Einstein and the Quantum

by Stone, A. Douglas

  A touching exchange ensued, in which Einstein insists he should not be a coauthor, “since you have performed the whole work; I feel like an ‘exploiter,’ as the socialists call it.” Schrödinger immediately demurs: “not even jokingly would I have … thought of you as an “exploiter” … one might say: ‘when kings go building, wagoners have work.’ ” On December 4 he sent Einstein a complete draft of the paper with the second author slot blank, but Einstein presented the paper to the Prussian Academy without his own signature. Einstein’s intuition was again right; there were subtle errors in the reasoning, and the approach itself turned out to be unwieldy, in contrast to Einstein’s first approach, which is still found in modern texts.

  However, only a couple of weeks later, Schrödinger had yet another ideal gas paper in the works, which led directly to the wave equation. Just prior to his letter to Einstein in early November, Schrödinger had finally managed to get hold of de Broglie’s thesis, which he had sought because of Einstein’s strong endorsement of it in his work on the quantum gas. Having now got hold of the thesis, Schrödinger tells Einstein that “because of it, section 8 [the wave-particle section] of your second [quantum gas] … paper has also become completely clear to me for the first time.” Two weeks later, writing to another physicist, he remarks that he is very much inclined toward a “return to wave theory.”

  His final ideal gas paper, submitted December 15, just before he left for his Christmas holiday, was a last effort to digest Einstein’s new quantum statistics; it is titled simply “On Einstein’s Gas Theory.” He sets out to derive the same answers as Einstein without accepting the strange state counting of Bose, which he thinks requires too great a “sacrificium intellectus.” The way out, he says, is “nothing else but taking seriously the De Broglie–Einstein wave theory of the moving particles, according to which the particles are nothing more than a kind of ‘white crest’ on a background of wave radiation.” He then formulates the mathematical problem differently from Einstein but shows how to reach exactly the same final equations. From the perspective of a modern physicist10 the two calculations are equivalent and have the same meaning, but Schrödinger felt that by interpreting the fundamental objects as waves instead of as particles the weirdness of indistinguishability was somehow made more palatable. He concludes by speaking of the particles as “signals” or “singularities” embedded in the wave, highly reminiscent of Einstein’s failed ideas of 1909–1910 and his later idea of “ghost fields” guiding the particles developed in the early 1920s. Except that Schrödinger clearly now thinks that the particles are the “ghosts,” the ephemera, since by viewing waves as fundamental he has “explained” Bose-Einstein statistics in a natural way. Later he would say, “wave mechanics was born in statistics.”

  A few days before Christmas 1925 Schrödinger set off to a familiar mountain lodge in the Swiss village of Arosa, determined to find a new equation to describe these matter waves. Although he took his skis (he was an expert alpinist) and was accompanied by an unnamed “old girlfriend,”11 it seems that this trip was really focused on wave equations. De Broglie, despite Einstein’s praise, had not produced a new governing equation, similar to Maxwell’s electromagnetic wave equation, that could predict or explain the remaining mysteries of the atom. He had produced some suggestive mathematical relations, which made contact with the old quantum theory of Bohr and Sommerfeld, but only at the most elementary level. The central mystery of quantum theory, quantization, was not really resolved by de Broglie’s work. Why wasn’t nature continuous? Why are only certain energies allowed for electrons bound to atomic nuclei?

  Schrödinger saw an answer. Classical waves, or vibrations, in a confined medium have certain natural constraints on their properties. Consider a violin string of a certain length, L, clamped down at each end. The notes it can play arise from vibrations of the strings; these vibrations are waves of sideways displacement in the string, but they cannot have a continuously varying wavelength as is possible for waves in an open (essentially infinite) medium. The longest wavelength, λ, they can have is twice the string length. Why is this the longest? First, consider that, for any wave on a string, the string’s displacement must be zero at the points where it is clamped down. The simplest form of displacement the string can have away from these fixed points is for it to be displaced everywhere in the same sideways direction (left or right) at a given instant, so that the maximum displacement is in the middle and it decreases back to zero at both clamped ends. This displacement then oscillates back and forth, causing sound waves of a certain pitch. Since we measure wavelength by the distance between points in the medium that take us through a peak and a trough, this shape corresponds to half the full wavelength, so the wavelength is λ = 2L. This will determine the lowest note the violin can play (for a given string tension). The next-lowest note will have λ = L, implying that there will be no displacement at the center of the string (even though that point isn’t clamped down). In general the only allowed wavelengths are λ = 2L/n, where n is a whole number (n = 1, 2, 3 …). That’s the point: for confined waves, nature produces whole numbers automatically. De Broglie had hinted at this, but now Schrödinger realized that this was the key to getting the quantum into quantum theory.

