Analytical and Quantum Mechanics

This UE represents the natural continuation of the UEs in classical Newtonian mechanics.

In the first part of the course, we take Classical Mechanics from the principle of least action to two new formulations: the Lagrangian formalism and the Hamiltonian formalism. We study the link between physical symmetries and conservation laws (E. Noether's theorem) and introduce Poisson brackets, which allow us to write the classical laws of temporal evolution of physical quantities in a form that already prefigures those of quantum mechanics.

In the second part of the course, starting from an examination of the experimental limits of classical mechanics, a new theory of mechanics is introduced: Quantum Mechanics. This is a theory that is conceptually completely different from previous classical theories, based on a description of physical phenomena in terms of probabilities and therefore no longer deterministic. It's a radical paradigm shift that has turned physics on its head over the last century, enabling a deeper understanding of physical nature, with fundamental and practical spin-offs that have radically changed the lives of mankind (atomic physics, chemistry, nuclear energy, transistors, LASERS, to name but a few). 

Provide the basic elements of Classical Mechanics in Lagrangian and Hamiltonian formulations, and introduce the basic concepts of the new theory of "non-classical" mechanics, i.e. quantum mechanics. This course will be a first exploration of the quantum world, and will provide the initial conceptual and mathematical tools needed for all quantum physics courses in L3, M1 and M2.

  - Newtonian mechanics

  - Mathematical Analysis

  - Linear algebra and matrix calculations

  - Electromagnetism

Recommended prerequisites* :

  - Statistical Physics  

   - Analysis and advanced matrix calculation

100% CT

Analytical Mechanics :

• Variational principle
• Lagrangian formalism
• Hamiltonian formalism
• Symmetries and conservation laws
• Fish hooks

Quantum Mechanics:

• Introduction, Planck's formula and solution of the black body problem, photoelectric effect, limits of the classical atomic model, Compton effect
• MQ postulates
• Wave-particle dualism
• Schrödinger equation
• Indeterminacy relations
• Elements of Hilbert spaces and wave functions
• Operators
• Switches
• One-dimensional problems (barrier/wells/harmonic oscillator)

CM: 31.5 h

TD: 31.5 h