"Why is the degenerate Fermi gas called "degenerate"?
Is it anything to do with degenerate energy levels?"
So, I went in search for answers.
The Wikipedia entry on Degenerate matter is a bit rambling and I found it unhelpful.
I then went to the library and looked at a few textbooks and found a range of answers. Some books use the term "degenerate" without any elaboration.
In the discussion below it seems looking at the Oxford dictionary is helpful:
Having lost the physical, mental, or moral qualities considered normal and desirable; showing evidence of decline.
technical: Lacking some usual or expected property or quality, in particular.Here are a few entries
degenerate. (This use of the word is completely unrelated to its other use to describe a set of quantum states that have the same energy).Daniel V. Schroeder, An Introduction to Thermal Physics, page 272.
[my favourite undergraduate text on statistical mechanics]
... a bit of terminology. At low temperature, quantum ideal gases behave very differently from the way a classical ideal gas behaves. ..... The quantum gases are said to be degenerate at low temperatures. This is not a moral judgement. Rather, the word "degenerate" is used in the sense of departing markedly from the properties of an "ordinary" classical gas.Ralph Baierlein, Thermal Physics, page 192.
Gas degeneration proper. The quantitative study of the deviations from the classical gas laws when xi is not very small ...Erwin Schrodinger, Statistical Thermodynamics (1936)
Here xi is the product of the particle density and the thermal deBroglie wave length cubed.
The definition of a quantum gas is one where xi becomes larger than one. (This occurs for "high" densities and "low" temperatures).
But then there is an interpretation in terms of degeneracy of energy levels because one can consider the case where each energy level has degeneracy g and condition for a non-degenerate gas (i.e. Maxwell-Boltzmann statistics to apply) is
g >> n_i ~ exp ((mu-Ei)/kB T) = number of particles in level i
I welcome comments.