Acronyms xiii

About the Authors xv

Preface xvii

Acknowledgements xix

About the Companion Website xxi

Introduction xxiii

1 Mathematical Preliminaries 1

1.1 Introduction 1

1.2 Generalising Vectors 2

1.2.1 Vector Spaces 2

1.2.2 Inner Product 5

1.2.3 Dirac Notation 7

1.2.4 Basis and Dimension 9

1.3 Linear Operators 10

1.3.1 Definition and Some Key Properties of Linear Operators 10

1.3.2 Expectation Value of Random Variables 12

1.3.3 Inverse of Operators 13

1.3.4 Hermitian Adjoint Operators 13

1.3.5 Unitary Operators 15

1.3.6 Commutators 15

1.3.7 Eigenvectors and Eigenvalues 17

1.3.8 Eigenvectors of Commuting Operators 18

1.3.9 Functions of Operators 18

1.3.10 Differentiation of Operators 19

1.3.11 Baker Campbell Hausdorff, Zassenhaus Formulae, and Hadamard Lemma 19

1.3.12 Operators and Basis State - Resolutions of Identity 20

1.3.12.1 Outer Product and Projection 20

1.3.12.2 Resolutions of Identity 21

1.4 Representing Kets as Vectors, and Operators as Matrices and Traces 22

1.4.1 Trace 24

1.4.2 Basis, Representation, and Inner Products 24

1.4.3 Observables 25

1.4.4 Labelling Vectors - Complete Sets of Commuting Observables - CSCO 25

1.5 Tensor Product 26

1.5.1 Setting the Scene: The Cartesian Product 26

1.5.2 The Tensor Product 27

1.6 The Heisenberg Uncertainty Relation 29

1.7 Concluding Remarks 32

2 Notes on Classical Mechanics 35

2.1 Introduction 35

2.2 A Brief Revision of Classical Mechanics 38

2.2.1 Lagrangian Mechanics 38

2.2.2 Hamiltonian Mechanics 41

2.3 On Probability in Classical Mechanics 45

2.3.1 The Liouville Equation 45

2.3.2 Expectation Values 48

2.4 Damping 50

2.5 Koopman-von Neumann (KvN) Classical Mechanics 53

2.6 Some Big Problems with Classical Physics 56

2.6.1 Atoms and Polarisers 56

2.6.2 The Stern-Gerlach Experiment 56

2.6.3 The Correspondence Principle - What It Is and What It Is Not 59

3 The Schrödinger View/Picture 63

3.1 Introduction 63

3.2 Motivating the Schrödinger Equation 64

3.2.1 Ehrenfest's Theorem, Poisson Brackets, and Commutation Relations 68

3.2.2 The Main Proposition 70

3.2.2.1 Summarising an Issue with the Above Argument 70

3.3 Measurement 71

3.3.1 Introducing Measurement 71

3.3.2 On the Possible Connection Between the State Vector and Probabilities 73

3.3.3 The Time-independent Schrödinger Equation 75

3.3.4 Measurement Outcomes 77

3.4 Representation of Quantum Systems 78

3.4.1 The Position and Momentum Representation 78

3.4.1.1 The One-dimensional Case 78

3.4.1.2 Three Dimensions 83

3.4.2 Spin 85

3.4.3 Spin and Position - The Spinor 88

3.5 Closing Remarks and the Axioms of Quantum Mechanics 89

4 Other Formulations of Quantum Mechanics 93

4.1 Introduction 93

4.2 The Heisenberg Picture 94

4.2.1 Background 94

4.2.2 Motivating the Heisenberg Equation of Motion 95

4.2.3 A Specific Example: the One-dimensional Harmonic Oscillator 100

4.2.4 The State, Representation, and Dynamics 101

4.2.5 Axioms of Quantum Mechanics Revisited 101

4.2.6 The Evolution Operator 102

4.2.7 Connection to the Schrö