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

Solution of the quantum initial value problem with transparent boundary conditions