QuantumPhysics.dvi

• Electro-magnetic: long-ranged; the mediator is the massless photon.


• Weak interactions: short-ranged; the mediator are the massiveW±andZbosons with masses
  on the order of 80GeV/c^2 and 90GeV/c^2 respectively.


• Strong interactions: short-ranged in the following sense.Although the mediator of the strong
  force in QCD is the massless gluon, confinement makes the range of the strong interactions
  short, the scale being set by the mass of the lightest strongly interacting particle, namely the
  π±andπ^0 , whose masses are approximately 135MeV/c^2.

12.6 The wave-function solution far from the target

In practice, one is interested in observing the scattering products far from the target, i.e. far from
the core of the potential. In this limit, the equations simplify considerably, and will yield physical
insight into the scattering process. The starting point is the integral equation (12.23), in the limit
of larger=|x|≫|y|. We use the approximation,

|x−y|=

√

(x−y)^2 =r−

x·y

r

+O(y^2 ) (12.29)

The propagator the becomes (we retain only the propagatorG=G+),

G(x,y;k^2 ) =−

eikr

4 πr

×e−ik

′·y

k′=k

x

r

(12.30)

The Lippmann-Schwnger equation is then solved by the following form of the wave function,

ψk(x) =

1

(2π)^3 /^2

{

eik·x−

eikr

r

f(k′,k)

}

(12.31)

where the functionfis given by

f(k′,k) =−

(2π)^3 /^2

4 π

∫

d^3 ye−ik

′·y

U(y)ψk(y) (12.32)

Note that from the definition off, we may readily read off that its dimension is a length, a remark
that will come in very useful later.

12.7 Calculation of the cross section

Clearly, one of the key quantities we are going to be interested in is the intensity of the outgoing
flux of particles, as a function ofkandk′. The corresponding physical quantity is thecross section.
To evaluate it, we begin by obtaining a formula for the probability densityρand for the probability
current densityj,

ρ(t,x) = ψ∗(t,x)ψ(t,x)

j(t,x) = −

i

2 m

(

ψ∗(t,x)∇ψ(t,x)−ψ(t,x)∇ψ∗(t,x)

)

(12.33)