Quantum Newton duality

Abstract

Newton revealed an underlying duality relation between power potentials in classical mechanics. In this paper, we establish the quantum version of the Newton duality. The main aim of this paper is threefold: (1) first generalizing the original Newton duality to more general potentials, including general polynomial potentials and transcendental-function potentials, 2) constructing a quantum version of the Newton duality, including power potentials, general polynomial potentials, transcendental-function potentials, and power potentials in different spatial dimensions, and 3) suggesting a method for solving eigenproblems in quantum mechanics based on the quantum Newton duality provided in the paper. The classical Newton duality is a duality among orbits of classical dynamical systems. Our result shows that the Newton duality is not only limited to power potentials, but a more universal duality relation among dynamical systems with various potentials. The key task of this paper is to construct a quantum Newton duality, the quantum version of the classical Newton duality. The quantum Newton duality provides a duality relations among wave functions and eigenvalues. As applications, we suggest a method for solving potentials from their Newtonianly dual potential: once the solution of a potential is known, the solution of all its dual potentials can be obtained by the duality transformation directly. Using this method, we obtain a series of exact solutions of various potentials. In appendices, as preparations, we solve the potentials which is solved by the Newton duality method in this paper by directly solving the eigenequation.