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First Semester 
Teaching style
Lingua Insegnamento

Informazioni aggiuntive

Course Curriculum CFU Length(h)
[60/60]  PHYSICS [60/00 - Ord. 2012]  PERCORSO COMUNE 12 96


1. Knowledge and understanding
Students will have a basic knowledge of the foundations, of the principles, of the mathematical tools and of the main applications of Quantum Mechanics. In particular, students should know Schrödinger equation and the principal methods used to solve it. The student should also have a basic knowledge of special relativity ant its phenomenological consequences , with some hints on the covariant formalism.

2. Applying knowledge
Successful students will be able to:
- solve simple quantum mechanics problems in both one and three dimensions;
- apply perturbative methods in solving quantum mechanics problems;
- solve simple special relativity problems;
- use quantum mechanics or special relativity approach in order to estimate order of magnitudes in physics problems.

3. Making judgements
Students will be able to
- establish when quantum mechanics or special relativity should be used;
- demonstrate an understanding of the significance of operators and eigenvalue problems in quantum mechanics;
- analyse the formal aspects of a problem in quantum mechanics or special relativity, with a critical attitude, owning theoretical tools to check properties and principles of the theory under study

4. Communication skills
Students will be able to coherently present and clearly illustrate the main concepts and the fundamental principles of quantum mechanics and of special relativity

5. Learning skills
- will be able to apply their knowledge also outside the context of quantum mechanics and special relativity
- will acquire understanding skills to proceed towards higher-level studies with a high degree of autonomy. In particular, students will be provided with a solid mathematical and theoretical background to successfully face and attend a subsequent Master Degree


The following pieces of knowledge are prerequisite: linear algebra, matrix calculus, Fourier transform, basic and advanced mathematical methods, classical mechanics and electromagnetism, Hamiltonian mechanics


1.Relativity Principle: absolute time and Galilei transformations
2.Basics of SR: Principle of inertia, clock synchronization, postulates of SR, invariance of the interval between two events, Lorentz transformations, simultaneity and causality; Minkowski diagrams; Michelson and Morley experiment
3.Phenomenology: length contraction, time dilation, proper time, velocity transformations, relativistic Doppler effect, muon lifetime, clock (twin) paradox
4.Relativistic dynamics: linear momentum, kinetic energy, total energy, momentum-energy relation, massless particles, energy and momentum Lorentz transformations, electric charge in electric and magnetic field
5.Space-time: rotations, vectors and tensors in R3, covariance of physics laws, Minkowski space, Lorentz transformations in covariant notation, four-vectors, four-tensors and scalars
6. Dynamics in covariant notation: velocity, acceleration, linear-momentum and force four-vectors

1.Limits of classical physics: black-body radiation, photoelectric effect, Compton effect. The Bohr atom, atomic spectra
2.Waves and particles: de Broglie hypothesis, optics and mechanics, experiments with electrons (Davisson and Germer), double-slit experiment. Plane waves and wave packets, the wave function and its interpretation. The Schrödinger equation. The superposition principle. Role of phases. Gaussian wave packet and its spreading. Uncertainty relation. Schroedinger equation from optics. Probability current and continuity equation. Expectation values. Ehrenfest theorem
3.The time independent Schrödinger equation. Eigenfunctions and eigenvalues. Expansion in series of eigenfunctions. Stationary states, energy measurements.
4.One-dimensional problems: Momentum eigenfunctions and the free particle. Potential step and barrier, (infinite) square well. Bound states and scattering states. Parity operator. The harmonic oscillator
5.Operators and vector spaces. Dirac notation. Coordinate and momentum representation. Unitary transformations. Matrix representation. Time evolution of operators. Symmetries and conservation laws: generators of translations and rotations. The harmonic oscillator with operator methods
6.Postulates of QM. Degeneracy and simultaneous observables. Uncertainty relations
7.Angular momentum: commutation relations; raising and lowering operators. Angular momentum in polar-spherical coordinates; spherical harmonics. Matrix representations
8.Schrödinger's equation in three dimensions: separation of variables. Harmonic oscillator in three dimensions. Central potentials: the hydrogen atom
9.Spin: Zeeman effect. interaction with an electromagnetic field; Uniform magnetic field. Stern and Gerlach experiment. Spin 1/2, Pauli spinors, Direct product of spaces. Rotations and spin
10.Identical particles. Indistinguishability. Slater determinant. Pauli exclusion principle
11.Addition of angular momenta: the Clebsch-Gordan coefficients, sum of two spins, orbital angular momentum and spin 1/2
12.Time independent perturbations: non degenerate perturbation theory, 1st and 2nd order perturbative corrections to the energy eigenvalues, 1st order corrections to the eigenstates; degenerate perturbation theory, application to the double degeneracy case, splitting of levels, hydrogen atom in a static electric field (Stark effect)
13.Fine structure of hydrogen atom: relativistic effects, 1st order corrections, Feynman-Hellmann theorem, spectroscopic notation
14.Time-dependent perturbation theory: transition probability, periodic perturbation, resonant emission and absorption, transition to continuum of states, Fermi Golden Rule, lifetime and widths of an unstable state

Teaching Methods

There will be 48 2-hour lectures and about 12 2-hour exercise sessions where quantum mechanics and special relativity problems will be fully worked out.

Verification of learning

The final exam has a written and an oral part.
Written part: solution of Quantum Mechanics (QM) exercises and solution of problems and discussion/derivation of formal aspects in Special Relativity (SR).
Oral exam (on Quantum Mechanics only): students may be asked to derive QM formulae, demonstrate QM theorems.
Students should also be able to show their understanding of the foundations of Quantum Mechanics and of Special Relativity and in particular should be able to identify the physics problems which require their use.
The final grade will be based upon:
1) knowledge of the principles and the foundations of Quantum Mechanics and Special Relativity;
2) ability in solving Quantum Mechanics and Special Relativity problems;
3) ability in deriving important Quantum Mechanics and Special Relativity formulae and results.
4) ability in connecting general concepts and their consequences
5) use of proper language and terminology
6) presentation skills
Special Relativity will roughly count as 25% of the final grade
In the winter session (January and February) students can take separately the RS and MQ parts (written and oral exams).


Quantum Mechanics
Suggested textbooks:

B.H. Bransden, C.J. Joachain: Quantum Mechanics (second edition); Pearson - Prentice Hall 2000

David J. Griffiths: Introduction to Quantum Mechanics (second edition); Pearson - Prentice Hall 2005

Stefano Forte, Luca Rotoli: Fisica Quantistica
Fisica Zanichelli

C. Cohen-Tannoudji, B. Diu, F. Laloe: Quantum Mechanics John Wiley & Sons 1977

Further textbooks:

J.J. Sakurai, Meccanica Quantistica Moderna; Zanichelli

Stephen Gasiorowicz: Quantum Physics (second edition); John Wiley & Sons 1996

Steven Weinberg: Lectures on Quantum Mechanics, (second edition) Cambridge University Press, 2015

Exercise books:
G. Passatore, Problemi di meccanica quantistica elementare; Franco Angeli
L. Angelini, Meccanica quantistica: problemi scelti; Springer

Special Relativity
Suggested textbooks:

Vincenzo Barone: Relatività: Principi e applicazioni; Bollati Boringhieri 2011

A.P. French: Special Relativity; W.W. Norton & Co. 1968

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