  Physicists have used billiards to understand and explore both classical and
quantum chaos. Recently, in 2001, a group at the University of Texas introduced
an experimental set up for modeling the wedge billiard geometry called optical
billiard in two dimensions. It is worth mentioning that this experiment is more
closely related with classical rather than quantum chaos. The motivation for
the present work was born from the idea of laying the foundations of a quantum
treatment for optical billiards, named "The Escape Problem", by presenting the
concept of a Transparent Boundary Condition. We consider a gas of particles
initially confined to a one dimensional box of length L, that are permitted to
escape. We find the solution of a Quantum Initial Value Problem using a
numerical method developed and entirely checked with an exact, analytic method.
The numerical method introduces a novel way to solve a Diffusion Type Equation
by implementing Discrete Transparent Boundary Conditions recently developed by
mathematicians.
