Overall objectives:

1 - Understand the main implications of the mathematical formalism of Quantum Mechanics in the description of physical systems.
2 - Understand that a quantum system can be described in terms of differential operators and wave function in continuous space, or in terms of matrix operators and wave function in Hilbert space
3 - Know how to use Dirac notation, calculate eigenvalues, temporal evolution, observation probabilities and expected values, both in continuous and matrix representations.
4 - Know how to apply matrix methods in systems such as: harmonic oscillator, spin-1/2 and angular momentum.
5 - Understand quantum entanglement and its consequences, particularly those associated with correlations between observation results and Bell inequalities.
6 - Know how to apply perturbation theory to quantum systems.
7 - Understand the basic principles of quantum computing.

Syllabus:

1 - Wave Quantum Mechanics. Stern-Gerlach experiment. Schrödinger equation. Operators. Dirac notation. Hamiltonian and Time Evolution. Overlapping states. Measurements and the collapse of the wave function.
2 - Matrix Quantum Mechanics. Quantum descriptions of systems with finite number of states. Correspondence with wave quantum mechanics. Spin-1/2 two-particle state. Angular momentum and rotation. Quantum entanglement.
3 - Quantum Mechanics free from paradoxes. The pilot wave and the quantum potential. Heisenberg's uncertainty principle, Schrödinger's cat, EPR paradox, Bell's inequalities and Bohm's interpretation.
4 - Harmonic oscillator. Bound states of central potentials.
5 - Perturbation theory. Degenerate and non-degenerate disorders. The Stark effect in hydrogen. The ammonia molecule in an electric field.
6 - Elements of quantum computing: Qubits and their manipulation. Quantum Gates and Algorithms. Quantum cryptography.

Literature/Sources:

John S. Townsend , 2012 , A Modern Approach to Quantum Mechanics , University Science Books
David Bohm, B.J. Hiley , 1993 , The Undivided Universe ? An ontological interpretation of Quantum Mechanics , Routledge London and New York
Marco Cardoso, Marta M. Correia, Samuel F. Martins et al. , 2017 , Mecânica Quântica , IST Press
Seiki Akama , 2015 , Elements of Quantum Computing , Springer International Publishing Switzerland
Michael A. Nielsen, Isaac L. Chuang , 2011 , Quantum Computation and Quantum Information , , Cambridge University Press

Assesssment methods and criteria:

Classification Type: Quantitativa (0-20)

Evaluation Methodology:
In theoretical classes the methodology is expository. The material is given on the board, all equations and formulas are derived from first principles and a strong emphasis is placed on the connection between the physical formulas and the real world. Theoretical-practical classes consist of problem solving. Some key problems are solved on the board, but the vast majority are solved by students independently, with individual help from teachers whenever necessary. Use of accessible online resources such as pedagogical simulations. Your exploration is appropriately tailored through questionnaires and guided activities. Examples of computational work: Calculation of energy levels in a poly-electronic atom with symmetrization of the wave function for identical particles. Construction of a quantum circuit with a given number of qubits and quantum gates and making measurements on the qubits. Verification of exercises with quantum circuits can be done at, for example: https://algassert.com/quirk. Computational tools that can be used: Python, C++, Fortran, MATLAB, COMSOL Multiphysics, OpenFoam. Assessment Model: B. Assessment Methodology: Components T and T-P: 2 tests. At the time of appeal, 1 or 2 tests can be improved. The weight of the computational work carried out in the final grade is 25%.