Quantum: Einstein, Bohr and the Great Debate About the Nature of Reality - Manjit Kumar (2009)

Part II. BOY PHYSICS

'Physics at the moment is again very muddled; in any case, for me it is too complicated, and I wish I were a film comedian or something of that sort and had never heard anything about physics.'

—WOLFGANG PAULI

'The more I think about the physical portion of the Schrödinger theory, the more repulsive I find it. What Schrödinger writes about the visualizability of his theory "is probably not quite right", in other words it's crap.'

—WERNER HEISENBERG

'If all this damned quantum jumping were really here to stay, I should be sorry I ever got involved with quantum theory.'

—ERWIN SCHRÖDIGNER

Chapter 7. SPIN DOCTORS

'One wonders what to admire most, the psychological understanding for the development of ideas, the sureness of mathematical deduction, the profound physical insight, the capacity for lucid, systematic presentation, the knowledge of the literature, the complete treatment of the subject matter, or the sureness of critical appraisal.'1 Einstein was certainly impressed by the 'mature, grandly conceived work' he had just reviewed. It was difficult for him to believe that the 237-page article, with 394 footnotes, on relativity was the work of a 21-year-old physicist who had been a student, and just nineteen, when asked to write it. Wolfgang Pauli, later nicknamed 'The Wrath of God', was acerbic and regarded as 'a genius comparable only with Einstein'.2 'Indeed from the point of view of pure science,' said Max Born, his one-time boss, 'he was possibly even greater than Einstein.'3

Wolfgang Pauli was born on 25 April 1900 in Vienna, a city still in the grip of fin de siècle anxiety while enjoying the good times. His father, also called Wolfgang, had been a physician, but abandoned medicine for science and in the process changed his family name from Pascheles to Pauli. The transformation was complete as he converted to Catholicism amid fears that the rising tide of anti-Semitism threatened his academic ambitions. His son grew up knowing nothing of the family's Jewish ancestry. At university, when another student said that he must be Jewish, Wolfgang junior was astonished: 'I? No. Nobody has ever told me that and I don't believe that I am.'4 He learnt the truth from his parents during his next visit home. His father felt vindicated by the decision to assimilate when, in 1922, he was appointed to a coveted professorship and became the director of a new institute for medical chemistry at Vienna University.

Pauli's mother, Bertha, was a well-known Viennese journalist and writer. Her circle of friends and acquaintances meant that Wolfgang and his sister Hertha, six years his junior, grew up accustomed to seeing leading figures from the arts as well as science and medicine at the family home. His mother, a pacifist and a socialist, exerted a strong influence on Pauli. The longer the First World War dragged on through his formative teenage years, 'the keener became his opposition against it and, generally, against the whole "establishment"', recalled a friend.5 When she died two weeks before her 49th birthday in November 1927, an obituary in the Neue Freie Presse described Bertha as 'one of the few truly strong personalities among Austrian women'.6

Pauli was academically gifted but far from a model pupil, finding school unchallenging. He began having private tuition in physics to compensate. Before long, when bored by a particularly tedious lesson at school, he began reading Einstein's papers on general relativity hidden under his desk. Physics had always loomed large in his young life in the form of the influential Austrian physicist and philosopher of science Ernst Mach, his godfather. For one who would later enjoy the company and friendship of the likes of Einstein and Bohr, Pauli said that contact with Mach, whom he last saw in the summer of 1914, was 'the most important event in my intellectual life'.7

In September 1918 Pauli left what he called the 'spiritual desert' that was Vienna.8 With the Austria-Hungarian empire on the verge of extinction and Vienna's past glories faded, it was the lack of top-flight physicists at the city's university that he was lamenting. He could have gone almost anywhere, but went to Munich to study with Arnold Sommerfeld. Having recently turned down a professorship in Vienna, Sommerfeld had already been in charge of theoretical physics at Munich University for a dozen years when Pauli arrived. From the beginning, in 1906, Sommerfeld set out to create an institute that would be 'a nursery of theoretical physics'.9 It was not as grand as the institute Bohr would soon create in Copenhagen, consisting as it did of only four rooms: Sommerfeld's office, a lecture theatre, a seminar room, and a small library. There was also a large laboratory in the basement where in 1912 Max von Laue's theory that X-rays were short-wavelength electromagnetic waves was tested and confirmed, bringing quick recognition to the 'nursery'.

Sommerfeld was an exceptional teacher with the uncanny knack of setting his students problems that tested, but did not exceed, their abilities. Having already supervised more than his fair share of talented young physicists, Sommerfeld soon recognised Pauli as someone of rare and exceptional promise. He was a man not easily impressed, but in January 1919 a paper on general relativity written by Pauli before leaving Vienna had just been published. In his 'nursery' he had a first-year student, not yet nineteen years old, who was already regarded by others as an expert in relativity.

