Unit PHYSICS OF MANY BODY SYSTEMS
- Course
- Physics
- Study-unit Code
- GP005480
- Curriculum
- In all curricula
- Teacher
- Andrea Orecchini
- Teachers
-
- Andrea Orecchini
- Matteo Rinaldi
- Hours
- 28 ore - Andrea Orecchini
- 28 ore - Matteo Rinaldi
- CFU
- 8
- Course Regulation
- Coorte 2024
- Offered
- 2025/26
- Learning activities
- Caratterizzante
- Area
- Microfisico e della struttura della materia
- Academic discipline
- FIS/04
- Type of study-unit
- Obbligatorio (Required)
- Type of learning activities
- Attività formativa monodisciplinare
- Language of instruction
- Italian
- Contents
- Quantum treatment of independent and correlated many particle systems. Variational methods with perturbative corrections. Paring phenomena and applications.
- Reference texts
- A.L. Fetter, J.D. Walecka “Quantum theory of many particle systems”,
McGraw-Hill, 1971; Dover 2002.
S. Boffi, “Da Heisenberg a Landau, introduzione alla fisica
dei sistemi a molte particelle”,
Bibliopolis, 2004.
A.G. Sitenko, V.K. Tartakovskii, “Lectures on the theory of the nucleus”,
Pergamon (1975).
N.H. March, W.H. Young and S. Sampanthar, “The many-body problem in Quantum
Mechanics”, Cambridge (1967); Dover (1995).
E. Lipparini, “Modern Many-Particle Physics”, WS (2008). - Educational objectives
- Knowledge of elementary techniques to face quantum many-body problems
- Prerequisites
- Knowledge in quantum mechanics, mathematical methods for physics, subatomic physics and solid state physics are essential.
Knowledge of elementary relativistic quantum mechanics can prove useful. - Teaching methods
- Face-to-face lectures
- Other information
- Learning verification modality
- Oral exam, consisting in two or three questions, aimed at establishing the acquisition of technical skills and verifying the level of understanding of the candidate. The exam has a total duration of about one hour.
- Extended program
- Generalities and formalism. The many-body problem. Nuclear Matter (NM) as a study example. The saturation of nuclear forces as a typical problem. Fermions and bosons. One- and two-body operators and density matrices, diagonal and non-diagonal. Computation of matrix elements of N-body operators for independent fermion systems described by Slater determinants. Fermi gas. Density matrices in the Fermi gas model. Statistical correlations. Fermi gas model for nuclei. NM in the Fermi gas model with perturbative corrections.
Introduction to scattering theory. Definition of scattering amplitude and phase shifts for scattering in a central potential. Optical theorem and Breit-Wigner formula. Relationship between phase shift, scattering amplitude and interaction potential in the Born approximation.
Independent particles. Variational methods. Notes of the use of variational methods in quantum mechanics. Examples of elementary applications such as the ground state of the deuteron. Hartree equations, self-compatible potential and perturbative correction. The harmonic oscillator. Hartree-Fock (HF) method and exchange term. HF method for NM: necessity to go beyond the independent particles model for interactions with hard-core.
Bose-Einstein condensates (BEC). Qualitative treatment and introduction to the scattering length.
Brillouin theorem for correlated particles. Perturbative methods for systems of correlated fermions: Bethe-Goldstone (BG) method for two-body correlations. Independent pairs model. Ground state energy of N pair-correlated fermions. Iterative BG series. Healing properties. Green's function for the Fermi sea. Application to nuclei (Brueckner-Bethe-Goldstone theory). Application to the study of NM with attractive potential and hard core: proof of nuclear stability. Notes on the application to finite nuclei.
Screening. Dielectric function and electrical susceptibility. Thomas-Fermi approximation and dielectric function. Effects of electron screening on electron-ion, electron-electron and ion-ion interactions.
Lattice vibrations in metals. Electrostatic interactions and screening of the ionic plasma frequency. Bohm-Staver relation. Dielectric function of a metal: general expression, combination of electronic and ionic screening in a simple model. Limits of validity. Application to the electron-electron interaction in a metallic solid: electron-phonon interaction and conditions for an attractive effective interaction between electron pairs.
Many-fermion systems. Slater determinants and representation in Fock space. Creation and annihilation operators, anticommutation rules, vacuum state.
Representation of operators by creation and annihilation operators. Number operator. Hamiltonian operator. One-body and two-body operators.
Fermi gas of electrons in second quantization. Expression of the Hamiltonian as a function of creation and annihilation operators (a+, a). Introduction of alternative creation and annihilation operators (c+, c) via canonical transformations. Expression of the Hamiltonian in terms of the new operators and discussion of the relevant contributions. Definition of quasi-particles and the new vacuum state. Number operator on quasi-particle states of the non-interacting Fermi gas.
The Bogoliubov-Valatin canonical transformation. The Bogoliubov-Valatin vacuum state (BCS vacuum) and its properties. Cooper pairs. Reduction of a Hamiltonian of interacting fermions to a Hamiltonian of non-interacting quasi-particles via the Bogoliubov-Valatin transformation. Lagrange multiplier method for the definition of the number of particles and BCS approach for the determination of the ground state. The case of non-interacting fermions.
Hamiltonian for interacting fermions. Derivation of the interaction contribution in second quantization. Wick's theorem for the contraction on the BCS vacuum of products of a and a+ operators in the case of electrons: expressions of the contracted products in the Bogoliubov-Valatin transformation. Application to the interacting Hamiltonian. BCS approach and reduction to a Hamiltonian of independent quasi-particles. Pair interaction potentials and gap. Gap equation: normal solutions vs. superconducting solutions. Overscreening and negative interaction potential for electrons. Population factors for electronic and quasi-particle states: gap formation for excited quasi-particle states. Origin of superconductivity. Spatial extension of the wave function of a Cooper pair, correlation between pairs, comparison with the Bose-Einstein condensate and superfluidity.
Phenomenology of superconductivity: critical temperature, resistivity, thermal conductivity, Meissner effect, critical magnetic field, specific heat, electrical conductivity for metallic junctions and for alternating currents, example of MgB2. - Obiettivi Agenda 2030 per lo sviluppo sostenibile