Einstein and the Quantum

by Stone, A. Douglas

  So our equally partitioned gas molecules are gliding along with the average kinetic energy 3kT/2, but their motion is not free of all forces. The presence of all that radiation generates a kind of frictional force, due to the Doppler effect. The Doppler effect, which is familiar for sound waves, is the observed change in frequency (pitch) that occurs when a sound source is moving toward or away from the receiver: when moving toward, its pitch is measured to be higher than when at rest, and similarly it is found to be lower when the sound source is moving away. The same effect occurs for light waves: when you move toward them their frequency increases, and when away from them it decreases. Actually the details are a bit different for light because, unlike sound, it always is measured to move at c, but Einstein had worked out the formulas for this while banishing the ether way back in the miracle year of 1905.

  Why does this effect cause friction? Imagine a situation where water waves of equal frequency are being generated in opposite directions at two ends of a swimming pool and you are standing in the middle, being buffeted forward and backward by those waves from each side. If you stand still, then on average you get an equal number of forward and backward shoves as the waves hit you, and you are not, on average, pushed toward either end of the pool. However, if you move at some reasonable speed (compared with the speed of the waves) toward one end, then you hit the incoming waves from that end more frequently than the waves generated behind you at the opposite end. This is the Doppler effect in a very concrete form: you are moving in a medium in which waves are also moving, such that you encounter more crests (at a higher frequency) when you move against the direction of wave propagation, and at a lower frequency when you move in the same direction as the waves. In this case the visceral effect is that you get knocked backward more than you get knocked forward, and you feel an effective force impeding your motion toward the end of the pool you are approaching. But if you turn around and start walking toward the other end, exactly the same thing happens, except now the force is pointing in the opposite direction; that is, it behaves like friction, slowing you down no matter which direction you go. It takes Einstein a dense three pages of algebra to work out the exact mathematical formula for this frictional force, but this is the essential idea.6

  But this is not the only force acting on the molecule; it cannot be, because if it were, over time the radiation field would extract all the kinetic energy from the molecules, leaving them at absolute zero temperature. (In our pool analogy, the walker gets too tired to walk against the current and just stands still.) Again we would have a version of the ultraviolet catastrophe. But Einstein knows how nature avoids this. The previous reasoning assumed that the absorption events occur in a perfectly regular sequence, whereas in actuality the molecule is being randomly buffeted by photons at irregular intervals, so that in any short interval it gets a net kick from the radiation that can push it in either direction, forward or backward. Einstein calculates the magnitude of this fluctuating force. And then he assumes that these two forces, the frictional one and the fluctuating one, must on average balance, precisely to avert the unobserved cooling of matter by radiation. But this balance equation depends on the mathematical form of the radiation distribution law, the infamous universal function ρ(υ, T). With great relish, Einstein shows that Planck’s law, and only Planck’s law, will make the two forces cancel each other on average.

  But central to all of Einstein’s reasoning is that each emission and absorption event is a directed process. “If we were to modify one of our postulates about momenta [forces], a violation of the [force balance] equation would be the consequence…. To agree with this equation—which is demanded by the theory of heat—in a way other than by our assumptions seems hardly possible.” He concludes, “If a molecule suffers a loss of energy in the amount hυ … then this process is a directional one. There is no emission of radiation in the form of spherical waves.”

  Not only has Einstein resolved a paradox in his own mind; he also has changed the nature of the evolving quantum theory. Werner Heisenberg, one of the founders of modern quantum mechanics, has pointed out that “[Einstein] himself, in his paper of [1917], … introduced such statistical concepts [into quantum theory].” Pascual Jordan, a key collaborator of Heisenberg’s, described Einstein’s paper as among the most important to influence the development of modern physics. From that point on, random, acausal processes would be integral to the theory. This was not a concept contained in the Bohr-Sommerfeld theory of the atom; it was Einstein who let this unwelcome genie out of the bottle. He would come to regret it.

