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 of 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 Heisenberg’s and Dirac’s 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 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 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

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).

The abilities a student must have include: applying the laws of mechanics and electromagnetism; applying differential and integral calculus in one and more than one dimensions; applying the basic concepts of linear algebra (matrices; vectors; linear operators).

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 cr.)
- random variables, probability distributions and their characteristics
- main distributions
- central limit theorem
- introduction to stochastic processes: from the Brownian motion to Google Page-rank

Classical statistical mechanics (2,5 cr.)
- diffusion and random walk
- phase space, analytical mechanics and Liouville Theorem
- micro- and macrostates, microcanonical ensemble, a-priori equiprobability principle
- thermal interaction and canonical ensemble
- relationships between thermodynamics and statistical mechanics
- the grand canonical ensemble
- paramagnetism
- non-interacting systems and the ideal gas
- example of an interacting system and phase transition

Quantum statistical mechanics (1 cr.)
- the Fermi- Dirac distribution
- the electron gas
- the Bose-Einstein's distribution
- the black body and Bose condensation
- specific heat of solids.

Modelling of complex systems: classical, quantum and biological applications (1,5 cr.)

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

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 discuss the quantization of physical systems for some specific case and/or to determine quantum effects and physical properties of interest emerging from this process.