Power law duality in classical and quantum mechanics

TL;DR

This paper explores the classical and quantum power-law duality symmetry, its limitations at the quantum level, and proposes methods to preserve it, providing new insights into energy spectra and dualities in various potentials.

Contribution

It introduces a duality symmetry framework for classical and quantum mechanics, addressing its breakdown at the quantum level and proposing procedures to maintain it.

Findings

Power-law duality is a classical symmetry that breaks down in quantum mechanics.

A method is proposed to preserve duality symmetry in quantum systems.

Semiclassical evaluation of energy spectra in fractional power potentials.

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 {the} Green function satisfying the radial Schr\"odinger 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.