The course (first semester, first year) essentially represents the continuation of the BS-program course ``Quantum Physics and Physics of complex systems" and aims to introduce advanced topics in quantum mechanics, the basis of quantum field theory, and the statistical-physics approach to quantum many-body systems. These will be subsequently exploited to study the condensed-matter and solid-state physics.
The course is essentially divided into three sections. The first part is focused on discussing the development of the canonical-ensemble description within the Fowler-Darwin method. This approach, particularly useful for quantum many-body systems, is then extended to the grand-canonical ensemble. In addition to study quantum systems typically considered in statistical-physics program, several applications concern more recent models and/or effects such as quantum phase transitions, ultracold atoms, and Bose condensates realized and investigated in the last two decades.
The second part, is devoted to discuss time-dependent Hamiltonian systems, the interaction picture, the use of coherent-state approach in various applications, and the adiabatic approximation. The third part, in addition to a short introduction to Special Relativity, discusses the fundamental aspects of quantum field theory, and, after introducing both the Lagrangian and Hamiltonian formulation of classical field theory, presents the second quantization of both relativistic and non relativistic field theories. Various models are considered which, together with the electromagnetic field, the Klein Gordon and Dirac models, and the second-quantized nonlinear Schroedinger equation, will include the derivation of both bosonic and fermionic lattice models and the discussion of their properties.
The knowledge transmitted by this course provides the background necessary to better approach subsequent first- and second-year courses of advanced Physics (such as Condensed matter theory, Statistical physics and Field theory and critical phenomena).
The knowledge transmitted by the course to students involves:
• the formalisms of classical and quantum field theory and of the second quantization
• the application to various models and the derivation of their significant properties.
• the approach of statistical mechanics to quantum many-body systems
• the application of the statistical-mechanics methods to several systems and the derivation
of their thermodynamcal properties
The transmitted abilities include:
• understanding the quantum-field theory in providing an effective description of complex systems of identical particles and solving related problems;
• applying quantum statistical mechanics to many-body systems;
• applying a correct quantum-field formalism to the study of quantum many-body systems;
• understanding how general is the formalism of statistical mechanics and how broad is the range of applicability of the techniques introduced in the course.
The students must know all the subjects of elementary Physics (mechanics, thermodynamics, electromagnetism, elements of structure of matter) and of elementary Quantum Physics and classical Statistical Mechanics.
Statistical mechanics of quantum many-body systems. Canonical-ensemble description within the Fowler-Darwin scheme. Extension to the grand-canonical ensemble [6 h].
Paramagnetic materials, Brillouin function and magnetization. Photon gas, black-body radiation and Planck formula. The Debye phonon model: heat capacity and other low-temperature properties [8 h].
Ideal Bose-Einstein gas. The mechanism of the BE condensation. Derivation of the critical properties at the transition point. Fermi-Dirac non-interacting gas: Fermi sphere, state equation and pressure [6 h].
Vortex field theory, derivation of the point-like vortex model. Mean-field approach to the Kosterlitz-Thouless transition [4 h].
The Lagrange formulation of classical field theory, least-action principle and Lagrange equations. The Hamilton formulation of classical field theory, and the quantization of classical fields in the canonical scheme [4 h].
Second quantization of nonrelativistic field theories: microscopic mode operators, commutator and anti-commutator algebra, Fock states. Properties of bosonic and fermionic fields; Bose-Einstein and Fermi-Dirac statistics [6 h].
Quantum field theory of the electromagnetic field and of the Klein-Gordon model. Fermionic field and Dirac equation [8 h].
Weakly interactiong Bose gases: Bogoliubov approach, low-energy spectrum, phonon excitations, vacuum fluctuations. Derivation of the Gross-Pitaevskii equation [4 h].
Lattice quantum models: derivation of the Bose-Hubbard and Fermi-Hubbard models. Superfluid-Mott transition: properties of the superfluid and insulating phases.
Perturbative approach and mean-field scheme [8 h].
Time-dependent systems, adiabatic approximation, Berry phase. Coherent-state approach to the Dicke model for the matter-radiation interaction. Fermi golden rule [6h].
Gauge transformations and quantum Mechanics. The case of a uniform magnetic field: Spectrum, Landau gauge, symmetric gauge, applications of symmetries and related wavefunctions [8 h].
Elements of Relativistic Quantum Mechanics. Introduction to Special Relativity. Relativistic aspects of Klein-Gordon and Dirac equations. The non-relativistic limit of the Dirac equation. Spin-orbit interaction. The massless limit of the Dirac equation. Effects of uniform magnetic field and scattering problems in relativistic quantum mechanics [12 h].
The teaching approach of this course is the traditional one based on frontal lectures. Each part of the program and the relevant calculations are discussed, step by step, and showed on the blackboard.
As shown in the syllabus, the course is essentially divided in three sections: quantum Statistical Physics, introduction to Field Theory and many body systems, and Elements of Relativistic Quantum Mechanics. In each section the introduction of new formalism, methods and new theoretical tools is always followed by their application to physical systems and models. The aim is to show how the theory allows one to achieve an effective modeling of quantum many-body systems and the derivation or the interpretation of physical properties and effects. Exercise are proposed in the form of simple applications of the theory that each student must solve on his own. Solutions are provided in the subsequent lectures.
Weekly meetings will be scheduled to give the students participting in this course the possibility to ask for questions, express doubts and clarify unclear points.
The PDF file containing the notes relevant to the course lectures are available in the web page of this course accessible through the Teaching Portal of Politecnico.
- W. Greiner, J. Reinhardt, Field Quantization, Springer Berlin, 1993
- W. Greiner, Quantum Mechanics, Special Chapters, Springer Berlin, 1998
- Kerson Huang, “Statistical Mechanics”, Wiley 1987;
- Frederick Reif, Fundamentals of Statistical and Thermal Physics, McGraw-Hill
- Richard C Tolman, “The Principles of Statistical Mechanics” Courier Corporation, 1938.
Modalità di esame: prova orale obbligatoria;
Exam: compulsory oral exam;
The exam is oral and consists of three questions concerning different topics of the course. Each question corresponds to a score of 10 out of 30 marks. The goal, in general, is to test the knowledge of the course program. These questions aim to evaluate the understanding of students concerning 1) the methodology and the mathematical tools used to quantize classical fields and to develop the ensemble theory for quantum systems, and 2) the significant physical properties and effects characterizing many-body quantum systems described within the statistical physics and the second-quantization formalism. To this end one of the three question is focused on discussing the methods and the mathematical formalism presented in the course, while the other questions are devoted to discuss quantum effects, their derivation, and physical properties of interest emerging from the applications of the theory to quantum-field models and many-body systems.