Semiclassical quantization using classical perturbation theory: Algebraic quantization of multidimensional systems

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Abstract
The method of algebraic quantization, a semiclassical analog of Van Vleck perturbation theory, is applied to multidimensional resonant, nonresonant, and nearly resonant systems. perturb, a special purpose program written in C, is utilized to implement classical perturbation theory efficiently to high order. States corresponding to both regular and chaotic classical regimes are quantized, and accurate eigenvalues obtained in both cases. Various quantization rules are compared, and a novel symmetry preserving rule is given which leads to good agreement with quantum mechanics. The method is able to reproduce purely quantum mechanical splittings to very good accuracy. Algebraic quantization combined with Padé resummation is used to determine energy eigenvalues for a resonant system with five degrees of freedom.