Planck's law and ultraviolet catastrophe

[kwuk]

Hi. I know this is a pretty basic principle, however I'm fairly new to the subject and was wondering if anyone is able to give a brief 'layman' explanation of why, as Planck's law states, at lower wavelengths the blackbody radiation falls to zero rather than continuing to climb as stated in the Rayleigh-Jeans law.

I have read a number of articles but none yet seem to have a basic enough explanation to allow me to 'picture' the principles involved.

Anyone that can help me with this would have my eternal gratitude!

Thanks.

 

[shouvikdatta8]

Latex Code I(\lambda,T) =\frac{2 hc^2}{\lambda^5}\frac{1}{ e^{\frac{hc}{\lambda kT}}-1}.

The above mathematical statement of Planck's law will clear the dirt. In the Rayleigh-Jeans law the function diverged for low \lambda (or in the limit \lambda -> 0), however this does happen over here. Planck assumed the quantization of energy the oscillators - they can only take up energies in bundles (or multiples of h\nu). This made all the difference for the mean energy of the dipole oscillators in his model. The Rayleigh-Jeans' Law can be derived from the Planck's law for high wavelengths.

 

[DeShark]

$$I(\lambda,T) =\frac{2 hc^2}{\lambda^5}\frac{1}{ e^{\frac{hc}{\lambda kT}}-1}$$

Sorry, I couldn't read it, so I put in the tex brackets and it all became clear. I'm quoting the tex, cause if it was useful to me, it might help someone else too..

 

[Ken G]

I think the "layman" explanation kwuk is looking for might go something like this. If we don't force the energy that goes into light to be in little h*nu bundles (quantized photons), then we expect there to be an amount of energy proportional to T to go into every "mode" of the system. That's pretty much the classical meaning of T-- it tells you the energy in each mode, sort of like when you pour water into an ice cube tray, each compartment fills to the same level and we can say that level is proportional to T. So then you get the Rayleigh-Jeans law just by asking how many "compartments" there are for each frequency, and filling them in proportion to T. However, there are a lot more such compartments at high frequency, all getting energy proportional to T, and that's the "ultraviolet catastrophe"-- there's no limit to this.

What saves you is that the water going into the compartments is not continuous, it is quantized, and even more importantly, it is quantized in a way that is proportional to the frequency corresponding to that compartment. Thus you reach a point where the energy that corresponds to T does not fulfill the requirements of even a single energy quantum, and that "cuts off" the distribution. There is still a probability that that compartment will get a "quantum of water" in it, despite the relatively low T, but the probability gets small as the frequency rises, and that is what rescues you and gives you a finite amount of water in the ice cube tray even when it has an infinite number of compartments in all.

Physically, what is happening here is that if the universe can just dump "T" worth of energy in every ice-cube compartment, it is happy (entropically speaking) to do so, and that leads to the catastrophe. But if you force it to put way more than "T" worth of energy to satisfy a single "quantum" in the higher-frequency bins, then it doesn't like that at all, and is so loathe to do so that the water in the tray becomes finite and the problem goes away.