What is so amazing about the Fermi energy?

In two weeks, I am giving two lectures about degenerate Fermi gases in a third-year undergraduate course on statistical mechanics. The textbook is the beautiful book by Schroeder. I want to highlight a few things that are amazing about the Fermi energy.1. It is given by an incredibly simple expression. EF = ℏ2 2m (3π2n)2/3 2. Besides fundamental constants, it is only determined by the number density of fermions n.3. It is relevant to diverse systems: electrons in a metal, liquid 3He, electrons in a white dwarf star, neutrons in a neutron star, ultracold fermionic atoms such as 6Li. The table below shows that for all of these systems many of them are in the regime where the temperature is much less than the Fermi temperature. In other words, the Fermi energy (temperature) is so "large" that "low" temperatures can be very "high". Note, how the temperatures of these systems span 18 orders of magnitude! System Typical Number Density (N/V) Typical Temperature (T) Fermi Temperature (TF)…