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

### Part II. BOY PHYSICS

### 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?

Werner Karl Heisenberg was born on 5 December 1901 in Würzburg, Germany. He was eight when his father was appointed to the country's only professorship of Byzantine philology at Munich University and the family moved to the Bavarian capital. For Heisenberg and his brother Erwin, almost two years older, home became a spacious apartment in the fashionable suburb of Schwabing on the northern outskirts of the city. They attended the prestigious Maximilians Gymnasium, where Max Planck had been a student 40 years earlier. It was also the school where their grandfather was now in charge. If the staff were tempted to treat the headmaster's grandsons more leniently than other pupils, then they quickly discovered there was no need. 'He has an eye for what is essential, and never gets lost in details', Werner's first-year teacher reported.__ ^{1}__ 'His thought processes in grammar and mathematics operate rapidly and usually without mistakes.'

August Heisenberg's father, forever the teacher, devised all manner of intellectual games for Werner and Erwin. In particular he always encouraged mathematical games and problem-solving. Pitting one brother against the other as they raced to solve them, it was evident that Werner was the more mathematically talented. Around the age of twelve he started learning calculus and asked his father to get him maths books from the university library. Seeing this as an opportunity to improve his son's grasp of languages, he started supplying him with books written in Greek and Latin. It was the beginning of Werner's fascination with the work of the Greek philosophers. Then came the First World War and the end of Heisenberg's comfortable and secure world.

The end of the war brought in its wake political and economic chaos throughout Germany, but few places experienced this more intensely than Munich and Bavaria. On 7 April 1919, radical socialists declared Bavaria a 'Soviet Republic'. As they waited for troops sent by Berlin to arrive and restore the deposed government, those opposed to the revolutionaries organised themselves into military-style companies. Heisenberg and some friends joined one of these. His duties were largely confined to writing reports and running errands. 'Our adventures were over after a few weeks,' Heisenberg recalled later, 'then the shooting died down and military service became increasingly monotonous.'__ ^{2}__ By the end of the first week in May the 'Soviet Republic' had been ruthlessly crushed, leaving over a thousand dead.

The harsh post-war reality led young middle-class teenagers like Heisenberg to embrace the romantic ideals of an earlier age as they flocked to join youth organisations such as the Pathfinders, the German equivalent of the Boy Scouts. Others, wanting more independence, set up their own groups and clubs. Heisenberg led one such group formed by younger pupils at his school. Gruppe Heisenberg, as they styled themselves, went hiking and camping in the Bavarian countryside and discussed the new world their generation would create.

In the summer of 1920, after graduating from the Gymnasium with such ease that he won a prestigious scholarship, Heisenberg wanted to study mathematics at Munich University. When a disastrous interview ended any chance of doing so, a despondent Heisenberg sought his father for advice. He made an appointment for his son to see an old friend, Arnold Sommerfeld. Although the 'small squat man with his martial dark moustache looked rather austere', Heisenberg did not feel intimidated.__ ^{3}__ He sensed that despite his appearance, here was a man with a 'genuine concern for young people'.

__August Heisenberg had already told Sommerfeld that his son was particularly interested in relativity and atomic physics. 'You are much too demanding', he told Werner.__

^{4}__'You can't possibly start with the most difficult part and hope that the rest will automatically fall into your lap.' Always eager to encourage and recruit raw talent to mould, he softened: 'It may be that you know something; it may be that you know nothing. We shall see.'__

^{5}

^{6}Sommerfeld allowed the eighteen-year-old to attend the research seminar intended for more advanced students. Heisenberg was lucky. Together with Bohr's institute in Copenhagen and Born's group in Göttingen, Sommerfeld's institute would form the golden triangle of quantum research in the years to come. When Heisenberg attended his first seminar he 'spotted a dark-haired student with a somewhat secretive face in the third row'.__ ^{7}__ It was Wolfgang Pauli. Sommerfeld had already introduced him to the portly Viennese during a tour around the institute on his first visit. The professor had been quick to tell Heisenberg, once Pauli was out of earshot, that he considered the boy to be his most talented student. Recalling Sommerfeld's advice that he could learn a great deal from him, Heisenberg sat down next to Pauli.

