TIME-DEPENDENT QUANTUM MECHANICS OF TWO-LEVEL SYSTEMS

by James P Lavine

Overview

With both industrial and teaching experience, the author explains the effects of time dependence in systems with two energy levels. The book starts with time-independent interactions and goes on to treat interactions with time-dependent electric and magnetic fields. Complete derivations are presented for each case, so the reader understands how the solutions are found. Both closed-form and numerical solutions are treated, and the calculations are compared with experimental data from the literature. Numerous plots are provided to show how the solutions depend on the parameters of the interactions.The book builds upon an undergraduate course in quantum mechanics and is useful for readers interested in magnetic resonance and quantum optics. In addition, this book is ideal for self-study by students or researchers starting on two-level systems. The detailed derivations and plots should ease readers into the study of two-level systems in a wide variety of settings.

Product Details

ISBN-13:	9789813272606
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	11/14/2018
Sold by:	Barnes & Noble
Format:	eBook
Pages:	256
File size:	16 MB
Note:	This product may take a few minutes to download.

Table of Contents

Preface v

1 Spin Precession and Rate Equations 1

1.1 Experiments and Time Dependence 2

1.2 Time Dependence and Time Evolution 3

1.3 Spin-½, the Pauli Matrices and Spin Precession 7

1.4 The Einstein Coefficients 16

1.5 Rate Equations 20

1.6 Two Rambles on Spin Precession for Spin-½ Particles 26

References 32

2 Two-Level Systems with a Time-Independent Interaction 35

2.1 Formulation of the Two-Level Problem 36

2.2 Finding the Eigenkets 40

2.3 The Occupation Probabilities 44

2.4 Examples with a Time-Independent Coupling Potential 48

2.5 The Sudden Approximation 53

2.6 Ramble 3 on Random Telegraph Signals 61

References 72

3 Two-Level Systems with a Time-Dependent Interaction 75

3.1 Developing the Coupled Equations 76

3.2 The Rotating Wave Approximation Enters 80

3.3 Exploring the Solutions for a Two-Level System 85

3.3.1 RWA Solutions versus Numerical Solutions 91

3.3.2 An Example of Rabi Oscillations 98

3.4 Two-Level System with Time Dependence and a Constant Field 102

3.5 Two-Level Systems in the Interaction Picture 106

3.6 Two-Level Systems with a Rotating Field 110

References 119

4 Applications and Examples of Two-Level Systems 121

4.1 Rabi Oscillations Observed with Atomic Beams 122

4.2 The Ammonia Maser 124

4.3 Magnetic Resonance 127

4.4 The NV^- Defect Center in Diamond 135

4.5 Ramble 4: The Pulse Area Theorem 142

References 144

5 The Density Matrix and the Relaxation of Two-Level Systems 147

5.1 Why the Density Matrix? 147

5.2 The Density Operator and the Density Matrix 149

5.3 An Equation for the Time-Dependent Density Matrix 151

5.4 The Density Matrix Applied to a Two-Level System 157

5.5 The Introduction of Relaxation into the Density Matrix 162

5.6 A Solvable Two-Level System with Relaxation Included 166

5.7 The Bloch Equations 173

5.8 A Solution to the Bloch Equations 179

5.9 More on the Density Matrix 184

5.10 Ramble 5: A Solution of the Time-Dependent Equations for the Density Matrix Elements 184

References 190

6 The Second-Order Correlation Function for Two-Level Systems 193

6.1 Two Levels versus Three Levels 195

6.2 The Second-Order Correlation Function and Resonance Fluorescence 198

6.3 The Second-Order Correlation Function for a Two-Level System 200

6.4 The Weak-Field Limit of the Second-Order Correlation Function for a Two-Level System 209

6.5 What is Next? 212

References 213

Appendices 217

Appendix 1 Physical Constants 217

Appendix 2 The Electromagnetic Field 219

References 222

Appendix 3 Diagonalization of a Matrix in Chapter 2 223

Appendix 4 The Time Evolution Operator and the Sudden Approximation 225

References 226

Appendix 5 Beyond the Rotating Wave Approximation 227

Reference 232

Appendix 6 The Second-Order Correlation Function for a Two-Level System 233

References 236

Index 237