On the action for a charged particle moving in a magnetic field

Abstract

Application of the Bohr-Wilson-Sommerfeld quantization condition to a charged particle in a uniform magnetic field requires knowledge of the canonical momentum of such a particle, which in turn requires students to know about the vector potential. The canonical momentum in this situation is conventionally obtained by introducing an appropriate additional term involving the corresponding vector potential in the Lagrangian. In this work, we take a different approach to this problem by analyzing it with Maupertuis principle of least action. We show that satisfying this principle for a charged particle moving in a uniform magnetic field requires that a term proportional to the flux passing through the area enclosed by the trajectory of the particle be included in its action. This additional term accomplishes two tasks. First, it facilitates applying the Bohr-Wilson-Sommerfeld quantum condition for a charged particle moving in a uniform magnetic field without using the Lagrangian approach, and, secondly, it gives the appropriate canonical momentum and Lagrangian for a charged particle in a general electromagnetic field in a straightforward manner.