I. INTRODUCTION The seminal model for understanding both classical and quantum chaos is the billiard. Depending on different considerations, such as shape and topology, a billiard model can present either stable or chaotic behavior. The most celebrated versions of classical billiards in two dimensions, in which chaos has been observed, are the Sinai billiard $^{1}$, Bunimovich stadium $^{2}$, and Polygonal billiards$^{3}$. In all of these examples, the shape of the boundaries plays a crucial role. In recent years, physicists have detected chaotic behavior at the nano scale 4 making an important impact in technology and, once again, billiards have helped to model quantum devices 5 such as nanotubes, quantum dots, etc. which are governed by quantum theory. A novel model of billiard is the one introduced in 1986 by Lehtihet and Miller 6 known as the “Wedge Billiard .” The model consists of a symmetrically inclined wedge of angle 2 θ with respect to the direction of a constant gravitational field g in which the particle is confined. They found surprising properties when the parameter θ is changed. Later, in 2001, Valery Milner $^{7}$, working in Mark Raizen’s laboratory at the University of Texas, introduced an experiment referencing the Wedge Billiard model geometry and confirming the properties mentioned above. This first experiment of billiards was named “ Optical Billiards .” It is worth mentioning that this experiment is more closely related to classical rather than “quantum” chaos . While very low by normal standards, the temperature they employed is not low enough to easily exhibit any quantum effects. Consequently, a classical theoretical model was adequate to obtain a good point of comparison. A key element in the experiment is that, by removing a small segment of the boundary, an escape route is provided at the billiard vertex. As time progresses, the atoms, moving under the influence of gravity, will eventually exit the billiard. A measure of the influence of "chaotic" orbits is provided by the mean lifetime that the atoms remain in the billiard. This scenario is mimicked in this work by setting up the Escape Problem ( EP ). Currently experimentalists are starting to probe the quantum regime. The idea of the work presented here springs from laying the foundation of a quantum treatment for optical billiards. However, the quantum mechanical problem is much more difficult since the system is two-dimensional (equivalent to four dimensions in the phase space), and not integrable. Thus analytical methods are not available, and a viable numerical 2