  He continues: “the molecule suffers a recoil … during this elementary process of emission of radiation; the direction of the recoil is, in the present state of theory, determined by ‘chance’ … the establishment of a quantumlike theory of radiation [appears] almost unavoidable. The weakness of the theory is … that it does not bring us closer to a link-up with the wave theory … [and] also leaves the time of occurrence and direction of the elementary processes a matter of ‘chance.’ Nevertheless, I fully trust in the reliability of the road taken [italics added].”

  Einstein was confident of his results not just because of the simplicity and elegance of the logic; he now believed he had attained the long-sought proof that light quanta were as “real” as any other elementary particles, not just a manner of speaking about the interaction of radiation with matter, as maintained by Planck, Lorentz, and others. He proclaimed as much in his next letter to Besso: “any such elementary process is an entirely directed process. Thus the light quanta are as good as established.”

  1 Circular orbits are allowed by classical mechanics but require a specific relationship between a planet’s orbital energy and its angular momentum, which never is precisely satisfied when a planet forms out of primordial matter. In our solar system, however, planetary orbits are quite close to being circular.

  2 This approach is now called Bohr-Sommerfeld-Wilson quantization; it will be discussed further below.

  3 Recall that Jeans’s discredited explanation for the blackbody radiation observations was based on the hypothesis that matter and thermal radiation were not in thermal equilibrium at high frequencies. Now there was general agreement that this was incorrect, and Einstein could base his new work on the assumption of equilibrium without fear of such criticism.

  4 This second paper did not become available until 1917 and is the one usually cited and discussed, so it is not widely appreciated that the key ideas were found between May and August of 1916, only six to nine months after the completion of general relativity.

  5 Eight years later Einstein would be the first to discover that the equipartition theorem can break down even for an atomic gas (see chapter 25), but those effects require such low temperatures that they would not become observable until the end of the twentieth century. Moreover, this fact does not invalidate the argument he is making in the current work.

  6 The analogy here is not perfect, because there is no ether in which light waves move, but as noted, there is a relativistic version of the Doppler effect which still leads to a frictional force on the gas.

  CHAPTER 22

  CHAOTIC GHOSTS

  “I have firmly decided to bite the dust with a minimum of medical assistance when my time has come, and up to then to sin to my wicked heart’s desire. Diet: smoke like a chimney, work like a horse, eat without thinking and choosing, go for walks only in really pleasant company, and thus only rarely, unfortunately, sleep irregularly, etc.” This was Einstein’s cheeky pronouncement to Elsa Einstein back in August 1913, before his arrival in Berlin and the monumental labors that occupied him between then and the completion of his new work on thermal radiation in 1916. Many historians regard the period of November 1915 to February of 1917 as Einstein’s second miraculous phase. During this period he produced fifteen papers, including the final form of the theory of general relativity, its first extensions into cosmology, as well as the next conceptual pillar in the emerging quantum t
heory, the ideas of spontaneous emission, intrinsic randomness, and the marriage of the Bohr atom with the Planck law, implying the reality of photons. And all this was accomplished in the midst of wartime and the steadily increasing hardship of daily life as hostilities dragged on and Germany’s prospects dimmed. By early 1917 an exhausted and ill Einstein would have to reconsider how seriously he intended to ignore the demands of his body.

