Solution of the Quantum Initial Value Problem with Transparent Boundary Conditions

TL;DR

This paper develops a numerical method to solve the quantum initial value problem with transparent boundary conditions, enabling accurate modeling of particle escape in quantum systems, and introduces a novel approach for diffusion equations.

Contribution

It presents a new numerical technique using discrete transparent boundary conditions for quantum initial value problems, bridging classical optical billiards and quantum chaos modeling.

Findings

Successfully solves quantum escape problem numerically

Validates the method with an exact analytic solution

Provides a new tool for quantum chaos research

Abstract

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.