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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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