  The winter of 1916, in which Einstein returned to quantum theory with renewed intensity, became known as the “turnip winter” in Berlin as the lowly turnip was fashioned into all manner of absent foodstuffs: bread, cake, coffee, and even something purporting to be “turnip beer.” The British were blockading food shipments, and as a result during that year of 1916 an estimated 120,000 Germans died from malnutrition. In February of 1917 Einstein, along with the rest of Berlin, was suffering through an unusually frigid winter, during which he fell ill with liver and bladder ailments that reached life-threatening severity, causing him to lose over fifty pounds in two months. Einstein had not suffered major privations during these years, thanks to packages of supplies sent to him by his friend Zangger in Switzerland and his relatives in southern Germany, so his illness was due to mainly to overwork, poor eating habits, and a chronically troubling digestive system, which Mileva referred to as his “famous complaint.” Having only just presented his new work on cosmology to the Prussian Academy on February 6, 1917, he took to his bed and on the fourteenth wrote to Paul Ehrenfest in Leiden canceling his planned visit to Holland. “I am quite infirm from a liver condition,” he explained, “which imposes on me a very quiet lifestyle and the strictest diet and regimen.” Two months later he wrote to Lorentz, “I have not been working much at all, and that under ideal circumstances.” By May he was singing a different tune than the exuberant overture he had sent to Elsa four years earlier. He told Besso he had resisted the doctor’s order to go for a “spa cure,” saying he could not “raise the necessary superstition”; but, he continued, “I am committing myself to do everything else—which is unbelievable—to abstain from drinking, etc., in short to perform the rites of medicine loyally and piously.”

  Having failed in a first attempt to obtain a divorce from Mileva, Einstein had maintained until this period a certain distance from his cousin Elsa, no longer being so eager to jump into a second marriage as in the heady early days of 1913. In the midst of his acute illness he would still write to Zangger, “I have come to know the mutability of all human relationships and have learned how to insulate myself against heat and cold, so the temperature is quite steadily balanced.” But Elsa now took the lead in nursing him back to health and regulating his convalescence; by the end of the summer of 1917 she had procured for him the apartment next to hers at Haberland Strasse 5 and had even moved his things into it while he was away traveling. By December he could report to Zangger, “my health is quite fair now…. I have gained four pounds since the summer, thanks to Elsa’s good care. She cooks everything for me herself, as this has proved necessary.” However, by January he was bedridden again for six weeks and did not feel fully healthy again until the following summer, despite “Elsa indefatigably cooking” his “chicken feed.” It was in that summer that Einstein finally received the consent to a divorce from Mileva, with its famous stipulation that she would receive the proceeds of his inevitable Nobel Prize (should he survive long enough to receive it). By the following June (1919), after the legal formalities had been concluded, Einstein would finally marry Elsa and fulfill her long-held desire to become Mrs. Albert Einstein. She would be a steady, reliable presence in his life for the next two decades but never the romantic companion he had imagined in his early love letters, written before actually moving to Berlin.

  FIGURE 22.1. Watercolor of Einstein and Paul Ehrenfest playing duets in Leiden during one of Einstein’s periodic visits. Original watercolor by Maryke Kamerlingh-Onnes, courtesy AIP Emilio Segrè Visual Archives.

  Einstein’s health problems, beginning in February 1917 and continuing well into 1918, along with the complex and draining personal issues of divorce and remarriage, made these two years less scientifically productive than the previous two had been. He continued to work on elaborations and popularizations of general relativity, but one senses that quantum theory and the new atomic mechanics remained paramount in his research ambitions. In March of 1917, while still too ill to do much, he wrote to Besso referring to his new paper on thermal radiation, which had only recently been published despite its provenance nine months earlier. “The quantum paper I sent out has led me back to the view of the spatially quantum-like nature of radiation energy. But I have the feeling that the actual crux of the problem posed to us by the eternal enigma-giver is not yet understood absolutely. Shall we live to see the redeeming idea?”

  The very next day he wrote to Zangger bemoaning his health and his lack of intellectual momentum: “Scientific life has dozed off, more or less; nothing is going on in my head either. Relativity is complete, in principle, and as for the rest, the slightly modified saying applies: … what he can do he does not want; and what he wants he cannot do.” Coming on the heels of his proclamation about the quantum enigma to Besso, his self-appraisal seems clear: what he wants most at the moment is to truly understand what is going on with atoms and their interaction with light.

