This lull provided a most opportune moment for another genius to step into the scene — a young clerk in the Swiss patent office in Bern. The great masterstroke of Albert Einstein, for indeed he is the person we mean, was to spot a key parallel between a black body and a different system that also has a spectrum determined by its overall temperature. In particular, we are alluding to an ideal gas trapped inside a container. Why did Einstein make this analogy, which no one else — or almost no one else — had imagined? And why did he trust it so deeply? We will shortly give some speculations.
Curiously enough, until the annus mirabilis, no totally convincing proof had been found of the existence of atoms (whether in solids, liquids, or gases). To be sure, certain audacious scientific spirits, such as Ludwig Boltzmann, from Austria, and James Clerk Maxwell, from Scotland, had conjectured that all gases consisted of myriads of small particles constantly bashing into each other, as well as bouncing off the walls that contained them, and from this “ideal gas” assumption they had been able to derive certain formulas that matched the empirical observations of real gases to an incredible degree of precision. Although this was a strong piece of evidence in favor of the atomic hypothesis, a good number of physicists, chemists, and philosophers remained skeptical.
It will help us here to draw another explanatory analogy — this time between an ideal gas and a frictionless pool table. On the pool table we imagine hundreds of tiny balls that have been set in motion by a violent explosion (such as the “break” at the beginning of a pool game), and that now are all madly bouncing off the table’s walls and off of one another. If someone were to tell us the amount of energy of the initial explosion, then we might wonder what the dominant speed (i.e., the most common speed) of the balls on the table would be, once things had settled down into a more or less stable state. More ambitiously, we might ask what the distribution of speeds of the balls will be. It might seem surprising, a priori, that there is a precise answer to such questions, but in fact there is. And in an analogous fashion, there is a precise formula — the so-called “Maxwell–Boltzmann distribution” — that gives, for any particular value of kinetic energy you wish, the percentage of molecules of gas that will have that energy (given the temperature of the gas). The location of the peak of this graph reveals the dominant kinetic energy of the gas molecules.
Albert Einstein had a hunch that these two types of system — the black body and the ideal gas — were deeply related despite their surface-level dissimilarities. In both cases, there is a container filled with energy, but beyond that, what would make anyone suspect that these two systems were deeply linked? Let’s shift the question back to the more familiar territory of our two analogues — the swimming pool and the pool table. It then becomes a question about the likelihood of there being a profound connection between the rippling surface of a swimming pool agitated by a splash, and hundreds of tiny balls bouncing about in a frenzied manner on a pool table, all set in motion by a sudden explosion. Both situations are filled with random motion and take place on horizontal, flat surfaces, but those very superficial facts would hardly seem to add up to a strong reason to make anyone suspect that a deep relationship links them.
Thus, for the vast majority of physicists at that time, Einstein’s analogy between the ideal gas (here likened to a seething billiard table) and the black body (here likened to a natatorium’s undulating surface) seemed utterly implausible. So why did Einstein see things differently? First of all, as he stated in the first of his articles of 1905, he had noticed a curious mathematical similarity linking the two formulas giving the energy distributions (for the blackbody spectrum, Einstein used a formula discovered by the German physicist Wilhelm Wien before Planck found his more precise one, and for the ideal-gas spectrum, he used the formula of Maxwell and Boltzmann), and this suggested to him that the physical similarity of the two systems might easily go well beyond the surface. All one can say here is that Einstein had an eagle eye; he almost always knew how to put his finger on just what mattered in a situation in physics.
It is fascinating to note that Wilhelm Wien, in his search for a formula for the blackbody spectrum in the mid-1890s, had had the excellent intuition — closely related to Einstein’s intuition some ten years later — to try using an analogy he had “sniffed”, linking the blackbody spectrum to Maxwell and Boltzmann’s ideal-gas spectrum. It is thus no coincidence that Einstein refound Wien’s analogy when he looked at the two formulas at the same time, for Wien’s formula was rooted in the Maxwell–Boltzmann formula. For Wien, however, the analogy between the two systems was solely formal; it did not suggest to him that the two systems had a deep physical link, and so in his mind he did not pursue it nearly as doggedly or as profoundly as did Einstein.
