In physics, when a mathematical model becomes really inoperative in regard with correct predictions, one is forced to replace it with a new one. It is precisely what happened with the emergence of quantum physics. Classical models were (progressively) superseded by quantum ones through quantization prescriptions. These procedures appear often as ad hoc recipes. I will describe well defined quantizations, based on integral calculus and Weyl-Heisenberg symmetry. They are described in simple terms through one of the most basic examples of mechanics. Starting from (probability or not) distribution(s) on the Euclidean plane viewed as the phase space for the motion of a point particle on the line, i.e., its classical model, I will show how to build corresponding quantum model(s) and associated probabilities (e.g. Husimi) or quasi-probabilities (e.g. Wigner) distributions. I will highlight the regularizing role of such procedures with the familiar example of the motion of a particle with a variable mass and submitted to a step potential. In quantum cosmology, I will show how the procedure allows to consider the quantum dynamics of the anisotropic part of the Mixmaster universe as a perturbation of the explicitly integrable periodic 3-particle Toda system.
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