# The Quantum Menagerie

**A tutorial introduction to the mathematics of quantum mechanics**

## About this book

Understanding quantum mechanics matters because it is the engine that powers the universe. This engine is fuelled by a few simple principles, but the consequences of those principles are both profound and strange.

In this richly illustrated book, quantum mechanics is explained using a finely balanced combination of words, diagrams and mathematics. The result is a tour of the most intriguing aspects of quantum mechanics, including Einstein's "spooky action at a distance", Bell's inequality, Schrödinger's cat, Heisenberg's uncertainty principle, and de Broglie's matter waves.

Supported by a comprehensive glossary, further readings, and tutorial appendices, this is an ideal introduction to the mathematics of quantum mechanics.

Published 2020

Paperback ISBN: 9781916279100

Hardback ISBN: 9781916279131

Download Chapter 1 (PDF, 1.2MB)

## Contents

**Preface**

**1. What is quantum mechanics?**

1.1 Introduction

1.2 Spooky action at a distance

1.3 The double slit experiment

1.4 Heisenberg's uncertainty principle

1.5 Schrödinger's cat

1.6 Interpreting quantum mechanics

1.7 Quantum Horizons

1.8 What makes quantum mechanics hard?

1.9 Summary

**2. Planck's act of desperation**

2.1 Introduction

2.2 Classical physics

2.3 Blackbody radiation

2.4 Planck's inspired guesswork

2.5 Counting standing waves

2.6 The ultraviolet catastrophe

2.7 The Boltzmann distribution

2.8 An act of desperation

2.9 Summary

**3. Einstein's unreasonable reality**

3.1 Introduction

3.2 Quantum filters

3.3 Single photons

3.4 The inverse quantum Zeno effect

3.5 The short version

3.6 Counting photon pairs: Bell's inequality

3.7 The hidden variable hypothesis

3.8 Summary

**4. Waves of light and matter**

4.1 Introduction

4.2 Matter waves

4.3 Heisenberg's uncertainty principle

4.4 Diffraction

4.5 Diffraction and uncertainty

4.6 Atomic models

4.7 Bohr's model

4.8 Quantised matter waves

4.9 Summary

**5. The double slit experiment**

5.1 Introduction

5.2 Interference in light and water

5.3 Flying clocks and oscillators

5.4 The travelling wave

5.5 Real interference

5.6 Complex interference

5.7 What is seen in light and water

5.8 Which slit for which photon?

5.9 Wheeler's delayed choice experiment

5.10 Summary

**6. Schrödinger's wave equation**

6.1 Introduction

6.2 Quantum guitar strings

6.3 The classical wave equation

6.4 Stationary classical waves

6.5 Schrödinger's wave equation

6.6 Wavefunctions in a box

6.7 Wavefunctions in a hydrogen atom

6.8 Summary

**7. Quantum interpretations**

7.1 Introduction

7.2 The Copenhagen interpretation

7.3 Objective collapse theories

7.4 Bohmian mechanics

7.5 The many-worlds interpretation

7.6 The Von Neumann-Wigner interpretation

7.7 Summary

**8. A history of quantum mechanics**

**Further reading**

**Appendices**

A. Glossary

B. Mathematical symbols

C. Complex numbers

D. The Boltzmann distribution

E. Pioneers of quantum mechanics

F. Fourier optics and Heisenberg

G. Wavefunctions and PDEs

H. Key equations

**References**

**Index**

## Reviews

"A wonderful resource for anyone who wants to teach themselves quantum mechanics for real. You will learn the necessary math, but the real emphasis is on conceptual understanding. Stone's book explains why things work a certain way, rather than just asserting that they do. It dispels the mystery from an intimidating subject."

**Sean Carroll, Research Professor, Caltech, author of ****Something Deeply Hidden**

"An informative and accessible introduction to the bizarre world of quantum mechanics and the different interpretations of the mathematical formalism. Stone's clear, organised style makes difficult concepts seem simple."

**Ian Stewart, Emeritus Professor of Mathematics, Fellow of the Royal Society, University of Warwick**

"The Quantum Menagerie is a wonderfully clear introduction to the notoriously demanding subject of quantum mechanics. Uniquely it blends the history of the field, including the work of Planck, Einstein, and others, with a splendidly lucid, step-by-step approach to the maths behind its key findings. James Stone explains milestone results that might appear abstract at first glance, such as Bell's inequality, in a delightfully visual manner. Recommended to anyone who would like to understand the formalism of quantum mechanics and needs the guidance of a seasoned explorer."

**Paul Halpern, author of ****Synchronicity and the Quantum Labyrinth**

"In this lively and entertaining book, James Stone traces the development of quantum mechanics, explaining how its salient features were born from pure guesswork, or, in some cases, sheer desperation, as scientists faced observations that refused to fit into the framework of classical physics. Dr Stone describes the problematic aspects of quantum mechanics, and the failed attempts to fix them, which in some cases led to the experimental confirmation of some of quantum mechanics more mind-blowing predictions. This book is written at a level suitable for beginning undergraduates."

**Richard Fitzpatrick, Professor of Physics, University of Texas at Austin**

"Stone's Quantum Menagerie is a terrific tour of the mathematical underpinnings of quantum mechanics. It provides a gentle introduction to unfamiliar mathematical concepts and their application to quantum physics. This book has been particularly useful to biology students on the Quantum Biology Doctoral Training Centre at the University of Surrey, most of whom have very little training in mathematics."

**Professor Johnjoe McFadden, Director, Leverhulme Quantum Biology Doctoral Training Centre, University of Surrey, UK**

Below are comments from PhD students at the Quantum Biology Doctoral Training Centre, University of Surrey.

"A great, no-nonsense, pragmatic, introduction to the foundations of quantum mechanics. I particularly enjoyed the the glossary and appendices, which were immensely useful."

**Louie Slocombe**

"This book offers a great overview of quantum mechanics, providing an informal relaxed experience for beginners, in addition to more advanced theory and maths for experienced readers."

**Lucy Ridout**

"Coming from a biological background with little knowledge of physics, I definitely fall within the target audience. The book is very well written and easy to follow. It uses a very good combination of diagrams, equations and most importantly comparisons that enabled me to visualise and understand the complex experiments and extraordinary behaviour of quantum mechanics. Additionally, the chapters are divided into numerous subheadings that allows the reader to go back and forth between sections with ease to recap on specific topics. Overall, a great book."

**Alejandro Sanchez-Pedreno Jimenez**

## Further reading

A short history of quantum mechanics (blog)