Environmental Engineering Reference
In-Depth Information
12.2 An electron and a positron can collide and produce a proton and an antipro-
ton, i.e.
e
−
+
e
+
→
p
. Find the minimum kinetic energy of the positron
in (a) a frame of reference in which the total momentum of the particles is
zero; (b) a frame of reference in which the positron collides with a stationary
electron.
[The masses of the electron and the positron are identical, and equal to
0.51 MeV/
c
2
. The masses of the proton and the antiproton are also identical,
and equal to 938.3 MeV/
c
2
.]
12.3 Prove that the minimum invariant mass of an arbitrary system of particles
is greater than or equal to the sum of the masses of the individual particles.
12.4 A photon with energy above 1.02 MeV has an energy greater than the rest
energy of an electron-positron pair. Nevertheless the process
p
+
e
−
+
e
+
γ
→
cannot occur in the absence of other matter or radiation. Why not?
12.5 This question is about the so-called transverse Doppler effect. Consider a
frame
S
in which a transverse wave
y(x,t)
=
sin
(kx
−
ωt)
propagates. As seen by an observer at rest in
S
, this wave is travelling along
the
x
direction with wavelength 2
π/k
and angular frequency
ω
. The speed
of propagation is
u
+
ω/k
.
Now consider a frame
S
which is moving at speed
V
in the
=
+
y
direction.
The origins of
S
and
S
coincide at
t
t
=
0. Show that an observer in
S
=
sees the following transverse wave:
1
γ
y
=−
Vt
+
sin
(
k
·
x
−
ω
t
).
Deduce
k
and
ω
, and hence show that
K
=
(ω/c,
k
)
transforms as a
four-vector.
What is the speed of propagation in
S
? Show that it reduces to the correct
values in the limits
u
c
.
12.6 A pion of momentum 32 MeV/
c
decays into a muon and a neutrino. Using
the conservation of four-momentum, and the fact that the neutrino is (to a
good approximation) massless, show that
→
c
and
u
(m
π
+
m
µ
)c
2
2
P
π
·
P
µ
=
,
where
P
π
and
P
µ
are the momentum four-vectors of the pion and muon.
If the outgoing muon travels at 90
◦
relative to the direction of the incoming
pion, use the above expression to determine the kinetic energy of the muon.
At what angle does the neutrino travel relative to the incoming pion?
[The mass of the pion is 140 MeV
/c
2
and that of the muon is 106 MeV/
c
2
.]
