QuantumPhysics.dvi

Note that the scale of the cross section is set by the momentumtransferq, and the mass.

This formula exhibits singular behavior for forward scattering whenθ∼0. In fact, if one tried
to integrate this formula to get the total cross section, onewould find that it diverges because
of the singularity atθ = 0. Since this divergence occurs for vanishing momentum transfer, it
amounts to an infrared (IR) problem. The fundamental cause for this problem is the fact that
the photon, which is the particle that mediates the Coulomb interaction, is massless. Massless
particles can be produced with very little momentum and energy. Thus, it is really not possible to
distinguish between a charged particle (say and electron) state and that state with an added very
low energy photon. Thus, the problem of Coulomb scattering is not strictly speaking one in which
the number of photons is conserved. A proper calculation would really include the effects of this
photon production, but this requires quantum field theory, and is beyond the scope of this course.

In practice, one does not necessarily scatter a charged particle off a target that exhibits Coulomb
potential behavior all the way out to spatial infinity. For example, if one scatters off an electrically
neutral atom, then the cloud of electrons effectively shieldsthe Coulomb potential of the nucleus.
Beyond a distance on the scale of the size of the atom, the electric potential will in fact fall off
exponentially, since the electron wave functions do so in the Coulomb problem.

Finally, since the first Born term was computed in perturbation theory, we cannot physically
take seriously its effects if these become large. Thus, the Rutherford formula for differential cross
section should be trusted only for anglesθaway from being to close to 0.

12.8.2 The case of the Yukawa potential

The typical potential for a short ranged potential is the Yukawa potential, given by

U(r) =−U 0

e−μr

r

(12.55)

where 1/μis the range of the potential. The first Born term is then foundto be

f(1)(q) =

2 mU 0

q^2 +μ^2

(12.56)

The finite range of this potential now makes its differential cross section finite atθ= 0, and this
effect may be interpreted as the fact that the Yukawa potentialarises fundamentally from massive
particle exchange.

12.9 The optical Theorem

A particularly simple and useful relation holds between thetotal cross section and theforward
scattering amplitude, which is referred to as the optical theorem (for historicalreasons). The
relation is expressed as follows,

σtot=

∫

4 π

dΩ

(

dσ

dΩ

)

=

4 π

k

Imf(k,k) (12.57)