Archimedes in the bathtub

With this new condition, Planck had imposed the quantization of the energy of the oscillators, "a purely formal assumption … actually I did not think much about it…" in his [Planck's] own words

The vacuum is finite precision.   So they knew what the energy levels were and worked backward to solve for the probability distribution in Schrodinger.  In the ball in the box, the wave mode determined the spectral lines, a finite set.  The quantization is not explicit in Schrodinger equation, it is in the operator modifying the distribution.  I think.

We have data scientists who solve Schrodinger using neural nets, and maximum entropy encoding is neural net done in batch mode.  Someone modeled a network of nodes passing wave packets using a variation of Schrodinger.

How did Planck solve the black body problem? Quantized and then used combinatorics, statistical mechanics.  All of it, a queueing problem I often think.