Bateman's dual system revisited: quantization, geometric phase and relation with the ground-state energy of the linear harmonic oscillator
Recall that the connection between classical and quantum descriptions, established by Ehrenfest's theorems, is exact only for systems with quadratic Hamiltonians. The authors deal with a system of another sort, the 2D system of damped harmonic oscillator coupled to its time-reversed image (Bateman's dual system), the quantization of which used to plague with many conceptual problems. Here the quantization of the Bateman's dual system is made by applying the Feynman-Hibbs prescription for evolution amplitude. The time-dependent quantum states are constructed and discussed entirely in the framework of the classical theory. The corresponding geometric (Pancharatnam) phase is calculated and found to be directly related to the ground state energy of the 1D linear harmonic oscillator to which the 2D system reduces under appropriate constraint.
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