Physics

This course unlocks the secrets of modern electronics. It explores semiconductor materials, quantum mechanics, and carrier transport, as well as the principles behind p-n junctions, transistors, and the devices that power today's technology.    

This course covers pre-relativistic physics, including Galilean transformations, the concept of the ether, and the Michelson-Morley experiment. It introduces the Principle of Relativity, Lorentz transformations, and their consequences. Topics include four-vectors, tensors, formal Lorentz transformations, particle dynamics and applications, relativistic electrodynamics, and the energy-momentum tensor.

This course is part of the Laurea Magistrale degree program and is intended for advanced level students. Enrollment is by permission of the instructor. This course covers the theoretical understanding and working knowledge of the principal gravitational phenomena determining the structure, the dynamics and the evolution of stellar systems, from open and globular clusters, to galaxies, to galaxy clusters. At the end of the course, the student is able to use in autonomy some of the advanced mathematical techniques needed in potential theory and in epicyclic theory. The course content is divided into 2 parts:

1. GENERALS

Gravitational field of point particles, principle of superposition. Integral representation for any distributions. Most important properties of the divergence operator and its coordinate-free representation starting from Gauss's Theorem. Operational introduction to the one-dimensional and multidimensional Dirac Delta in Cartesian and curvilinear coordinates. Calculation of the divergence of the field of extended distributions, Poisson's equation for the field. Direct proof of the First and Second Newton's theorem (homogeneous spherical shells). Alternative demonstration using Gauss's theorem. Coordinate-free representation of the gradient, curl, and Laplacian operators. Notes on differential forms. Exact fields and their properties, potential and work. Closed fields. Stokes' theorem, closed fields in simply and non-simply connected domains. Existence of the potential and its connection with the total energy of a particle. Potential difference as a line integral. Formal calculation of the potential of a point mass. Potential of extended distributions, general expression and discussion of the meaning of the additive constant. Poisson and Laplace equations. First and second Green's identities, uniqueness of the solution of the Poisson equation in bounded volumes with prescribed boundary conditions. Field inside cavities with equipotential boundary. Helmholtz Decomposition Theorem. Definition of concentric and similar ellipsoids. Definition of homoeoid. Statement of the Third Newton's Theorem for finite homoeoids. Field inside a heterogeneous hollow homoeoid from the principle of superposition. Co-area theorem, relationship with the field of homoeoids. Definition of confocal ellipsoidal coordinates. Classification of the three families of associated quadrics. Ellipsoidal coordinates: orthogonality, gradient, Laplacian. Application to the problem of the ellipsoidal layer with zero internal field. Potential of the heterogeneous ellipsoid. Chandrasekhar's formula. Introduction to the multipole expansion of potential in the far field. Monopole, dipole and quadrupole terms.

Introduction to the concept of Green's function for linear differential operators and their use in solving nonhomogeneous problems. The potential of a material point as an explicit example of a Green's function for the Laplacian. Separation of variables for the Laplacian in Cartesian coordinates. Fourier transform and inverse transform in Rn, the case of the Dirac Delta. Green's function in Cartesian coordinates. Green's function in spherical coordinates. Separation of variables. Rotational invariance and the azimuthal quantum number m. Orthogonality of azimuthal functions. Associated Legendre equation for the latitude angle, transformation into an algebraic equation. Outline with examples of singularities of ODEs, both mobile and fixed. Fuchs' theorem, regular points, regular singularities, and essential singularities. Classification for the Legendre equation. Frobenius method and polar quantum number. Legendre functions and associated functions P and Q. Legendre polynomials. Rodrigues formulas, norm of associated polynomials. Orthogonality of solutions with Sturm-Liouville theory. Spherical harmonics as eigenfunctions of the angular part of the Laplacian. Systems with cylindrical symmetry. Generating function for Legendre polynomials, multipole moments. Gegenbauer polynomials. Addition theorem for spherical harmonics. Separation of variables for the vacuum solution of the Laplacian in cylindrical coordinates. Bessel equation and its properties: orthogonality of solutions, singular points. Asymptotic analysis of Bessel functions for large values of the argument. Closure relation and Hankel transform. Green's function in cylindrical coordinates for the Laplacian. Any density potential with Fourier-Bessel transforms. Case of axisymmetric systems. Infinitely thin axisymmetric disks, potential in the plane of the disk, homogeneous rings. Thin disk rotation curve. Mestel's disc and exponential, implications for the dark matter halos. Potential of axisymmetric systems using elliptical integrals.

