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Physics 4230 Spring 2006 Homework 4
1. Consider three identical particles of mass m in a one dimensional box of length L. The potential is
constant within the box and infinite at the two ends. Quantum mechanics gives the energy levels as
 2l 2
2
l 
2mL2
 l 2 where l=1,2,3,4,…
a) Write down the total energy of the lowest few states for the case of bosons. Write down properly
symmetrized wavefunctions for the two lowest energy states. Write down the first few terms of the three
particle canonical partition function for this system.
b)Write down the total energy of the lowest few allowed states for the case fermions. Write down properly
antisymmetrized wavefunctions for the two lowest allowed energy states. Write down the first few terms of
the three particle canonical partition function for this system. Indicate which states are forbidden for
fermions but allowed for bosons.
c) Treat the three particles using Boltzmann statistics, ie.

Z Boltzmann(T,L,3) 
Z(T,L,1) 3
3!
Show that the terms in the partition function are the same as the terms in the Bose and Fermi cases except
that the coefficients of the Boltzmann factors are different.
d)Evaluate the two lowest order terms of the internal energy at low temperature (>>1) for the three cases
and
note the differences. For example show that the internal energy in the Fermi case is approximately
U  14  7 exp 7

e) What is the ground state energy and the first excited state energy for N bosons the box? What about N
fermions? Assume both the number or particles and the size of the box are both large but the number of
particles per unit length N/L is kept constant. How does the Fermi energy (the energy of the highest
occupied level) depend on the number of particles per unit length?
2. Use the grand canonical ensemble to analyze the case of N identical particles of mass m in a one
dimensional box of length L. The potential is constant within the box and infinite at the two ends. Quantum
mechanics gives the energy level as
n 
2
 2l 2
2
2mL
 l 2 where l=1,2,3,4,…
a)Show that the grand canonical partition function for bosons is

(T,L, )  

l 0
1
1 exp( (l 2  ))
.
b)Show that the grand canonical partition function for fermions is


(T,L, )   1 exp( (l 2  )).
l 0
c)The grand potential is defined by
(T,L, )  kB T ln (T,L, )


and the average number of particles is
 
N    .
 T ,L
Differentiate the grand potential both case to show that the Bose-Einstein and Fermi-Dirac occupancies
are give by

nl
Bose

1
e
 ( l   )
1
and
nl
Fermi

1
e
 ( l   )
1
Sketch both occupancies as a function of  and describe their behaviors. What is the maximum occupancy
for fermions? What about bosons.
d)Show for fermions at zero temperature that the chemical potential is the same as the Fermi energy.
e)The
Boltzmann statistics is
grand partition function for

(T,L, )   e 
N 0
Z(T,L,1) N
N!
where Z(T,L,1) is the one particle canonical partition function

Z(T,L,1)   e l .
2

l 0
Take the derivative of the grand potential for this case to show the average number of particles in the box is
of the form


N   nl
.
Boltzmann
l 0
What is the average Boltzmann occupancy as a function of temperature and chemical potential? Show that
if the average Bose and Fermi occupancy is small compared to unity then the Bose and Fermi occupancies
reduce to the Boltzmann occupancy.

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
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. Use statistical mechanical methods to show that Bose-Einstein condensation occurs
for a non-interacting gas of bosons held in a three dimensional harmonic oscillator potential at sufficiently
low temperatures.
a) Show that 87Rb is a boson.
b)The energy of the single particle in a three-dimensional harmonic oscillator
is 
3

  lx  l y  lz   where lx,. ly. and lz are non-negative integers. Since l  l x  ly  lz is a

2 
non-negative integer one can express the average number and the average energy as functions of l. Show
that the degeneracy gl of the level with energy
case of large l ( gl 
 3
l 2 3l
   l   is gl    1 . Specialize to the

2 
2
2
2
l
)which dominates for large N.
2

c)The grand potential is given by


(T, )  k B T  gl ln 1 exp  (l  ).
l 0
(Let’s shift the energy scale to get rid of the zero point energy for the rest of the problem.) Show that the

average number of particles in the trap is given by

N 
l 0

gl
exp  (   )l 1



0
 
N  
 
 . Show this can be written
 T
l2
dl
1 k T 
  B 
2 exp  (   )l 1 2   
3 

0
x 2 dx
.
e x e  1
d)Sketch the behavior of the Bose-Einstein occupancy factor to argue that for fixed N, as T decreases the
chemical potential must increase. Since the chemical potential cannot lie above the ground state eventually
the chemical potential reaches the level of the ground state as T decreases. This is called the critical
temperature. Determine the critical temperature as a function of N and  . You will need to evaluate the
following integral.
1
2


0
x 2 dx 1

e x 1 2


0
y 2ey dy
function by numerically summing the series.
 m

m1

1
3


1
  3 . Estimate the value of the zeta
m3
m1
4. Bose-Einstein condensation continued
a) Determine
 the critical temperature in the Cornell and Wieman experiment for 1000 atoms in the trap
with oscillator frequency f=200Hz. Repeat for the cases N=10 6 and 109.
b) 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.
c) Determine the internal energy per particle and the heat capacity per particle as functions of T for T<T c.
d) For the case N=109, how many atoms are in the ground state level of the oscillator when T/T c=0.9? How
many atoms are in the first excited state? Use the fact the chemical potential is approaching the ground
state energy. Show that while the ground state contains several hundred million atoms, the first excited
state contains less than one thousand.
e) Show that the Bose-Einstein critical temperature is the roughly the point where the average level
occupation per particle is unity, i.e. the number of particles in the trap and the number of energy states
below kBTc are about the same.