Compléments de mathématiques

1) Legendre transformation, differential and implicit derivation

2) Dynamical systems (modelling, phase plane, linear and nonlinear systems, Hamiltonian systems, Liouville theorem)

3) Series and integral representations

4) Series and integral solutions of differential equations (change of variable, symmetries, Frobenius method)

5) Hilbert spaces (square integrable functions, orthonormal basis, linear operators, eigenvalues and eigenvectors, commuting operators)

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

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

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

9) Hermite polynomials and quantum harmonic oscillator (Hermite polynomials and functions, properties)

10) Spherical harmonic functions (laplacian in spherical coordinates, spherical harmonics, eigenfunctions of spherical laplacian, parity and recurrence relations, 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 the 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 lattice 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

• Open question with short answer
• Open question with developed answer
• Closed question with multiple choices (MCQ)
• Visual question
• Closed question True or False (T/F)

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 )