6
Pauli, Heisenberg, and the Great Quantum Breakthrough
EVERYONE AGREED with Pauli that there should be four not three quantum numbers. His exclusion principle had shown that no two electrons in an atom could have the same four quantum numbers. Beyond that his colleagues could see that the principle must have huge implications. But no one could yet see what they were.
By the beginning of 1925 it was clear that Bohr’s theory of the atom as a miniature solar system no longer provided even an adequate basis for understanding the atom, let alone for the exclusion principle or the anomalous Zeeman effect. Bohr’s theory by now was under attack from all sides.
The demise of Bohr’s theory of the atom
Pauli, despite his best intentions, had been one of the key wielders of the knife. As he had discovered, the theory had failed to produce a realistic model of either a hydrogen-molecule ion or a helium atom. Then new data appeared showing that the hydrogen atom did not respond to being hit by light as if it were a tiny solar system. This model produced spectral lines for the struck light that did not agree with those found in the laboratory.
Bohr fought back with a variation of his theory in which the invisible orbits of the invisible electrons were replaced by invisible electrons on springs, each emitting light at the frequency of an observed spectral line. To emphasize that these invisible electrons were an intermediate kind of reality, he referred to them as “virtual oscillators.” Pauli wanted nothing to do with them. He had had enough of bizarre models and was totally discouraged.
Heisenberg, who was twenty-four, thought otherwise. Throughout spring 1925 he pushed Bohr’s theory of virtual oscillators to its limits. But it failed. It seemed that Bohr’s theory barely worked even for the hydrogen atom and even then no one really understood why. Atomic physics lay in ruins.
Many physicists spoke of their despair. Pauli did not respond well to crises and was becoming more and more depressed. He joked bitterly that physics was all wrong and wrote to Kronig, “I wish I were a film comedian or something similar and had never heard of physics.” He hoped, he added, that “Bohr will rescue us with a new idea.”
Around this time, Pauli wrote to Bohr about Heisenberg, “I always feel strange with him. When I think about his ideas, they seem dreadful to me and inwardly I swear about them. For he is very unphilosophical, he pays no attention to expressing clearly the fundamental assumptions and their connection with existing theories. But when I talk to him he pleases me very much and I see that he has all sorts of new arguments…. I believe that some time in the future he will greatly advance science.” Pauli was to be proved right.
Unlike Pauli, Heisenberg thrived in periods of chaos. Far from despairing, he would go all out to find a solution. He welcomed the stretch of the imagination required by Bohr’s virtual oscillators. He used his immense experience in every aspect of atomic physics, together with his natural audaciousness, spurred on by Pauli’s critical comments—among them that he should deal only with quantities that can be measured in the laboratory, such as the energy and momentum of electrons, and avoid abstract concepts such as orbits of electrons. “We must adjust our concepts to experience,” was the approach Pauli suggested. Heisenberg worked day and night and came up with a whole new atomic physics that was to become known as quantum mechanics. Full of excitement, Pauli wrote that Heisenberg’s work gave him “new hope and a renewed enjoyment in life.”
“We must adjust our concepts to experience”
Like every highly creative scientist of his era, Pauli was a philosophical opportunist. He picked and chose from whatever philosophy had to offer to tackle the problem at hand. Scientists use philosophy when they ask the deepest of questions, such as What constitutes a scientific theory? What sort of physical objects should it consider and how should it treat them? What is physical reality?
At the beginning of the twentieth century these questions became crucial when scientists had to contend with objects—such as electrons and atoms—that they could not actually see. Classical ways of understanding the world suddenly seemed insufficient. An intellectual tidal wave—the avant-garde—swept across Europe.
Scientific concepts, ways of thinking, and ways of knowing were all being re-examined. Einstein did so when he discovered his special theory of relativity in 1905. This upheaval in thinking pervaded the world outside science too. In 1907 Pablo Picasso launched cubism with his “Les Demoiselles d’Avignon” and in 1910 Wassily Kandinsky unveiled abstract expressionism. In 1913 Igor Stravinsky ruptured all the conventions of classical ballet with his “Rite of Spring.” The postwar 1920s produced the twelve-tone music of Arnold Schönberg, Bauhaus architecture, and James Joyce’s extraordinary novels, which encompassed everything from relativity to cubism. Meanwhile Freud and Jung were investigating the unconscious.
