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From Classical to Quantum Models: The Regularising Rôle of Integrals, Symmetry and Probabilities


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 Classical models were (progressively) superseded by quantum ones through quantization prescriptions.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.

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  1. An apodization function (also called a tapering function or window function) is a function used to smoothly bring a sampled signal down to zero at the edges of the sampled region


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The author is indebted to the Centro Brasileiro de Pesquisas Físicas (Rio de Janeiro) and CNPq Agency (Brazil), and the Institute for Research in Fundamental Sciences (IPM, Tehran) for financial support. He also thanks the CBPF and the IPM for hospitality. He is grateful to Evaldo M.F. Curado (CBPF) for valuable comments on the content of this work.

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Correspondence to Jean-Pierre Gazeau.

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Gazeau, JP. From Classical to Quantum Models: The Regularising Rôle of Integrals, Symmetry and Probabilities. Found Phys 48, 1648–1667 (2018).

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  • Quantum Model
  • Heisenberg Weyl (WH)
  • Phase Space Geometry
  • Integrable Quantum
  • Variable Mass Particle