By this time Schrödinger had also visited Berlin. Einstein found him most amiable. Schrödinger was Viennese, cultured, warm, and sophisticated. Both men were married, because they liked having someone to take care of them, but both found their pleasures elsewhere and convinced themselves their wives took this in good spirit. Though he spent years in Berlin, Einstein never felt at home among the “cool, blond Prussians.” Heisenberg was by birth a southern German, from Bavaria, but his family was northern in its culture and habits. He inclined to formality and polite manners, which to Einstein came across as stiffness and reserve. Schrödinger, by contrast, was a man Einstein could feel at ease with.
But congeniality did not prevent Einstein from seeing the flaws in Schrödinger’s ambitions for physics. Lecturing in Berlin, Schrödinger expanded on his hope that the waves of his equation would turn out to be direct physical pictures of electrons and other entities—not particles as such, but concentrations in space of mass and charge. Einstein was sympathetic but wary. Schrödinger was clearly expressing a hope, not a demonstrable argument. It might, Einstein could easily see, be mere wishful thinking.
Heisenberg put it more crudely. Of Schrödinger’s physics, he wrote to Pauli, “the more I think about [it] the more repulsive I find it…to me, it’s crap…but excuse this heresy and speak of it no more.”
Schrödinger had published a brief argument in support of his interpretation. The waveform corresponding to a particle sailing through empty space, he showed, would hold together indefinitely. This physical integrity, Schrödinger argued, made the bunched-up wave an acceptable stand-in for a traditional particle.
But this result was the exception, not the rule. Max Born used wave mechanics to ponder a more complicated case, the collision of two particles, and came to a very different conclusion. After collision, he found, the waves corresponding to the rebounding particles spread out something like ripples on a pond, which by Schrödinger’s interpretation would seem to mean that the particles themselves had become smeared out in all directions. That made no sense. A particle, even if it were a concentrated wave motion, must ultimately be identifiable in a classical sense. In Bohr’s language, this was an instance of the correspondence principle, that the quantum description of a collision had to pass over, in the aftermath, to a suitable classical description. More fundamentally, it was just a question of common sense. A particle had to be somewhere; it couldn’t disperse uniformly throughout space. The end result of a collision had to amount to two distinct particles moving off in well-defined directions. That’s what happened in the Compton effect.
Thinking along these lines, Born came to a neat conclusion. The spreading waves leaving the collision site described, he proposed, not actual particles but their probabilities. In other words, a direction where the wave was strong was a direction in which rebounding particles were likely to emerge. Where the wave was weak, by contrast, particles were less likely to be seen.
If this was so, Schrödinger’s equation generated not a classical wave but something wholly new. In the case of an electron in an atom, the wave must represent not some physically spread-out mass or charge, but rather the chance of finding an electron here, there, or somewhere else.
This depiction, odd as it was, harmonized with matrix mechanics. Heisenberg had defined the position of an electron in a backward manner, expressing it as a composite of the atom’s electromagnetic characteristics. In a sense, Heisenberg had thereby depicted the electron’s physical presence as a combination of things it might be doing, rather than some specific indication of where it was.
Born’s recognition of wave mechanics as dealing with probability didn’t just clarify what Schrödinger’s equation meant. It also fleshed out the physical as opposed to the purely mathematical connection between wave mechanics and matrix mechanics. The price to be paid for this recognition was the intrusion into physics of probability in a new form.
Yet this conclusion slipped into the tight world of the quantum physicists with no fanfare. No one seemed to take any special note of Born’s argument. That his result excited little immediate attention gave Born, in later years, further cause for bitterness. Other physicists were inclined to say in retrospect that of course they knew Schrödinger’s ideas on the meaning of the waves were clearly wrong, and, yes, they could see that the waves connoted probability. Heisenberg in particular would say that the meaning of matrix elements as probabilities had been evident to him from the outset—although he didn’t trouble to write this down anywhere. Textbooks on quantum mechanics, even those written soon after the subject’s genesis, tended to state the definition of probability but give it no particular attribution, as if it were a step too obvious to warrant further explanation.
