Uncertainty

by David Lindley

  Arriving in Copenhagen, Slater found his embryonic hypothesis warmly received. Bohr and Kramers particularly seized on the suggestion of an underlying field that somehow interacted with atoms, determining how and when they would emit or absorb light. They were not so keen, however, on Slater’s idea that the radiation field would also guide the passage of light quanta. They went to work on the young visitor, reiterating by ceaseless tag-team argument how his clever idea could be reshaped into an acceptable theory. The three of them began working on a paper together. That is, Bohr mused out loud, Kramers jotted notes as best he could, and Slater stood expectantly by. In letters home, Slater said how thrilled he was that his ideas were being taken seriously by no less than Bohr. He was confident he would see the finished paper fairly soon, he added. By the end of January 1924 it had been sent out for publication—an astonishingly fast piece of work for anything with Bohr’s name on it. Bohr, Kramers, and Slater was the order of the authors.

  Characteristically, the BKS paper offers not a tightly constructed quantitative model but a nonmathematical sketch, an outline of a possible theory. It contains but one exceedingly simple equation. Instead, the paper describes in purely qualitative terms a new kind of radiation field that surrounds atoms, influences their absorption and emission of light, and also transports energy between them.

  There’s another new ingredient, not original to BKS but adapted from an earlier suggestion. As Bohr had explained to his audience at Yale, the idea of electrons orbiting nuclei in planetary style could no longer be taken seriously, but no one had come up with any better account. So BKS used a subterfuge. They pictured an atom as a set of “virtual oscillators,” each one corresponding to a particular spectroscopic line. In elementary terms, all simple oscillators—a pendulum, a weight on a spring, an electron zipping around and around—obey basically the same mathematical law. To avoid specifics, BKS made use of the standard physics of oscillating systems without trying to connect the presumed oscillators to some explicit picture of how electrons actually moved within an atom. This was all in keeping with the spirit of BKS, which clearly aimed to offer a blueprint of a possible theory, not a finished model.

  A summary sentence from the BKS paper expresses both the vague nature of their proposal and the frustratingly elusive style of Bohr’s prose: “We will assume that a given atom in a certain stationary state will communicate continually with other atoms through a time-spatial mechanism which is virtually equivalent with the field of radiation which on the classical theory would originate from the virtual harmonic oscillators corresponding with the various possible transitions to other stationary states.”

  Bohr seems to think, as lawyers do, that punctuation can only create ambiguity. What’s also remarkable, on close inspection, is how nebulous this language is. Crucial arguments are expressed in conditional tenses and rely on locutions of deliberately vague intent: communicate with, time-spatial mechanism, virtually equivalent…The finicky care with which each phrase was written and rewritten and written again is apparent, yet the curious result is that the more carefully Bohr tries to express himself, the more his meaning recedes. As Einstein once put it, Bohr “utters his opinions like one perpetually groping and never like one who believes he is in possession of the definite truth.” This was meant as praise, evidently, but a later collaborator admitted that Bohr’s manner had its downside: “You could never pin Bohr down to any statement; he would always give the impression of being evasive, and to an outsider who didn’t know him, he would make a very poor showing.”

  Among the maddeningly vague proposals BKS freely dispensed, one blunt conclusion stood out: according to their theory, energy was not absolutely conserved. Because the emission and absorption of energy run according to rules of probability, energy can disappear from one place and reappear somewhere else—or vice versa—without the one event being strictly connected, by old-fashioned cause and effect, to the other. The mysterious radiation field acts as a sort of escrow account for energy, so that the sums always add up in the long run, but in the short term there can be temporary deposits and overdrafts.

  So eager was Bohr to banish all reference to Einstein’s light quanta, because he wanted to preserve classical wave theory, that he ended up throwing classical energy conservation out the window instead. Clearly, there would be no easy reconciliation of these contradictory ideas.

