This course applies basic quantum mechanics principles (Schrödinger wave equation and perturbation theory, classical and quantum free-electron theories, band theory for solids) to understand the functional properties of materials (including electrical, optical, optoelectronic, magnetic and topological properties etc.). Topics in this course include single-particle Schrodinger equation and its applications in several typical quantum mechanical systems and realistic quantum materials and advanced characterization techniques; non-degenerate and degenerate (time-independent) perturbation theories, and their applications in the ground state of helium atom, Stark effect, and Zeeman splitting; classical free-electron gas model, quantum free-electron theory, quantum density of states, Fermi-Dirac distribution, Maxwell Boltzmann distribution, Fermi energy, and Fermi surface; Bloch’s theorem, approaching band model through Schrödinger equation, nearly free-electron model, tight binding model, Kronig-Penney model for deriving the formation of discrete energy levels and band structures of crystalline solids; apply band structures to classify materials and understand electrical, optical, and topological properties of recently emerging materials systems (two-dimensional materials and topological insulators etc.).