One other factor that might have contributed to Einstein’s faith in his analogy between the physics of the ideal gas and that of the black body (not just between the mathematical formulas for their spectra) was the fact that only a few months earlier, he had found and deeply exploited an analogy between an ideal gas and another physical system — namely, a liquid containing colloidal particles whose nonstop, apparently random hopping-about could be observed through a microscope. This analogy had allowed him to argue persuasively for the existence of extremely tiny invisible molecules that were incessantly pelting the far larger colloidal particles (like thousands of gnats bashing randomly into hanging lamps) and giving them their mysterious hops, known as “Brownian motion”. It is thus probable that two distinct forces in Einstein’s mind — the mathematical similarity of the formulas and also his recent Brownian-motion analogy — gave him great trust in his analogy between a black body and an ideal gas.
In any case, building on the bedrock of his latest analogy, Einstein undertook a series of computations, all based on thermodynamics, the branch of physics that he thought of as the deepest and most reliable of all. First he calculated the entropy of each of the systems and then he transformed the two entropy formulas so that they would look as similar as possible to each other; in fact, at the end of his ingenious manipulations, they wound up exactly identical except for the algebraic form of one simple exponent. This provocative maneuver made it clear that the two systems were far more intimately related than Wilhelm Wien had ever suspected.
In the key spot in the formula for the ideal gas’s entropy, the letter “N” appeared, standing for the number of molecules in the gas; in “the same” spot in the formula for the black body’s entropy, the expression “E/hν” appeared. (The letter “h” stands for Planck’s constant, and the Greek letter “ν” — “nu” — for the frequency of the electromagnetic waves, always inversely proportional to their wavelength.) Einstein had thus compressed the entire distinction between these two vastly different physical systems down into one tiny but telling contrast: an integer N in one case, and the simple expression “E/hν” in the other.
But what did this precision pinpointing of their difference mean? Well, E/hν represents the act of dividing up the total energy E (a large number of ergs, an erg being a standard energy unit) into many minuscule chunks all having energy hν (a tiny fraction of one erg). This ratio tells how many small chunks make up the larger chunk; and thanks to the cancellation of the units (ergs in both numerator and denominator), it is a “pure” number: its value is independent of the system of units used. Einstein’s analogy now plays a key role, telling us that this number in the blackbody system maps onto the number of molecules N in the ideal gas. The dividing-up of E into identical pieces all having size hν (a “measurement” of E, to echo the term used in Chapter 7 for one of the naïve analogies to division), was an unmistakable clue, for Einstein, that the radiation in the cavity was composed, as is a gas, of discrete particles. For any given wavelength, all “light chunks” carried the same tiny load of energy.
Even for its finder, this was a monumentally shocking idea, because to him, just as to all physicists of his day, electromagnetic radiation was synonymous with light (along
with light’s cousins having longer and shorter wavelengths), and Einstein was very aware, as were all his colleagues, that the ferocious battle between advocates of light as corpuscles and advocates of light as undulations had finally been conclusively won, a century earlier, by the undulatory side. Furthermore, ever since then, thanks especially to Maxwell’s fundamental equations, discoveries in physics had reinforced over and over the view of light as continuous waves and not as discrete particles. How, then, could a corpuscular view of light possibly stage a comeback a hundred years after its demise? And yet, this is exactly what seemed to be happening, thanks to a very simple analogy.
