Particle in a Box: A Basic Paradigm in Quantum Mechanics — Part 2

The problem of a particle moving freely inside a region with rigid, perfectly reflecting walls serves as a paradigm to illustrate numerous aspects of quantum mechanics (QM). In Part 11 of this two-part article, we discussed several of these aspects using, respectively, a line segment, a ring, and a square as the region concerned. In the present part, we shall consider the cases of a circular region and a surface of constant positive curvature (a sphere). We shall then comment on the general case of a dynamical billiard. Finally, we turn to the inverse problem: given the energy spectrum of a particle moving freely inside a region, what can be deduced about the geometrical properties of the region? Some important results in this regard will be described in brief.