This course aims to equip graduate students with advanced knowledge of quantum mechanics necessary to conduct research and understand literature. It will consist of four different parts: (i) The theory of angular momentum; (ii) Symmetries in quantum mechanics; (iii) Perturbation theory in quantum mechanics; (iv) Introduction to modern many-body theory. This course will mainly focus on the applications of quantum mechanics in condensed matter physics and materials science. In particular, this course will expose the students to some of the latest developments in topological phases of matter, including the physics of topological insulators and topological superconductors.