Uncertainty

by David Lindley

  Pauli had come across the same awkwardness. He wrote to Heisenberg about it, using p as a standard notation for momentum, while the adjacent letter q then stood for position. “You can look at the world with the p-eye,” he said, “and you can look at it with the q-eye, but if you want to open both eyes at the same time, you will go crazy.”

  Quantum particles wouldn’t reveal themselves clearly. They yielded up contradictory pictures. That was the conundrum Heisenberg wrestled with. How could he find a way to force quantum mechanics to give up its secrets, to let him see what was going on inside?

  He couldn’t! That was the answer that flashed into his mind one evening as he plodded around the park, lost in thought. Just as, on Helgoland, he had realized that it would never be possible to describe quantum jumps in the continuous language of classical physics, so now the same lesson bore in on him, in a yet larger way. There was no way to force a quantum system to yield up a description that would make unambiguous sense in classical terms.

  Well, yes, but wasn’t that what he had been trying to tell Bohr for months now? Except now he began to see Bohr’s point of view. You might not be able to come up with an unambiguous account—but that didn’t mean, as Heisenberg had thought until now, that you just gave up trying and moved on. You had to find some way of talking about quantum systems.

  At last Heisenberg was able to grasp a point that neither he nor Bohr had understood so far. The crucial question was not a theoretical one, still less, as Bohr often seemed to think, a philosophical one. It was in the end a practical matter.

  You might not be able to talk about the position and momentum of quantum objects in a way that would make sense under the old rules. But what you can still do, Heisenberg now saw, is what physicists have always done—you can attach meaning to position and momentum by measuring them. The way to cut through theoretical confusion was to pay attention to practicalities.

  He just needed to think of a simple example to make his insight plain. And so, perhaps with Compton’s pretty experiment of a few years earlier in the back of his mind, he hit upon the disarmingly straightforward example that has made his name iconic. An electron flies through space. An observer shines light upon it, then detects the light that bounces off the speeding particle. By measuring this scattered light—its frequency and direction—the observer can try to deduce the position and momentum of the electron at the moment the light hit it. And that, as Heisenberg discovered, is where things get interesting.

  Light consists of quanta—or photons, as they had recently been dubbed by the American physical chemist Gilbert Lewis. The encounter between one of these photons and the flying electron is a quantum event. That encounter, as Born had proved, doesn’t yield definite outcomes, but a range of possible outcomes, with various probabilities. Reversing the logic, Heisenberg now realized that an observer cannot infer a single unique event that would have led to the measured outcome. Instead, a range of possible electron-photon encounters could have happened. Which must mean, he saw, that it would be impossible to infer uniquely what the position and momentum of the electron was.

  Pauli had said you could look at position or you could look at momentum, but you can’t look at both at once. Heisenberg, thinking the matter carefully through, realized it wasn’t as simple as that. It wasn’t either-or, but an inescapable compromise. The more an observer tried to extract information about the electron’s position, the less it was possible to know about its momentum, and vice versa. There would always be, as Heisenberg put it, an “inexactness” (Ungenauigkeit) in the conclusions.

  It was during Bohr’s absence that Heisenberg persuaded himself of this tidy but startling result. He had learned to be wary of Bohr’s intense scrutiny of new ideas. He wrote Pauli a long letter explaining what he had come up with, but to Bohr he sent only a brief note saying that an interesting development awaited his return. By the time Bohr got back, Heisenberg had already sent his paper off for publication. Bohr read it, grew fascinated, then gravely troubled.

  Heisenberg described an encounter between two particles, a photon and an electron, and found an inexactness that derived from the unpredictability of that collision. Bohr—inevitably, exasperatingly—came up with another way to look at the matter. An observer detecting the photon measures it not as a particle but as a little bundle of waves. And in classical optics, he reminded Heisenberg, waves have limited resolving power. That is, light of a certain wavelength cannot render clear images of any object smaller than that wavelength. The picture becomes blurry. And that, Bohr said, was the explanation for what Heisenberg had found. It was in the act of using information from a wave measurement to infer the properties of a particle that inexactness sprang up.

