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Solution
ID: past-exam-of-the-short-options-paper-shared-by-prelims-and-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2017/s18/2/a/solution
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Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford
/
2017
/
S18
/
2
/
a
/
Solution
by
Codex
0
2026-10-25
Let
b
=
a
/2
,
q
=
2
m
(
U
0
−
∣
E
∣
)
/ℏ
and
κ
=
2
m
∣
E
∣
/ℏ
. Bound levels have
−
U
0
<
E
<
0
.
Up to
normalization
,
parity
gives
ψ
e
(
x
)
=
{
A
cos
(
q
x
)
,
A
cos
(
q
b
)
e
−
κ
(
∣
x
∣
−
b
)
,
∣
x
∣
≤
b
,
∣
x
∣
>
b
,
(1)
ψ
o
(
x
)
=
{
A
sin
(
q
x
)
,
A
sgn
(
x
)
sin
(
q
b
)
e
−
κ
(
∣
x
∣
−
b
)
,
∣
x
∣
≤
b
,
∣
x
∣
>
b
.
(2)
The respective inverse squared
normalization constants
are
b
+
sin
(
2
q
b
)
/
(
2
q
)
+
cos
2
(
q
b
)
/
κ
and
b
−
sin
(
2
q
b
)
/
(
2
q
)
+
sin
2
(
q
b
)
/
κ
. These include both exponential tails.
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:
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