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Providing a solid theoretical background in photon-matter interaction, Nonrelativistic Quantum X-Ray Physics enables readers to understand experiments performed at XFEL-facilities and x-ray synchrotrons. As a result, after reading this book, scientists and students will be able to outline and perform calculations of some important x-ray-matter interaction processes. Key features of the contents are that the scope reaches beyond the dipole approximation when necessary and that it includes short-pulse interactions. To aid the reader in this transition, some relevant examples are discussed in detail, while non-relativistic quantum electrodynamics help readers to obtain an in-depth understanding of the formalisms and processes.
The text presupposes a basic (undergraduate-level) understanding of mechanics, electrodynamics, and quantum mechanics. However, more specialized concepts in these fields are introduced and the reader is directed to appropriate references. While primarily benefiting users of x-ray light-sources, the material is equally of relevance to researchers in various disciplines, such as life sciences, biology, materials science, physics, and chemistry that plan on applying these new facilities in their respective fields.



Table of Contents

Preface XIII

Part I Introduction 1

1 Introduction 3

1.1 Motivation 3

1.2 Comparing X-Rays with Optical Radiation 3

1.3 Novel X-Ray Sources 5

1.4 Unit Systems 6

1.5 Overview of Lagrangian and Hamiltonian Mechanics 9

1.5.1 Lagrangian Mechanics 9

1.5.2 Hamiltonian Mechanics 10

1.6 Approximations 12

1.6.1 Semiclassical Approximation 12

1.6.2 Dipole Approximation 13

2 Review of Some Concepts in Quantum Mechanics 15

2.1 Introduction 15

2.2 Dirac's Bra-Ket (Bracket) Notation 15

2.3 Eigenvalues and Eigenfunctions 16

2.4 Functions of Operators 18

2.5 Point Particle in a Radially Symmetric Potential 19

2.5.1 Radial Schrödinger Equation 19

2.5.2 Bound States in a Modified Attractive Coulomb Potential 21

2.5.3 Unbound States in a Coulomb Potential 21

2.5.4 Pure Coulomb Potential 22

2.6 Mixed States 23

2.6.1 Isolated Systems 23

2.6.2 Coupled Systems 25

2.7 Schrödinger and Heisenberg Pictures of Quantum Mechanics 26

2.7.1 Evolution Operator in the Schrödinger Picture 26

2.7.2 Equivalent Pictures of Quantum Mechanics 28

2.7.3 Schrödinger Picture 28

2.7.4 Heisenberg Picture 29

2.8 Representing Quantum Mechanics in Position and Momentum Space 29

2.9 Transition from Classical Mechanics to Quantum Mechanics 31

2.10 Molecular Orbital Approximation 31

2.10.1 Derivation of the Hartree-Fock Equations 32

2.10.2 Interpretation of Orbital Energies 38

2.10.3 Post-Hartree-Fock Methods 40

Part II Quantization of the Free Electromagnetic Field 41

3 Classical Electromagnetic Fields 43

3.1 Introduction 43

3.2 Maxwell's Equations 43

3.3 Electromagnetic Potentials 44

3.3.1 Field Equations 44

3.3.2 Gauge Transformation 45

3.3.3 Coulomb Gauge 45

3.3.4 Lorenz Gauge 46

3.4 Transverse and Longitudinal Maxwell's Equations 46

3.4.1 Helmholtz Decomposition of Maxwell's Equations 47

3.4.2 Decomposition of the Field Equations in the Coulomb Gauge 47

3.5 The Free Electromagnetic Field as a Sum of Mode Oscillators 48

3.5.1 Density of States of the Radiation Field 53

3.5.2 Radiation Cavity in Thermodynamic Equilibrium 54

