Quantum mechanics is one of the most successful scientific theories. It describes systems on the atomic scale and its development is a key step in the ongoing electronic revolution.

This lecture will start with a brief introduction to the fundamental principles of quantum theory: Origins; wave functions in Hilbert space; stationary and time-dependent Schrödinger equations, uncertainty principle.

Then we will take a quantum leap and discuss quantum spin systems, that are key to our understanding of condensed matter physics. Specifically, we will review spin operators; tensor products of Hilbert spaces; Heisenberg models; infinite-volume systems; phase transitions; graphical representations of quantum spin systems.

Mathematically, there will be some analysis, PDEs, Fourier analysis, functional analysis, algebra, and probability theory. But no prerequisites beyond 2nd year core modules will be assumed; this lecture will be self-contained.

Chapters 3 and 4.

Chapter 5.

Chapter 6.

Chapter 7 and index.

Here are the full revised notes.