Ph135c. Solution set #2, 4/15/10
1.
First consider the bound state. As shown in the book, its energy is given by solving the following
transcendental equation:
K
cot(
Kr
o
) =

k
where:
K
=
p
2
μ
(
V
o

E
B
)
k
=
p
2
μE
B
here
μ
is the reduced mass for the protonneutron system,
μ
≈
m
p
2
.
The scattering length
a
t
is defined by:
a
t
=

lim
k
→
0
δ
0
k
where
δ
0
is the phase shift in the
s
wave scattered off the potential. For a square well with depth
V
0
and radius
r
0
,
δ
0
is given by solving:
k
tan(
Kr
o
) =
K
tan(
kr
0
+
δ
0
)
where now:
K
=
p
2
μ
(
V
0
+
E
)
This gives:
δ
0
= tan

1
(
k
K
tan(
Kr
0
))

kr
0
To obtain the scattering length, we need to extract the lowest order (in
k
) term of this expression.
Since the argument of the inverse tangent function is small for small
k
, we can approximate it by
tan

1
x
≈
x
, and we find:
δ
0
≈
k
(
tan(
K
0
r
0
)
K
0

r
0
)
where
K
0
=
√
2
mV
0
. The scattering length is then:
a
t
=
r
0

tan(
K
0
r
0
)
K
0
1
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Plugging in the known values
E
B
= 2
.
22
MeV
and
a
t
= 5
.
4
fm
, these two equations can be solved
numerically, and we find:
r
0
= 2
.
1
fm
V
0
= 34
.
2
MeV
Next we need to extract the effective range of the scattering. To do this, we’ll need to determine
δ
0
to the next order in the energy. This is tedious to do by hand, but using Mathemtica we find:
δ
0
=
k
K
0
tan(
K
0
r
0
)

K
0
r
0
+
k
3
6
K
0
3
3
K
0
r
0

3 tan(
K
0
r
0
) + 3
K
0
r
0
tan
2
(
K
0
r
0
)

2 tan
3
(
K
0
r
0
)
+
...
Or, if we note from above that tan(
K
0
r
0
) =
K
0
(
r
0

a
t
) and
K
0
2
= 2
mV
0
, this can be rewritten
as:
=

a
t
k
+
k
3
a
t
4
μV
0
+
1
6
r
0
3

1
2
a
t
2
r
0
+
1
3
a
t
3
We write this as
δ
0
=

a
t
k
+
bk
3
, where numerically
b
≈
27
fm
3
. Then we have:
k
cot(
δ
0
) =
k
cot(

a
t
k
+
bk
3
) =

1
a
t
+
k
2
a
t
3

b
a
t
2
+
...
The expression in parentheses is then half the effective range, which we can compute to be:
r
e
= 1
.
7
fm
2.
a)
Typically the ground state is associated with
‘
= 0, as a nonzero
‘
introduces an extra positive
contribution to the potential which tends to raise the allowed energies. However,
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 Spring '11
 Smith
 Physics, Angular Momentum, Energy, Mass, Neutron, Quantum Field Theory, Rotational symmetry, spin state, −H −

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