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We consider a quantum spinless nonrelativistic charged particle moving in the$xy$ plane under the action of a time-dependent magnetic field, described bymeans of the linear vector potential ${\bfA}=B(t)\PG{-y(1+\alpha),x(1-\alpha)}/2$, with two fixed values of the gaugeparameter $\alpha$: $\alpha=0$ (the circular gauge) and $\alpha =1$ (the Landaugauge). Although the magnetic field is the same in all the cases, the systemswith different values of the gauge parameter are not equivalent fornonstationary magnetic fields due to different structures of induced electricfields, whose lines of force are circles for $\alpha=0$ and straight lines for$\alpha=1$. We derive general formulas for the time-dependent mean values ofthe energy and magnetic moment, as well as for their variances, for anarbitrary function $B(t)$. They are expressed in terms of solutions to theclassical equation of motion $\ddot\vep +\omega_{\alpha}^2(t) \vep=0$, with$\omega_1=2\omega_0$. Explicit results are found in the cases of the suddenjump of magnetic field, the parametric resonance, the adiabatic evolution, andfor several specific functions $B(t)$, when solutions can be expressed in termsof elementary or hypergeometric functions. These examples show that theevolution of the mentioned mean values can be rather different for the twogauges, if the evolution is not adiabatic. It appears that the adiabaticapproximation fails when the magnetic field goes to zero. Moreover, the suddenjump approximation can fail in this case, as well. The case of slowly varyingfield changing its sign seems especially interesting. In all the cases,fluctuations of the magnetic moment are very strong, frequently exceeding thesquare of the mean value.

Energy and magnetic moment of a quantum charged particle in time dependent magnetic and electric fields of circular and plane solenoids