Even better, Schrödinger hinted in one of the several papers he published in 1926, it might now be possible to understand a quantum jump, a transition from one state to another, not as an abrupt and discontinuous change but as a fluid transformation of one standing wave pattern into another, with the wave reconfiguring itself rapidly but nonetheless smoothly.
The old guard was delighted. Einstein wrote enthusiastically to Schrödinger, scribbling in the margin of his letter that “the concept of your paper shows real genius.” He and Planck quickly invited Schrödinger to Berlin. Einstein wrote again to tell Schrödinger, “I am convinced that you have made a decisive advance…just as I am convinced that the Heisenberg-Born way is going in the wrong direction.” Classical order, it suddenly seemed, had been restored.
What Einstein called the Heisenberg-Born way was by contrast an exotic, intricate, forbidding mathematical system that had blossomed quickly from Heisenberg’s Helgoland inspiration. On July 19, Max Born, still struggling to resolve the glimmer of familiarity that Heisenberg’s strange calculus had evoked, had taken a train to Hannover for a meeting of the German Physical Society. As he sat in his compartment reading and scribbling, recognition dawned: what Heisenberg was doing, in his extemporized way, belonged to an arcane branch of mathematics that went by the name of matrix algebra. Born remembered learning something about it years ago, when he still had thoughts of becoming a pure mathematician. Until now he had never seen it put to any practical use.
A matrix is an array of numbers set out in rows and columns. Matrix algebra is a set of arithmetical rules for combining and manipulating matrices in a systematic fashion. The elements of Heisenberg’s calculations could likewise be written out, Born now saw, in the form of square arrays, with each position in the array denoting a transition from one atomic state to some other state. Crucially, the multiplication rule that Heisenberg had so painfully devised was precisely the multiplication rule for matrices already known to a select band of mathematicians. Heisenberg had known none of this, of course. It was his acute insight into physics that led him to the answer he needed.
Born now realized that an entire branch of mathematics already existed, ready-made for quantum mechanics. At some point Pauli, coming down from Hamburg, joined the same train and came across Born thrilled with his discovery and eager to explain what he now understood. Pauli was not just unimpressed but floridly caustic. “I know you are fond of tedious and complicated formalism,” Born recalled him saying. “You are only going to spoil Heisenberg’s physical ideas by your futile mathematics.” Thus was welcomed into the world the subject that soon became known as matrix mechanics.
But Born didn’t let his former pupil’s sarcasm deter him. Back in Göttingen he and his new assistant, Pascual Jordan, worked up a full account of Heisenberg’s system in the formal language of matrix algebra. Then Heisenberg, returning after a trip to Cambridge and a restorative jaunt with his Pfadfinder brethren, joined with Born and Jordan on what became known as the Dreimännerarbeit—the three-man paper—which further refined and extended matrix mechanics. Heisenberg, though gratified that his physical intuition had served him well, nonetheless shared at least a twinge of his friend Pauli’s skepticism. He disliked the name “matrix mechanics,” thinking it too redolent of pure mathematics, of a kind that was moreover unfamiliar and off-putting to most physicists.
A simmering dispute took root here. All his life, Born nursed a resentment that his and Jordan’s contributions to quantum mechanics were undervalued or even overlooked. It was “awfully clever of Heisenberg,” he admitted, to have come up with matrix algebra without knowing what it was, but at the same time he seemed unable to grasp the magnitude of Heisenberg’s conceptual leap. Only when he and Jordan had fleshed out the idea with necessary mathematical rigor, he believed, could it really be called a theory. That was characteristic of Born. Not a man given to physical insight, he failed to appreciate the power of scientific intuition in others. Saying that Heisenberg was “awfully clever” seems to imply that he thought his young colleague was some sort of idiot savant struck by lightning.
In any case, matrix mechanics did not win a rapturous reception from the community of physicists. They first had to learn this new branch of mathematics, then, having done so, struggled to understand what, physically, the matrices represented. Quantum mechanics, in matrix algebra disguise, was horribly complicated. At the same time, it seemed to be largely a formal achievement. The mathematical physicists claimed it was logically sound, and that it captured neatly many of the puzzling propositions that infested quantum theory. That was all very well, but what could you do with it?
