The particle in the circular box

Abstract

The particle in the box in Cartesian coordinates was analytically and numerically described in a recent text through the Schrödinger equation. Changing the shape of the box not only modifies the solution to the Schrödinger equation but produces different ways of approaching the particle-in-the-box problem and introduces new mathematical challenges. The level of complexity of the Schrödinger equation can change significantly depending on the coordinate system used. In the case of polar coordinates, an immediate consequence corresponds to the natural appearance of orbital degeneracy, compatible with the description of certain molecular systems, such as the pi electrons of the benzene ring. Another important aspect is that the particle in the ring corresponds to part of the solution of the Schrödinger equation for the rigid rotor and the hydrogen atom. This text addresses alternatives for solving the Schrödinger equation for the simplest circular system, the particle in the ring.