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.
Taylor J. R.: Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover 2006
Friedrich H.: Theoretical Atomic Physics, Springer Verlag, Heidelberg 1991
Kukulin V.I., Krasnopolsky V.M., Horáček J.: Theory of Resonances, Kluwer-Academia, Praha 1989
Oral exam. Before coming to exam, student must solve theoretical excercise which will be given to all students in the middle of semester. Oral exam consists of two questions. First question is to explain in detail the theoretical background of the solution of the excercise. The second question will be selected from the topics covered by sylabus of the lecture.
I. INTRODUCTION TO CLASSICAL SCATTERING THEORY:
Trajectory, asymptote. Deflection function, differential cross section.
Hard sphere. Typical interatomic potential: rainbow, glory and orbiting.
II. GENERAL FORMULATION OF QUANTUM SCATTERING THEORY:
Free dynamics and interaction. Trajectory, asymptote.
Asymptotic condition and Moller operator.
Orthogonality, asymptotic completeness, S-operator.
Properties of S-operator, energy conservation. K-matrix.
Differential cross section.
General definitions shown on example of particle scattering on potential in 3D.
III. TIME INDEPENDENT FORMULATION:
Green operator and resolvent and their properties.
Connection to Moller operators, T-matrix and expression for S-operator.
Lippmann-Schwinger equation for stationary states and T-operator.
Asymptotics of stationary scattering solution.
IV. SCATTERING FROM SPHERICALLY SYMMETRIC POTENTIAL:
Angular momentum conservation for scattering quantities. Eigenphases.
Partial scattering amplitude and partial wave expansion of cross section.
Expansion of Green function and stationary states.
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 theorem.
VI. ANALYTIC PROPERTIES NEAR E->0 AND NEAR RESONANCE.
Scattering length and its behavior.
Resonance. Phaseshift behavior.
Breit-Wigner and Fano formula.
VII. INTRODUCTION TO MULTICHANNEL SCATTERING
Channels, channel hamiltonian and interaction.
Moller operators and S-matrix. T-operators.
Stationary scattering states and multichannel L-S equation.
Projection method and optical potential.
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
Use on nonspherical and nonlocal potentials.
Boundary conditions and cross sections.
XI. INTRODUCTION IN QUANTUM DEFECT THEORY
Rydberg states and quantum defect.
Threshod behavior and Seaton's theorem.