Blackbody radiation - quantum to classical

[spaghetti3451]

I have a question regarding the parameters that reduces the Planck distribution to the Rayleigh-Jeans distribution.

According to the Planck distribution, the average energy in a unit volume in the $\nu$ frequency mode of a blackbody radiation field is $<U> = \frac{h\nu}{e \frac{h\nu}{KT} - 1}$. And , I see that in both the limits $\nu \rightarrow 0$ and $T \rightarrow \infty$, the expression reduces to the classical Rayleigh-Jeans form.

Are these two limits a part of the correspondence principle?

 

[Jano L.]

What do you mean by correspondence principle exactly?

Mathematically, the above formula approaches $kT$ in both limits.

In practice, we check the limit by keeping temperature close to common temperatures and look at low frequencies.

The limit $T\rightarrow\infty$ is hard to realize.

I think the comparison to $kT$ makes little sense as a check on the correctness of the Planck formula in this limit. Even in classical theory, the average energy should be lower than $kT$ for high temperatures.

This is because the assumptions of Rayleigh and Jeans are very implausible for high temperatures; radiation has to be enclosed in a box with perfectly reflecting walls, but this is very unlikely to be possible, as the known metals melt down for temperatures higher than few thousand K. The Rayleigh-Jeans derivation has restricted validity even from the viewpoint of classical theory, although this is often being forgotten today.