Statistical mechanics: Total number of photons in a cavity

[cdot]

Homework Statement

Show that the number of photons in equlibrium at tempertaure t in a cavity of volume V is,
N=[2.404 V (t/ħc^3]/Pi^2

The total number of photons is the sum of the average number of photons over all modes n->∑<s>

Homework Equations

n=Sqrt[nx^2+ny^2+nz^2]
ωn=(n Pi c)/L

The Attempt at a Solution

So I actually have the solution to this problem thanks to the internet but I don't quite understand everything about it.
To start, I don't undertsand how the total number of photons is the sum of the AVERAGE number of photons in each mode. Obviousely the total number of photons in the cavity will be the sum of the photons at each mode so the question must be assuming that the average number of photons in each mode IS the number of photons in each mode? Does this follow from the sharpness of the multiplicity function and the fact that we're unlikely to find other numbers of photons in a particular mode other than the average (in thermal equilibrium)?

But anyway, assuming the total number of photons is indeed given by ∑<s> over all n (where n=Sqrt[nx^2+ny^2+nz^2]),

<s>=1/(exp(ħω/t)-1) where ω=(n*Pi*c)/L for n=(0,1,2,...)

Summing the expression for <s> over all n is actually a sum over nx,ny,nz:
∑∑∑<s> where each sum is over nx,ny, or nz

Now I don't understand the next step which involves turning this sum into an integral over the volume element dnx*dny*dnz "in the space of the mode indices"

I don't understand why you can turn that sum into an integral and I don't understand how it works out o be
(1/8)∫ [4*Pi*n^2] dn ?

I think this may be a math trick that I haven't been taught yet or forgot? What is the space of the mode indices?

 

[DrClaude]

So I actually have the solution to this problem thanks to the internet but I don't quite understand everything about it.
To start, I don't undertsand how the total number of photons is the sum of the AVERAGE number of photons in each mode. Obviousely the total number of photons in the cavity will be the sum of the photons at each mode so the question must be assuming that the average number of photons in each mode IS the number of photons in each mode? Does this follow from the sharpness of the multiplicity function and the fact that we're unlikely to find other numbers of photons in a particular mode other than the average (in thermal equilibrium)?

The number of photons in the cavity is fluctuating, so the question is asking for the average number of photons, even though it is not mentioned explicitly.

I don't understand why you can turn that sum into an integral and I don't understand how it works out o be
(1/8)∫ [4*Pi*n^2] dn ?

You assume that the modes are closely spaced, such that you can transform the sum into an integral. As for the factor, have you heard about the density of states?

 

[cdot]

Thanks for your reply. I didn't think about how the number of photons is not constant. That makes sense now. I also now understand how you can approximate the sum as an intergal (especially for closely spaced modes). However, I still don't understand how the integral works out to be what it is. The integral would be a triple integral over nx,ny,nz. I thought the expression we're integrating is the same expression that we're summing...so we have ∫∫∫ ħω/(exp[ħω/t]-1) dn where ω= (n*Pi*c)/L and n =Sqrt[nx^2+ny^2+nz^2] and the limits of integration for each are from n=0 to n=Infinity. I can see that the factor of 1/8 would come out because we only care about positive nx,ny, and nz but I don't undrstand how the sum ∑(...) turns into 1/8 ∫ 4 Pi n^2 dn (...). Where did that 4 Pi n^2 come from? Am I forgetting something from calc 3?

 

[DrClaude]

It comes from the density of states. See for example the derivation of eq. (1.6) in http://tdqms.uchicago.edu/sites/tdqms.uchicago.edu/files/uploads/ReferenceMaterial/Density%20of%20States.pdf