I was sitting in a lecture the other day and the professor was sort of reviewing the basics of QM for the beginning of a new semester. He was discussing the difference between classical and quantum mechanics and he mentioned that quantum mechanics leaves the potential energy of classical mechanics untouched, and adjusts only the dynamics to account for physical behavior. This is something I've thought about before and have never quite understood. Why is it that the potential energy remains unchanged in the quantum picture? As a naive example, why not try to account for the quantized energy levels of hydrogen by inventing a potential energy function with a succession of "wells" of varying depths at different radial distances, each corresponding to an energy value? I'm quite certain that this wouldn't work, but I mean to say, why not something like it? In short, why is it that the creators of quantum mechanics formulated the theory as a rectification of hamiltonian dynamics, rather than seeking a solution in, say, the form of the coulomb potential. Is there a deep reason that QM leaves the classical potential untouched and concerns itself only with the equations of motion? Is there a historical reason? Have there been attempts to account for the behavior of quantum systems by rethinking the potential energy?