Adiabatische Invarianz im Mittel / Adiabatic Invariance in the Average

Abstract

Abstract We study the possibility to construct an adiabatic invariant for charged particles in electromagnetic fields, that vary strongly in space but only slowly in time (measured by a small parameter ε). We specialize to those fields, where the motion of the particle is described by a Hamiltonian that corresponds to a one-dimensional oscillator with a potential exhibiting two minima. This type of potential can occur in the case of one-dimensional fields with neutral sheets. The action integral I=∮ p dq is no longer an adiabatic invariant for particles with an energy near the value of the inner maximum of the potential, nor is it a continuous function of the energy. The second difficulty can be removed by proper redefinition. Then by using the area conservation in the phase plane it can be shown that for any arbitrary particle ensemble in the phase plane of the one-dimensional oscillator the mean value of the action integral is adiabaticly invariant to order ε ln4 ε. Under certain circumstances it can happen that a coherent ensemble splits up into two separated groups. It will be shown that this also can be understood essentially as an adiabatic process.