Einstein and the Quantum

by Stone, A. Douglas

  As already mentioned, by 1903 Einstein had settled into his routine in Bern, working six days a week at the patent office, giving private lessons, and nonetheless finding time to pursue research in fundamental physics. Later he would refer to this period as “those happy Bernese years.” With his charisma and joie de vivre he had very quickly acquired a group of comrades who would share this idyllic interlude with him. The first of these new companions was a Romanian philosophy student, Maurice Solovine, who showed up at his flat in response to Einstein’s earnest advertisement for private physics lessons. A typical Einsteinian episode ensued. Following an enthusiastic invitation to enter his humble abode, Solovine was immediately “struck by the extraordinary brilliance of his large eyes.” Two and a half hours passed in a twinkling as the men discussed science and philosophy, and by the next session Einstein, having quite forgotten the original profit motive, declared physics lessons too much of a bother and proposed instead that they should meet freely to discuss ideas of all sorts. Very soon they added to their ranks another young aspiring intellectual, Conrad Habicht, a mathematics student, who had attended the Poly a bit ahead of Einstein and whose acquaintance Einstein had made during his vagabond years after graduation.

  Habicht had the most jovial and high-spirited relationship with Einstein of all his peers; their letters to each other are rife with playful sarcasm. Together with Solovine, the two men founded a reading and discussion group, which they satirically dubbed the “Olympia Academy.” Habicht graciously allowed Einstein the esteemed position of president, complete with a commemorative (cartoon) bust and a grandiloquent dedication in Latin, celebrating his unerring command of “those fabulous molecules.” It was also Habicht who dubbed our Valiant Swabian “Albert Ritter von Steissbein,” which loosely translates as “Knight of the Tailbone,” presenting him with an engraved tin plate bearing this title. Far from being offended, Einstein and Mileva “laughed so much they thought they would die,” and henceforth Albert occasionally signed letters to Habicht with this sobriquet. The heraldic crest above his bust is aptly chosen: a link of sausages, one of the few foodstuffs the Olympians could afford to eat at their august gatherings.

  Despite the evident joviality of the meetings, the members, along with occasional guests such as the attentive but silent Mileva, took their studies very seriously, and Einstein acquired many of his lasting philosophical views during the two years of meetings. The group would convene at the apartment of one of the members, and over a frugal repast would debate the meaning and merits of the assigned works, which included philosophy (David Hume and John Stuart Mill), history and philosophy of science (Henri Poincaré and Ernst Mach), and occasionally great literature (Don Quixote and Antigone).

  FIGURE 8.1. (a) Hand-drawn cartoon by Maurice Solovine celebrating Einstein as President of the Olympia Academy, with his bust garlanded in hanging sausages. (b) Satirical inscription in Latin that accompanied the cartoon. It translates as “The man of Hechingen, expert in the noble arts, versed in all literary forms – leading the age towards learning, a man perfectly and clearly erudite, imbued with exquisite, subtle and elegant knowledge, steeped in the revolutionary science of the cosmos, bursting with knowledge of natural things, a man with the greatest peace of mind and marvelous family virtue, never shrinking from civic duties, the powerful guide to those fabulous receptive molecules, infallible high priest of the poor in spirit.” Courtesy the Albert Einstein Archive.

  Solovine recalled that these gatherings brimmed over with merriment, although woe to him who would slight the gravity of the occasion. On one memorable night, Solovine, having skipped out on a meeting at the last minute to attend a concert, returned to his apartment to find his bed piled high with furniture and household items, his room enveloped in thick cigar smoke, and a scolding note in Latin pinned to his wall. Einstein remembered these get-togethers with the greatest fondness. In 1953 he wrote to Solovine:

  FIGURE 8.2. The principals of the Olympia Academy circa 1903; on the left is Conrad Habicht, center is Maurice Solovine, and on the right is Einstein. ETH-Bibliothek Zurich, Image Archive.

  To the immortal Olympia Academy!

