Change of energy and magnetic moment of a quantum charged particle after a fast jump of the magnetic field in solenoids of arbitrary cross sections

Abstract

We consider a quantum charged particle moving in the xy plane under the action of a time-dependent magnetic field described by means of the most general linear vector potential of the form A=B(t)−y(1+α),x(1−α)∕2. Such potentials with α≠0 can be created inside infinite solenoids with non-circular cross sections. The physical properties, such as energy and magnetic moment, do not depend on the choice of gauge in the thermodynamic equilibrium state. But systems with different values of α are not equivalent for nonstationary magnetic fields, due to different structures of induced electric fields. Using the approximation of the stepwise variation of the magnetic field, we obtain explicit formulas describing the change of the energy and magnetic moment from the initial equilibrium state, created with the aid of a weak anisotropic harmonic potential. The results are quite different for different values of the gauge parameter α, as well as for the initial high temperature and low temperature states. A strong amplification of the magnetic moment can happen even for rapidly decreasing magnetic fields. In addition, the magnetic moment becomes a strongly oscillating function of time after the jump of the field. Strong fluctuations of the magnetic moment (described in terms of the variance) are discovered in all regimes, including the initial equilibrium state. These fluctuations are sensitive to the shape of the trap confining the particle initially. Moreover, these fluctuations do not depend on the Planck constant in the high temperature case, and the magnetic moment variance is much bigger than the square of mean magnetic moment.