The course consists of two modules A) Introduction to Quantum Mechanics The module will cover standard topics for an introduction to Quantum Mechanics. Starting from the definitions of states, operators and observables, and the relative operations between them, the student will be introduced to the Schrödinger equations and to the most important systems in one- and three-dimensions. The course will be concluded with an introduction to various approximate methods and the study of particles in a magnetic field. B) Ecology and evolution The goals of this module are both to teach students fundamental concepts in ecology and evolution and provide them with basic notions and tools in dynamical systems and stochastic processes. The module will be divided into two main parts. The first part will be devoted to introducing the students with classic models in population and community dynamics. The second part will focus on evolutionary theory and population genetics. We will discuss classic and modern experimental evidence of Darwinian (micro)evolution and the observational pieces of evidence of macroevolution. The students will study the effect of selection, mutations, and drift on the fixation of neutral, beneficial, and deleterious mutations in the context of simple stochastic models of population genetics.

A) Introduction to Quantum Mechanics At the end of the course, the student is expected to be able to manage concepts like Hilbert spaces, solutions of Schrödinger equation with various methods, including approximate ones. B) Ecology and evolution At the end of the first part of the module, the students are expected to be able to quantitatively model community dynamics in the presence of different interaction types, identifying and justifying the important assumptions. They should also be able to perform stability analysis, identify the presence of bifurcations and have basic notions of limit cycles and chaos. At the end of the second part of the module, the student will know how to quantify the relative importance of the fundamental mechanisms of evolution (reproduction, mutation, selection, genetic drift and recombination) using the tools of stochastic processes.

A) Introduction to Quantum Mechanics Notions and tools of basic classical physics, linear algebra, vector spaces, eigenvalue problems, Fourier transforms, differential equations are needed. B) Ecology and evolution Calculus, Basics of Linear Algebra, Basics of Probability

A) Introduction to Quantum Mechanics [A. Rosa] Topics are divided in two groups. The first group contains very basic concepts, that the student will read from the notes, while frontal lectures will mainly focus on applications and exercises. The second group will be more in the form of “traditional” frontal lectures, with theory and exercises presented back-to-back. GROUP 1 1.1) The Rise of Quantum Mechanics • The Black Body Problem • The Hypothesis of Photons and the Photoelectric Effect • Optical Spectra of Atomic Elements • The Quantization of Angular Momentum: the Stern-Gerlach Experiment 1.2) Mathematical Foundations: Vector Spaces • bra|’s and ket’s: Definitions • Operators in Quantum Mechanics 1.3) Hermitian Operators • Definitions and Results • Eigenvectors and Eigenvalues 1.4) Physical Observable • Hermitian Operators and Physical Observables, Commuting and Non-commuting Observables • Measuring Observables and Quantum averages: A physical Interpretation 1.5) Representations in Quantum Mechanics • Representations of States and Operators • Representations and Probabilities in QM • Change of Representations 1.6) Eigenvectors and Eigenvalues of the Momentum; Uncertainty Principle 1.7) Schrödinger Equation, Conserved Quantities and Stationary States 1.8) Time Evolution in Quantum Mechanics GROUP 2 2.1) One-Dimensional Problems - A recap 2.2) Algebraic methods: harmonic oscillator and angular momentum 2.3) Schrödinger Equation in Three Dimensions; Central Potentials 2.4) Approximate methods I - Variational methods 2.5) Approximate methods II - Perturbative methods 2.6) Approximate methods III - Semi-classical (WKB) approximation 2.7) Motion of particles in a magnetic field B) Ecology and evolution [J. Grilli] 1. Single population dynamics (exponential growth, logistic growth, Allee effect) 2. Two species dynamics. Fixed points and stability. Lotka-Volterra 3. Multispecies communities 4. Experimental and observational evidence of evolution 5. Genetic Drift 6. Mutation and Selection 7. Coevolution

A) Introduction to Quantum Mechanics For the topics from 1.1 to 1.8 the student should read the notes before the frontal lecture; during the lecture the student will be exposed to applications and exercises. The topics from 2.1 to 2.7 will be presented in the form of more traditional frontal lectures, exercises will always be proposed and solved. B) Ecology and evolution Theory classes

A) Introduction to Quantum Mechanics Merzbacher, E.; Quantum Mechanics (J. Wiley & Sons, Inc.) Messiah, A.; Quantum Mechanics (Dover) Picasso, L. E.; Lectures in Quantum Mechanics (Springer) d’Emilio, E., Picasso, L. E.; Problems in Quantum Mechanics: with Solutions (Springer) (6) Sakurai, J. J.; Modern Quantum Mechanics (Addison-Wesley) B) Ecology and evolution Murray, Mathematical Biology vol I Strogatz, Non-linear dynamics and Chaos

Dispense;

Modalità di esame: Prova scritta (in aula); Prova orale obbligatoria;

Exam: Written test; Compulsory oral exam;

... A) Introduction to Quantum Mechanics The exam is written only, the student will be asked to solve exercises. The maximum score of the exam is 30/30, as per the Italian system. B) Ecology and evolution The exam is only oral. The students will be assigned homework that will be discussed during the oral exam.

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.