Homework Set # 12 – Not Due – Solutions will be provided on December 4

1. Consider an ideal Bose gas confined to a region of area A in two dimensions. Express the number of particles in the excited states, N_e, and the number of particles in the ground state, N₀, in terms of z, T, and A, and show that the system does not exhibit Bose-Einstein condensation unless T->0K.

2. Problem 7.14. Do not calculate C_P. Hint: use that the density of states in n-dimensions is given by a(ε)dε=V2π^n/2p^n-1dp/[hⁿΓ(n/2)].

3. Use the Debye approximation to find the following thermodynamic functions of a solid as a function of the absolute temperature T:

   1. ln Z, where Z is the partition function.

   2. the mean energy E.

   3. the entropy S.

Express your answer in terms of the function D(x₀)=(3/x₀³)∫₀^x0(x³/(e^x-1))dx.

4. Evaluate the function D(x₀) (given above) in the limits when x₀>>1 and x₀<<1. Use these results to express the thermodynamic functions ln Z, E, and S calculated in the previous problem in the limiting cases in which T<<Θ_D and when T>>Θ_D.

5. Problem 7.34

6. An ideal Fermi gas is at rest at absolute zero and has a Fermi energy μ. The mas of each particle is m. If v denotes the velocity of a molecule, find <v_x> and <v_x²>.

7. Consider an ideal gas of N electrons in a volume V at absolute zero.

• a) Calculate the total mean energy <E> of this gas.

• b) Express <E> in terms of the Fermi energy μ.

• c) Show that <E> is properly an extensive quantity, but that for a fixed volume V, <E> is not proportional to the number N of particles in the container. How do you account for this last result despite the fact that there is no interaction potential potential between the particles?

8. Derive Eq.(8.1.9) Hint: use Eq.(8.1.4) and recurrence formula (E.6).

9. Problem 8.14 (solve it only for T=0.

10. Problem 8.15