Course

The aim of this course is to  explore some of the many ways in which symmetries play a role in  physics. We’ll start with an overview of the concept of symmetries and their description in the  language of  group theory. We will then discuss continuous symmetries and infinitesimal  symmetries, their fundamental role in Noether’s theorem, and their formalisation in terms of Lie  groups and Lie algebras. In the last part of the course we will focus on symmetries in quantum  theory and introduce representations of (Lie) groups and Lie algebras.

This course has two main goals: (1) to introduce some key models from condensed matter physics;  and (2) to introduce some numerical approaches to studying these (and other) models.  As a  precursor to these objectives, we will carefully understand many-body states and operators from  the perspective of condensed matter theory.  (However, I will cover only spin models.  We will not  discuss or use second quantization.)

Once this background is established, we will study the method of exact diagonalization and write  simple python programs to find ground states, correlation functions, energy gaps, and other  properties of the transverse-field Ising model.  We will also discuss the computational limitations  of exact diagonalization.  Finally, I will introduce the concept of matrix product states, and we will  see that these can be used to study ground state properties for much larger systems than can be  studied with exact diagonalization.

Each 90-minute session will include substantial programming exercises in addition to lecture.  Prior programming experience is not expected or required, but I would like everyone to have  python (version 3) installed on their computer prior to the first class, including Jupyter notebooks;  see “Resources” below.

The goal of this course is to introduce the path integral formulation of quantum mechanics and a  few of its applications. We will begin by motivating the path integral formulation and explaining its  connections to other formulations of quantum mechanics and its relation to classical mechanics.  We will then explore some applications of path integrals. Each 90-minute session will include roughly equal amounts of lecture time and activities. The  activities are designed to enhance your learning experience and allow you to assess your own level  of understanding.

The aim of this course is to understand the thermodynamics of quantum systems and in the  process to learn some fundamental tools in Quantum Information. We will focus on the topics of  foundations of quantum statistical mechanics, resource theories, entanglement, fluctuation  theorems, and quantum machines.