Lectures in Scattering Theory (eBook)

International Series of Monographs in Natural Philosophy

A. G. Sitenko (Autor)

P. J. Shepherd (Herausgeber)

eBook Download: PDF

2013 | 1. Auflage
280 Seiten
Elsevier Science (Verlag)
978-1-4831-8682-5 (ISBN)

Infos

Lectures in Scattering Theory discusses problems in quantum mechanics and the principles of the non-relativistic theory of potential scattering. This book describes in detail the properties of the scattering matrix and its connection with physically observable quantities. This text presents a stationary formulation of the scattering problem and the wave functions of a particle found in an external field. This book also examines the analytic properties of the scattering matrix, dispersion relations, complex angular moments, as well as the separable representation of the scattering amplitude. The text also explains the method of factorizing the potential and the two-particle scattering amplitude, based on the Hilbert-Schmidt theorem for symmetric integral equations. In investigating the problem of scattering in a three-particle system, this book notes that the inapplicability of the Lippman-Schwinger equations can be fixed by appropriately re-arranging the equations. Faddeev equations are the new equations formed after such re-arrangements. This book also cites, as an example, the scattering of a spin-1/2 particle by a spinless particle (such as the scattering of a nucleon by a spinless nucleus). This text is suitable for students and professors dealing with quantum mechanics, theoretical nuclear physics, or other fields of advanced physics.

Inhalt

Front Cover 1
Lectures in Scattering Theory 4
Copyright Page 5
Table of Contents 6
Preface 10
Chapter 1. Quantum-mechanical Description and Representations 12
1.1. Quantum-mechanical Description of Physical Systems 12
1.2. The Schrödinger Representation 14
1.3. The Heisenberg Representation 15
1.4. The Interaction Representation 16
Chapter 2. The Scattering Matrix and Transition Probability 21
2.1. The Scattering Matrix 21
2.2. The Time-shift Operator in the Interaction Representation 24
2.3. Integrals of Motion and Diagonalization of the 5-Matrix 28
2.4. The Transition Probability per Unit Time 29
2.5. An Integral Equation for the /-Matrix 32
2.6. Transformation of the Scattering Matrix. Cross-sections 34
Chapter 3. Stationary Scattering Theory 43
3.1. The Scattering Amplitude 43
3.2. The Lippmann-Schwinger Equation 47
3.3. The Relation between the Scattering Amplitude and the Transition Matrix 50
3.4. Inelastic Scattering and Reactions 51
3.5. The Born Approximation 54
Chapter 4. Wave Function of a Particle in an External Field 62
4.1. Scattering in a Central Field. Expansion in Partial Waves 62
4.2. The Rectangular Potential Well 69
4.3. The Coulomb Field 72
Chapter 5. The Optical Theorem 80
5.1. The Relation between the Total Cross-section and the Elastic Scattering Amplitude 80
5.2. The Unitarity Relation for the Elastic Scattering Amplitude 81
Chapter 6. Time Reversal and the Reciprocity Theorem 85
6.1. Transformation of the Wave Functions and Operators on Time Reversal 85
6.2. The Time-reversal Operator for Specific Systems 88
6.3. The Time-reversed Wave Function 89
6.4. The Reciprocity Theorem and Detailed Balance 92
Chapter 7. Analytic Properties of the Scattering Matrix 97
7.1. Analytic Properties of the Radial Wave Functions 97
7.2. The Case of Non-zero Angular Momenta 103
7.3. Zeros of the Jost Function and Bound States 105
7.4. The Symmetry and Location of the Scattering Matrix Singularities in the Complex Plane 109
7.5. Bound States and Redundant Zeros 113
7.6. Quasi-stationary States and Resonances 117
7.7. Virtual States 124
7.8. The Scattering Matrix in the Case of a Rectangular Potential Well 126
Chapter 8. Dispersion Relations 145
8.1. Integral Representations of the Jost Functions 145
8.2. Levinson's Theorem 149
8.3. The Complex Energy Surface 150
8.4. Analyticity of the Scattering Matrix and the Causality Principle 152
8.5. Dispersion Relations for the Scattering Amplitude 154
Chapter 9. Complex Angular Momenta 160
9.1. Analytic Properties of the Scattering Matrix in the Complex Angular Momentum Plane 160
9.2. Poles of the Scattering Matrix in the Complex Angular Momentum Plane 165
9.3. Asymptotic Behaviour of the Scattering Amplitude when COS ..8 171
Chapter 10. Separable Representation of the Scattering Amplitude 177
10.1. The Scattering Amplitude off the Energy Surface 177
10.2. The Hilbert-Schmidt Expansion for the Scattering Amplitude 180
10.3. Properties of the Eigenvalues and Eigenfunctions of the Kernel of the Lippmann-Schwinger Equation 182
Chapter 11. Scattering in a Three-particle System 198
11.1. The Faddeev Equations 198
11.2. Positions and Momenta in a Three-particle System 203
11.3. The Momentum Representation 205
11.4. Expansion in Partial Waves 208
11.5. Separable Expansion of the Two-particle /-Matrix and Reduction of the Faddeev Integral Equations to Onedimensional Form 211
Chapter 12. Scattering of Particles with Spin 222
12.1. The Spin Wave Function and Density Matrix 222
12.2. Expansion of the Density Matrix in Spin-tensors 229
12.3. The Scattering Amplitude of Particles with Spin 235
12.4. Coupling of Spin and Orbital Angular Momenta and Diagonalization of the S-Matrix 241
12.5. Scattering of a Spin-½ Particle by a Spinless Particle 247
12.6. Scattering of a Spin-1 Particle by a Spinless Particle 256
Bibliography 274
Index 276

Produkt-Details

• Erscheint lt. Verlag: 22.10.2013
• Sprache: englisch
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Quantenphysik
• Themenwelt: Sozialwissenschaften ► Ethnologie
• Themenwelt: Sozialwissenschaften ► Politik / Verwaltung
• Themenwelt: Sozialwissenschaften ► Soziologie
• ISBN-10: 1-4831-8682-2 / 1483186822
• ISBN-13: 978-1-4831-8682-5 / 9781483186825