  The mathematics of the old quantum theory had not done this in a natural way. Bohr and his followers had taken mathematical expressions that are continuous (i.e., don’t involve whole numbers exclusively) and had simply restricted them to whole numbers by fiat. Schrödinger, by contrast, was looking for an equation that simply did not have continuous solutions, one in which each solution would be connected to a whole number organically.

  As noted above, wave equations, through the quantization of the wavelength, have this property. They are differential equations, describing continuous change in space and time; but when the waves are confined, only certain wavelengths are physically possible. So Schrödinger was looking for a matter-wave differential equation to describe electrons: it would have to describe a matter field and an extended “disturbance” of the field that varied in space and time but was confined to the vicinity of the atomic nucleus by the electron’s attraction to the nucleus.

  Maxwell’s electromagnetic wave equation was the only fundamental wave equation of physics at that time, but because photons are massless, it had no safe place for the introduction of Planck’s constant (as Einstein had learned to his dismay fifteen years earlier). Schrödinger’s challenge was to fashion a wave equation, modeled on Maxwell’s, that included Planck’s constant as well as the physical constants e and m, representing the charge and mass of the electron. Maxwell’s wave equation does, however, contain the wavelength, λ, of the EM waves which then lead to quantized values and whole numbers, when it is written for “standing waves” confined to a specific region. So the key idea was to write an equation similar to the electromagnetic wave equation but for a new wave field, a “matter wave” described by a mathematical expression now known as the “wavefunction,” and to replace λ using de Broglie’s relation λ = h/p = h/mv.

  With this approach Schrödinger had an equation containing h, Planck’s constant, and m, the electron mass. The electron velocity, v, can be eliminated from the equation in favor of the difference between its total energy and potential energy. The potential energy of the electron orbiting the nucleus of course depends on its charge, e. The resulting “time-independent Schrödinger equation” for hydrogen contains the “holy trinity,” h, m, and e. And now the moment of triumph: since only certain wavelengths are allowed, that implies that the only unknown in the equation, the total energy of the electron, can only take on certain allowed values. The energy of electrons in an atom is quantized, not by fiat, but due to the fundamental properties of waves confined to a fixed region in space.12

  After returning from his “ski” trip, Schrödinger immediately subjected his new equation to the acid test: could it reproduce the energy levels, grouped into different “shells,” for the case of atomic hydrogen, wh
ich were known from spectral measurements for decades and “explained” in an ad hoc manner by the old quantum theory? The answer was a resounding yes. The details of the calculations took only a few weeks, and on January 27, 1926, the first of Schrödinger’s seminal papers was received at the Annalen der Physik. It states the breakthrough thus: “in this paper, I wish to consider … the simple case of the hydrogen atom … and show that the customary quantum conditions can be replaced by another postulate, in which the notion of ‘whole numbers’ … is not introduced…. The new conception is capable of generalization and strikes, I believe, very deeply at the true nature of the quantum rules.”

  After presenting the detailed solution of his hydrogen equation, he briefly touches on its interpretation and its origins. “It is, of course, strongly suggested that we try to connect the [wavefunction] with some vibration process in the atom, which would more nearly approach reality than the electron orbits [of Bohr-Sommerfeld theory],” but he feels that this is premature, since the theory needs further development. However, “Above all, I wish to mention that I was led to these deliberations by the suggestive papers of M. Louis de Broglie … I have lately shown that the Einstein gas theory can be based on [such] considerations … the above reflections on the atom could have been represented as a generalization from those on the gas model.” Later Schrödinger would say, “My theory was inspired by L. de Broglie … and by brief, yet infinitely far-seeing remarks of A. Einstein.”

