MATHEMATICAL METHODS FOR PHYSICS

Learning objectives
This course is offered during the first semester of the first year, so it is the first course that all students will be exposed to upon their entrance into the Master program in Science and Technology of Bio and Nanomaterials. As the background of the incoming students may (and indeed is) diverse, this course is designed to provide the advanced mathematical skills that will enable all students to tackle more advanced courses of the first and second years. While is not a corse (i.e. required) course, it will be strongly recommended to all students whose original degree did not provide a sufficient mathematical background. Particular emphasis will be devoted to problem solving techniques of Quantum Mechanics, as for instance partial differential equations, eigenvalues and eigenfunctions, perturbation theory, and algebra of commutators, rather than to formal methods.

Expected students learning results
During the course the students will develop the following important abilities:
1. Single out the main points of a complex problem
2. Decompose it into simpler sub-problems that can be more conveniently solved
3. Perform non-trivial calculations on their own
Upon exit from this course, students will then be able to:
1. Identify the correct technique to be used to solve a physics problem
2. Solve the most common ordinary and partial differential equations in the physical sciences
3. Use Fourier transforms
4. Compute eigenvalues and eigenvectors

Course prerequisites
The course is designed to be as self-consistent as possible. A standard Calculus course covering up to partial derivatives, integrals and series of functions is required. Useful, but not necessary, are the knowledge of introductory physics concepts in mechanics and electromagnetism, at the same level of those offered at any first level BS degree.

Course layout
INTRODUCTIONS (Taylor’s series, Planar polar coordinates, Complex numbers, Chain rule in differentiation, Hyperbolic functions)
ORDINARY DIFFERENTIAL EQUATIONS (General definitions and simple concepts, 1st order, 1st degree differential equations , Separation of variables, Exact differential equations, Linear 1st order differential equations, Bernoulli equation, Higher order differential equations homogeneous and with constant coefficients, Particular solution evaluation)
FOURIER ANALYSIS AND DIRAC δ-FUNCTION (Basic idea of the Fourier expansion, Fourier series, Conjugate variables, Advantages, uses of Fourier series, Complex form, Fourier transform, Dirac δ-function and representations, Proprieties of the Dirac δ-function, Relation between Dirac δ-function and Fourier transform, Convolution theorem, Parseval theorem. Fourier transform of a derivative, Fourier transform of a real function, Gaussian integrals, Fourier transform of a Gaussian, Solution of diffusion equation)
VECTOR ANALYSIS (Basic concepts in Matrices, Orthogonal transformations, Scalar, vector and tensor fields, Differential operators, Gauss theorem and divergence, Stokes theorem and Curl)
VECTOR SPACES (Linear independence and basis, Linear transformations, Inverse linear transformations and existence conditions, Matrix representation with respect to a basis, Special classes of matrices, Determinant and relative properties, Similarity transformations, System of linear equations, Eigenvalue problem, Examples of eigenvalues and eigenvectors, Normal modes of a triatomic molecule)
HILBERT SPACES (Scalar product and pre-Hilbert spaces, Hilbert spaces, Hermitian operators, Unitary operators, Dirac ket-bra formalism, {X} e {P} representations and L2 Hilbert space, Commutators and uncertainty principle, Functions of operators and commutator algebra, Translation operator T)
Additional topics
GREEN FUNCTIONS AND INTEGRAL EQUATIONS
CALCULUS OF VARIATIONS

Suggested reading
G. B. Arfken and H.Weber Mathematical Methods for Physicists (Elsevier 2005) [BAS]

D. McQuarrie Mathematical Methods for Scientists and Engineering (University Science Books 2003) [BAS]

Assessment methods
Written and oral exams
Detailed description of the assessment methods
Final grade will be the average of an oral exam (worth 50% of the final grade) and of the average grade reported on homeworks that will be assigned during the semester (and worth the additional 50% of the final grade). All homeworks must be handed in within the due date. Failure to do that will result into the impossibility of taking the oral exam. Later turning in will be penalized in terms of grades. The allotted time for each homework will be on average three weeks.

Teaching methods
Lectures at the blackboard with all calculations spelled out in details. Lecture notes as well as supporting materials will also be available at the instructor moodle learning platform

English

written and oral