Planck's assumptions for treatment of blackbody radiation

Question

Planck, in his treatment to solve blackbody radiation, considered that the electromagnetic waves inside the cavity of the blackbody are standing waves due to oscillating charges on the wall of the cavity. At equilibrium, the frequency of oscillation of the oscillating charge is equal to the frequency of the electromagnetic wave produced by it i.e. at equilibrium, energy of the oscillating charge is equal to the energy of the produced electromagnetic wave.

He proposed that, the energy of the oscillating charge is integral multiple of $h\nu$ i.e. $\epsilon_n=nh\nu$ where $n=0,1,2,3,4,...$

But, we know that quantum harmonic oscillators have energy $\epsilon_n=(n+\frac{1}{2})h\nu$ where $n=0,1,2,3,4,...$

Then, are not the oscillating charges on the wall of the Cavity supposed to be quantum harmonic oscillators?

Answer 1

Planck did not go as far. He assumed a classical view of the EM field, and thought the oscillators have some special quantized way of emitting light. Quantum theory with its harmonic oscillator model was not developed yet.

Quantum harmonic oscillator is a general model. It does not necessarily have electric charges in it.

The moment one adds electric charges to harmonic oscillator, complications arise, because the two or more charges will interact and this interaction complicates the model. But in principle if the interaction between the charges is low enough, one could have charged oscillator model that would be very close in its behaviour to a standard quantum harmonic oscillator model.

Whether energy of the wall oscillator is $n h\nu$ or $(n+\frac{1}{2})h\nu$ does not matter much. For any single oscillator, this just changes its lowest energy from $0$ to $\frac{1}{2}h\nu$. Also total energy and energy per frequency interval changes accordingly, but this is not something that would have obvious measurable effect. It's just shifting the numbers. Changes of oscillator energy when changing states are the same in both variants.

Answer 2

About the options available to Planck:

At the time (around 1900) the (natural) assumption was that the energy of electromagnetic waves can continuously accumulate.

Planck's supposition that the oscillation modes for oscillating charges in the walls of the cavity must be multiples of some quantity: it seems to me Planck came up with that because to him that was the only supposition available.

It was only later that it became clear that it is not properties of wall material, but properties of light itself that act to produce the distribution of the black body spectrum.

We have that for every frequency of the spectrum the luminosity is a multiple of some quantity that is proportional to the frequency of the light.

In order for light of a particular frequency to be emitted, for example an ultraviolet frequency, the emission event must be a single event. The frequency of the light cannot accumulate; only the luminosity accumulates.

The spectrum of these emission events is a continuous spectrum, because this type of emission is a highly randomized process.

The reason that in the spectrum the luminosity tapers off towards the ultraviolet is statistical: the large energy jump of generation of ultraviolet light has a lower probability of occurring.

So: while Planck was under the impression he was working with a supposition about properties of oscillators in the walls, the reason that Planck arrived a good fit with the experimental data goes back to properties of light.