1. 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

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

3. 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.

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

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

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

10. The final expression:

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

12. 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….

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

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

15. 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!

16. 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!

17. 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!

18. 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!

19. Spheric Symmetric potentials:
Total Cross Section:

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

21. 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

22. 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