“The idea suggested itself” was how Heisenberg put it years later—but it suggested itself to him and to no one else. Heisenberg’s leap here is reminiscent of the leap Einstein made when, by reexamining the apparently self-evident notions of time and location, he was led to his theory of relativity. A judicious questioning of the obvious may well be a mark of genius.
But genius also requires fortitude. It was not difficult for Heisenberg to write down, in a formal mathematical way, equations that expressed an electron’s position and velocity as combinations of an atom’s fundamental oscillations. But when he inserted these composite expressions into standard equations of mechanics, what he created was an almighty mess. Single numbers became lists of numbers; straightforward algebra exploded into pages of confused, repetitive formulas. For weeks Heisenberg tried different calculations, played algebraic games with Fourier series, floundered uselessly, then ground to a halt when a monstrous attack of hay fever clogged his brain.
On June 7 he took an overnight train to the northern coast of Germany. So red and inflamed was his face that the landlady of an inn where he stopped for breakfast the next morning thought he had been beaten up—not an outlandish possibility in the Germany of the mid-1920s. He then boarded a ferry for the small, barren island of Helgoland, about fifty miles out into the North Sea. A military outpost during the First World War, Helgoland was by this time a resort, frequented by those in search of fresh sea air and isolation.
Heisenberg stayed for a week and a half, clambering about the rocky shore, resting, reading Goethe, talking to hardly a soul but thinking, always thinking. Refuge for Heisenberg always meant a retreat to nature, to mountains, forests, and water. Slowly his head cleared. In this lonely place he could let his mind dwell on physics.
What had brought Heisenberg to a dead stop was not any grand conceptual puzzle but a basic problem of multiplication. He had turned position and velocity from single numbers into multicomponent sums. Multiplying two numbers together produces another number. Multiplying two lists of numbers together creates a page full of possible terms, consisting of each member of the first list multiplied by each member of the second. Which terms were important, and how should they be added to generate a meaningful product?
Wrangling this mess into order, Heisenberg found his answer by concentrating on physics, not mathematics. The elements of his algebra were oscillations, each representing a transition from one state to another. The product of two such elements, he saw, must represent a double transition, one state to a second, then from the second to a third. The way to arrange his multiplication table, Heisenberg now deduced, was to put together elements corresponding to the same initial and final state, summing over all possible intermediaries. This realization—after a little work, to be sure—gave him the key by which he could devise a multiplication rule that was both manageable and sensible.
At three o’clock one morning, lying sleeplessly on his bed in a small hostel, Heisenberg knew that he had the tool enabling him to perform calculations in his new mechanics. He could write down, for example, a mathematical formula for the mechanical energy of some system, expressed in his strange calculus. There was no guarantee he would get a useful answer. His elaborate method might deliver gibberish.
So he rose from his bed and started figuring. In his feverish state he made endless slips and errors and had to start over again and again. But finally he got an answer, and it was more than he could have dreamed for. With joy and bewilderment he discovered that his strange mathematics indeed yielded a consistent result for the energy of a system—but only so long as that energy was one of a restricted set of values. His new form of mechanics was, in fact, a quantized form of mechanics.
This was remarkable but altogether inexplicable. In all previous attempts at the quantum theory of atoms, the physicist had to plug in, somewhere along the way, Planck’s original quantization rule or some close variant of it. Heisenberg had done no such thing. He wrote down the standard equations for a simple mechanical system, inserted his strange composite expressions for position and velocity, applied his novel rule of multiplication—and found that the transformed mathematics held together only when the energy took on certain values.
His system, in other words, quantized itself, with no further prompting from him. As Planck, a quarter of a century earlier, had seen that radiation must be quantized, so now Heisenberg, in an utterly different way, had discovered that the energy of a mechanical system must likewise be quantized. This was as wonderful as it was mystifying.
Elated, unable to sleep, Heisenberg went out to the shore in what was now the early light of morning and climbed onto a rock while the sun rose on a new day. What he had found was a gift from above, he thought, a discovery of unwarranted and unexpected proportions. He lay on the rocks in the warming sunlight, marveling at the beautiful consistency of his strange calculations, and thought to himself, he recalled later, “Well, something has happened.”
One thing disturbed him. His multiplication rule was not reversible. That is, x times y was not necessarily the same as y times x. This was nothing Heisenberg had ever encountered before. But it was what he needed; it was what the new physics demanded.
Passing through Hamburg on the way back to Göttingen, Heisenberg consulted excitedly with Pauli, who urged him to write up his ideas quickly. In letters to Pauli in the following weeks, Heisenberg complained that things were going slowly, that it was all very unclear to him, that he didn’t know how it was going to turn out—but at the same time he passed on to Pauli his latest results, a set of ideas and conclusions that would form the backbone of his developing view of quantum mechanics. By early July, he had written what he called a “crazy paper” setting out his discovery. He sent a copy to Pauli, eager for his friend’s judgment but wary too. He was convinced, he told Pauli, that in doing away with the classical notions of position and velocity, he was on the right track; he was still not sure that his transformed versions of these things were right. That part of the paper, he confided, seemed “formal and feeble; but perhaps people who know more can make something reasonable out of it.” He begged Pauli to respond to his draft in a couple of days, because “I must either finish it or burn it.”
