Physics 7230 Homework 4. Due Friday April 4, 2003
3. Analytical analysis of Bose Condensation in a Harmonic Oscillator Potential using the Grand
Canonical Ensemble.
In June 1995 an experimental atomic physics group headed by Eric Cornell and Carl Wieman in JILA
87
demonstrated the first observation of Bose-Einstein condensation in a vapor of Rb . (See Science 269,
198 (1995) and Physics Today, p. 17, Aug. 1995.) The vapor was first cooled using laser cooling
techniques and then held in a three-dimensional harmonic oscillator potential with a resonant frequency of
f=200 Hz. The shape of the potential was slowly changed to allow the highest energy atoms to “evaporate”
from the potential well, lowering the temperature of the remaining atoms. The temperature of the gas was
on the order of 170 nK and about a thousand atoms were left in the potential when the Bose-Einstein
condensation occurred. Show using statistical mechanical methods that Bose-Einstein condensation occurs
for a non-interacting gas of bosons held in a three dimensional harmonic oscillator potential at sufficiently
low temperatures.
The energy of the single particle in a three-dimensional harmonic oscillator is
where lx,. ly. and lz are non-negative integers. Since

   lx  l y  lz  
2 
3
l  l x  ly  lz is a non-negative integer one can
express the average number and the average energy as functions of l. Determine the degeneracy gl of the
level with energy
 3
   l  . Specialize to the case of large l which dominates in the

2 
thermodynamic limit.
Use the quantity

(T, )  (2s 1)k BT  gl ln 1 exp  ( l  ).
l 0
Show that the average number of particles in the trap is given by
 
N  
 
 .
 T
Determine the critical temperature by setting the chemical potential to zero. Determine the numerical value
for the transition temperature for 1000 atoms in a potential with a resonant frequency f=200Hz.
Determine the condensate fraction n0 =N0/N (the fraction of the atoms in the ground state) as a function of
T for T<Tc and plot.
Determine the internal energy per particle and the heat capacity per particle as functions of T and
z  exp() for fixed N. You will need to numerically determine the chemical potential as a function of
T/Tc at fixed N for T>Tc. Use this to plot the internal energy and heat capacity per particle at fixed N as a
function of temperature. Show the heat capacity is discontinuous at the critical temperature.
4. Consider the following single particle dispersion relation:

 k  0 ka
,
where  0 and a are microscopic parameters.
a) Determine the density of states for spin-s particles in d-dimensions.
b) Assuming the particles are bosons, determine the Bose-Einstein condensation temperature. Show that for
some range of d and  that the condensation temperature is T=0.
c) For cases in which the condensation temperature is nonzero, determine the dependence of the constant
volume heat capacity per particle on temperature in the condensed phase.
5. During the first few moments of the universe the temperature of the black-body photons was large
enough to create electron-positron pairs. For temperatures much higher than mc 2/kB the electrons and
positrons that are created are extremely relativistic, i.e.
 k  ck
Using the theory of black-body radiation as a guide, calculate the number density and the energy density of
the electrons and positrons as functions of temperature. Since the numbers of electrons and positrons are
not conserved and the electrons and positrons are present in equal numbers the chemical potentials for both
species is zero. Treat the electrons and positrons as spin-1/2 non-interacting fermions. Show that the
combined energy density of the electrons and positrons is 7/8 of the energy density of the photons.
Determine the ratio of the number density of electrons to the number density of photons. Show that the
Coulomb energy between the electrons and positrons is small compared to their kinetic energy since the
fine structure constant is a small number.
1. The pressure of an ideal Fermi gas is
-(-) 


.
p(T,) = kBT  d g() ln1 + e






Derive the expressions for the number density, the internal energy density, the
1 n 
entropy density, the isothermal compressibility ( KT  2 
 
 ), and the specific
n  T
 s
heat at constant chemical potential, c=T   , all using derivatives of the
 T
pressure. Do not assume any particular form for g().
Determine the low temperature (kBT<<) dependence of the specific heat and
compressibility.
For the free Fermi gas show the specific heat and compressibility are much smaller
than a classical ideal gas values at the same temperature and density.
2. Estimate the Fermi temperature of liquid 3He. Helium-3 is spin-1/2, liquifies at about
g
3K and has a density of 0.079 3 . Is liquid 3He well described as a degenerate Fermi
cm
gas in this temperature range? Estimate the specific heat at 0.1 Kelvin.
3 Analytical
analysis of a Fermi Gas in a Harmonic Oscillator Potential using the Grand
Canonical Ensemble. Read the article: “Onset of Fermi degeneracy in a trapped Fermi
gas”, Science 285 1703 (1999). by B. DeMarco and D.S. Jin. See also Physics Today p.
17, Oct. 1999. The trap has a radial frequency r and an axial frequency z .
 Show the energy density of single particle states in the trap is of the form
g()   dl l  dlz (   rl   zlz ) and determine the functional form.
 The authors use two different spins states of K40 to facilitate cooling. Determine the
2
1/ 3
Fermi temperature for one species is F  (6 r  zN ) . Determine the numerical


value of the Fermi temperature for the 7x105 atoms in their trap for a choice of r.in
their experimental range.
Show for T=0 that U=<E>= 3/4 N EFermi.
Extra credit: By numerically determining the chemical potential as a function of
temperature for fixed N you should be able to reproduce the theoretical curve shown
in figure 4.