  In an odd display of timorousness, probably because he knew what the answer would be, Bohr didn’t approach Einstein directly but asked Pauli to find out what the old man thought of BKS. “Quite artificial” and even “dégoûtant” (he used the French word) was how Einstein judged the proposal, Pauli reported, adding for good measure that he also completely disapproved of it. And to Max Born, Einstein wrote, “I would rather be a cobbler or even a casino worker than a physicist” if this was where theory was headed. Born himself, asked many years later about BKS, turned the question back to his interviewer: “Can you explain to me what the BKS theory was? It was a thing I never grasped properly in my whole life.”

  It had but a brief existence. Bohr, Kramers, and Slater were obliged to argue that Compton’s results proved only a statistical truth. Individual collisions between X-rays and electrons would not necessarily conserve energy, but in bulk any discrepancies would cancel out. But new experiments, by Compton and others, quickly proved this assertion false. Individual collisions obey precisely the expected rule and conserve energy exactly.

  By the spring of 1925, Bohr admitted that BKS was a bust. Slater remained embittered all his life at the way his idea had been mangled, he said later, into something he didn’t truly endorse. For Kramers, the failure of BKS, following on the forcible suppression of his discovery of Compton scattering, seemed to mark the end of any ambition that he would one day produce truly great physics. He fell into a mild depression, according to his biographer, and was thereafter more subdued in the employment of his scientific imagination.

  The BKS proposal marks, despite all this, a turning point. Depending on one’s interpretation of what the theory actually was, it was either the last gasp of attempts to rest quantum theory on some sort of classical foundation or else the first proof that all such efforts were doomed.

  The most influential ingredient of BKS, in retrospect, was one that entered into the argument—not unlike Planck’s original proposal of the energy quantum—as a sort of trick to get around other difficulties. That was the use of ill-defined virtual oscillators as a means to talk about how an atom emitted and absorbed light while deliberately avoiding any discussion of what precisely the electrons in the atom were doing.

  Developing this idea, Kramers proved a little later—in a rigorous mathematical way, as contrasted to the woolly conceptualizing of BKS—that the oscillator picture was far more than a convenient dodge. An atom’s interaction with light of any frequency, Kramers demonstrated, could be calculated in its entirety from the appropriate set of virtual oscillators. All the necessary physics was in there.

  But did that mean that the old imagery of electron orbits could be dispensed with altogether? Kramers, apparently, thought not. The virtual oscillators were merely an interim substitute, he believed, for the details of an underlying atomic model that would work on more or less traditional lines.

  Others took an opposite view. Writing to Bohr, Pauli posed the crucial issue: “It seems to me the most important question is this: to what extent it’s allowable to speak at all of definite orbits of electrons…In my view Heisenberg has taken exactly the right position on this point, in that he doubts it’s possible to speak of definite orbits. Kramers has thus far never admitted to me any such doubt as reasonable.”

  Seeing Kramers’s theory of virtual oscillators, Heisenberg had indeed quickly perceived its revolutionary implications, and just as quickly determined to release the idea from its traditional moorings. It was he who would transform this bold conceptual innovation into a wholly new theory of atoms—in fact, of physics.

  Chapter 9 />
  SOMETHING HAS HAPPENED

  When Sommerfeld came back from Madison in the spring of 1923, Heisenberg returned to Munich from Göttingen to finish his doctorate. To that end he had pursued a project in mathematical fluid dynamics, unrelated to quantum theory but a steady topic. His doctoral examination was nonetheless a struggle. Because he had to show mastery of physics in general, experimental as well as theoretical, Heisenberg had grudgingly enrolled in a laboratory course under the supervision of Wilhelm Wien, professor of experimental physics at Munich. Wien was a distinguished researcher whose careful measurements of the spectrum of electromagnetic radiation had been crucial to Planck’s 1900 introduction of the quantum hypothesis. But the curmudgeonly Wien, conservative in science as well as politics, was skeptical about Planck’s innovation and openly detested the quantum theory of the atom that his colleague Sommerfeld was forging.