  Bohr’s reinterpretation outraged Heisenberg. First, because Bohr was dragging waves back into it, which bore the taint of Schrödinger’s name, and second, because Bohr’s argument seemed to be about the limitations of classical optics, not the unpredictability of quantum events.

  But no, Bohr retorted, that wasn’t it either. It was precisely because of the mixing of incommensurable concepts—particles and waves, quantum collisions and optical resolving power—that inexactness crept in. It was the outward sign of the internal mismatch between quantum and classical principles. This interpretation, as it happened, fell in very neatly with ideas that Bohr had been pondering while he skied alone in Norway. He had evolved a broad new principle, soon to be christened “complementarity,” according to which both the wave aspect and the particle aspect of quantum objects had necessary but contradictory roles to play. Depending on the problem, one aspect or another might come to the fore, but neither could be neglected entirely. Heisenberg’s inexactness, he declared, was the demonstrable evidence of this unavoidable disharmony.

  Heisenberg was aghast. He had worked out an elegant result in a straightforward way. Now Bohr wanted to smother it in the thick metaphorical garb that he loved but that Heisenberg found so oppressive. Heisenberg wanted to go ahead and publish his discovery. Bohr wanted him to contact the journal and hold the paper up while they worked out together the best presentation of the physics. Heisenberg refused. Bohr then found a technical error in Heisenberg’s analysis that was, to Heisenberg’s enormous chagrin, reminiscent of the error he made years ago in his thesis defense as he had tried to answer Willy Wien’s questions about standard optical theory. Heisenberg insisted it wasn’t a big problem and pushed on. Eventually, in May, he reluctantly agreed to add an endnote to his paper, just before it went to press, thanking Bohr for clarifications and allowing that the precise source of observational “uncertainty”—he now used that word, which Bohr preferred—was perhaps not as evident as the author’s presentation implied.

  It was in this painful, quarrelsome way that Heisenberg’s famous uncertainty principle entered the world. As Bohr and Heisenberg wrestled back and forth over how best to express it, the unavoidable difficulty, said Heisenberg, was that “our words don’t fit.”

  Certain words caused particular difficulty. Writing wearily to Pauli, Heisenberg remarked that “all the results in the paper are certainly correct and Bohr and I are in agreement about them—only between Bohr and myself there are considerable differences in taste over the word ‘anschaulich.’” This adjective has caused problems for German-speaking physicists, still more for those faced with translating it into English. Heisenberg titled his paper on inexactness “Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik,” which has been rendered by one author as “On the Perceptual Content of Quantum Theoretical Kinematics and Mechanics,” by another as “On the Physical Content…,” while a third translates anschaulich as “intuitive.” It is as if a single word can mean both “concrete” and “abstract.”

  The verb anschauen means “to look at”; something anschaulich is therefore something capable of being looked at. Heisenberg means to speak about phenomena that the physicist can in principle observe, hence the translation of anschaulich as “perceptual”—t
hat is, perceivable. Hence too its rendering as “physical”—meaning quantities that are empirically meaningful in the traditional way. And thence, with a hop and jump, comes “intuitive,” because the quantities that make sense to physicists are those, such as position and momentum, that have familiar or commonsense meaning. (The flaw here is that no one thought of momentum as intuitive until Newton invented it and made it part of every later scientist’s common sense.)

  Equally tricky is the more famous word that came into physics and thence into wider circulation. In talking about experimental measurements, Heisenberg consistently used the word Ungenauigkeit, “inexactness.” But in one section of his paper, referring to the theoretical point both Dirac and Pauli had made about ambiguity in the theoretical description of a system, he switched to Unbestimmtheit, from the verb bestimmen, “to determine.” He made a distinction, that is, between the inexactness of experimental outcomes and the indeterminacy of mathematical descriptions. Only in his endnote does there appear abruptly the word Unsicherheit, “uncertainty,” which was Bohr’s choice and which through Bohr made its way into the vocabulary of English-speaking physicists.

