear gradient function can be solved by three linearly independent weight functions.Further,based on the weakened weak formulation,a system of linear equation with the smoothed gradient is established for NS-FEM-L with PML or TBC.Some numerical examples also demonstrate that the presented method possesses more stability and high accuracy.It turns out that the modified gradient makes the NS-FEM-L-PML and NS-FEM-L-TBC possess an ideal stiffness matrix,which effectively overcomes the instability of original NS-FEM.Moreover,the convergence rates of L 2 and H1 semi-norm errors for the two NS-FEM-L models are also higher.

Green Function Method for Quantum Transport Based on the Generalized Fourier Transform

The rigorous relations between the propagators of transient Schr¨odinger equations and stationary Green functions are established.Based on the generalized Fourier transform,non-singular transparent boundary condition for transient problem is proposed in a representation of Green functions.The unified framework of Green function method is presented for converting an open boundary problem into a bounded boundary problem.Numerical scheme for time-dependent Schr¨odinger equation with non-singular transparent boundary condition is designed to simulate the propagations of a free Gaussian wave packet and the resonant tunnelling through double barriers.Numerical results validate the effectiveness of non-singular transparent boundary condition.