5 credits
30.0 h + 30.0 h
Q1
Teacher(s)
Duhr Claude; Hagendorf Christian;
Language
French
Prerequisites
LMAT1121
The prerequisite(s) for this Teaching Unit (Unité d’enseignement – UE) for the programmes/courses that offer this Teaching Unit are specified at the end of this sheet.
The prerequisite(s) for this Teaching Unit (Unité d’enseignement – UE) for the programmes/courses that offer this Teaching Unit are specified at the end of this sheet.
Main themes
The purpose of this teachingunit is to familiarize the student with the mathematical tools and methods ofquantum physics. The covered topics are elements of theory of the classical linear partial differential equation theory of physics (heat equation, wave equation and Laplace equation), Fourier series andtheFourier transformation, elements of the theory of Hilbert spaces and orthogonal polynomials over finite and infinite intervals.
Aims
At the end of this learning unit, the student is able to : | |
| 1 | a. Contribution the teaching unit to the learning outcomes of the programme
1.1, 1.3, 1.4, 2.1, 3.1, 3.2, 3.3, 3.4, 3.5, 3.6
b. Specific learning outcomes of the teaching unit
¿ find the solutions of the classical partial differential equations of physics in simple geometries; ¿ determine the Fourier series of a given function; ¿ apply the abstract theory of Fourier series in Hilbert spaces; ¿ construct the classical orthogonal polynomials and apply them to the solution of differential equations; ¿ apply the Fourier transform to solve partial differential equations. |
The contribution of this Teaching Unit to the development and command of the skills and learning outcomes of the programme(s) can be accessed at the end of this sheet, in the section entitled “Programmes/courses offering this Teaching Unit”.
Content
1. Fourier series :periodic functions, trigonometric polynomials, Fourier series, Bessel's inequality, Parseval's theorem, convergence and Dirichlet's theorem, applications.
2. Partial differential equations :classification of linear partial differential equations of second order, heat equation, wave equation, Laplace equation, existence and uniqueness of solutions, solution methods.
3. Hilbert spaces :pre-Hilbert spaces, completeness and Hilbert spaces, Hilbert bases, examples (sequence and function spaces), abstract theory of Fourier series.
4. Orthogonal polynomials :definition on finite and infinite intervals, recurrence relations, Rodriguez' formula and the classical orthogonal polynomials (Jacobi, Chebyshev, Legendre, Laguerre, Hermite), second order differential equations, application of Legendre polynomials and spherical harmonics in physics.
5. The Fourier transformation :definition and properties, convolution product, Poisson summation formula, applications to the solution of linear differential equations, distributions and their Fourier transformation.
2. Partial differential equations :classification of linear partial differential equations of second order, heat equation, wave equation, Laplace equation, existence and uniqueness of solutions, solution methods.
3. Hilbert spaces :pre-Hilbert spaces, completeness and Hilbert spaces, Hilbert bases, examples (sequence and function spaces), abstract theory of Fourier series.
4. Orthogonal polynomials :definition on finite and infinite intervals, recurrence relations, Rodriguez' formula and the classical orthogonal polynomials (Jacobi, Chebyshev, Legendre, Laguerre, Hermite), second order differential equations, application of Legendre polynomials and spherical harmonics in physics.
5. The Fourier transformation :definition and properties, convolution product, Poisson summation formula, applications to the solution of linear differential equations, distributions and their Fourier transformation.
Teaching methods
The learning activities consist of lectures and exercisesessions.
The lectures introduce the concepts and ideas of mathematical methods that are necessary for understanding modernphysics (such as quantum physics), establish rigorous results and present computational techniques and strategies. Furthermore, the connection with other teaching unitsof the Bachelor's programme in physics are emphasized.
The main objective of the exercisesessions is the application of the theory to concrete examples.
The lectures introduce the concepts and ideas of mathematical methods that are necessary for understanding modernphysics (such as quantum physics), establish rigorous results and present computational techniques and strategies. Furthermore, the connection with other teaching unitsof the Bachelor's programme in physics are emphasized.
The main objective of the exercisesessions is the application of the theory to concrete examples.
Evaluation methods
The evaluation is based on a written exam. It deals with the application of calculation techniques of mathematical methods of physics. It tests the student's knowledge and hisunderstanding of the notions seen in the theoretical course, the mastery of calculation techniques and the coherent presentation of this analysis.
Online resources
The MoodleUCL website of this teaching unit contains a detailed plan of the covered topics, a complete bibliography, exercise sheets and a collection of exam subjects from past years.
Bibliography
' W. Appel 'Mathématiques pour la physique ' et les physiciens', Éditions H & K, Paris (2008).
' C. Aslangul 'Des mathématiques pour les sciences, De Boeck (2011).
' C. Aslangul 'Des mathématiques pour les sciences, De Boeck (2011).
Faculty or entity
PHYS
