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Power Law Duality in Classical and Quantum Mechanics
Version 1
: Received: 24 January 2021 / Approved: 25 January 2021 / Online: 25 January 2021 (16:42:35 CET)
A peer-reviewed article of this Preprint also exists.
Inomata, A.; Junker, G. Power Law Duality in Classical and Quantum Mechanics. Symmetry 2021, 13, 409. Inomata, A.; Junker, G. Power Law Duality in Classical and Quantum Mechanics. Symmetry 2021, 13, 409.
Journal reference: Symmetry 2021, 13, 409
DOI: 10.3390/sym13030409
Abstract
The Newton-Hooke duality and its generalization to arbitrary power laws in classical, semiclassical and quantum mechanics are discussed. We pursue a view that the power-law duality is a symmetry of the action under a set of duality operations. The power dual symmetry is defined by invariance and reciprocity of the action in the form of Hamilton’s characteristic function. We find that the power-law duality is basically a classical notion and breaks down at the level of angular quantization. We propose an ad hoc procedure to preserve the dual symmetry in quantum mechanics. The energy-coupling exchange maps required as part of the duality operations that take one system to another lead to an energy formula that relates the new energy to the old energy. The transformation property of Green function satisfying the radial Schrödinger equation yields a formula that relates the new Green function to the old one. The energy spectrum of the linear motion in a fractional power potential is semiclassically evaluated. We find a way to show the Coulomb-Hooke duality in the supersymmetric semiclassical action. We also study the confinement potential problem with the help of the dual structure of a two-term power potential.
Subject Areas
Power-law duality; Classical and quantum mechanics; Semiclassical quantization; Supersymmetric quantum mechanics; Quark confinement
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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