Solution of the Quantum Initial Value Problem with Transparent Boundary Conditions

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.