Einstein, recognizing that there was nothing to do but accept the image so clearly suggested by his analogy, came to the staggering hypothesis, flying in the face of the most solidly established facts, that the electromagnetic radiation in a blackbody cavity consisted of small corpuscles — small packets of energy, analogous to the N molecules in an ideal gas (and, to show how far ahead of his times Einstein was, we point out that at that moment in physics, even the existence of atoms and molecules was still considered suspect by some skeptical holdouts!). Each of these mysterious “lumps of radiation” would necessarily possess exactly the energy hν, which thus had to be the minimal amount of energy associated with the frequency ν. Called “light quanta” by Einstein, such particles are known today as “photons”.
Light Quanta Are Scorned While Sound Quanta Are Welcomed
Although not in the least controversial today, Einstein’s bold suggestion in 1905 that light must consist of particles was harshly and unanimously dismissed by his colleagues. Later in life, he declared this hypothesis, based on but the shakiest of analogies, to be the most daring idea of his entire career; indeed, it was so daring that it unleashed, among his colleagues, a barrage of scorn and hostility whose magnitude, duration, and ferocity he surely could not have anticipated.
In the conclusion of his light-quantum article, the young “Technical Expert, Third Class” (the lowest rank at the Swiss patent office) had both the cleverness and the courage to suggest three possible experimental ways to confirm or refute his theory, thus taking the risk of handing weapons to his enemies, with which they could potentially shoot him down! In particular, the second of his suggestions involved looking at the photoelectric effect, in which, when electromagnetic radiation (such as light) falls on a piece of metal, some electrons come flying out of the metal. It was an odd little effect but was considered of no great moment for physics, and had been observed for the first time, but only very crudely, in 1887 by the German physicist Heinrich Hertz, in a series of experiments in which he conclusively demonstrated the existence of electromagnetic waves, thus brilliantly confirming Maxwell’s equations.
Einstein realized that his theory of light quanta yielded precise predictions for the photoelectric effect. In particular, in a very simple equation, it predicted the rate of ejection of electrons as a function of the wavelength of the incident light, and this prediction was in stark contradiction with predictions based on Maxwell’s universally accepted equations. Einstein could not know, nor could any other physicist of the time, what would be revealed by precise measurements of the photoelectric effect, but it was clear to him that such experiments would be decisive and might lead to a great battle, because if his prediction turned out to be correct, the world of physics would be forced to reject Maxwell’s equations as the basis of electromagnetism. This was among the most paradoxical moments in the entire history of physics, for Hertz’s experiments, which had so triumphantly confirmed Maxwell’s equations, were also the source of the tiny anomaly that now threatened to undermine those very equations. However, the investigation that Einstein suggested in his conclusion was very difficult to carry out, and it took quite a number of years before the experiments yielded clear results.
In 1905, though, no one paid the least attention to the light-quantum hypothesis, as everyone but Einstein was completely convinced of the validity of Maxwell’s equations. Light was made of waves; that was that. To doubt it was simply insanity. Even Max Planck, who had dreamt up the idea of quanta of energy of vibrating atoms, proclaimed that the new hypothesis of quanta of light was senseless. (It is of note that Planck, some years earlier, had also declared that the hypothesis of atoms was senseless, but by 1905 he greatly regretted having done so.) Despite his colleagues’ unanimous scorn, the young Einstein had an unshakable faith in his own ideas, and was not discouraged. (Actually, calling them “his colleagues” is a bit of a stretch, since until 1908, Einstein was merely an amateur physicist, his official job being that of Technical Expert in the patent office.)
In 1907, Einstein pushed his quantum ideas yet further. He proposed a new analogy that built both on Max Planck’s idea of energy quanta in vibrating atoms and on his own idea of light quanta. This analogy had to do with sound waves inside solids. Essentially, Einstein came up with the idea of sound quanta, although he never used this terminology. (Today, the quanta making up sound waves are called “phonons”, echoing “photon”; they play a key role in the physics of matter.) With his new way of conceiving of vibrations inside solids, Einstein was able to resolve a major mystery concerning the heat capacity of solids. This time, most curiously, the world of physics, even as it disdainfully continued to reject light quanta, unanimously accepted the validity of Einstein’s explanation of the heat capacity of solids, based on sound quanta, and in 1909 the Dutch physicist Peter Debye deepened Einstein’s theory and created a very powerful theory of heat capacities, which physicists quickly and warmly welcomed, all while still giving the cold shoulder to Einstein’s light-quantum hypothesis.
