Sitenko / Shepherd Lectures in Scattering Theory

International Series of Monographs in Natural Philosophy
1. Auflage 2013
ISBN: 978-1-4831-8682-5
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 280 Seiten, Web PDF

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.

Autoren/Hrsg.

Sitenko, A. G.

Shepherd, P. J.

Inhaltsverzeichnis

1;Front Cover;1
2;Lectures in Scattering Theory;4
3;Copyright Page;5
4;Table of Contents;6
5;Preface;10
6;Chapter 1. Quantum-mechanical Description and Representations;12
6.1;1.1. Quantum-mechanical Description of Physical Systems;12
6.2;1.2. The Schrödinger Representation;14
6.3;1.3. The Heisenberg Representation;15
6.4;1.4. The Interaction Representation;16
7;Chapter 2. The Scattering Matrix and Transition Probability;21
8;2.1. The Scattering Matrix;21
9;2.2. The Time-shift Operator in the Interaction Representation;24
10;2.3. Integrals of Motion and Diagonalization of the 5-Matrix;28
11;2.4. The Transition Probability per Unit Time;29
12;2.5. An Integral Equation for the /-Matrix;32
13;2.6. Transformation of the Scattering Matrix. Cross-sections;34
14;Chapter 3. Stationary Scattering Theory;43
14.1;3.1. The Scattering Amplitude;43
14.2;3.2. The Lippmann-Schwinger Equation;47
14.3;3.3. The Relation between the Scattering Amplitude and the Transition Matrix;50
14.4;3.4. Inelastic Scattering and Reactions;51
14.5;3.5. The Born Approximation;54
15;Chapter 4. Wave Function of a Particle in an External Field;62
15.1;4.1. Scattering in a Central Field. Expansion in Partial Waves;62
15.2;4.2. The Rectangular Potential Well;69
15.3;4.3. The Coulomb Field;72
16;Chapter 5. The Optical Theorem;80
16.1;5.1. The Relation between the Total Cross-section and the Elastic Scattering Amplitude;80
16.2;5.2. The Unitarity Relation for the Elastic Scattering Amplitude;81
17;Chapter 6. Time Reversal and the Reciprocity Theorem;85
17.1;6.1. Transformation of the Wave Functions and Operators on Time Reversal;85
17.2;6.2. The Time-reversal Operator for Specific Systems;88
17.3;6.3. The Time-reversed Wave Function;89
17.4;6.4. The Reciprocity Theorem and Detailed Balance;92
18;Chapter 7. Analytic Properties of the Scattering Matrix;97
18.1;7.1. Analytic Properties of the Radial Wave Functions;97
18.2;7.2. The Case of Non-zero Angular Momenta;103
18.3;7.3. Zeros of the Jost Function and Bound States;105
18.4;7.4. The Symmetry and Location of the Scattering Matrix Singularities in the Complex Plane;109
18.5;7.5. Bound States and Redundant Zeros;113
18.6;7.6. Quasi-stationary States and Resonances;117
18.7;7.7. Virtual States;124
18.8;7.8. The Scattering Matrix in the Case of a Rectangular Potential Well;126
19;Chapter 8. Dispersion Relations;145
19.1;8.1. Integral Representations of the Jost Functions;145
19.2;8.2. Levinson's Theorem;149
19.3;8.3. The Complex Energy Surface;150
19.4;8.4. Analyticity of the Scattering Matrix and the Causality Principle;152
19.5;8.5. Dispersion Relations for the Scattering Amplitude;154
20;Chapter 9. Complex Angular Momenta;160
20.1;9.1. Analytic Properties of the Scattering Matrix in the Complex Angular Momentum Plane;160
20.2;9.2. Poles of the Scattering Matrix in the Complex Angular Momentum Plane;165
20.3;9.3. Asymptotic Behaviour of the Scattering Amplitude when COS ..8;171
21;Chapter 10. Separable Representation of the Scattering Amplitude;177
21.1;10.1. The Scattering Amplitude off the Energy Surface;177
21.2;10.2. The Hilbert-Schmidt Expansion for the Scattering Amplitude;180
21.3;10.3. Properties of the Eigenvalues and Eigenfunctions of the Kernel of the Lippmann-Schwinger Equation;182
22;Chapter 11. Scattering in a Three-particle System;198
22.1;11.1. The Faddeev Equations;198
22.2;11.2. Positions and Momenta in a Three-particle System;203
22.3;11.3. The Momentum Representation;205
22.4;11.4. Expansion in Partial Waves;208
22.5;11.5. Separable Expansion of the Two-particle /-Matrix and Reduction of the Faddeev Integral Equations to Onedimensional Form;211
23;Chapter 12. Scattering of Particles with Spin;222
23.1;12.1. The Spin Wave Function and Density Matrix;222
23.2;12.2. Expansion of the Density Matrix in Spin-tensors;229
23.3;12.3. The Scattering Amplitude of Particles with Spin;235
23.4;12.4. Coupling of Spin and Orbital Angular Momenta and Diagonalization of the S-Matrix;241
23.5;12.5. Scattering of a Spin-½ Particle by a Spinless Particle;247
23.6;12.6. Scattering of a Spin-1 Particle by a Spinless Particle;256
24;Bibliography;274
25;Index;276