I know the Rayleigh-Jeans law and how the formula predicts UV catastrophe. Without getting into the exact derivation, I am trying to get some intuitive understanding of it by using some of the broad arguments that they used, based on whatever I found on the net and various books looks like they applied statistical physics method to waves.

They assumed that total energy of radiation is equally distributed among all possible frequencies. I get that. I guess I am struggling with the 2nd argument which was there is no limit on the number of modes of vibrations that can excited? Why would there be no limit? Can someone please explain?

  • $\begingroup$ Why would there be a limit? $\endgroup$ Dec 15 '18 at 19:54

They assumed the hypotheses of kinetic theory and considered waves of light inside a metallic cage. These waves are standing waves and so the frequencies that the light takes are quantized and so they have an integer number $ n $ associated. If you consider $ n_x $, $ n_y $ and $ n_z $ as the integer numbers for every coordinate then they found that the frequency was given by the sum of the squares of these numbers $ n_x^2 + n_y^2 + n_z^2 $ which forms a sphere. The distribution of frequencies then is given by the surface of the sphere of a given radius $ f $ and so the distribution is given by the square of the frequency $ f^2 $. This is why the Rayleigh-Jeans distribution is a parabola.

It's not very intuitive why this is the case but their derivation goes along those lines. The very important fact is that they found that the distribution of frequencies is not homogeneous and depends on the square of the frequencies, so higher frequencies are likelier than lower frequencies. That's how I picture it at least. If higher frequencies contribute more to the distribution then it is easy to see how at higher frequencies the energy that the blackbody realeases goes to infinity.

  • $\begingroup$ that is where I am strugging. How did they conclude distribution of frequencies is not homogeneous.? you right once that is cae it can be easily seen how it would lead of UV catastrophe. $\endgroup$
    – user31058
    Dec 15 '18 at 21:42
  • $\begingroup$ Well because the distribution of frequencies lie in the surface of a sphere of radius f. The surface of the sphere is $ 4\pi f^2 $ so you see how it depends on $ f^2 $. You may ask why is it a sphere? Well from the fact that the frequencies are quantized (which is a common consequence of waves) they found the relation $ f=\frac{c}{2L}\sqrt{n_x^2+n_y^2+n_z^2} $ which describes this sphere. I recommend you read Eisberg's "Fundamentals of Modern Physics" or "Quantum Physics of Atoms, Molecules, Solids, Nuclei & Particles" in where he talks about this. $\endgroup$
    – Kirtpole
    Dec 15 '18 at 22:02

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