electiveHQ

This module introduces Quantum Mechanics in its modern mathematical setting. Several canonical, exactly-solvable models are studied, including the harmonic oscillator, and the Hydrogen atom. 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 [The Hydrogen atom] 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, [Piecewise constant one-dimensional potentials] Bound and unbound states, wells and barriers, scattering, transmission coefficients, tunneling, [The harmonic oscillator] Creation and annihilation operators, coherent states, [Approximation methods] Time-independent perturbation theory: the non-degenerate case, variational methods for estimating the ground-state energy [Introduction to Quantum Information] Qubits and quantum logic gates Further topics may include: Spin coherent states, how to build a microwave laser, the Dyson series for time-evolution for time-dependent Hamiltonians, one-dimensional Dirac potentials, time-independent perturbation theory for degenerate eigenstates, the fine structure of Hydrogen, numerical (spectral) methods for solving the Schrodinger equation