Chapter 8 question 11,

Chapter 9 problem 28,

four problems below

1) Consider conduction electrons in aluminum at room temperature (20oC).

a) Compare the fraction of electrons that have 10%
more energy than the fermi energy to the fraction that have energy equal
to the fermi energy, by taking a ratio n(E)/n(E_{F}).

b) Repeat for electrons having 1% more energy than
the fermi energy.

c) Comment on how quickly the distribution drops
off above E_{F}.

2) Using Fig. 5.11 in handout

a) Approximate DE=E(^{4}F_{3/2})-E(^{4}I_{11/2}).
Check that this would yield a wavlength of 1.06mm.

b) What wavelength light would be need to
optically pump the electron from the ^{4}I_{9/2} state?

3)

a) Derive an expression for the "density" of a black
hole as a function of its Schwartzschild radius *only *(only Rsch
and constants should appear). By "density," I mean the mass of the black
hole divided by the (Euclidean) volume within the Schwartzschild radius.

b) Plot this "density" as a function of Schwartzschild
radius, for radii in the range from 5X1025 m - 5X1026 m. On your plot draw
a rectangle that spans the range of possible values for the density and
radius of our universe (or at least our current best estimate.
You can take the range of radii of the observable universe to be 1-2X10^{10}cyr,
and the range of densities to be 2X10^{-27} -1X10^{-26}
kg/m^{3}.) Is it possible our universe is one big black hole?

4) Given data in the Origin file chaos301.opj,
which is available in the \t folder on computers numbered 5-8 in room 204.
The data represents the differences in angle for two pendulums with nearly
identical initial conditions.

a) Find the average Lyapunov multiplier for the
data. Does this indicate chaos?

b) Graph the difference data and fit to an exponential.
What is the Lyapunov exponent? Does this indicate chaos?