Einstein and the Quantum (30 page)

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 theory, 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 interference phenomenon is formed, consistent with the classical theory.