In order to derive an expression for the average particle number, recall that
In the thermodynamic limit, we may take the sum over to an integration:
But
Therefore, it proves useful to change variables of integration
from n to 
, using the above relation:
Thus,
In order to derive an expression for the average energy, recall that the energy eigenvalues were given by
Therefore, the average energy is given by
At T=0, this becomes
If the same change of variables is made, one finds that
Thus, the average energy can be seen to be related to the average particle number by
which is clearly not 0 (as it would be classically).
Note that the pressure can be obtained simply in the following way: Recognize that
The energy is given by
Thus,
Comparing these two equations for the energy and pressure shows that
Note, that just like the energy, the pressure at T=0 is not zero. The T=0 values of both the energy and pressure are:
These are referred to as the zero-point energy and pressure and are
purely quantum mechanical in nature. The fact that the pressure does
not vanish at T=0 is again a consequence of the Pauli exclusion principle
and the effective repulsive interaction that also showed up in the low
density, high temperature limit.
Using the expansion for 
, we can derive the thermodynamics
in this limit.