3.1 EARLY ATTEMPTS TO DEAL WITH THE FAILURES OF THE CORPUSCULAR THEORY

3.1.1 An Act Of Desperation: The Quantum Object Announces Itself

Max Planck constructed an electromagnetic theory of radiant heat which had led him to a derivation of Wien’s law for the spectral distribution of the energy emitted by a black body. Once the experimental results obtained by Rubens and Kurlbaum indicated that Wien’s law was a poor representation of how the spectral density depended on temperature for fixed values of the wavelength, Planck was confronted with the task of understanding where his theory went wrong and of trying to modify it accordingly. Over an extremely short period of time he did three things. First, he proposed a simple formula for the functional dependence of the spectral density of a black body on the product $λ T$ which would fit the Rubens and Kurlbaum data. Second, he modified his own theory to produce that functional dependence rather than Wien’s law. And third, since the second step seemed to him to be designed to obtain, rather than to derive, his fit to the data, he found a way of deriving it.

A few days before Rubens and Kurlbaum published their results, Planck pointed out that their data is much better fitted by the following:

J (λ T) = \frac{C λ^{- 5}}{e^{\frac{c}{λ T}} - 1}

rather than by the Wien formula into which it transforms when $λ T$ is very small. In fact, as Rubens and Kurlbaum showed, the Wien formula for the spectral density displayed a marked concave curvature when plotted against the temperature at a fixed wavelength, while the Planck formula is a straight line following the data almost exactly. For instance, for a wavelength of 24 microns, Wien’s formula provided a value for the spectral density at 500 degrees Celsius which was 50% higher than the data and a value which was 25% lower at 1,500 degrees Celsius; Planck’s formula, by contrast, provides values which differ from the data by less than three percent at both temperatures.

Planck realized of course that simply fitting the experimental data, while informative, was not an answer. He therefore returned to his electromagnetic theory in which he derived a relationship between the entropy and the energy density of black body radiation and showed that a slight generalization of that relationship would produce his, rather than Wien’s, formula. He published that result in October of 1900. But this manner of obtaining an explanation of the Rubens and Kurlbaum data appeared to him too much as a back-fit of his electromagnetic theory of radiant heat to these new data. He continued therefore to seek a modification of the theory which would not be subject to that criticism. By December he found it. We cannot here describe Planck’s procedure because it is technically well beyond the scope of our essay; nevertheless, we shall endeavor to illuminate the main assumption on which it rests. Planck considers, as always, a medium enclosed within mirrored walls and in which he places a large number of linear, monochromatic, oscillating resonators at suitable distance from each other. He then makes the totally unexpected assumption that the energy of a monochromatic resonator can only take on values which are multiples of a specified energy $ε$ . Consequently, the total energy invested in the resonators under consideration must itself be a finite multiple of $ε$ , say $P$ . Those $P$ parts can be distributed over the number $N$ of resonators of the specified wavelength in a number of different ways which he calls "complexions" and which he calculates to be given by:

n = \frac{{(N + P)}^{N + P}}{N^{N} P^{P}}

Then, he observes that, just as in the statistical physics of Boltzmann, the entropy of a system of resonators with a given energy should be proportional to the natural logarithm of the total number of complexion, which amounts to assuming that all complexions are equally likely to occur. From these two assumptions—that resonator energy is discrete and that complexions are equally probable—he derives the spectral distribution of the black body radiation with which the resonators are in equilibrium at temperature $T$ :

J (λ T) = \frac{8 π ε λ^{- 4}}{e^{\frac{ε}{k T}} - 1}

But, according to Wien, that spectral distribution must quite generally be a function of $λ T$ only. Therefore $ε$ must be inverse proportional to the resonator wavelength:

ε = \frac{h c}{λ}

where $c$ is the speed of light in the medium enclosed by the walls and $h$ is a universal constant thereafter referred to as the Planck constant. Eventually, he obtains for the spectral density of a black body:

J (λ T) = \frac{8 π h c}{λ^{5}} \frac{1}{e^{\frac{h c}{k λ T}} - 1}

where $k$ is the Boltzmann constant. This result manifestly reproduces Rubens and Kurlbaum.

What Planck did was nothing short of revolutionary: it did not follow from anything that preceded it. In fact, he himself described this step as an act of desperation in a 1931 letter to R.W. Wood. How he came to perform such an act of desperation is neither totally clear nor is it important. What is relevant for us here is that by doing so he has changed the ontology of the resonator. The electron which oscillates in it is not some small particle driven to move back and forth by an elastic force according to Newtonian mechanics but an entirely new entity. By changing the rules which applied to it, he actually changed the object. And thus the quantum object announced itself.