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s = ∫ 1 θ r +  dr  2 dθ
2
d 
0 θ  θ
c. Use the result of (b) above to derive the length of the cardioid r =
a(1 + cos(θ)) between the angles 0 and π.
Pb. 4.36 In Pb. 3.27, you plotted the Fermi-Dirac distribution. This curve
represents the average population of fermions in a state with energy ε (ignore
for the moment the internal quantum numbers of the fermions). As you would
have noticed, this quantity is always smaller or equal to one. This is a mani-
festation of Pauli’s exclusion principle, which states that no two fermions can
be simultaneously in the same state. This, of course, means that even at zero
absolute temperature, the momentum of almost all fermions is not zero; that
is, we cannot freeze the thermal motion of all electrons at absolute zero. This
fact is fundamental to our understanding of metals and semiconductors, and
will be the subject of detailed studies in courses on physical electronics.
In nature, on the other hand, there is another family of particles that
behaves quite the opposite; they are called Bosons. These particles are not
averse to occupying the same state; moreover, they have a strong affinity,
under the proper conditions, to aggregate in the lowest energy state avail-
able. When this happens, we say that the particles formed a Bose condensate.
This phenomenon has been predicted theoretically to occur both on the labo-
ratory scale and in some astrophysical objects (called neutron stars). The phe-
nomena of superconductivity, superfluidity, and pion condensation, which
occur in condensed or supercondensed matter, are manifestations of Bose
condensates; however, it was only recently that this phenomenon has been
observed to also occur experimentally in gaseous systems of atoms that were
cooled in a process called laser cooling. The details of the cooling mechanism
do not concern us at the moment, but what we seek to achieve in this problem
is an understanding of the fashion in which the number density (i.e., the
number per unit volume) of the condensate can become macroscopic. To
achieve this goal, we shall use the skills that you have developed in numeri-
cally integrating and differentiating functions.
The starting point of the analysis is a formula that you will derive in future
courses in statistical physics; it states that the number of particles in the con-
densate (i.e., the atoms in the gas that have momentum zero) can be written,
for a noninteracting Bosons system, as:

n = n − 1 g z ()
condensate λ 3 32
/
T
where λ is a quantity proportional to T –1/2 , n is the total number density, and
T
the second term on the RHS of the equation represents the number density of
the particles not in the condensate (i.e., those particles whose momentum is
not zero). The function g 3/2 (z) is defined such that:

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