Physics 512
Winter 2003
Homework Set #6 – Due Monday, March 3
1. One dimensional scattering. Consider scattering from a potential
V
(
x
) =
−
gδ
(
x
). If
we send in a particle from the left
ψ
inc
(
x
) =
e
ikx
x <
0
there will be an amplitude for both reﬂection and transmission. The transmitted part
may be written
ψ
trans
=
S
(
E
)
e
ikx
=
√
Te
i
(
kx
+
δ
)
x >
0
where
T
is the (real) transmission coeﬃcient and
δ
the phase shift (
E
is the energy).
a
) Find the transmission coeﬃcient and phase shift and show that
T
is insensitive to
the sign of
g
. What are the limiting values of
δ
for
E
→
0 and
E
→ ∞
(consider
both positive and negative
g
)?
Although the incident wavefunction is
e
ikx
, there must be a reﬂected one as well.
Thus the complete wavefunction may be given as
ψ
(
x
) =
ψ
<
=
e
ikx
+
Be
−
ikx
x <
0
ψ
>
=
Se
ikx
x >
0
We must satisfy the continuity and jump conditions at
x
= 0
:
ψ
<
(0) =
ψ
>
(0)
,
ψ
<
(0) =
ψ
>
(0) +
2
mg
¯
h
2
ψ
(0)
This gives a set of equations
1 +
B
=
S,
ik
(1
−
B
) =
ikS
+
2
mg
¯
h
2
S
which may be solved to give
S
(
E
) =
1
−
img
k
¯
h
2
−
1
,
E
=
¯
h
2
k
2
2
m
(1)
The transmission coeﬃcient and phase shift is obtained by rewriting
S
(
E
)
in
terms of a magnitude and phase,
S
=
√
T e
iδ
. The result is simply
T
=
1
1 +
(
mg
k
¯
h
2
)
2
,
δ
= tan
−
1
mg
k
¯
h
2
Since
g
enters squared in
T
, the transmission coeﬃcient is insensitive to the sign
of
g
. However this is not the case for the phase shift. As for the limiting behavior
of
δ
, we note that there is always a
2
π
phase ambiguity present. However we can
1
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define the phase to be
0
when
g
→
0
.
This corresponds to taking
tan
−
1
to lie
between
−
π/
2
and
π/
2
. The limiting phase shifts are then given by
E
→
0
E
→ ∞
g >
0
π/
2
0
g <
0
−
π/
2
0
In general, we may understand the sign of the phase shift as being related to the
attractive versus repulsive nature of the potential.
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 Winter '03
 Unknow
 Physics, Work, Phase Shift, SCATTERING, Fundamental physics concepts, Merzbacher, incident wavefunction

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