Pauli’s ambivalence continued. Shortly after the Born-Jordan paper appeared, he wrote to a colleague that “the immediate task is to save Heisenberg’s mechanics from being drowned further in formal Göttingen scholarliness and to more clearly bring out its physical essence.” Heisenberg at one point lost his cool with Pauli’s scathing attitude and angrily wrote to him that “your endless griping about Copenhagen and Göttingen is an utter disgrace. Surely you will allow that we are not deliberately trying to ruin physics. If you’re complaining that we’re such big jackasses because we haven’t come up with anything physically new, maybe you’re right. But then you’re just as much of a jackass, since you haven’t achieved anything either.”
Stung, Pauli set to work, and in less than a month had managed to use matrix mechanics in all its pure glory to derive the Balmer series of spectral lines for hydrogen—the same thing Bohr had done so many years earlier with his first simple model. Pauli’s calculation was a tour de force, a powerful and convincing demonstration that matrix mechanics was more than mathematical formalism. “I hardly need tell you,” a mollified Heisenberg now wrote, “how thrilled I am about the new hydrogen theory, and how amazed that you worked it out so quickly.”
On the other hand, Pauli’s proof was no walk in the park. The fiendish mathematics still frightened most physicists, and the claim that matrix mechanics was intellectually profound meant nothing if you couldn’t follow the reasoning.
Further confusion arrived in November 1925, in the form of an elegant paper by Paul Dirac, a young physicist at Cambridge. Dirac, it appears, hadn’t met Heisenberg on his recent visit to Cambridge, but saw a copy of the paper that Heisenberg dropped off. Dirac digested Heisenberg’s insight and came up with his own rigorous mathematization of quantum mechanics, similar to what Born and Jordan had worked out but with a different foundation. Dirac reached back into an obscure corner of classical mechanics to find a differential operator that also obeyed the Heisenberg multiplication rule. Matrix-like elements appeared in Dirac’s calculus, but in a secondary way.
It all fit together, apparently. Yet it was profoundly confusing that quantum mechanics could be dressed up as two different although evidently related systems of mathematics. In Göttingen, naturally, they liked matrices, but in Copenhagen Dirac’s elegant and, as it turned out, broader and more powerful analysis won approval.
Physicists outside these select circles, meanwhile, wondered if anyone would produce a version of quantum mechanics they could understand. And that was why Schrödinger’s wave equation, when it appeared in early 1926, was so gratefully received. It contained no funny algebra, just old-fashioned differential equations. Schrödinger himself made no bones about his attitude to matrix mechanics. He “was scared away,” he wrote, “if not repulsed, by its transcendental algebraic methods, which seemed very difficult to me.”
Sommerfeld also saw the advantages of the wave equation. Matrix mechanics, he thought, was “extremely intricate and frighteningly abstract. Schrödinger has now come to our rescue.”
But Schrödinger had a larger agenda. He wanted not simply to promote an easier version of quantum mechanics but to undo some of the damage quantum mechanics had wrought. In his Nobel Prize lecture from 1933, Schrödinger talked of how, as he wrestled to create his wave equation, it had been uppermost in his mind to save “the sou
l of the old system” of mechanics.
Schrödinger insisted that a particle was not a tiny billiard ball but a tightly gathered packet of waves that created the illusion of a discrete object. Everything, fundamentally, came down to waves. There would be an underlying continuum, with no discontinuities, no discrete entities. There would be no quantum jumps, but instead smooth transformations from one state to another.
None of this followed directly from Schrödinger’s equation. It was what Schrödinger hoped his wave equation would lead to. In July 1926 he lectured in Munich on his wave vision of quantum mechanics. Heisenberg was in town, having come from Copenhagen with the double purpose of visiting his parents and listening to Schrödinger in person. He admired the practical utility of wave mechanics, the way it made simple calculations possible. But he didn’t like Schrödinger’s broader assertions and rose from the audience to express a few objections. If physics was to be once again entirely continuous, he asked, how was it possible to explain the photoelectric effect or Compton scattering, both of which by this time amounted to direct experimental evidence for the proposition that light came in discrete, identifiable packets?
