Description
Possibly useful information:
Z ∞
−∞
dxe−ax2+bxdx =
r
π
a
e
b
2
4a (for a > 0)
Γ(n) = (n − 1)! =
Z ∞
0
dxxn−1
e
−x
ln N! ≈ N ln N − N (for N 1)
ζ(m) =
∞
∑
n=1
n
−m =
1
Γ(m)
Z ∞
0
dx x
m−1
e
x − 1
ζ(1) = ∞ ζ(2) = π
2
6
ζ(3) = π
4
90
1. (5 pts) Consider an ensemble with fixed energy and number, but in which
the volume is allowed to fluctuate. The ensemble average of the volume is
constrained to be hVi. Find the probability of microstate i of the system, pi
,
and define the partition function for this ensemble, ZH, by maximizing the
Gibbs entropy.
2. (10 pts) For free bosons in a D-dimensional box with an energy-momentum
relation e = aps
, where a and s are positive constants, what is the dimension
at which Bose-Einstein condensation begins to occur at low temperatures, in
terms of D and s?
3. (10 points) Consider a one-dimensional solid of length L at temperature T containing N nuclei in a chain. Each nucleus contributes one spin-half conduction
electron (so the rest of the electrons can be neglected). Model excitations of
the lattice using a one-dimensional version of the Debye model, so that the
density of states in frequency space is given by g(ω) = L/(2πcs), where cs
is
the sound speed, k = ω/cs
is the wavenumber, and the energy of a phonon is
given by e = h¯ ω. Since motion is only possible in the x-direction, the waves
can have only one polarization. Treat the electrons as a free, non-relativistic
gas confined to move in one-dimension.
(a) What is the electron Fermi energy of this system?
(b) What is the electron contribution to the energy of the system at zero temperature?
(c) What is the Debye frequency for the lattice?
(d) What is the phonon contribution to the energy for T small compared to
the Debye temperature?