Quantum Mechanics - Bjergstrom, Kieran Niels

. .

Résumé :

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 Schr?dinger View/Picture 63

3.1 Introduction 63

3.2 Motivating the Schr?dinger 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 Schr?dinger 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 Schr?dinger Picture, and Revisiting Issues with Ehrenfest's Theorem 105

4.3 Wigner's Phase-space Quantum Mechanics 106

4.3.1 Background 106

4.3.2 Motivating the Phase-space Equation of Motion 107

4.3.3 Quantising Phase S...

Biographie:

Mark Julian Everitt is the Director of Studies for Physics at Loughborough University, UK, where he has led a comprehensive revision of the physics degrees, increasing the level of challenge, guided by principles of authenticity and industry requirements. The first seven chapters of the book are derived from his second-year introductory core Quantum Physics module in these new degrees. His research includes the engineering of quantum circuits and devices for quantum technologies, feedback, measurement and control of quantum systems, and pioneering the field of quantum systems engineering as a distinct discipline.

Kieran Niels Bjergstrom began his career as a theoretical physicist researching the realistic modelling of open quantum systems, the reliability of quantum devices, and early notions of Quantum Systems Engineering (QSE) - which was the topic of his PhD. He has worked in academia, industry and business developing QSE principles, tools and methodologies for maturing commercially relevant quantum technologies. He is director of a technology and strategy consultancy advising on areas of innovation including the strategic impact of quantum technologies, methods and policies for realising quantum technology's commercial potential, and applied QSE.

Stephen Neil Alexander Duffus is a university teacher within the Physics Department at Loughborough University. He has an established reputation of communicating complex ideas in an engaging and accessible fashion. During his PhD, his main area of research was in open quantum systems....

Sommaire:

From classical analytical mechanics to quantum mechanics, simulation, foundations & engineering

Quantum mechanics is a fundamental and conceptually challenging area of physics. It is usually assumed that students are unfamiliar with Lagrangian and Hamiltonian formulations of classical mechanics and the role played by probability. As a result, quantum physics is typically introduced using heuristic arguments, obscuring synergies with classical mechanics.

This book takes an alternative approach by leveraging classical analytical mechanics to facilitate a natural transition to quantum physics. By doing so, a solid foundation for understanding quantum phenomena is provided.

Key features of this textbook include:

• Mathematics and Classical Analytical Mechanics: The necessary mathematical background and classical analytical mechanics are introduced gradually, allowing readers to focus on one conceptual challenge at a time.
• Deductive Approach: Quantum mechanics is presented on the firm foundation of classical analytical mechanics, ensuring a logical progression of concepts.
• Pedagogical Features: This book includes helpful notes, worked examples, problems, computational challenges, and problem-solving approaches to enhance understanding.
• Comprehensive Coverage: Including advanced topics such as open quantum systems, phase-space methods, and computational methods for quantum physics including good programming practice and code design. Much of the code needed to reproduce figures throughout this book is included.
• Consideration of Foundations: The measurement problem and correspondence principle are addressed, including an open and critical discussion of their interpretation and consequences.
• Introduction to Quantum Systems Engineering: This is the first book to introduce Quantum Systems Engineering approaches for applied quantum technologies development.

This textbook is suitable for undergraduate students in physics and graduate students in mathematics, chemistry, engineering, and materials science....