Aim of the course (2nd semester, 2nd year) is to provide the students with theoretical basis of modern physics in the two fields of quantum mechanics and statistical physics, to be subsequently exploited to study the physical structure of matter and solid-state physics, with particular emphasis on applications to the area of ICTs.

The course is divided into two sections: in the first one, fundamental aspects of quantum physics are treated in depth, in both the Schroedinger and Heisenberg formalisms. In the second half, the students will be taught the very basics of statistical mechanics and of physics of complex systems.

The knowledge transmitted by this course provides the background necessary to understand subsequent 3rd year courses of advanced Physics; in addition, the concepts introduced here will serve as a basis for the understanding of all physics-oriented courses in MSc programs where the students of Physical Engineering can enroll without academic debts after graduation.

The transmitted abilities mostly concern the formalization of elementary-to-intermediate level problems of quantum and statistical mechanics, and the related problem-solving activities.

The main objective of the Physics of Complex System part is to offer an introduction to probability, thermodynamics, statistical mechanics, and Markov chains. Numerous examples are used to illustrate a wide variety of both physical and non-physical phenomena (e.g. through examples drawn from biology, computer science, etc.).

The knowledge transmitted by the course to students involves:
• the general principles of quantum theory and related formalisms;
the quantum behaviour of atomic and sub-atomic systems.
• the phenomena connected to the statistical mechanics (classical and quantum) of many-body systems
• the formalism of statistical mechanics for treating complex physical systems

The transmitted abilities include:
• applying the quantum theory to fully describe simple physical systems in one to many dimensions, and solving related problems;
• applying classical and quantum statistics to many-body systems;
• applying a correct formalism to the study of complex 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, wave optics, elements of structure of matter) and the basic mathematics (Calculus I, Calculus II, Geometry).

1) The Lagrangian and Hamiltonian approaches to classical mechanics. (0.5 cr)
2) The postulates of Quantum Theory. Time evolution of quantum systems. The Eherenfest theorem. The canonical-quantization scheme. (0.5 cr.)
3) The Hilbert space of quantum states. Scalar product, completeness relation. Properties of Hermitian operators and eigenvalue equations. (0.7 cr.)
4) Raising and lowering operators, solution of the harmonic-oscillator problem. The Heisenberg uncertainty relation. Coherent states and semiclassical picture of quantum systems. (0.6 cr.
5) Spectrum and eigenstates of the angular momentum. The two-body problem. The hydrogen atom. (1.0 cr.)
6) Dirac's formulation of quantum states and operators. The Schroedinger and Heisenberg representations of quantum Mechanics. (0.6 cr)
7) The spin operator and spin states. Addition of angular momenta. Time-independent perturbation theory. (0.7 cr)
8) Charged particles in the electromagnetic field. The Zeeman effect for Hydrogen atoms (weak and strong-field limit) (0.8 cr)
9) Symmetric and antisymmetric states of identical particles. Bosons, fermions and symmetrization principle. The exclusion principle. The Helium atom. (0.6 cr)

• Introduction to Probability and Statistics (1.0 cr.)
o A reminder on probability
o Ensembles and probability
o Conditional probability and Bayes Theorem
o Application to Bayesian Theorem: data and inference.
o Probabilistic approach to classification.
o Application in Bioinformatics: modeling DNA motives and position weight matrices.
o General properties of probability distributions
o Law of Large Numbers and the Central Limit Theorem (without derivation)
• Diffusion (0.5 cr)
o Diffusion and Heath equation
o Random Walks and relation to Diffusion equation
o Diffusion in presence of external forces
o Einstein’s Relation
• Ensembles (2.5 cr.)
o Lagrange and Hamilton formulation (quick review)
o Systems and Ensembles
o Liouville’s Theorem and conservation of probability
o Poincaré Recurrence Theorem
o Microcanonical Ensemble
o Canonical Ensemble
o Legendre Transforms
o Grand-Canonical Ensemble and NPT ensemble
o Virial
• Interacting Models and Phase Transitions (0.75 cr.)
o Ising model and lattice-gas
o Mean-field theory by: saddle point, combinatorial approach, and variational methods
o Spontaneous symmetry breaking
o One dimensional ferromagnets: transfer matrix theory
• Markov Chains (0.25 cr.)
o Introduction to Discrete Markov Processes
o Molecular Dynamics and Monte Carlo Markov Chains
o PageRank: a random walk on Web Pages

Class exercises include simple problem-solving activities, with strict connections to theoretical lectures. In some cases, scientific calculators (students' personal property) may be required.

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.

- Franz Schwabl, Quantum Mechanics, Springer-Verlag 2007
- David J. Griffiths, Introduction to Quantum Mechanics, Addison-Wesley 2005

- Frederick Reif, Fundamentals of Statistical and Thermal Physics, McGraw-Hill
- C. Kittel, "Elementary statistical physics". Courier Corporation. 2004; Part 1: Chapter 1,2,3;
- Kerson Huang, "Statistical Mechanics", Wiley 1987;
- Richard C Tolman, "The Principles of Statistical Mechanics" Courier Corporation, 1938.

For the quantum mechanics part, 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 used to quantize classical systems, 2) the significant properties and effects that characterize physical systems as a consequence of the quantization process and 3) the mathematical tools used to develop quantum mechanics. To this end one of the three question is focused on proving/discussing some theorem or general property characterizing quantum systems and the mathematical formalism of quantum mechanics. The other questions are devoted to discussing the quantization of physical systems for some specific case and/or to determine quantum effects and physical properties of interest emerging from this process.

With respect to the Complex Systems part, the exam consists in a written test (one exercise on non-interacting system + 2 theoretical questions).