Pauli quickly became known, and feared, for his sharp and incisive criticism of new and speculative ideas. Some would later call him the 'conscience of physics' for his uncompromising principles. Stout with bulging eyes, he was every inch the Buddha of physics, albeit one with a biting tongue. Whenever he was lost deep in thought, Pauli unconsciously rocked back and forth. It was acknowledged far and wide that his intuitive grasp of physics was unmatched among his contemporaries and probably not even surpassed by Einstein. He judged his own work even more harshly than that of others. At times Pauli understood physics and its problems too well, and that hampered the free exercise of his creative powers. Discoveries that he might have made if his imagination and intuition had roamed a little more freely went instead to colleagues less talented and unconstrained.

The only person towards whom he was, and remained, diffident was Sommerfeld. Even as a celebrated physicist, whenever Pauli found himself in the presence of his former professor, those who had been on the receiving end of his sharp judgements were always amazed to see the 'Wrath of God' responding with 'Ja, Herr Professor', 'Nein, Herr Professor'. They hardly recognised the man who had once ticked off a colleague: 'I do not mind if you think slowly, but I do object when you publish more quickly than you think.'10 Or on another occasion saying of a paper he had just read: 'It is not even wrong.'11 He spared no one. 'You know, what Mr Einstein said is not so stupid', he told a packed lecture theatre while still a student.12 Sommerfeld, sitting in the front row, would not have tolerated such a remark coming from any of his other students. But then he knew none of them would have uttered it. When it came to evaluating physics, Pauli was self-confident and uninhibited even in the presence of Einstein.

In a clear sign of the high regard in which he held Pauli, Sommerfeld asked him to help write a major article on relativity for the Encyklopädie der Mathematischen Wissenschaffen. Sommerfeld had accepted the task of editing the fifth volume of the Encyklopädie that dealt with physics. After Einstein declined, Sommerfeld decided to write on relativity himself but found he had little time to do so. He needed help and turned to Pauli. When Sommerfeld saw the first draft, 'it proved to be so masterly that I renounced all collaboration'.13 It was not only a brilliant exposition of the special and general theories of relativity, but an unrivalled review of the existing literature. It remained for decades the definitive work in the field and drew Einstein's wholehearted praise. The article appeared in 1921, two months after Pauli received his doctorate.

As a student, Pauli preferred to spend his evenings enjoying the Munich nightlife in some café or other, returning to his lodgings to work through much of the night. He rarely attended lectures the following morning, turning up only around noon. But he attended enough to be drawn to the mysteries of quantum physics by Sommerfeld. 'I was not spared the shock which every physicist accustomed to the classical way of thinking experienced when he came to know Bohr's basic postulate of quantum theory for the first time', Pauli said more than 30 years later.14 But he quickly got over it as he set about tackling his doctoral thesis.

Sommerfeld had set Pauli the task of applying the quantum rules of Bohr and his own modifications to the ionised hydrogen molecule, in which one of the two hydrogen atoms that make up the molecule has had its electron ripped off. As expected, Pauli produced a theoretically impeccable analysis. The only problem was that his results did not agree with the experimental data. Used to one success after another, Pauli was despondent at this lack of agreement between theory and experiment. However, his thesis was regarded as the first strong evidence that the outer limits of the Bohr-Sommerfeld quantum atom had been reached. The ad hoc way in which quantum physics was bolted onto classical physics had always been unsatisfactory, and now Pauli had shown that the Bohr-Sommerfeld model could not even deal with the ionised hydrogen molecule, let alone more complex atoms. In October 1921, armed with his doctorate, Pauli left Munich for Göttingen to take up the post of assistant to the professor of theoretical physics.

Max Born, 38, a key figure in the future development of quantum physics, had arrived in the small university town from Frankfurt just six months before Pauli. Growing up in Breslau, capital of the then Prussian province of Silesia, it was mathematics, not physics, that attracted Born. His father, like Pauli's, was a highly cultured medical man and academic. A professor of embryology, Gustav Born advised his son not to specialise too early once he enrolled at Breslau University. Dutifully, Born settled on astronomy and mathematics only after having attended courses in physics, chemistry, zoology, philosophy and logic. His studies, including time at the universities of Heidelberg and Zurich, ended in 1906 with a doctorate in mathematics from Göttingen.

Immediately afterwards he began a year of compulsory military service that was cut short because of asthma. After spending six months in Cambridge as an advanced student, where he attended the lectures of J.J. Thomson, Born returned to Breslau to begin experimental work. But quickly discovering that he possessed neither the patience nor the skills required to be even a competent experimenter, Born turned to theoretical physics. By 1912 he had done enough to become a privatdozent in the world-renowned mathematics department at Göttingen, where they believed that 'physics is much too hard for physicists'.15

Born's success in tackling a string of problems by harnessing the power of mathematical techniques unknown to most physicists led in 1914 to an extraordinary professorship in Berlin. Just before war broke out, another newcomer arrived at the epicentre of German science: Einstein. Before long the two men, who shared a passion for music, became firm friends. When war came, Born was called up for military service. After a spell as a radio operator with the air force, he spent the rest of the war conducting artillery research for the army. Fortunately stationed near Berlin, Born was able to attend seminars at the university, meetings of the German Physical Society, and musical evenings at Einstein's home.

