Mecânica Quântica Matemática

Objectives

Introduce several mathematical topics in quantum mechanics, with emphasis on non-perturbative aspects and the role of different types of symmetries.

Syllabus

Basic Principles: Operators, Hilbert space, quantization, observables, probability; Wave function, Schroedinger eq.; Bosonic, grassmannian, super/conformal oscillators; Witten index; Elliptic potentials; Bound-states vs scattering; Feynman integral, propagators. Symmetries: Wigner's theorem, self-adjoint operators, spectral theory; Angular momentum; Representations, quantum statistics; Addition of angular momenta; Quantum groups. Resurgence and Summability: Perturbation theory; Instantons, tunneling; Multiple wells; Complex/exact WKB approximation; Borel summability, resurgence, transseries. Coherence & Decoherence: EPR, quantum entanglement; Coherent states; Decoherence; Quantum-to- classical transition; Quantum information, quantum chaos, black holes. Quantization: Dirac constraints; Schroedinger eq., oscillators, scattering; Bogoliubov transformations; Geometric, deformation quantization; Non-commutative geometry.

Cross Competence Component

The UC allows the development of transversal competences on Critical Thinking, Creativity and Problem Solving Strategies, in class, in autonomous work and in the several evaluation components. The percentage of the final grade associated with these competences should be around 15%.

Ethical Principles

All members of a group are responsible for the group’s work. In any assessment, every student shall honestly disclose any help received and sources used. In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.