Einstein and the Quantum

by Stone, A. Douglas

  While Thomson was receiving this rude awakening, his former protégé Rutherford had already established experimentally that Thomson’s plum pudding was indigestible. In radioactive decay, doubly ionized helium atoms, known as alpha particles, are ejected from the decaying atom; these can be collimated into a beam that can probe the structure of the atom. Starting in 1909, experiments by Hans Geiger3 and Ernest Marsden, under Rutherford’s supervision, found that alpha particles in such a beam were occasionally knocked completely sideways by a thin foil of metal atoms. As one might imagine from the pudding analogy, a diffuse sphere of positive charge as envisioned by Thomson could never result in such violent redirection of the alpha particle. As Rutherford put it, it was as if one had fired an artillery shell at tissue paper and found it ricocheted back at oneself. Thus there had to be something hard at the center of the atom, and since the electrons were known to be light, it could only be a highly localized and relatively heavy ball of positive charge, which he called the atomic nucleus. (Except for hydrogen, atomic nuclei contain equally heavy neutral particles, neutrons, with no charge, but that fact would not be established for two more decades.)

  Thus when Bohr sat down to explain the mechanics of the atom in 1913, he knew several very important things. One could try to picture the atom as a miniature solar system, with light, negatively charged electrons orbiting around a localized, heavy, positively charged nucleus. The electrons had to orbit, because if they were ever stationary they would be attracted directly into the nucleus. Yet they couldn’t radiate energy while they orbited, as required by classical electrodynamics; otherwise they would eventually collapse into the nucleus anyway. But that was not a problem; that was a feature. He had grasped Lorentz’s point, that his theory could only be successful if it violated some aspect of classical physics, and if it somehow got Planck’s constant, h, into the equations. Blackbody theory had succeeded by using classical physics where it worked and changing the rules where it didn’t. Why not take the same approach to the atom?

  Bohr wrote his seminal paper on atomic spectra in early 1913, having returned to Denmark from England, and he sent it to Rutherford for his criticism. Rutherford was mightily impressed by the conclusions but objected to its length, as well as its wordy and ponderous style. “I really think you should abbreviate some of the discussions to bring it into a more reasonable compass,” he wrote; “as you know, it is the custom in England to put things very shortly and tersely in contrast to the German method, where it appears to be a virtue to be as long-winded as possible.” Far from being cowed by this reaction, Bohr traveled to Manchester and fought successfully for every single word of his manuscript. This semantically rich but equation-poor scientific writing style would be a lifelong characteristic of the Dane.

  Bohr begins by noting that while his mentor, Rutherford, has established experimentally the plausibility of a “solar system” model of the atom, “in an attempt to explain some of the properties of matter on the basis of this atom-model we meet … with difficulties of a serious nature arising from the apparent instability of the system of electrons [due to radiative energy loss]…. The way of considering the problem has, however, undergone essential alterations in recent years owing to the development of the theory of energy radiation, and … affirmation of the new assumptions introduced in this theory, found by experiments … such as specific heat, photoelectric effect, Röntgen rays [x-rays], etc. The result … seems to be a general acknowledgement of the inadequacy of the classical electrodynamics in describing the behavior of systems of atomic size.” Here, after mentioning, Einstein’s two great predictions of experimental quantum behavior, the photoelectric effect and specific heat, he cites the proceedings of the First Solvay Congress. He continues, “Whatever the alteration of the laws of motion of the electrons may be, it seems necessary to introduce into [them] a quantity foreign to the classical electrodynamics, i.e. Planck’s constant.”

  He realizes he is treading in Einstein’s footsteps. “The general importance of Planck’s theory for the discussion of the behavior of atomic systems was originally pointed out by Einstein [he cites Einstein’s 1905 and 1906 papers on light quanta, and his 1907 paper on specific heats]. The considerations of Einstein have been developed and applied to a number of different phenomena, especially by Stark, Nernst, and Sommerfeld.” Note that, as with Nernst in 1910, Einstein, not Planck, is seen as the promulgator of the quantum atom. But unlike Einstein in the years 1908–1911, Bohr will not try to put the constant h into Maxwell’s equations. He will introduce it into classical mechanics by means of two stand-alone postulates, in a move very similar to Planck’s original restriction ε = nhυ (which in Einstein’s hands became quantization of vibrational energy).

