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Chapter 4. Particle scattering
4.1 Classical particle kinematics
The generic two-particle scattering problem assumes a single particle with vector kinetic momentum p, incident on a second particle
which is initially at rest in the lab frame. The first particle transfers
momentum and energy to the second particle, and both particles
exit to a final state. The final state is governed by conservation of
total momentum and energy, as well as the details of the interaction. The first particle is assumed to come from a distance which
is large compared with the dimensions of the interaction volume
of the two particles. Similarly, both particles exit to a large distance in the final state. At large distances the potential energy of
the interaction can be ignored. In this section we investigate the
constraints imposed by momentum and energy conservation. This
general topic is referred to as kinematics.
We assume the incident particle has rest mass m 1 , vector kinetic
momentum p 1 , and total energy E 1 . We assume the stationary
particle has rest mass m 2 , vector kinetic momentum p 2 , and total
energy E 2 . These quantities are related by
2 2
2 4
E
2 = p
+ m
1
1 c
1 c
2 2
2 4
E
2 = p
+ m ,
(4.1)
2
2 c
2 c
consistent with special relativity. We assign the value p 1 = p for
the incident particle, where p is assumed to be known a priori,
along with the rest masses m 1 and m 2 . We assign the value p 2 = 0
for the stationary target particle. We assume that neither particle has internal degrees of freedom. By implication, we ignore the
effects of spin in the following analysis. We further assume that
each particle retains its original rest mass through the collision.
Since the only force is that which acts between the two particles, it follows that the center of mass moves at constant velocity. The motion of the center of mass is therefore uninteresting.
We seek a frame of reference such that the total momentum of
the two-particle system is zero. We call this system the center-ofmomentum or CM frame. The scattering is most simply analyzed
Chapter 4. Particle scattering
4.1 Classical particle kinematics
The generic two-particle scattering problem assumes a single particle with vector kinetic momentum p, incident on a second particle
which is initially at rest in the lab frame. The first particle transfers
momentum and energy to the second particle, and both particles
exit to a final state. The final state is governed by conservation of
total momentum and energy, as well as the details of the interaction. The first particle is assumed to come from a distance which
is large compared with the dimensions of the interaction volume
of the two particles. Similarly, both particles exit to a large distance in the final state. At large distances the potential energy of
the interaction can be ignored. In this section we investigate the
constraints imposed by momentum and energy conservation. This
general topic is referred to as kinematics.
We assume the incident particle has rest mass m 1 , vector kinetic
momentum p 1 , and total energy E 1 . We assume the stationary
particle has rest mass m 2 , vector kinetic momentum p 2 , and total
energy E 2 . These quantities are related by
2 2
2 4
E
2 = p
+ m
1
1 c
1 c
2 2
2 4
E
2 = p
+ m ,
(4.1)
2
2 c
2 c
consistent with special relativity. We assign the value p 1 = p for
the incident particle, where p is assumed to be known a priori,
along with the rest masses m 1 and m 2 . We assign the value p 2 = 0
for the stationary target particle. We assume that neither particle has internal degrees of freedom. By implication, we ignore the
effects of spin in the following analysis. We further assume that
each particle retains its original rest mass through the collision.
Since the only force is that which acts between the two particles, it follows that the center of mass moves at constant velocity. The motion of the center of mass is therefore uninteresting.
We seek a frame of reference such that the total momentum of
the two-particle system is zero. We call this system the center-ofmomentum or CM frame. The scattering is most simply analyzed