Thermal Physics: Photon Statistics on Bose Particles

[Andrew Lewis]

Homework Statement

[/B]
I have solve the rest of this problem pretty easily and see no problems with working with Indistinguishable particles, Distinguishable particles, fermions and Bosons. Part c has me very confused though about what it is even asking.

Suppose a system with equally spaced energy levels (see diagram), with values e, 2e, etc., and that there are 3 noninteracting particles which can occupy these levels. (a) Find the ground state configuration and energy, and the degeneracy of the ground state, for the case that the 3 particles are (i) indistinguishable spinless bosons, (ii) indistinguishable spin-1/2 fermions, (iii) distinguishable particles. (b) Repeat this procedure for the 1st and 2nd excited state, finding energy and degeneracies for the three cases listed in (a). (c) Consider also the case of Bose particles with photon statistics: what is the lowest-energy configuration and energy, and that of the next two lowest energy configurations. How many ways can these be rearranged?

Homework Equations

Bose-Einstein Distribution n(ε)=g/(e^(ε-μ)/(kT) - 1)
Planck Distribution n(ε)=1/(e^ε/(kT) -1)[/B]

3. Attempt at a Solution

Photons in a box are not countable. I was thinking about using the partition function but that just leaves me with the Planck distribution and working with that formula doesn't tell me the lowest energy configuration or the exact number of particles instead it gives me the average energy or average number of particles. How do Bose particles behaving under photon statistics even behave when energy levels get involved? My only other idea is to treat the system exactly as I did in the standard Bose particle case but this time account for polarization but my gut is telling me that is not correct at all.

My current thought is to ignore the idea of there being 3 photons and following the standard treatment of photons in a box depending on the energy of said box. The lowest energy of the box follows E=ħω and counts following the natural numbers. This is all fine until i need to consider the idea of what the problem means by configuration and the energy of that configuration. I know the energy of the 3 smallest modes photons in a cavity. Could somebody clarify what configuration in this instance means? My only clue as to what this might mean is I need to imagine 3 photons in the box each one in the smallest energy mode just like a regular boson. If that is the case though the statistics would follow standard boson statistics rather than giving any new insight on how a photon behaves. My gut is telling me that is an incorrect.

 

[Andrew Lewis]

I think I may have a bit more understanding after walking away and reading a bit more literature but i would like to clarify my thinking and make sure I am on the right track. I am thinking that rather than try to imagine singular bosons we just treat this like standard cavity radiation and say the smallest mode of vibration for our photons is ħω = ε for the fundamental frequency of our box and instead just follow the Plank Distribution to the letter. If this is the case then the 3 smallest modes for the energy are E_n= nħω. Though I'm still a bit puzzled on what the problem means by energy configuration. Does it still want me to imagine 3 photons? If i imagine 3 photons the solution reduces to the simplest indistinguishable boson case which I have already done in part a and b of the problem.