7.1.2 Absorbing Boundary Conditions

The scattering of an incoming particle by a potential can be modeled by using transparent boundary conditions [Arn01] in the Schrödinger equation. In this picture a particle represented by an infinite coherent wave packet fixed by a wave vector $k$ is scattered from the potential $V$. The incoming waves are assumed to be plane waves of the form $e^{\imath kx}e^{-\imath \frac{E}{\hbar} t}$. The assumption of an infinite wave trail is an idealization. A more realistic model consists in modeling the particle as an incoming coherent wave packet with finite coherence length. The main reason for the above assumption seems to be that it allows for an analytic solution in simple cases and leads to easily implementable boundary conditions. For the Schrödinger equation absorbing boundary conditions can be discretized as ``transparent BCs'', for the Riccati equation we get Dirichlet type BCs.