As an illustration of the use of occupation numbers in the evaluation
of the quantum partition function, let us consider the simple
case of Boltzmann statistics (ignoring spin statistics or treating
the particles as distinguishable). The canonical partition function
Q(N,V,T) can be expressed as a sum over the quantum numbers
for each particle:
In terms of occupation numbers, the partition is
where 
is a factor that tells how many different physical
states can be represented by a given set of occupation numbers 
.
For Boltzmann particles, exchanging the momentum labels of two particles
leads to a different physical state but leaves the occupation numbers
unchanged. In fact the counting problem is merely one of determining
how many different ways can N particles be placed in the different
physical states. This is just
For example, if there are just two states, then the occupation numbers
must be 
and 
where 
. The above formula gives
which is the expected binomial coefficient.
The partition function therefore becomes
which is just the multinomial expansion for
Again, if there were two states, then the partition function would be
using the binomial theorem.
Therefore, we just need to be able to evaluate the sum
But we are interested in the thermodynamic limit, where 
.
In this limit, the spacing between the single-particle energy levels becomes
quite small, and the discrete sum over 
can, to a very good approximation,
be replaced by an integral over a continuous variable:
Since the single-particle eigenvalues only depend on the magnitude of ,
this becomes
where 
is the thermal deBroglie wavelength.
Hence,
which is just the classical result. Therefore, we see that an ideal gas of distinguishable particles, even when treated fully quantum mechanically, will have precisely the same properties as a classical ideal gas. Clearly, all of the quantum effects are contained in the particle spin statistics. In the next few lectures we will see just how profound an effect the spin statistics can have on the equilibrium properties.