Fermi energy at zero temprature

Question

If I have a beaker that is at zero temperature in thermal equilibrium with its surrounding. If I start filling fermions (say electrons) in it, then according to the Fermi-Dirac statistics, the energy of the beaker after I have filled it with $N$ electrons will be related to the number of particles and the volume of the beaker, I am confused in which form is the Fermi energy stored in the electrons? If it its in form of translational kinetic energy, then wouldn't those electrons strike the beaker at high speeds and increase the temperature of the beaker from 0 kelvin to some finite amount? I am a beginner and I think I am missing something here.

Answer 1

I will assume that there is no electro-magnetic interaction between the electrons and talk about fermions in what follows. The fermions have kinetic energy $E=\frac{p^2}{2m}$ and at zero temerature they fill up all the low energy states (each fermion takes a phase space volume of $(2 \pi \hbar)^3$) up to the fermi energy $E_F=\frac{p_F^2}{2m}$. So the fermi energy is the kinetic energy of the highest energy fermions.

If it its in form of translational kinetic energy, then wouldn't those electrons strike the beaker at high speeds and increase the temperature of the beaker from 0 kelvin to some finite amount?

This should not happen:

1. If there are no other processes that compensate for the Energy transfer from the fermions to the beaker, the fermions and the beaker are not in thermal equilibrium.

2. If the fermions are at zero temperature, they should not be able to increase the temperature of the beaker, as this would lower their own temperature, which is at $T=0$ already.

3. Suppose a fermion with incoming energy $E$ hits the beaker and has energy $E^\prime$ afterwards. If there is an energy transfer from the fermion to the beaker $E^\prime < E \leq E_F$. But since all states with energy below $E_F$ are filled up with other fermions this energy transfer cannot happen.

Answer 2

[...] in which form is the Fermi energy stored in the electrons? If it its in form of translational kinetic energy, then wouldn't those electrons strike the beaker at high speeds and increase the temperature of the beaker from 0 kelvin to some finite amount? I am a beginner and I think I am missing something here.

What seems misunderstood here is that 'Fermi energy' is not a form of energy, but a specific energy value - the topmost energy of the filled fermionic levels at zero temperature. It thus coincides with chemical potential, although at non-zero temperature or for more complex energy spectra the two are clearly not the same (confusingly, chemical potential is also called Fermi level)