Learning objectives
Students are supposed to get to master the mathematical skills which are essential for the study of the most advanced subjects in Physics, in particular Quantum Physics. They should understand the power of mathematical tools for tackling a variety of problems in different fields: what they learn should put them in a position to solve problems in many different circumstances. Students will be involved in solving problems in front of their colleagues during the lessons; they will be also asked to present solutions to problems they will be assigned for being worked out at home. All this is intended as a training of their communication skills (they should be able to argue in public).
Prerequisites
Basic notions of real analysis, calculus, geometry and algebra
Course unit content
The course aims at providing a final framework of the mathematical skills which are fundamental for a physicist. We want to combine mathematical rigor and fluency in acquiring new tools, with the final goal of ability in problem solving.
First goal is to complete a basic preparation in classical real and complex analysis, with the theory of analytic functions (residues, power series, integration in the complex plane).
Main part of the course is devoted to the theory of linear operators in finite dimensional spaces (aiming at a sufficiently rigorous knowledge of the spectral theory), with the due insight in algebra and metric topology. The extension to functional spaces L1 and L2 will be obtained through approximation problems, orthogonal functions, Fourier series and transform, presented in the perspective of a first approach to quantum mechanics.
In this frame, there will be a short course on differential equations in the complex field, with applications to the Schroedinger equation.
Full programme
Numerical Fields.
Complex Analysis. Analytic functions, basic instruments.
Residues, power series, definite integrals.
Linear manifolds, abstract vector spaces. Linear dependence. Dimension.
Real and compolex spaces. Isomorphism.
Scalar product. Orthogonality.
Metric spaces. Basic notions in topology.
Basis, orthogonal systems, orthogonalization.
Basis transformation.
Linear functional and Riesz Theorem Dirac formalism.
Sequences and convergence.
Linear applications and matrices.
Abstract linear operators.
Diagonalization.
Adjoint operators.
Eigenvalues and eigenvectors.
Hermitian, unitary and normal operators.
Projectors. Function of operators.
Complete sets of hermitian operators.
Polynomials and orthogonal functions.
Approximation. L1 and L2 spaces.
Fourier series and transforms.
Differential equations in the complex field.
Special functions.
Applications to the Schroedinger equation.
Bibliography
There are many excellent books on the subjects covered. A (partial) list includes
V. Smirnov, Corso di Matematica superiore, vol.III,2 (MIR)
E. Onofri, Teoria degli Operatori lineari, http://www.fis.unipr.it/home/enrico.onofri/#Lezioni
F.G.Tricomi, Metodi Matematici della Fisica (Cedam)
M.Spiegel, Variabili Complesse (Schaum, Etas)
E.Kolmogorov, S.Fomin, Elementi di teoria delle funzioni e dell'analisi funzionale (ER)
Teaching methods
Lessons and exercises in the classroom (with students involved in working out the solutions). Exercises will be assigned for being worked out at home.
Assessment methods and criteria
Final written and oral tests.
The written test consists in exercises aiming to check the skill in calculus, trating problems which are variations of exercises already developed in the lessons.
The oral test consists in the discussion of some typical subjects, showing the methodological and conceptual mastership of the tudent on the fundamental topics.
Other information
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