Nonlinear systems driven by an external field can exhibit chaotic behavior when treated classically. In this work, a quantum-mechanical study of driven nonlinear systems is presented, with a particular attention to the generation and break up of resonance zones and their relation to the chaotic behavior. The systems studied are a particle in a single square-well potential, in a double square-well potential and in the one-dimensional hydrogen atom.
Resonance zones are generated in a nonlinear system when it is subjected to a sinusoidal external field. In a classical treatment of a particle in a single square-well potential, a resonance zone is broken and phase-space trajectories become chaotic when the field strength exceeds a critical value. In a quantum treatment, the breaking of resonance zones leads to delocalization of the particle resulting in the spreading of probability amplitudes over a large number of energy levels. For the case of a double square-well potential, the delocalization occurs more easily and the probability amplitudes spread more widely. This can be understood when the difference in the energy level structure between the single and double square-well potentials is considered. The study presented also provides a basic framework in which many aspects of field-induced resonance and delocalization that occur in the hydrogen atom reported in the literature can be understood.
