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21.
(a) and (b) Schr¨
odinger’s equation for the region
x>L
is
d
2
ψ
dx
2
+
8
π
2
m
h
2
[
E
−
U
0
]
ψ
=0
,
where
E
−
U
0
<
0. If
ψ
2
(
x
)=
Ce
−
2
kx
,then
ψ
(
x
)=
C
0
e
−
kx
,where
C
0
is another constant satisfying
C
0
2
=
C
.Thu
s
d
2
ψ/dx
2
=4
k
2
C
0
e
−
kx
=4
k
2
ψ
and
d
2
ψ
dx
2
+
8
π
2
m
h
2
[
E
−
U
0
]
ψ
=
k
2
ψ
+
8
π
2
m
h
2
[
E
−
U
0
]
ψ.
This is zero provided that
k
2
=
8
π
2
m
h
2
[
U
0
−
E
]
.
The quantity on the righthand side is positive, so
k
is real and the proposed function satisFes
Schr¨
odinger’s equation. If
k
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 Fall '08
 SPRUNGER
 Physics

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