This booklet contains notes from a course that was developed by L. D. Faddeev at the beginning of the 70s and taught for many years by O. A. Yakubovskii to students of Leningrad University. As indicated by the title, lectures are prepared mainly for mathematically oriented students requiring a higher level of rigour than is usual in courses given by physicists. The main structure of quantum mechanics is explained gradually starting with a thorough discussion of states and their evolution in classical mechanics, followed by a notion of a state in quantum mechanics and a description of relations between quantum and classical mechanics. After a discussion of one-dimensional examples, several three-dimensional problems are treated using tools from representation theory of the group of rotations. The last part of the book is devoted to scattering theory, a discussion of the spin of particles, multi-electron atoms and the Mendeleev periodic system of elements. The book is very well-written and can be easily understood by students only having knowledge of basic undergraduate mathematics.