After the war, in the spring of 1919, Max von Laue, an ordinary professor at Frankfurt, suggested to Born that they swap posts. Laue had won the 1914 Nobel Prize for the theory behind the diffraction of X-rays by crystals, and wanted to work with Planck, his former supervisor and a scientist he idolised. Born, encouraged by Einstein to 'definitely accept', agreed, as the exchange meant promotion to a full professorship and independence.16 Less than two years later, he moved to Göttingen to head the university's institute of theoretical physics. It consisted of one small room, one assistant, and a part-time secretary, but Born was determined to build on these humble beginnings an institute to rival Sommerfeld's in Munich. High on his list of priorities was getting Wolfgang Pauli, whom he described as 'the greatest talent in the physics area that has emerged in the last years'.17 Born had already tried once before and failed, as Pauli opted to stay in Munich to finish his doctorate. This time he got his man.

'W. Pauli is now my assistant; he is amazingly intelligent, and very able', Born wrote to Einstein.18 Soon he discovered that the hired help had his own way of doing things. Pauli might have been brilliant, but he put in long hours of hard thinking as he continued his practice of working into the middle of the night and sleeping late. Whenever Born was unable to give his eleven o'clock lecture, the only way he could ensure Pauli would be there to teach in his place was by sending the maid to wake him up at 10.30am.

It was clear from the beginning that Pauli was an 'assistant' in name only. Born admitted later that he learnt more from Pauli, despite his bohemian ways and poor time-keeping, than he was able to teach the 'infant prodigy'. He was sad to see him go when in April 1922 Pauli left to become an assistant at Hamburg University. Swapping the quiet life of the small university town that he could hardly bear for the bustling nightlife of the big city was not the only reason he left so quickly. Pauli trusted his sense of physical intuition in pursuit of a logically flawless argument when tackling any physics problem. Born, however, turned much more readily to mathematics and allowed it to lead his search for a solution.

Two months later, in June 1922, Pauli was back in Göttingen to hear Bohr's celebrated lecture series and met the great Dane for the first time. Impressed, Bohr asked Pauli if he would come to Copenhagen for a year as his assistant to help edit work in progress for publication in German. Pauli was taken aback by the offer. 'I answered with that certainty of which only a young man is capable: "I hardly think that the scientific demands which you will make on me will cause me any difficulty, but the learning of a foreign tongue like Danish far exceeds my abilities." I went to Copenhagen in the fall of 1922, where both my contentions were shown to be wrong.'19 It was also, he recognised later, the beginning of 'a new phase' in his life.20

Aside from helping Bohr, Pauli made a serious effort in Copenhagen to explain the 'anomalous' Zeeman effect – a feature of atomic spectra that the Bohr-Sommerfeld model could not explain. If atoms were exposed to a strong magnetic field, then the resulting atomic spectra contained lines that were split. It was quickly shown by Lorentz that classical physics predicted a splitting of a line into a doublet or a triplet: a phenomenon known as the 'normal' Zeeman effect which Bohr's atom could not accommodate.21 Fortunately, Sommerfeld came to the rescue with two new quantum numbers and the modified quantum atom resolved the problem. It involved a series of new rules governing electrons jumping from one orbit (or energy level) to another based on three 'quantum numbers', n, k, and m, that described the size of the orbit, the shape of the orbit, and the direction in which the orbit was pointing. But the celebrations were short-lived when it was discovered that the splitting of the red alpha line in the spectrum of hydrogen was smaller than expected. The situation grew worse with the confirmation that some spectral lines actually split up into a quartet or more instead of just two or three lines.

Although called the 'anomalous' Zeeman effect because the extra lines could not be explained using either existing quantum physics or classical theory, it was in fact far more common than the 'normal' effect. For Pauli it signalled nothing less than the 'deep seated failure of the theoretical principles known till now'.22 Having set himself the task of rectifying this miserable state of affairs, Pauli could not come up with an explanation. 'Up till now I have thoroughly gone wrong', he wrote to Sommerfeld in June 1923.23 Consumed by the problem, Pauli later admitted that he was in complete despair for some time.

One day another physicist from the institute met him while strolling around the streets of Copenhagen. 'You look very unhappy', said his colleague. Pauli turned on him: 'How can one look happy when he is thinking about the anomalous Zeeman effect?'24 The use of ad hoc rules to describe the complex structure of atomic spectra was just too much for Pauli. He wanted a deeper, more fundamental explanation of the phenomena. Part of the problem, he believed, was the guesswork involved in Bohr's theory of the periodic table. Did it really describe the correct arrangement of electrons inside atoms?

