Odalisque Posted:

This second step was only possible due to a certain amount of luck (or skill, even though Planck himself called it “a fortuitous guess at an interpolation formula”Log in to see images!. It was during the course of polishing the mathematics of his formula that Planck stumbled upon the beginnings of Quantum Theory. Briefly stated, he had two mathematical expressions:

* (i) from the previous work on the red parts of the spectrum, he had x;

* (ii) now, from the new infrared data, he got x².

Combining these as x(a+x), he still has x, approximately, when x is much smaller than a (the red end of the spectrum); but now also x² (again approximately) when x is much larger than a (in the infrared). The formula for the energy E, in a single mode of radiation at frequency λ, and temperature T, can be written

E = \frac{h \lambda}{e^{\frac{h \lambda}{k T}} – 1}

This is (essentially) what is being compared with the experimental measurements. There are two parameters to determine from the data, written in the present form by the symbols used today: h is the new Planck’s constant, and k is Boltzmann’s constant. Both have now become fundamental in physics, but that was by no means the case at the time. The “elementary quantum of energy” is hλ. But such a unit does not normally exist, and is not required for quantization.