Uncertainty

by David Lindley

  He was a shy, careful child. During the war, his character began to emerge. Charged with adult responsibilities in the local militia, Werner discovered a certain charisma, an ability to lead and command respect, if not affection. Away from his straitlaced family, he found room to breathe among loose organizations of young men who tramped the mountains, hiked the countryside, and lost themselves in earnest adolescent discussions of art and science, music and philosophy. Such groups went back a couple of decades and belonged to larger associations with such names as Pfadfinder (pathfinder) and Wandervogel (migratory bird). Modeled on the Boy Scout movement recently started by Baden-Powell in Great Britain, the German groups tended to be more romantic in spirit than their hearty, practical British counterparts. After the war especially they became a repository for all kinds of wistful, wishful thinking about a new and peaceful society. As Heisenberg put it, “the cocoon in which home and school protect the young in more peaceful periods had burst open in the confusion of the times, and…by way of substitute, we had discovered a new sense of freedom.”

  The youth movement was at heart adolescent and middleclass, an indulgence available only to the fortunate. Thomas Mann, in Doctor Faustus, described similarly earnest bucolic pilgrimages by young students and commented sharply that “such a temporary style of life, when a city-dweller who engages in intellectual pursuits becomes a stopover guest at some primitive rural spot of Mother Earth’s…has something artificial, patronizing, dilettantish about it, a trace of the comic.”

  In some of these youth organizations lay seeds that would grow into the strident and violent Hitler Youth of a decade or so later. But Heisenberg’s group remained assiduously apolitical, and his wanderings (there were trips as far afield as Austria and Finland) represented for him a solace that he clung to even as his scientific career blossomed. All his life Heisenberg wanted to believe that he could deal with the nasty exigencies of political strife by looking the other way and retreating into nature.

  In 1920 Heisenberg’s father arranged for Werner to interview with Ferdinand Lindemann, an elderly mathematics professor at Munich. Years before, Lindemann had opposed Sommerfeld’s appointment on the grounds that an applied mathematician who dabbled in physics was a wretched creature indeed. He occupied a gloomy office crammed with furniture of dated design. On the desk sat a small black dog, which glared at the young supplicant and proceeded to yap ever more loudly as Lindemann attempted to probe Heisenberg’s interests and knowledge. Through the din, Heisenberg managed to confess nervously that he had been reading about relativity. “In that case you are completely lost to mathematics,” said Lindemann, concluding the interview.

  So Heisenberg went to see Sommerfeld instead and got a warmer, though not uncritical, reception. Sommerfeld was impressed by Heisenberg’s command of mathematics and his interest in current physics, but disturbed by the candidate’s apparent concern for philosophical questions rather than the scientific fundamentals of experiment and theory, which Heisenberg seemed to find insufficiently grand. Walk before you run was the essence of Sommerfeld’s advice: if you want to tackle the deep questions, you have to gain mastery of the subject first. Heisenberg went away thinking that physics might turn out to be a little tedious. With his yough movement friends he wrangled over big issues: What is knowledge? How can we be sure of it? What constitutes progress? Sommerfeld wanted him to learn about the fine structure of spectral lines in hydrogen and the anomalous Zeeman effect in the alkali metals. Nonetheless, Heisenberg signed on to study physics with Sommerfeld.

  For his thesis work, he took up a safe problem in the classical physics of fluid flow, but that was a mere sideshow compared to his rapid immersion in quantum theory. Heisenberg was not nearly as well schooled in physics as Pauli, but perhaps for that very reason he was less hidebound, less apt instantly to see the difficulties rather than the possibilities of strange but promising suggestions.

  Pauli told Heisenberg that once you had the right mathematics, you had all you needed; you could pose problems and calculate answers. But Heisenberg wanted something more, a more elemental or visceral understanding. Of the quantum atom they were trying to elucidate, he told Pauli, “I have grasped the theory with my brain, but not yet with my heart.” The Bohr-Sommerfeld atom of those days was, he said, a “peculiar mixture of incomprehensible mumbo-jumbo and empirical success.”

