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) Introduction to systems and computational neuroscience The module covers a range of fundamental ideas in systems and computational neuroscience, with a particular focus on sensory perception.

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) Introduction to systems and computational neuroscience Students will be familiar with the core ideas in the neuroscience of vision and touch, and in the Bayesian theory of perception. They will be able to navigate and interpret the modern research literature on vision, touch, and Bayesian models of perception

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) Introduction to systems and computational neuroscience No special requirements. The course is self-contained, and all relevant prerequisites in mathematics, probability, and biology are introduced during the lectures.

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) Introduction to systems and computational neuroscience [D. Zoccolan, M. Diamond, E. Piasini] Part 1. Physiology and functions of the mammalian visual system Introduction to anatomy and physiology of the visual system A systems/computational approach to the study of the visual system; Anatomy of the visual system Classic findings about physiology of lower-level visual areas Data analysis approaches in Systems Neuroscience Classic findings about physiology of higher-level visual areas Descriptive models of visual neurons How to build models of visual neuronal responses (i.e., stimulus/response maps) Mechanistic models of the visual system Inferring the mechanisms underlying the response properties of visual neurons Functional models of the visual system Understanding neuronal population codes Part 2. Sensory Systems: Tactile Perception 1. Introduction to the study of the cerebral cortex 2. Sensory maps in the cerebral cortex 3. Transduction 4. Somatosensory system and pain 5. Methods for computational neuroscience of perception 6. Encoding and decoding 7. Perceptual memory 8. Neuroscience of perceptual knowledge Part 3: Bayesian modeling of perception Perception as Bayesian inference Bayesian inference under sensory noise Cue combination and evidence accumulation Discrimination, detection and classification

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) Introduction to systems and computational neuroscience Mainly frontal lectures, except 3 hours of computational demonstrations in the Bayesian modeling module

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) Introduction to systems and computational neuroscience No additional reading material is necessary beyond what is provided during lessons. However some of the teachers’ material comes from these sources and students are invited to seek additional background there: · Dayan, P. & Abbott, L. F. Theoretical Neuroscience. (MIT Press, 2001). · Martin, A. R., Brown, D. A., Diamond, M. E., Cattaneo, A. & De-Miguel, F. F. From Neuron to Brain, Sixth Edition. (Oxford University Press, 2021). · Rieke, F., Warland, D. & Bialek, W. Spikes: exploring the neural code. (The MIT Press, 1999). · Wichmann, F. A. & Hill, N. J. The psychometric function: I. Fitting, sampling, and goodness of fit. Perception & Psychophysics 63, 1293-1313, doi:10.3758/BF03194544 (2001). · Ma, Koerding and Goldreich (2022). Bayesian Models of Perception and Action (https://www.cns.nyu.edu/malab/bayesianbook.html) · Richard McElreath (2nd ed 2020). Statistical Rethinking. (https://xcelab.net/rm/ )

Dispense;

Modalità di esame: Prova scritta (in aula);

Exam: Written test;

... 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) Introduction to systems and computational neuroscience Written 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.