Foundations of Quantum Mechanics

(3 Credit Hours) · Undergraduate

Taught as MAPH 30210 "Foundations of Quantum Mechanics" at host institution. This module starts with the formulation of Quantum Mechanics in its modern mathematical setting. Then several traditional models, including tunnelling and the Hydrogen atom are treated. Some time-dependent perturbation theory is introduced. The module ends with the notion of time evolution in Quantum Mechanics. Course outline: Mathematical Structure: Hilbert spaces, self-adjoint and unitary operators, spectral measures. Postulates of Quantum Mechanics: States, observables and measurements, the correspondence principle, the Heisenberg uncertainty principle. One-dimensional systems: The harmonic oscillator, creation and annihilation operators, potential barriers and wells, tunnelling. Angular momentum and the hydrogen atom. Approximations: Elements of time-independent perturbation theory, the WKB approximation. Time evolution in the Schrödinger and Heisenberg pictures. When taught in Dublin, Ireland the course PHYC 30030 Quantum Mechanics at UCD; Postulates of Quantum Mechanics. Operators, observables and eigenfunctions. Co-ordinate and momentum representations. Hermitian operators. Matrix methods. Uncertainty Principle. Ehrenfest's theorem. Harmonic Oscillator, Ladder operators. Angular momentum. Schrödinger theory of the hydrogen atom. Degeneracy. Fine-structure. Normal Zeeman effect. Pauli theory of electron spin. Stern-Gerlach experiment, Spin-orbit interaction. Total angular momentum. Clebsch-Gordan coefficients. Anomalous Zeeman effect. Time independent perturbation theory, charged harmonic oscillator, Stark effect. Variation method. When taught in Trinity: PY3P01 Quantum Mechanics at TCD; Mathematical foundations of quantum mechanics: description of quantum states and dynamical variables, eigenvalue equations, superposition principle, expectation values. Solution of Schrödinger equation for 1-dimensional systems: SHO using ladder operators, Kronig-Penney model. Angular momentum: calculation of spectrum using ladder operators, orbital angular momentum and parity, spin, addition of angular momenta. Solution of Schrödinger equation in 3 dimensions: the hydrogen spectrum, relativistic corrections and spin-orbit coupling. Time independent perturbation theory.