Mathematical methods
This second-year course introduces the basic mathematical machinery needed for quantum mechanics and for solving the most important differential equations that arise in undergraduate physics.
Reading
I strongly recommend Fabian Essler’s notes and problems for this course from MT2009. My lectures cover the same topics, but in a slightly different order and with different mistakes.
Mathematical methods for physics and engineering by Riley, Hobson & Bence covers practically all of the material in this course and – most importantly – offers plenty of exercises.
Mathematics for physicists by Dennery & Krzywicki has no exercises, but it provides accessible accounts of the concepts developed in this course.
Lectures
There are 20 hours of lectures, delivered in ten 2-hour bursts. Here is an outline of the topics to be covered with an approximate timetable.
Week 1 Vectors: vector spaces; inner products; dual vector space Week 2 Linear operators Week 3 Functions as vectors: Fourier series; Fourier transforms Week 4 Linear operators on functions: Sturm–Liouville Weeks 5,6 Partial differential equations