Physics 31
Spring, 2007
Solution to HW #3
Problem A
When we considered the classical analog of
Compton scattering, we wrote the following conservation
equations for the
x
and
y
components of momentum and
for the energy:
mv
=
mv
±
cos
φ
+
Mu
±
cos
θ
(1)
0=
mv
sin
φ
−
±
sin
θ
(2)
1
2
mv
2
=
1
2
mv
±
2
+
1
2
±
2
(3)
(a) Compare Eqs. (1) and (2) with the corresponding equa
tions in the text for Compton scattering [Eqs. (2.16)
and (2.17)]. How do they diFer?
(b) Write the relativistic energy conservation equation for
Compton scattering, that is, the case of a photon (fre
quency
ν
) incident on an electron (mass
M
) at rest. You
will need Eq. 1.24 from page 76 of the text.
Hint:
If the
mass of the electron is
M
, and its momentum after the
collision is
p
±
e
, then its energy after the collision is
E
=
p
M
2
c
4
+(
p
±
e
)
2
c
2
.
See if you can massage your energy equation to get
Eq. (2.19) in the text. (Note that the text is a little
sloppy about using the symbol “
p
” for several diFerent
things. In Eq. (2.19),
p
is the electron momentum after
the collision, the same quantity denoted by
p
±
e
here.)
(c) (Optional) Solve Eqs. (1)–(3) to verify the result pre
sented in class,
1
E
±
−
1
E
≈
4
1
−
cos
φ
Mv
2
.
You should ±nd the solution to an exact quadratic equa
tion for
v
, and then look for simplications using results
valid for
m/M
±
1.
²or part (a): the equations in the text use the momentum
p
e
of the electron, rather than
mv
. In that formulation, you
don’t have to worry about the factors of
γ
.
²or part (b), equate the initial energy to the ±nal:
hν
+
Mc
2
=
hν
±
+
q
(
2
)
2
p
±
e
c
)
2
Rearrange the terms:
hν
−
hν
±
+
2
=
q
(
2
)
2
p
±
e
c
)
2
.
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 Spring '07
 Hickman
 Physics, Energy, Kinetic Energy, Mass, Momentum, Photon, Compton scattering

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