The course consists of two modules ADVANCED QUANTUM MECHANICS The course aims to discuss advanced concepts of quantum mechanics and several applications relevant for the future study of condensed matter physics, many-body theory and statistical mechanics. ADVANCED NUMERICAL METHODS Introduction to numerical methods applied to molecular dynamics simulations. The course covers the theoretical foundations of molecular dynamics and Monte Carlo simulations, starting from fundamental concepts from Newtonian dynamics, statistical mechanics, and variational principles. The course includes practical sessions where students implementing and applying the algorithms.

ADVANCED QUANTUM MECHANICS The student will have to manage advanced concepts like: quantum problems for identical particles, the basics of the path-integral formulation of quantum mechanics, the basics of second quantization. ADVANCED NUMERICAL METHODS The student will be able to understand the basic concepts of statistical mechanics, and will be capable of executing molecular dynamics simulations in different ensembles.

ADVANCED QUANTUM MECHANICS The knowledge of basic concepts and tools of quantum mechanics are needed (see the topics of the ”Introduction to Quantum Mechanics” module). ADVANCED NUMERICAL METHODS Basic knowledge of the Python programming language. Basic concepts in mechanics (equations of motion, Poisson brackets) and calculus.

ADVANCED QUANTUM MECHANICS (A. Rosa) Definition of the Propagator and Formulation of Quantum Mechanics Using Path Integrals 1.Time-Dependent Perturbation Theories 2.Identical Particles in Quantum Mechanics 3.Symmetries in Quantum Mechanics 4.The density matrix operator 5.Fermi and Bose distributions 6.Introduction to second quantization ADVANCED NUMERICAL METHODS (G.Bussi and C. Micheletti) 1.Basic concepts of Newtonian dynamics and Statistical Mechanics: energy conservation, time reversibility and phase-space incompressibility, Liouville Theorem, Ergodicity. Derivation of equilibrium statistical mechanics from variational principles (definition of entropy, ideal gas, microcanonical, canonical and grandcanonical statistical ensembles, law of mass acrtion) [10h] 2.Integration schemes for molecular dynamics: Verlet, Trotter splitting, Velocity Verlet. Dependence of the results on the time step. [10h] 3.Sampling the canonical ensemble with Monte Carlo: Metropolis-Hastings rule, balance and detailed balance, hybrid Monte Carlo. [10h] 4.Sampling the canonical ensemble with molecular dynamics: velocity rescaling, Berendsen thermostat, Andersen thermostat, Langevin dynamics, stochastic velocity rescaling. Stochastic equations: Itoh rule, Fokker-Planck equation. [10h]

ADVANCED QUANTUM MECHANICS Frontal theory lectures and classes of applications and exercises ADVANCED NUMERICAL METHODS 30h are devoted to theory classes. 10h are devoted to the correction of computational exercises done independently by the student at home.

ADVANCED QUANTUM MECHANICS Dirac, P. A. M.; The Principles of Quantum Mechanics (Oxford University Press) Merzbacher, E.; Quantum Mechanics (J. Wiley & Sons, Inc.) Messiah, A.; Quantum Mechanics (Dover) Picasso, L. E.; Lectures in Quantum Mechanics (Springer) d’Emilio, E., Picasso, L. E.; Problems in Quantum Mechanics: with Solutions (Springer) Sakurai, J. J.; Modern Quantum Mechanics (Addison-Wesley) ADVANCED NUMERICAL METHODS Kerson Huang; Statistical mechanics Julia Yeomans; Statistical mechanics of Phase transitions Frenkel – Smith; Understanding Molecular Simulation: From Algorithms to Applications Tuckerman; Statistical Mechanics: Theory and Molecular Simulation

Dispense; Esercizi risolti; Video lezioni tratte da anni precedenti;

Modalità di esame: Prova scritta (in aula); Prova orale obbligatoria;

Exam: Written test; Compulsory oral exam;

... ADVANCED QUANTUM MECHANICS The exam is written only, the student will be asked to solve exercises. The maximum score of the exam is 30/30, as per the Italian system. ADVANCED NUMERICAL METHODS Oral exam.

Gli studenti e le studentesse con disabilità o con Disturbi Specifici di Apprendimento (DSA), oltre alla segnalazione tramite procedura informatizzata, sono invitati a comunicare anche direttamente al/la docente titolare dell'insegnamento, con un preavviso non inferiore ad una settimana dall'avvio della sessione d'esame, gli strumenti compensativi concordati con l'Unità Special Needs, al fine di permettere al/la docente la declinazione più idonea in riferimento alla specifica tipologia di esame.