Pauli first encountered this ferment of ideas through his godfather, the positivist Ernst Mach. As a boy he was spellbound by the scientific equipment in Mach’s apartment. Its ultimate purpose, said Mach, was to eliminate unreliable thinking—to demonstrate that the only thing that was really out there was what you can experience with your senses. The rest was all metaphysics—quite literally beyond physics and not worth considering, mere illusion.
Atoms could not be experienced with the senses. Did that mean they were merely “metaphysical,” in Mach’s pejorative sense? Were they not part of the elaborate scientific theories which made predictions that could be proved in the laboratory? According to Einstein’s theory of relativity—Pauli’s first scientific love—time turned out to depend on the motion of a clock and our world was four-dimensional, not three as everyone had always thought. The message of relativity theory seemed to be that scientists should look beyond what was immediately perceptible by the senses. It was to Einstein’s disappointment that he failed ever to convince Mach to accept relativity theory.
In the light of relativity theory Mach’s view seemed too restrictive. A group of young philosophers with strong scientific backgrounds began to meet in the coffeehouses of Vienna to discuss how to correct this situation, how to bring positivism into line with relativity theory. They called themselves the Vienna Circle and came up with a sophisticated version of positivism that they dubbed “logical positivism.” Then they renamed it “logical empiricism”: the word “empiricism” refers to experimental data (empirical data). Logical empiricism emphasized the role of mathematics in that a theory required a consistent logical or mathematical structure. Mach, on the other hand, regarded mathematics as merely an economical way to summarize experimental data.
In the view of the Vienna Circle a scientific theory had to be built on empirical data with the help of mathematics and had to generate predictions that could be tested in the laboratory. Science was a two-way street, beginning with data and ending with predictions that could be verified by data in the laboratory. Logical empiricism also insisted that every concept in a scientific theory must be measurable. Distance could be measured with a ruler, time by clocks, and so on. Thus they claimed that Einstein’s discovery of relativity theory was actually in accordance with positivism.
As for atoms, this was just a name for a list of experimental results. The rays emerging from cathode-ray tubes—primitive television tubes—were assumed to be a sort of light ray with an electric charge. Actually, every scientist knew that cathode rays were made up of electrons. Both Mach and the logical empiricists declared that atoms were not real as they could not be seen or measured individually. But the logical empiricists were able to see a way around Mach’s rejection of Einstein’s theory of the relativity of time in that it emerged from a consistent mathematics and experiments had been done to illustrate it in the laboratory. Mach’s philosophical heirs made the important point that the criterion “to observe something in the laboratory” had to be replaced by “to ascertain it or measure it in the laboratory.”
Pauli was well read in philosophy and introduced himself to the then-doyen of the Vienna Circle, the German-born Moritz Schlick. Schlick was twice his age and an esteemed professor at the University of Vienna, where he had taken over Mach’s position. Schlick was impressed with Pauli’s philosophical acumen. Pauli did not let the fact that he was a mere postdoctoral student hinder him from giving Schlick his blunt assessment of positivism. He had no objection to it, he wrote in 1922, “But, of course, it is not the only [philosophical approach].”
Indeed it was not. With the rise of psychoanalysis scientists began to look into how they had come up with their discoveries. Einstein wrote, “There is no logical path to these laws; only intuition, resting on sympathetic understanding of experience, can reach them.” In this way, he added, scientists could glimpse the “pre-established” harmony of the universe. Logical empiricists, however, saw this as aimless babble conjured up by scientists years after the fact.
In their view scientists constructed theories by moving logically—mathematically—from experimental data to a theory. They churned out equation after equation until they had solved the problem at hand. Einstein considered this wrongheaded. Scientists were unanimous in agreeing that their methods of research bore no resemblance to the proposals of positivists and logical empiricists. The key point for creative scientists such as Einstein was the delicate balance they had to maintain between the information obtained from experimental data and the laws of the theory as expressed in mathematics.
Pauli undoubtedly read Einstein’s views as well as the famous polemic in the first decade of the twentieth century between Mach and the discoverer of quantum theory, Max Planck, whose opinions were similar to Einstein’s. Planck accused Mach of degrading physics whereupon Mach simply withdrew in disgust: “I cut myself off from the physicist’s mode of thinking.”
