# How is the degenerate electron gas state “degenerate”?

What is "degenerate" in the degenerate electron gas state?

Why is it called degenerate?

• Good question. I think of degenerate as meaning having the same energy as in degenerate energy levels. However the the phrase degenerate matter means any matter whose properties are dominates by quantum statistics (en.wikipedia.org/wiki/Degenerate_matter). It isn't obvious to me why the same word is used in two different contexts. – John Rennie Aug 29 '13 at 10:41
• @JohnRennie I think the word degenerate in this context refers to the possibility of multiple particles occupying the same single-particle state (cf. multiple states with the same energy). Obviously in a Fermi system this possibility does not exist, whether or not the system is quantum degenerate. However the Pauli principle only becomes relevant at temperatures where fermions would "try" to occupy the same state in the absence of the exchange statistics. – Mark Mitchison Jul 9 '15 at 14:00

A degenerate gas is one where more than one electron (in fact two, one in each spin state) occupies each possible low-energy state up to the Fermi energy. I suppose the term "degenerate" comes from the multiple occupancy of each energy level.

A normal gas consists of particles that do not interact much except for elastic collisions.

Often, describing a gas in a simplified way ignoring the other interactions completely is good enough.
This simplified model is the "ideal gas".

This simplified description fits for the electron gas also, as long as the pauli exclusion principle does not become relevant.
This would be called an ideal Fermi gas.

But there are cases where the pauli principle is very relevant, when not enough low energy levels are free for electrons.

When electrons are confined into a limited volume, the pauli principle limits the number of electrons that can occupy that space for a given maximal energy level.

Now, the electron gas is degenerate in that there is another interaction apart from what normally makes a gas - and this can be the dominating property of the degenerate gas.
For example, the pressure of a degenerate electron gas is higher than one would expect from a gas.

The degenerate electron gas does not only behave differently from an ideal gas, it has fundamentally different properties based on quantum effects.

• Degenerate gases are often considered to be ideal gases (point-like, non-interacting particles). What you term an ideal gas, I would call a "perfect gas". – Rob Jeffries Jul 9 '15 at 15:16
• Not sure - at least it seems like perfect gas is mainly used in the context of thermodynamics, and ideal gas has quite some uses on quantum mechanics level. But I feel I miss your point. – Volker Siegel Jul 9 '15 at 15:31
• My point is just that ideal gases can be degenerate. en.wikipedia.org/wiki/Ideal_gas#Ideal_Bose_and_Fermi_gases and that degeneracy pressure is not the result of any force between the particles. – Rob Jeffries Jul 9 '15 at 15:55
• I understand this description of an ideal Fermi gas to be a simplification and approximation assuming no interacton. That is, it ignores the effects that lead to degeneraction. It's just simplified and helpful as long as it's a "simple" case - just as the classic ideal gas is a simplification that is helpful in many "simple" cases - but not in some extreme conditions. – Volker Siegel Jul 9 '15 at 21:26
• The term ideal - means, point-like, non-interacting particles with only elastic collisions. It does not ignore the Pauli exclusion principle, which is not a force, and is not deemed to be an "interaction" in the same way as say coulomb repulsion between electrons or Thomas-Fermi corrections, both of which could render a degenerate electron gas non-ideal. – Rob Jeffries Jul 9 '15 at 22:23