Compléments de mathématiques

1) Legendre transformation and differential 

2) Probabilities (probability density, position and dispersion of a variables, sum of variables)

3) Dynamical systems (modelling, phase plane, linear systems, behavior near equilibrium, Hamiltonian systems, Liouville theorem)

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

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

6) Fourier transformation (definition, properties, interpretation, gaussian function, inversion formula, sampling, spectrogram, relation with Fourier series, Heisenberg uncertainty relation)

7) Convolution (definition, interpretation, properties, link with probability, Dirac delta, convolution in imaging, autocorrelation)

8) The diffusion equation (heat equation, fundamental solution, solution on the line, half-line, plane, separation of variables)

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

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

11) Spherical harmonic functions (laplacian in spherical coordinates, spherical harmonics, eigenfunctions of spherical laplacian, parity and recurrence relations, Legendre polynomials)

12) Hydrogen atom (Laguerre polynomials, Schrödinger equation, central potential, bound states of the hydrogen atom)

At the end of this teaching unit, a student will be able to
1) comprenhend and manipulate Legendre transformations and the differential
2) understand the meaning of probability density functions,
3) model a temporal evolution with a dynamical system, solve a linear dynamical system, understand the phase plane and Liouville theorem
4) understand the Frobenius method
5) write the Fourier series of a simple function, draw a lattice and reciprocal lattice in 2D
6) manipulate some Fourier integrals and draw some Fourier transforms in 2D
7) understand the importance of the Fourier transformation in theoretical and experimental chemistry
8) understand the role of convolution in experimental science and probability
9) understand the mathematical description of diffusion
10) verify if a function is an eigenfunction of a linear operator
11) manipulate Hermite polynomials
12) use spherical harmonic functions and Legendre polynomials
13) separate variables in the Schrödinger equation in spherical coordinates

In the context of the course subjects, a student will be able to interpret and produce mathematical content (such as texts, diagrams, and formulas) using a variety of notations (lowercase and uppercase letters, italics, boldface, Latin and Greek letters, numbers, symbols, etc.), presented in common fonts and sizes.

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

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 exercises. Exceptionally (pandemic, open session, ...) the written exam can be replaced by an oral exam.

The course instructor reserves the right to invite any student to a supplementary oral interview to verify their personal mastery of the material covered in their submitted examination, particularly where inconsistencies in the examination prevent the instructor from properly assessing the student’s mastery of the knowledge and skills addressed in the course.

Mark calculation method (including weighting of intermediary marks)

No partial marks are given, only a single global mark out of 20.

The purpose of any assessment is to verify the student’s personal mastery of the learning outcomes. It is therefore the student’s responsibility to demonstrate such mastery. This principle applies in all circumstances and at every stage of the assessment process, both during and after the examination, including in the event of a challenge to the grade by the student, an irregularity, suspected academic misconduct, etc. The grade exclusively reflects the personal mastery that the course instructor can reasonably consider to have been established in light of all the available information.

Language(s) of evaluation

• french
• (if applicable english )