By 1922 the electrons in the Bohr-Sommerfeld model were believed to move in three-dimensional 'shells'. These were not physical shells, but energy levels within atoms around which electrons seemed to cluster. A vital clue in helping Bohr construct this new electron shell model was the stability of the so-called noble gases: helium, neon, argon, krypton, xenon and radon.25 With atomic numbers of 2, 10, 18, 36, 54 and 86, the relatively high energies required to ionise any noble gas atom – to rip away an electron and turn it into a positive ion – together with their reluctance to chemically bond with other atoms to form compounds, suggested that the electron configurations in these atoms were extremely stable and consisted of 'closed shells'.

The chemical properties of the noble gases were in stark contrast to the elements that preceded them in the periodic table – hydrogen and the halogens: fluorine, chlorine, bromine, iodine, and astatine. With atomic numbers 1, 9, 17, 35, 53 and 85, all of these elements easily formed compounds. Unlike the chemically inert noble gases, hydrogen and the halogens united with other atoms because in the process they picked up another electron and thereby filled the single vacancy in the outermost electron shell. By doing so, the resulting negative ion had a completely full or 'closed' set of electron shells and acquired the highly stable electronic configuration of a noble gas atom. Mirroring the halogens, the alkalis group – lithium, sodium, potassium, rubidium, caesium and francium – were quick to lose an electron as they formed compounds and became positive ions with the electron distribution of a noble gas.

The chemical properties of these three groups of elements formed part of the evidence that led Bohr to propose that the atom of each element in a row of the periodic table is built up from the previous element by the addition of another electron to the outer electron shell. Each row would end with a noble gas in which the outer shell was full. Since only electrons outside the closed shells, called valence electrons, took part in chemical reactions, atoms with the same number of valence electrons shared similar chemical properties and occupied the same column in the periodic table. The halogens all have seven electrons in the outermost shell, requiring just one more electron to close it and acquire an electron configuration of a noble gas. The alkalis, on the other hand, all have one valence electron.

It was these ideas that Pauli heard Bohr outline during the Göttingen lectures in June 1922. Sommerfeld had greeted the shell model as 'the greatest advance in atomic structure since 1913'.26 If he could mathematically reconstruct the numbers 2, 8, 18 … of the elements in the rows of the periodic table, then it would be, Sommerfeld told Bohr, 'the fulfilment of the boldest hopes of physics'.27 In truth, there was no hard mathematical reasoning to back up the new electron shell model. Even Rutherford told Bohr that he was struggling 'to form an idea of how you arrive at your conclusions'.28 Nevertheless, Bohr's ideas had to be taken seriously, especially after the announcement in his Nobel lecture in December 1922 that the unknown element with atomic number 72, later called hafnium, did not belong to the 'rare earth' group of elements was later confirmed to be correct. However, there was no organising principle or criteria behind Bohr's shell model. It was an ingenious improvisation based on an array of chemical and physical data that could in large part explain the chemical properties of the various groupings of elements in the periodic table. Its crowning glory was hafnium.

As he continued to fret over the anomalous Zeeman effect and the shortcomings of the electron shell model, Pauli's time in Copenhagen came to an end. In September 1923 he returned to Hamburg, where the following year he was promoted from assistant to privatdozent. But with Copenhagen a short train journey and a ferry across the Baltic Sea, Pauli was still a regular visitor to the institute. He concluded that Bohr's model could work only if there was a restriction on how many electrons could occupy any given shell. Otherwise, in contradiction of the results of atomic spectra, there seemed nothing to prevent all the electrons in any atom from occupying the same stationary state, the same energy level. At the end of 1924 Pauli discovered the fundamental organising rule, the 'exclusion principle', that provided the theoretical justification that had been missing in Bohr's empirically devised electron shell atomic model.

Pauli was inspired by the work of a Cambridge postgraduate student. Edmund Stoner, 35, was still working on his doctorate under Rutherford when in October 1924 his paper 'The Distribution of Electrons Among Atomic Levels' was published in the Philosophical Magazine. Stoner argued that the outermost or valence electron of an alkali atom has as many energy states to choose from as there are electrons in the last closed shell of the first inert noble gas that follows it in the periodic table. For example, lithium's valence electron could occupy any one of eight possible energy states, exactly the number of electrons in the corresponding closed shell of the gas neon. Stoner's idea implied that a given principal quantum number n corresponds to a Bohr electron shell which would be completely full or 'closed' when the number of electrons it contains reaches twice its number of possible energy states.

