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Category: PHY 831

Description

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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?

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