Quiz #5
1.
Consider a particle of mass
m
, incident on a step
potential of height
V
0
, located at
x
=0
. The
energy of the particle is above the step height,
E
>
V
0
.
•
Make the ansatz:
•
Determine the wave vectors
k
1
and
k
2
in terms of
E
and
V
0
.
•
Solve for
r
and
t
using the appropriate boundary
conditions
!
"
#
>
<
+
=
$
0
0
)
(
2
1
1
x
te
x
re
e
x
x
ik
x
ik
x
ik
%
Quantum Reflections
•
Consider scattering from a step potential, with
an Energy above the barrier height:
E
x
V
0
I
II
E
x
k
i
x
k
i
I
re
e
x
1
1
)
(
!
+
=
"
x
k
i
II
te
x
2
)
(
=
!
h
mE
k
2
1
=
(
)
h
0
2
2
V
E
m
k
!
=
t
r
II
I
=
+
=
1
)
0
(
)
0
(
!
!
t
ik
r
ik
II
I
2
1
)
1
(
)
0
(
)
0
(
=
!
"
=
"
#
#
)
1
(
1
2
1
r
k
k
r
!
=
+
r
k
k
r
k
k
1
1
2
2
!
=
+
2
1
2
1
k
k
k
k
r
+
!
=
2
1
1
2
k
k
k
t
+
=
(
)
2
2
1
2
2
2
1
2
1
2
2


k
k
k
k
k
k
r
R
+
+
!
=
=
Probabilities:
(
)
2
2
1
2
1
2
4


k
k
k
t
T
+
=
=
(
)
1
2
5
2
2
1
2
2
2
1
2
1
!
+
+
"
=
+
k
k
k
k
k
k
T
R
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Probability Current
•
Clearly we must have
R
+
T
= 1
•
So
R
= 
r

2
and
T
= 
t

2
must not be correct
•
The problem is that in this case the velocity in
region I is not the same as in II
•
This suggests that we need to think in terms of a
probability current
•
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 Fall '08
 MichaelMoore
 mechanics, Current, Energy, Mass, Trigraph, plane wave, k2, step potential

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