If each electron in an atom is assigned the quantum numbers n, k, m, and each unique set of numbers labels a distinct electron orbit or energy level, then according to Stoner, the number of possible energy states for, say, n=1, 2 and 3 would be 2, 8 and 18. For the first shell n=1, k=1 and m=0. These are the only possible values the three quantum numbers can have and they label the energy state (1,1,0). But according to Stoner, the first shell is closed when it contains 2 electrons, double the number of available energy states. For n=2, either k=1 and m=0 or k=2 and m=–1,0,1. Thus in this second shell there are four possible sets of quantum numbers that can be assigned to the valence electron and the energy states it can occupy: (2,1,0), (2,2,-1), (2,2,0), (2,2,1). Therefore, the shell n=2 can accommodate 8 electrons when it is full. The third shell, n=3, has 9 possible electron energy states: (3,1,0), (3,2,-1), (3,2,0), (3,2,1), (3,3,-2), (3,3,-1), (3,3,0), (3,3,1), (3,3,2).29 Using Stoner's rule, the n=3 shell can contain a maximum of 18 electrons.

Pauli had seen the October issue of the Philosophical Magazine, but ignored Stoner's paper. Not known for his athleticism, Pauli ran to the library to read it after Sommerfeld mentioned Stoner's work in the preface to the fourth edition of his textbook Atomic Structure and Spectral Lines.30 Pauli realised that for a given value of n, the number of available energy states, N, in an atom that an electron could occupy was equivalent to all the possible values that the quantum numbers k and m could take, and was equal to 2n2. Stoner's rule yielded the correct series of numbers 2, 8, 18, 32 … for the elements in the rows of the periodic table. But why was the number of electrons in a closed shell twice the value of N or n2? Pauli came up with the answer – a fourth quantum number had to be assigned to electrons in atoms.

Unlike the other numbers n, k, and m, Pauli's new number could have only two values, so he called it Zweideutigkeit. It was this 'two-valuedness' that doubled the number of electron states. Where there had previously been a single energy state with a unique set of three quantum numbers n, k, and m, there were now two energy states: n, k, m, A and n, k, m, B. These extra states explained the enigmatic splitting of spectral lines of the anomalous Zeeman effect. Then the 'two-valued' fourth quantum number led Pauli to the exclusion principle, one of the great commandments of nature: no two electrons in an atom can have the same set of four quantum numbers.

The chemical properties of an element are not determined by the total number of electrons in its atom but only by the distribution of its valence electrons. If all the electrons in an atom occupied the lowest energy level, then all the elements would have the same chemistry.

It was Pauli's exclusion principle that managed the occupancy of the electron shells in Bohr's new atomic model and prevented all of them from gathering in the lowest energy level. The exclusion principle provided the underlying explanation for the arrangement of the elements in the periodic table and the closing of shells with chemically inert rare gases. Yet despite these successes, Pauli admitted in his paper, 'On the Connection between the Closing of Electron Groups in Atoms and the Complex Structure of Spectra', published on 21 March 1925 in Zeitschrift für Physik: 'We cannot give a more precise reason for this rule.'31

Why four quantum numbers, and not three, were needed to specify the position of electrons in an atom was a mystery. It had been accepted since the seminal work of Bohr and Sommerfeld that an atomic electron in orbital motion around a nucleus moves in three dimensions and therefore requires three quantum numbers for its description. What was the physical basis of Pauli's fourth quantum number?

In the late summer of 1925 two Dutch postgraduate students, Samuel Goudsmit and George Uhlenbeck, realised that the property of 'two-valuednes' that Pauli had proposed was not just another quantum number. Unlike the three existing quantum numbers n, k, and m that specified the angular momentum of the electron in its orbit, the shape of that orbit, and its spatial orientation respectively, 'two-valuedness' was an intrinsic property of an electron that Goudsmit and Uhlenbeck called 'spin'.32 It was an unfortunate choice of name that conjured up images of spinning objects, but electron 'spin' was a purely quantum concept that solved some of the problems still besetting the theory of atomic structure while neatly providing the physical justification of the exclusion principle.

George Uhlenbeck, 24, had enjoyed his time in Rome as a private tutor to the son of the Dutch ambassador. He had secured the position in September 1922 after having gained the equivalent of a bachelor's degree in physics from Leiden University. No longer wishing to be a financial burden to his parents, it was the perfect opportunity for Uhlenbeck to be self-sufficient as he worked towards his master's degree. With no formal lectures to attend, he learned most of what he needed from books, with only the summer back at the university. Unsure whether to pursue a doctorate when he returned to Leiden in June 1925, Uhlenbeck went to see Paul Ehrenfest, who had succeeded Hendrik Lorentz as professor of physics, in 1912, after Einstein chose Zurich.

