I Relationship between Planck Distribution and Quantization of Energy

Summary
I got stuck with the correlation between Planck Distribution and Quantization of Energy.
I came to understand that Planck Distribution is necessary to explain UV catastrophe. With that necesity in the background, the distribution equation eventually suggests that the energy emitted by black body has discret values. But I wonder how that's related to E=nhv. I understand that "n" also implies the discrete energy values but how does "n" ultimately contribute to deriving distribution equation? I might need a mathematical explanation.
 
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As indicated, to me, it sounds like this experiment shows that energy is quantized or has discrete amounts when it's tranferred and Planck Distribution is an explanation for that.
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vanhees71

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The crucial point is that radiation in a cavity comes to thermal equilibrium through interaction with the walls of the cavity. Kept at constant temperature, the radiation must come to thermal equilibrium at this temperature, which means that the rate of absorption and emission of radiation energy are the same. In the classical picture each mode of the radiation field (i.e., the solution of Maxwell's equation harmonic in time with (angular) frequency ##\omega##) can exchange any quanitity of energy, but this leads to the validity of the equipartition theorem, i.e., each mode would contain on average ## k T## of energy, which clearly leads to the UV catastrophe since you have infinitely many modes.

According to QT (i.e., in this case quantum electrodynamics) each mode can only exchange an integer number of energy portions of the size ##\hbar \omega##, i.e., you can excite only an integer number of "photons" of each mode. Together with the Bose nature of photons, which is due to the integer spin of 1 of the em. field, you get Planck's radiation law, which leads to a finite total radiation energy ##\propto T^4##, i.e., it solves the problem with the UV catastrophe.
 
i.e., each mode would contain on average ## k T## of energy, which clearly leads to the UV catastrophe since you have infinitely many modes.

(i.e., in this case quantum electrodynamics) each mode can only exchange an integer number of energy portions of the size ##\hbar \omega##, i.e., you can excite only an integer number of "photons" of each mode.
So is it safe to say that according to QT, it limits the number of mode that can exchange energy rather than allowing infinite number of mode of energy exchange so that the total radiation energy becomes finite rather than infinite?
 

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