Analytical and Quantum Mechanics

This unit is a natural continuation of the units on classical Newtonian mechanics.

In the first part of this unit, we cover classical mechanics, starting with the principle of least action and arriving at two new formulations: the Lagrangian formalism and the Hamiltonian formalism. We study the connection between physical symmetries and conservation laws (E. Noether’s theorem) and introduce Poisson brackets, which allow us to write the classical laws of time evolution of physical quantities in a form that already foreshadows those of quantum mechanics.

In the second part of the course, beginning with an examination of the experimental limits of classical mechanics, a new theory of mechanics is introduced: quantum mechanics. This theory is conceptually entirely different from previous classical theories, based on a description of physical phenomena in terms of probabilities and is therefore no longer deterministic. This is a radical paradigm shift that revolutionized physics in the last century and has enabled a deeper understanding of the physical world, with fundamental and practical implications that have radically changed human life (atomic physics, chemistry, nuclear energy, transistors, and lasers, to name just a few). 

To provide the fundamentals of classical mechanics in both the Lagrangian and Hamiltonian formulations, and to introduce the basic concepts of the new “non-classical” theory of mechanics, i.e., quantum mechanics. This course will offer an initial exploration of the quantum world and provide the basic conceptual and mathematical tools necessary for all quantum physics courses at the L3, M1, and M2 levels.

  - Newtonian Mechanics

  - Mathematical Analysis

  - Linear Algebra and Matrix Calculus

  - Electromagnetism

Recommended prerequisites*:

  - Statistical Physics  

   - Advanced matrix analysis and computation

100% CT

Analytical Mechanics:

• Variational Principle
• Lagrangian Formalism
• Hamiltonian Formalism
• Symmetries and Conservation Laws
• Fishing Hooks

Quantum Mechanics:

• Introduction, Planck's formula and the solution to the blackbody problem, the photoelectric effect, limitations of the classical atomic model, the Compton effect
• Assumptions of the MQ
• Wave-Particle Duality
• Schrödinger Equation
• Indeterminacy Relations
• Elements of Hilbert Spaces and Wave Functions
• Operators
• Switches
• One-Dimensional Problems (Barrier/Well/Harmonic Oscillator)

CM: 31.5 hours

TD: 31.5 hours