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

TL;DR

This paper demonstrates that a particle in a time-dependent, spatially homogeneous electromagnetic field can be transformed into a harmonic oscillator with a time-varying frequency, allowing exact solutions in certain cases.

Contribution

It establishes a canonical and unitary equivalence between a charged particle in a time-dependent electromagnetic field and a harmonic oscillator, enabling integration of the system.

Findings

Eigenstates are entangled states of the harmonic oscillator.

Exact integration possible for specific time dependencies of the magnetic field.

Unitary transformations represent canonical transformations in Hilbert space.

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.