Einstein believed, as did many scientists, in a world beyond perceptions in which electrons actually existed. Philosophers called this view “scientific realism.” There were scores of hybrid philosophies besides scientific realism and positivism which asserted that in fact there was nothing “out there.” Pauli counted himself a “‘heretic,’ not bowing down to any God, authority or ‘ism.’”
As a philosophical opportunist, Pauli saw that positivism offered a way out of the morass of 1925, when Bohr’s theory of the atom had collapsed with nothing to replace it. He thus advised Heisenberg to drop the unmeasurable concept of electron orbits and focus instead on measurable concepts like energy and momentum. This meant dropping the reassuring visual image of the atom as a solar system. Pauli’s belief was that once the “systems of concepts are settled,” that is, once the new atomic theory had been worked out, then “will visual imagery be regained,” as he wrote to Bohr. At Bohr’s Institute, Heisenberg and Bohr shared all correspondence from Pauli and eagerly awaited it.
Quantum mechanics—the new atomic physics
Heisenberg’s quantum mechanics identified individual electrons within atoms by the radiation they emitted while jumping between different stationary states, that is, the condition of an electron characterized by four quantum numbers as well as its momentum and energy, measurable as spectral lines. The transitions, or jumps, of the electrons maintained the flavor of the discontinuous quantum jumps in Bohr’s theory of the atom. Discontinuities were a fact of life in the world of the atom, especially in a theory based on electrons as particles.
Pauli was convinced that Heisenberg’s quantum mechanics would make it possible to solve problems that he had been unable to solve with the old Bohr theory. Late in 1925, he set out to calculate the stationary states for the simplest atom—hydrogen—using quantum mechanics. It involved juggling very complex mathematics but he came up with the answer with amazing speed.
Werner Heisenberg in 1925, when he discovered quantum mechanics.
Bohr applauded Pauli’s “wonderful results.” Heisenberg complained he was “a bit unhappy” that he had not solved the problem himself, but was full of admiration and surprise that Pauli had done it “so quickly.”
Irked that Pauli had stolen a march on him, just a month later Heisenberg, along with Pascual Jordan, another brilliant young physicist, tried applying quantum mechanics to the problem that had driven everyone to despair—the anomalous Zeeman effect. Just as Kepler’s ellipses had eliminated the cumbersome circles moving on circles, so Pauli’s new concept of spin—part of Heisenberg’s quantum mechanics—at a stroke swept away the concepts of massive inert cores with their two-valuedness and strange forces which had cluttered up Bohr’s theory. The problem had finally been put to rest, and the solution also helped set Heisenberg’s quantum mechanics on a firm basis. This time they had the theory right.
Physicists applauded these calculational breakthroughs. But no one—including Bohr, Heisenberg, and Pauli—really understood the theory itself, because the properties of atomic entities were so impossible to imagine. Not only was it unfamiliar and difficult to use, the mathematics of Heisenberg’s quantum mechanics lacked any helpful visual image. Being a hybrid version of Bohr’s virtual oscillators, it was like trying to visualize infinity. Its fundamental particles were also unvisualizable. But this was fine with Heisenberg, who felt the time was not ripe for a return to visual images which, in the past, had always turned out to be misleading.
Then the French physicist Louis de Broglie suggested that electrons might be waves—in other words, that material objects, such as ourselves, might be considered as waves. His inspiration was Einstein’s discovery, made some two decades previously, that light—traditionally thought of as a wave—could also be a particle, dubbed a light quantum. Perhaps electrons as well as light might be both wave and particle at the same time—simply beyond imaginable.
In spring 1926, the flamboyant Erwin Schrödinger, at the University of Zürich, burst on the scene. At thirty-nine, Schrödinger was an outsider in age, temperament, and thought to the group of impetuous twenty-something quantum physicists who clustered around Bohr in Copenhagen, Sommerfeld in Munich, and Born in Göttingen.
