Introduction to formal non-relativistic quantum scattering theory. Problem solving in scattering theory and basics of numerical solution of scattering problems. Analytic properties of scattering quantities. Aimed at graduate students of theoretical physics, mathematical modelling and chemical physics.
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Classical scattering theory: trajectory, deflection function, differential cross sections.
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Elements of quantum theory of collisions: trajectories in Hilbert space, bound and scattering spectrum, scattering operator, scattering experiment and observable cross sections.
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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.
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Scattering from a spherically symmetric potential: conservation of angular momentum and decoupling of partial waves. Scattering phase shift, partial cross sections.
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Practical applications: numerical implementation of a method for solving radial Schrödinger equation. Application to electron scattering from atoms.
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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.
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Partial wave method: application to non-spherical and non-local problems. Boundary conditions, relations between different bases (K-matrix, T-matrix, S-matrix).
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Multichannel scattering: definition of channels, stationary scattering states and coupled-channels approach.
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Variational methods: Kohn approach, Schwinger method for scattering amplitude, R-matrix approach, pole expansion of the R-matrix.
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Quantum defect theory: Rydberg states and quantum defect. Threshold behavior and Seaton's theorem.