Statistical Mechanics - Physics at Oregon State University

Chapter 5
Fermi and Bose systems of
free, independent particles.
5.1 Fermions in a box.
All the formulas we have derived for the thermodynamic variables of an independent
Fermi gas contain a sum over orbitals. One would like to convert these
sums to integrals in order to use standard analytical techniques. In order to do
this we need to know the details of the single particle energies. Without such
further knowledge we cannot derive results in more detail.
Free, independent particles.
We still assume that the particles are independent. Also, in the simplest
case we assume that there are no external forces acting on the particles. This
is easy when we assume that the energy of the orbital is given by the energy
levels of a particle in a box. Of course, there are the implicit forces due to the
pressure on the sides of the box, but these forces are taken into account by our
boundary conditions. When we are done with our calculations, the pressure
needed to keep the box at a constant volume will follow automatically. We also
assume that the box is a cube. This is not essential, but makes the mathematics
easier. The sides of the cubic box are of length L and the energy levels are
ɛ(nx, ny, nz) = 2
2M
π
2 (n
L
2 x + n 2 y + n 2 z) (5.1)
As we have done before, we define a wave vector k by k = π
L (nx, ny, nz). We
use this wave vector to label the single particle orbitals. We also need a spin
label, but at this point we assume that there are no magnetic effects. The set
of wave vectors is discrete, with step sizes ∆kx = ∆ky = ∆kz = π
L . Therefore,
any sum over orbitals is equal to
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