Syllabus
I. INTRODUCTION TO CLASSICAL SCATTERING THEORY:
Trajectory, asymptote. Deflection function, differential cross section.
EXAMPLES: Hard sphere. Typical interatomic potential: rainbow, glory and orbiting phenomena.
II. GENERAL FORMULATION OF QUANTUM SCATTERING THEORY:
Separation of free and non-free dynamics. Trajectory, asymptote. Asymptotic condition and Moller operator. Orthogonality, asymptotic completeness, S-operator. Properties of S-operator, energy conservation, optical theorem. Differential cross section.
III. TIME INDEPENDENT FORMULATION:
Green’s operator and resolvent and their properties. Lippmann-Schwinger equation for stationary states and T-operator. Asymptotics of stationary scattering solution in different bases: K-matrix, T-matrix, S-matrix.
IV. SCATTERING FROM SPHERICALLY SYMMETRIC POTENTIAL:
Angular momentum conservation for scattering quantities. Scattering phase shift. Partial scattering amplitude and partial wave expansion of cross section. Expansion of the Green’s function and stationary states.
EXAMPLES: Practical applications. Numerical implementation of a method for solving radial Schrödinger equation. Application to electron scattering from atom.
V. ANALYTICITY IN p AND E
Transformation of L-S equation to equation of Voltera type. Jost function and Jost solution and their properties. Interpretation of S-matrix poles. Levinson’s theorem.
VI. ANALYTIC PROPERTIES NEAR E -> 0 AND NEAR A RESONANCE.
Scattering length and its behavior. Resonance. Phaseshift behavior. Breit-Wigner and Fano formula.
VII. INTRODUCTION TO MULTICHANNEL SCATTERING
Channels, channel hamiltonian and interaction. Stationary scattering states and multichannel L-S equation. Method of coupled-channels. Cross sections.
VIII. VARIATIONAL PRINCIPLES IN SCATTERING
Kohn method and its usage in basis. Schwinger variational principle.
IX. R-MATRIX METHOD
Basic principles and derivation of the method. Use of basis. Pole expansion of R-matrix.
X. PARTIAL WAVE METHOD
Application to nonspherical and nonlocal potentials. Boundary conditions and cross sections.
XI. INTRODUCTION TO QUANTUM DEFECT THEORY
Rydberg states and quantum defect. Threshod behavior and Seaton's theorem.
Annotation
Introduction to formal non-relativistic quantum scattering theory. Analytic properties of scattering quantities.
Solved problems in scattering theory and basics of numerical solution of scattering problems. For the graduate students of the theoretical physics, mathematical modelling and chemical physics.