  This first paper was followed in rapid succession by five more in just six months, in which Schrödinger, the consummate craftsman, working alone, determined essentially all the known properties of atomic spectra from the solutions of his wave equation. It was a breathtaking display, about which even his competitor Born would later remark, “what is more magnificent in theoretical physics than Schrödinger’s six papers on wave mechanics?” The normally reserved Sommerfeld called Schrödinger’s equation “the most astonishing among all the astonishing discoveries of the twentieth century.” Hence, by June of 1926, physicists had uncovered most of the new laws and mathematical methods necessary to describe physics on the atomic scale; they just didn’t yet know what they meant. However, an interpretation was soon to emerge, one that would challenge the philosophical principles that both Schrödinger and Einstein held dear.

  1 When Einstein held the theoretical physics position is was only at the level of an associate professorship (extraordinarius); it was subsequently upgraded to a full professorship (ordinarius).

  2 Schrödinger’s interest in philosophy was so great that in 1918, before the war ended, he had been planning to “devote himself to philosophy” more than physics, only to find that the chair he expected to receive, in Czernowitz, Ukraine, had disappeared along with Austrian control of the region.

  3 While Schrödinger was discreet in this final public document, in 1933, to his diary, he confided that he never slept with a woman “who did not wish, in consequence, to live with me for all her life.” There is some evidence to back this up.

  4 An English theoretical physicist, Paul Adrien Maurice Dirac, was the last of the trio of wunderkind to play a founding role in quantum mechanics, along with Pauli and Heisenberg. Born in 1902, he was even younger than Heisenberg, and upon hearing Heisenberg speak at Cambridge in July of 1925, he shortly afterward invented his own, mathematically elegant version of the quantum equations. A few years later he discovered the “Dirac Equation,” the quantum wave equation that takes into account the effects of relativity. However, he interacted little with Einstein during the period 1925–26 and so is not very relevant to our historical narrative; Einstein remarked of him in August 1926, “I have trouble with Dirac. This balancing on the dizzying path between genius and madness is awful.”

  5 First in class.

  6 “That is the Schrödinger.”

  7 It is likely that it is at this time he studied and understood Einstein’s 1917 reformulation of the Bohr-Sommerfeld theory, which he later praised so highly.

  8 This is always a great compliment from one physicist to the other, unless the former is angling for priority.

  9 For expert readers, this is the “division by N!” in the partition function (state counting), which is approximately correct at high temperature and is still used in modern texts.

  10 Einstein immediately saw this equivalence, writing to Schrödinger after his paper appeared, “I see no basic difference between your work on the theory of the ideal gas and my own.”

  11 Hermann Weyl, a distinguished physicist and a friend of Schrödinger’s, later famously commented that Schrödinger “did his great work during a late erotic outburst in his life.”

  12 This simple argument, while certainly part of Schrödinger’s reasoning, was not how he first presented the equation and omits his failed initial attempt to come up with an equation consistent with Einstein’s relativity theory.

  CHAPTER 28

  CONFUSION AND THEN UNCERTAINTY

  If we are still going to have to put up with these damn quantum jumps, I am sorry that I ever had anything to do with quantum theory.

  —SCHRÖDINGER TO BOHR, OCTOBER 1926

  “I am convinced that you have made a decisive advance with your formulation of the quantum condition, just as I am equally convinced that the Heisenberg-Born route is off the track.” Thus Einstein wrote to Schrödinger in late April of 1926. The Heisenberg-Born route, a different approach to the “quantum conditions,” introduced the term “quantum mechanics” as a more rigorous replacement for the nebulous conceptual structure of “quantum theory.” This method had begun to bear fruit six months earlier than Schrödinger’s, and unlike his work it arose independently of Einstein’s recent successes with the quantum gas.