  Wien was thus naturally disposed to show some hostility toward Sommerfeld’s latest wunderkind, and the young man’s ill-concealed disdain for experimental matters only made things worse. At Heisenberg’s oral exam in July, Wien pelted the candidate with questions about his laboratory work that he should have been able to answer easily enough but, through his own neglect and indifference, was not. Wien wanted to know the resolving power of a certain optical device. Heisenberg couldn’t recall the textbook formula, tried to work it out on the spot, and got it wrong. Wien was appalled. Only after tense negotiation with Sommerfeld would he reluctantly affirm that Heisenberg had shown an adequate knowledge of the broad range of physics. The brilliant young man got his doctorate, but with a grade barely above a mere pass.

  Momentarily abashed, Heisenberg went quickly to Göttingen to confirm that his previously agreed plan to spend the following year there was still acceptable to Born. It was, and Heisenberg immediately left for Finland with his Pfadfinder comrades to refresh his spirit among the northern lakes and forests. By September he was in Göttingen, eager to put his doctorate behind him, along with all pretense of any interest in experimental physics, and apply himself to the perplexing array of puzzles that threatened to bring quantum theory to a standstill.

  Heisenberg was gathering clues. In March of the following year, he made a short visit to Copenhagen—his first—where he found Bohr and Kramers in the thick of their enthusiasm over the BKS proposal. Though he balked at that idea in its entirety, one part of the sketchy theory—the virtual oscillators—lodged in his brain. There was at this time not even a halfway reasonable account of how electrons in atoms behaved. So it seemed like an ingenious stratagem, and perhaps more than just a stratagem, to put aside all those irksome technical concerns about electron orbits, and instead think of the atom as a collection of oscillators tuned to the appropriate spectroscopic frequencies.

  No one could say what these oscillators were supposed to be, in detailed physical terms. But that was precisely the point. The oscillators were intended to capture the observed characteristics of atoms, not their internal structure, which—as Bohr had been cryptically hinting for some time—might not be amenable to model building in the traditional way. By thinking in terms of these oscillators, theorists gave themselves some breathing room.

  Back in Göttingen, meanwhile, Born was hatching his own plan. He published a paper calling for a new system of “quantum mechanics”—the first appearance of that term—by which he meant a structure of quantum rules obeying their own logic, not necessarily following the time-honored dictates of classical, Newtonian mechanics. For Born, sketchy hypotheses and far-flung analogies of the BKS type were no good. He relied on mathematics to light the way ahead, and he had a particular trick in mind.

  The language of classical physics is the differential calculus devised by Newton and independently by Leibniz to deal with continuous variation and incremental change. But in trying to understand the workings of atoms, physicists came up against phenomena that were abrupt, spontaneous, and discontinuous. An atom was in one state, then it was in another. There was no smooth passage between the two. Traditional calculus could not cope with such discontinuities. So Born, making a virtue of necessity, proposed instead to substitute a calculus of differences—a mathematical system that would take for its basic elements the differences between states rather than the states themselves.

  This, Heisenberg could see, bore some relation to what Kramers was doing with his virtual oscillators. Both approaches brought the transitions between states to center stage and pushed the underlying states into the wings. Digesting these ideas, Heisenberg came up with an ingenious argument that justified theoretically one of the peculiar half-quantum formulas he and Landé had divined empirically some time ago. A small step, of uncertain significance, but perhaps a step in the right direction.

  Then came a lull. During the summer of 1924 Heisenberg went off again with the Pfadfinder, this time to Bavaria. Bohr, worn out by the years of effort establishing and now running his new institute, took the summer off to relax in the Swiss Alps and at his country cottage outside Copenhagen. Rutherford, no lazybones, commended Bohr for having the sense to take a long break. For the remainder of 1924, and into the following year, quantum mechanics stayed hidden.

  Einstein had already told Born he would rather be a cobbler than deal with the kind of physics Bohr, Kramers, and Slater were peddling. He was not the only one threatening to quit the business. Writing to a colleague in May 1925, Pauli moaned that “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 and had never heard of physics. I only hope now that Bohr will save us with some new idea.” (Charlie Chaplin’s movies were all the rage in Germany at the time.)

  But Bohr had his own ideas about where the urgently needed new idea might come from. “Now everything is in Heisenberg’s hands—to find a way out of the difficulties,” he remarked to an American scientist visiting Copenhagen about this time.

