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1) As a mathematical convenience he assumed that the oscillators in the walls of black-body cavity could only have energies that were multiples of a minimum energy ∆E. This was ultimately described as the break-through that moved his analysis from the classical to the quantum realm.

2) He then used a statistical analysis to assign the correct number of oscillators with energy ∆E to a given frequency.

I have been through the mathematics of Planks formula, versus Rayleigh-Jeans. Plank used exactly the same relationship to determine the number of standing wave modes at a given frequency. This varied as the frequency-squared and resulted in the "ultra-violet catastrophe".

It was not the assignment of minimum energy increments that saved Plank from the same result. It was the fact that his energy increments had the relationship ∆E = hf/KT, versus Rayleigh-Jeans use of ∆E = KT.

The question is, what motivated Plank to come up with this frequency-dependent relationship for mode-energy?