  It represented the radical point of view that since atoms were, practically speaking, impossible to observe in space and time, one should stop attempting to describe them by space-time orbits as in classical mechanics. Instead one should develop a description in terms of observable atomic variables, which might not themselves be easily visualized, such as the absorption frequencies for light incident on the atom, and how strongly each frequency was absorbed. The first breakthrough using this approach had come from the twenty-three-year-old prodigy Werner Heisenberg, who formulated his method in July of 1925. By the time of Schrödinger’s work, Einstein had been ambivalently struggling with this new framework for quite some time already, since Heisenberg was working in the research group of his close friend Max Born in Göttingen. Born had immediately informed him of Heisenberg’s initial sighting of a New World of the atom, writing in a letter dated July 15, 1925, that Heisenberg’s paper “appears rather mystifying, but is certainly true and profound.”

  Heisenberg was not the only young genius to find his way to Born’s research team in Göttingen. Six of Born’s research assistants and one of his PhD students would go on to win the Nobel Prize,1 and three of them—Enrico Fermi, Wolfgang Pauli, and Heisenberg—would contribute cornerstones to the rising quantum edifice. Born, only three years younger than Einstein, was from Prussian Silesia, and was of Jewish descent (like many of Einstein’s closest friends). He had been appointed associate professor at Berlin from 1915 to 1919, arriving just in time to observe Einstein’s awe-inspiring success with general relativity theory. He, and his wife Hedwig, formed a lifelong friendship with Einstein, although Born maintained as well a certain reverence for his friend, whom he would refer to, after his death, as “my beloved master.” Born made seminal contributions to physics and eventually won the Nobel Prize himself, but he was not an imposing intellect, and he sometimes had trouble keeping up with his brilliant wards. Of Pauli, who was renowned for his critical brilliance, he said, “I was from the beginning quite crushed by him … he would never do what I told him to do.” Heisenberg, he recalled, was quite different: “he looked like a simple peasant boy, with short, fair hair, clear bright eyes and a charming expression” when he arrived, “very quiet and friendly and shy…. Very soon I discove
red he was just as good in the brains as the other one.”

  After a few months spent visiting Niels Bohr in the fall of 1924, Heisenberg returned to Göttingen with the germ of an idea for a completely new quantum theory of the atom, distinct from the old Bohr-Sommerfeld approach. This approach, while it worked for hydrogen and a few other atoms, appeared to be breaking down for more complicated atoms and molecules. In fact, by 1924 more than a decade had passed since Bohr’s pathbreaking work, and a full quarter century since that of Planck; many physicists were beginning to wonder if the fundamental laws of the atom were simply beyond human ken. In May of 1925 the enfant terrible, Pauli, wrote despairingly to a friend, “right now physics is very confused once again—at any rate it’s much too difficult for me and I wish I were a movie comedian or some such.” However, Heisenberg was just about to shake the field out of its malaise.

  FIGURE 28.1. Werner Heisenberg circa 1927. AIP Emilio Segrè Visual Archives, Segrè Collection.

  Heisenberg’s idea was to take the continuous trajectory of a particle, which in classical physics is represented by the three Cartesian coordinates x, y, z that vary continuously with time, and replace each coordinate with a list of numbers arrayed in rows and columns, rather like a Sudoku puzzle. Each number in the list is not fixed, but oscillates in time sinusoidally, with a characteristic frequency. When applied to electrons in an atom, the frequencies corresponded to the observable “transition frequencies” at which the atom would absorb and emit light. First, however, Heisenberg considered the most basic “toy problem” of mechanics, the familiar linear harmonic oscillator (mass on a spring). He was able to show that using his new definition for position, and a similar one for momentum, the energy of the oscillator was conserved; that is, it didn’t change in time as long as the energy took the special values found by Planck so long ago, quantized in steps of hυ (where υ is the frequency of the oscillator). So here, in Heisenberg’s new arithmetic, the whole numbers of quantum theory also arose naturally from the math and were not imposed externally, just as they would later appear naturally in Schrödinger’s wave approach. Heisenberg first discovered this while recovering from an allergy attack on the North Sea island of Helgoland, and he was so excited that he stayed up all night working, and then, lying on a rock watching the sun rise, he thought to himself, “well something has happened.”