PERIOD: MARCH - APRIL The very nature of the course is interdisciplinar. In fact, its main aim is to give PhD students, regardless of their previous the background knowledge of the Hamiltonian formalism, whose importance in all areas of physics is widely recognized. It is also well-known that this formalism represents a close and natural link to the development of quantum mechanics. At present this subject is not offered to all students in the Bachelor and Master courses at Politecnico.

DISAT, Politecnico di Torino, E-mail: francesco.raa@polito.it Prerequisites: Linear algebra, Maxwell equations, Schrodinger equation. PROGRAM  The variational formalism for discrete classical systems Kinetic and potential energies, the Lagrangian L. Hamilton's principle, Lagrangian equations of motion. The Legendre transformation, the Hamiltonian H. Simmetry properties, cyclic coordinates, conserved quantities. Hamiltonian equations of motion, Poisson brackets and their properties.  Quantization of classical discrete systems From Poisson brackets to commutators, quantization of a classical system, Heisenberg equations of motion. The quantum harmonic oscillator in terms of ladder operators a, ay and number operator ^n.  A summary of quantization of the electromagnetic eld Maxwell equations, the four-potential A. Fourier expansion of the four-potential in classical and quantum electromagnetic theory.  The quantum optical Jaynes-Cummings model Two-level (or spin 1 2 ) systems. Interaction between a two-level atom and a single-mode quantized radiation eld, evaluation of the free Hamiltonian and interaction Hamiltonian. Physical predictions of the model (atomic population inversion, Rabi frequency). An introduction to the general quantum Rabi model. 1 Textbooks mainly on classical systems 1. H. Goldstein, C. P. Poole, J. L. Safko, Classical Mechanics (Pearson, 3rd edition, 2002) 2. D. J. Griths, Introduction to Electrodynamics (Pearson, 4th edition, 2013) 3. C. Lanczos, The Variational Principles of Mechanics (Dover, 1986) 4. D. S. Lemons, Perfect Form: Variational Principles, Methods and Ap- plications in Elementary Physics (Princeton University Press, 1997) 5. L. Meirovitch, Methods of Analytical Dynamics (McGraw-Hill Classic Textbook Reissue Series, 1988) 6. Moiseiwitsch B. L. Variational Principles (Wiley, 1966) Some textbooks on quantum systems 1. C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics (Hermann, 1977) 2. R. P. Feynman, R. B. Leighton, M. Sands, The Feynman's Lectures on Physics, vol. III, various editions available. 3. I. N. Levine, Physical Chemistry (McGraw-Hill, 6th edition, 2009) 4. J. J. Sakurai, Modern Quantum Mechanics. Revised Edition (Addison- Wesley, 1994). Textbooks/Further reading on quantum optics 1. G. S. Agarwal, Quantum Optics (Cambridge University Press, 2012) 2. L. Allen, J. H. Eberly, Optical resonance and two-level atoms (Dover, 1987) 3. L. Mandel, E. Wolf, Optical coherence and quantum optics (Cambridge University Press, 1995) 4. D. F. Walls, G. J. Milburn, Quantum Optics (Springer, 1994) 2

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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.