2. COLLISIONLESS SYSTEMS

Introduction to the epicyclic approximation. Notes on curvilinear coordinates, velocity and acceleration in cylindrical coordinates. Newtonian equations of motion in general axisymmetric potentials, conservation of energy and Jz. Deduction of equations from the Euler-Lagrange equations. The meridional plane, its motion, and effective potential. Equations of motion in the meridional plane, orbital families, circular orbits and their (equivalent) equations. Interpretation of total energy as energy for motion in the meridional plane, extremum properties for the energy of circular orbits, centrifugal barrier, zero-velocity curves. Development of the effective potential to second order. Frequency of vertical and radial epicycles. Radial and vertical motion on the epicycle in the case of stable orbits, zero-velocity ellipses. Rayleigh criterion and examples of applications. First-order angular motion, coordinates on the equatorial plane referred to the deferent, equation of the epicycle on the equatorial plane, and determination of the axes for the epicyclic ellipse. Epicycles in Coulomb, harmonic, and flat rotation potentials: frequency and shape. Relation of Oort constants to the radial epicyclic frequency. Closed, rosette, and open orbits: closure conditions, pattern angular velocity, Lindblad kinetic waves, and the dynamical phenomenology of disks.

Students will gain a thorough grounding in the life cycle of stars. Students learn to describe the stages of nucleosynthesis in stars; calculate the equations of hydrostatic equilibrium; use the equations of energy transport to calculate basic properties of stars; describe in detail the evolutionary stages different classes of stars are thought to go through; and describe in detail the end stages of the life cycle of a star and the different types of stellar remnants.

This course investigates material properties from the perspective of condensed matter physics. We will cover the mechanical, thermal, electrical, magnetic and optical properties of materials, learn to apply qualitative and quantitative analysis and understand these properties based on the interaction between atoms and molecules, crystalline structure, electronic and magnetic structures and morphology. Through the course of study, we will know the most dominant factors affecting material properties and know how to estimate material properties in a certain range.

This course develops a level of competence in Python, a modern programming language currently used in many physics research labs for mathematical modelling. No prior experience is required. The course starts with a grounding in the use of Python and discusses numerical methods. The main focus is then on the ways in which Python can be used for problem solving in physics and astrophysics.

This course covers the following subjects: representations of numbers and arithmetic error (floating point math), functions and roots, linear and non-linear systems of equations, interpolation and approximative representations of functions, numerical differentiation and integration, ordinary and partial differential equations, eigenvalue problems (wave equations), molecular dynamics simulations (planet systems, Lennard-Jones liquids, molecular chaos), stochastics, Monte-Carlo integration, Monte-Carlo metropolis simulation (lattice spin model), optimization of non-linear problems, steepest descent,  conjugate gradient, simulated annealing (traveling salesman problem), Fourier transforms, spectral analysis (analysis of acoustic signals, audio synthesis), networks, infection models, random walks, reaction-diffusion systems, predator-prey population dynamics, cellular automata (Game of Life), and artificial neural networks.

This course provides a fundamental introduction to a wide range of modern biophysics. This is a multidisciplinary scientific area where a number of theoretical and experimental methods from physics are used to understand and examine biological systems. The course begins from the fundamental biological building blocks, including proteins, DNA/RNA, and membranes. It discusses their structure and interactions both on a molecular level and their role in large systems such as the structure of the cell, the movement of organisms and the signaling of nerves. The course describes the fundamental physical mechanisms for interaction and transport that biological organisms use, and introduces modern experimental techniques for obtaining structural and thermodynamical biophysical information at the nanoscale.

This course provides an introduction to classical mechanics covering vectors, applications of Newton’s laws, conservation laws and forces, motion in a plane, circular motion, equilibrium and elasticity, rotational motion, simple harmonic motion, energy and power; mechanical and sound waves, temperature, heat and the first law of thermodynamics. Prerequisite: first semester of differential calculus.