Ehrenfest, born in Vienna in 1880, had been a student of the great Boltzmann. Together with his Russian wife, Tatiana, who was a mathematician, Ehrenfest had produced a series of important papers in statistical mechanics as he eked out a living as a physicist in Vienna, Göttingen and St Petersburg. Over the twenty years as Lorentz's successor, Ehrenfest established Leiden as a centre of theoretical physics and in the process became one of the most respected figures in the field. He was renowned for his ability to clarify difficult areas of physics, rather than for any original theories of his own. His friend Einstein later described Ehrenfest as 'the best teacher in our profession' and one 'passionately preoccupied with the development and destiny of men, especially his students'.33 It was this concern for his students that led Ehrenfest to offer the wavering Uhlenbeck a two-year post as an assistant while he set about getting a doctorate. The offer proved irresistible. Ehrenfest, who ensured whenever possible that his trainee physicists worked together in pairs, introduced him to another graduate student, Samuel Goudsmit.

A year and a half younger than Uhlenbeck, Goudsmit had already published well-received papers on atomic spectra. He had arrived in Leiden in 1919 not long after Uhlenbeck, who called Goudsmit's first paper at only eighteen 'a most presumptuous display of self-confidence' but 'highly creditable'.34 Given his doubts, a clearly talented younger collaborator might have intimidated others, but not Uhlenbeck. 'Physics,' Goudsmit said towards the end of his life, 'was not a profession but a calling, like creative poetry, music composition or painting.'35 However, he had chosen physics simply because he had enjoyed science and mathematics at school. It was Ehrenfest who kindled a real passion for physics in the teenager as he set him tasks related to analysing and finding order in the fine structure of atomic spectra. While he was not the most studious, Goudsmit possessed an uncanny skill at making sense out of empirical data.

By the time Uhlenbeck returned to Leiden from his time in Rome, Goudsmit was spending three days a week in Amsterdam working in Pieter Zeeman's spectroscopy laboratory. 'The trouble with you is I don't know what to ask, all you know is spectral lines', Ehrenfest complained as he fretted about setting Goudsmit a much-delayed exam.36 Despite concerns that his flair for spectroscopy was having a detrimental impact on his all-round development as a physicist, Ehrenfest asked Goudsmit to teach Uhlenbeck the theory of atomic spectra. After Uhlenbeck was brought up to date on the latest developments, Ehrenfest wanted the pair to work on the alkali doublet lines – the splitting of spectral lines due to an external magnetic field. 'He knew nothing; he asked all those questions which I never asked', said Goudsmit.37 Whatever his shortcomings, Uhlenbeck had a thorough knowledge of classical physics that led him to pose intelligent questions that challenged Goudsmit's understanding. It was an inspired piece of pairing by Ehrenfest that ensured that each would learn from the other.

Throughout the summer of 1925 Goudsmit taught Uhlenbeck everything he knew about spectral lines. Then one day they discussed the exclusion principle, which Goudsmit thought was no more than another ad hoc rule that brought a little more order to the unholy mess of atomic spectra. However, Uhlenbeck immediately hit upon an idea that Pauli had already dismissed.

An electron could move up and down, back and forth, and side to side. Each of these different ways of moving physicists called a 'degree of freedom'. Since each quantum number corresponds to a degree of freedom of the electron, Uhlenbeck believed that Pauli's new quantum number must mean that the electron had an additional degree of freedom. To Uhlenbeck, a fourth quantum number implied that the electron must be rotating. However, spin in classical physics is a rotational motion in three dimensions. So if electrons spin in the same way, like the earth about its axis, there was no need for a fourth number. Pauli argued that his new quantum number referred to something 'which cannot be described from the classical point of view'.38

In classical physics, angular momentum, everyday spin, can point in any direction. What Uhlenbeck was proposing was quantum spin – 'two-valued' spin, spin 'up' or spin 'down'. He pictured these two possible spin states as an electron spinning either clockwise or anti-clockwise about a vertical axis as it orbits the atomic nucleus. As it did so, the electron would generate its own magnetic field and act like a subatomic bar magnet. The electron can line up either in the same or in the opposite direction as an external magnetic field. Initially it was believed that any allowed electron orbit could accommodate a pair of electrons provided that one had spin 'up' and the other spin 'down'. However, these two spin directions have very similar but not identical energies, resulting in the two slightly different energy levels that gave rise to the alkali doublet lines – two closely spaced lines in the spectra instead of one.

