UV catastrophe - why a problem? Ultra violet catastrophe happens when we integrate the Rayleigh Jeans spectral density over all frequencies to calculate the total energy, which yields:
$$ \int_0^{+\infty} \frac{8\pi \nu^2}{c^3} k_B T \,\text{d}\nu \rightarrow +\infty.$$
This divergence caused problems as it was not observed in experiments, and was avoided when Bose-Einstein statistics for photon modes was used, so as to yield a finite result.
Apologies if this is a very dumb question, but would it not make sense that, for infinitely many frequencies, the energy tends to infinity? So it just the case that they did not observe this experimentally, or was there a deeper theoretical motivation as to why the UV catastrophe caused problems?
 A: My understanding is that the conventional explanation is somewhat oversimplified. If there were some hypothetical system that obeyed the Rayleigh-Jeans formula, then it would contain infinite energy if it were in thermal equilibrium (each mode would have an energy on the order of $k T$, and this would sum up to infinity). That doesn't mean that such a system would ever actually have infinite energy, because it would need to get that energy from somewhere in the first place. The ultraviolet catastrophe instead implies that such systems would never reach thermal equilibrium. They might keep absorbing energy endlessly, for example, which isn't observed experimentally.
A: How could any experiment even logically measure infinite energy, in agreement with the classical theory?  If a blackbody at arbitrary temperature radiated infinite energy, it would incinerate everything.  It would be like an atomic bomb, but ... well ... infinitely more powerful.  And yet there are (very good) blackbodies in the real world, and we're still here.
See also my question Ultraviolet catastrophe in a classical world.