Schrödinger had found the equation that converted de Broglie’s vision of matter as waves into a coherent theory. His version of atomic physics, which he called wave mechanics, was based on treating light and electrons as waves. “My theory was inspired by L. de Broglie,” he wrote. “No genetic relation whatever with Heisenberg is known to me. I knew of his theory, of course, but felt discouraged, not to say repelled, by the methods of the transcendental algebra, which appeared very difficult to me, and by the lack of visualizability.”
Schrödinger’s wave mechanics sprang from a preference for a mathematics that was more familiar and beautiful, as opposed to what he referred to as Heisenberg’s ugly “transcendental algebra.” The “Schrödinger equation” offered great advantages in calculations over Heisenberg’s quantum mechanics, added to which it enabled the electron in an atom to be visualized as a wave surrounding the nucleus. It had taken Pauli twenty-odd pages to solve the hydrogen atom problem. Schrödinger did it in a page.
Schrödinger pointed out that the wave nature of matter promised a return to classical continuity. The passage of an electron between stationary states could be envisioned as a string passing continuously from one mode of oscillation to another.
One year earlier there had been no viable atomic theory. Now there were two: Heisenberg’s quantum mechanics and Schrödinger’s wave mechanics.
Heisenberg was furious about Schrödinger’s work and even more so about its rave reviews from the physics community. “The more I reflect on the physical portion of Schrödinger’s theory, the more disgusting I find it,” he wrote to Pauli. “What Schrödinger writes on the visualizability of his theory I consider crap.”
Heisenberg saw wave functions—that is, the solutions to Schrödinger’s wave equation—as nothing more than a means to expedite calculations. To demonstrate this he applied them to the problem that had driven Born, Pauli, and himself to despair: to find a mathematical way to describe the properties of the helium atom. No one had been able to deduce stable orbits, or stationary states, for the two electrons in the helium atom using Bohr’s theory of the atom. This being the case, they could not move on to deduce spectral lines for the helium atom because these resulted from its electrons dropping down from a higher to a lower orbit. Instead, the electrons’ orbits remained unstable, meaning that an electron could be knocked out of the helium atom by the smallest of disturbances.
But in Heisenberg’s quantum mechanics there were no orbits. The problem became one of deducing the atom’s spectral lines from its stationary states expressed directly in terms of the electrons’ energy and momentum in the atom. If the spectral lines turned out to be wrong, it would show that there were serious problems with the way quantum mechanics defined stationary states, that is, the energy levels of electrons in atoms. The spectral lines of the helium atom were particularly interesting to physicists because, as had been observed in the laboratory, they fell into two distinct groups. But why should this be the case?
Insight into the exclusion principle
The helium atom has two electrons. Using his quantum mechanics Heisenberg showed how the two sets of spectra arise. To elucidate his result and speed up his calculation of numerical values for the spectral lines, he used Schrödinger’s wave functions—the solutions to the Schrödinger equation—for both the spins and positions of these two electrons. The total wave function is the result of multiplying these two wave functions together. But there are many possible ways of constructing the total wave function for these two electrons.
Heisenberg found that only one sort produced the two distinct groups of spectral lines characteristic of the helium atom. This particular wave function had a unique property. It changed its sign when the spins and positions of the electrons were swapped. It was antisymmetrical, which also meant that it went to zero if the electrons had the same spins or positions.*
What was nature’s selection device for choosing these two sets of wave functions for the two spectra out of the several possible ones? Heisenberg was stumped. Something strange was going on here. Perhaps it related to Pauli’s exclusion principle, according to which no two electrons could have the same spin and position. If they did then one of the two wave functions that make up the total wave function—either for their positions or for their spins—would have to become zero. Perhaps that was the way nature selected the wave function suitable for a particular system of electrons. Thus Heisenberg realized that the exclusion principle was related to the symmetry property of the wave functions for a collection of electrons, in this case two electrons. It was a step forward in exploring its implications beyond making sense of the periodic table of elements.
It was a typical Heisenberg project. He chose a fundamental problem—in this case to understand the spectrum of the helium atom—and then let his intuition lead him into new terrain: the symmetry property of wave functions whether they are symmetric or antisymmetric. Thus he realized how essential the exclusion principle was for quantum mechanics: without it quantum mechanics could not be complete.
