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Here I have a question about the Derivation of Planck's formula in this text. On page 5 is derived the differential expression

$dN= 2 \times \frac{4\pi \nu^2 V}{c^3} d\nu $

for the number of modes of oscillation in the frequency interval from $\nu $ to $\nu + d\nu $ inside a quadratic box of volume $V=L^3$ ($L$ edge length).

Essentially, it is obtained from $d\widetilde{N}$ which counts the number of of lattice points inside the $\vec{k}$-space having length from $k$ to $k +dk$ corresponding via dispersion relation to $\nu $ to $\nu + d\nu $
and then multiplying it by factor $2 \times$, because for every fixed lattice point $\vec{k}$ there are two linearly independent polarizations.

Question: The last part I not understand. Why do we have to multiply $d\widetilde{N}$ by this factor $2$? Or in other words, why to suffice to count only the number of linearly independent modes (for each fixed $\vec{k}$ and not all modes?

Thinking naively, if we fix a $\vec{k}$ inside $[k,k+dk]$ satisfying boundary conditions of the box (ie such one which contributes to $d\widetilde{N}$), there should be a lot of ways this wave can be polarized, it can be plane-polarized in any direction, circular-polarized, unpolarized and so on. Indeed there are should be infinitely many ways a mode get polarized for each $\vec{k}$, see eg this animation. But why only the miximal number of independent modes contribute to the sum $dN$? Isn't this rather unnatural?

Of course it's clear that on the other hand any polarized mode can be represented as linear combination of exactly two. In mathematical terms: the vector space of all polarizations with respect to fixed $\vec{k}$ is $2$-dimensional.

But my question is why $dN$ should count for every $\vec{k}$ only the maximal number of linear independent modes (ie here $=2$) and not all? Isn't this more "natural"?
Recall that our final goal is to calculate the energy density $u(T)$ which in turn is obtained by itegration of $\langle E \rangle dN = \langle E \rangle \cdot \frac{2 \times 4\pi \nu^2 V}{c^3} d \nu$ over the total frequence spectrum, where $\langle E \rangle=kT$.

But then intuitively it seems more natural integrate over all modes, not only the maximal number of linear independent, because we want the total amount of energy for fixed $T$, or not?

Therefore the factor $2$ in $dN$ looks rather cumbersome to me. My intuition on $u(T)$ is that to obtain the energy density we have to sum / integrate over all possible modes, because we want to obtain the complete energy density, which is in certain way a "total" quantity. But the procedure of picking only the maximal linear independent modes, looks like rather unnatural, because finally we want to obtain in some sense a "complete amount" of energy, but calculate it via picking sparesely out only some cadidates, instead to "summing over all."

Or do I missing here the right intuition behind the meaning of the energy density?

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there should be a lot of ways this wave can be polarized, it can be plane-polarized in any direction, circular-polarized, unpolarized and so on. Indeed there are should be infinitely many ways a mode get polarized

Yes.

But my question is why dN should count for every k⃗ only the maximal number of linear independent modes (ie here =2=2) and not all? Isn't this more "natural"?

Fourier expansion and corresponding modes are not based on the "natural" or "intutitive" thinking. They are based on mathematics of Fourier series, where we expand function as linear combination of orthonormal functions that form basis. These functions are all linearly independent. Each additive term in the simplest expression of the Fourier series is linearly independent of each other.

There is at most two such terms for any $\mathbf{k}$. Using more terms for given $\mathbf k$ would make the third and other terms for that $\mathbf k$ linearly dependent on the first two terms, and thus it would not be a Fourier series in the simplest form. Those other terms would be terms that are linearly dependent on the first two ones.

One could then take these superfluous terms, express them as linear combinations of the first two ones, and end up with two linearly independent terms for given $\mathbf k$, only with different coefficients than the original first two terms had.

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    $\begingroup$ What I not completely understand is how in this derivation (I should better say that's essentially the derivation Rayleigh-Jeans law) the Fourier methods are explicitly involved. Do I understand your idea correctly, that it's finally a matter of terminology, that that what is called "modes" form by definition members Fourier basis (and as consequence must be linear indepenend, but general property of Fourier basis? Is this your point? $\endgroup$
    – user267839
    Jan 14, 2023 at 23:31
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    $\begingroup$ hmmm, let me try to understand this philosophy: essentially the goal of Fourier theory is to endow a wide class of functions $f(x)$ with the property to be expandable with respect to a Fourier basis (in same spirit like in linear algebra every vector can be expanded with respect a fixed basis), $f(x)= \sum \hat{f}_n e^{i2 \pi kx} $ (or the "extension" to integral version $... = \int \hat{f}(z) e^{i 2 \pi z x}dz$). $\endgroup$
    – user267839
    Jan 15, 2023 at 0:00
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    $\begingroup$ So roughly the philosophy is that to understand $f(x)$ it suffice to understand the "weights or coefficients" (Fourier trafos) with respect to Fourier basis (= so only certain choices of linear indep modes, and the other modes just "not matter", ie not provide "new" information). And the proof plays just such game with $f(x)=u(\nu)$? $\endgroup$
    – user267839
    Jan 15, 2023 at 0:03
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    $\begingroup$ That is, when we want in the linked derivation to know the "total" energ $u(T)$, it suffice to work with weights of these "choosen" modes (aka Fourier basis), instead to use - what I called before the more "natural" way - all the modes as "input"? That's the point? $\endgroup$
    – user267839
    Jan 15, 2023 at 0:06
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    $\begingroup$ Alright, thank you, I think I understand it now. So the issue is that this (from naive point of view) a bit misleading formulation on page 5 that "The final result is that the number of modes of oscillation in the frequency interval [...]" really means usually "the linearly independent modes" in sense of Fourier basis, right? $\endgroup$
    – user267839
    Jan 15, 2023 at 0:31

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