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## Main Question or Discussion Point

Not many people understood his proof in class, and the textbook's proof wasn't very clear so we went by with other derivations online. Then he filled half the midterm with his method, so I'm trying to understand how he did things.

Looking back it seems very similar to the proofs we found online but different in a lot of places.

The goal is to come to the spectral energy density form of

$$u(f)df=\frac{8\pi h f^3}{c^3}\frac{1}{e^\frac{hf}{k_B T}-1}df$$

The closest proof to his method was this: qsp_chapter10-plank.pdf which divides the law into two parts: the number volume density ##\frac{8\pi f^2 df}{c^3}## and the average energy per state ##\frac{hf}{e^\frac{hf}{k_B T}-1}##. I'll refer to this proof as the PDF proof.

The professor divided Planck's Law into three parts instead: number volume density, energy per state ##hf## and state occupation ##\frac{1}{e^\frac{hf}{k_B T}-1}##.

Then he derived the number volume density. The PDF proof gives that since ##n## in all directions is positive we're only concerned with only the positive octant of the reciprocal space, resulting in a factor of 1/8.

The professor does not have that factor. Instead, he starts the proof with ##k=\frac{n\pi}{L}## from the wave in a box scenario, then all of a sudden switches to ##k=\frac{2\pi n}{L}##. The form then looks like:

$$N = 2\big( \frac{L}{2\pi}\big)^3\int_0^{4\pi}\int_0^{k_{max}}k^2 dk d\Omega$$

where ##\frac{L}{2\pi}## is, in his words, the density per dimension and the integral is the volume of the k-space. He specifically put them down separately, raising the power of ##\frac{L}{2\pi}## for each dimension instead of going through (or rather, skipping steps for) the volume integral like the PDF proof.

So from all of this I can't understand:

Looking back it seems very similar to the proofs we found online but different in a lot of places.

The goal is to come to the spectral energy density form of

$$u(f)df=\frac{8\pi h f^3}{c^3}\frac{1}{e^\frac{hf}{k_B T}-1}df$$

The closest proof to his method was this: qsp_chapter10-plank.pdf which divides the law into two parts: the number volume density ##\frac{8\pi f^2 df}{c^3}## and the average energy per state ##\frac{hf}{e^\frac{hf}{k_B T}-1}##. I'll refer to this proof as the PDF proof.

The professor divided Planck's Law into three parts instead: number volume density, energy per state ##hf## and state occupation ##\frac{1}{e^\frac{hf}{k_B T}-1}##.

Then he derived the number volume density. The PDF proof gives that since ##n## in all directions is positive we're only concerned with only the positive octant of the reciprocal space, resulting in a factor of 1/8.

The professor does not have that factor. Instead, he starts the proof with ##k=\frac{n\pi}{L}## from the wave in a box scenario, then all of a sudden switches to ##k=\frac{2\pi n}{L}##. The form then looks like:

$$N = 2\big( \frac{L}{2\pi}\big)^3\int_0^{4\pi}\int_0^{k_{max}}k^2 dk d\Omega$$

where ##\frac{L}{2\pi}## is, in his words, the density per dimension and the integral is the volume of the k-space. He specifically put them down separately, raising the power of ##\frac{L}{2\pi}## for each dimension instead of going through (or rather, skipping steps for) the volume integral like the PDF proof.

So from all of this I can't understand:

- Does state occupation refer to the average state that the photon occupies? How does this state occupation translate from a photon in a box to unbound free space?

- How did he go from ##k=\frac{n\pi}{L}## to ##k=\frac{2\pi n}{L}## and eliminate the 1/8 factor?
- He says ##\frac{L}{2\pi}## is density per dimension, and that it's cubed because there's three dimensions. Am I right in interpreting this as the maximum allowed length per state? Or is this a volume of some sort instead of density?
- He separated physical dimensions ##\frac{L}{2\pi}## from the volume integral. Then he manipulated the integral in another problem as if it referred to physical space instead of reciprocal space. Taken by itself, what does the k-space integral ##\int_0^{4\pi}\int_0^{k_{max}}k^2 dk d\Omega## mean?