Computation of Leaky Quantum Carpets : Implementing SBP-SAT Methods with the Sommerfeld Radiation Boundary Condition for the Schrödinger Equation

Sammanfattning

A numerical solver for the one-dimensional time-dependent Schrödinger equation using an SBP-SAT scheme has been constructed, imposing Dirichlet, Neumann, and absorbing boundary conditions. The focus is on the absorbing boundary condition, which is based on the Sommerfeld radiation condition. The solutions for all boundary conditions have been illustrated using quantum carpets when implementing different potentials. The solver works as expected for the Dirichlet and Neumann boundary conditions, as the total probability for a particle in a box, is conserved within the box. The precision is, in the Dirichlet case, further verified by a convergence study against an analytical solution. Regarding the Sommerfeld radiation boundary condition, the solver exhibits zero reflection for high-energy states, whereas reflections and, in some cases, numerical divergence occur for low-energy states.