THE BREAKTHROUGHS IN QUANTUM THEORY DURING 1925, including the one made by Fermi toward the end of the year, were brilliant. The characters involved were fascinating and the source of endless, often amusing anecdotes. To appreciate these achievements and how they affected Fermi’s own work, it is important to understand the state of theoretical and experimental physics at the time and the contributions of two quantum theorists in particular—Wolfgang Pauli and Paul Dirac.
IT WAS MAX PLANCK IN THE 1890S WHO FIRST PROPOSED THAT ENERGY was not “continuous” but came in discrete, tiny “packets,” or “quanta.” (“Quantum” is the singular of quanta and gives the theory its name.) This was the only way Planck could explain the odd problems encountered by German engineers who were trying to build a more energy-efficient light bulb to compete with the American engineers at General Electric and Westinghouse. Building on Planck’s solution, Einstein later proposed that light was composed of such packets, which he dubbed “photons.” Further experimentation showed, rather confusingly, that photons appeared to display the characteristics of both particles and waves—a very strange duality that remains at the heart of quantum theory to this day.
The concept that energy seemed to come in discrete quanta was great news for scientists who were working with spectroscopes. For over a century it was known that when elements were isolated, heated, and the light produced put through a spectroscope (essentially a high-precision prism), the spectrum that resulted was not a continuous rainbow of light at all but consisted of distinct lines of color. Each element had its own unique pattern of lines, corresponding to different frequencies of color. That energy came in discrete packets and that spectra were not actually continuous suggested a connection between the two.
The early twentieth century had more than its share of geniuses. Planck and Einstein were two. A third, the Danish physicist Niels Bohr, had an idea based on the Planck-Einstein quantum work and on the experimental discoveries of a fourth genius, Ernest Rutherford, which suggested the underlying mechanism that produced these lines. Rutherford and his colleagues in Manchester, England, conducted elegant experiments to show that gold atoms seemed to have a very specific structure: they seemed to consist of a “cloud” of light, negatively charged matter surrounding a core of heavy, positively charged matter buried deep within that cloud. In 1897, Rutherford’s colleague J. J. Thomson isolated a negative particle, the electron, which led to the conclusion that the negative cloud consisted of electrons. Physicists began to call the heavy central core of the atom the “nucleus.” (Rutherford discovered the positive particle in the nucleus, the proton, in 1919. Neutral particles in the nucleus, called neutrons, were only discovered in 1932. They played an important part in Fermi’s story in due course, but when Fermi arrived in Florence, no one yet knew that they existed.)
With Rutherford’s discoveries in mind, Bohr began playing with a compelling idea. Perhaps electrons were confined to specific “orbits” around a nucleus and could not exist in the spaces between these orbits. All electrons had a minimum orbit, called a ground state, below which they could not go. Otherwise, an electron with negative charge would be attracted to a nucleus with positive charge and all matter would collapse into itself. Electrons could, though, “leap” from one orbit to another, stimulated by the absorption or emission of a particle of light, that is, Einstein’s photon. The leap from one orbit to another would correspond to the emission or absorption of a specific frequency of light, depending on the frequency of the electron’s orbit around the nucleus. Bohr imagined this movement to be similar to the way in which planets orbit the sun.
Bohr’s model was the work of a true genius and was perhaps the critical breakthrough (although in most important details it was wrong). Good as it was, though, it left some important questions unanswered. It seemed clear that when an electron was stimulated by a photon, it would absorb that photon and jump to a higher energy orbit, but it was not at all clear that it would always jump the same way. Of equal interest: What would determine whether it jumped back down to a lower energy level, thus emitting a photon? What determined the actual energy level of each orbit? These puzzles defied explanation, but they were reflected and magnified by some spectroscopic phenomena.
One such phenomenon was called the Zeeman effect, named after the Dutch physicist Pieter Zeeman who first observed it. In a magnetic field, the spectral lines produced by an element split. Another phenomenon was also puzzling: even without a magnetic field, some lines in a given spectrum are notably more intense—that is, they shine brighter—than other lines. Were there laws that determined why some lines would be brighter and others dimmer?
For physicists these puzzles were far from trivial. They were deeply disturbing. A full theory would be expected to explain all observed phenomena and to predict with precision any particular observation. The classical mechanics theory of Isaac Newton and the classical electromagnetic theory developed by James Clerk Maxwell each produced specific and precise predictions about the phenomena they addressed and were the standards against which physicists judged a new theory’s success. From this vantage point, quantum theorists were proving distinct failures.
Fermi was certainly aware of these problems. The nine months he spent in Göttingen exposed him firsthand to the frustrations of those struggling with these issues, including Born, Heisenberg, and Jordan. That he continued to keep abreast of their work is apparent in his work during his subsequent stint in Leiden, where he published his paper on the problems of predicting the intensity of spectral lines in November 1924. Fermi cited work by Sommerfeld, Heisenberg, and Born, so he certainly was aware of the progress being made during 1925. His own contribution came toward the end of the year, based largely on the work of a young theoretical physicist from Vienna named Wolfgang Pauli.
