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

Scattering Theory: 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

Number of scattered particles into : per unit time: Differential Cross Section: Total Cross Section: Dimension: Area Interpretation: Effective area for scattering. Dimension: None Interpretation: ”Probability” for scattering into The Scattering Cross Section (To be corrected, see Endre Slide)

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

Quantal Scattering - No Trajectory! (A plane wave hits some object and a spherical wave emerges ) 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. Procedure:

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

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

Example from 1D: Forward scattering Reflection In this case (since potential is discontinuous) we can find f excactly by ”gluing”

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

The final expression:

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

Integral equation 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!

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

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

We obtains: 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

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

The Born approximation: The scattering amplitude is then: :) The momentum change Fourier transform of the potential! Use incomming wave instead of Under integration sign: 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: Depends on angles - but roughly independent when qR << 1 Thats it!