There was also the problem that had been Pauli’s original bête noir from his PhD thesis, in which he showed that Bohr’s theory of the atom failed to produce a stable hydrogen-molecule ion, H+2, even though it existed in nature. This problem vexed Born and Heisenberg as well. Pauli wrote to his friend Wentzel, “In Copenhagen sits a gentleman who is calculating H+2 according to Schrödinger.” The “gentleman” was the Danish physicist Øvind Burrau who, as Pauli pointed out, started directly with Schrödinger’s wave mechanics as opposed to starting from the quantum mechanics as Heisenberg had done and used Schrödinger’s wave mechanics only for calculations. As a result he was able to solve the problem simply. Heisenberg wrote to Pauli that, in his opinion, Burrau had straightened out the situation and mentioned the symmetry properties of the wave functions that Burrau had deduced. Perhaps Heisenberg had hoped to find a solution starting from his quantum mechanics. But these once-key problems had become mere calculations now that the correct atomic physics had been worked out.
Although problem after problem that had resisted solution using the old Bohr theory was now being solved by atomic physics, the meaning of the theories used—Heisenberg’s quantum mechanics and Schrödinger’s wave mechanics—was still not understood. And the tension between the two factions was growing.
To the Schrödinger faction Burrau’s successful result, as well as Heisenberg’s for the helium atom (despite his assertion that he had used Schrödinger’s theory merely to facilitate calculations) was proof that Schrödinger’s theory offered a solution to every problem of atomic structure, whereas Heisenberg’s was daunting to use and ugly. This of course greatly pleased Schrödinger.
Heisenberg’s uncertainty principle
In fall 1926, Bohr summoned Heisenberg to his institute in Copenhagen to hammer out a resolution to the dispute with Schrödinger. They struggled for days over numerous cups of tea and bottles of Carlsberg beer. Heisenberg wrote to Pauli: “What the words ‘wave’ or ‘particle’ mean we know not any more; [we are in a] state of almost complete despair.”
The crux of the problem was this: how could ordinary language, with its visual connotations, be used to describe a realm of nature that defied the imagination?
While Bohr and Heisenberg were deliberating in Copenhagen, Pauli had an idea. He immediately wrote it up and mailed it to Heisenberg. It was based on an insight Born had had, looking into Schrödinger’s wave equation.
Born had suggested that the wave function was a wave of probability for an electron moving between stationary states. Pauli pushed the idea further. He realized that the wave function gave the probability of an electron being detected in a certain region of space. In his usual way, he didn’t bother to write a paper to publish this idea and in the end Born took the credit.
But as he was working out the mathematics for Heisenberg, he came up with another extraordinary discovery: if he could determine a particle’s position accurately, he could not determine its momentum with the same accuracy. Pauli was puzzled as to why this should be so. Why couldn’t he determine both with the same degree of accuracy? Heisenberg was struck by the insight. He was, he wrote to Pauli, “more and more inspired by the content of your last letter every time I reflect on it.”
By February 1927 Bohr and Heisenberg had hit an impasse in their discussions on the deep meaning of the quantum theory, which seemed to be riddled with ambiguities. Bohr took a skiing break. Left to his own devices, Heisenberg set to work. The result was a paper that he called “On the Intuitive Content of the Quantum-Theoretical Kinematics and Mechanics.” Hidden behind this daunting title was one of the most earth-shattering discoveries of modern physics: the uncertainty principle.
Heisenberg realized that the supposed ambiguities of the quantum theory were essentially a problem of language. The issue was how to define words such as “position” and “velocity” in the ambiguous realm of the atom, a world in which “things” can be both wave and particle at the same time. He used the term “intuitive” in the title of his paper, for his goal was to redefine the word in the world of the atom.
Certain concepts in quantum physics, Heisenberg claimed, such as “position” and “momentum” (mass times velocity), were “derivable neither from our laws of thought nor from experiment.” Instead we would have to look into the peculiar mathematics of quantum mechanics, which should have alerted us all along that in the realm of the atom we would have to apply such words with great care.
In his paper Heisenberg made the amazing assertion that the more accurately we measure an electron’s momentum in a certain experiment, the less accurately we can measure its position in that experiment. This quickly became known as the uncertainty principle. It was earth-shattering in that it questioned our understanding of the inherent nature of the physical world as completely as Einstein’s relativity theory.
