Dynamical Billiards

Introduction

We now consider the classical and quantum dynamics of a point particle in a 2D enclosure with perfectly reflecting walls. Specifically, we will focus on the Bunimovich Stadium. This deceptively simple system is interesting because:

  1. Classically, it exhibits chaotic behavior.
  2. Quantum mechanically, its eigenmode structure is rich and interesting.
  3. The transition between the classical and quantum behaviors is curious and, to a some extent, mysterious.
Moreover, both the classical and quantum billiard problems illustrate an important lesson about scientific computation: Generic methods can only take you so far. Whether we are constructing a model or analyzing data, the most fruitful approaches are formulated with the specific problem squarely in mind.

The Bunimovich Stadium

Classical Stadium Billiards

Quantum Eigenfunctions of the Stadium