B5654 - MATERIA CONDENSATA

Learning outcomes

At the end of the course, the student is familiar with the fundamentals of quantum mechanics for the study of condensed matter. They are capable of addressing problems using Schrödinger's equation for a periodic potential, its solving methods, and its main applications in condensed matter, particularly band structure. The student will apply methods in exercises and laboratory sessions to study the electronic properties of solids.

Course contents

1. Physical and historical introduction

100 years of Quantum Mechanics. From Planck's black body radiation law to modern technologies.

Photoelectric and Compton effects. Particle behaviour of light.

Material waves and De Broglie's theory. Wave-particle duality; Davisson-Germer experiment.

Bohr-Sommerfeld atomic model; Franck-Hertz experiment. Correspondence principle.

2. Mathematical formalism

Hilbert space, wave vectors and their representation (coordinates, Dirac, momenta); linear operators as description of physical observables and of states transformations (*); change of basis (*); time evolution.

Quantum superposition of states. Probabilistic interpretation of the wave function and measurement processes in Quantum Mechanics. Expectation values and matrix elements (*). Commutation rules (*) and uncertainty relations (Heisenberg).

3. Energy eigenvalues and eigenfunctions properties

Schroedinger equation. Hermitian operators. Energy eigenvalue problem and time-independent Schroedinger equation (*).

Second-order differential equations and their solutions; role of boundary conditions.

One-dimensional problems: particle in a potential box, harmonic oscillator (*), potential barrier. Tunnel effect.

4. Periodic potentials

Bloch theorem and notion of wavevector in reciprocal space. First Brillouin zone and bands formation. Tight-binding approximation. Kronig-Penney model.

5. Atoms and their orbitals

Hydrogen and hydrogen-like atoms. Hamiltonian and wave function separability for the H atom. Angular part and spherical harmonics; Legendre equation and associated functions.

Quantum properties of the angular momentum; commutatation between L and its components; solutions of the radial wave equation; total wave function; quantum numbers nlm and orbitals (*).

6. Spin

Electron spin and Pauli principle. Stern-Gerlach experiment.

7. Final complements

Mention to many-electrons wave functions.

Mention to perturbation theory in Quantum Mechanics and examples.

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For non-attending students, the list of topics covered up to that point will be updated weekly on the course's VIRTUALE page, along with specific references in the bibliography and/or lecture notes where their treatment can be found.

(*) Contact points with the other joint course "Atoms, Molecules and Symmetries"

Readings/Bibliography

“Molecular Quantum Mechanics”, fifth edition, P. Atkins, R. Friedman (Chapters Intro, 1, 2, 3, 6).

To complete and deepen the exposition of the first book (according to student's needs with possible suggestions by the teacher):

“Quantum Mechanics”, B. H. Bransden and C. J. Joachain

“Quantum Physics”, S. Gasiorowicz

“Quantum Mechanics – an introduction”, W. Greiner

“Introduction to the quantum theory”, D. Park

D. Tong’s Lectures on Quantum Mechanics (web University of Cambridge)

Notes and other studying support during the course.

Teaching methods

Front lectures for theory; exercises worked out together to be re-visited in a critical way during personal study or in some workgroups.

Assessment methods

The exam for the integrated course in Quantum Theory of Matter will focus on the topics covered by the modules "Condensed Matter" as well as "Atoms, molecules and symmetries".

As far as the topics covered by the module "Condensed Matter" are concerned, the exam will be composed by a written stage, in which the students will be asked to solve some exercises on the operatorial formalism, eigenvalue problems in most simple or significant cases, and an oral stage in which the written part will be discussed, and the theoretical or applied skillnesses in quantum Physics of single particles subject to various interaction potentials will be further evaluated.

There is no threshold to surpass in the written stage, all students will have to go through also the oral one.

The final mark of the integrated course will be computed as the artimetic mean of the marks collected in the two modules. Passing of the integrated course will be acknowledged only if the student reaches at least 18/30 in each of the modules.

 

Students with Specific Learning Disabilities (SLD) or temporary/permanent disabilities are advised to contact the University Office responsible in a timely manner (https://site.unibo.it/studenti-con-disabilita-e-dsa/en). The office will be responsible for proposing any necessary accommodations to the students concerned. These accommodations must be submitted to the instructor for approval at least 15 days in advance, and will be evaluated in light of the learning objectives of the course.

Teaching tools

1) Personal computer, blackboard, projector, openboard, lecture notes.

2) Exercises worked out together.

3) The teaching documents will be delivered through the VIRTUALE platform.

Office hours

See the website of Cristian Degli Esposti Boschi