Quantum scattering theory
This course is taught in English and is aimed at graduate and postgraduate students of theoretical physics, mathematical modelling and chemical physics. The material is divided into two blocks, the first one taught by Zdeněk Mašín, the second one by Roman Čurík (Academy of Sciences of Czech Republic).
The main aim of the course is to introduce the students to the formalism of non-relativistic theory quantum collisions, to analytic properties of scattering quantities and to various methods of finding scattering solutions of the Schrödinger equation including numerical methods and their practical implementation.
- Classical scattering theory: trajectory, deflection function, differential cross sections.
- Elements of quantum theory of collisions: trajectories in Hilbert space, bound and scattering spectrum, scattering operator, scattering experiment and observable cross sections.
- Time-independent formulation: connection between the time-dependent scattering of wave packets and the stationary formulation. Lippmann-Schwinger equation. Asymptotic form of the stationary states. Green’s operator.
- Scattering from a spherically symmetric potential: conservation of angular momentum and decoupling of partial waves. Scattering phase shift, partial cross sections.
- Practical applications: numerical implementation of a method for solving radial Schrödinger equation. Application to electron scattering from atoms.
- Analytic properties of the scattering amplitude: Jost function and poles of the S-matrix. Interpretation of S-matrix poles. Levinson’s theorem. Resonances and phase-shift behavior. Breit-Wigner and Fano formulas.
- Partial wave method: application to non-spherical and non-local problems. Boundary conditions, relations between different bases (K-matrix, T-matrix, S-matrix).
- Multichannel scattering: definition of channels, stationary scattering states and coupled-channels approach.
- Variational methods: Kohn approach, Schwinger method for scattering amplitude, R-matrix approach, pole expansion of the R-matrix.
- Quantum defect theory: Rydberg states and quantum defect. Threshold behavior and Seaton’s theorem.