PC4245 PARTICLE PHYSICS
HONOURS YEAR
Tutorial 3
1. a. Consider the scattering of particles 1 and 2 in the CM frame
1+2  3 + 4
Derive the following expression
[( p 1 p 2 ) 2  (m1 m2 c 2 )]1 / 2  ( E1  E 2 )  p / c
~1
b. Obtain the corresponding formula for the lab frame (particle 2 at rest)
[Answer : ( p 1  p 2 ) 2  (m1 m2 c 2 ) 2  m2 p c ]
~1
2. Consider the case of elastic scattering, A + B  A + B, in the lab frame (B
initially at rest), assuming the target B is so heavy ( mB c 2  E A ) that its recoil is
negligible. Show that the differential scattering cross section is given by
(d / d)  ( / 8m B c) 2 M
2
3. Consider the collision 1+2  3 + 4 in the lab frame (2 at rest), with particles 3
and 4 massless. Obtain the formula for the differential cross section.
d

[ Answer :
 ( )2
d
8
SM
2
p
~ 3
]
m2 p ( E1  m2 c  p c cos
2
~1
~1
4. Analyze the problem of elastic scattering (m3  m1 , m4  m2 ) in the lab frame
(particle 2 at rest). Derive the formula for the differential cross section.
5. [The purpose of this problem is to demonstrate that particles described by the
Dirac equation carry “intrinsic” angular momentum ( S ) in addition to their orbital
~
angular momentum ( L) , neither of which is separately conserved, although their
~
sum is. It should be attempted only if you are reasonably familiar with quantum
mechanics.]
a) Construct the Hamiltonian, H, for the Dirac equation. [Hint: Solve equation
(7.19) for p  / c Solution: H  c  (  p  mc) , where p  ( / i) is the
~
momentum operator.]
~
~
b) Find the commutator of H with the orbital angular momentum L  x x p .
~
~
~
[Solution: [ H , L]  ic ( x p)]

~
~
~
Since [ H , L] is nor zero, L by itself is not conserved. Evidently there is some
~
~
other form of angular momentum lurking here. Introduce the “spin angular
momentum,” S , defined by the equation S  ( / 2)  .
~
~
~
c) Find the commutator of H with the spin angular momentum, S  ( / 2)  .
~
~
[Solution: [ H , S ]  ic ( x p)]

~
~
~
It follows that the total angular momentum, J  L  S , is conserved.
~
~
~
d) Show that every bispinor is an eigenstate of S , with eigenvalue  2 s(s  1) ,
2
~
and find s. What, then, is the spin of a particle described by the Dirac
equation?
6. Construct the normalized spinors u(+) and u(-) representing an electron of
momentum p with helicity  1 . That is find the u’s that satisfy the Dirac
~
equation with positive energy p  , and are eigenspinors of the helicity operator

( p/ p ) with eigenvalues  1 .
~
~
~
Solutions:


W ()


,
u (  )  p   mc   p
~
() 

 p   mc W 


 3

1
p  p
()
W 
~ 

3  p 1  ip 2 
2 p( p  p )

~
7.
~
Evaluate the amplitude M for electron-muon scattering in the CM system,
assuming the e- and  approach one another along the z axis, repel, and return
back along the z axis. Assume the initial and final particles all have helicity +1.
[ Answer : M  2 g e2 ]
A(oh):Tutorial3