Einstein and the Quantum (28 page)

Bohr's explanation of the Pickering-Fowler spectrum was the clincher; the theory wasn't just a tautology designed to fit the known hydrogen spectrum. It had predictive and explanatory power. When he became aware of this, Einstein was mightily impressed. Einstein's old Zurich acquaintance, the chemist George Hevesy, was the first to tell him the news. He met Einstein at the German Physical Society conference in Vienna in 1913 and, as he wrote Rutherford, “
we came to speak of Bohr's theory
, he told me that he had once similar ideas, but did not dare to publish them. ‘Should Bohr's theory be right, it is of the greatest importance' [he said]. When I told him about the Fowler spectrum the big eyes of Einstein looked even bigger and he told me, ‘Then it is one of the greatest discoveries.' ” In writing to Bohr, Hevesy recounted a further highly revealing comment from Einstein: “
I told him [the explanation
of the Fowler spectrum]…. When he heard this he was extremely astonished and told me: ‘
Then the frequency of the light does not depend at all on the frequency of the electron
[italics added] … this is an enormous achievement. The theory of Bohr must then be right.' ”

FIGURE 20.1.
Albert Einstein and Niels Bohr in discussion, circa 1925–30. Photograph by Paul Ehren fest, courtesy AIP Emilio Segrè Visual Archives.

So this was the revolutionary step that Einstein, who had mused on spectral lines as early as 1905, had not been willing to take. The
frequency of light had nothing to do with the frequency of motion of the electron in the atom—who would have guessed? Einstein, the originator of so many “crazy” leaps of intuition himself, could recognize one when he saw it. What impressed him were not Bohr's calculations, which were simple, but the insight to guess what one should keep and what one should drop from the laws of classical physics. Much later he lauded Bohr's achievement thus:

All my attempts, however, to adapt
the theoretical foundation of physics to this [quantum] knowledge failed completely. It was as if the ground had been pulled out from under one, with no firm foundation to be seen anywhere, upon which one could have built. That this insecure and contradictory foundation was sufficient to enable a man of Bohr's unique instinct and tact to discover the major laws of the spectral lines and of the electron-shells of the atoms, together with their significance for chemistry, appeared to me like a miracle—and appears to me as a miracle even today. This is the highest form of musicality in the sphere of thought.

Inspired by the establishment of the Bohr postulates for the atom, Einstein was able to write a crucial coda to Bohr's composition shortly after he returned to quantum cogitation in 1916, one that would add a new framework to the insecure foundation of the developing quantum theory.

 

¹
Clara was reportedly particularly disturbed by the death of Haber's brilliant young collaborator, Otto Sackur, who was killed in a laboratory explosion while trying to improve gas weapons.

²
Football here refers to soccer of course. In fact Bohr was by all accounts a skilled goalkeeper, although not the athletic equal of his brother, Harald, who became a noted mathematician; Harald won a silver medal in soccer competing for Denmark in the 1908 Summer Olympics.

³
Inventor of the eponymous radiation counter.

⁴
As long as the electron is bound to the nucleus (i.e., doesn't have enough energy to escape from the nuclear attraction).

⁵
Bohr got this answer by considering the average frequency of all circular orbits starting infinitely far away from the nucleus, where
f
= 0, all the way in to the final orbit, with frequency
f
. Bohr's restriction can be alternatively phrased as the constraint that the angular momentum of an electron in a Bohr orbit is equal to
nh
/2
π
, which is the form that generalizes to other force laws and is emphasized in modern accounts. This idea was expressed but not strongly emphasized in his original paper.

⁶
The ratios of the distances in frequency between pairs of lines corresponded to the difference of the inverse of the squares of integers.

CHAPTER 21

RELYING ON CHANCE

Fundamental as [the relativity theory] of Einstein
has proved to be for the development of the principles of physics, its applications are for the present still at the very limit of the measurable. His tackling of other questions that are at the moment at the center of attention has proved to be much more significant for applied physics. Thus he was the first to demonstrate the significance of the quantum hypothesis also for the energy of atomic and molecular motions…. That he might sometimes have overshot the target in his speculations, for example in his light quantum hypothesis, should not be counted against him too much.

—PLANCK, NERNST, RUBENS, AND WARBURG, LETTER NOMINATING EINSTEIN FOR THE PRUSSIAN ACADEMY OF SCIENCES, JUNE 12, 1913

“
Why Planck and I engaged him
just as you take on a butler, and now look what a mess he's made of physics; one can't turn one's back for a minute.” This was Nernst's sardonic appraisal of Einstein's triumph with general relativity. The esteemed professors of the Prussian Academy had not brought Einstein to Berlin to monkey around with the law of gravity and the geometry of space-time; they were expecting him to lead them to victory in the race to understand the atom. Instead he had ceded the inside rail to the Dane Niels Bohr, and his English collaborators, just as the Great War had begun. Moreover they, along with most of the physics community, had never accepted Einstein's notion of a new theory in which wave-particle duality would naturally emerge and light would really have particulate properties.

One might have thought that Bohr's theory would have advanced the belief in photons. Bohr's second postulate stated that when an electron makes a transition between two of its allowed orbits, the result is “the emission of homogeneous radiation, for which the relation between the frequency and the amount of energy emitted is the one given by Planck's theory [
ε
=
hυ
].” If one squinted just a little at that sentence (as indeed one often had to do with Bohr's writing), it sure seemed like he was saying that the atom emitted one Einsteinian light quantum every time an electron changed its energy state. But Bohr did
not
mean to say this and in fact remained opposed to the particulate concept of light for the next twelve years. He believed that quantum mechanics governed the atom but that the light emitted in the transition between allowed orbits, while it contained the fixed quantum of energy,
hυ
, was nonetheless solely an electromagnetic wave.

