On October 19,1900, Planck presented a function that described the expected spectral energy density for temperatures in the case of an ideal black-body.
We should describe this function verbally without recourse to Planck’s constant (it is concealed in the “arbitrary constants” alpha and beta), because – while the function worked as a phenomenological guide for understanding experimental results – no one knew what it meant or what the “arbitrary constants” implied about anything except “ideal resonators” and even in that case, things were not clear, as Kuhn says in Black-Body Theory and the Quantum Discontinuity, 1894-1912.
Anyway, the function is:
The density (rho) for each wave frequency for a particular temperature is
Alpha times (the wave frequency cubed) times ((one/e with the exponent (beta (the wave frequency)/the temperature) -1).
Which, even with alpha and beta expanded to have the dimensions and values of some constants, is not a particularly complicated function. What it generates is a curve that, for any particular temperature for an ideal radiator, gives the expected amount of energy packed into each wavelength of the electromagnetic spectrum, all the way from super-high-energy x-rays to extremely-low-energy microwaves. Curiously enough, this is an extremely vital way of summarizing what happens in the process of having a universe since it shows you what one form of more-or-less pure energy looks like and acts like. Now the “acts like” thing is tricky. The nature of this “action” certainly was not obvious to Planck, but we will look into how he got to his function and how it came to be understood.
First – the immediate stimuli for the stated function were two:
- Experimental measurements in a new range of wavelengths by Lummer and Pringsheim in September of that year had showed that all earlier attempts to arrive at a function for an ideal temperature-spectral energy distribution had failed, including Planck’s own, developed in March of 1900
- To clarify Planck’s earlier work in the area of such problems as why the classical equipartitioning of energy among modes in stable equilibrium seemed not to be working – in fact classical functions predicted a curve for spectral intensity that is generally referred to as the “ultraviolet catastrophe” – in Classical terms, beyond a certain temperature there can be absolutely huge amount of energy packed into the shorter wavelengths beyond the violet end of the visible spectrum– which is not observed experimentally. Apparently (says Wikipedia – which also notes that Paul Ehrenfest was the first to use the term “ultraviolet catastrophe”) that would mean all that energy would be radiated away very quickly and things would cool off very fast. Still, even without the classical functions, energy had to be distributed somehow into an equilibrium state while preserving the overall logic of thermodynamics.
Second – it took quite a while to work out the physical and theoretical implications of all the elements of Planck’s function and his new constant – so I will look into some of that process here.
So, I’ll note again that the function Planck gave on October 19, 1900 was a marvelous work of phenomenology – it accounted for all the known experimental results, but as a mathematical tool rather than as the application of a “theory.” Since, however, Planck was supposed to be doing fundamental work in the theory of thermodynamics, he worked extremely hard for the next two months to figure it all out and he sort of did and in December he set out to explain how the function related to thermodynamics and why it had to have a new constant ‘h’ to possess this fundamental relation.
As Kuhn describes this first explanation or “derivation” of the function, Planck said he had to use the relation epsilon = h times nu (and that should be little epsilon for a little tiny energy element, not big E for just plain Energy and h is the new constant and nu is the wave frequency) where h is 6.55 times 10 to the negative 27 erg-seconds to distribute the energy (well, the little ε-energies anyway) to the “resonators” (ie what would later be seen as “emitters” more-or-less). Notice that the constant, h ( with the relation ε= h times nu) , has all kinds of dimensionality built in: energy, time and in fact space as well since the frequency of an electromagnetic wave is in terms of the passage of the wave over some distance at the speed of light, so h relates all of those dimensions and sort of locates them all as intersecting in the imaginary space of one tiny leap. But, again, no one at the time had a clear idea of what that tiny leapiness at the heart of thermodynamics might really be or imply, even if it was very useful both phenomenologically in accounting for experimental results and theoretically in maintaining the logic of thermodynamics and relating it to electromagnetic properties.