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
is scattered from the potential
.
The incoming waves are assumed to be
plane waves of the form
.
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.