  But this mumbo-jumbo was clearly the most exciting part of physics to be involved in. Sommerfeld introduced Heisenberg to the fourth quantum number he had recently devised, and asked his new student to see if he could expand the scheme to embrace more of the oddities of the anomalous Zeeman effect. Ingenious and resourceful, showing both technical skill and scientific imagination, Heisenberg did what his teacher asked—and came up with a result that startled them both. In trying to account for a greater variety of spectral lines, Heisenberg devised a clever formula that worked nicely as long as he gave the already mysterious fourth quantum number half values: 1/2, 3/2, 5/2, and so on. (It would do no good to multiply by 2 to get rid of the fractions, because then the sequence would be 1, 3, 5…missing the even numbers.)

  This Sommerfeld was not prepared to contemplate. A half quantum went against the whole point of the enterprise. Pauli agreed. Once you allowed halves, you would open the door to quarters and eighths, he said, and soon there would be no quantum theory left.

  While Heisenberg and Sommerfeld wrestled over this bizarre proposal, they were taken aback to see essentially the same idea put into print by another young German, Alfred Landé. Landé had first learned of quantum theory as a student in Göttingen, during Niels Bohr’s prewar visit. Like Heisenberg, Landé offered no justification for the half-quantum trick, except that it seemed to account for a couple of interesting puzzles.

  Irked by his loss of priority, Heisenberg now tried to recapture the lead by attempting a theory of the half quantum. Sommerfeld had suggested that the fourth number had something to do with the rotation of an outer electron relative to the atom’s core. Heisenberg insouciantly went further and proposed that this rotation could somehow be split into half units, one part belonging to the electron, the other to the core. When this outer electron made a transition, only a half quantum of rotation would come into play.

  Heisenberg was enraptured by his own ingenuity, but neither Sommerfeld nor Pauli took to his idea. It was adventurous and imaginative, to be sure—or speculative and unfounded, to put it another way. Even so, Sommerfeld consented to let this paper go to a journal, where it became Heisenberg’s first published work. Landé didn’t think much of the idea either, and wrote to Heisenberg pointing out that his theory effectively threw away the sacred principle of conservation of angular momentum. Heisenberg didn’t much care. All the old rules were up for grabs. As Landé put it many years later, Heisenberg’s strategy, when he ran up against a difficult problem, was not to look strenuously for a solution within the confines of known physics but immediately to search for something wholly new, something radical. This attitude would bring the young man great success, but it could mis-fire too.

  Sommerfeld likewise judged Heisenberg truly smart but alarmingly cavalier. In a word, immature. He thought enough of his young pupil’s work to write to Einstein about it, praising Heisenberg’s attempted theory but admitting to reservations. “It works fine, but the foundation of it is quite unclear,” he said. “I can only press on with the technicalities of quanta; you must make your philosophy.”

  Remarkable or foolish or both, Heisenberg’s first attempt at theoretical physics upended his earlier attitude. He now saw that progress came not from pondering weighty philosophical issues but from trying to solve specific problems. And it was good to keep one’s mind open to new thinking. Pauli’s jibe had some truth. Heisenberg didn’t know enough physics to see how absurd his half-quantum theory was. But then, Heisenberg already suspected, Sommerfeld was inclined to be too cautious, and Pauli too skeptical. Many years later Heisenberg met the great American physicist Richard Feynma
n, who lamented that young physicists were no longer allowed the luxury of making mistakes. Their teachers and colleagues jumped ferociously on any unsound reasoning before it had a chance to blossom. But sometimes, Feynman told Heisenberg, he might have an idea that he knew didn’t make sense, but “damn it, I can see that it’s right.”

  Under Sommerfeld’s guidance, Heisenberg had the profoundly valuable experience of seeing his first idea in physics, inspired but controversial, thrown into the open and left to fend for itself. It was exhilarating. Criticism only spurred Heisenberg to keep at it. He had found his path. The classical order was disintegrating, and Heisenberg would join the search for a new system. In physics as in politics, the young man had no nostalgia for old certainties.

  Chapter 7

  HOW CAN ONE BE HAPPY?

