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Chris Parkes
4th Year Particle Physics I Option
Question Sheet
Section I – Relativistic Quantum Mechanics
1. Starting from the relativistic expression for energy, E2-p2c2=m2c4,make the
standard Quantum Mechanical substitutions to derive the Klein-Gordon
Equation.
2. Show that the wave equation for a free particle,
 ( x, t )  N exp[ i ( p.x  Et ) / ] satisfies the Klein-Gordon equation,
 2  ( x, t )
  2 c 2  2  ( x, t )  m 2 c 4  ( x, t ) and determine the Energy E.
2
t
Show that the complex conjugate of the above solution also satisfies the K-G
equation and determine the energy in this case. Comment on the energies
obtained.
 2
3. A beam of positively charged pions is incident on a one-dimensional potential
step of height eVo, i.e.
V(x)=0
for x<0
V(x)= Vo
for x>0
p  p'
Show that the reflection coefficient,R, for the pions is R 
p  p'
2
where
1
pc  ( E 2  m0 c 4 ) 2
1
p' c  [( E  eVo ) 2  m0 c 4 ] 2
2
Discuss the physical interpretation of the case when
eVo > E+moc2 and p’<0
What experimental situation does the choice
eVo > E+moc2 and p’>0 represent ?
 ( x, t )
 H ( x, pˆ ) ( x, t ) , where H, the
t
3

Hamiltonian, is H  ic i
 mc 2 and p̂  i is equivalent to the
xi
i 1
Klein-Gordon equation providing
 i 2  1;  2  1; i   i   0; i j   j i  0(i  j )
4. Show that the Dirac Equation, i
5. The relativistic wave equation for pions has two solutions. Why does the
corresponding equation for electrons have four solutions ?
6. Describe the process e+e- e+e- in terms of hole theory.
Section II – Reaction Kinematics
1.
Show that in a fixed target experiment with particle a incident on
particle b, the square of the centre of mass energy, S, is given by the
expression S  ma2  2 E a mb  mb2
2.
In a colliding beam experiment, particles a and b, of same mass and
equal but opposite momenta interact to give particles c and d. Show
that the centre of momentum of particle c, p c* , is given by the
1
expression p c*  p d*  {[W 2  (mc  md ) 2 ][W 2  (mc  md ) 2 ]} 2 / 2W
and that the total energy of particle c is given by
Ec*  (W 2  mc2  md2 ) / 2W where W is the total centre of mass energy.
What is the corresponding expression for the centre of mass energy of
particle d?
3.
Show that the invariant mass of a pair of photons of energies, E1 , E2
with angle  between their directions is given by the expression
m 2  4 E1 E2 sin 2 ( / 2) . Consider the decay of a particle of mass m
energy E into 2 photons, and show that the minimum opening angle
between the photons is given by sin(  / 2)  m / E . A particle of
energy 10 GeV decays to two photons with opening angle 2. Could
this particle be an  meson (of mass 0.549 GeV) or a  meson (of
mass 0.135 GeV)?
4.
Define four momentum transfer. In an elastic scattering experiment,
particle a is scattered through an angle * in the centre of mass system.
Show that the four momentum transfer, q, is given by the expression
q 2  2 p *2 (1  cos  * ) , where p* is the centre of mass momentum of a.
Show that in the limit that the masses of the particles may be
neglected, q is proportional to the total centre of mass energy.
5.
The experimental cross section for e   e        in a colliding
ring experiment at high energy can be fitted to the equation
d
2

(
)(1  cos 2  * ) where d * is an element of solid angle
d *
16 E 2
into which the - emerges, * is the angle the - makes with the
incident e- and E is the energy of the particles in each beam. Show that
4 2
the total cross section, , is given by  
where S  Ecm and
3S
 is the fine structure constant.
Calculate the total cross section for e   e        at CM
energies of 2 GeV and 40 GeV. What are the corresponding cross-
sections for e   e        ? The masses of the electron, muon and
tau lepton are 0.511 MeV, 105.6 MeV and 1777 MeV respectively.
6.
In a colliding beam experiment, define luminosity and explain how it is
related to the interaction rate. Obtain an expression for luminosity in
terms of the design features of an accelerator and discuss how the
features can be optimised to improve overall performance.
Section III – Feynman Diagrams
1. Draw the topologically distinct QED Feynman diagrams which contribute to
the following processes in lowest order:
a.   e     e 
b. e   e   e   e 
2. In lowest order, the process e   e        is given by the following
Feynman diagram. Estimate the distance between the vertices at energies
much larger than the masses of the particles in (a) the rest frame of the
electron (b) the centre of mass frame. Check the consistency of these estimates
by considering the Lorentz contraction in going between the electron rest
frame and the centre of mass frame.

e-
e+

3. Which of the following annihilation processes are possible:
with n=1,2,3,4
e  e   n
For each invalid process, give a reason why this is so. For each valid process,
draw at least one Feynman diagram to illustrate it. At a given interaction
energy, how would you expect the annihilation rates to compare for the
different valid processes ?
4. In electron positron colliders, leptons scatter freely from each other and we do
observe free leptons. In high energy proton colliders, quarks also freely scatter
from each other but yet we do not observe free quarks. Explain this paradox.
5. Draw Feynman / quark flow diagrams for the following processes and classify
them as strong, electromagnetic or weak decays:
a.    p        n
b.    n     p
c.  0  e   e   e   e 
d. D   K       
Section IV – Symmetries, Invariances and Conservation Laws
1. Use the standard commutation relations for angular momentum operators to
show that L and S remain good quantum numbers if the spin dependent forces
arise from a simple spin-orbit interaction i.e. if
ˆ  Sˆ
H  H 0  L
where
ˆ  H , Sˆ  0
H0 , L
0
and  is a constant.
2. The deuteron is a bound state of two nucleons with spin-1 and positive parity.
Show that it may only exist in the 3S1 and 3D1 states of the pn system.
3. Electric dipole transitions between atomic states are characterized by the
selection rule l  1 . Hence find the parity of the photon.
4. The intensity of the electrons emitted in the decay of polarized cobalt-60
nuclei is found to be consistent with the form:
I ( , v)  1   vc cos
where v is the magnitude of the electron velocity and theta is the angle
between its direction and the cobalt-60 spin. Deduce the value of the
coefficient alpha by considering events in which the electron is emitted in the
direction of the decaying nuclei. The spins of the cobalt and Nickel nuclei are
J=5, J=4 respectively. Orbital angular momentum may be neglected.[Hint:
consider the relativistic limit for v and compare with neutrinos]
5. Draw a Feynman diagram to show that D 0  D 0 mixing may occur.
6. Define Helicity and show that it is a pseudo-scalar quantity.
7. Show that the total decay rates for the reactions K 0    e  e and

 

K 0    e  e are equal if CP is conserved.