Einstein himself had been chastened by his failure to find quanta in a modified Maxwellian electrodynamics and since 1911 had spoken of photons only sotto voce. In May of 1912 he expressed his position to Wien: “
One cannot seriously believe
in the existence of countable quanta, since the interference properties of light emitted by a luminous point in different directions are not compatible with it. Nevertheless, I still prefer the “honest” theory of quanta to the hitherto found compromises meant as its replacements.” Einstein here identifies a puzzle that he had been struggling with for several years already in 1912. In classical electromagnetism a point source of radiation (such as an atom) emits a uniform spherical wave of light traveling outward in all directions; nonetheless if one refocuses some of these diverging rays to a screen, they will show standard interference patterns. Einstein and Lorentz had already agreed that each single light quantum was capable of interfering with itself, so such a picture suggested that light quanta from a point source must “break up” in some manner, belying their existence as “countable quanta.” Another issue Einstein had pondered was the phenomenon of radioactivity, in which one or more particles is emitted from the nucleus, apparently at a random time and in a random direction. Already in 1911 he saw a parallel here with electromagnetic radiation, writing to Besso, “
the process of absorption
[of
light] … really does have similarity with a radioactive process.” His close friend and future Nobel laureate Otto Stern recalls how much this problem vexed him during his Prague years: “
Einstein always wracked his brain
about the law of radioactive decay. He constructed such models.” After Bohr's eye-opening atomic theory, it struck him that the new picture of emission and absorption could permit these troublesome puzzle pieces to fit together.

In February of 1916 Einstein had already put general relativity aside and begun to catch up on the quantum theory of atoms. Sommerfeld had written Einstein in December of 1915 to ask him to look at an improvement he had made in Bohr's formulas that Einstein would find particularly interesting. Sommerfeld had realized that there was no need for Bohr to restrict the electrons to circular orbits; they could also move on elliptical orbits (as do all the planets
¹
in our solar system). He then used a variant on Bohr's approach, which he had invented, to determine the quantized energies of the allowed orbits.
²
He found a more general formula than Bohr's that initially gave the same results for the hydrogen spectra. But in his new approach he was able to take into account an additional effect that Bohr could not. Einstein's special relativity theory predicts that the measured mass of an electron will increase with increasing velocity. Electrons whizzing around the nucleus were calculated to be moving at a significant fraction of the speed of light, and so this increase in mass should have a measurable effect on the electron's orbital frequency. Including this effect, as Sommerfeld now had done, caused the spectral lines in the hydrogen series to split into closely spaced groups of lines (“fine structures”), their number depending on the final state of the electron after light has been emitted. Sommerfeld wondered in his letter to Einstein whether the newly minted general theory of relativity would affect
his calculations, but Einstein assured him that these effects were too small to matter in this context. The “fine-structure” effect had been previously seen and, just at that time, had been measured carefully by the noted German spectroscopist Friedrich Paschen (after whom one of the original hydrogen series was named). In late December of 1915 he wrote to Sommerfeld, “
My measurements … agree everywhere
most beautifully with your fine structures.” The experimentalist was so delighted by this transformation of an experimental anomaly into an important discovery that he is reported to have loudly exclaimed, “
Now I believe in relativity theory
!”

It was this beautiful marriage of Bohr's atomic quantum theory and relativity principles that so impressed Einstein, once he had digested it; this is the work that in February of 1916 he called “
a revelation
” in an ecstatic letter to Sommerfeld. Later, in August, in the midst of his own work, inspired by Bohr's theory, he wrote Sommerfeld again to say, “
your spectral analyses
are among my finest experiences in physics. It is just through them that Bohr's idea becomes entirely convincing. If only I knew which little screws the Lord is adjusting there!” By then he had already shown, in a paper submitted on July 17, that Bohr's single postulate, that electrons make transitions between allowed stationary energy states via absorption and emission of radiation of energy
hυ
, had remarkable implications.

In this first paper of 1916, titled “Emission and Absorption of Radiation in Quantum Theory,” he returns to the theme he first expounded in 1909, that Planck's derivation of the blackbody law is contradictory because it
uses
classical electrodynamics to relate the mean energy of an oscillator to the energy density of the radiation field but then
departs
from classical physics to calculate this mean energy according to a quantum prescription. He again praises Planck's courage to leap into the unknown—”
his derivation was of unparalleled boldness
—but adds, “however it remains unsatisfactory that the electromagnetic-mechanical analysis [used] is incompatible with quantum theory.” He continues: “Since Bohr's theory of spectra has achieved its great successes, it seems no longer doubtful that the basic idea of quantum theory must be maintained.” In the interests of consistency, he
says, Planck's classical assumptions must be “replaced by quantum-theoretical contemplations on the interaction between matter and radiation. In this endeavor I feel galvanized by the following consideration, which is attractive both for its simplicity and generality.”

The simple, general consideration he mentions flows from the concept of thermal equilibrium, which we have encountered earlier. The blackbody law holds for radiation in contact with matter, so that the entire system (radiation + matter) has settled into the most probable thermodynamic state. In the thermal equilibrium state the entropy of the system is at its maximum value, its temperature is no longer changing, and the average energy of both the radiation and the matter is not changing in time.
³
But since the matter and radiation are continually exchanging energy, this state of equilibrium is not to be conceived of as the absence of interaction but rather as continually compensating change. One can imagine the two systems (matter and radiation) as two swimming pools connected by pipes, so that water is flowing from one pool into the other through certain pipes and being pumped back into the first pool through other pipes, and on average the level of water [energy] in each pool does not change.