3.6 Charged Particle in an Electromagnetic Field and the Minimal-Coupling Hamiltonian 56

4 Harmonic Oscillator 59

4.1 Introduction 59

4.2 Classical Harmonic Oscillator with One Degree of Freedom 59

4.3 Quantum Mechanical Harmonic Oscillator 60

4.4 N-Dimensional Quantum Mechanical Harmonic Oscillator 64

5 Quantization of the Electromagnetic Field 67

5.1 Introduction 67

5.2 Transition to a Quantum Mechanical Description 67

5.3 Photon Number States (Fock States) 71

5.4 Photons 73

5.4.1 Photon Momentum and Poynting Vector 73

6 Continuous Fock Space 77

6.1 Introduction 77

6.2 Three-Dimensional Continuum Field 77

6.2.1 Number States in the Continuum Field 80

6.3 One-Dimensional Treatment 84

6.3.1 Intensity 85

6.3.2 Description in the Time Domain 86

7 Coherence 89

7.1 Introduction 89

7.2 Review of Classical Coherence Theory 89

7.2.1 First-Order Coherence 90

7.2.2 Second-Order Coherence 92

7.2.3 Chaotic Light 93

7.3 Quantum Coherence Theory 96

7.3.1 Coincidence Detection Using an Ideal Photon Detector 96

7.3.2 Field Correlations 98

7.3.3 Coherence 101

8 Examples for Electromagnetic States 103

8.1 Introduction 103

8.2 Quantum Phase of Radiation Fields 103

8.2.1 Dirac's Phase Operator 104

8.2.2 Quantum Sine and Cosine Operators 105

8.2.3 Phase State Projectors 108

8.3 Single-Mode States 109

8.3.1 Pure Single-Mode States 110

8.3.2 Statistical Mixtures of Single-Mode States 112

8.3.3 Coherent States 113

8.4 Multimode States 117

8.4.1 Multimode Fock States 117

8.4.2 Multimode Coherent States 119

8.4.3 Localized Radiation (Wave Packets Describing Localized Photons) 120

8.4.4 Chaotic Light 123

8.5 One-Dimensional Continuum Mode States 124

Part III Interaction of X-Rays with Matter 125

9 Interaction of the Electromagnetic Field with Matter 127

9.1 Introduction 127

9.2 Tensor Product of Matter and Radiation Hilbert Spaces 127

9.3 Interaction Hamiltonian for the Electromagnetic Field and Matter 128

10 Time-Dependent Perturbation Theory 133

10.1 Introduction 133

10.2 Interaction Picture 134

10.2.1 Pure States 134

10.2.2 Mixed States 136

10.3 Transition Probabilities 137

10.3.1 Time Dependence of Perturbations 137

10.3.2 Transition Probabilities 139

10.4 Perturbative Expansion of Transition Amplitudes 141

10.4.1 Transition Amplitude in First Order 144

10.4.2 Transition Amplitude in Second Order 145

10.4.3 Transition Between Discrete States 148

10.4.4 Transition from Discrete to Continuous States 149

10.4.5 Transition Between Continuous States 152

10.4.6 Scattering (^S) and Transition ( ^ T) Matrices 153

10.5 Time-Dependent Perturbation Theory for Mixed States 154

10.5.1 Isolated System 154

10.5.2 Coupled Systems 155

11 Application of Perturbation Theory to the Interaction of Electromagnetic Fields with Matter 159

11.1 Introduction 159

11.2 Feynman Diagrams 160

11.3 Mixed States 161

11.3.1 Transition Probabilities 162

Part IV Applications of X-Ray-Matter-Interaction Theory 165

12 X-Ray Scattering by Free Electrons 167

12.1 Introduction 167

12.2 Energy and Momentum Conservation 167

12.2.1 Scattering of Photons by Free Electrons 167

12.2.2 A Free Electron Cannot Absorb a Photon 170

12.3 Scattering Cross Section 171

12.4 Scattering From an Electron at Rest 176

12.4.1 Kinematics 176

12.4.2 Nonrelativistic Scattering Cross Section 177

12.4.3 Polarization 178

12.4.4