Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

Chapter 3
Scattering from a potential
Until now we have considered the discrete energy levels of simple, one-electron
Hamiltonians, corresponding to bound, localized states. Unbound, delocalized
states exist as well for any physical potential (with the exception of idealized
models like the harmonic potential) at sufficiently high energies. These states
are relevant in the description of elastic scattering from a potential, i.e. processes
of diffusion of an incoming particle. Scattering is a really important
subject in physics, since what many experiments measure is how a particle is
deflected by another. The comparison of measurements with calculated results
makes it possible to understand the form of the interaction potential between
the particles. In the following a very short reminder of scattering theory is
provided; then an application to a real problem (scattering of H atoms by rare
gas atoms) is presented. This chapter is inspired to Ch.2 (pp.14-29) of the book
of Thijssen.
3.1 Short reminder of the theory of scattering
The elastic scattering of a particle by another is first mapped onto the equivalent
problem of elastic scattering from a fixed center, using the same coordinate
change as described in the Appendix. In the typical geometry, a free particle,
described as a plane wave with wave vector along the z axis, is incident on the
center and is scattered as a spherical wave at large values of r (distance from the
center). A typical measured quantity is the differential cross section dσ(Ω)/dΩ,
i.e. the probability that in the unit of time a particle crosses the surface in the
surface element dS = r 2 dΩ (where Ω is the solid angle, dΩ = sin θdθdφ, where
θ is the polar angle and φ the azimuthal angle). Another useful quantity is the
total cross section σ tot = ∫ (dσ(Ω)/dΩ)dΩ. For a central potential, the system
is symmetric around the z axis and thus the differential cross section does not
depend upon φ. The cross section depends upon the energy of the incident
particle.
Let us consider a solution having the form:
ψ(r) = e ik·r + f(θ)
r eikr (3.1)
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