Physics 253a
1
Problem Set 3 Solutions
October 6, 2010
1. (a) In the center of mass frame, the
e
+
and
e

approach each other with opposite
momenta and total energy
E
cm
:
p
e
+
=
E
cm
2
(1
,
1
,
0
,
0) and
p
e

=
E
cm
2
(1
,

1
,
0
,
0).
The muon and antimuon look the same, but with their momenta rotated by
θ
:
p
μ
+
=
E
cm
2
(1
,
cos
θ,
sin
θ,
0) and
p
μ

=
E
cm
2
(1
,

cos
θ,

sin
θ,
0).
Now,
s
= (
p
e
+
+
p
e

)
2
= (
E
cm
,
0
)
2
=
E
2
cm
t
= (
p
μ


p
e

)
2
=

2
p
μ

·
p
e

=

E
2
cm
2
(1

cos
θ
)
u
= (
p
μ
+

p
e

) =

2
p
μ
+
·
p
e

=

E
cm
2
(1 + cos
θ
)
(b) Note that
s
+
t
+
u
= 0.
(c) In class we found
dσ
d
Ω
=
e
4
64
π
2
E
2
cm
(1 + cos
2
θ
)
Note that
4
tu
s
2
= 1

cos
2
θ
. So we can rewrite the above as
dσ
d
Ω
=
e
4
32
π
2
s
±
1

2
tu
s
2
²
=
e
4
32
π
2
s
3
(
t
2
+
u
2
)
where we’ve used
s
2
= (
t
+
u
)
2
=
t
2
+ 2
tu
+
u
2
.
(d) We have
s
+
t
+
u
= (2
m
2
e
+ 2
p
e

·
p
e
+
) + (
m
2
e
+
m
2
μ

2
p
e

·
p
μ

)
+(
m
2
e
+
m
2
μ

2
p
e

·
p
μ
+
)
= 4
m
2
e
+ 2
m
2
μ

2
p
e

·
(
p
μ
+
+
p
μ


p
e
+
)
= 4
m
2
e
+ 2
m
2
μ

2
p
e

·
p
e

= 2(
m
2
e
+
m
2
μ
)
Where we’ve used conservation of momentum in the last line. In general, we’d
have
s
+
t
+
u
=
∑
i
m
2
i
.
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View Full DocumentPhysics 253a
2
2. (a) (Almost) straight oﬀ Wikipedia:
dσ
d
Ω
= (2
Z
)
2
±
α
2
mv
2
²
2
1
sin
4
(
θ/
2)
where
m
is the mass of the
α
particle,
v
is its incoming velocity,
Z
is the nucleon
charge, and
α
=
e
2
4
π
is the ﬁne structure constant (the Wikipedia article forgets
the product of the
α
particle and nucleon charge 2
·
Z
). This formula treats the
nucleus
N
as having inﬁnite mass, so that the
α
particle scatters with the same
kinetic energy
E
kin
=
1
2
mv
2
as it started with. It also assumes the
α
particle is
a nonrelativistic point particle moving in a classical Coulomb potential.
(b)
i. Let the
α
particle have initial momentum
p
i
= (
E,
p
i
) and ﬁnal momentum
p
f
= (
E,
p
f
). Note that the initial and ﬁnal energies are the same in the limit
that the nucleon mass is inﬁnite. Also,

p

=
√
2
mE
kin
in the nonrelativistic
limit. The relevant Feynman diagram is
γ
N
α,p
i
N
α,p
f
By momentum conservation, the virtual photon has
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 '10
 SCHWARTZ
 Electron, Center Of Mass, Energy, Mass, Momentum, Fundamental physics concepts, Feynman diagram

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