This brought an irritated response from Willy Wien, who no doubt still cherished warm memories of Heisenberg’s abysmal performance at his doctoral defense just three years earlier. Jumping in before Schrödinger could speak, Wien said, as Heisenberg remembered it, “that while he understood my regrets that quantum mechanics was finished, and with it all such nonsense as quantum jumps, etc., the difficulties I had mentioned would undoubtedly be solved by Schrödinger in the very near future.”
But Sommerfeld, after hearing Schrödinger, also began to have doubts. “My overall impression,” he wrote to Pauli shortly afterward, “is that ‘wave mechanics’ is certainly an admirable micromechanics, but that it doesn’t come close to solving the fundamental quantum puzzle.”
Heisenberg’s objection to wave mechanics was not merely technical. He didn’t approve of its style. In formulating the concept behind matrix mechanics, Heisenberg had overtly put observational elements—the frequency and strength of atomic transitions—in a central role, while the undetectable motion of individual electrons remained behind the scenes. Schrödinger’s waves sought to restore the older perspective. Particles, according to Schrödinger, were merely manifestations of underlying waves, but while these waves were fundamental, they were not, so it appeared, directly detectable. Wave mechanics promoted a veiled quantity to theoretical primacy, and this was not, Heisenberg profoundly believed, the right way to construct quantum mechanics.
The apparent simplicity of Schrödinger’s waves was highly misleading, Heisenberg thought, and physicists were fooling themselves if they thought the Schrödinger method represented a restoration of classical values. It was not long before that suspicion was borne out.
Chapter 11
I AM INCLINED TO GIVE UP DETERMINISM
Göttingen produced matrix mechanics. Wave mechanics came from Zurich. Other voices chimed in from Copenhagen and Cambridge. From their Olympian perch in Berlin, meanwhile, Albert Einstein and Max Planck surveyed the scene. Einstein was a few years short of fifty, Planck almost seventy. Both were by now essentially conservative figures. As long as there was confusion over the apparently contradictory mathematical forms of quantum mechanics, and concomitant mystification about the physical import of the theory, both men could cling to the hope that something closer in spirit to classical thinking might yet emerge.
One aspect of the confusion dissolved with surprising ease and rapidity. In the spring of 1926 Schrödinger found that wave mechanics and matrix mechanics were not fundamentally different after all. Despite their seemingly contradictory appearances, they were in effect the same theory dressed up in strikingly different mathematics. In a nutshell, Schrödinger’s waves can be used to calculate numbers that obey matrix algebra, while matrix algebra, applied to the appropriate quantities, can be made to yield Schrödinger’s equation. Schrödinger was not alone in finding this remarkable equivalence. Pauli had proved it too, in a letter to Jordan, although apparently the proof didn’t rise to his exacting standard of publishability, and just a little later the same argument appeared in the Physical Review, in a paper written by Carl Eckart, a German-American theorist from a young but promising establishment calling itself the California Institute of Technology.
But these demonstrations of the mathematical equivalence of the two versions of quantum mechanics only made it that much harder to understand how two such different portrayals of physics could arise from the same source. Physicists continued to find Schrödinger’s waves comfortably familiar, while matrix mechanics remained inscrutably alien. Was there one best way to talk about the physics, or did it come down to questions of taste and convenience?
