Aperçu des semaines

  • Généralités

  • 28 septembre - 4 octobre

  • 5 octobre - 11 octobre

      • Deflection function
      • Collision cross section
      • Hard sphere scattering, Coulomb scattering
      • Conditions on validity of the classical approach
      • Examples of quantum vs classical regime
      • Scattering cross section and its relation to reaction rate coefficients
  • 12 octobre - 18 octobre

      • Importance of using symmetry
      • Standing wave basis of solutions
      • Scattering solutions as an expansion into standing wave basis
      • Application to 1D potential well scattering
      • Scattering phase shift
      • Ramsauer-Townsend minimum
  • 19 octobre - 25 octobre

      • Scattering boundary conditions
      • Scattering amplitude and the definition of the differential scattering cross section
      • Partial wave expansion and the standing-wave basis in 3D
      • Expansion coefficients for the scattering state
      • Riccati functions
      • Asymptotic phase shift and its relation to attractive and repulsive potentials
      • Cross sections and the unitary limit on partial cross sections
      • K-matrix, S-matrix and T-matrix
      • Wigner's threshold law
      • Applications: low-energy and high energy behavior of hard sphere scattering, square well scattering. The role of energy vs angular momentum.
  • 26 octobre - 1 novembre

  • 2 novembre - 8 novembre

      • Scattering of wavepackets vs plane-waves
      • Definition of quantum cross section using probabilities and integration over impact parameters
      • Expansion of TD solution into basis of stationary scattering states
      • Use of stationary phase approximation to calculate the shape of scattered wavepacket
      • Equivalence of the time-dependent result with the stationary result
      • Assumptions on the geometry of the scattering experiment
  • 9 novembre - 15 novembre

      • Integral form of Schrodinger equation
      • Derivation of the free Green's operator of Schrodinger equation
      • Regularization of the Green's function and the correspondence with boundary conditions
      • Integral form of the scattering amplitude
      • Definition of the T-operator (and T-matrix)
      • Explicit solution of the LS equation using full Green's function of Schrodinger equation
      • LS equation for Green's operator and the T-operator
      • Partial-wave expansion of the momentum-space T-matrix elements: reduction to partial wave T-matrix elements
      • Born approximation and its validity. Example of use for electron-atom scattering, the atomic form factor.
  • 16 novembre - 22 novembre

      • Expression for S-matrix in terms of Jost functions
      • Regular solution and its relation to Jost functions
      • Analyticity of the Regular solution
      • Analytic continuation of Jost functions into complex plane
      • Regions of analyticity of Jost functions and the S-matrix
      • Extensions of analyticity for short-range potentials
      • Distribution of poles of the S-matrix in the complex plane
      • Resonances, virtual states, bound states as poles of the S-matrix
      • Breit-Wigner form of resonant scattering cross section and phase shift
      • Signatures of resonances and virtual states in scattering cross sections and phase shifts
      • Riemann surface of S-matrix in complex energy plane: physical vs unphysical sheets.
      • Classical scattering asymptotes and orbits vs quantum in/out asymptotic and scattering states
      • Unitary vs Isometric operators
      • Asymptotic condition, assumptions on the scattering potentials, orthogonality theorem
      • Intertwining relations for Moeller operators and implication for conservation of energy in scattering.
      • S-matrix and its decomposition using scattering amplitude
      • Calculation of quantum cross section in momentum space
      • Optical theorem
      • Derivation of T-matrix elements and scattering amplitude from TD point of view and connection with the stationary theory
  • 23 novembre - 29 novembre

  • 30 novembre - 6 décembre