There are basically two questions:
- How is the fermi energy an energy difference, and at the same time the energy of the highest occupied state (implying that the lowest state is 0)?
This comes from the confusion based on the not full or incorrect citation, "...the energy difference between the highest and lowest occupied single-particle states in a quantum system of non-interacting fermions at absolute zero temperature."
Now the correct statements come as follows:
the Fermi energy is the energy difference between the highest and lowest single particle energy state at absolute zero
in a Fermi gas, the lowest state is known to have zero kinetic energy, but in a metal, lowest occupied state is the bottom of the conduction band, thus non zero kinetic energy
So the answer to your question is that in a Fermi gas, yes, the energy difference is equal to the energy of the highest occupied state, because the lowest is zero, but in a metal, the answer is no, because the lowest energy level is non zero energy.
Why is the Fermi energy a kinetic energy?
- the Fermi energy is an energy difference, usually kinetic energy, whereas the Fermi level is kinetic and potential energy
As a consequence, even if we have extracted all possible energy from a Fermi gas by cooling it to near absolute zero temperature, the fermions are still moving around at a high speed. The fastest ones are moving at a velocity corresponding to a kinetic energy equal to the Fermi energy. This speed is known as the Fermi velocity.
https://en.wikipedia.org/wiki/Fermi_energy
The Fermi momentum and velocity:
${\displaystyle p_{\mathrm {F} }={\sqrt {2m_{0}E_{\mathrm {F} }}}},$
${\displaystyle v_{\mathrm {F} }={\frac {p_{\mathrm {F} }}{m_{0}}}}.$
They correspond to the Fermi energy, thus the Fermi energy is regarded as a kinetic energy (the speed of the electrons).
