Analytical and Quantum Mechanics

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

In the first part of the course, we cover classical mechanics, starting with the principle of least action and arriving at two new formulations: Lagrangian formalism and 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 foreshadows those of quantum mechanics.

In the second part of the EU, starting from an examination of the experimental limits of classical mechanics, a new theory of mechanics is introduced: quantum mechanics. This theory is conceptually completely different from previous classical theories, based on a description of physical phenomena in terms of probabilities and therefore no longer deterministic. This is a radical paradigm shift that revolutionized physics in the last century and led to a deeper understanding of physical nature, with fundamental and practical implications that have radically changed the lives of humanity (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 provide an initial exploration of the quantum world and will provide the basic conceptual and mathematical tools necessary for all quantum physics courses in the third year of undergraduate studies and the first and second years of graduate studies.

  - Newtonian mechanics

  - Mathematical Analysis

  - Linear algebra and matrix calculations

  - Electromagnetism

Recommended prerequisites:

  - Statistical Physics  

   - Advanced matrix analysis and 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 to the black body problem, photoelectric effect, limitations of the classical atomic model, Compton effect
• Assumptions of MQ
• Wave-particle duality
• Schrödinger equation
• Relationships of indeterminacy
• Elements on Hilbert spaces and wave functions
• Operators
• Switches
• One-dimensional problems (barrier/well/harmonic oscillator)

CM: 31.5 hours

Tutorial: 31.5 hours