1. Explain briefly (in about 50 words for each) what is meant by each of the following.

(a) The partition function

(b) The quantum volume

(c) The Fermi energy

(d) The entropy of a black hole

2. Explain briefly (in about 50 words for each) what is meant by each of the following. (a) The density of states

(b) The grand partition function

(c) The ultraviolet catastrophe

(d) Hawking radiation of a black hole

3. Imagine you are in possession of 5 identical coins that can be either Heads or Tails.

(a) How many microstates and macrostates are there?

(b) What is the probability of getting all Heads or all Tails?

(c) In the context of the coin problem above, what do you understand by the term multiplicity?

4. Study the figure above and answer the following questions:             (a) What is being described here and why is the concept important?

(b) How would you use these figures to explain the origin of the Boltzmann distribution?

(c) Imagine a closed box with a very large number atoms. If all of the gas atoms (blue) start with the same kinetic energy K, what happens to the distribution of K as time passes?

5. This is is a slight variant of the previous question.

(a) What is the Bose-Einstein distribution and why is it important?

(b) Explain how the Boltzmann distribution arises in a system of N interacting particles that all start out with the same kinetic energy.

(c) We know that entropy averaged over a system never decreases in a confined gas, but can it ever be constant with time? Use a formula to clarify your answer.

6.  The nearest star to Earth is the Sun with a surface temperature of 5800 K, a dwarf star that lives for billions of years creating light through nuclear fusion at its core.

(a) How much energy is contained in the electromagnetic radiation filling a cubic metre of space at the Sun’s surface?

(b) Sketch the Sun’s spectrum as seen Earth orbit (i.e.   without any correction for absorption by the Earth’s atmosphere) as a function of photon energy and mark the region visible to the human eye (400 − 700 nm).

(c) What fraction of the Sun’s total energy is visible to the human eye?

7.  Planck solved the “UV catastrophe” by summing over quantized energy states with wavelength λ for a blackbody with temperature T.  The emitted spectrum is known as the Planck function such that

P(λ,T) =                                                                   (1)

(a) Derive the Planck function in terms of frequency ν (c = λν) using the relation

dλ

dν

(b) In the figure above (left), P(λ,T) has a maximum brightness that peaks at a chang- ing wavelength as a function of temperature, λ 1T  = c1 , known as Wien’s First

Law. If you plot the same spectrum using P(ν,T), it looks similar in shape except that the peak frequency ν2  corresponds to a wavelength λ2  that obeys a different form, λ2T = c2 , known as Wien’s Second Law.  (c1  and c2  are different constants.) Comment on why there are two versions of Wien’s Law for a fixed temperature T.

(c) In the figure above (right), the two stars Acrux and Gacrux in the Southern Cross are bluer and redder than the Sun, respectively.  The surface temperatures of the stars are as follows: 24000 K (Acrux), 5800 K (Sun), 3600 K (Gacrux).

(i) Determine the peak wavelengths for all three stars (in microns) using both Wien’s Laws where c1   =  2.8 × 10 −3   and c2   =  5.0 × 10 −3   in SI units  (six numbers).

(ii) Determine the ratio of the total energies emitted by both Southern Cross stars when compared to the Sun (two numbers).

(d) Give an example other than stars where we can witness Wien’s Law in action in the natural world, i.e. where cooler objects are red and hotter objects are blue.

8. In 1941, astronomers were studying starlight passing through a cold, dense, cyanogen (CN) molecular cloud.  From spectral absorption lines in the star’s spectrum, they cal- culated that, for every ten CN molecules in the ground state, there were three molecules in the first excited state with energy 5 × 10 −4  eV above the ground state. In fact, there are three excited states at that energy, with excited molecules in each of the states on average. Given the coldness of the cloud, this was a surprise and led the astronomers to claim that the cloud was in thermal equilibrium with a mysterious “reservoir.”

(a) Write down a formula for the relative probability of being in an excited state com- pared to the ground state, and derive the temperature of that external radiation bath. Explain the scientific significance of this result.

(b) In the figure above, compute the average energy of the 5 atoms,  = P  E(s)P(s)

i=1

where P(s) is the probability of an atom being in state s, and E(s) is the energy of the state.

(c) Compute the deviation from the mean, ∆Ei  = Ei −  for i = 1 − 5, and the average of the squares of the five deviations, (∆Ei )2 . This quantity is the variance σ 2 .

(d) Prove that an equivalent formula for the variance is given by σ 2  = E2  − ( )2 , i.e. the variance is equal to the average of the squares minus the square of the average. Demonstrate that this is true for the 5 atoms above.

9. Consider a single harmonic oscillator with allowed energies 0, hν, 2hν , ···

(a) Evaluate the partition function Z for this simple system; express the answer as an expansion and in its most compact form.

(b) Write down an expression for the average energy of this simple system at a temper- ature T.

(c) Derive an expression for the total energy of this system if it has N identical oscillators at a temperature T.

(d) Derive an expression for the heat capacity of the system in (c).

10. Study the figure above and answer the following questions.