Russell's Blog

There is no class today on account of President's day. Instead, here is the summary for Wednesday's lecture, which I neglected to post.

We began examining the geometry of the scattering problem. The scattering geometry is this: A target sits at the origin of a spherical coordinate system. A beam of particles is directed at the target. The target is treated as a macroscopic object, so the beam is expressed in terms of a "probability current." This is simply a way of treating a continuous stream of traveling wave functions. Of course, each wave function must be normalized, but because the beam is switched on for an arbitrary length of time (forever) the beam itself is not. Thus, the probability current formalism describes the flux of probability through the surface. Typically, one would choose the surface to be the beam-facing surface of the target.

Some sort of interaction occurs among the particles in the target and the particles in the beam; usually, the particles in the beam are scattered off of a potential in the target. The Rutherford scattering experiment is an example of a beam scattered by a potential. Alternatively, interaction with the beam could change the state of the target; a photon beam could excite the valence electrons in the target, which would eventually return to the unexcited state and re-emit the photons, or beam of positrons could annihilate with electrons in the target, or a beam of high-energy neutrons could smash the target nuclei into assorted atomic debris. If the beam is scattered by the target without changing the state of the target, this is called elastic scattering.

The scattering problem is useful because because beam-target interactions will divert the beam into a characteristic angular distribution of beam particles. The scattering problem can be used to study the properties of the target, the beam, the interaction, or any combination thereof.

Nonrelativistic elastic scattering

To introduce the technique, we consider a beam of nonrelativistic, spinless particles scattered elastically by a fixed potential. Because the number of collisions in a given time will vary with the beam intensity, it is convenient to eliminate the time interval from the equations by expressing the problem in terms of a cross section. The term is an analogy to the classical collision problem; one may imagine the target as a cloud of spheres with a given radius. The classical cross section of such a system would be the sum of the two-dimensional area presented by each sphere to the beam per unit volume of the target. One can think of it roughly as the "effective area" per unit area of the target, or as the "opacity" of the target.

For Coulomb scattering, the beam and the target will not actually contact one another (otherwise we would have to take other forces into account, and it wouldn't be Coulomb scattering any more). The cross section is then the area within which a beam particle will "significantly" interact with a target particle. Rather than specifying some arbitrary definition of "significantly," it is better to define the problem in terms of a differential cross section which scatters the the beam into a differential solid angle. This is better illustrated by a picture, so I'll stop blathering about the geometry.

As I mentioned above, the beam is not normalizable. Generally, one treats it as a plane wave. The scattering problem amounts to calculating over what solid angle the probability current passing through a given differential cross section will be scattered by a potential.