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Physics 512
Winter 2003
Homework Set #7 – Solutions
1. Consider the scattering of a beam of spinless particles of momentum ¯
hk
initially
traveling along the +ˆ
z
direction by a potential of the form
V
(
~r
)=
V
0
±
δ
(
x
)
δ
(
y
−
b
)
δ
(
z
)+
δ
(
x
)
δ
(
y
+
b
)
δ
(
z
)
²
a
) Calculate the scattering amplitude and diFerential cross section in the Born ap
proximation.
Using the Born approximation, we compute
f
Born
(
~q
−
m
2
π
¯
h
2
Z
V
(
~r
0
)
e
−
i~q
·
~r
0
d
3
~r
0
=
−
mV
0
2
π
¯
h
2
±
e
−
iq
y
b
+
e
iq
y
b
²
=
−
mV
0
π
¯
h
2
cos
q
y
b
where
~q
=
~
k
0
−
~
k
. Since the beam initially travels in the
+ˆ
z
direction, we must
have
k
y
=0
. Hence we may write
f
Born
(
~q
−
mV
0
π
¯
h
2
cos
k
0
y
b
so that
dσ
d
Ω
=

f
Born

2
=
³
mV
0
π
¯
h
2
´
2
cos
2
k
0
y
b
Using spherical coordinates and conservation of energy (

~
k
0

=

~
k

) this may be
written as
dσ
d
Ω
=
³
mV
0
π
¯
h
2
´
2
cos
2
(
kb
sin
θ
sin
ϕ
)
b
) This potential represents scatterers located at
y
=
−
b
and
y
=+
b
. How does the
quantum result diFer from what one would expect classically?
The quantum scattering cross section has the form
dσ
d
Ω
∼
cos
2
(
scattering direction
)
so it results in an interference pattern (very similar to that of a double slit).
Classically, particles would scatter from either the Frst scatterer or the second, but
not both simultaneously (multiple scattering could occur, but only sequentially).
Hence there would be no interference classically (which sounds pretty obvious
when one thinks about it).
1
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View Full Document2. Merzbacher, Exercise 13.11. If
V
=
C/r
n
, obtain the functional dependence of the
Born scattering amplitude on the scattering angle. Discuss the reasonableness of the
result qualitatively. What values of
n
give a meaningful answer?
Since the potential is spherically symmetric, we may use the expression
f
Born
(
θ
)=
−
2
m
¯
h
2
Z
∞
0
V
(
r
)
sin
qr
r
2
dr
=
−
2
mC
q
¯
h
2
Z
∞
0
sin
r
n
−
1
dr
where
q
=2
k
sin
θ
2
. Before attempting to integrate this, we may obtain the functional
dependence of this expression on
q
by simply performing a change of variables,
x
=
.
The result is
f
Born
(
θ
−
2
mC
¯
h
2
I
n
q
n
−
3
=
−
2
mC
¯
h
2
I
n
(2
k
)
n
−
3
sin
n
−
3
θ
2
where
I
n
=
Z
∞
0
sin
x
x
n
−
1
dx
(1)
Ignoring for the moment the convergence of
I
n
, we Fnd the general behavior
f
Born
(
θ
)
∼
sin
n
−
3
θ
2
Note that we assume
n
is nonnegative (but not necessarily to be an integer), otherwise
the potential would grow indeFnitely for large
r
, in violation of the scattering assump
tions. ±or
n<
3
, this blows up in the forward direction, corresponding to enhanced
small angle scattering. However there is an opposite e²ect of suppression for
n>
3
.
This at least seems reasonable, since the larger
n
is, the steeper (and more highly
localized) the potential is. ±or a very sharp potential, most of the time the incoming
particles would miss it altogether. But in the instances when they hit the potential,
they would most likely get scattered by large angles.
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 Winter '03
 Unknow
 Physics, Momentum, Work

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