There are several explanations for how Planck used quantization to explain blackbody radiation correctly without the ultraviolet catastrophe. I will follow this explanation.
For a cavity, the mode density per volume is 8πν2/c3. Classically, the probability of occupation is equal for all modes and each mode has an energy kT. Thus, one obtains the Rayleigh-Jeans formula for blackbody radiation
I=8πν2c3kT.
This blows up for large ν. Planck's law instead says that the probability of occupation of a mode is 1ehν/kT−1
For instance, I understand that the Boltzmann distribution (which works for continuous distributions) has probability of occupation that goes as p(E)=e−E/kT. Let energy be proportional to frequency (but not quantized) and this also averts the ultraviolet catastrophe and qualitatively produces the same shape as Planck's law.
EDIT
Nope, the above statement is wrong. I realized that if you actually do the integration correctly, you get the law below which still blows up for large μ I=8πν2c3kT.
So where does the requirement of quantization exactly come in?
This question is closely related to this one and this one but is not a duplicate since those questions were asked at a more basic level.
Answer
From the classical Boltzmann theory you have the probability that the mode has energy E p(E)=Ae−E/kT
From this distribution you get the average energy ˉE=∫∞0p(E)EdE=kT
Planck found that instead of equation (2) the average energy of a mode needs to be ˉE=hνehν/kT−1
There was no physical explanation for equation (3) available until this time. But Planck saw (you may call it mathematical intuition) that pn=Ae−nhν/kT,with n=0,1,2,…∞
Now, in equations (4) and (5) n and En=nhν obviously have discrete values, in contrast to E in equations (1) and (2) having continuous values. The physical interpretation of this is that the mode doesn't have arbitrary energies E, but only discrete energies En=nhν. Or saying it more vividly: Light of frequency ν consists of particles, each having energy hν.
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