On the other hand, Born himself, in a later interview, acknowledged that perhaps he didn’t see at the time just how revolutionary his result was. Physicists in those days all knew about the statistical physics of the nineteenth century, and many had dabbled with the idea that such statistical uncertainty might run deeper still. There was the link, first made clear by Einstein, that the intensity of emission lines from an atom had to do with the likelihood of one internal transition versus another. There had been, too, the intermittently appealing suggestion that perhaps conservation of energy would turn out to be only statistically true. As Born put it, “we were so accustomed to making statistical considerations, and to shift it one layer deeper seemed to us not very important.”
Yet this later sentiment is belied by Born’s own words from his 1926 paper. There he noted that it was no longer possible to say what the specific outcome of a collision would be. You could only specify the probabilities of a range of outcomes. “Here the whole problem of determinism arises,” he then wrote. “[In] quantum mechanics there exists no quantity which in an individual case determines the result of a collision…I myself am inclined to give up determinism in the atomic world.”
Determinism was the linchpin of classical physics, the crucial principle of causality. Born was now putting into words Einstein’s greatest fear, one he had expressed repeatedly for years. In classical physics, when anything happens, it happens for a reason, because prior events led up to it, set the conditions for it, made it inevitable. But in quantum mechanics, apparently, things just happen one way or another, and there is no saying why.
If Born evidenced confusion about the meaning of his discovery, Einstein decidedly did not. Toward the end of 1926, he wrote to Born in words that have become famous through repetition, not least by their author, who liked his phrasing so much he would trot it out at every opportune moment. “Quantum mechanics is very imposing,” he told Born. “But an inner voice tells me that it is not the real McCoy. The theory delivers a lot but hardly brings us closer to the secret of the Old One. I for one am convinced that He does not throw dice.” If probability were to replace causality, then as far as Einstein was concerned the rational basis for constructing theories of physics had been swept away.
But younger physicists, as usual, blithely disdained such metaphysical fretting and quickly latched onto the identification of Schrödinger’s waves as a measure of probability. Bohr, still the guiding spirit, approved. But others chose to bow out, notably the men who had invented wave mechanics, Louis de Broglie and Schrödinger himself. Following his precocious insight that particles must have wave properties, de Broglie duly collected a Nobel Prize in 1929 but made no further significant contributions to quantum mechanics. All his life he insisted that the probability interpretation was wrong.
Schrödinger likewise became from this time on more of a critic of quantum mechanics than a contributor to it. In September 1926, he visited Copenhagen, not long after Heisenberg had succeeded Kramers as Bohr’s assistant. Bohr wanted, so he said, to hear Schrödinger’s views firsthand, to understand them better. In the event, Bohr pressed and hounded Schrödinger to explain himself from the moment he arrived, questioning his visitor in his standard relentless manner, a style that Bohr took to be the n
atural habit of scientific inquiry but which seemed to Schrödinger like an interrogation of Kafkaesque inescapability. Schrödinger became tired and ill and took to his bed at the institute. Mrs. Bohr fussed over him with tea and cakes while Bohr perched at the end of the bed, day and night it must have seemed, saying, “but Schrödinger, you must at least admit that…”
Heisenberg took only a modest part in this banging of heads. He recalled Schrödinger wistfully suggesting that some way might still be found to obtain Planck’s 1900 formula for the spectrum of electromagnetic radiation without the need for quanta. “There is no hope of that,” Bohr told him, speaking crisply for once. Schrödinger tried to resist, telling Bohr that “the whole idea of quantum jumps leads to nonsense,” and that “if we are going to have to put up with these damn quantum jumps, I am sorry that I ever had anything to do with quantum theory,” at which Bohr soothed him by saying that “the rest of us are very thankful” for wave mechanics, because of its clarity and simplicity.
