On the virial theorem for a particle in a box: Accounting for Cauchy's boundary condition

Abstract

Most introductory books on quantum mechanics discuss the particle-in-a-box problem through solutions of the Schrödinger equation, at least, in the one-dimensional case. When introducing the virial theorem, however, its discussion in the context of this simple model is not considered and students ponder the question of the validity of the virial theorem for a system with, apparently, no forces. In this work, we address this issue by solving the particle in a finite box and show that the virial theorem is fulfilled when the appropriate Cauchy boundary conditions are taken into account. We also illustrate how, in the limit of the infinite potential box, the virial theorem holds as well. As a consequence, it is possible to determine the averaged force exerted by the walls on the particle. Finally, a discussion of these results in the classical limit is provided.