Statistical Mechanics - Physics at Oregon State University

Chapter 4
Statistics of independent
particles.
4.1 Introduction.
Particle statistics.
If you think about the consequences of assuming a low density limit for a gas,
it comes to mind immediately that the interactions between the particles must
be very small. An ideal gas is defined as a gas of non-interacting particles in the
low density limit. If we leave out the low density, but still assume that there are
no interactions between the particles, quantum effects will start to play a role
and in that case the gas is a Bose or Fermi gas. Although one might object that
in this case it it not correct to ignore the inter-particle interactions, there are
enough systems where these interactions are small and a non-interacting gas is
a very good first approximation. This explains why it is useful at all to discuss
Fermi or Bose gases.
The treatment of the ideal gas in the previous chapter is useful, but it is not
a good starting point for more precise calculations. In this chapter we discuss
an approach which is very versatile, and can be used to include the effects of
particle statistics and interactions.
Orbitals.
The energy levels used in the previous section were the energy levels of a single
particle in a large box. What happens when we add a second particle to this
box? If the particles are really independent, the presence of this second particle
will not alter the energy levels of the first. Neither will the first particle influence
the energy levels of the second, and hence both particle will have the same set
67