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When I was studying thermodynamics, I came across Wien's law and Stefan Boltzmann law. When I checked for a derivation, I found that these could be proved by using Planck's law. But I am unable to understand it's derivation. I am not clear with the part which derives the number of states per unit volume. The derivation asserts that the minimal cell volume is $h^3$ and then the number of states are found by dividing the volume element by $h^3$.

  1. My question is how minimum cell volume is $h^3$? If we use "minimum" cell volume then doesn't it give us the maximum number of states and not the exact number of states? If that's the case, how can we find the exact density of states?

  2. The derivation seems to use the term states for photon. I am not comfortable with such ideas.

I am not familiar with such ideas (like density of states). So, could someone help?

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There is a minimum cell volume in wave-vector space, this is due to quantization of the eigen modes of the EM field inside a cavity with given boundary conditions. For example, for boundary condition of vanishing wave on the boundary of a 3D cube with length L, the eigen modes will be of the form: $$\vec{E} = A \sin (\frac{\pi}{L}n_x x) \sin (\frac{\pi}{L}n_y y) \sin (\frac{\pi}{L}n_z z)$$ Where $n_x,n_y,n_z$ are three natural numbers which determine the wavevector (and frequency) of the EM wave. As you can see, in $k$ each mode can be separated from other modes with a "cell" of volume $(\frac{\pi}{L})^3$. So for these boundary conditions it is easy to find this $h^3$, but the specific boundary conditions don't really change the fact that there will be some kind of quantization in the modes which will let you divide $\vec{k}$ space into cells.

About the photon part of your question, I think you're confusing (photons are a description of energy states in the EM field, but most of the time using states in this context one usually refers to the $\vec{k}$ modes - like you said - "density of states"). Can you please add a quotation?

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  • $\begingroup$ Thanks for your answer . Could you elaborate more on the fact that the specific boundary conditions don't really change the fact that there will be some kind of quantization in the modes which will let you divide space into cells? $\endgroup$
    – user262378
    May 5, 2020 at 12:36
  • $\begingroup$ Also how can I illustrate the same for an arbitrarily shaped cavity? $\endgroup$
    – user262378
    May 5, 2020 at 12:39
  • $\begingroup$ You have great questions, and I don't remember the answer :/ . I think they are worth a new question on Stack Exchange. $\endgroup$ May 6, 2020 at 6:38

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