Understanding the Ultraviolet Catastrophe: The Role of Planck's Quanta

[gsingh2011]

I'm just wondering what the specific problem and solution was for the ultraviolet catastrophe. I understand that it was because scientists were calculating that in an oven, the radiation was showing infinite energy. Is this due to their being an infinite number of waves or what? And whatever the reason was for getting this calculation, why did they assume that.

Then when Planck came along and suggested the idea of a quanta, how did this fix the problem?

 

[turbo]

You can google "ultraviolet catastrophe". It's an old problem and there is a lot written about it.

 

[Born2bwire]

From a recent post of mine:

At the time, the limiting cases of the power spectrum were known via both experiment and theory. The Rayleigh-Jeans fit lower frequency radiance while the Wien fit the higher frequency (though Wien seemed to derive his equations more from empirical fitting than strong theoretical footing). The Rayleigh-Jeans distribution can be found using classical electrodynamics and classical statistical mechanics. However, as mathman stated, you end up with the ultraviolet catastrophe where the energy density suffers from an ultraviolet divergence. It was theorized at the time, by Rayleigh and others, that the fault laid in the classical equipartition theory.

Planck looked at the Wien and Rayleigh-Jeans results and proposed an interpolation between the two results. This was the same as the resulting Planck distribution. It then took him several weeks to find a physical and theoretical reasoning behind this and this was done by throwing out the classical equipartition theory and devising a new one that required the energy to be quantized. Actually, quantization of energy was done by Boltzmann as a tool for derivations but with Boltzmann the quantization did not affect the final results. However, removing the quantization in Planck's derivation simply results in the Rayleigh-Jeans distribution again. Thus, Planck's use of quantization was essential. In addition to the quantization, Planck used a different method for counting the elements which is consistent with what is now called Bose-Einstein statistics (as opposed to the Maxwell-Boltzmann statistics that gave rise to the Rayleigh-Jeans).

So basically Planck found a way to fit an equation that matched the Wien and Rayleigh-Jeans distributions and was able to a posteriori derive this distribution by using a new equipartition theorem. This matched the suspicions of other physicists at the time that the classical statistical equipartition theory may be the problem.

Milonni has a few sections in his Quantum Vacuum book that discusses this in detail.

Basically, when classical statistics and thermodynamics were applied to the problem of the black body radiator then the energy density diverged as the frequency went up. The Wien distribution worked for higher frequencies but it wasn't really developed on theory but rather empirical evidence. Planck was able to match up both the limiting cases of the Rayleigh-Jeans and Wien distributions from a theoretical standpoint and this required him to make changes to statistical physics at the time and to assume quantized energy.

 

[gsingh2011]

From a recent post of mine:



Basically, when classical statistics and thermodynamics were applied to the problem of the black body radiator then the energy density diverged as the frequency went up. The Wien distribution worked for higher frequencies but it wasn't really developed on theory but rather empirical evidence. Planck was able to match up both the limiting cases of the Rayleigh-Jeans and Wien distributions from a theoretical standpoint and this required him to make changes to statistical physics at the time and to assume quantized energy.

OK, so pretty much Planck's equation only worked when energy was quantized. And the original problem was that scientists believed that the higher the frequency went the more energy it had (according to their equations), which wasn't the case. So did there equations also say that black body radiation could produce ever increasing frequencies? Is that why it resulted in infinite energy?

 

[Born2bwire]

OK, so pretty much Planck's equation only worked when energy was quantized. And the original problem was that scientists believed that the higher the frequency went the more energy it had (according to their equations), which wasn't the case. So did there equations also say that black body radiation could produce ever increasing frequencies? Is that why it resulted in infinite energy?

Well, even Planck's distribution predicts emission at all frequencies so that isn't the problem. But in the Planck and Wien distribution, the amount of energy for the limit of infinite frequency drops down fast enough so that the energy spectrum for these high frequencies is negligible. In the Wien limit, the energy density is
$$\rho(\nu) = C\nu^3e^{-D\nu/T}$$
but the Rayleigh-Jeans distribution is
$$\rho(\nu) = A\nu^2 T$$
The total energy density is found by integrating the above,
$$u = \int_0^\infty \rho(\nu)d\nu$$
We can easily see that the Rayleigh-Jeans blows up as the frequency increases. The Wien does not blow up because it is exponentially convergent.

But basically what they found was that by using the classical theory of they day, they could come up with a distribution that matched experiments for the low frequency spectrum. As experiments improved and they were able to measure increasingly higher frequencies, they found that the higher frequencies diverged from predicted behavior. They then found ways to fit functions for these higher frequencies but these distributions failed at mid and low frequencies. Essentially, they could predict the low and high spectrums but they did not have a way to predict the entire spectrum or the region that transitions between the two extremes. Planck came up with a solution and to do so he had to make changes to the statistical theory of the day. But this was not too unexpected as it seems that some physicists suspected that problems with the theory were to blame.