Abstract
This thesis is related to the field of didactics of university mathematics. It aims at studying the didactic transposition of mathematical knowledge involved in undergraduate quantum mechanics. More precisely, this work is at the confluence of two groups of studies: on the one hand, research on the didactic phenomena occurring when mathematics is taught in another field, in particular theoretical physics, and their consequences on teaching and learning; on the other hand, research on the conditions and constraints that can influence the evolution of the still prevalent paradigm of visiting works towards the paradigm of questioning of the world, which is in the process of emerging. In order to carry out these analyses, we relied on the general model of teaching and learning provided by the Anthropological Theory of the Didactic (ATD). Indeed, the historical development of quantum mechanics, its current state as well as its teaching, have been or are realized within scholarly or schooling institutions of several disciplines, which we had to distinguish and then characterize, in order to trace as precisely as possible the diffusion of knowledge within them. In this thesis, we therefore use a certain number of tools of the ATD, in an epistemological context where no one has yet resorted to it, which is the interface between mathematics and theoretical physics. Quantum mechanics is the result of a truly co-disciplinary conceptual development between physics and mathematics, which has led to the shaping of objects whose didactic ecology is quite particular, and therefore deserves an in-depth study. Ours begins with a wide-range research in epistemological history on the interaction between mathematics and physics during the establishment of the theory (1900-1930). Then, based on interviews with four professors at the University of Montpellier as well as praxeological analyses of lecture notes and homework sets, we proceed to a study of the internal and external didactic transpositions of mathematical knowledge involved in the first undergraduate course on quantum mechanics (third year). Then, we present the setting up, the unfolding and our textit{a posteriori} analyses of a Study and Research Path set up at the interface between mathematics and quantum mechanics, in third year of bachelor's degree. Following our analyses of the transpositive phenomena at the interface in the traditional paradigm of visiting the works, we thus study in this last part the conditions and constraints weighing, in the paradigm of questioning the world, on the existence of mathematical knowledge at the interface with quantum physics.