## Brazilian Journal of Physics

*Print version* ISSN 0103-9733

#### Abstract

JESUS, V.L.B. de; GUIMARAES, A.P. and OLIVEIRA, I.S.. **Classical and quantum mechanics of a charged particle in oscillating electric and magnetic fields**.* Braz. J. Phys.* [online]. 1999, vol.29, n.3, pp. 541-546. ISSN 0103-9733. http://dx.doi.org/10.1590/S0103-97331999000300019.

The motion of a particle with charge *q* and mass m in a magnetic field given by **B** = k*B*_{0} + *B*_{1 }[**i***cos*(w*t*) + **j***sin*(w*t*)] and an electric field which obeys Ñ ´**E** = -¶**B**/¶*t* is analysed classically and quantum-mechanically. The use of a *rotating coordinate system* allows the analytical derivation of the particle classical trajectory and its laboratory wavefunction. The motion exhibits two resonances, one at w = w_{c} = -*qB*_{o}*/m*, the *cyclotron frequency*, and the other at w = w* _{L}* = -

*qB*

_{o}

*/2m*, the

*Larmor frequency*. For w at the first resonance frequency, the particle acquires a simple closed trajectory, and the effective hamiltonian can be interpreted as that of a particle in a

*static*magnetic field. In the second case a term corresponding to an effective static electric field remains, and the particle orbit is an open line. The particle wave function and eigenenergies are calculated.

**Keywords
:
**magnetic resonance; ion trapping; isotope separation; magnetic confinement.