Uhlenbeck and Goudsmit showed that electron spin could be either plus or minus half, values that satisfied Pauli's restriction for the fourth quantum number to be 'two-valued'.39

By the middle of October, Uhlenbeck and Goudsmit had written a one-page paper and showed it to Ehrenfest. He suggested that the normal alphabetical order of names be reversed. Since Goudsmit had already published several well-received papers on atomic spectra, Ehrenfest was concerned that readers would think that Uhlenbeck was the junior partner. Goudsmit agreed, as 'it was Uhlenbeck who had thought of spin'.40 But as to the soundness of the concept itself, Ehrenfest was unsure. He wrote to Lorentz asking for 'his judgement and advice on a very witty idea'.41

Although 72, retired and living in Haarlem, Lorentz still travelled to Leiden once a week to teach. Uhlenbeck and Goudsmit met him one Monday morning after his lecture. 'Lorentz was not discouraging', said Uhlenbeck. 42 'He was a little bit reticent, said that it was interesting and that he would think about it.' A week or two later, Uhlenbeck went back to receive Lorentz's verdict and was given a stack of papers full of calculations in support of an objection to the very notion of spin. A point on the surface of a spinning electron, Lorentz pointed out, would move faster than the speed of light – something forbidden by Einstein's special theory of relativity. Then another problem was discovered. The separation of the alkali doublet spectral lines, predicted using electron spin, was twice the measured value. Uhlenbeck asked Ehrenfest not to submit the paper. It was too late. He had already sent it to a journal. 'You are both young enough to be able to afford a stupidity', Ehrenfest reassured him.43

When the paper was published on 20 November, Bohr was deeply sceptical. The following month he travelled to Leiden to participate in the celebrations to mark the 50th anniversary of Lorentz receiving his doctorate. As his train pulled into Hamburg, Pauli was waiting on the platform to ask Bohr what he thought about electron spin. The concept was 'very interesting', said Bohr. His well-worn put-down meant he believed that electron spin was flawed. How, he asked, could an electron moving in the electric field of the positively-charged nucleus experience the magnetic field necessary for producing the fine structure? When he arrived at Leiden, two men impatient to know his views on spin met Bohr at the station: Einstein and Ehrenfest.

Bohr outlined his objection about the magnetic field and was amazed when Ehrenfest said that Einstein had already resolved the problem by invoking relativity. Einstein's explanation, Bohr admitted later, was a 'complete revelation'. He now felt confident that any remaining problems surrounding electron spin would all sooner rather than later be overcome. Lorentz's objection was based on classical physics, of which he was a master. However, electron spin was a quantum concept. So this particular problem was not as serious as it first appeared. The British physicist Llewellyn Thomas solved the second. He showed that an error in the calculation of the relative motion of the electron in its orbit around the nucleus was responsible for the extra factor of two in the separation of the doublet lines. 'I have never since faltered in my conviction that we are at the end of our sorrows', Bohr wrote in March 1926.44

On the return leg of his trip, Bohr met more physicists eager to hear what he had to say about quantum spin. When his train stopped at Göttingen, Werner Heisenberg, who just a few months earlier had finished his stint as Bohr's assistant, and Pascual Jordan were waiting at the station. Electron spin, he told them, was a great advance. He then travelled to Berlin to attend the 25th anniversary celebrations of Planck's famous lecture to the German Physical Society in December 1900 that was the official birthday of the quantum. Pauli lay in wait at the station, having travelled from Hamburg to quiz the Dane once again. As he feared, Bohr had changed his mind and was now the prophet of electron spin. Unmoved by initial attempts to convert him, Pauli called quantum spin 'a new Copenhagen heresy'.45

A year earlier he had dismissed the idea of electron spin when a 21-year-old German-American, Ralph Kronig, had first proposed it. On a two-year odyssey around some of Europe's leading centres of physics after gaining his PhD at Columbia University, Kronig arrived in Tübingen on 9 January 1925, prior to spending the next ten months at Bohr's institute. Interested in the anomalous Zeeman effect, Kronig was excited when his host, Alfred Landé, told him that Pauli was expected the following day. He was coming to talk to Landé about the exclusion principle before submitting his paper for publication. Having studied under Sommerfeld and later served as Born's assistant in Frankfurt, Landé was highly regarded by Pauli. Landé showed Kronig a letter Pauli had written to him the previous November.

In the course of his life, Pauli wrote thousands of letters. As his reputation grew and the number of correspondents increased, his letters were highly prized and passed around and studied. For Bohr, who saw past the sarcastic wit, a letter from Pauli was an event. He would slip it into his jacket pocket and carry it around for days, showing it to anyone remotely interested in whatever problem or idea Pauli was dissecting. Under the cover of drafting a reply, Bohr would conduct an imaginary dialogue as though Pauli were seated in front of him smoking his pipe. 'Probably all of us are afraid of Pauli; but then again we are not so afraid of him that we dare not admit it', he once playfully declared.46

Kronig later recalled that as he read Pauli's letter to Landé his 'curiosity was aroused'.47 Pauli had outlined the need to label every electron inside an atom with a unique set of four quantum numbers and its consequences. Immediately Kronig began thinking about the possible physical interpretation of the fourth quantum number, and came up with the idea of an electron rotating about its axis. He was quick to appreciate the difficulties attached to such a spinning electron. However, finding it 'a fascinating idea', Kronig spent the rest of the day developing the theory and doing the mathematics.48 He had worked out much of what Uhlenbeck and Goudsmit would announce in November. When he explained his findings to Landé, both men were impatient for Pauli to arrive and give his seal of approval. Kronig was taken aback when Pauli ridiculed the notion of electron spin: 'That is surely quite a clever idea, but nature is not like that.'49 So fervent had Pauli been in rejecting the proposal, Landé tried to soften the blow: 'Yes, if Pauli says so, then it is not like that.'50 Dejected, Kronig abandoned the idea.