  During the summer of 1922, Germany enjoyed a momentary calm. Food was scarce, but few starved. Money was tight, but the hyperinflation that obliged people to cart around billions of marks’ worth of scruffy notes in wheelbarrows to buy bread and milk had not yet caught fire. In Göttingen, the weather was gorgeous, and it was there that theorists gathered in June to listen to a series of lectures on quantum theory from the subject’s acknowledged guide and master, Niels Bohr. Sommerfeld naturally went, and he insisted that his precocious and already controversial pupil Heisenberg come too. Even in the relatively well-to-do Heisenberg family there was little spare money to go around, so Sommerfeld paid for Werner’s trip to Göttingen out of his own pocket. Heisenberg slept on someone’s couch and was constantly hungry. But that was nothing unusual for students in those days, he recalled.

  Pauli was also there. After taking his doctorate at Munich the previous autumn, he had spent the winter semester in Göttingen, then had moved on to a position in Hamburg. Now he journeyed south to meet Bohr for the first time.

  Bohr’s visit was significant politically as well as scientifically. Like Einstein, Bohr detested German militarism and imperialism but did not approve of the postwar attempt to isolate German science from the rest of the world. A furious insistence on vengeance would not engender peace.

  Bohr had already begun to reestablish contacts in Germany. He had visited Berlin in 1920 at the invitation of Planck and Einstein. It was Bohr’s first meeting with these two towering figures, and both found the young Dane admirable. Einstein and Bohr subsequently exchanged little mash notes. “Not often in life has a person delighted me so by his mere presence,” wrote Einstein to Bohr. “I am now studying your great papers and—if I happen to get stuck somewhere—I have the pleasure of seeing your friendly young face before me, smiling and explaining.” “It was one of the greatest experiences in my life to meet you and speak with you,” Bohr responded. “I shall never forget our conversation on the way from Dahlem to your home.”

  Two years later, when Bohr visited Göttingen, some of the old stuffiness of that university had dissipated. The new head of theoretical physics was Max Born, who eight years earlier, during Bohr’s prewar visit, had been one of the eager young scientists in the back of the audience. Born possessed a measure of the Göttingen fondness for mathematical rigor, but he had embraced the wondrous new physics despite its sloppiness and inconsistency.

  In the lovely June weather of 1922, Bohr delivered, in his discursive, Delphic style, a series of lectures setting out the view of quantum theory from Copenhagen. The Bohr Festspiele this luminous week was later dubbed, echoing a Handel Festspiele that played in Göttingen about the same time.

  The windows were open, and summer noises drifted into the hushed seminar room. One local member of the audience complained that the senior Göttingen faculty members, as before, bagged the best seats at the front, leaving the junior scientists to gather at the back, where they struggled to catch Bohr’s slow, indistinct words. But Heisenberg was entranced. He had learned his quantum physics from Sommerfeld, whose technical style emphasized simple models and elementary calculations. Of the master’s voice, by contrast, Heisenberg said that “each one of his sentences revealed a long chain of underlying thoughts, of philosophical reflections, hinted at but never fully expressed…It all sounded quite different from Bohr’s lips.”

  Bohr talked about some recent ideas that he and an assistant had been developing in Copenhagen. Heisenberg, who had read the papers and criticized them with Pauli, had the temerity to speak up from the back of the room with his objections. This caused the gentlemen at the front to turn their heads. Bohr knew of Heisenberg by name, from his work on the rather detestable half-quantum idea, and after the lecture was over he invited the young man on a long walk. They strolled up the Hainberg, the modest hill that overlooks Göttingen, and, sitting in a coffee shop, dissected quantum theory. “My real scientific career only began that afternoon,” Heisenberg said years later.

  He wanted to know, Heisenberg told Bohr, what quantum theory meant. Beyond the ingenious calculations and fitting of complex spectral lines to peculiar systems of quantum numbers and rules, what, he wanted to know, was the underlying conception, the true physics of it all? Bohr did not insist on the need for detailed classical models that could be translated systematically into quantum terms. Rather, he told Heisenberg, the point of models was to capture as much as one could hope to say about atoms, given the inadequacies of the ideas with which physicists were fumbling along. “When it comes to atoms,” Bohr concluded enigmatically, “language can be used only as in poetry. The poet, too, is not nearly so concerned with describing facts as with creating images and establishing mental connections.”

