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PROBLEM SET 10.1
1.
The Lennard-Jones potential between two molecules is
r
12
r
6
V
=
4
ε
−
where
areconstants, and
r
is the distance between the molecules. Use the
functions
goldBracket
and
goldSearch
to find
ε
and
σ
σ/
r
thatminimizes the potential
and verify the result analytically.
2.
One wavefunction of the hydrogenatomis
C
27
e
−
σ/
3
2
ψ
=
−
18
σ
+
2
σ
where
σ
=
zr
/
a
0
z
a
0
2
/
3
1
81
√
3
C
=
π
z
=
nuclear charge
a
0
=
Bohr radius
r
=
radial distance
Find
σ
where
ψ
is at a minimum. Verify the result analytically.
3.
Determine the parameter
p
that minimizes the integral
π
sin
x
cos
px dx
0
Hint
:use numericalquadraturetoevaluate the integral.
4.
R
1
= 2
R
2
= 3.6
i
1
i
2
R
5
= 1.2
E
= 120 V
R
i
1
i
2
R
3
= 1.5
R
4
= 1.8
Kirchoff'sequationsfor the two loops of the electricalcircuit are
R
1
i
1
+
R
3
i
1
+
R
(
i
1
−
i
2
)
=
E
R
2
i
2
+
R
4
i
2
+
R
5
i
2
+
R
(
i
2
−
i
1
)
=
0
Find the resistance
R
that maximizes the powerdissipatedby
R
.
Hint
: solve
Kirchoff'sequations numericallywith one of the functions in Chapter2.
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