  In September 1924, Heisenberg at last went to Copenhagen for an extended stay of several months. He chose to arrive at a time when he knew Kramers would not be there. Older by seven years, and in manner and appearance older still, Kramers was the only young physicist who intimidated Heisenberg a little. Heisenberg played the piano; Kramers played cello and piano. Heisenberg struggled to master Danish and English; Kramers spoke several languages with ease. He was not merely knowledgeable but opinionated too. Where Pauli thought Kramers sharply amusing, Heisenberg found him patronizing. Kramers, Heisenberg said years later (one can almost hear the gritted teeth), “was always a perfect gentleman in every way; he was too much of a gentleman.”

  And, of course, Kramers had been with Bohr for years, a close relationship that Heisenberg could only envy.

  In Copenhagen, Heisenberg started on a research project but soon tangled with both Bohr and Kramers, who scrutinized every publication emerging from the institute and presented Heisenberg with a long list of the deficiencies of the paper he wanted to send out. Heisenberg was “completely shocked,” he recalled. “I got quite furious.” But he fought back. In personal matters he may still have been shy, but in defense of his science he was truculent and determined. He beat back the objections and wrote a paper that Bohr agreed (not without a further exasperating round of revision) should be published. Heisenberg gained confidence from this experience. He also learned that he might be wise sometimes to keep his ideas to himself for a while.

  A paragraph from Pauli to Bohr, written just before Heisenberg’s first trip to Copenhagen, is worth quoting for its insight into Heisenberg’s scientific character:

  Things always go very oddly with him. When I think about his ideas, they strike me as dreadful and I curse to myself over them. He is quite unphilosophical—he pays no attention to working out clear principles or connecting them with existing theory. But when I speak to him I like him very much and I see that he has all kinds of new arguments—in his heart, at least. So then I realize—apart from the fact that he’s personally such a nice fello
w—that he’s really outstanding, a genius even, and I think he can truly move science forward again…Hopefully you and he together can take a big step forward in atomic theory…Hopefully also Heisenberg will come home with a philosophical attitude in his thinking.

  So he did, at least a little. During a brief, awkward collaboration with Kramers, Heisenberg took to heart ever more strongly the picture of an atom as a set of tuned oscillators. His perspective on what a theory of atoms ought to do was evolving rapidly. Forever gone now was the old Sommerfeld style of model, with electrons following well-defined orbits governed by classical mechanics. Of course, Heisenberg had nothing yet to put in place of such thinking. But his focus was inexorably shifting. Worry less about what atoms are. Think more about what they do.

  But a shifting perspective is only helpful if it leads to a real theory. Heisenberg had to find some way to give logical shape to his evolving thoughts. Back in Göttingen again, turning the matter over and over, he found a way forward by jumping into the past. What his roving mind now latched onto was the century-old mathematical machinery of the Fourier series.

  In the classic but pertinent example, any vibration of a violin string, no matter how harsh or discordant, is equal to some weighted combination of the string’s pure tones, its fundamental and harmonics. Heisenberg was already thinking of an atom as a set of oscillators. Now it occurred to him to take that imagery to its fullest conclusion. “The idea suggested itself,” he said in a lecture three decades later, “that one should write down the mechanical laws not as equations for the positions and velocities of the electrons but as equations for the frequencies and amplitudes of their Fourier expansion.”

  Heisenberg’s bland phrase doesn’t begin to convey the bizarre and radical nature of what he was aiming to do. In classical physics, a particle’s position and velocity are its defining characteristics, the basic elements to which the laws of mechanics applied. For electrons in atoms, however, Heisenberg was now proposing to make the frequencies and intensities of the still-hypothetical oscillators the primary elements of a new calculus, so that the position and velocity of electrons would then be defined only secondarily, in terms of the oscillator strengths. This was a revolutionary reversal. Starting with Bohr and Sommerfeld, the central idea of old quantum theory had been to figure out how electrons move in an atom and deduce from those motions the atom’s spectroscopic frequencies. Heisenberg turned this logic exactly backward. The characteristic frequencies would be the basic elements of his atomic physics, and the motion of electrons would be expressed only indirectly.