  In your short active life dear Academy, you took delight, with childlike joy, in all that was clear and intelligent. Your members created you to make fun of the long-established sister academies. How well your mockery hit the mark I have learned to appreciate through long years of careful observation…. Even if we have become somewhat decrepit, a glimmer of your bright, vivifying radiance still lights our lonely pilgrimage…. To you our fidelity and devotion until the last learned gasp!

  A. E.—now corresponding member

  Besides Habicht and Solovine, Einstein had another intellectual confidant in Michele Besso. He had met Besso, who was six years older and already a graduate of the Poly, while he was a student in Zurich. Besso, of middle-class Jewish origin, was an engineer, who had applied for and obtained a position in the patent office in Bern at Einstein’s suggestion. Einstein had also introduced him to his eventual wife, Anna Winteler, who was a daughter of Einstein’s host family in Aarau. Besso and Einstein became lifelong and intimate friends. If Einstein occasionally exhibited the absentmindedness of a starry-eyed dreamer, he was an absolute model of Swiss efficiency compared with the impractical Besso, whose boss had once pronounced him “completely useless and almost unbalanced.” Einstein, while granting that in many respects Michele was “an awful schlemiel,” nonetheless enjoyed and profited from their exchanges: “[Besso] has an extraordinarily fine mind whose working, though disorderly, I watch with great delight.” Later, when reflecting upon his achievements of 1905, he said that he “could not have found a better sounding-board in the whole of Europe.” Besso and Einstein walked home together daily during that period, and Einstein shared with him his developing ideas about physics. Although Besso is often mentioned as the first to hear about relativity theory, apparently the main subject of their conversations was something else: Einstein’s new hypothesis about the nature of light.

  This hypothesis surely would have been presented to the Olympia Academy; however, by the end of 1904, Habicht had obtained a post as a mathematics and physics teacher in Schiers, quite a distance from Bern, and in 1905 Solovine moved from Bern to Paris, where he worked on a journal, the Revue Philosophique. Hence the academy was out of session as Einstein was producing his string of hits. In May of 1905 he sent a typical jocular missive to Habicht, whose absence he had felt:

  Dear Habicht, such a solemn air of silence has descended between us that I almost feel as if I am committing a sacrilege when I break it now with some inconsequential babble….

  So what are you up to you frozen whale, you smoked, dried, canned piece of soul…? Why have you still not sent me your dissertation? Don’t you know that I am one of the 1½ fellows that would read it with interest and pleasure, you wretched man?

  I promise you four papers in return, the first … deals with radiation and the energy properties of light and is very revolutionary…. The fourth paper is still only a rough draft at this point, and is an electrodynamics of moving bodies which employs a modification of the theory of space and time.

  This lively discourse indicates Einstein’s contemporaneous valuation of his great works of 1905; they were all exciting, and he was proud of them, but one was actually revolutionary, the one on the quantum properties of light. This paper, titled “On a Heuristic Point of View Concerning the Production and Transformation of Light,” grew out of his last work on statistical mechanics, in which he had focused for the first time on Planck’s theory of radiation, the theory that required the critical but unappreciated introduction of a smallest energy element, ε = hν.

  Recall that Planck had avoided the question of whether he was allowed to apply statistical mechanics to radiation by relating the radiation energy, which he wanted to calculate, to the average energy of vibrating molecules in the blackbody. But instead of directly calculating this energy, he took the odd detour of calcul
ating the entropy of the molecules, and, with full knowledge of the answer he had to get in order to agree with experiment, he wrestled the entropy into the correct form, using the outré assumption of a minimal energy element. Given Einstein’s focus on statistical mechanics, which, unaware of Gibbs’s prior work, he thought he was extending in a novel manner beyond gases, he would naturally focus on the place at which Planck took his detour. Why not just calculate the average energy of a molecule?

