Probabilities as a bridge between classical and quantum-mechanical treatments

There are significant differences in the structures of course presentations of classical mechanics and quantum mechanics, but also essential links that connect the two approaches. A large part of the differences in approach do not involve quantization, but rather the traditional use of instantaneous values to describe macroscopic objects and position probability densities to describe microscopic objects. A pedagogic reformulation of the classical problem to bridge this gap is presented here. The method is numerical in nature, but for illustrative purposes it is herein applied to two specific cases for which analytic solutions also exist, the Kepler–Coulomb potential and the isotropic harmonic oscillator. Using this approach, classical systems can be examined in the presence of perturbations in the same way as is used in quantum mechanics courses. Semiclassical quantization can also be introduced to extend the connection to a quantum-mechanical treatment.