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Action Quantization, Energy Quantization, and Time Parametrization

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Abstract

The additional information within a Hamilton–Jacobi representation of quantum mechanics is extra, in general, to the Schrödinger representation. This additional information specifies the microstate of \(\psi \) that is incorporated into the quantum reduced action, W. Non-physical solutions of the quantum stationary Hamilton–Jacobi equation for energies that are not Hamiltonian eigenvalues are examined to establish Lipschitz continuity of the quantum reduced action and conjugate momentum. Milne quantization renders the eigenvalue J. Eigenvalues J and E mutually imply each other. Jacobi’s theorem generates a microstate-dependent time parametrization \(t-\tau =\partial _E W\) even where energy, E, and action variable, J, are quantized eigenvalues. Substantiating examples are examined in a Hamilton–Jacobi representation including the linear harmonic oscillator numerically and the square well in closed form. Two byproducts are developed. First, the monotonic behavior of W is shown to ease numerical and analytic computations. Second, a Hamilton–Jacobi representation, quantum trajectories, is shown to develop the standard energy quantization formulas of wave mechanics.

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Acknowledgements

I heartily thank A. E. Faraggi and M. Matone for their invited, helpful comments on an earlier version, especially on Möbius transformations. Although we differ on compactification, their correspondence has always been cordial. I also thank one of the referees whose incisive comments contributed to significant improvements in the paper.

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    Edward R. Floyd

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Floyd, E.R. Action Quantization, Energy Quantization, and Time Parametrization. Found Phys 47, 392–429 (2017). https://doi.org/10.1007/s10701-017-0067-6

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