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Here we prove that the classical (respectively, quantum) system, consistingof a particle moving in a static electromagnetic field, is canonically(respectively, unitarily) equivalent to a harmonic oscillator perturbed by aspatially homogeneous force field. This system is canonically and unitarilyequivalent to a standard oscillator. Therefore, by composing the twotransformations we can integrate the initial problem. Actually, the eigenstatesof the initial problem turn out to be entangled states of the harmonicoscillator. When the magnetic field is spatially homogeneous buttime-dependent, the equivalent harmonic oscillator has a time-varyingfrequency. This system can be exactly integrated only for some particular casesof the time dependence of the magnetic field. The unitary transformationsbetween the quantum systems are a representation of the canonicaltransformations by unitary transformations of the corresponding Hilbert spaces.

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