# Polarization and degeneracy

#### jaobyccdee

In photon statistics, g is defined as the internal degeneracy per particle, and the text gives the example that photon have two possible polarization states in three space dimensions, thus g=2. Why is the number of possible polarization equals the internal degeneracy of the particle?

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#### Ken G

Gold Member
Degeneracy just means different states that have the same energy. In most applications, the different polarization states of the photon have the same energy, so that is a degeneracy. I'm not sure why they call it an "internal" degeneracy, I wasn't aware there were any external degeneracies! Perhaps they view spin as something internal, whereas orbital degrees of freedom are external.

#### Cthugha

I'm not sure why they call it an "internal" degeneracy, I wasn't aware there were any external degeneracies!
They are terming it internal degeneracy as opposed to configurational degeneracy. The latter happens when you have a larger number of macroscopic, equivalent configurations. This is interesting for example in systems showing residual configurational entropy like spin ice.

#### Ken G

Gold Member
They are terming it internal degeneracy as opposed to configurational degeneracy. The latter happens when you have a larger number of macroscopic, equivalent configurations. This is interesting for example in systems showing residual configurational entropy like spin ice.
OK, so there is a formal meaning to the term, thank you. It sounds like the distinctions live in an interesting crossover region between microscopic quantum mechanics and macroscopic degrees of freedom that nevertheless associate with quantized action.

#### Cthugha

Maybe one of the best known examples is ice made from simple water. It is well known that the distance between two Oxygens is roughly 2.76 Angströms, while the OH-bond is roughly 0.95 Angströms. So in ice there are two possible positions for the hydrogen atom along the O-O line: Either at 0.95 Angströms to oxygen atom one and 1.81 Angströms to oxygen atom two or the other way round.

As a consequence in an ice crystal each oxygen atom will have two associated hydrogen atoms at the short distance and two at the long distance, but the exact arrangement is not set as each arrangement fulfilling the above rule leads to the same energy. Therefore you do not get a single ground state when going to 0K, but a huge number of degenerate fround states. You see similar results in spin ice materials like dysprosium titanate.

For some materials the effect can indeed be shown to exist in measurements of the specific heat. However, I suppose, the measurements are no fun.

#### Ken G

Gold Member
That's quite interesting, I did not know that! Counting those states must be an interesting exercise in combinatorics.

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