Physics

1) Crisis of classical physics, wave-particle duality, double slits experiment. The wave function interpretation as probability amplitude. 2) Time-dependent and stationary Schroedinger equation, expectation values of observables, momentum operator and position operator, conservation of the norm in the quantum evolution, continuity equation. 3) Dirac formalism, Hilbert space, added and self-adjoint operators; change of basis. 4) Canonical commutation relations, commutation relation between position and momentum. 5) Evolution of the expectation values ​​of the observables; Ehrenfest theorem. Quasi-classical states. 6) Measurement of an observable: collapse of the wave function. Continuous spectrum and non-normalizable states. 7) Eigenfunctions of position operators; representation of coordinates and momentum; position operator in the momentum representation. 8) Uncertainty relations. Wave functions with minimum product of the uncertainty between position and momentum: temporal evolution in the case of a free particle. Relationship between the uncertainty of energy and the characteristic time. 9) Common basis of observables that commute. Complete sets of observables that commute. 10) Separation of variables. One-dimensional problems. General properties of the solutions of the one-dimensional Schroedinger equation. 11) Particle on a segment. Rectangular potential barrier, discrete spectrum and continuous spectrum. 12) Momentum operator as a generator of translations. Spatial inversion and parity operator. 13) Harmonic oscillator: operators of creation and destruction, eigenvalues ​​and eigenfunctions of energy. 14) Operator of time evolution. Representations of Schroedinger and Heisenberg. Particle in a box. Three-dimensional harmonic oscillator. 15) Tensorial product of vector spaces. Angular momentum: commutation relations among its components. Eigenvalues ​​of operators total angular momentum J and Jz with the algebraic method. The orbital angular momentum; spherical harmonics. 16) The angular momentum operators as a generators of the rotations; transformation under rotations. Symmetry, invariance conservation laws. 17) Rotationally invariant Hamiltonian, radial equation. 18) Coulombian potential: eigenvalues ​​and eigenfunctions of the discrete spectrum. 19) Problem of the two bodies: motion of the center of mass and relative motion, central problems: hydrogen atom and three-dimensional harmonic oscillator. 20) Composition of angular moments: eigenstates of total angular momentum and coefficients of Clebsch-Gordan. 21) Stern and Gerlach experiment, electron spin. Pauli matrices. Rotations in the spin space. Pauli equation. 22) Identical particles. Properties of states under the action of the exchange operator: bosons and fermions. Basis for states describing identical particles. 23) Time independent perturbation theory: non-degenerate case. Extension to the degenerate case. 24) Theory of time-dependent perturbations. First order theory. Transition probability. Case of sinusoidal perturbation.

Fundamental knowledge of the following subjects: linear algebra, vectorial spaces and operators, Fourier transform, mechanics and electrodynamics, Maxwell equations and electromagnetic waves, special relativity, lagrangian and Hamiltonian formalism, equations of motion and their invariance, conserved quantities.

J.J Sakurai J. Napolitano, Modern quantum mechanics L. Landau and E. Lifshitz, Quantum mechanics C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (2 Vol set)

Lectures and training sessions in classroom.

Facoltativa

Written and oral exam.

J.J Sakurai J. Napolitano, Modern quantum mechanics L. Landau and E. Lifshitz, Quantum mechanics C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (2 Vol set)

Lectures and training sessions in classroom.

