  In physics, one is often misled in thinking that the mathematical model of a
system is part of or is that system itself. Think of expressions commonly used
in physics like "point" particle, motion "on the line", "smooth" observables,
wave function, and even "going to infinity", without forgetting perplexing
phrases like "classical world" versus "quantum world".... On the other hand,
when a mathematical model becomes really inoperative with regard to 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(progressively) superseded by quantum ones through quantization prescriptions.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. In the present paper, well
defined quantizations, based on integral calculus and Weyl-Heisenberg symmetry,
are described in simple terms through one of the most basic examples of
mechanics. Starting from (quasi-) probability 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, we will show how to build corresponding quantum
model(s) and associated probabilities (e.g. Husimi) or quasi-probabilities
(e.g. Wigner) distributions. We 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.