  “Inexactness,” in truth, is a poor word to describe what Heisenberg found, since it fails to distinguish the new inability he pinpointed from the ubiquitous and long-standing difficulty of making any measurement exactly. A few old-fashioned physicists still prefer to speak, in English, of the indeterminacy principle, which is a better way of putting it. (In the afterword to his play Copenhagen, Michael Frayn suggests, a little more pointedly still, “indeterminability.”) German-speaking physicists today refer to die Unschärfe Relation, a nice choice. In German as in English, sharpness is the quality of a well-made photograph, so unscharf means “blurred.” To speak of the blurriness principle suggests a pleasant connotation, that the more you squint and peer, the less you can make out whatever it is you are trying to see. But “blurriness” is no doubt an insufficiently grand word to enter the English scientific lexicon at this late hour.

  “Our words don’t fit,” Heisenberg told Bohr, and perhaps he switched from one word to another because he reckoned no word would perfectly capture his idea. But Bohr seemed to think he could find the right words or phrases, if only he kept trying. Only by expressing quantum mechanics in familiar terms, he insisted, could physicists hope to make sense of it as something more than a set of mathematical relationships.

  In June 1927, Pauli visited Copenhagen, hoping to act as a mediator between the warring principals. Heisenberg had been driven to tears at one time by Bohr’s unceasing interrogation. On other occasions his frustration caused him to snap back harshly and angrily. Bohr, in all this, as in his earlier encounter with Schrödinger, seems to have maintained a serene, insufferable calm. Pauli soothed Heisenberg a little, but the dispute saw no tidy conclusion.

  Heisenberg, in any case, was about to leave Copenhagen in order to take up a professional chair at the University of Leipzig. There, away from Bohr’s vexing presence, Heisenberg reflected on the previous few months, and after a time wrote ruefully to Bohr, regretting how ungrateful he must have seemed. A brief visit to Copenhagen later in the year helped mend fences.

  Never again, though, would these two have so close, difficult, or intense an intellectual engagement as they had during Heisenberg’s time as Bohr’s assistant. Heisenberg, still only twenty-six, had achieved security as professor in his own right, which among other things at last assuaged his father’s frequent concern that he was squandering his intellectual talents on frivolous matters. Meanwhile Bohr, perhaps a little put out that Heisenberg, working alone, had hit upon a daring and perplexing new argument that seemed to threaten principles physicists had long cherished, took as his next task the formulation of a sound philosophy for understanding this strange concept of uncertainty.

  Chapter 13

  AWFUL BOHR INCANTATION TERMINOLOGY

  For all its subsequent notoriety, the uncertainty principle’s arrival did not trigger instant unrest and rioting in the halls of physics and philosophy. Born, recognizing Schrödinger’s waves as representations of probability, had already said that determinism must go. Pauli and Dirac had seen that there was something strange about the way quantum physics manifested itself to the outside world. Heisenberg’s uncertainty pinned down that strangeness, put a number on it, and—perhaps most important to Heisenberg—dashed any lingering hope that Schrödinger with his waves could restore some sort of classical reality to physics.

  But this discussion, to the select few engaged in it, concerned the inner workings of quantum mechanics. It was Bohr, developing his new philosophy of complementarity, who grappled overtly with the way that the phenomena of quantum mechanics must make themselves known in a broader context. Complementarity, for Bohr, flowed from his idea of correspondence, that the quantum world must transform seamlessly into the classical world, which is what we continue to see all around us. Complementarity was supposed to make quantum mechanics comprehensible and practical to the great mass of working physicists. It was in this attempt at translation that the truly revolutionary aspects of quantum physics burst onto a larger stage.