[1] Introduction: crisis of classical mechanics and birth of Quantum Mechanics -Matter-wave duality, double slit experiment, wave function as a probability amplitude. [2] Axiomatic definition of Quantum Mechanics - Hilbert space, self-adjoint operators, Dirac formalism, change of basis; -Uncertainty relation. Wave functions with minimum product of uncertainty between position and momentum -Common basis of commuting observables. Complete set of commuting observables. - Expectation values of observables, measurement of observables: collapse of the wave function (Copenhagen interpretation) - Position and momentum operators, continuous spectrum of eigenvalues and non-normalizable states, eigenfunctions of the position operator; position and momentum representations; position operator in momentum representation - Symmetries in QM and in Hamiltonian mechanics, generators and conserved quantities, Noether theorem - Momentum as generator of spatial translations. Canonical commutation rules, commutator of position and momentum operators. Spatial inversion and parity. - Hamiltonian as generator of time evolution. Time-dependent Schroedinger equation, Heisenberg dynamical picture, Ehrenfest theorem. - Angular momentum as generator of rotations, commutation rules of angular momentum. Symmetry groups and their properties. [3] Problems in 1 dimension - Free particle: plane waves and wave packets, energy spectrum. Degeneracy theorem. - Free particle on a segment. Square potential well, discrete and continuous energy spectrum. Non-degeneracy theorem. - General properties of the solutions of Schroedinger equation in 1-dimension. Quantization of energy levels. - Harmonic oscillator: raising and lowering operators, energy eigenvalues and eigenfunctions. Coherent states. - Scatterings in 1 dimension, reflection and transmission coefficients, tunnel effect. Probability conservation, continuity equation. [4] Time-independent perturbation theory (Rayleigh-Schroedinger) - Theory for discrete and non-degenerate spectra, extension to the degenerate case. Validity of the approximation. [5] 3-dimensional systems and Theory of Angular Momentum - Tensor product of vectorial spaces. Separation of variables. Particle in a box. 3-dimensional harmonic oscillator. Two-body problem in QM: center-of-mass and relative motions. - Rotationally invariant systems, radial equation. Algebraic method for the computation of the eigenvalues of angular momentum. Orbital angular momentum, spherical harmonics. 3-dimensional harmonic oscillator. - Coulomb potential and hydrogen atom: discrete energy eigenvalues and eigenfunctions. - Spin as an internal degree of freedom. Pauli matrices, spin rotations. - Charged particle in external electromagnetic field. Dipole moment, precession of angular momentum, Pauli Hamiltonian. - Stern e Gerlach experiment and measurement of the electron spin - Composition of angular momenta: eigenstates of total angular momentum and Clebsch-Gordan coefficients - Gauge invariance [6] Identical Particles. - Particle exchange operator and its action on the state vector. Bosons and fermions. Hilbert space for identical particles and its properties.

Fundamental knowledge of the following subjects: linear algebra, vectorial spaces and operators, Fourier transform, mechanics and electrodynamics, Maxwell equations and electromagnetic waves, special relativity, lagrangian and Hamiltonian formalism, equations of motion and their invariance, conserved quantities.

S. Weinberg, Lectures on Quantum Mechanics (seconda edizione), Cambridge Univ. Press L. Picasso, Lezioni di Meccanica Quantistica (seconda edizione), edizioni ETS S. Forte, L. Rottoli, Fisica Quantistica, Zanichelli editore R. Shankar, Principles of Quantum Mechanics (second edition), Springer D. Griffiths, Introduction to Quantum Mechanics, Addison-Wesley ed.

Lectures and training sessions in classroom.

Lectures and training sessions in classroom.

Written and oral exam.

S. Weinberg, Lectures on Quantum Mechanics (seconda edizione), Cambridge Univ. Press L. Picasso, Lezioni di Meccanica Quantistica (seconda edizione), edizioni ETS S. Forte, L. Rottoli, Fisica Quantistica, Zanichelli editore R. Shankar, Principles of Quantum Mechanics (second edition), Springer D. Griffiths, Introduction to Quantum Mechanics, Addison-Wesley ed.

Lectures and training sessions in classroom.

1) The crisis of the classical physics 2) Schroedinger equation 3) Dirac formalism and Hilbert spaces. 4) Canonical commutation relations 5) Time evolution 6) Measurement in quantum mechanics 7) Coordinate and momentum representations 8) Indeterminacy relations 9) Complete sets of commuting observables. 10) One-dimensional problems. 11) Finite-depth square potential. 12) Translations and parity. 13) Harmonic oscillator in one and three dimensions. 14) Temporal evolution. 15) Tensor product of vector spaces. Angular momentum. Eigenvectors of J2 and Jz. Spherical harmonics. 16) Rotations. 17) Rotationally invariant hamiltonians 18) Coulomb potential: spectrum 19) Two-body problem 20) Angular momentum composition and Clesch-Gordan coefficients 21) Spin. Pauli metrices 22) Identical particles: bosons and fermions. 23) Time-independent perturbation theory. 24) Time-dependent perturbation theory.

Basic notions of classical physics and linear algebra.

L. Picasso, Lezioni di Meccanica Quantistica, ETS, Pisa Patri', Testa, Fondamenti di Meccanica Quantistica, Nuova Cultura.

The course consists in theoretical lectures in which the theory is presented and in recitation classes in which problems are discussed and solved.

Optional attendance

The exam consists is: i) Written test. The student should solve two problems analogous to those presented in the course, to those assigned in previous exams (a collection can be found in the e-learning site) and also collected in the book by Testa and Patri'. ii) Oral test. The student will be required to derive two results/theorems/arguments presented in the course. The assigned mark will be directly proportional to the knowledge of the subject.

The course consists in theoretical lectures in which the theory is presented and in recitation classes in which problems are discussed and solved.