  In this context, what is needed is the average “thermal energy” of the molecule. If we imagine the molecule as consisting of several atoms connected by chemical bonds, which can vibrate just like macroscopic springs, and if it were possible to direct energy to a particular isolated molecule, then according to the classical view one can tune the energy to any desired value (until it gets so large that the molecule breaks apart). However, Planck had appended the ad hoc restriction that each vibration could only have certain energy values, constrained to be a whole number times hν (although it wasn’t really clear he believed wholeheartedly in this constraint). Nonetheless, in either view, the energy of any specific molecule was something that could and would vary in time, whatever the temperature of the surroundings. But each molecule is typically in contact with other molecules through collisions (in the gaseous state) or mutual vibrations in the solid state, so that if the material’s temperature is fixed, each molecule has a definite average energy that does depend on the temperature of the surrounding environment and is termed the thermal energy.

  Several times in his papers on statistical mechanics Einstein had already done a calculation for the kinetic part of the average thermal energy, the contribution of molecular velocity to the total thermal energy. It was a trivial generalization of those calculations to consider a molecule that vibrates back and forth and hence also has potential energy. Potential energy is just that—energy stored by doing work against a force. Continuing the spring analogy, when a mass on a spring is pulled and then clamped at a greater extension, potential energy is stored in the mass, which can be released when the mass is unpinned and oscillates back and forth. When a molecule vibrates, its chemical bonds are compressed and stretched like a spring, alternately storing and releasing potential energy. In this case, according to Maxwell/Boltzmann/Einstein statistical mechanics, the average potential and kinetic energies are equal and have a simple relation to temperature, Emol = kT, where k is Boltzmann’s constant (the same one that comes into the formula for the entropy, S = k log W), and T is the temperature.1 Note that this average thermal energy is independent of the frequency.

  It may initially seem counterintuitive that two identical masses connected to springs that vibrate at very different frequencies (i.e., have different degrees of stiffness) can have on average the same energy. The high-frequency vibrating mass will oscillate faster, and thus will have higher kinetic energy, which, for an oscillator, also must be equal on average to its potential energy. Thus it should just have more total energy, right? No, because the two springs will also vibrate different distances from their unstretched positions. The relation Emol = kT tells us that the high-frequency molecule must make smaller-amplitude vibrations than the low-frequency molecule, in just such a way that the average energy of the two is equal. This statement of classical statistical mechanics, that all vibrating structures have the same average energy, has a fancy name: the equipartition theorem; it implies that the total energy of the system is equally shared by each microscopic part. This theorem will be very important in Einstein’s reasoning going forward.

  Since Einstein had by now developed a great understanding of classical statistical mechanics, he surely would have leaned initially toward the “obvious” equipartition formula: Emol = kT, which via Planck’s reasoning would immediately yield a hypothesis for the radiation law. However, as we shall soon see, using the hypothesis so obtained gives a paradoxical result, so Einstein at some point rejected this approach. Planck’s route, on the other hand, involved an obvious fudge, the ad hoc hypothesis of a minimal energy element, with no fundamental justification at all. This dead end is the point at which ordinary mortals would have thrown up their hands and given up. Instead, in one of the greatest demonstrations of flexible thinking ever, Einstein abandoned his beloved classical statistical mechanics and opened his mind to a new and bizarre possibility, the possibility that the hallowed Maxwell equations, whose perfection he had long admired, were not the final word on the nature of light.

  1 Here and elsewhere I assume that temperature is measured using the absolute (kelvin) scale, where it is always a positive number.

  CHAPTER 9

  TRIPPING THE LIGHT HEURISTIC

  “The wave [Maxwell] theory of light … has proved itself splendidly in describing purely optical phenomena and probably will never be replaced by another theory…. [However] it is conceivable that … the [wave] theory of light may lead to contradictions with experience when it is applied to the phenomena of production and transformation of light. Indeed it seems to me that the observations regarding ‘blackbody radiation,’ photoluminescence, and [the photoelectric effect] … can be understood better if one assumes that the energy of light is discontinuously distributed in space. According to the assumption to be contemplated here, when a light ray is spreading from a point, the energy is not distributed continuously over ever-increasing spaces, but consists of a finite number of energy quanta that are localized in points in space, move without dividing, and can be absorbed or generated only as a whole” (italics added).