PAULI’S ENORMOUS GIFTS IN MATHEMATICS AND PHYSICS WERE recognized early and cultivated by those around him. He studied in Munich with the great theorist Arnold Sommerfeld, who famously said that by the time Pauli arrived in Munich there was little that Sommerfeld could teach him. Like Fermi, Pauli made his first contributions in relativity theory, writing a treatise on the subject at the age of sixteen. A subsequent treatment of the subject, written for a German encyclopedia on mathematics, established his international reputation while he was still at university.
Like Fermi, he was a child prodigy, and like Fermi he was short, about five-foot-five. In practically every other way he was Fermi’s antithesis. Physically, Fermi was solid but fit and unremarkable in appearance. Pauli was round, tending toward obesity, but he was darkly attractive, with soulful eyes and sensuous lips. Fermi was not a drinker; Pauli drank heavily and struggled with alcoholism throughout his life. Fermi habitually retired early; Pauli was a night owl, enjoying the sybaritic life at local cafés and cabarets wherever he happened to be. While at university in Munich, Pauli mixed with artists, writers, musicians, and other bohemians in the Schwabing district of the city. The most bohemian activity Fermi would indulge in was a hike in the mountains.
Fermi was gifted as both a theorist and an experimentalist. As a theorist Pauli was perhaps even more gifted than Fermi, but as an experimentalist he was a disaster. Physicists joked that if a piece of equipment wasn’t working properly, Pauli must be in the vicinity. Pauli also had a legendary mean streak, something Fermi completely lacked. Pauli gave new meaning to the phrase “acid-tongued.” He is said to have described a somewhat undistinguished colleague of his with the incredibly dismissive “So young and already so unknown.” He once derisively called Fermi a “quantum engineer.” Of a particularly murky and speculative theoretical paper, he famously exclaimed, “It’s so bad it’s not even wrong.” He delighted in calling his close friend Werner Heisenberg a “fool.” As a young man he informed the eminent British astrophysicist Arthur Eddington that work the older man was pursuing in general relativity theory was “meaningless for physics.” When Einstein came to lecture at the University of Munich, Pauli was the first to speak after the great man’s talk. “What Professor Einstein has just said is not really as stupid as it may have sounded,” he explained helpfully.
As noted previously, Fermi was not a philosopher and rarely showed any interest in intellectual or cultural matters beyond physics. He had little time for religion or spirituality. Pauli was born Catholic and took his religion seriously, rejecting the opportunity to join his irreligious colleagues in disparaging religious beliefs. He had a deep mystical streak, which he indulged later in life in conversations and in a decade-long correspondence with the great psychoanalyst Carl Jung.
He was also a man of obsessions, and by the time he was doing a post-doc in Copenhagen under Niels Bohr, over the winter of 1922–1923, Pauli became obsessed with the “anomalous” Zeeman effect, a problem with which he would struggle over the next several years. At this time a colleague ran into him walking the streets of Copenhagen in a state of despair. “You look very unhappy,” his friend commented. “How can one look happy when he is thinking about the anomalous Zeeman effect?” came Pauli’s reply.
As mentioned above, the Zeeman effect involved the splitting of spectral lines in the presence of a magnetic field. Bohr’s model explained the normal tripling of spectral lines, but often more than three lines would appear—sometimes four, sometimes six. By early 1925, as a professor in Hamburg, Pauli developed a solution of sorts but wasn’t completely satisfied.
In the Bohr model, electron orbitals required three numbers to specify their location, their frequency, and their orientation. Location corresponded to their distance from the nucleus. Electrons could only be in certain specific orbitals and could not exist in the spaces between orbitals. The frequency related to the speed of their orbits. The orientation number referred to how the specific orbital oriented to the axis of the nucleus. Each one of these three numbers was a quantum “number” that, taken together, identified the quantum “state” of an electron.
Pauli realized that if a new quantum number was added to the mix, a number that could have only one of two opposite values, the anomalous Zeeman effect could be explained. He also posited—because the mathematics explained the phenomenon so beautifully—that no two electrons could share all four quantum numbers. This may not seem terribly subtle, but the implication is astonishing: two electrons can be in exactly the same place at the same time, moving at the same speed and in the same orientation, as long as their fourth quantum number is different. Thus was born Pauli’s “exclusion” principle.
Pauli tried to find physical interpretations of this fourth number, but eventually gave up. One idea, first suggested by a young colleague of his, Ralph de Laer Kronig, was that electrons actually “spin,” and that the spin can either be “right-handed” or “left-handed,” producing intrinsic angular momentum in opposite directions. Pauli ridiculed the idea, because it implied, at least to him, that if you measured the speed of the electron at its “equator” (think of a spinning globe) that speed would have to be faster than light, which was impossible. Suitably chastened, Kronig dropped the idea. It was picked up again at the end of the year by Fermi’s good friends Uhlenbeck and Goudsmit, who did not know of Pauli’s objection. If they had, they might not have published their idea, but as it turned out, the two of them did not know of Kronig’s spin proposal and are thus often given credit for the idea of electron spin. Pauli still resisted the idea. It was Dirac who, in 1926, pointed out that a relativistic interpretation permitted—even required—electron spin.