  Following his “perfectly quantic” derivation of the Planck law, Einstein’s period of vacillation on the reality of light quanta, begun in 1911, was over. He was convinced that light quanta were full-fledged particles, which were localized in space, moved along directed trajectories, and carried momentum as well as energy. This conviction simply renewed the challenge of how to reconcile their reality with the interference properties exhibited by electromagnetic radiation, which seemed to require that light extend over large regions of space. While no idea had arisen, either from him or from the expanding community of quantum physicists, for a new mathematical theory of light that could encompass these two conflicting aspects, around this time Einstein began to develop a conceptual framework that could serve as a stopgap measure on the way to a fuller theory.

  He hypothesized that light is emitted in a twofold process. While a guiding wave obeying the classical Maxwell equations is generated, at the same time some number of localized light quanta are ejected from the atom in specific directions, carrying all the energy. He mentions this idea briefly in a letter to Sommerfeld: “I am convinced that besides the directed energetic process, a kind of spherical wave is emitted, because of the possibility of interference for large-aperture angles. But … I am not convinced that what is being emitted immediately (the directed process) has an oscillatory character.” He apparently had lengthy exchanges with Ehrenfest and Lorentz detailing his views, although he neither published nor spoke publicly of them. Lorentz himself included them (with credit to Einstein) in lectures at Caltech in 1922, and a letter from Lorentz to Einstein in November of 1921 survives, in which he recapitulates Einstein’s proposal.

  Basic idea: … Upon the emission of light there are two sorts of radiation. They are:

  1. An interference radiation, which occurs according to the normal laws of optics but does not transmit any energy…. Consequently, they themselves cannot be observed; they just show the way for the energetic radiation. It is like a dead pattern that only comes to life through the energetic radiation.

  2. The energetic radiation. It is composed of indivisible quanta of [energy] hυ. Their path is given by the (vanishingly small) flow of energy from the interference radiation, and therefore they can never reach a spot where this flow is zero…. full interference radiation is formed [even if] … only a single quantum is emitted, which thus also can reach the receiving screen at only one spot. But this elementary instance is repeated countless times…. The various quanta now distribute themselves statistically … [so] that their average number at each point on the screen is proportional to the intensity of the incident interference radiation there. In this way the observed interfe
rence phenomenon is formed, consistent with the classical theory.

  Einstein had realized that making sense of the behavior of light requires that that we describe radiation by an extended field or disturbance, which would determine the measurement of energy transfer at specific points in space, while at the same time the energy transfer would itself be composed of localized, and quantized, units. He came up with the picturesque name of “ghost fields” (Gespensterfeld) for his guiding disturbance, and it seemed that he was inclined to regard the particulate quanta as the “real” things while the extended, wavelike entity was relegated to a secondary, spectral existence. Nonetheless he took the idea quite seriously, and from 1918 on discussed it widely, so that not just Einstein’s inner circle of confidants knew of it but also young students, such as Eugene Wigner,1 who recalled that Einstein was “quite fond of it.” Moreover, according to Lorentz, Einstein also foresaw that the ghost field would be used to determine probabilities for the light quanta. Continuing his description of Einstein’s idea, he states: “It has to be assumed that at each reflection and diffraction, whenever an incident light beam is split into two or more beams, the probability that a light quantum takes one or the other path is proportional to the intensities of the motions of light along these various paths, calculated according to the classical laws” (italics added). “Classical laws” here refers to Maxwell’s equations, which determine the reflection and diffraction of electromagnetic waves, and hence determine the probability that the quanta will go one way or the other. For individual photons, it seems, there are no deterministic laws of motion; only the properties of the guiding field are set by deterministic laws. These two ideas—that quantum particles are guided by an extended field, which allows them to “interfere with themselves,” and that this extended field obeys definite laws but doesn’t determine the fate of individual quanta—are key concepts in the modern quantum theory of radiation (and of matter). Usually they are attributed to Max Born, who applied them first to electrons, but he always credited Einstein as their source, even on the Nobel podium in Stockholm.