Syllabus
* introduction to scattering theory
- time dependent picture of scattering, asymptotic condition, in- and out- states, S-matrix, cross sections
- time independent formalism of scattering theory: Lippmann-Schwinger equation, T-matrix, Green operators, Born series
- applications: transmission/reflection probabilities for 1D problems, scattering cross sections in 3D
- scattering resonances: shape, Feshbach, light induced
* nonhermitian scattering theory
- analytic continuation of the S-matrix into the complex energy/momentum plane
- classification of the poles of the S-matrix: (anti)bound states, (anti)resonances
- Siegert pseudostate formalism interconnecting the nonhermitian world with the conventional scattering theory
- application: scattering problems in 1D
* complex scaling methods for the calculation of resonances
- wavefunction of metastable states in the x-representation
- complex transformations of Hamiltonian as implicated by its Laplace-Fourier representation
- numerical applications of the complex transformations of the Hamiltonian
* atoms in strong electromagnetic field
- Hamiltonian for laser-matter interaction
- application of the Floquet theory for the interaction of atoms with electromagnetic field in the classical dipole approximation
* incomplete spectrum in the case non-hermitian Hamiltonian
- exceptional point (branch point)
- quantum dynamics when encircling an exceptional point
- PT symmetric waveguides
Annotation
This course is suitable for students who have passed through an introductory quantum mechanics, and who wish to delve deeper into more sophisticated and subtle parts of quantum theory (scattering, nonhermitian formalism, light-matter interaction). The lectures are aimed not only at highlighting fundamental physics insights, but also at introducing a broad range of powerful mathematical and computational techniques.
At the end of the course, the students can choose either an open book exam or a final project (which can possibly end up with an original scientific paper).