Quantum

by Manjit Kumar

  In the late summer of 1925 two Dutch postgraduate students, Samuel Goudsmit and George Uhlenbeck, realised that the property of ‘two-valuedness’ 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.

  Chapter 8

  THE QUANTUM MAGICIAN

  ‘On a quantum-Theoretical Reinterpretation of Kinematics and Mechanical Relations’ was the paper that everyone had been waiting for and some had hoped to write. The editor of the Zeitschrift für Physik received it on 29 July 1925. In the preamble that scientists call an ‘abstract’, the author boldly stated his ambitious plan: ‘to establish a basis for theoretical quantum mechanics, founded exclusively on relationships between quantities which, in principle, are observable.’ Some fifteen pages later, his goal achieved, Werner Heisenberg had laid the foundations for the physics of the future. Who was this young German wunderkind and how he had succeeded where all others had failed?