Compléments de mathématiques

1) Legendre transformation, differential and implicit functions

2) Dynamical systems (modelling, phase plane, linear and nonlinear systems, Liouvuille theorem, numerical solutions)

3) Series and integral representations

4) Series and integral solutions of differential equations (Frobenius method)

5) Euclidean, hermitian and Hilbert spaces (Kronecker symbol, basis, linear operators, eigenvalues and vectors, commuting operators)

6) Fourier series (periodic functions, lattice and reciprocal lattice (also in 3D), properties, convergences, forbidden symmetries, numerical computation)

7) Fourier transformation (properties, gaussian function, inversion formula, sampling, relation with Fourier series)

8) Partial differential equations and diffusion (laplacian operator, heat equation, fundamental solution, separation of variables)

9) Hermite polynomials and quantum harmonic oscillator

10) Spherical harmonic functions (laplacian in spherical coordinates, harmonic polynomials, properties, Legendre polynomials)

11) Hydrogen atom (Laguerre polynomials, Schrödinger equation, central potential, complete solution)

At the end of this teaching unit, a student will be able to
1) comprenhend and manipulate Legendre transformations and the differential
2) model a temporal evolution with a dynamical system, solve a linear dynamical system, understand the phase plane and Liouville theorem
3) manipulate power series (exponential, geometric and binomial)
4) understand Frobenius method
5) verify if a function is an eigenfunction of a linear operator
6) write the Fourier series of a simple function, draw a latticed and reciprocal lattice in 2D
7) manipulate some Fourier integrals and draw some Fourier transforms in 2D
8) understand the notion of laplacian and a mathematical description of diffusion
9) manipulate Hermite polynomials
10) use spherical harmonic functions and Legendre polynomials
11) separate variables in the Schrödinger equation in spherical coordinates

Required and Corequired knowledge and skills

General mathematics (cartesian coordinates, functions, derivatives, integrals, matrices, determinants)

Required and corequired courses

Cours ayant celui-ci comme co-requis

Theoretical courses and exercises

Contribution to the teaching profile

– Acquire, assimilate and exploit basic knowledge of mathematics, physics, chemistry, biology and geo-sciences

– Develop transversal knowledge

– Collect, analyse and synthesize knowledge

– Identify problems and formulate scientific questions

– Solve problems

– demonstrate intellectual openness

References, bibliography, and recommended reading

Syllabus for sale at PUB and available on UV (Moodle)

Course notes

• Syllabus
• Université virtuelle

Contacts

Prof. Ignace Loris: Ignace.Loris@ulb.be, local 2.O.7.107, Teams, ...

Campus

Plaine

Method(s) of evaluation

• written examination

written examination

One integrated written exam of theory and exercices. Exceptionally (pandemic, open session, ...) the written exam could be replaced by an oral exam.

Mark calculation method (including weighting of intermediary marks)

No partial marks. One mark out of 20.

Language(s) of evaluation

• french
• (if applicable english )