00691 - Quantum Mechanics

Learning outcomes

Al termine dell'insegnamento lo studente conosce i fondamenti della meccanica quantistica per lo studio dello stato condensato. È in grado di trattare problemi tramite l'equazione di Schroedinger per un potenziale periodico, i suoi metodi di risoluzione e le sue principali applicazioni in fase condensata in particolare la struttura a bande. Lo studente applicherà in esercitazioni e in laboratorio i metodi per lo studio delle propertia' elettroniche dei solidi.

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, etc.); 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.

Ehrenfest theorem.

Kronig-Penney model. Tight-binding model.

4. 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.

 

5. Spin

Electron spin and Pauli principle. Stern-Gerlach experiment. Two-level problem. Rabi formula.

6. Final complements

Mention to many-electrons wave functions.

Mention to perturbation theory in Quantum Mechanics. Stark effect. Zeeman effect.

 

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(*) 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 and exercises worked out together.

Assessment methods

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

As far as the topics covered by the module "Quantum Mechanics" 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.

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