28891 - Quantum Physics

Learning outcomes

At the end of the course the student: - knows the limits of classical physics and the experiments that point to them; - knows the general structure of wave mechanics and operatorial methods in quantum mechanics; - knows the Schroedinger equation in 3 dimensions and the hydrogen atom solution; - is able to use variational methods and time-independent perturbation theory to solve some basic problems.

Course contents

Crisis of Classical Mechanics and beginning of Quantum Mechanics - Wave and particle theories of light - Photoelectric Effect, Compton Effect. - Bohr quantization, Bohr atom. - Interference experiments: wave-particle duality. - De Broglie hypotesis and wave function. - Schrödinger Equation for the time evolution.
Mathematical Formalism - Hilbert spaces, linear operators, eigenstates and eigenvalues. - Commutativity. - Complete sets of eigenstates. - Eigenfunction expansion. - Square summable functional spaces. - Orthogonal Polinomials. - Series and Fourier transforms. - Distribution Theory.
General Principles of Quantum Mechanics - Postulates. - Eigenvalue problem for the Hamiltonian. - Measure theory and development in eigenstates. - Mean Values. Ehrernfest theorem. - Position-momentum commutation rules. - Heisenberg Indetermination relations. - Fourier transform and momentum representation. 1D Problems - Parity. - Harmonic Oscillator: polynomial and à la Dirac solutions. - Potential wells. - Potential barriers, tunnel effect. - Delta Potential.
Approximate Methods - Time independent perturbation theory, non-degenerate case: first and second orders; degenerate case - Time dependent perturbation theory. - Variational methods.
Symmetries - Symmetries, infinitesimal transformations and their generators. - Translations and momentum. Rotations and angular momentum. - Parity.
Angular momenta - 3D spatial rotations and angular momentum in QM, its eigenvalues and eigenvectors. - Parity and angular momentum. - Complete sets of commuting operators - Half-integer eigenvalues and electron spin - Pauli matrices algebra - Combination of angular momenta.
Central symmetry problems - Central potentials. - Azimutal and accidental degenrations. - Coulomb problem (hydrogen atom without spin) - Links with Bohr atom - Stark-LoSurdo effect - 3D isotropic harmonic oscillator.
Identical particles and statistics - Bosons and fermions - Pauli Exclusion principle.
Interaction of an electron with the electromagnetic field - Normal and anomalous Zeeman effect - Supplementary terms in the Hydrogen atom hamiltonian. - Fine and hyperfine structures of the hydrogen atom.

Readings/Bibliography

Notes of the course are available onAMS Campus.
For the written exam preparation, the student is strongly encouraged to see the exam problems of the past years, also present onAMS Campus
Other books suitable for deepening some specific arguments:
* Cohen-Tannoudji C., Diu B., Laloe F. - Mécanique Quantique - Dounod (in French) - Quantum mechanics, - Wiley Ed. (in English)
* Galindo Pascual, Mecanica Cuantica (in Spanish) Eudema (Madrid)
* L.D. Landau, E.M. Lifshitz - Theoretical Physics vol.3: Quantum Mechanics, Non-relativistic Theory - MIR Eds.
* J. Chahoud, Meccanica Quantistica (in Italian) - Ed. Pitagora

Teaching methods

* Lectures at the blackboard and with projection of transparencies, with the participation of students.
* Exercises solved together with the students with comments.
* Further exercises proposed as homework with correction in the class. Although not compulsory, they may reveal crucial for the preparation of the written exam.

Assessment methods

Written Exam:
* time: 3 hours.
* from one to four problems for a total of about 8 questions.
* sheets for scketchy calculations etc... will be provided, not to be given back at the end. Only the offical copy, possibly ordered and well commented will be given. * any text, notes etc... can be used without restriction.
* the result of the proof has 14 months validity.
Oral Exam:
* can be done only in presence of a positively evaluated written exam
* 3 questions chosen by the teachert
* the final score is (approximatively) an average between the written and oral results.

Teaching tools

The "Avvisi" section of the teacher website will be used for commuincations.
The learning material will be made available on the AMS Campus website.

Office hours

See the website of Francesco Ravanini

See the website of Sandro Turrini