In Göttingen, Heisenberg presented a draft to Born, saying he didn’t trust his own judgment enough to know whether it was worth submitting for publication. Born was immediately enthusiastic and sent the paper off to the Zeitschrift für Physik. To Born’s mathematically acute mind, Heisenberg’s strange calculus, awkwardly expressed, provoked surprise, excitement, and an elusive glimmer of recognition that he couldn’t at first trace. He conveyed the news to Einstein a few days later, warning him that although Heisenberg’s work “looks very mystical, it is certainly correct and profound.”
Chastened by his experience in Copenhagen, Heisenberg waited until the end of August before letting Bohr in on the news. “As Kramers has perhaps told you, I have committed the crime of writing a paper on quantum mechanics,” he wrote, uninformatively. Kramers, by chance, had been in Göttingen for a few days when Heisenberg returned from Helgoland. He and Heisenberg talked, evidently, but Kramers relayed nothing of their conversation to Bohr. It’s entirely possible that Heisenberg, still unsure of his ideas and already wary of Kramers, said too little for Kramers to grasp.
Heisenberg began his paper with a bold declaration. “An attempt is made,” he wrote, “to obtain foundations for quantum-theoretical mechanics based exclusively on relationships between quantities that are in principle observable.” Observability: it was the coming principle of this new mechanics. Forget about trying to account for the behavior of electrons directly; instead, express what you would like to know in terms of what you can see—the spectroscopic characteristics of an atom.
For all its revolutionary implications, Heisenberg’s paper was a curiously abstract presentation. It talked only of simple mechanical systems defined in formal terms. Nowhere was there discussion of actual atoms and electrons. It was
a foundation for quantum mechanics, not the thing itself. Whether this new approach would lead to a genuine physical theory remained to be seen.
Pauli, writing to another physicist some weeks later, said that Heisenberg’s idea “has given me new joie de vivre and hope…it’s possible to move forward again.” Einstein, when he saw the short paper, had a very different reaction. He wrote immediately to a colleague that “Heisenberg has laid a large quantum egg. In Göttingen they believe it (I don’t).”
Perhaps Heisenberg had indeed, as he put it, committed a crime in writing on quantum mechanics. The verdict was not yet in. In any case, as he soon discovered, he was not the only perpetrator.
Chapter 10
THE SOUL OF THE OLD SYSTEM
In November 1924, the science faculty of the University of Paris gathered to hear a doctoral thesis defense. The candidate, Louis de Broglie, was thirty-two years old, having been delayed in his scientific career first by family tradition and then by the war. The de Broglies, over the generations, had provided France with a succession of statesmen, politicians, and military officers. Louis’s father was a member of parliament, and Louis had studied history at the Sorbonne with a view to becoming a diplomat. But he had a considerably older brother, Maurice, who got caught up in the 1890s mania for X-rays and decided, against the wishes of their father and grand father, on the life of a scientist. Maurice filled his younger brother’s head with compelling talk of radiation and electrons. Louis too switched to science.
During the war, the younger de Broglie served with a mobile radiotelegraphy unit, learning firsthand the practical value of classical electromagnetic wave theory. From his brother he heard about the controversial notion of light quanta. He was hardly the only scientist to be aware of the seeming inconsistency of these two views of light, but he came at the problem from an angle no one else had thought of.
Late in 1923 an elementary idea crossed his mind. If light, in the form of Einstein’s quanta, could act in ways that made it look at least notionally like a stream of particles, might not particles also display some of the properties of waves?
Cobbling together a makeshift but ingenious argument that combined Planck’s quantization rule for radiation with Einstein’s famous E = mc2 for moving objects, de Broglie was able to set out a logically consistent case associating a wavelength with any speeding particle. The faster the particle, the smaller this wavelength.
But was this more than a mere algebraic formula? Did the implied wavelength actually connote any physical wavelike behavior? Unencumbered by any deep understanding of quantum theory, de Broglie applied his idea to the hopelessly outdated Bohr atom and hit upon a striking result. For an electron circling the nucleus in the innermost orbit, he calculated a wavelength exactly equal to the orbit’s circumference. For an electron in the next orbit—higher in energy, with a bigger radius—he found that the circumference was twice the electron’s wavelength. The third orbit was three wavelengths around, and so on, in simple progression.
Just as the fundamental note and harmonics of a violin string correspond to those vibrations for which a whole number of wavelengths fit into the string’s length, so the allowed orbits of the Bohr atom were those for which a whole number of electron wavelengths fit around the orbit’s circumference. Perhaps quantization was no more mysterious, after all, than the physics of vibrating strings.
