Mathematics

General skills The course aims to transmit to students a deep knowledge of the mathematical structure of Quantum Mechanics, of the historical and conceptual path leading to its formulation, and of its relations with other mathematical subjects (as e.g. functional analysis, operator theory, theory of Lie groups and their unitary representations). Specific skills A) Knowledge and understanding After the conclusion of the course, successful students will know and understand the fundamental concepts of Fourier theory, the mathematical analogy between classical mechanics and geometric optics, the historical and conceptual path which led to overcome Classical Mechanics in favour of the more general Quantum Mechanics, and the mathematical structure of Quantum Theory, with a particular emphasis on dynamical aspects (time evolution) and on the analysis of the symmetries of a quantum system (representation of the symmetry group). B) Applying knowledge and understanding The general knowledge will be complemented by the application of general concepts to some specific models, and by the ability to analyze symmetries and dynamics of simple quantum systems. Specific simple systems will be analyzed in detail, including the case of a quantum particle in a linear potential, in a harmonic potential, in a uniform magnetic field, and in a Kepler potential (hydrogenoid atom). Successful students will be potentially able to apply the general concepts also to other more complex systems, including non-hydrogenoid atoms, molecules and crystalline solids. C) Making judgements The analysis of the historical and conceptual path which led to overcome Classical Mechanics in favour of the more general Quantum Mechanics will make successful students able to autonomously judge the epistemological foundations of a physical theory, and hence to understand its natural range of application and validity. This critical judgement will lead students to privilege an epistemological apophantic approach, with respect to an apodictic one. Moreover, successful students will be able to autonomously judge the validity of a mathematical statement, through a critical analysis of the hypotheses and of the deductive steps leading to the proof of the statement itself, and to autonomously formulate counterexamples to mathematical statements whenever one of the hypotheses is denied. D) Communication skills Successful students will acquire the ability to communicate what has been learned through written themes and oral exams, and to formulate a logically structured speech, with a clear distinction between hypotheses, deduction and thesis. E) Learning skills Successful students will acquire the ability to identify the most relevant topics in a subject and to make the logical connections between the topics covered.