There was no rapprochement. Schrödinger became angry, Heisenberg recalled, but had no answer to Bohr’s soft-spoken but unending assault. Exhausted, he retreated to Zurich with his views unchanged.
Einstein, unhappy, continued to press his objections. Toward the end of 1926 he was writing to Sommerfeld that the great technical successes coming from Schrödinger’s equation tended to obscure the deeper question of whether it genuinely offered a complete picture of what he quaintly insisted on calling “real events.” “Are we really closer to a solution of the puzzle?” he plaintively asked.
Increasingly, Einstein spoke and wrote in the suggestive, allusive way he later became famous for. Other physicists heard more than they wanted to know about the secrets of the Old One, about the God who doesn’t play dice, about the Lord being subtle but not malicious. Einstein talked as if he alone could know the inner truths of nature. His unhappiness was for this reason unanswerable. He objected to the presence of probability in physics but had found no way of getting rid of it. And the problem was about to get worse.
Chapter 12
OUR WORDS DON’T FIT
As Heisenberg and Schrödinger, along with their allies and critics, tussled over the meaning of the physics they were creating, the forty-one-year-old Niels Bohr held on to his role as guide and guru. Increasingly, though, other physicists questioned his judgments and fretted over his opaque pronouncements. Schrödinger, recovering from his ordeal in Copenhagen, confessed to frustration in dealing with Bohr. “The conversation is almost immediately driven into philosophical questions,” he wrote to a friend. “Soon you no longer know whether you really take the position he is attacking, or whether you really must attack the position he is defending.”
In September, Paul Dirac arrived in Copenhagen for a six-month visit. Of Bohr’s famously allusive way of lecturing, Dirac observed that audiences were “pretty well spellbound,” but as for himself he complained that “[Bohr’s] arguments were mainly of a qualitative nature, and I was not able to really pinpoint the facts behind them. What I wanted was statements which could be expressed in terms of equations, and Bohr’s work very seldom provided such statements.”
Dirac, a cool, laconic loner, could hardly have been more different from the gregarious Bohr. Dirac’s legendary taciturnity came about because his father, a naturalized Briton of Swiss origin, used to insist that his boy speak French at the dinner table, and, as Dirac explained later, “since I found I couldn’t express myself in French, it was better for me to stay silent than to talk in English. So I became very silent at that time—that started very early.” On top of that, his parents apparently had no friends, never went out, and never invited anyone to their house, so young Paul had few opportunities for small talk in English either.
Dirac respected Bohr but failed to become starry-eyed and worshipful in his presence. Perhaps for that very reason Bohr found the tall, silent Englishman strangely admirable. While Bohr struggled to put broad philosophical concepts into words, Dirac said little but sought a terse clarity in the pure logic of mathematics and unveiled his formulations—precise, if a little arid—only when he was sure of every detail. He recognized, though, that a full and systematic mathematical expression of quantum theory was not the whole story. As he said in his dry way, “getting the interpretation proved to be rather more difficult than just working out the equations.”
Dirac was generally happy to play his part and leave matters of interpretation to others. That sort of laissez-faire attitude did not suit Heisenberg, who found himself increasingly at odds with his mentor Bohr. The two of them became embroiled in a tense, delicate dispute that neither man could let alone. Heisenberg had invented quantum mechanics, after all; he could hardly fail to assume some proprietorial right over the way it was portrayed and used. Bohr, on the other hand, could not altogether shake off his first impression of Heisenberg as a somewhat callow scientific thinker, piercingly imaginative but just as often wayward and impetuous. At this point in the game, Bohr thought, wisdom was required, and who was the man for that?
In Copenhagen, the two men would spend hours together during the day, Bohr talking as always in his unrelenting, insistent way while Heisenberg, animated and agitated, struggled to interrupt. In the evenings they would often continue the haggling as they took a turn around the pleasant grassy park that adjoined the institute. Often, too, even late into the night, Bohr would knock on the door of the attic room at the institute where Heisenberg was staying, offering just a small clarification or emendation of what he had been trying to say earlier. Not infrequently these footnotes to the day’s discussion would sprawl into the small hours. Bohr would stick to no fixed schedule. Whatever had to be said had to be said there and then. Clashing like this for weeks, both men grew weary of the argument, and of each other.
