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In this site they arrive to a condition on the tuple of integers specifying a mode of the cavity. In the equation $$ n_x^2 + n_y^2 + n_z^2 = \frac{4L^2}{\lambda^2} $$ it is not clear what $\lambda$ is ? I know that the superpostion of plane waves (in the cavity) with wave vector $(\vec k_1, \vec k_2, \vec k_3)$ is not a plane wave with $\vec k = \vec k_1 + \vec k_2 + \vec k_3$ (if the wavevectors are not colinear) If it is a wavelength of some wave which one is it ?

In the next box they want to evaluate the number of modes (solutions I think) which can meet that condition. To this end they compute a volume in the $n$-space. Why not (a part of) the area of a sphere ? For example they count $(n_x,n_y,n_z) = \frac{4L^2}{\lambda^2}(\frac{1}{10},\frac{1}{10},\frac{1}{10})$ while it doesn't satisfy the condition above.

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  • $\begingroup$ Reverse the equation $\lambda=2L/\sqrt{ n_x^2 +n_y^2+n_z^2}$, this means that only waves that satisfy that conditions are allowed in the cavity. Wavelength is quantized. Does this answer your question? $\endgroup$
    – Mauricio
    Mar 23, 2022 at 13:09
  • $\begingroup$ Also you should stick to a single question. $\endgroup$
    – Mauricio
    Mar 23, 2022 at 13:09
  • $\begingroup$ Could you give an example of such a wave that vanishes at the boundaries ? can it be plane wave ? if not how do you define its wavelength ? Otherwise you don't answer my question. And the second question is not a very separate question $\endgroup$
    – Physor
    Mar 23, 2022 at 13:11
  • $\begingroup$ It is a stationary wave. Something proportional to sines and cosine, like $\sin(n_x \pi x/ L)\sin(n_y \pi y/ L)\sin(n_z \pi z/ L)$ (depends on the origin of the coordinates system). $\endgroup$
    – Mauricio
    Mar 23, 2022 at 13:19

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  1. What is $\lambda$?

$\lambda$ is the wavelength of the standing wave. You can look at the equation $n_x^2+n_y^2+n_z^2=\frac{4L^2}{\lambda^2}$ as a condition that tells you what values of $\lambda$ are allowed to exist in your cavity. For each triple of integers $(n_x, n_y, n_z)$, you calculate $\lambda$, and that wavelength will fit in the box. Of course, there are degeneracies; some triples will lead to the same value of $\lambda$, such as $(1, 0, 0)$ and $(0, 1, 0)$. We can say that the wave corresponding to this value of $\lambda$ is an allowed mode in the cavity.

  1. Why look at the volume?

First, the idea of looking at volumes and areas in $n$-space instead of counting discrete points is only valid in the large $L$ limit, when you can think of the allowed values of $\lambda$ as forming a continuum. (Your head may be spinning that we started off by pointing out that only discrete values of $\lambda$ are allowed, and then immediately make an approximation that lets $\lambda$ be continuous. Just stick with it, it will pay off.)

Second, the language on the website you linked to is not very precise. If you want to satisfy the equation $n_x^2+n_y^2+n_z^2=4 L^2 / \lambda^2$, then in $n$-space you are looking for a spherical surface with radius $2 L / \lambda$.

The calculation that is being done (even though this is not explained well) is to say that there is a cap on the energy that a mode can have. Saying there is a maximum energy is equivalent to saying that there is a minimum wavelength (because $E=h f = h c / \lambda$), which means in $n$-space there is a maximum radius. The volume of a sphere with that radius counts the number of modes that do not exceed this maximum energy. Actually you have to multiply the volume of the sphere by $\frac{1}{8}$ to account for the condition that $n_x, n_y, n_z$ are all non-negative, and by $2$ to account for the fact that a photon has two polarization states.

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  • $\begingroup$ Do you know some good source on this ? in Wiki it s**ks more, there is no derivation for Rayleigh–Jeans law. Also if the wave $\sin(n_x \pi x/ L)\sin(n_y \pi y/ L)\sin(n_z \pi z/ L)$ is such a wave that is supposed to have that wavelength $\lambda$, how do get $\lambda$ from that wave if you encounter it e.g. on the street. Since this wave doesn't seem to meen a plane wave so it is not easy to get it wavelength $\endgroup$
    – Physor
    Mar 23, 2022 at 13:30
  • $\begingroup$ It's been a long time since I've studied this, although from what I remember I did not think any of the textbooks I read did a good job of explaining what's going on. I've found Richard Fitzpatrick's notes are usually good: farside.ph.utexas.edu/teaching/sm1/Thermalhtml/node103.html, as are David Tong's (see Chapter 3): damtp.cam.ac.uk/user/tong/statphys/statphys.pdf $\endgroup$
    – Andrew
    Mar 23, 2022 at 13:33
  • $\begingroup$ @Physor Why do you say the wave is not a plane wave? I think of it as ${\rm Im}\ e^{i \pi / L (n_x x + n_y y + n_z z)}$, so it effectively is a plane wave in the direction of $n_x \hat{e}_x + n_y \hat{e}_y + n_y \hat{e}_z$, where $\hat{e}_i$ are unit vectors in the $i$ direction. $\endgroup$
    – Andrew
    Mar 23, 2022 at 13:36
  • $\begingroup$ Yep! it seems like that if one considers it in that way (I did that too) but does it vanish on the boundary ? that was my problem with that. Did I miss something ? $\endgroup$
    – Physor
    Mar 23, 2022 at 13:43
  • $\begingroup$ @Physor That's just a detail. If you look at the dark and light spots in the heat map of $\sin(k_x x) \sin (k_y y)$, you can see they form the pattern of a traveling wave moving in the direction of $\vec{k}=k_x \hat{e}_x + k_y \hat{e}_y$ (I'm just talking about 2d because it's easier to plot, the same thing is true in 3d). Click here for an example $\endgroup$
    – Andrew
    Mar 23, 2022 at 14:12

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