A strong friendship and great mutual respect developed between Albert Einstein and Max Planck, and in 1913, the latter nominated Einstein for membership in the Prussian Academy of Sciences, which was one of the most distinguished scientific societies in the world. In his nomination letter, Planck sang Einstein’s praises, but when it came to the subject of light quanta, which Einstein had continued to champion, Planck commented, “That he may sometimes have missed the target in his speculations, as, for example, in his hypothesis of light quanta, cannot be held too much against him, for it is not possible to introduce really new ideas even in the most exact sciences without sometimes taking a risk.”
In the decade from 1906 to 1915, the distinguished American physicist Robert Millikan carried out a long and very careful series of experiments on the photoelectric effect. From the start, he was convinced that Einstein’s ideas on the subject were worthless, since they directly contradicted the century-old finding, due to Thomas Young in England and Augustin Fresnel in France, and spectacularly confirmed in 1887 by Heinrich Hertz in Germany, that light consists of waves, and this fact precluded particles of light. For Millikan as for nearly everyone else, the idea of light being both particulate and wavelike was inconceivable. Nonetheless, his experiments wound up confirming Einstein’s predictions perfectly, which plunged Millikan into deep cognitive dissonance. In a major book summarizing his work, published in 1917, Millikan admitted that his results supported Einstein’s revolutionary predictions to the hilt, but he insisted that one should beware of Einstein’s “reckless” ideas about light because they had no theoretical underpinning. Put otherwise, although Einstein’s conjectural explanation of the photoelectric effect had furnished impeccable predictions, one should give it no credence because it had not been rigorously derived from previously known physical laws. Millikan even had the temerity to declare in his article that Einstein himself no longer believed in his own “erroneous theory” about light (a pure speculation on Millikan’s part, without the slightest basis in fact).
To add insult to injury, although the 1921 Nobel Prize in Physics was awarded to Albert Einstein, it was not for his theory of light quanta but “for his discovery of the law of the photoelectric effect”. Weirdly, in the citation there was no mention of the ideas behind that law, since no one on the Nobel Committee (or in all of physics) b
elieved in them! Light quanta had been unanimously rejected by the members of the community of physicists, even the most adventurous among them. For example, the following year, Niels Bohr, the great Danish physicist and admirer of Einstein, in his acceptance speech for his own Nobel Prize, which had just been awarded to him for his contributions to quantum theory, brusquely dismissed Einstein’s ideas about the corpuscularity of light as “not able to throw light on the nature of radiation”.
And thus Albert Einstein’s revolutionary ideas on the nature of light, that most fundamental and all-pervading of natural phenomena, were not what won him the only Nobel Prize that he would ever receive; instead, it was just his little equation concerning the infinitely less significant photoelectric effect. It’s as if the highly discriminating Guide Michelin, in awarding its tiptop rank of three stars to Albert’s Auberge, had systematically ignored its chef’s consistently marvelous five-course meals and had cited merely the fact that the Auberge serves very fine coffee afterwards.
Vindication of Einstein’s Boldest Analogy
The turning point when light quanta at last emerged from the shadows came only in 1923, when the American physicist Arthur Holley Compton astonished the world of physics with his experimental discovery that when an electromagnetic wave approaches an electrically charged particle (an electron in an atom, for instance), it transfers to the particle some of its kinetic energy and momentum, but does not do so as Maxwell’s equations predicted. In fact, Compton found that the wave–particle “collision” that takes place in such a situation obeyed the long-known mathematical rules of collisions between two particles, with the energies of the incoming and outgoing waves matching exactly what Einstein had predicted in his 1905 paper about light quanta. And thus, at long last, light became particulate!
Surfaces and Essences Page 79