Quantum scattering theory
Section outline
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- 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
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- 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
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- 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.
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- Derivation of Numerov method
- Computation of scattering phase shifts
- Search for bound states and numerical instabilities of the Numerov method
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- 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
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- 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.
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- 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.
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- 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
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- Kohn variational principle
- Schwinger variational principle
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- Eigenchannel formulation
- Variational principle
- Eigenchannel formulation - channel decomposition
- Spectral decomposition
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- Numerical solution of a multichannel problem
- Completion of this task is required to be eligible for the final exam