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

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 $Q=(-{ħ}^2/2m)\Delta R/R$ to the classical Hamilton-Jacobi equation. The Bohr’s correspondence principle is thus fulfilled exactly without taking the limits of high quantum numbers, of $\overrightarrow{ħ}0,$ 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.