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Filling the Box with Fermions

If we fill a cold box with
fermions, they will all go into different low-energy states.
In fact, if the temperature is low enough, they will go into the lowest energy
states.

If we fill up all the states up to some energy, that energy is called the **Fermi energy**.
All the states with energies lower than
are filled, and all the states with energies
larger than
are empty.
(Non zero temperature will put some particles in excited states, but, the idea of the
Fermi energy is still valid.)

Since the energy goes like
,
it makes sense to define a radius
in
-space out to which the states are filled.

The number of states within the radius is

where we have added a factor of 2 because fermions have two spin states.
This is an **approximate counting of the number of states**
based on the volume of a sphere in
-space.
The factor of
indicates that we are just using one eighth of the sphere in
-space
because all the quantum numbers must be positive.
We can now **relate the Fermi energy to the number of particles in the box**.

We can also integrate to get the **total energy** of all the fermions.

where the last step shows how the total energy depends on the number of particles
per unit volume
.
It makes sense that this energy is proportional to the volume.
The step in which
and
is related to
is often useful.

Jim Branson
2013-04-22