The Advent Calendar of Physics: Science Works

Moving along in our countdown to Newton’s birthday, we come to 1900, and one of the most revolutionary moment in the history of physics, represented in today’s equation:

This is Max Planck’s formula for the spectrum of the “black-body” radiation emitted by a hot object at temperature T. It’s also the equation highlighted on what might be the most famous xkcd cartoon (albeit in different notation).

This is a fitting next step in the countdown not only for reasons of chronology, but also because it’s a nice bridge from thermodynamics and statistical mechanics. After all, the red glow of a hot object is a very universal phenomenon, and seems like it ought to be straightforward to explain in some way using what we know about thermal physics. But it’s not.

The importance of this equation is not so much the specific physics it describes, but the steps that had to be taken to get it.

Planck worked out the general form of this equation by looking at the shape of the spectra recorded by experimental colleagues, but struggled to come up with a derivation of it from first principles. Eventually, he hit on a mathematical trick, which involved saying that the energy contained in light waves was not a continuous function of their amplitude (as you would expect for classical waves), but depended on the frequency (the Greek letter nu (ν) in the above equation). He set up his equations using a constant (now given the symbol h) multiplying the frequency to determine the energy of a particular frequency of light, intending to take a limit as h went to zero (a type of mathematical trick that gets used elsewhere, so it’s not a completely ad hoc approach). He found, though, that the equations only worked out correctly if he kept the h around as a very small but definitely non-zero constant.

There’s absolutely no reason to expect this relationship to hold, and Planck always hoped that there would prove to be some more elegant alternative to this ugly approach. A hundred and ten years of subsequent experiments have confirmed, though, that this is absolutely the correct approach to understanding the black-body radiation, and the deep structure of the universe.

The graph accompanying that xkcd cartoon represents data from the WMAP satellite, which measured the spectrum of the cosmic microwave background radiation, which is a relic of the Big Bang. The spectra they measured fit the equation above perfectly, and the tiny variation in the temperature associated with the spectrum of microwaves coming from different points in the sky is the best source of information we have about the very early universe.

So, take a moment today to appreciate Max Planck’s big leap, which was the opening shot of the quantum mechanical revolution that completely changed physics. And come back tomorrow to see the next step in the process.