Homework 1
20 + 5 (bonus) = 25 points
Problem 1 (1 point)
At what velocity, the relativistic expression for particle momentum p=mv
γ
becomes 1% different form the
non-relativistic formula p=mv.
Problem 2 (2 points)
Show that a photon of energy
ε
and scattered off an electron of mass m at rest (γ + e
–
→ γ + e
–
) would
change its wavelength
λ
depending on the scattering angle
θ
according to the Compton scattering
formula:
∆λ
/
λ
=
ε
/
m
⋅(1-cos
θ
)
Problem 3 (2 points)
A neutral particle decayed into proton p and negatively charged pion π
-
. The momenta of charged
particles were measured from the curvature of their tracks in magnetic field (in units of GeV):
p
x
p
y
p
z
p -0.488 -0.018 2.109
π
-
-0.255 -0.050 0.486
Find the mass and energy of the neutral particle. (2 points)
Problem 4 (5 + 3 bonus = 8 points)
A particle of mass M at rest decays into three particles A, B, and C of different masses m
A
, m
B
, m
C
.
Unlike in the case of a two-body decay, the energies of these three particles will not be the same in all
decays, i.e., they can and will vary from decay to decay (of course, the total energy will be equal to M
and total momentum—zero).
Case I: The particle A has its minimum allowed energy.
a) Draw the momentum vectors of all three particles in this case? (1 point)
b) What is this minimum energy for the particle A? (1 point)
c) What would the energies of the other two particles B and C be in this case? (2 points)
Case II: The particle A has its maximum allowed energy.
a) Draw the momentum vectors of all three particles in this case? (1 point)
b) What is this maximum energy for the particle A? (1 bonus point)
c) What would the energies of the other two particles B and C be in this case? (2 bonus points)
Hint: Recall a problem of a two-body decay that we considered in the class. Once you have figured out
what needs to happen for a particle A to be at its minimum/maximum energy, the rest trivially follows
from the two-body decay equations.