De Broglie published his idea in two papers that appeared toward the end of 1923. They attracted little attention. A year later, presenting a more complete version at his thesis defense, he got a wary response. His examiners found the notion of electron waves too simplistic and at the same time too fantastic. They couldn’t argue with his algebra. Whether it meant anything physically they couldn’t decide. Still, one of the examiners sent a copy of de Broglie’s work to Einstein, who was fond of simple ideas with huge implications. His verdict was unambiguous. The fog has begun to lift, he commented.
But no one else took much notice.
Born in 1892, de Broglie was a decade older than Heisenberg, Pauli, and the other youthful adventurers who were creating Knabenphysik—lads’ physics—at Göttingen and elsewhere. Older still was Erwin Schrödinger, born in Vienna in 1887 to an affluent, somewhat raffish family with English as well as Austrian ancestry. An only child, Erwin grew up in a splendid apartment in central Vienna. The Schrödingers had little taste for music but a passion for the racy, erotic theater of late-nineteenth-century Vienna. Erwin was raised by women—his delicate mother and her two sisters. Even at Gymnasium he stood out as much for his confident, charming, slightly louche manner as for his evident intellectual capabilities.
Schrödinger enrolled at the University of Vienna in the autumn of 1906, in the weeks following Boltzmann’s suicide. Later, during the war, he saw fighting and won a medal. His most influential teacher, Fritz Hasenöhrl, died in battle. Hasenöhrl’s demise was one of the chief reasons Pauli, a few years later, left his hometown to study in Munich. After enjoying a number of infatuations during his twenties, Schrödinger married in 1920 a woman who adored him and looked after him. He came to depend on her, as a substitute for the women who had raised him, but saw no reason why marriage should impede his instincts. He eventually had three children with three different women, but none with his wife.
In 1921 Schrödinger accepted a comfortable position in Zurich, where life was much easier than in postwar Vienna. By this time he had published work on electron theory, on the atomic properties of solids, on cosmic rays, on diffusion and Brownian motion, on general relativity—all of it well regarded, none of it spectacular. Although he worked on contemporary problems, Schrödinger was something of a traditionalist. He found repellent the idea that electrons, in the Bohr-Sommerfeld atom, jumped abruptly from one orbit to another. This sort of discontinuity, he thought, did not belong in physics, because—as Einstein also complained—it brought with it a degree of unpredictability, of things happening for no discernible reason.
As word began to leak out of de Broglie’s reinterpretation of electron orbits as standing waves, Schrödinger realized that a theoretical result of his, published a year or two earlier, had in a far more opaque way been hinting at the same thing. Where de Broglie was a dabbler theoretically, Schrödinger was equipped with a sophisticated mastery of mathematical techniques. He latched onto de Broglie’s intuitive sketch with the idea that it ought to be possible to make a real theory out of it.
In the middle of 1925, when Heisenberg was on his rocky outpost in the North Sea devising his peculiar new calculus, Schrödinger wrote a paper enlarging on de Broglie’s electron waves, in which he threw out in passing the suggestion that particles are not really particles at all but, as he put it, “whitecaps” of an underlying wave field. That suited Schrödinger’s view of the physical world. Once you accept the existence of particles, of discrete packets of energy, you cannot avoid discontinuity, spontaneity, and all such related ills. But if what we think of as particles are in reality the superficial manifestation of underlying waves and fields, then continuity may be restored.
As Schrödinger incessantly talked up the wonders of de Broglie’s waves, a more skeptical Zurich colleague challenged him. If these were, so to speak, the waves of the future, then where was the wave equation? De Broglie’s argument merely affixed a wavelength to an electron moving with a certain speed. It said nothing about what these waves were, what determined their form, what if anything was their physical meaning. For all respectable classical wave motions—electromagnetic waves, ocean waves, sound waves—a mathematical equation relates the thing oscillating to the force or influence that makes it oscillate. For de Broglie’s waves no such equation existed. They were not, at this point, actual waves so much as a disembodied or abstracted idea of wave motion.
Over the Christmas holiday of 1925, Schrödinger took himself away from his wife and spent some days at a resort near Davos, Switzerland, with a girlfriend whose name has been lost to history. In what one physicist later described as
“a late erotic outburst in his life,” Schrödinger (close to forty years old by now) found what he was looking for—a wave equation that captured de Broglie’s intuition in a formal manner. (In truth, this was but one of many erotic outbursts in Schrödinger’s life, although it is the only one that gave rise to great physics.)
Schrödinger’s equation described a field governed by a mathematical operator that embodied a kind of energy function. Applied to an atom, the equation yielded a limited number of solutions in the form of static field patterns, each one representing a state of the atom with some fixed energy. Quantization came about in what seemed like a pleasingly classical way. To obtain representations of atomic states, Schrödinger stipulated that the solution ought to go to zero, as the mathematicians say, at large distances—otherwise it wouldn’t correspond to an object localized in space. With this condition, his equation yielded up only a finite set of stable configurations, each possessing some discrete amount of energy. This was no more mysterious, he thought, than getting a finite set of vibrations for a violin string fixed in place at both ends.
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