Canonical equivalence of a charge in a time dependent, spatially-homogeneous electromagnetic field to a time-dependent perturbed oscillator

Abstract

Here we prove that the classical (respectively, quantum) system, consisting of a particle moving in a static electromagnetic field, is canonically (respectively, unitarily) equivalent to a harmonic oscillator perturbed by a spatially homogeneous force field. This system is canonically and unitarily equivalent to a standard oscillator. Therefore, by composing the two transformations we can integrate the initial problem. Actually, the eigenstates of the initial problem turn out to be entangled states of the harmonic oscillator. When the magnetic field is spatially homogeneous but time-dependent, the equivalent harmonic oscillator has a time-varying frequency. This system can be exactly integrated only for some particular cases of the time dependence of the magnetic field. The unitary transformations between the quantum systems are a representation of the canonical transformations by unitary transformations of the corresponding Hilbert spaces.