Homework questions on quantum theory (blackbody radiation)

Strangedaze

Hello.. i need help with two homework questions if anyone would be so kind.
I do not know where to begin on solving these questions.
The questions are as follows:

1. If the angular frequencies of waves in a three-dimensional box of sides L generalize to:

Ω = πc/L (nx²+ny²+nz²)^1/2

Where all n are integers, show that the number of distinct states in the frequency interval f to f + Δf
(f=Ω/2π) is given by (where f is large)

dN = 4
π(L^3/c^3)f²df

2. Let the energy density in the frequency interval f to f + df within a blackbody at temperature T be dU(f,T). Show that the power emitted through a small hole of area
ΔA in the container is

c/4dU
(f,T)ΔA

THANK YOU!

We have
f=c/2L (nx²+ny²+nz²)^1/2 (1)
So with common f we have a (1/8) of a sphere with radius r = (nx²+ny²+nz²)^1/2=(2L/c)f
the area of this (1/8) of a sphere is πr^2/2 = (1/2)π(f 2L/c)^2
Now, as we want to count the triples (nx, ny, nz) for which f given by (1) falls between f and f+df we need to to find the "thikness" of the sphere, which is dr=(2L/c)df. Thus we get
number of states = πr^2/2 dr = (1/2)π(f 2L/c)^2(2L/c)df = 4π(L/c)^3f^2df
best regards

Last edited:

I decided to enclose a picture of the situation . Hope this help as well as equations!

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