Hilbert Spaces

Introduction

Welcome to the Hilbert Spaces notes!

Hilbert Spaces were introduced to formalise the mathematics behind quantum mechanics. Bristol’s most famous alumnus, Paul Dirac, used Hilbert spaces to show the equivalence between the wave mechanics of Schroedinger to the matrix mechanics of Heisenberg, thus creating the unified theory of quantum mechanics. We will look at the mathematics, philosophy, and history of these ideas.

These notes are a short summary of the key axioms and derivations for the Hilbert Spaces section of the Advanced Mathematics for Physicists course. They are very much intended to accompany, not replace, the lectures!

Course Overview

The chapters leading up to Chapeter 9  Hilbert Spaces introduce the mathematics of Hilbert spaces and a more mathematical look at the underpinning of quantum mechanics.

In Chapter 10  Dirac vs Von Neumann we take a detailed look at the respective positions of Dirac and von Neumann about the relevance of Hilbert spaces to quantum mechanics.

We then use this newfound understanding to look at orthogonal polynomials in Chapters 11  Fourier Series to 17  Sturm Liouville Theory. In this way you will see the deep connections between waves, discrete states, and their description in Hilbert spaces.