  After Heisenberg left Copenhagen for Leipzig, Bohr began the slow and painful process of composing his own interpretation of the uncertainty principle. With his new assistant, Oskar Klein, as amanuensis, Bohr thought out loud, practiced his pronouncements, then each morning discarded what Klein had struggled to write down the previous day, and started over. When the Bohrs went to their country cottage on the Danish coast, north of Copenhagen, for the summer, Klein tagged along. The painfully slow composition continued. Margrethe Bohr, normally cheerful and stoic, was reduced on occasion to tears—not, as Heisenberg had been, because she disputed her husband’s angle on physics, but because he had gone on an extended mental absence from what should have been their family vacation. The Bohrs, by this time, had a lively collection of five children, all boys; a sixth boy would arrive the following year.

  For all his dithering over phraseology, Bohr never wavered from his underlying conviction. Any practical description of a quantum object’s properties or behavior must ultimately be couched in classical terms. That was unarguable. The result of any experiment was necessarily a concrete datum, not a cloud of probabilities.

  Uncertainty and complementarity, Bohr thought, shed light on why Schrödinger’s waves were by no means the classical constructs their author had hoped for. Formally, Schrödinger’s equation is deterministic in the old-fashioned sense. That is, if you know the wave function for some system at a certain time, you can calculate it exactly and unambiguously at any later time—provided, that is, you don’t attempt any observation in the interim. Measurement is what causes Born’s probability interpretation of the wave to swing into action: different results are possible, with different likelihoods.

  Heisenberg’s uncertainty nailed down the inescapability of the discord between one possible measurement and another. An observer can choose to measure this, that, or the other, but has to put up with resulting incommensurabilities. And that uncertainty feeds into the future development of the system. The quantum wave function changes to reflect the fact that one particular measurement outcome occurred and other possibilities didn’t—and that in turn influences the possible outcomes of subsequent measurements that might be made. Complementarity was Bohr’s way of trying to keep all these conflicting possibilities under one roof.

  Bohr presented his overarching philosophy in September 1927, at a meeting in Como, northern Italy, to mark the centenary of the death of Alessandro Volta, the Italian pioneer of electricity. Historically, his lecture there marks as well as anything the formal introduction into science of the idea that measurements are not passive accountings of an objective world but active interactions in which the thing measured and the way it is measured contribute inseparably to the outcome. At the time, though, Bohr’s tortured and tortuous remarks mostly fell flat. Those who were not utterly baffled felt that Bohr was for some
reason trying to say what they already knew, only in a needlessly mysterious way.

  Bohr prepared an account of his Como talk for the scientific journal Nature. The process consumed many months in agonized redrafting, entreaties from the editor, assistance from Pauli, abject apologies from Bohr, followed by further delays. The result, printed finally in April of the following year, was accompanied by an editorial comment lamenting that Bohr had destroyed any last possibility that classical principles of physics might be restored, but hoping, by way of a poor substitute, that Bohr’s elusive phrases were not “the last word on the subject, and that [physicists] may yet be successful in expressing the quantum postulate in picturesque form.”

  Bohr said, for example, that he and his colleagues were “adapting our modes of perception borrowed from the sensations to the gradually deepening knowledge of the laws of Nature,” a statement that only with difficulty passes grammatical scrutiny and no doubt caused the average reader of Nature and interpreter of nature to gape helplessly.

  The idea emerged, so often said and so little understood, that measurement disturbs the system being measured. But as Bohr tried to explain, all measurements amount to disturbances of what’s being measured. The new thing about quantum mechanics, he wanted to get across, is really that measurement defines what is being measured. What you get from a measurement depends on what you choose to measure, which is nothing new, but as Heisenberg had now proved, measuring one aspect of a system closes the door on what else you can find out, and thus fatally restricts the information that any future measurement might yield.

  At Como, Born stood up to say briefly that he agreed, mostly, with Bohr. Crucially, so did Heisenberg. Only one or two insiders knew of his and Bohr’s fierce, tense standoff during the previous months. Now, apparently, all that was done with, and Heisenberg had nothing but praise and thanks for his mentor.