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