  This, to Heisenberg, was strange and revelatory. Only a generation earlier, Boltzmann and his allies had argued strenuously for the atom as a concrete thing, not a theoretical abstraction, still less a poetical allusion. Was Bohr now saying that physicists couldn’t hope to describe atoms concretely, that they must make do with analogies and metaphors? That the intrinsic reality of an atom was inaccessible to them? That perhaps it was meaningless even to talk about the intrinsic reality of an atom?

  It is not clear how far the reader can trust Heisenberg’s account of these and other meetings. Writing many years after the events, he pretends to reconstruct long, intense conversations, set out in complex, thoughtful paragraphs. It is hard to shake the feeling that Bohr, in Heisenberg’s recollection, says things that suit a view of Bohr’s physics fashioned by Heisenberg over the many intervening years. What’s undeniable is that Heisenberg’s first meeting with Bohr truly changed his view of what quantum theory was about.

  Although Bohr understood that quantum theory might not follow classical rules, he also insisted, from the earliest days, that the language of classical physics—which so successfully described the everyday world—remained indispensable. His bridge across that gap was an overarching idea he called the correspondence principle, which said that the quantum theory of the atom ought to match seamlessly on to classical analyses of atomic behavior, when the latter are known to work. For example, electron jumps between low-lying orbits, close to a nucleus, involve large and abrupt changes of energy, whereas in transitions between states with large quantum numbers—distant orbits in the far reaches of an atomic system—the energy change is small compared with the energies of the orbits themselves. The more modest the quantum jump, the more closely it resembles the kind of incremental change amenable to classical treatment. The correspondence principle meant that in such instances, quantum and classical behavior should tend to the same outcome, and indeed Bohr had used reasoning of this sort to flesh out details of his atomic model.

  In general, though, putting the correspondence principle to good use in complicated situations demanded a certain finesse on the part of the practitioner. A textbook published in the early 1920s said that the correspondence principle “cannot be expressed in exact quantitative laws [but] in Bohr’s hands it has been extraordinarily fruitful.” Abraham Pais, who has written extensively on this period of physics, comments mysteriously that “it takes artistry to make practical use of the corresponde
nce principle.” Emilio Segrè, another physicist reminiscing about the old days, agrees that the correspondence principle was hard to formulate precisely, and explains that in practice it amounted to saying, “Bohr would have proceeded in this way.”

  Thus arises the Bohr mystique. In his schematic, intuitive way, Bohr saw how to build quantum theory, and other physicists were supposed to follow his lead, even though they couldn’t quite see what he was doing. Bohr acquired a reputation for lecturing in slow, rambling, painfully constructed sentences pregnant, so it seemed, with great meaning that lay just a little beyond the audience’s reach. Somehow it was always the listener’s responsibility to discern what Bohr meant, not Bohr’s job to speak more clearly. Like any guru worthy of the name, Bohr was cryptic and indirect.

  Writing home to his parents after his first encounter with Bohr, Heisenberg evidently felt he had made an impression. Although Bohr as well as Sommerfeld expressed grave reservations about the half-quantum idea, they had to concede, according to Heisenberg, that they couldn’t prove him wrong and that their objections came down to “generalities and matters of taste.” Bohr apparently described Heisenberg’s work in one of his lectures as “very interesting,” which the young man, as yet unfamiliar with Bohr’s idioms, took as an endorsement. At the end of it all, Bohr indicated to Heisenberg that he should find a way to spend some time in Copenhagen.

  Bohr had a new disciple.

  In a sign of the rapidly increasing maturity of the American scholarly scene, Sommerfeld was invited to spend the academic year beginning in September 1922 in far-flung Madison, Wisconsin. He was happy to spread the quantum gospel to an eager, fresh audience, and the chance to earn some foreign income as the German mark became ever more worthless was nothing to scoff at. While he was away, he arranged for Heisenberg, who had not yet graduated, to continue his studies with Born in Göttingen.