Statistical Mechanics - Physics at Oregon State University

68 CHAPTER 4. STATISTICS OF INDEPENDENT PARTICLES.
of energy levels available. These energy levels are called single particle states.
In order to avoid using the word state in two different meanings, we will follow
a standard approach and call these single particle states orbitals. One is used
to the word orbital for systems of electrons, but here it is generalized to systems
of arbitrary particles.
An example will help to describe this. Consider an atom, for which the
states of the electrons are given by the quantum numbers n, l, m. We lump these
quantum numbers together and use the symbol o, for orbital. The corresponding
energies depend, however, on the states of all other electrons. In Helium, for
example, the energy of an electron in the 1s state is different in the 1s 2 2s 0
configuration and in the 1s 1 2s 1 configuration. It is larger (more negative) in
the second case, because the 2s electron does not screen the nucleus as well as
the 1s electron. In this case there are many body effects, often described by the
word correlation. The energy levels are of the form ɛ s o, where s stands for the
state of the whole system.
Orbital energy of independent particles.
In an ideal gas we have a special situation. Here the particles do not interact,
and the energy of a particle is independent of the states of the other particles.
We say that there is no correlation, or no screening, and we can write
ɛ s o = ɛo
(4.1)
for all orbitals o and in all states s of the total system. We say that the particles
are independent.
This is a useful approximation to make, but is it realistic? In a Helium
answer the answer is clearly no. But it turns out that for electrons in a metal
the independent electron approximation is not a bad starting point. Well, more
or less, as long as we replace the electrons by quasiparticles, that is electrons
with a corresponding empty space around them, out of which other electrons
have been repelled. For details, see books on solid state physics.
Total energy of independent particles.
A general many body state of a system with N particles is completely known
if we know how many particles are in each orbital. This defines a function of
the single particle orbital index o for each many body state s, ns o. The energy
of this many body state in terms of the single particle orbital energies ɛo is
E(state s) =
(4.2)
and of course we have
o
N(state s) =
o
n s oɛo
n s o
(4.3)