  With these comments Einstein began his paper that presented a “heuristic point of view” (not a theory) on the production and transformation of light. A noted Einstein biographer has called the last statement of this quotation “the most revolutionary sentence written by a physicist of the twentieth century.” There had been no significant work on a particulate view of light since Newton’s theory of light as granular was definitively refuted in the early part of the nineteenth century. Thus Einstein’s work has no real antecedents; it is a bolt from the blue. In contrast, what became special relativity theory was very much in the air by 1905, and when Einstein wrote his first paper on that topic, four months later, it involved a new derivation of mathematical properties of space and time that had already been written down by Lorentz (albeit without the radical interpretation given to them by Einstein).

  The paper hypothesizing that light could be conceived of as a stream of particles (which Einstein called “quanta” and which we now refer to as “photons”) is seventeen pages long and consists of an introduction and nine numbered sections. It is clearly written and would have been relatively easy to understand for an expert like Planck (who was theory editor for the journal). The paper makes a number of assertions, both qualitative and quantitative, about experimental phenomena that can be understood from the new point of view it puts forward but are puzzling when viewed from the vantage point of Maxwell’s theory. Astoundingly, every statement in this paper is correct when assessed on the basis of our modern quantum theory of radiation and its interaction with matter.

  While Einstein discusses experiments related to light extensively in the paper, this is not how he begins. Instead, he starts with the theoretical consideration already mentioned, that classical statistical mechanics fails to describe blackbody radiation. In this first section, titled “On a difficulty encountered in the theory of blackbody radiation,” he explains, in a manner similar to Planck’s, that to analyze the energy density of blackbody radiation, one does not need to assume statistical mechanics applies to thermal radiation. Instead, since the radiation is always exchanging energy with matter, one needs only to figure out the average energy per molecule of a gas in contact with electrons, which then emit light (here, ironically, he refers to the “sad specimen” of Drude and his electron theory for support). Having established this, he goes right to the point: classical statistical mechanics insists that the energy per molecule is kT. End of story. He then invokes Planck’s own equation,
which says that the blackbody energy density at frequency ν, which he denotes by the Greek letter ρ, is proportional to this molecular energy, for which Planck had substituted, not kT, but a totally different answer, which he had gotten by working backward from his entropy calculation. But, Einstein points out, if one instead sticks with the answer Emol = kT, the only answer that makes sense according to classical reasoning, there is a problem with the total energy of the radiation. Actually, a really big problem: the total energy summed over all the frequencies of radiation is infinite. Even though physicists toss around infinities nonchalantly, we are still not allowed to have an infinite answer for something you can measure. The total energy coming out of a finite-size blackbody had been measured; it obeys a known relation, called the Stefan-Boltzmann law,1 and it is most assuredly finite. So the obvious approach gives an impossible result.

  The source of the impossible answer was easy to trace. In a gas there is a very large but finite number of molecules; thus each molecule can have a fixed amount of energy, kT, and the total energy will be some large but finite quantity. In a box with trapped radiation, however, there are an infinite number of wavelengths of radiation that fit inside the box (remember that the wavelength of Maxwell radiation can be arbitrarily small). If each wavelength of radiation carries the same energy, then the total energy adds up to infinity, just as Einstein found. Is it possible that only some wavelengths actually get their share of the energy, leaving others below their thermal quota? No, it is not possible. Since the radiation and electrons are always interacting, if some wavelengths didn’t get their “fair share” of the energy, then radiation would continually suck energy out of matter until all the matter cooled to absolute zero. Much later the physicist Paul Ehrenfest2 came up with a catchy name for this phenomenon: the ultraviolet catastrophe (almost all the energy should flow into the shorter, ultraviolet wavelengths). Not to worry, it doesn’t happen.