Elementi di meccanica dei continui e meccanica quantistica

The course is worth 6 credits in total (5 credits of lectures and 1 credit of problem-solving sessions), it consists of 47 hours of classroom teaching. It explores advanced topics in mechanics, highlighting the connections between the methods of Hamiltonian mechanics, quantum mechanics, and continuum mechanics. 

Knowledge and understanding: Students will acquire knowledge of the fundamental concepts of Hamiltonian mechanics, non-relativistic quantum mechanics, and continuum mechanics. In particular, they will develop an understanding of the role of symmetries and conservation laws in the mathematical formulation of physical systems, the meaning of classical and quantum observables, the mathematical structure of the Schrödinger equation and of the main quantum-mechanical models, as well as the principles governing the kinematic and dynamic description of continuous media. Students will also become familiar with the mathematical language used in the formulation of physical models, understanding the relationships among geometric structures, differential equations, and conservation principles.

Applying knowledge and understanding: Students will be able to formulate and analyze mathematical models of physical systems using the tools introduced throughout the course. They will learn to use Poisson brackets and conservation laws in the study of Hamiltonian systems, apply the operator formalism and the Schrödinger equation to elementary problems in quantum mechanics, determine eigenvalues and eigenstates of simple observables, and interpret their physical meaning. In the context of continuum mechanics, students will be able to derive and apply balance equations, interpret the physical meaning of the Cauchy stress tensor, and use fundamental fluid dynamics models to address elementary problems, recognizing the mathematical structures common to the various contexts studied.

Making judgements: Students will develop the ability to identify the mathematical tools most appropriate for the description of a physical phenomenon and to critically assess the assumptions underlying the models employed. They will be able to distinguish between classical, quantum, and continuum descriptions of a physical system, recognizing their domains of validity, fundamental assumptions, and possible generalizations. Students will also acquire the ability to analyze simple theoretical problems independently and to establish connections among topics belonging to different areas of mathematical physics.

Communication skills: Students will be able to present the fundamental concepts of Hamiltonian mechanics, quantum mechanics, and continuum mechanics with clarity and mathematical rigor, using the specialized terminology of the discipline appropriately. They will be capable of presenting proofs, arguments, and computational procedures both orally and in written form, justifying the methodological choices adopted and discussing the results obtained in a coherent and precise manner.

Learning skills: Students will acquire the conceptual and methodological tools required to pursue independently the study of advanced topics in mathematical physics, differential equations, and mathematical modelling. They will be able to deepen their understanding of specialized subjects through the study of advanced textbooks and research articles, developing the ability to integrate knowledge from different areas of mathematics and theoretical physics.

Lectures will be delivered in person, unless regulations require a blended or fully remote mode of instruction. To ensure equal opportunities and compliance with current regulations, students requiring accommodations may request an individual meeting in order to arrange appropriate compensatory and/or dispensatory measures, taking into account the learning objectives of the course and their specific needs. Students may also contact the Department's CInAP (Centre for Active and Participatory Inclusion) representative, Prof. Patrizia Daniele.

Basic knowledge of calculus, linear algebra, and mechanics.

Part I. Elements of Hamiltonian Mechanics

Phase space and the Hamiltonian formulation of classical mechanics. Poisson brackets and their fundamental properties. Hamiltonian flows and first integrals. Symmetries and Noether's theorem. Actions of Lie groups on symplectic manifolds and the momentum map. Linear momentum and angular momentum as conserved quantities associated with symmetry groups. Symplectic reduction. 

Part II. Fundamentals of Quantum Mechanics

From classical mechanics to canonical quantization. Quantum observables and canonical commutation relations. Position and momentum operators. Quantum states, eigenvalues, eigenvectors, and the spectral interpretation of observables. Energy and angular momentum operators. Time-dependent and time-independent Schrödinger equations. Time evolution of quantum systems. Schrödinger and Heisenberg representations. Symmetries and conservation laws for observables. Fundamental applications and examples: the free particle, the one-dimensional harmonic oscillator, and the method of creation and annihilation operators. Central-force problems in three dimensions. The rotation group and its representations. Angular momentum operators and spherical harmonics. The radial Schrödinger equation. The hydrogen atom: energy levels, eigenvalue degeneracy, and quantum numbers. 

