According to the documents I have read, Plank made two changes to Rayleigh-Jeans approach in order to produce an equation that matched the black-body radiation, experimental curves:
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?

According to the documents I have read, Plank made two changes to Rayleigh-Jeans approach in order to produce an equation that matched the black-body radiation, experimental curves:
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?
There was already a catastrophic dependency in the sense that energy tended to infinity as frequencies got higher according to the classical calculations, but that was obviously not observed besides being absurd, so he came up with a relation between energy and frequency that approximated observations, he did it basically in a heuristic way, that is just trying different formulas to find the simplest that worked, even if he rejected for a long time the implications.

There was already a catastrophic dependency in the sense that energy tended to infinity as frequencies got higher according to the classical calculations, but that was obviously not observed besides being absurd, so he came up with a relation between energy and frequency that approximated observations, he did it basically in a heuristic way, that is just trying different formulas to find the simplest that worked, even if he rejected for a long time the implications.
Thanks. The interesting thing is that Wien used a similar approach and came very close. His formula fit experimental results at wavelengths shorter than 10E06 meters and deviated only slightly at longer wavelengths. Down to 10E2 meters, you need a log-intensity scale to see it.

I'm still not sure why Plank's approach was considered a quantum-breakthrough, unless it was the fact that "h" had to be a specific value (quanta) to fit the curve?

vanhees71
Thanks. The interesting thing is that Wien used a similar approach and came very close. His formula fit experimental results at wavelengths shorter than 10E06 meters and deviated only slightly at longer wavelengths. Down to 10E2 meters, you need a log-intensity scale to see it.
It was precisely Wien's other formula, the displacement law(that had already been empirically validated) what demanded the frecuency dependence of the energy discrete elements E(E=hv) in the final Planck's formula.
I'm still not sure why Plank's approach was considered a quantum-breakthrough, unless it was the fact that "h" had to be a specific value (quanta) to fit the curve?
It was mostly the implications of this for both radiation(Einstein photons) and matter(Bohr "jumps" and later Schrodinger and Heisenberg full QM).