Phys 119B
Due: March 3 before 5:00 on Gradescope
HW#7
Reading: Schroeder 7.4, 7.5
1) Derive an expression for ⟨𝛿𝑛𝑖 𝛿𝑛𝑗 ⟩ for the photon gas. 𝛿𝑛𝑗 = 𝑛𝑗 − ⟨𝑛𝑗 ⟩ is the
fluctuation of the occupation number from average, so ⟨𝛿𝑛𝑖 𝛿𝑛𝑗 ⟩ is either the variance
(𝑖 = 𝑗) or covariance (𝑖 ≠ 𝑗 )of the photon occupation numbers. Leave your answer
in terms of 𝑇, 𝜀𝑖 , 𝜀𝑗 .
2) Schroeder 7.41
3) Schroeder 7.45
4) Estimate the ratio of electron to phonon heat capacities in copper at room temperature
(298K). Assume the Fermi temperature for the electrons is 80,000 K and the Debye
temperature for the phonons is 315 K. (Since room temperature and the Debye
temperature are comparable, you will have to evaluate an integral numerically.)
5) Schroeder 7.64
6) Derive an expression for the low temperature heat capacity for a 2D Debye solid
(phonon contribution only).
7) Prove that Bose-Einstein condensation will not occur for ideal gasses in two
dimensions at finite temperature and finite density. However, prove that it will occur
for a relativistic gas with E = cp in 2D and provide an expression for the critical
temperature for the onset of condensation as a function of density.
© Frank Brown 2023