State space and operator algebra. Spectral resolution. Measurement process in quantum mechanics. Deterministic evolution and time-dependent Schrodinger equation. Time-evolution operator, basic properties and series representation. Dynamical derivatives. Schrodinger, Heisenberg and Dirac pictures. Time-dependent perturbation theory. Applications: free-particle, wavepacket for direct and reciprocal space representation, Heller wavepacket evolution, frozen and thawed Gaussians, two-level system, dynamic polarizability and photoabsorption.
O1 Mixed states, density operator, Liouville-von Neumman equation. Interaction with an environment, Markov approximation, Lindblad equation (in brief). Linear response theory, response function, Kubo-Martin-Schwinger relation, fluctuation-dissipation theorem.
Born-Oppenheimer approximation. Time-dependent transition probability. Diabatic framework and diabatic electronic probability transition. Adiabatic framework and the Landau-Zener formula. The surface hopping.
Numerically exact solution of the Schrodinger equation: exact diagonalization, split-operator, Lanczos method. Approximation solutions: the time-dependent variational principleand the variational methods. Configurational methods: TDSCF, MCTDH, ML-MCDTH.
O2 The electronic problem: wave function vs. electron density. Hartree-Fock and post-Hartree-Fock methods: perturbation theory, multiconfigurational methods, configuration itneraction. Density functional theory: Thomas-Fermi model, Hohenberg-Kohn theorems and Khon-Sham method. Exchange and correlation functionals. Pseudopotentials.
Macroscopic derivation of the rate constant. Microscopic derivation of the rate constant. Collisional and reactive cross-section. Transition State Theory. Unimolecular reations. Marcus theory for electron transfer.
Semiclassical theory. Feynman's path integrals.
[O1 e O2 are mutually exclusive and agreed with the students, on the basis of their interests.]
Prerequisites for admission
Attending Chimica Fisica A, Chimica Quantistica e Metodi matematici applicati alla chimica is strongly recommended
Traditional. The course is organized through a series of lectures on blackboard. To support the lessons, a rich teaching material is made available, consisting of lesson handouts written by the instructors.
- D. Tannor, Introduction to Quantum Mechanics: A Time-Dependent Perspective, University Science Books, Sausalito, CA, 2007 - R.D. Levine, Molecular Reaction Dynamics, Cambridge University Press, Cambridge, 2005 - A. Messiah, Quantum mechanics, Dover Publications, New York, 2000 - A. Nitzan, Chemical Dynamics in Condensed Phases: Relaxation, Transfer and Reactions in Condensed Molecular Systems, Oxford University Press, 2006
For most of the topics addressed lectures notes can be provided upon request.
Assessement methods and criteria
The examination consists of an oral interview, roughly 40' long for the two parts of the course. The student will be asked to reproduce the physical proofs of theorems shown during the classroom lectures. To this end, the student will be prompted to solve simple exercises under the instructors' supervision on topics addressed during the classroom lectures.