  So what was the problem with “Plancking” electron motion in the atom anyway? The problem is expressed by Einstein in his first series of letters to Lorentz: “For me the only difficulty consists in the fact that Planck’s foundation (the introduction of the hυ-quanta) does not apply to the elementary foundation of the theory, but only to the special case of oscillating structures [with a single frequency]. We do not know therefore, and cannot deduce … what kind of electrical and mechanical laws we have to introduce for the free electron … there is no contradiction here, but only the difficulty of generalizing Planck’s approach.” As already noted, Planck had taken the simplest possible mechanical system, a mass on a spring (what physicists call a linear oscillator) and appended the ad hoc rule that its energy could only be an integer times hυ, a quantization rule with no justification in Newtonian physics.

  The reason that a linear oscillator is so simple is that it makes a periodic motion (going back and forth on exactly the same path) and the frequency of this motion is independent of how far you stretch the spring. As discussed earlier, when you stretch the spring farther, you add more total energy to the motion; the extra energy exactly compensates for the extra distance it must travel in one period, so it oscillates over the longer distance in exactly the same time, that is, with the same frequency. So for a linear oscillator the frequency of oscillation is a constant, independent of the energy of the oscillator, and the rule ε = nhυ determines what energies are allowed. When a Planck oscillator emits a light quantum, hυ, it loses one quantum of energy, now having ε = (n –1)hυ, but the frequency of motion doesn’t change, because it is a linear oscillator. The same frequency, υ, describes both the frequency of the electronic motion and the frequency of the emitted light, just as one would expect in classical electrodynamics: oscillating charges produce radiation at their oscillation frequency.

  But this single-frequency behavior of the linear oscillator is due to its simple force law; the restoring force of the spring (which pulls it back and forth) is proportional to the distance the spring is stretched. The force on an electron in a nuclear atom is not of that kind; Bohr assumed (correctly) that it was the inverse square law of electrical attraction between oppositely charged particles. At least for the simplest case, the hydrogen atom (one electron moving around one proton), this force law leads to periodic orbital motion,4 but the frequency of the motion changes continuously as the energy is changed. For example, two electrons in circular orbits of different orbital radii not only have different “binding energies” to the nucleus but also have different orbital frequencies. Since the math involved is exactly the same as for a planet orbiting the Sun, all the equations for how this works were well known from classical celestial mechanics. In particular, planets farther away from the Sun are less strongly attracted to it, have less binding energy, and hence orbit the Sun with a longer period (lower frequency); Pluto’s year, for instance, is 248 Earth years. Bohr was prepared to use the same principles of classical mechanics for electron orbits; but then how does one implement a quantum rule for the allowed energies when, unlike for an oscillator, the frequency changes when the energy does?

  Let’s call the electron orbital frequencies f to distinguish them from the frequencies of light the atom might em
it, which we will still call υ. By a rather tenuous argument, Bohr came up with a surprising answer. The allowed energies for electrons orbiting the nucleus were given by a whole number times Planck’s constant times half the final orbital frequency.5 Planck’s ε = nhf for oscillators became Bohr’s ε = nhf/2, for electron orbits in hydrogen. The unique circular orbits picked out by this rule were termed “stationary states,” which were postulated to remain stable forever, without radiating energy. Since, as noted, the frequencies of the orbits themselves vary with the energy of orbit, this rule does not lead to equally spaced energy levels, in contrast to what Planck had found for oscillators.

  In coming up with this less-than-obvious generalization of Planck’s rule, Bohr was guided by remarkable intuition, and by a critical observation about the spectra of light emitted from hydrogen gas. It had been known for a very long time that the light emitted from atomic gases does not consist of a continuous “rainbow” of colors but instead has a few very pure colors. When this light was focused onto an optical device called a diffraction grating, which functions essentially like a prism, the different colors could be separated from one another in space and then projected onto a screen, giving rise to narrow bands, or “spectral lines,” of color. Since different elements give different sequences of colors (lines), one could use such spectra to identify which elements were emitting the light. This was, of course, of great importance for disciplines such as physical chemistry and also for astronomy, where it could be used to identify the elements present in distant stars. But why certain lines appeared for a certain element, and why the lines were spaced in certain patterns, were completely mysterious from the point of view of fundamental physics.

