Knowledge: basics of quantum mechanics mastering the mathematical formalism at a rigorous level; interpretation of quantum mechanics and related contemporary debate; applications to devices, from transistors to contemporary single-particle technology. Skills: solving problems on spectra of quantum observables, probability transitions, from spin systems to atomic systems. Write down quantum mechanical models for simple systems and discuss them. Reading autonomously classics of quantum mechanics as well as modern treatises.

Linear algebra, matrices, unitary and hermitian transformations. Diagonalization, quadratic forms, spectral theorem for finite-dimensional spaces.

First part: Introduction 1. Double-slit experiment ([P],[G],[F]). The wave model and the photon model. Notion of superposition. 2. Double-slit experiment with damper and dephaser ([dEP]). Heuristic construction of the space of the states. Probability of transition between two states ([P],[G],[I]). 3. Polarization of light, polarization of photons ([P],[G]). Malus’ Law. Probablistic interpretation. Notion of linear, circular, elliptic polarization of light and of photons. Construction of the mathematical space of the polarization states of a photon ([P],[G]). 4. The measurement and the collapse of the state, with its mathematical description. 5. Entanglement of photons: the Einstein-Podolski-Rosen phenomenon and Bell’s inequality in the form of Ghirardi-Mermin ([G]). Notion of tensor product and mathematical description of the entanglement. 6. Greenberger-Horne-Zeilinger states ([G]) and the role of non-locality. 7. Bennet-Brassard protocol for the Quantum Key Distribution for Cryptography ([G]). Second part: Conceptual and Mathematical Structure of Quantum Mechanics ([I],[T]) 1. Axioms of quantum mechanics: mathematical structure of the theory. Hilbert spaces. Dirac’s notation. 2. Space of states of a composed system (more on tensor products in Dirac’s notation). 3. Observables (self-adjoint operators). 4. Hamiltonian operator. 5. Schroedinger’s Equation (time-dependent). Time propagator and unitary groups. Stone’s Theorem. 6. Measurament. Collapse of the quantum state if the result of a measurement is an element of the discrete spectrum of the observable. Orthogonal projections. Third part: Systems with a finite dimensional space of states ([I]). 1. Examples with polarization and spin systems. 2. Density matrix. Link with entanglement. 3. Property of self-adjoint and unitary matrices. 4. Spectral theorem for finite-dimensional self-adjoint operators, with proof. 5. Compatibility of two observables. Commutators. 6. Theorem of simultaneous diagonalizability of two finite-dimensional self-adjoint operators, with proof. 7. Examples and exercises. Fourth part: Systems with an infinite dimensional space of states ([P],[T]) 1. Heisenberg’s Uncertainty Principle. 2. Why Heisenberg’s Uncertainty Principle cannot be fulfilled in a finite-dimensional space. 3. Linear operators in infinite-dimensional spaces: boundedness and continuity. 4. Why Heisenberg’s Uncertainty Principle cannot be fulfilled by a couple of bounded operators. 5. Unbounded operators. Closedness and closability (with examples). Resolvent and spectrum. 6. Symmetric (or Hermitian) and Self-adjoint operators. Role of the operator domain. 7. Discrete and continuous spectrum. 8. Examples: position and momentum operators. Difficulties in defining the momentum operator on the halfline. 9. Spectral Theorem for unbounded, self-adjoint operators (without proof). 10. Self-adjointness and boundary conditions. 11. Examples: Dirichlet, Neumann, Robin. Spectral resolution of the Laplacian in the presence of Boundary conditions. 12. Some exactly solvable models: spectrum, eigenfunctions, generalized eigenvalues, generalized eigenfunctions for Dirichlet, Neumann, Robin’s condition on an interval, for Dirac’s delta potential on the line. References [dEP] E. d’Emilio, L. E. Picasso, Problems in Quantum Mechanics with Solutions, 2017. [G] G. C. Ghirardi, Sneaking a Look at God’s Cards: Unraveling the Mystery of Quantum Mechanics, 2007. [I] C. Isham, Lectures on Quantum Theory, 1995. [P] L. E. Picasso, Lectures on Quantum Mechanics: A Two-Term Course, 2016. [T] A. Teta, A Mathematical Primer on Quantum Mechanics, 2018. 1.Introduction: Nonlocality of nature, Photons, Polarization, Bell’s Inequality and its experimental violation. Two-slit experiments and interpretation in terms of photons. Occurrence of Probability. Space of the polarization states of photons. Action of Polaroids and of birifrangent crystals. Interpretation of Malus’ law in terms of photons. 2. Conceptual and Mathematical Structure of Quantum Mechanics. Axioms of the theory. Hilbert space of states. Dirac’s notation. Space of states of a composed system: tensor product. Observables and Self-adjoint operators. Hamiltonian operator. Schroedinger Equation, Stone’s Theorem, Propagators. Measurement. Collapse of the state. Orthogonal projectors. 3. System with finite dimensional space of states. Polarization, qubits. External product, density matrix, link with entanglement. Evolution of density matrices. Properties of self-adjoint operators. Spectral Theorem for finite-dimensional self-adjoint operators with proofs. Compatible observables. Commutators. Link between commutativity and simultaneous diagonalizability, with proof. 4. Heisenberg’s Uncertainty Principle. Why it cannot be fulfilled in a finite dimensional space. Linear operators in infinite-dimensional Hilbert spaces. Boundedness and continuity. Why Heisenberg’s Uncertainty Principle cannot be fulfilled by any couple of bounded operators. Unbounded operators. Closedness and closability. Resolvent and Spectrum. Symmetric (or Hermitian) and Self-adjoint operators. Role of the operator domain. Discrete and continuous spectrum. Weyl’s sequences. Explicit examples: Laplacian on the line, self-adjoint Laplacians on the halfline and on a finite interval. Self-adjointness and boundary conditions. Spectral theorem for unbounded operators. Examples in more dimensions. Dirac’s delta potentials. Potential wells. Bessel functions. Hydrogen atom. Harmonic oscillator. Historical introduction: Black-body radiation, Photoelectric effect, Radiative collapse, Bohr’s atom. Two-slit experiment. Superposition principle. Two state systems: polaroid, spin, qubit. Schroedinger Equation for finite-dimensional systems. The problem of measurement. Structure of finite-dimensional Hilbert spaces. Dirac’s Notation. Self-adjointness and observables. Spectrum. Spectral theory, simultaneous diagonalization and compatible observables. Born’s rule. Collapse. Statistical mixtures, density matrix. Bell inequalities, locality, EPR, interpretation. Examples: quantum algorithms (Grover), quantum teleportation, quantum cryptography. Ininite-dimensional Hilbert spaces. Time-dependent and stationary Schroedinger Equation. Born’s rule and collapse for infinite-dimensional systems. Continuous spectrum Potential well, Dirac potential, Harmonic oscillator, Hydrogen atom. Creation and annihilation of particle: Fock spaces. Some examples from quantum electrodynamics and mean-field theories.

