  We consider a quantum spinless nonrelativistic charged particle moving in the
$xy$ plane under the action of a time-dependent magnetic field, described by
means of the linear vector potential ${\bf
A}=B(t)\PG{-y(1+\alpha),x(1-\alpha)}/2$, with two fixed values of the gauge
parameter $\alpha$: $\alpha=0$ (the circular gauge) and $\alpha =1$ (the Landau
gauge). Although the magnetic field is the same in all the cases, the systems
with different values of the gauge parameter are not equivalent for
nonstationary magnetic fields due to different structures of induced electric
fields, 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 of
the energy and magnetic moment, as well as for their variances, for an
arbitrary function $B(t)$. They are expressed in terms of solutions to the
classical 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 sudden
jump of magnetic field, the parametric resonance, the adiabatic evolution, and
for several specific functions $B(t)$, when solutions can be expressed in terms
of elementary or hypergeometric functions. These examples show that the
evolution of the mentioned mean values can be rather different for the two
gauges, if the evolution is not adiabatic. It appears that the adiabatic
approximation fails when the magnetic field goes to zero. Moreover, the sudden
jump approximation can fail in this case, as well. The case of slowly varying
field changing its sign seems especially interesting. In all the cases,
fluctuations of the magnetic moment are very strong, frequently exceeding the
square of the mean value.
