Bohr’s correspondence principle: The cases for which it is exact

Abstract

Two-dimensional central potentials leading to the identical classical and quantum motions are derived and their properties are discussed. Some of zero-energy states in the potentials are shown to cancel the quantum correction $Q=(\ensuremath{-}{\ensuremath{\Elzxh}}^{2}/2m)\ensuremath{\Delta}R/R$ to the classical Hamilton-Jacobi equation. The Bohr's correspondence principle is thus fulfilled exactly without taking the limits of high quantum numbers, of $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\Elzxh}}0,$ or of the like. In this exact limit of $Q=0,$ classical trajectories are found and classified. Interestingly, many of them are represented by closed curves. Applications of the found potentials in many areas of physics are briefly commented.