'Doesn't he look the typical Hussar officer?' whispered Pauli as Sommerfeld entered.__ ^{8}__ It was the beginning of a lifelong professional relationship that never quite blossomed into a closer personal friendship. They were simply too different. Heisenberg was quieter, friendlier, less outspoken and critical than Pauli. He romanticised nature and loved nothing more than hiking and camping with his friends. Pauli was drawn to cabarets, taverns and cafes. Heisenberg had done half a day's work while Pauli still slept soundly in his bed. Yet Pauli exerted a strong influence on Heisenberg and never passed up a chance to tell him, with tongue in cheek: 'You are a complete fool.'

^{9}In the middle of writing his dazzling review of relativity, it was Pauli who steered Heisenberg away from Einstein's theory and towards the quantum atom as a more fertile area of research in which to make his name. 'In atomic physics we still have a wealth of uninterpreted experimental results,' he told Heisenberg; 'nature's evidence in one place seems to contradict that in another, and so far it has not been possible to draw an even halfway coherent picture of the relationship involved.'__ ^{10}__ It was likely, thought Pauli, that everyone would still be 'groping about in a thick mist' for years to come.

__As Heisenberg listened, he was inexorably drawn into the realm of the quantum.__

^{11}Sommerfeld soon assigned Heisenberg a 'little problem' in atomic physics. He asked him to analyse some new data on the splitting of spectral lines in a magnetic field and to construct a formula that replicated the splitting. Pauli warned Heisenberg that Sommerfeld hoped that deciphering such data would lead to new laws. It was an attitude that for Pauli bordered on 'a kind of number mysticism', but then he admitted, 'no one has been able to suggest anything better'.__ ^{12}__ The exclusion principle and electron spin still lay in the future.

Heisenberg's ignorance of the accepted rules and regulations of quantum physics allowed him to tread where others, wedded to a more cautious and rational approach, feared to. It enabled him to construct a theory that appeared to explain the anomalous Zeeman effect. Having dismissed an earlier version, Heisenberg was relieved when Sommerfeld sanctioned the publication of his latest effort. Although it was later shown to be incorrect, his first scientific paper brought Heisenberg to the attention of Europe's leading physicists. Bohr was one of those who sat up and took notice.

They first met in Göttingen in June 1922 when Sommerfeld took some of his students to hear Bohr's series of lectures on atomic physics. What struck Heisenberg was how precise Bohr was in his choice of words: 'Each one of his carefully formulated sentences revealed a long chain of underlying thoughts, of philosophical reflections, hinted at but never fully expressed.'__ ^{13}__ He was not alone in sensing that Bohr reached his conclusions more by intuition and inspiration than by detailed calculations. At the end of the third lecture, Heisenberg rose to point out some difficulties that remained in a published paper that Bohr had praised. As people began to mingle after the question-and-answer session, Bohr sought out Heisenberg and asked the twenty-year-old if he would like to accompany him on a walk later that day. Their hike to a nearby mountain lasted some three hours, and Heisenberg later wrote 'that my real scientific career only started that afternoon'.

__For the first time, he saw 'that one of the founders of quantum theory was deeply worried by its difficulties'.__

^{14}__When Bohr invited him to Copenhagen for a term, Heisenberg suddenly saw his future as one 'full of hope and new possibilities'.__

^{15}

^{16}Copenhagen would have to wait. Sommerfeld was due to go to America and in his absence had arranged for Heisenberg to study with Max Born in Göttingen. Although he looked 'like a simple farm boy, with short fair hair, clear bright eyes, and a charming expression', Born quickly discovered that there was much more to him than met the eye.__ ^{17}__ He was 'easily as gifted as Pauli', Born wrote to Einstein.

__When he returned to Munich, Heisenberg finished his doctoral thesis on turbulence. Sommerfeld had chosen the topic to broaden his knowledge and understanding of physics. During the oral examination his inability to answer simple questions, such as the resolving power of a telescope, almost cost him his doctorate. Wilhelm Wien, the head of experimental physics, was dismayed when Heisenberg struggled to explain how a battery worked. He wanted to fail the upstart theorist, but reached a compromise with Sommerfeld. Heisenberg would get his doctorate, but would be awarded the second-lowest mark – grade III. Pauli had passed with grade I.__

^{18}Feeling humiliated, that evening he packed his bags and caught the overnight train. He could not bear to stay in Munich a minute longer and fled to Göttingen. 'I was astonished when, one morning long before the appointed time, he suddenly appeared before me with an expression of embarrassment on his face', recalled Born later.__ ^{19}__ Heisenberg anxiously recounted the tale of his oral exam, worried that his services would no longer be required as an assistant. Eager to cement Göttingen's growing reputation for theoretical physics, Born was confident that Heisenberg would bounce back and told him so.

