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.

E_F =

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". 

4. Quantum effects can occur at surprisingly high temperatures. Most experiments that probe quantum effects are performed at low temperatures, often on the scale of a few Kelvin, or even less. Yet, inside a star, the quantisation of energy levels and quantum statistics can dominate the physics of the system, even at temperatures of millions or billions of Kelvin.

Note, how the temperatures of these systems span 18 orders of magnitude!

System	Typical Number Density (n/V)	Typical Temperature (T)	Fermi Temperature (TF)
Electrons in a metal	~1022 – 1023 cm-3	~300 K (Room temp)	~5 × 104 K
3He atoms in liquid 3He	1.6 ×1022 cm-3	< 1 K (Cryogenic)	~2 – 5 K
Nucleons in atomic nuclei	1.6 ×1038 cm-3	~0 K (Ground state)	4.4 × 1011 K
Electrons in a white dwarf star	~1029 – 1030 cm-3	~107 K (Interior)	~108 – 109 K
Neutrons in a neutron star	~1038 – 1039 cm-3	~108 – 109 K	~1011 – 1012 K
Fermionic atoms in an ultracold gas	~1012 – 1014 cm-3	~10 – 100 nK	~100 – 1000 nK

The table was made with Google Gemini.

Two asides.

A. An old blog post considered the interesting question of why people call these systems "degenerate."

B. A more profound question is why neglecting the interactions between the fermions (which involves energies comparable to the Fermi energy in most of these systems) can give a theory that is so successful, both qualitatively and reasonably quantitatively. Landau's Fermi liquid theory provides the answer.