Scattering Theory: Revised and corrected 2004-2007 Classical Scattering: Differential and total cross section Examples: Hard sphere and Coulomb scattering Quantal Scattering: Formulated as a stationary problem Integral Equation Born Approximation Examples: Hard sphere and Coulomb scattering

The Scattering Cross Section out N out Number of scattered particles into : Differential Cross Section: Total Cross Section:

Quantal Scattering - No Trajectory Quantal Scattering - No Trajectory! (A plane wave hits some object and a spherical wave emerges) Procedure: Solve the time independent Schrödinger equation Approximate the solution to one which is valid far away from the scattering center Write the solution as a sum of an incoming plane wave and an outgoing spherical wave. Must find a relation between the wavefunction and the current densities that defines the cross section.

Current Density: Incomming current density: Outgoing spherical current density:

Example - Classical scattering: ® Hard Sphere scattering: = Geometrical Cross sectional area of sphere! Independent of angles!

The Schrödinger equation - scattering form: Now we must define the current densities from the wave function…

The final expression:

Summary Write the Schrödinger equation as: Asymptotics: Then we have: …. Now we can start to work

Integral equation where we require: because…. With the rewritten Schrödinger equation we can introducea Greens function, which (almost) solves the problem for a delta-function potential: Then a solution of: can be written: where we require: because….

This term is 0 This equals Integration over the delta function gives result: Formal solution: Useless so far!

Note: solves the problem! Must find G(r) in and: Then: The function: ”Proof”: The integral can be evaluated, and the result is:

Inserting G(r), we obtain: implies that: Inserting G(r), we obtain: At large r this can be recast to an outgoing spherical wave….. The Born series: And so on…. Not necesarily convergent!

Write the Schrödinger equation as: Asymptotics: SUMMARY We obtains: At large r this can be recast to an outgoing spherical wave….. The Born series: And so on…. Not necesarily convergent!

Asymptotics - Detector is at near infinite r The potential is assumed to have short range, i.e. Active only for small r’ : 1) 2) Asymptotic excact result: Still Useless!

The Born approximation: Use incomming wave instead of Under integration sign: The scattering amplitude is then: :) The momentum change Fourier transform of the potential! Valid when: Weak potentials and/or large energies!

Spheric Symmetric potentials: Total Cross Section:

Summary - 1’st. Born Approximation: Best at large energies!

Example - Hard sphere 1. Born scattering: Classical Hard Sphere scattering: Quantal Hard Sphere potential: Thats it! Depends on angles - but roughly independent when qR << 1

Approximation methods in Quantum Mechanics Kap. 7-lect2 Introduction to Approximation methods in Quantum Mechanics Time dependent Time-independent methods: Methods to obtain an approximate eigen energy, E and wave function: Golden Rule perturbation methods Methods to obtain an approximate expression for the expansion amplitudes. Ground/Bound states Continuum states Perturbation theory Variational method Scattering theory Non degenerate states Degenerate states