The double pendulum is one of the best tools for demonstrating classical chaos and yet its quantization has somehow escaped attention. An attempt is made to fill this gap. We show that the constraint-induced Riemann curvature scalar plays an important role in the proper quantization and evaluation of the energy spectra of this system. The quantum energies are compared with the semiclassical results obtained earlier by resumming the Birkhoff–Gustavson normal form series. The general agreement between the two sets of energies demonstrates how a proper resummation makes it possible to use a divergent normal form series for semiclassical analyses.
We explain the dynamics of cold atoms, initially trapped and cooled in a magneto-optic trap, in a monochromatic stationary standing electromagnetic wave field. In the large detuning limit the system is modeled as a nonlinear quantum pendulum. We show that wave packet evolution of the quantum particle probes parametric regimes in the quantum pendulum which support classical period, quantum mechanical revival and super revival phenomena. Interestingly, complete reconstruction in particular parametric regime at quantum revival times is independent of potential height.
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