Meccanica quantistica
Prerequisiti
Basic Linear Algebra
Fundamental concepts such as vector spaces, orthonormal bases, linear operators, eigenvalues, and eigenvectors.
Classical Electrodynamics
Basic principles of electric and magnetic fields, Maxwell’s equations, electromagnetic potentials, and gauge concepts.
Analytical Mechanics
Lagrangian and Hamiltonian formulations of mechanics, principle of least action, generalized coordinates, and conservation laws.
Programma
Mathematical Methods
Algebraic and analytical tools used in the formulation of quantum theory.
Axiomatic Introduction to the Theory
Postulates of quantum mechanics and the physical meaning of operators, observables, and states.
Temporal Evolution
Schrödinger equation, time evolution operators, Schrödinger and Heisenberg pictures.
1D Systems
Infinite square well, potential wells and barriers, harmonic oscillator.
Characteristic Function and Wigner Distribution
Phase-space formalism, quasi-probability representations of quantum states.
Magnetic Fields
Quantization in the presence of magnetic fields, magnetic moment, and gauge invariance.
Aharonov-Bohm Effect
Topological effects and the role of the vector potential in quantum mechanics.
Angular Momentum
Orbital and intrinsic angular momentum, operator algebra, quantization.
Central Potentials
Spherically symmetric problems, radial equation, hydrogen atom.
Spin
Spin-½ systems, spin operators, Stern–Gerlach experiment.
Discrete Symmetries (Parity, Time-Reversal)
Fundamental symmetries and their implications on dynamics and state structure.
Approximation Methods
Perturbation theory (non-degenerate and degenerate), WKB method.
Identical Particles
Indistinguishability principle, state symmetrization, Bose–Einstein and Fermi–Dirac statistics.
Obiettivi formativi
Develop a Deep Understanding of the Foundations of Quantum Theory
Students will understand the axiomatic formulation of quantum mechanics, including the role of Hilbert spaces, operators, observables, and the probabilistic interpretation of measurement outcomes.
Master Mathematical Techniques for Quantum Systems
Students will gain proficiency in the mathematical methods (linear algebra, differential equations, operator algebra) required to analyze quantum systems rigorously.
Analyze and Solve Canonical Quantum Systems
Students will be able to solve standard one-dimensional problems (e.g., infinite square well, harmonic oscillator, potential barriers) and interpret their physical implications.
Understand Quantum Dynamics
Students will learn to describe and compute the time evolution of quantum states using both the Schrödinger and Heisenberg pictures.
Explore Phase-Space Representations
Students will be introduced to alternative formulations of quantum mechanics, such as the Wigner distribution and characteristic function, and understand their use in connecting classical and quantum descriptions.
Investigate Quantum Phenomena in Magnetic Fields
Students will analyze quantum behavior in the presence of magnetic fields and understand the physical significance of gauge invariance and the Aharonov-Bohm effect.
Apply the Theory of Angular Momentum
Students will understand the algebra of angular momentum operators and apply it to problems involving orbital and spin angular momentum, including systems with spherical symmetry.
Develop Skills in Approximation Techniques
Students will learn key approximation methods (perturbation theory, WKB, variational method) and apply them to systems that cannot be solved exactly.
Understand the Role of Discrete Symmetries
Students will study the effects of symmetry operations such as parity and time-reversal on quantum systems and their relevance to fundamental physical laws.
Analyze Systems of Identical Particles
Students will understand the principles governing identical particles in quantum mechanics and the distinction between bosons and fermions through symmetrization postulates and quantum statistics.
Foster Physical Intuition and Problem-Solving Skills
Students will develop a strong physical intuition for quantum phenomena and strengthen their ability to formulate and solve problems using the formal tools of quantum theory.
Riferimenti bibliografici
J.J. Sakurai, “Meccanica Quantistica Moderna” (Zanichelli, Bologna 1985)
L. Ballantine, “Quantum Mechnics” (World Scientific, Singapore 1998)
A. Messiah, “Quantum Mechanics” (Dover, New York 1999)
Moduli
| Modulo | Ore | CFU | Docenti |
|---|---|---|---|
| Meccanica quantistica | 40 | 6 | Vittorio Giovannetti |
| Didattica integrativa | 10 | 0 | Vasco Cavina |