Scattering Theory

1. Quantum Mechanical Description and Representations.- 1.1 Quantum Mechanical Description of Physical Systems.- 1.2 Schrödinger Representation.- 1.3 Heisenberg Representation.- 1.4 Interaction Representation.- 1.5 Time-Dependent Green Functions.- Problems.- 2. The Scattering Matrix and Transition Probability.- 2.1 The Scattering Matrix.- 2.2 Time Shift Operator in the Interaction Representation.- 2.3 Integrals of Motion and S Matrix Diagonalization.- 2.4 Transition Probability per Unit Time.- 2.5 Integral Equation for die t Matrix.- 2.6 Transformation of the Scattering Matrix. Cross Sections.- Problems.- 3 Stationary Scattering Theory.- 3.1 The Scattering Amplitude.- 3.2 The Lippmann-Schwinger Equation.- 3.3 The Möller Operators ?+ and ?-.- 3.4 The Green Functions G0 and G.- 3.5 The Scattering Amplitude and the Transition Matrix.- 3.6 Inelastic Scattering and Reactions.- 3.7 Born Approximation and Perturbation Theory.- 3.8 High Energy Approximation.- Problems.- 4. Particle Wave Functions in the External Field.- 4.1 Partial Wave Expansion.- 4.2 Square Well Potential.- 4.3 Coulomb Field.- 4.4 Partial Green Functions and the Scattering Matrix.- 4.5 Variable Phase Approach.- Problems.- 5. Optical Theorem.- 5.1 The Total Cross Section and the Elastic Scattering Amplitude.- 5.2 Unitarity Relation for the Elastic Scattering Amplitude.- Problems.- 6. Time Inversion and Reciprocity Theorem.- 6.1 Transformation of Wave Functions and Operators Under Inversion of Time.- 6.2 Time Inversion Operators for Particular Systems.- 6.3 Time-Inversed Wave Function.- 6.4 The Reciprocity Theorem and Detailed Balance.- Problems.- 7. Analytic Properties of the Scattering Matrix.- 7.1 Analytic Properties of Radial Wave Functions.- 7.2 Generalization to Include Nonzero Angular Momenta.- 7.3Jost Function Zeros and Bound States.- 7.4 Symmetry and Dislocation of Scattering Matrix Singularities in the Complex k Plane.- 7.5 Bound States and Extra Zeros.- 7.6 Quasistationary States and Resonances.- 7.7 Virtual States.- 7.8 The Scattering Matrix in the Case of a Square Well Potential.- Problems.- 8. Dispersion Relations.- 8.1 Integral Representations of the Jost Functions.- 8.2 Levinson Theorem.- 8.3 Complex Energy Shell.- 8.4 Analyticity of the Scattering Matrix and the Causality Principle.- 8.5 Dispersion Relations for the Forward Direction Scattering Amplitude.- 8.6 Dispersion Relations for the Arbitrary Direction Scattering Amplitude.- Problems.- 9. Complex Angular Momenta.- 9.1 Analytic Properties of the Scattering Matrix in the Complex Angular Momentum Plane.- 9.2 Poles of the Scattering Matrix in the Complex Angular Momentum Plane.- 9.3 Analytic Properties of the Scattering Amplitude in the Complex z Plane.- 9.4 Asymptotic Behavior of the Scattering Amplitude for Large z.- 9.5 Momentum Transfer Dispersion Relations.- Problems.- 10. Double Dispersion Relations.- 10.1 Mandelstam Representation.- 10.2 Spectral Density and Unitarity Condition.- Problems.- 11. The Inverse Problem of Scattering Theory.- 11.1 Integral Representation of the Solutions of the Scattering Problem.- 11.2 Reproducing the Potential by the Scattering Phase Shifts.- Problems.- 12. Separable Representation of the Scattering Amplitude.- 12.1 The Scattering Amplitude off the Energy Shell.- 12.2 Hilbert-Schmidt Expansion of the Scattering Amplitude.- 12.3 Properties of Eigenvalues and Eigenfunctions of the Kernel of the Lippmann-Schwinger Equation.- Problems.- 13. Three-Particle Scattering.- 13.1 The Faddeev Equations.- 13.2 Coordinates and Momenta in the Three-Particle System.- 13.3 Momentum Representation.- 13.4 Partial Wave Expansion.- 13.5 Separable Expansion of the Two-Particle t Matrix and One-Dimensional Form of the Faddeev Equations.- Problems.- 14. Scattering of Spin-Possessing Particles.- 14.1 The Spin Wave Function and the Density Matrix.- 14.2 Spin-Tensor Expansion of the Density Matrix.- 14.3 The Scattering Amplitude in the Case of Spin-Possessing Particles.- 14.4 Addition of Spin and Angular Momentum and Diagonalization of the S Matrix.- 14.5 Spin ½ - Spin 0 Particle Scattering.- 14.6 Spin 1 - Spin 0 Particle Scattering.- Problems.- References.- General Reading.