Max Planck and the birth of quantum theory, part 2

While a student at Berlin from 1877 to 1878, Max Planck had been taught by Gustav Kirchhoff, who among other things laid down some rules about how electrical circuits work, and studied the spectra of light given off by hot substances. In 1859 Kirchhoff proved an important theorem about ideal objects that he called black bodies. A black body is something that soaks up every scrap of energy that falls upon it and reflects nothing, hence its name. It's a slightly confusing name, however, because a black body isn't just a perfect absorber, it's a perfect emitter as well. In one form or another, a black body gives back out every bit of energy that it takes in. If it's hot enough to give off visible light, then it won't be black at all. It might glow red, orange or even white. Stars, for example, despite the obvious fact that they're not black, act very nearly as black bodies, so too do furnaces and kilns because of their small openings that allow radiation to escape only after it's been absorbed and re-emitted countless times by the interior walls. Kirchhoff proved that the amount of energy a black body radiates from each square centimeter of its surface hinges on just two factors, the frequency of the radiation and the temperature of the black body. He challenged other physicists to figure out the exact nature of this dependency, what formula accurately tells how much energy a black body emits at a given temperature and frequency. Experiments were carried out using apparatus that behaved almost like a black body, a hot hollow cavity with a small opening, and equations were devised to try to match theory to observation. On the experimental side, the results showed that if you plotted the amount of radiation given off by a black body with frequency, it rose gently at low frequencies or long wavelengths, then climbed steeply to a peak before falling away less precipitantly on the high frequency or short wavelength side. The peak drifted steadily to higher frequencies as the temperature of a black body rose. For example, a warm black body might glow brightest in the infrared and be almost completely dark in the visible part of the spectrum, whereas a black body at several thousand degrees radiates the bulk of its energy at Scientists knew this was how perfect black bodies behaved because their lab data based on apparatus that closely approximated perfect black bodies told them so. The sticking point was to find a formula rooted in known physics which matched these experimental curves across the whole frequency range. Planck believed that such a formula might provide the link between irreversibility and the absolute nature of entropy, his scientific holy grail. Matter seemed to be moving in a promising direction when in 1896 Wilhelm Wien of the Physical Technical Institute or PTR in Berlin gave one of the strongest replies to the Kirchhoff challenge. Wien's law agreed well with the experimental data that had been gathered up to that point and it drew the attention of Planck, who time and again tried to reach Wien's formula using the second law of thermodynamics as a springboard. It wasn't that Planck didn't have faith in the formula that Wien had found, he did, but he wasn't interested in a law that was merely empirically correct or an equation that had been tailored to fit experimental results. He wanted to build Wien's law up from pure theory and thereby hopefully justify the entropy law. In 1899 Planck thought he'd succeeded by assuming that blackbody radiation is produced by lots of little oscillators like miniature antennas on the surface of the blackbody. He found a mathematical expression for the entropy of these oscillators from which Wien's law followed. Then came a hammer blow. Several of Wien's colleagues at the PTR Otto Lohmer, Ernest Pringsheim, Ferdinand Kirchbaum and Heinrich Rubens did a series of tests that undermined the formula. By the autumn of 1900 it was clear that Wien's law broke down at lower frequencies in the far infrared and beyond. On that fateful afternoon of October the 7th, hair doctor Rubens and his wife visited the Planck home and inevitably the conversation turned to the latest results from the lab. Rubens gave Planck the bad news about Wien's law. After his guests had left, Planck set to thinking where the problem might lie. He knew how the blackbody formula first sought by Kirchhoff four decades earlier had to look mathematically at the high frequency end of the spectrum, given that Wien's law seemed to work well in this region, and he knew from the experimental results how a blackbody was supposed to behave in the low frequency regime, so he took the step of putting these relationships together in the simplest possible way. It was a guess, no more, a lucky intuition as Planck put it, but it turned out to be absolutely dead on. Between tea and supper Planck had the formula in his hands that told how the energy of blackbody radiation is related to frequency. He let Rubens know by postcard the same evening and announced his formula to the world at a meeting of the German Physical Society on October the 19th. One of the myths of physics which is echoed time and again in books and in college courses even today is that Planck's blackbody formula had something to do with what's called the ultraviolet catastrophe. It didn't. This business of the ultraviolet catastrophe is a bit of a red herring, if you don't mind mixing colorful metaphors, worthwhile mentioning here only to set the record straight. In June 1900 the eminent English physicist Lord Raleigh, known as John Strutt before he became