چكيده فارسي :
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 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 Classical models were (progressively) superseded by quantum ones through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. 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 rôle of such procedures with the
familiar example of the motion of a particle with a variable mass and submitted to a step potential..