What they argued over during these never-ending days in late 1926 was, in one form or another, the question of continuity versus abruptness. Schrödinger, of course, wanted it all to come down to waves, with discrete particles and their capricious behavior merely an illusion. That, Heisenberg and Bohr could at least agree, was a lost cause. But Heisenberg, having enthusiastically jettisoned the old ways, characteristically wanted to run to the opposite extreme and embrace the most radical thinking at all costs. Quantum mechanics forced physicists to think in new ways, to learn a new language. Too bad, said Heisenberg. They would have to get used to it.
To Bohr, that attitude was cavalier—or, what was worse, shallow. As he pointed out repeatedly and forcefully, position and velocity and all the other old reliables of classical mechanics had not suddenly lost all their utility. In the world outside the atom, the old concepts continued to serve well. There had to be, Bohr insisted, a connecting up. You had to be able to get from the discontinuity and discreteness of the quantum world to the smooth continuity of the familiar classical world.
Heisenberg found Bohr’s attitude frustrating, almost deliberately so, as if frustration were a desirable state, a commendable aspiration. It was as if Bohr wanted to find a way to talk about quantum mechanics in classical language while at the same time freely admitting it couldn’t be done—not, at least, without contradiction and inconsistency. But Bohr positively reveled in contradiction; it constituted his own internal Socratic discourse.
Whenever Heisenberg claimed he understood how quantum mechanics worked, or at least that he could reliably make use of it, Bohr would just as reliably find an obscure point, a lack of logical clarity. “Sometimes,” Heisenberg recalled, “I had the impression that Bohr really tried to lead me onto Glatteis, onto slippery ground…I remember that sometimes I was a bit angry about it.” But he ruefully acknowledged that if Bohr could so dependably put his finger on subtle problems, then perhaps they were on slippery ground after all.
This banging of heads could go on only so long. By early 1927, Bohr and Heisenberg had stated and restated their opinions so often that they were talking past each other, reduced to helpless frustration that neither one could or
would acknowledge what the other was saying. In February, Bohr went to Norway, to spend some time skiing. Originally he had planned this as a trip for the two of them, but now it seemed better to go alone. Heisenberg, meanwhile, could trudge by himself around the park in the early evening, without Bohr dogging his every step.
But the nagging echo of Bohr’s voice stayed with him. Suppose it was true, as Bohr asserted, that position and velocity must continue to have meaning, even if it was not the traditional meaning physicists had always assumed. What would that new meaning be? How could he get at it?
In their wrangling thus far, Heisenberg and Bohr had treated the issue as a theoretical one. Classical mechanics worked with one set of precepts, quantum mechanics with another, and how were the two to be reconciled? It was, to borrow Dirac’s phrase, a matter of getting the interpretation, of hearing what the mathematics was trying to say. Dirac, in fact, had provided an important clue, though Heisenberg hadn’t immediately latched onto it.
While in Copenhagen, Dirac had put the finishing touches to his magisterial presentation of quantum mechanics, in which he showed in a perfectly general way how to take some problem in classical mechanics and define its quantum equivalent. He could also do the reverse—that is, he could show how some quantum mechanical system would look if you insisted on describing it in classical terms. But in that translation, he found, a curious discrepancy arose. Beginning with some quantum system of particles, for example, you could work out a classical picture in which the positions of the particles were the primary elements, or you could choose instead to speak in terms of particle velocities—or rather, in terms of momentum, mass times velocity, which to physicists is the more fundamental quantity. Strangely, though, these position and momentum portraits didn’t match up as they should, if they were merely alternative portrayals of a single underlying system. It was as if the position-based account and the momentum-based account were somehow depicting two different quantum systems, not the same one in different ways.
Uncertainty Page 13