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A novel hybrid approach to the dynamics of electron interacting with time-dependent electromagnetic fields, namely, the Maxwell-Schrödinger approach, has been employed to study a system of electron confined in single- and multi-well electrostatic potentials subjected to pulsed laser fields. A comparison of the results of simulation to those obtained by the conventional Maxwell-Newton approach has been made by calculating the time response of the current density and the time-evolution of the electric field at an observation point. The results obtained by these two distinct approaches agree very well for the single-well potential while disagree qualitatively for the multi-well potential. This clearly demonstrates limitation of applicability of the conventional Maxwell-Newton scheme to study electron dynamics in electromagnetic fields.