Homework problems #4
1. Computer problem: Approximate the value of Riemann function ζ(3/2) a) using a numerical integration
of
ζ(n) =
1
Γ(n)
∞
0
xn−1
ex −1
dx,
b) by calculating sum of
ζ(n) =
∞
∑
k=1
1
kn
.
2. From the Landau potential of extremely relativistic bosonic gas
Ω = −
8πgV
(2π¯h)3
(kBT)4
c3
B4
µ
kBT
(1)
determine the number of particles, the entropy, and energy of the gas. In the limit of very high temperatures,
determine the specific heat cV,N and the state equation p = p(N,V,T).
3. Let us consider ideal Fermi-Dirack gas with particle energy proportional to the momentum via ε ∝ ps.
The gas is closed in a box with energy V in n dimensional space. Show that the pressure P is
PV =
s
n
E, (2)
and that the adiabatic equation (S and N is constant) is
PV1+ s
n = const. (3)
Show that for T → ∞ the heat capacity becomes
cV,N =
n
s
N. (4)
4. Let us assume that our Universe is a spherical cavity with radius 1028 cm in thermal equilibrium and
opaque walls.
(a) If the cavity temperature is 3K, estimate the total number of photons and their energy in the cavity.
(b) If the temperature of the cacity is 0K and the Universe contains 1080 electrons, estimate the Fermi
momentum of these electrons.
The solution should be submitted not later than on May 14th.
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