OVERVIEW

This course is the natural continuation of the Quantum Physics course from the third year of the undergraduate degree. The applications of quantum mechanics to systems of physical interest require the development of techniques that allow the handling of many degrees of freedom. This is necessary both for the description of non-relativistic systems, such as statistical quantum systems and solid-state systems, and in a relativistic context. The course focuses on the first of these two aspects. Therefore, techniques for the quantum description of many-body systems will be introduced from a theoretical point of view, focusing on their application to physical systems in a non-relativistic context, such as superfluidity and superconductivity.

AIMS AND CONTENT

LEARNING OUTCOMES

To provide the student with the foundations of many-body quantum mechanics as treated within the framework of second quantization, and to familiarize them with the applications of this method to some of the main physical phenomena in modern physics.

AIMS AND LEARNING OUTCOMES

•  Learn the basic theoretical techniques of second quantization for non-relativistic many-body quantum systems.
•  Introduce the concept of canonical transformations and the consequent concept of quasiparticles.
•  Familiarize with the concept of phase transitions and spontaneous symmetry breaking.
•  Introduce mean-field approximation techniques for the description of interacting quantum systems.
•  Learn to describe relevant physical phenomena using second quantization techniques.

PREREQUISITES

• Classical Physics: fundamentals of analytical mechanics, statistical mechanics, and classical electromagnetism.
• Non-relativistic Quantum Mechanics: formalism, perturbation theory.

TEACHING METHODS

Lectures delivered at the blackboard. Approximately 30% of the teaching hours are dedicated to applying the theoretical concepts through exercises.

SYLLABUS/CONTENT

1. Basics of quantum mechanics
2. Second quantization - bosons
   • Quantum systems of identical particles
   • The occupation number representation
     • Single particle operators
   • The fock space and the creation and annihilation operators
     • The dilute non-interacting bosons gas
     • Many-particles operators
   • The Wick's theorem
     • The Wick's theorem for thermal states
   • Linear canonical transformations
3. Theory of non-interacting and interacting bosons
   • Non-interacting bosons: the Bose-Einstein condensate
   • Interacting bosons at zero temperature
     • The ideal Bose gas
     • Bogoliubov's approximation
     • The Bogoliubov transformation
     • The Landau criterion for superfluidity
     • Ground State and low-lying excitations
     • Ground state correlation functions
     • Spontaneous symmetry breaking
     • Depletion of the condensate
4. Elastic excitations: phonons and electro-magnetic field
   • Phonons
     • Lattices in one dimensions
     • Phonons in three dimensions
     • Polarization states and helicity
     • The infinite volume limit of phonon fields
     • From field to phonons
   • Quantization of the lectromagnetic field
   • Interaction of the electromagnetic field with matter
     • First order perturbations: single photon events
5. Second quantization -  fermions
   • The Fock's space and the canonical anticommutation relations algebra
     • Many particles operators
     • The Wick's theorem for fermions
   • Canonical transformations for fermions
     • Bogoliubov transformations
6. Theory of non-interacting and interacting fermions
   • The free fermion gas: particles and holes
   • Turning on interactions
     • The Coulomb interactions between electrons
     • The electron-phonon interaction
     • Combining Coulomb and electron-phonon interactions
7. Mean field theory
   • The art of mean field
   • The Hrtree-Fock approximation
   • Broken symmetries
8. Microscopic theory of superconductivity
   • the Coooper instability
     • The Cooper's argument
   • The BCS reduced Hamiltonian
   • The BCS mean field theory
     • The BCS groundstate
     • Quasi-particles excitations
     • The gap equation
     • The specific heat

RECOMMENDED READING/BIBLIOGRAPHY

• L. Landau, Lifsitz, Vol. 2, Field theory
• L. Landau, Lifsitz, Vol.  3, Quantum mechanics
• H. Bruus, K. Flensbeg, Many-body quantum theory in condensed matter physics
• P. Phillips, Advanced solid state physics
• A. Amoretti, Lecture notes

TEACHERS AND EXAM BOARD

Exam Board

ANDREA AMORETTI (President)

GIOVANNI RIDOLFI

NICOLA MAGGIORE (Substitute)

LESSONS

EXAMS

EXAM DESCRIPTION

The exam consists of a written test and an oral examination. The written test will include two exercises to be completed within a maximum of three hours. During the written test, students are allowed to consult texts or notes.

The oral examination (about 20/30 minutes) covers the theoretical topics discussed in the lectures and is based on the written test.

ASSESSMENT METHODS

The written test aims to assess the student's practical skills. The two exercises proposed are inspired by those carried out during the exercise sessions in class.

During the oral test, which draws on the written test, the goal is to verify the student's understanding of the fundamental concepts underlying the study of many-body quantum systems.

Exam schedule

FURTHER INFORMATION

Students who have valid certification of physical or learning disabilities on file with the University and who wish to discuss possible accommodations or other circumstances regarding lectures, coursework and exams, should speak both with the instructor and with Professor Sergio Di Domizio (sergio.didomizio@unige.it), the Department’s disability liaison.