Derivation of a Hamiltonian for a Charged Particle without Recourse to Electromagnetic Potentials

Abstract

The Hamiltonian for a charged particle in an electromagnetic field is usually presented in terms of the electromagnetic potentials A and φ. However for discussions of a large number of atomic and molecular properties, it is more convenient to have available a Hamiltonian which is expressed directly in terms of the more familiar electric field E and magnetic induction B. Here it is shown that one can construct a generalized velocity-dependent potential function of E and B, which yields the correct expression for the Lorentz force (truncated after the electric quadrupole, magnetic dipole terms). From this generalized potential the corresponding classical Lagrangian and Hamiltonian are obtained directly in terms of E and B. The transition to a quantum mechanical [E, B]-dependent Hamiltonian operator is considered, and we list a variety of, static as well as time-dependent, atomic and molecular properties which can be discussed in a unified way using this Hamiltonian as a starting point.