a baron, pointed out that if you assume something known as the equipartition of energy, which has to do with how energy is distributed among a bunch of molecules, then classical mechanics blows up in the face of blackbody radiation. The amount of energy a blackbody emits just shoots off the scale at the high frequency end, utterly in conflict with the experimental data. Five years later Raleigh and his fellow countryman James Jeans came up with a formula afterward known as the Raleigh-Jeans law which shows exactly how blackbody energy is tied to frequency if you buy into the equipartition of energy. The name ultraviolet catastrophe, inspired by the hopelessly wrong prediction at high frequencies, wasn't coined until 1911 by the Austrian physicist Paul Ehrenfest. None of this had any bearing on Planck's blackbody work. Planck hadn't heard of Raleigh's 1900 comments when he came up with his new blackbody formula in October. In any case, it wouldn't have mattered. Planck didn't accept the equipartition theorem as fundamental, so the ultraviolet catastrophe, which sounds very dramatic and as if it were a turning point in physics, doesn't really play a part in the revolution that Planck ignited. Planck had his formula in October 1900 and it was immediately hailed as a major breakthrough. But the 42-year-old theorist, methodical by nature, wasn't satisfied simply by having the right equation. He knew that his formula rested on little more than an inspired guess. It was vital to him to be able to figure it out, as he'd done with Wien's law, logically, systematically from scratch. So began, as Planck recalled, To achieve his fundamental derivation, Planck had to make what was for him a major concession. He had to yield ground to some of the work that Boltzmann had done. At the same time, he wasn't prepared to give up his belief that the entropy law was absolute, so he reinterpreted Boltzmann's theory in his own non-probabilistic way. That was a crucial step and it led him to an equation that has since become known as the Boltzmann equation, which ties entropy to molecular disorder. We're all familiar with how at the everyday level, things tend to get more disorganized over time. The contents of houses become more and more randomized unless energy is injected from outside the system to tidy them up. What Planck found was a precise relationship between entropy and the level of disorganization in the microscopic realm. To put a value on molecular disorder, Planck had to be able to add up the number of ways a given amount of energy can be spread among a set of blackbody oscillators. And it was at this juncture that he had his great insight. He brought in the idea of what he called energy elements, little snippets of energy into which the total energy of the blackbody had to be divided in order to make the formulation work. By late 1900, Planck had built his new radiation law from the ground up, having made the extraordinary assumption that energy comes in tiny indivisible lumps. In the paper he wrote, presented to the German Physical Society on December the 14th, he talked about energy as made up of a completely determinate number of finite parts, and introduced a new constant of nature, h, with the fantastically small value of about 6.6 times 10 to the minus 27 ergs per second. This constant, now known as Planck's constant, connects the size of a particular energy element to the frequency of the oscillators associated with that element. Something new and extraordinary had happened in physics, even if nobody immediately caught on to the fact. For the first time someone had hinted that energy isn't continuous. It can't, as every scientist had assumed up to that point, be traded in arbitrarily small amounts. Energy comes in indivisible bits. Planck had shown that energy like matter can't be chopped up indefinitely. It's always transacted in tiny parcels, or quanta. And so Planck, who was anything but a maverick or an iconoclast, began the transformation of our view of nature, and the birth of quantum theory. It was to be a slow delivery. Physicists, especially Planck, the reluctant revolutionary as one historian called him, didn't quite know what to make of this bizarre suggestion of the graininess of energy. In truth, compared with all the attention given to the new radiation law, this weird quantisation business at its heart was pretty much overlooked. Planck certainly didn't pay it much heed. He said he was driven to it in an act of despair, and that a theoretical interpretation had to be found at any price. To him, it was hardly more than a mathematical trick, a theorist's sleight of hand. As he explained in a letter written in 1931, the introduction of energy quanta was a purely formal assumption, and I really did not give it much thought, except that no matter what the cost, I must bring about a positive end. Far more significant to him than the quantum discontinuity was the impressive accuracy of his new radiation law, and the new basic constant it contained. This lack of interest in the strange energy elements has led some historians to question whether Planck really ought to be considered the founder of quantum theory. Certainly he didn't see his work at the time as representing any kind of threat to classical mechanics or electrodynamics. On the other hand, he did win the 1918 Nobel Prize in Physics for his discovery of energy quanta. Perhaps it would be best to say that Planck lit the spark and then withdrew. At any rate, the reality of energy quanta was definitely put on a firm footing a few years later, in 1905, by the greatest genius of the age, Albert Einstein.