Homework 2

NOTE: acceptable formats for turning in your homework are as follows:-
(1) handwritten clearly in blue or black ink
(2) handwritten clearly in 2B or darker pencil
(3) typewritten.
NO other types of work will be marked. In addition, if any of your working / logic is unclear, points will be docked -- in physics, it's not only getting the right answer, it's knowing how to get it using the correct method that matters.

Total marks: 80. [Note: even though HW#1 was nominally out of 50, both assignments are still worth 15% each of your term grade.]

[5 marks] Question 1: Give a concise definition for the following terms, in about 3 sentences, showing that you understand the meaning and basic physics of the terms:
(a) Helmholtz free energy
(b) Classical regime
(c) Spectral density
(d) Kirchhoff law
(e) Grand partition function

[5 marks] Question 2 : dimensional analysis practice
Consider our friend the box of phonons. Take the speed of sound v, Planck's constant h, the number of atoms in the crystal N , the box volume V, and the Boltzmann constant k_B, and derive the Debye temperature by dimensional analysis. Do NOT perform the derivation from lecture; take the quantities I have listed above and produce the object with the dimensions of absolute temperature. This should be the Debye temperature, up to numerical factors. Show ALL steps in your working (incomplete logic => incomplete marks!).

[10 marks] Question 3:
KK 3.3 Free energy of a harmonic oscillator
[Notice the relation to Q3 from HW#1]

[10 marks] Question 4:
KK 3.11 One-dimensional ideal gas
(*)

[20 marks] Question 5:
(a) What does CMB stand for?
(b) KK 4.1 Number of thermal photons;
(c) KK4.18 Isentropic expansion of photon gas.
[Note: these two problems form part of the story of the expansion of our universe and cooling of the CMB]

[10 marks] Question 6:
KK 4.5 Surface temperature of the Earth

[10 marks] Question 7:
KK 4.13 Energy fluctuations in a solid at low temperatures.

[10 marks] Question 8:
Consider a photon gas in 3 dimensions (the usual). Start from the differential form of the First Law of Thermodynamics.
(a) Write S(T,V) and find expressions for both first partial derivatives.
(b) Derive a Maxwell relation, and use it to find a differential equation for U/V. Integrate this to get the Stefan-Boltzmann proportionality U/V ~ T⁴
If it helps, you may use the fact that p=(1/3)(U/V) in your derivations.

(*) Bonus points on Q4: starting from the fundamental derivation of the equilibrium pressure p, derive an equation for the r.m.s. deviation of the pressure from its equilibrium value. It will be helpful to relate the r.m.s. pressure deviation to the r.m.s. energy deviation, and use the partition function Z. You should find that the r.m.s. pressure deviation is proportional to (k_B T² / V) times the partial derivative of the equilibrium pressure w.r.t. temperature.