Unable to contain his anger when electron spin was quickly embraced, in March 1926 Kronig wrote to Bohr's assistant Hendrik Kramers. He reminded Kramers that he had been the first to suggest electron spin and had not published because of Pauli's derisive reaction. 'In future I shall trust my own judgement more and that of others less', he lamented, having learnt the lesson too late.51 Disturbed by Kronig's letter, Kramers showed it to Bohr. No doubt remembering his own dismissal of electron spin when Kronig had discussed it with him and others during his stay in Copenhagen, Bohr wrote to express his 'consternation and deep regret'.52 'I should not have mentioned the matter at all if it were not to take a fling at the physicists of the preaching variety, who are always so damned sure of, and inflated with, the correctness of their own opinion', replied Kronig.53

Despite feeling robbed, Kronig was sensitive enough to ask Bohr not to mention the whole sorry affair in public, since 'Goudsmit and Uhlenbeck would hardly be very happy about it'.54 He knew they were entirely blameless. However, both Goudsmit and Uhlenbeck became aware of what had happened. Uhlenbeck later openly acknowledged that he and Goudsmit 'were clearly not the first to propose a quantized rotation of the electron, and there is no doubt that Ralph Kronig anticipated what certainly was the main part of our ideas in the spring of 1925, and that he was discouraged mainly by Pauli from publishing his results'.55 It was proof, a physicist told Goudsmit, 'that the infallibility of the Deity does not extend to his self-styled vicar on earth'.56

In private, Bohr believed that Kronig 'was a fool'.57 If he was convinced of the correctness of his idea, then he should have published no matter what others thought. 'Publish or perish' is a rule not to be forgotten in science. In his heart, Kronig must have reached a similar conclusion. His initial outburst of bitterness towards Pauli amid the disappointment of missing out on electron spin had dissipated by the end of 1927. At only 28, Pauli was appointed professor of theoretical physics at the ETH in Zurich. He asked Kronig, who was once again spending time in Copenhagen, to become his assistant. 'Every time I say something, contradict me with detailed arguments', Pauli wrote to Kronig after he accepted the offer.58

By March 1926 the problems that had led Pauli to reject electron spin had all been resolved. 'Now there is nothing else I can do than to capitulate completely', he wrote to Bohr.59 Years later, most physicists assumed that Goudsmit and Uhlenbeck had received the Nobel Prize – after all, electron spin was one of the seminal ideas of twentieth-century physics, an entirely new quantum concept. But the Pauli-Kronig affair meant that the Nobel committee shied away from giving them the prestigious award. Pauli always felt guilty for discouraging Kronig. Just as he did for receiving the Nobel Prize in 1945 for the discovery of the exclusion principle while the Dutchmen were denied. 'I was so stupid when I was young!' he said later.60

On 7 July 1927, Uhlenbeck and Goudsmit received their doctorates within an hour of each other. Flouting convention, the ever-thoughtful Ehrenfest had arranged it that way. He had also secured both of them jobs at the University of Michigan. With few positions then available, Goudsmit said towards the end of his life, the post in America 'was for me a far more significant award than a Nobel Prize'.61

Goudsmit and Uhlenbeck provided the first concrete evidence that existing quantum theory had reached the limits of its applicability. Theorists could no longer use classical physics to gain a foothold before 'quantising' a piece of existing physics, because there was no classical counterpart to the quantum concept of electron spin. The discoveries of Pauli and the Dutch spin doctors brought to a close the achievements of the 'old quantum theory'. There was a sense of crisis. The state of physics 'was from a methodological point of view, a lamentable hodgepodge of hypothesis, principles, theorems, and computational recipes rather than a logical, consistent theory.'62 Progress was often based on artful guessing and intuition rather than scientific reasoning.

'Physics at the moment is again very muddled; in any case, for me it is too complicated, and I wish I were a film comedian or something of that sort and had never heard anything about physics', wrote Pauli in May 1925, some six months after discovering the exclusion principle.63 'Now I do hope nevertheless that Bohr will save us with a new idea. I beg him to do so urgently, and convey to him my greetings and many thanks for all his kindness and patience towards me.' However, Bohr had no answers to 'our present theoretical troubles'.64 That spring, it seemed that only a quantum magician could conjure up the yearned-for 'new' quantum theory – quantum mechanics.