  In 1888 the Swedish physicist Johannes Rydberg had noted that the frequencies of hydrogen spectral lines seemed particularly simple: they formed a series that could be generated by a simple arithmetic formula.6 Bohr realized that the specific generalization of Planck’s rule that he had come up with, combined with standard classical formulas relating the frequency of an orbit to its energy, would give rise to Rydberg’s spectral series. That looked extremely promising.

  But he had to add one more critical step to explain Rydberg’s observations. Rydberg did not directly observe electron energies; he observed the frequencies of the light emitted from the atom, presumably as a consequence of the change of energy state of the orbiting electron. Bohr postulated that when the electron absorbed or emitted energy by changing its orbital state, the frequency, υ, of the light that was emitted or absorbed was determined by the Planck/Einstein rule: ε1 − ε2 = hυ, where ε1 is the initial energy of the electron and ε2 is its final energy. This assumption, combined with his rule for quantizing electron energies described above, implied that only certain frequencies of light could be emitted or absorbed by hydrogen atoms, exactly those observed by Rydberg!

  That was the good news. There was also bad news. With this argument Bohr did something so radical that even Einstein, the Swabian rebel, had found it inconceivable; Bohr disassociated the frequency of the light emitted by the atom from the frequency at which the electron orbited the atom. In the Bohr formula, ε1 − ε2 = hυ, there are two electron frequencies, that of the electron in its initial orbit and that of the electron in its final orbit; neither of these frequencies coincides with the frequency, υ, of the emitted radiation! This was a pretty crazy notion to a classical physicist, for whom light was created by the acceleration of charges and must necessarily mirror the frequency of the charge motion. Bohr admitted as much: “How much the above interpretation differs from an interpretation based on the ordinary electrodynamics is perhaps most clearly shown by the fact that we have been forced to assume that a system of electrons will absorb radiation of a frequency different from the frequency of vibration of electrons calculated in the ordinary way.” However, he noted, using his new rule, “obviously, we get in this way the same expression for the kinetic energy of an electron ejected from an atom by photo-electron effect as that deduced by Einstein.” So, as a final justification, he relied on exactly the experimental evidence that motivated Einstein’s light-quantum hypothesis and inaugurated the search for a new atomic mechanics.

  But Bohr’s atomic theory was hardly the new mechanics for which Einstein had been searching. There was still no underlying principle to replace classical mechanics, just another ad hoc restriction on classical orbits, a variant on Planck’s desperate hypothesis. So be it, Bohr decided; his gambit was not elegant, but it worked! Not only did it explain the hydrogen spectrum for visible light, which consisted of two different series named for their discoverers, the Balmer and Paschen series; it also predicted a series of ultraviolet lines that had not been observed. Unknown to Bohr, the American physicist Theodore Lyman had already begun detecting this series during the preceding few years, and his results, shortly to be published, would be found to agree with the Bohr formula. But the pièce de résistance was the application of Bohr’s formula to puzzling astrophysical spectral lines. A new series of spectral lines had been observed in the blue supergiant star ζ-Puppis by the astrophysicist E. C. Pickering and subsequently produced on earth, in helium-hydrogen mixtures, by another astrophysicist, Alfred Fowler. These lines corresponded to every other line in the Balmer series for hydrogen and, for want of a better explanation, had been tentatively assigned to hydrogen. Bohr realized that these lines were explained perfectly by his formula. All he had to do was assume that they arose from singly ionized helium atoms in the star’s photosphere. Singly ionized helium has only a single electron, like hydrogen, but of course it has a larger nuclear charge and mass; simply plugging into Bohr’s formula the new values of charge and mass gave the observed spectral lines.

  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 mirac
le—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/2π, 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.