  We consider a charged particle which is driven by a time-dependent flux
threading a circular ring system. Various approaches including classical
treatment, Fourier expansion method, time-evolution method, and
Lewis-Riesenfeld method are used and compared to solve the time-dependent
problem. By properly managing the boundary condition of the system, a
time-dependent wave function of the charged particle can be obtained by using a
non-Hermitian time-dependent invariant, which is a specific linear combination
of initial angular-momentum and azimuthal-angle operators. The eigenfunction of
the linear invariant can be realized as a Gaussian-type wave packet with a peak
moving along the classical angular trajectory, while the distribution of the
wave packet is determined by the ratio of the coefficient of the initial angle
to that of the initial canonical angular momentum. In this topologically
nontrivial system, we find that although the classical trajectory and angular
momentum can determine the motion of the wave packet; however, the peak
position is no longer an expectation value of the angle operator. Therefore, in
such a system, the Ehrenfest theorem is not directly applicable.
