Charged Particle in a Magnetic Field

Ram K. Varma

This chapter is titled: “Construction of classical Hamiltonian function of a charged particle in an electric and a magnetic field.” Here we first obtain expression for Lagrangian function for a charged particle in an electric field and a magnetic field. Then we obtain Hamiltonian function from Legendre transformation of the Lagrangian function. To solve the problem of the book at non-zero magnetic field, we need Hamiltonian operator for an electron in presence of electric and a magnetic field. To get the operator, we first need classical Hamiltonian function of a charged particle in an electric and a magnetic field. We can replace operators for observables in the Hamiltonian function to get Hamiltonian operator.

Walter C. Henneberger, Mojtaba Jafarpour

Manoj K. Harbola

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's 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 particle's trajectory 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, second, it gives the appropriate canonical momentum and Lagrangian for a charged particle in a general electromagnetic field in a straightforward manner.