Pauli’s exclusion principle was audacious. To understand his achievement and his particular way of doing physics, one must appreciate that he proposed a mathematical solution that had no physical interpretation at all—it was simply the only way he could imagine to solve the problem he was confronting. It was, in a sense, like Planck’s invention of the quantum, a mathematical solution without an underlying physical interpretation. Like Planck’s idea, it took Pauli enormous courage to propose it and even more to resist the efforts of colleagues to propose the most obvious physical interpretation. That he was wrong in resisting the idea of electron spin takes nothing away from the achievement itself, a beautiful one which bespeaks his own particular genius.
THE EXCLUSION PRINCIPLE ALONE WAS A GREAT STEP FORWARD. IT provided an explanation of the anomalous Zeeman effect and gave a more complete understanding of how electrons “fill up” the orbits surrounding the nucleus. More breakthrough work was done that year. A young German named Werner Heisenberg, working in Göttingen with Max Born and a young theorist named Pascual Jordan, formulated a method to analyze the way electrons move between orbits, using a mathematical technique called matrix multiplication that successfully explained the varying intensity of spectral lines. Late in the year, a Viennese physicist named Erwin Schrödinger, closeted away in a ski chalet with one of his numerous lovers, came up with a differential “wave” equation that did much the same thing, using techniques more familiar to the average physicist. Physicists now had two highly effective ways of delving into previously incomprehensible physical phenomena.
Why did both approaches, so different in form and function, provide the same answer to the thorny questions raised by quantum theory? It took yet another genius to show that matrix mechanics and wave mechanics were two sides of the same coin.
HE WAS, IN THE WORDS OF HIS BIOGRAPHER GRAHAM FARMELO, “the strangest man.”
Paul Adrien Maurice Dirac was the baby of that generation of quantum pioneers, a year younger than Fermi and Heisenberg, two years younger than Pauli. Born in Bristol, England, he was the son of a strict father whose severe discipline had a lasting effect on the young man. Tall, thin, and extremely reserved, he rarely spoke in sentences longer than two or three words. His teachers at the Merchant Venturer’s Technical College considered him brilliant, and he easily obtained a place at St. John’s College, Cambridge, but his family finances prevented him from attending. Instead, he attended the University of Bristol for his undergraduate education and St. John’s for his graduate degree, the first to be given in the field of quantum theory. In his thesis he provided an independent derivation of Heisenberg’s quantum mechanics based on his observation that the underlying mathematical structure of the matrix algebra, in particular its noncommutative property, was analogous with a particular form of the mathematics physicists used to express classical mechanics.
Dirac was, in some sense, the anti-Pauli. We might think of him today as suffering from Asperger’s syndrome. He was socially awkward in the extreme and literal-minded to an exasperating degree. He once asked Heisenberg why people danced. Heisenberg replied, “When there are nice girls, it is a pleasure.” Dirac considered this for a moment and blurted out, “But how do you know beforehand that the girls are nice?” Once during a lecture a student raised his hand and said that he did not understand an equation that Dirac had written on the blackboard. Dirac remained silent because, as he explained later, the person in the audience had not asked a question. His colleagues at Cambridge reportedly defined a “dirac” as a unit of one word per hour. He was also aggressively irreligious. In one famous exchange at the 1927 Solvay conference, talk among the younger physicists ventured into the area of philosophy and religion. Heisenberg recounts that Dirac made what was for him an impassioned plea to the effect that religion had no place in the world of a physicist. Pauli, who was silent for much of the discussion, reportedly ventured, “Well, our friend Dirac has got a religion and its guiding principle is ‘There is no God and Dirac is His prophet.’” Heisenberg reported that everyone had a good laugh, none more so than Dirac himself.
Among the brilliant young theorists of his generation, Dirac may well have been the most brilliant. His PhD thesis was impressive and attracted the attention of the physics world at large. Born was stunned that a doctoral student could master the field so completely, especially since the final Born-Heisenberg-Jordan paper had not yet been completed. Others shared Born’s surprise. Almost immediately, Dirac was propelled into the highest ranks of theoretical physicists. His was the first PhD degree ever granted in the new field of quantum theory and heralded the arrival of a superstar. Later in the year, Dirac followed it up during a post-doc in Copenhagen with an even more impressive paper, which demonstrated mathematically the underlying unity between the Heisenberg and Schrödinger approaches, seeing both as special cases of something called transformation theory. It was a significant finding, but Dirac’s best was yet to come. In 1927, he produced the first paper to develop the concept of a quantum field using the electromagnetic field as his focus. The paper had an historic impact on physics and laid the groundwork for Fermi’s second great contribution.
IN 1925, HOWEVER, ALL OF DIRAC’S MAJOR CONTRIBUTIONS WERE still in the future. As the year came to an end, while Schrödinger was holed up in the Austrian Alps with his girlfriend during Christmas wrestling with his equation and putting it in the proper form and while young Dirac was grinding out his thesis, Fermi was also thinking deeply about quantum problems. But the problems he considered were slightly different. He had been contemplating them for several years and now the work of Pauli, in particular, showed him the way toward a solution.