Università degli Studi di Perugia

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Study-unit Code
In all curricula
Alessandro Paciaroni
  • Alessandro Paciaroni - Didattica Ufficiale
  • 56 ore - Didattica Ufficiale - Alessandro Paciaroni
Course Regulation
Coorte 2018
Learning activities
Microfisico e della struttura della materia
Type of study-unit
Obbligatorio (Required)
Type of learning activities
Attività formativa monodisciplinare
Language of instruction
Many body systems with Coulom interaction. Perturbative techniques. Second quantization and condensed matter physics. Electron gas. Density functional and applications. Introduction to magnetism in condensed matter and link to the electron states. Lattice vibrations and their properties.
Reference texts
General, medium level: Solid State Physics, N. W. Ashcroft e N. D. Mermin
Specific: Condensed Matter in a Nutshell, G. D. Mahan
Advanced: Many-Particle Physics, G. D. Mahan
Tecnic: Quantum Theory of Solids, C. Kittel
Educational objectives
The student should obtain the basic knowledge on the many electron systems and the relationship between the electron states and the main phenomena in condensed matter physics. Finally the student should acquire the basis of the modern calculation techniques to calculate the properties of matter by means of the Density Functional theory.
Good knowledge of the basis of Quantum Mechanics and elements of Statistical Physics.
Teaching methods
Other information
Learning verification modality
Oral exam. The test, lasting 45-60 minutes, is devoted to the discussion with the student to define the ability in solving problems of condensed matter physics.
Extended program
Basis of the many body problem in physics. Generalities of simple solution methods.
Condensed matter as a system made up of N (point like) nuclei and N Z electrons with electrostatic interaction only. Hamiltonian for condensed matter. Non-existence of exact solutions. Approximation with massive nuclei.
Recall the role of the symmetry of the many electron wavefunction. Independent particle approximation in the case of an atom and approximation limits. Discussion and criticism about the atomic shell model
Slater determinants and some properties. The Hartee-Fock approximation as a 1st order of an perturbative procedure. The atomic case. Matrix elements of one and two body operator. Use of a series of Slater determinants in the case of an atom and Configuration Interaction method. Short description of the method and results.
Introduction to second quantization. Fock space and relationship to the Slater determinants representation in the case of fermions. Operators of creation and annihilation for fermions and bosons. Commutation rules. Quantum statistics and commutation rules.
Field operators. Commutarion rules. Density operator and field operators. Construction of the hamiltonian in the second quantization form.
Equation of motion of the field operator using a general hamiltonian. Auto-consistent Hartree-Fock equation as an approximation for the equation of motion of the field operator.
Analytic Hartree-Fock approximation for the homogeneous electron gas. Electron gas energy as a function of density in the Hartree-Fock approximation. Intrinsic limit of the approximation: zero density of states at the Fermi level.
Perturbative approach to the electron gas. Identity of the 1st order approximation and the Hartree-Fock approximation. Divergence of the higher order terms. Impossibility of a perturbative procedure in systems with Coulomb interaction. Possibility of pertubative approach in the case of short-range forces, e.g. nuclear matter.
Introduction to the dielectric response. The Thomas-Fermi approximation. Dielectric screening of Thomas-Fermi. The auto-consistent field approximation. Relationship between the dielectric response and the dynamic structure factor. Collective response (plasmon) and the single particle response. Screening.
The density functional method. The Hohenberg e Kohn theorem and consequences. Elementary discussion of the variational principle. Derivation of the Kohn and Sham equation. Ground state energy and density as the only result of the method derived form the theory. Potential of the method. Application of the method to elements and typical results.
Introduction to magnetism in the mcondensed matter. Basic information. The electron spin and the anomalous gyromagnetic ratio and magnetism, the Einstein de Haas experiment. Wannier orbitals and derivation of the Hubbard hamiltonian in second quantization. Local approximation of the hamiltonian and Hartree-Fock approximation. Partial description of the ferromagneti behaviour of the transition 3d metals. Interpretation of the non-integer magnetic moments and trend in the case of ferromagnetic alloys. Basic notion on the antiferromagnetism. Remarks on the magnons for ferromagnetic and antiferromagnetic systems. The bosonic nature of magnons. The Heisenberg hamiltonian as derived from the Hybbard hamiltonian and limits.
The density functional in the case of magnetic systems. Local spin density approximation and comparison of the results with experimental findings.
Finite nuclear mass and its effect. Validity limit and Migdal parameter. Harmonic approximation. Classic solution for the normal modes of a N-particle system. Quantization of lattice vibrations. Harmonic hamiltonian in the second quantization. The bosonic nature of the lattice vibrations as consequence of the commutation rules.
Application to crystals, existence of acoustic modes. Introduction to the thermal Green’s function and equation of motion. Green’s function and dynamic structure factor. Anharmonicity and consequences. Mention to the electron-phonon interaction and Fröhlich hamiltonian.
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