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. For the temperature range that was explored, 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. We call it ``The Escape Problem'', and approach it by applying the concept of a Transparent Boundary Condition (TBC). Since a four-dimensional phase space is computationally very difficult to investigate, here we will explore a pair of one-dimensional examples. First, as a benchmark, we will consider the classical regime by analyzing a "gas of particles'' limited to stay inside a one dimensional box of length L. The focus of our effort is the solution of the corresponding Quantum Initial Value Problem (QIVP). We employ a recently developed numerical method and test it for a simple situation with an exact, analytic solution. The numerical method introduces a novel way to solve a diffusion type equation by implementing discrete transparent boundaries conditions (DTBCs) recently developed by mathematicians. The method is then extended to include a linear, external potential.
| Identifer | oai:union.ndltd.org:TCU/oai:etd.tcu.edu:etd-10162009-115824 |
| Date | 16 October 2009 |
| Creators | Puga, Alejandro |
| Contributors | Bruce N. Miller |
| Publisher | Texas Christian University |
| Source Sets | Texas Christian University |
| Language | English |
| Detected Language | English |
| Type | text |
| Format | application/pdf, application/octet-stream |
| Source | http://etd.tcu.edu/etdfiles/available/etd-10162009-115824/ |
| Rights | unrestricted, I hereby certify that, if appropriate, I have obtained and attached hereto a written permission statement from the owner(s) of each third party copyrighted matter to be included in my thesis, dissertation, or project report, allowing distribution as specified below. I certify that the version I submitted is the same as that approved by my advisory committee. I hereby grant to TCU or its agents the non-exclusive license to archive and make accessible, under the conditions specified below, my thesis, dissertation, or project report in whole or in part in all forms of media, now or hereafter known. I retain all other ownership rights to the copyright of the thesis, dissertation or project report. I also retain the right to use in future works (such as articles or books) all or part of this thesis, dissertation, or project report. |
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