In the classical physics of Newton we can measure the position and momentum of an object with the same degree of accuracy by observing how it moves. Using a telescope and a clock we can measure both the location of a falling stone and how fast it is moving with an accuracy limited only by the width of the telescope’s crosshairs and the clock’s mechanism. If we make these errors as small as possible we can deduce very precisely the stone’s position and momentum. In principle, the product of the errors in measurements of position and momentum can both be zero. In quantum mechanics this is not possible.
Heisenberg wrote all this down in a detailed fourteen-page letter to Pauli. He asked for “severe criticism” after all, it was Pauli who had given him the key idea. Pauli was elated. “It becomes day in the quantum theory,” he declared.
Bohr’s complementarity and beyond
Bohr, however, was furious. He refused to let Heisenberg publish his paper on the subject, saying that Heisenberg had not provided any firm foundation for his argument. Furthermore, Heisenberg had based the argument entirely on the assumption that light and electrons behaved like particles.
Bohr insisted that electrons and light be understood as both wave and particle, even though this could not be imagined. One could visualize electrons and light as either a wave or a particle so long as one remembered the restrictions required by quantum mechanics, among them Heisenberg’s uncertainty principle understood within the larger context of waves and particles.
This meant that electrons in experiments could exhibit one aspect or the other, but not both at once. If one experimented on an electron as if it were a wave, that was what it would be for the duration of the experiment, and similarly if one treated it as a particle. Bohr called this “complementarity.”
Bohr was convinced that complementarity was relevant not only to physics but also to psychology and to life itself. Its basic idea, he wrote, “bears a deep-going analogy to the general difficulty in the formation of human ideas, inherent in the distinction between subject and object.” As in the Chinese concept of yin and yang, complementary pairs of concepts defined reality. There is nothing paradoxical about an electron having the characteristics of both a wave and a particle until an experiment is performed on it. It dawned on Bohr that in the weird quantum world there need not be only yes and no, an electron need not actually be either particle or wave. There could be in-betweens as well as ambiguities. An electron’s wave and particle aspects complement each other, and their totality makes up what the electron is. Thus the electron is made up of complementary pairs—wave and particle, and position and momentum. Similarly it is the tension between complementary pairs—love and hate, life and death, light and darkness—that shapes our everyday existence.
Bohr sent the manuscript of his article on complementarity to Pauli for corrections and critical remarks. Pauli replied immediately. Apart from certain comments on details, he entirely agreed with Bohr’s thesis.
Only the more philosophically inclined scientists took complementarity seriously. Pauli was one. He began to look to complementarity as another way to study consciousness as in the various ways of “knowing” practiced in the East and West. He was growing more and more interested in the conscious and the unconscious, the rational and the irrational, and in how physics could be used to understand these complementarities. He was beginning to suspect that this was to be his life’s work. The only problem was how to approach it.
Paul Dirac and quantum electrodynamics
The previous autumn the eccentric twenty-five-year-old English physicist Paul Dirac had visited the Bohr Institute. Dirac had already made important contributions to atomic physics and was eager to rub shoulders with other physicists of his generation whose papers he had studied in detail, Heisenberg and Pauli among them.
Dirac had been privy to the intense conversations between Bohr and Heisenberg on the issue of whether light and matter could be both wave and particle. In 1927 he was able to provide the vital clarification through a mathematical method he had developed for moving between the two and thus brought about “complete harmony between the wave and light quantum descriptions.” Dirac’s mathematical method ultimately concerned the way in which electrons and light interact. It formed the basis for a whole new subject, which scientists dubbed quantum electrodynamics. Pauli and Heisenberg worked enthusiastically to develop this new field.
Dirac’s equation
The following year Dirac came up with a crucial equation—the Dirac equation. It described how electrons interacted with light and also agreed with relativity theory. The equations in Heisenberg’s and Schrödinger’s theories did not agree with relativity, and while the spin of the electron had to be inserted into these theories, it popped right out of Dirac’s, thus underscoring the relationship of spin to relativity.
But Pauli and his colleagues were dissatisfied. Among other problems, Dirac’s equation for the electron predicted that objects with negative energy should actually exist. Physicists believed particles of negative energy to be like negative time: they simply could not exist. Heisenberg commented to Pauli that Dirac’s equation was the “saddest chapter in modern physics.” Furthermore, it did nothing to elucidate Pauli’s exclusion principle.
