Mathematics for quantum physics (M. Fischer) :
In quantum mechanics, the state of a physical system is described by a wave function. This function can be seen as a point in a (vector) space as in classical mechanics except that this point (or vector) actually lives in a space of infinite dimension, since it is a (linear) space of functions. It turns out that the classical Euclidean vision can be extended to the spaces of quantum mechanics, thanks to the central concept of Hilbert space: a complete normalized vector space whose norm (metric) is derived from a scalar product as for the Euclidean spaces. The usual Hilbert spaces are built using the Lebesgue integral theory. A first work will be to see how a wave function in a Hilbert space can be described by an infinite (but often discrete) system of coordinates thanks to the notion of orthonormal basis or Hilbert basis. This will also make it possible to interpret in the formalism of quantum mechanics a wave function as an infinite superposition of states with a priori probabilities (apart from any measure) more or less important. The coherence of such a formalism will be ensured by the Bessel-Parseval relation. In addition, Fourier analysis will appear to be a very particular case. A second work will focus on simply describing a sufficiently rich class of linear applications (or operators on a Hilbert space) in terms of spectral decomposition as in the case of finite dimension. In the formalism of quantum mechanics, this will make it possible to analyze the result of an observable or measurement since the latter is "naturally" associated with a self-adjoint operator... Finally, the example of the quantum harmonic oscillator will be studied in detail to illustrate the different concepts, in particular of symmetry and compactness of an operator.
Introduction to Quantum Mechanics (R. Charrière) :
The course Introduction to quantum mechanics presents the theoretical and conceptual bases necessary for the description of phenomena on the atomic scale. The course will start with a reminder of the historical experiences that led to the development of quantum mechanics. The formalism will then be presented, with the fundamental concepts (bra, ket, operators, eigenvectors and eigenvalues ...) and the various basic principles (superposition principle, Shrödinger's equation, principle of quantum measurement ...) . The second part of the course will be devoted to the calculation of quantum states and energy levels of the hydrogen atom, with the presentation of the quantum angular moment operator and its properties. The fine structure of the hydrogen atom will also be mentioned, calculated using the perturbations method.
| On completion of the unit, the student will be capable of: | Classification level | Priority |
|---|---|---|
| understand geometry in Hilbert spaces as an extension of usual 3D geometry | 2. Understand | Important |
| decompose a vector on a Hilbert basis | 3. Apply | Essential |
| analyze the spectral decomposition of some self-adjoint operators | 4. Analyse | Useful |
| study a simple case of operator from quantum mechanics | 4. Analyse | Useful |
| Understand the concepts and formalism of quantum mechanics | 2. Understand | Essential |
| Understanding the calculation of quantum states and energy levels of the hydrogen atom | 2. Understand | Essential |
| Calculate quantum states and energy levels of simple systems | 3. Apply | Essential |
| Know the perturbation theory and the fine structure of the hydrogen atom | 1. Knowledge | Important |
| Percentage ratio of individual assessment | Percentage ratio of group assessment | ||||
|---|---|---|---|---|---|
| Written exam: | 100 | % | Project submission: | 0 | % |
| Individual oral exam: | 0 | % | Group presentation: | 0 | % |
| Individual presentation: | 0 | % | Group practical exercise: | 0 | % |
| Individual practical exercise: | 0 | % | Group report: | 0 | % |
| Individual report: | 0 | % | |||
| Other(s): 0 % | |||||
| Type of teaching activity | Content, sequencing and organisation |
|---|---|
| Course | Mathematics for quantum physics : Introduction to Quantum Mechanics : |
| Tutorials |
Mathematics for quantum physics : Introduction to Quantum Mechanics : |