1. The problem statement, all variables and given/known data 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> 2. Relevant equations n=Sqrt[nx^2+ny^2+nz^2] ωn=(n Pi c)/L 3. 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?