Pauli’s anti-Dirac equation
In 1934 Pauli set out to find an equation to supplant Dirac’s. His colleague in this was his assistant Victor Weisskopf. Weisskopf had been a student of Born’s and Bohr’s. Like Pauli, he was Viennese. The two shared a deep appreciation for literature and music, in particular Mozart’s operas. Weisskopf was also a concert-level pianist. Well over six feet tall, athletically built, and with a cultured air, he stands out in group photographs.
Viki, as Weisskopf became known to his friends, loved to tell the story of his first meeting with Pauli in the fall of 1933:
The first time I came to see him, I knocked at the door—no answer. He was in a very bad mood at that time, the whole period was a difficult one for him for personal reasons. When he didn’t answer, after a few minutes I opened the door. ‘You are Weisskopf; yes, you will be my assistant. I will tell you that I wanted to take [Hans] Bethe but he works on solid state [physics]. I don’t like this kind of physics although I started it.’ He gave me some problem…and after a few weeks I showed him what I had done; he was very dissatisfied with it and he said, ‘I should have taken Bethe.’
This was Pauli’s idea of humor, but it also shows what a difficult, sharp-tongued man he could be. Many people could not handle his sardonic sense of humor and this led, no doubt, to some physicists thinking twice before taking on the job of his assistant. But despite the inauspicious start Pauli’s collaboration with Weisskopf was to be both fruitful and memorable.
The two came up with an equation that had many of the same properties as Dirac’s and agreed with relativity theory. But while Dirac’s was for any particle with half a unit of spin, theirs was for particles with no spin. None had been detected at the time. However, when they included spin in their equation it no longer agreed with relativity.
But why? As he was fiddling about with his equations Pauli realized something entirely new: that particles with no spin differ fundamentally from particles with half a unit of spin. Particles with half a unit of spin—1/2, 3/2 and so on (known as Fermions after the Italian physicist Enrico Fermi)—obeyed Fermi-Dirac statistics (discovered by Fermi and Dirac in 1926), meaning that the overall wave function (that is, the solution of the Schrödinger equation) for a collection of Fermions exhibited antisymmetry.*
The only known such particles at the time were the electron, neutron, neutrino, and proton. Particles with whole-number spin—zero, one, and so on (called Bosons after the Indian physicist Satyendra Nath Bose)—obey Bose-Einstein statistics (discovered by Bose and Einstein in 1924), meaning that they have an overall wave function that remained the same when their positions and spins are exchanged; it was symmetrical. In other words, the two sets of particles have different symmetry properties. From this Pauli deduced that the exclusion principle applies to particles with half a unit of spin and not to particles with a whole unit of spin.
There was no obvious reason for this. The only conclusion was that nature had spoken. Something more than mathematics was involved in wave equations. Physicists now began investigating the properties of wave equations for particles of any sort of spin. The difficult mathematics involved was very much to Pauli’s taste.
Six years later, in 1940, he summarized and extended this work. Instead of using any particular wave equation such as Dirac’s, his own, or any other, he used the mathematical properties of wave functions to explore how they behaved when relativity theory, spin, statistics, and the exclusion principle were applied to them. He cut through all the mathematics to deduce a conclusion that was highly significant. The exclusion principle, he discovered, cannot be applied to any theory that includes relativity and applies to particles with whole-number spin; but it is essential to theories dealing with particles with half-units of spin.
The “connection between spin and statistics is one of the most important applications of special relativity theory,” he wrote. He had been trying for a long time to find a connection between spin, statistics, and relativity and at last he had done it. Compact yet rigorous in its mathematical presentation, the paper he wrote on the subject was Pauli at his best. Many physicists regard it as his most brilliant.
Finally, some sixteen years after Pauli had first come up with the exclusion principle and with the concept of spin—and had first realized that there were four, not three quantum numbers—he had managed to discover some of the key implications of his first great discovery. From the start everyone had realized that the exclusion principle explained the periodic table of elements. Now it was known that it could be used as a tool to explore the behavior of every particle with half a unit of spin and that it had no connection with any other sort of particle.