Angular Momentum and the Two-Dimensional Free Particle

The model system consisting of a free particle is revisited in order to fully exploit it as a teaching resource. Because of this pedagogical aim, we chose the two-dimensional free particle system as the best choice between the opposite requests of algebraic simplicity and physical interest. Indeed, this is the simplest system possessing angular momentum which is thus introduced without undue reference to rotational motion. After briefly reviewing the classical mechanics, the quantum mechanics of a free particle is described in some detail. It is shown how two different sets of eigenfunctions naturally arise from requiring the particle to have a well-defined linear momentum or a well-defined angular momentum. The reasons for the different constant of motions and quantization schemes appearing in the two sets are discussed. The complex eigenfunctions are plotted and the significance of such plots is critically assessed. Finally, it is shown how the two alternative descriptions of the free particle are related to each other.