Part III. Elements of Continuum Mechanics

Lagrangian and Eulerian descriptions of motion. Continuum kinematics. Deformations and the deformation gradient. Velocity, spin, and vorticity. Reynolds transport theorem and its consequences. Balance principles for continuous systems. Conservation equations for mass, linear momentum, angular momentum, and energy in both integral and differential form. Body forces and surface forces. Cauchy's theorem and the Cauchy stress tensor. Constitutive equations and the principle of material frame-indifference. Perfect fluids and Newtonian fluids. Euler and Navier-Stokes equations. Two-dimensional motions of incompressible and irrotational fluids. Complex potential theory and applications of complex analysis to planar fluid dynamics. Point-vortex dynamics and the Hamiltonian formulation of the system. Connections between the geometric structures of classical mechanics and models of fluid dynamics.

Hamiltonian Mechanics

1. R. Abraham, J.E. Marsden, Foundations of Mechanics, Benjamin/Cummings.
2. V.I. Arnold, Mathematical Methods of Classical Mechanics, Springer.
3. J.E. Marsden, T.S. Ratiu, Introduction to Mechanics and Symmetry, Springer.

Quantum Mechanics

1. L.D. Faddeev, O.A. Yakubovskiĭ, Lectures on Quantum Mechanics for Mathematics Students, American Mathematical Society.
2. B.C. Hall, Quantum Theory for Mathematicians, Springer.
3. L.A. Takhtajan, Quantum Mechanics for Mathematicians, American Mathematical Society.
4. V. Moretti, Spectral Theory and Quantum Mechanics, Springer.
5. D. Bohm, Quantum Theory, Dover Publications.

Continuum Mechanics

1. M.E. Gurtin, An Introduction to Continuum Mechanics, Academic Press.
2. C. Truesdell, W. Noll, The Non-Linear Field Theories of Mechanics, Springer.

Assessment will consist of an oral examination. If necessary, the examination may also be conducted remotely through online platforms.

Hamiltonian Mechanics

• Define Poisson brackets and prove their main properties.
• Explain the connection between symmetries and conserved quantities in the Hamiltonian formulation of mechanics.
• Describe the concept of momentum map and determine it in simple examples associated with translations and rotations.
• Compute the Hamilton equations associated with a given mechanical system.
• Discuss the geometric meaning of symplectic reduction and illustrate it with an example.

Quantum Mechanics

• Introduce the concept of a quantum observable and explain the physical meaning of eigenvalues and eigenstates.
• Derive the canonical commutation relations between position and momentum operators.
• Discuss the Schrödinger and Heisenberg representations, highlighting their similarities and differences.
• Determine the energy eigenvalues of the harmonic oscillator using creation and annihilation operators.
• Solve the free-particle problem and interpret its solutions.
• Define the angular momentum operators and compute their commutation relations.
• Describe the role of the rotation group in quantum mechanics.
• Determine the energy levels of the hydrogen atom and discuss their degeneracy.

Continuum Mechanics

• Explain the difference between the Lagrangian and Eulerian descriptions of motion.
• Define the deformation gradient and discuss its geometric meaning.
• State and prove the Reynolds Transport Theorem.
• Derive the local balance equations for mass and linear momentum.
• Define the Cauchy stress tensor and describe its physical interpretation.
• Discuss the principle of material frame-indifference and its role in the formulation of constitutive equations.
• Derive the Euler equations for a perfect fluid.
• Describe two-dimensional incompressible and irrotational flows and their connection with holomorphic functions.
• Formulate the point-vortex system and illustrate its Hamiltonian structure.