Eager to stay abreast of the developing drama, Einstein and Planck invited the principal actors to Berlin. Heisenberg came first to what he called “the chief citadel of physics in Germany,” though he surely knew that in quantum mechanics the provinces flourished while the capital languished. His lecture to the distinguished Berlin professors did not seem to stick particularly in Heisenberg’s mind. Far more memorable was his first searching conversation with Einstein. He had hoped to see the great man four years earlier in Leipzig, but Einstein had stayed away from that meeting after the assassination of Foreign Minister Rathenau, and Heisenberg had fled after being robbed. Back then, Heisenberg had been a mere twenty-one-year-old, still a little shy, wrestling with the dubious half-quantum business. Four years on, Einstein was still Einstein, well on his way to becoming the shaggy-maned, shabbily dressed figure of popular legend, but Heisenberg was not at all the same young man. He had held his own in disputes with Sommerfeld, Pauli, and Bohr. He had found the key to quantum mechanics. In appearance he was still the same clean-cut, unassuming figure he had always been (he looked like a peasant boy, Born had remarked on first seeing him in Göttingen; like a carpenter’s apprentice, someone in Copenhagen said), but his confidence had grown. In quantum mechanics, he was the expert, Einstein the critic.
After the lecture, the two walked through the streets to Einstein’s home, arguing back and forth. Einstein objected sharply to the obscurities of matrix mechanics, the way it sent position and velocity to the back of the room and brought enigmatic, unfamiliar, abstrusely mathematical quantities to the fore. Heisenberg protested that these strange developments had been forced upon him because he was trying to build a theory on what the physicist can actually observe about an atom, not on its unknown and perhaps unknowable internal dynamics. In any case, Heisenberg asked, wasn’t this essentially the same strategy that Einstein had used with such stunning success years before, when he came up with special relativity?
To which Einstein could only grumble in response, as Heisenberg tells the story, that “possibly I did use that kind of reasoning…but it is nonsense all the same.”
In devising relativity, Einstein reinvented space and time. His starting point had been to inquire closely into the meaning of simultaneity. In Newtonian mechanics, time was absolute. If two events happened in different places at the same time, then their simultaneous occurrence was an objective fact, an indisputable datum. But Einstein had the wit to ask how observers of these two events could know that they happened at the same time. They would have to synchronize their watches, as characters in war movies used to say. That meant exchanging signals—by flashes of light, by talking on the radio. But these signals travel, at most, at the speed of light, and by scrupulously following how different observers would in practice establish the times and locations of events, Einstein showed that in general they could not agree on simultaneity. Two events that happened, according to one observer, at the same time would be seen by another as happening one after the other.
In much the same way, Heisenberg insisted, it was no good imagining you could construct an absolute, God’s-eye view into the inside of an atom. You could only observe in various ways the atom’s behavior
—the light it absorbed and emitted—and infer as best you could what was going on inside.
Einstein wasn’t buying it. In relativity, although observers may disagree, events retain a distinct and unarguable physicality. A collection of observers comparing notes could arrive at a mutually acceptable consensus on what they had all seen, because special relativity accounts for the discrepancies between their individual stories. An underlying objectivity persists.
That, as Einstein saw it, was far from the case with quantum mechanics. Heisenberg appeared to be saying, he thought, that it was foolish even to ask for a consistent depiction of an atom’s structure and behavior. Matrix mechanics especially, so it seemed to him, high-handedly ruled out of order questions about an electron’s disposition that physicists had always given themselves the right to ask. And, Einstein firmly believed, were perfectly entitled to continue asking.
Heisenberg pushed back. Relativity had been controversial because it undermined the old questions physicists had always asked about space and time, and forced them to ask new ones. That did not mean that space and time became meaningless. He and his colleagues were trying to do the same thing for atoms—figure out the right questions to ask. Old kinds of knowledge would be lost, to be sure, but new ones would come in their place.
He had to admit, though, that he hadn’t yet worked it all out. Quantum mechanics was still a work in progress. The conversation trailed off inconclusively.
Schrödinger’s wave mechanics, by contrast, seemed to Einstein to offer hope. The standing wave picture of an electron in an atom had a tangible air about it. Not long after his meeting with Heisenberg, Einstein wrote to Sommerfeld that “of the recent attempts to obtain a deeper formulation of the quantum laws, I like Schrödinger’s best…I can’t help but admire the Heisenberg-Dirac theories, but to me they don’t have the smell of reality.”
Uncertainty Page 12