Quantum scattering theory
Weekly outline


 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



 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
 RamsauerTownsend minimum



 Scattering boundary conditions
 Scattering amplitude and the definition of the differential scattering cross section
 Partial wave expansion and the standingwave 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
 Kmatrix, Smatrix and Tmatrix
 Wigner's threshold law
 Applications: lowenergy and high energy behavior of hard sphere scattering, square well scattering. The role of energy vs angular momentum.



 Derivation of Numerov method
 Computation of scattering phase shifts
 Search for bound states and numerical instabilities of the Numerov method


 Scattering of wavepackets vs planewaves
 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 timedependent result with the stationary result
 Assumptions on the geometry of the scattering experiment



 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 Toperator (and Tmatrix)
 Explicit solution of the LS equation using full Green's function of Schrodinger equation
 LS equation for Green's operator and the Toperator
 Partialwave expansion of the momentumspace Tmatrix elements: reduction to partial wave Tmatrix elements
 Born approximation and its validity. Example of use for electronatom scattering, the atomic form factor.



 Expression for Smatrix 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 Smatrix
 Extensions of analyticity for shortrange potentials
 Distribution of poles of the Smatrix in the complex plane
 Resonances, virtual states, bound states as poles of the Smatrix
 BreitWigner 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 Smatrix 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.
 Smatrix and its decomposition using scattering amplitude
 Calculation of quantum cross section in momentum space
 Optical theorem
 Derivation of Tmatrix elements and scattering amplitude from TD point of view and connection with the stationary theory



 Kohn variational principle
 Schwinger variational principle

 Eigenchannel formulation
 Variational principle
 Eigenchannel formulation  channel decomposition
 Spectral decomposition



 Numerical solution of a multichannel problem
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