Blackboard lectures and slides, with exercises. Participation and interaction is strongly encouraged.

Notes from lectures (not covering the whole programme) and from exercise sessions. C. Isham, Lectures on Quantum Theory, 1995. A. Teta, “A Mathematical Primer on Quantum Mechanics”, Springer 2018. Luigi E. Picasso, “Lectures in Quantum Mechanics”, Springer 2015 Luigi E. Picasso, E. d’Emilio, “Problems in Quantum Mechanics with Solutions”, Springer 2015. Griffiths, Schoeter, “Introduction to Quantum Mechanics”, 1995 Preskill, Lecture Notes on Quantum Information and Computation, online.

Lecture slides; Lecture notes;

Exam: Written test; Optional oral exam;

A written exam with two or three exercises on the programme of the course. Every exercise consists of three/four open questions. The duration of the written exam is three hours. The maximal grade is 30 cum laude. Upon request by the student or the teacher, an oral exam is delivered, with at least two questions on the programme of the course. In the oral exam, a first question has theoretical character, a second question consists in an exercise. Further questions can be addressed if needed to reach a precise definition of the grade.

In addition to the message sent by the online system, students with disabilities or Specific Learning Disorders (SLD) are invited to directly inform the professor in charge of the course about the special arrangements for the exam that have been agreed with the Special Needs Unit. The professor has to be informed at least one week before the beginning of the examination session in order to provide students with the most suitable arrangements for each specific type of exam.