Foundations of Quantum Mechanics

This module introduces Quantum Mechanics in its modern mathematical setting. Several canonical, exactly-solvable
models are studied, including one-dimensional piecewise constant potentials, Dirac potentials, the harmonic
oscillator, and the Hydrogen atom. Three calculational techniques are introduced: time-independent perturbation
theory, variational methods, and numerical (spectral) methods. The postulates of Quantum Mechanics,
[Mathematical background] Complex vector spaces and scalar products, linear forms and duality, the natural scalar product derived from linear forms, Hilbert spaces, linear operators, commutation relations, expectation values, uncertainty, [Time evolution and the Schrodinger equation] Derivation of the Schrodinger equation for time-independent Hamiltonians, the position and momentum representations, the probability current, the free particle [Piecewise constant one-dimensional potentials] Bound and unbound states, wells and barriers, scattering, transmission coefficients, tunneling, [The harmonic oscillator] Solution by power series, Hermite polynomials, creation and annihilation operators, coherent states, [The Hydrogen atom] Solution by separation of variables, quantization of energy and angular momentum, general treatment of central potentials in terms of spherical harmonics, [Angular momentum] Motivation: angular momentum in the hydrogen atom, as derived from spherical harmonics, angular momentum in the abstract setting, intrinsic angular momentum, addition of angular momenta, Clebsch-Gordan coefficients,
[Approximation methods] Time-independent perturbation theory: the non-degenerate case, variational methods for
estimating the ground-state energy