Born was convinced that physics had to be rebuilt from the ground up. The mish-mash of quantum rules and classical physics that was at the heart of the Bohr-Sommerfeld quantum atom had to give way to a logically consistent new theory that Born called 'quantum mechanics'. None of this was new for physicists trying to disentangle the problems of atomic theory. However, it signalled the awareness of a creeping sense of crisis in 1923 at the inability of physicists to cross the atomic Rubicon. Pauli was already loudly proclaiming to anyone who would listen that the failure to explain the anomalous Zeeman effect was evidence 'that we must create something fundamentally new'.__ ^{20}__After meeting him, Heisenberg believed that Bohr was the one most likely to make the breakthrough.

Pauli had been in Copenhagen as Bohr's assistant since the autumn of 1922. He and Heisenberg kept each other informed about the latest developments at their respective institutes through a regular exchange of letters. Heisenberg, like Pauli, had also been working on the anomalous Zeeman effect. Just before Christmas 1923, he wrote to Bohr about his latest efforts and received an invitation to spend a few weeks in Copenhagen. On Saturday, 15 March 1924, Heisenberg stood in front of the three-storey neo-classical building with its red tiled roof at Blegdamsvej 17. Above the main entrance he saw the sign that greeted every visitor: 'Universitetets Institut for Teoretisk Fysik'. Better known as the Bohr Institute.

Heisenberg soon discovered that only half of the building, the basement and the ground floor, was used for physics. The rest was set aside for accommodation. Bohr and his growing family lived in an elegantly furnished flat that occupied the entire first floor. The family maid, the caretaker, and honoured guests were housed on the top floor. On the ground floor, besides the lecture hall with its six long rows of wooden benches, was a well-stocked library and offices for Bohr and his assistant. There was also a modest-sized workroom for visitors. Despite its name, the institute had two small laboratories on the first floor, with the main laboratory housed in the basement.

The institute was struggling for space with a permanent staff of six and almost a dozen visitors. Bohr was already making plans to expand. Over the next two years the adjacent land was bought and two new buildings were added that doubled the capacity of the institute. Bohr and his family moved out of their flat into a large purpose-built house next door. The extension meant a substantial renovation of the old building that included more office space, a dining room, and a new self-contained three-room flat on the top floor. It was here that Pauli and Heisenberg often stayed in later years.

There was one thing that no one at the institute wanted to miss: the arrival of the morning post. Letters from parents and friends were always welcome, but it was correspondence from far-flung colleagues and the journals that were seized upon for the latest breaking news from the frontiers of physics. However, not everything revolved around physics, even if much of the talking did. There were musical evenings, games of table tennis, hiking trips, and outings to watch the latest motion picture.

Heisenberg had arrived with such high hopes, but his first few days at the institute left him feeling frustrated. Expecting to spend time with Bohr almost as he stepped through the front door, he had hardly seen him. Used to being the best, Heisenberg was suddenly faced with Bohr's international posse of brilliant young physicists. He was intimidated. They all spoke several languages, while he sometimes struggled to express himself clearly in German. Enjoying nothing more than walks in the countryside with his friends, Heisenberg thought that everyone at the institute possessed a worldliness that he did not. However, nothing left him as despondent as the realisation that they understood much more of atomic physics than he did.

As he tried to shake off the blows to his self-esteem, Heisenberg wondered if he would ever get the chance to work with Bohr. He had been sitting in his room when there was a knock on the door and in strode Bohr. After apologising for being so busy, he proposed that the two of them go on a short walking tour. There was little chance, Bohr explained, of him being left alone long enough at the institute for the pair of them to talk at any length. What better way of getting to know one another than a few days of walking and talking? It was Bohr's favourite pastime.

Early the following morning they caught the tram to the northern outskirts of the city and began their walk. Bohr asked Heisenberg about his childhood and what he remembered about the outbreak of war ten years earlier. As they headed north, instead of physics they talked about the pros and cons of war, Heisenberg's involvement in the youth movement, and Germany. After spending the night at an inn, they walked to Bohr's country cottage in Tisvilde, before heading back to the institute on the third day. The 100-mile walk had the effect that Bohr desired and Heisenberg craved. They got to know each other more quickly.

They had talked about atomic physics, yet when they finally returned to Copenhagen, it was Bohr the man, rather than the physicist, that had captivated Heisenberg. 'I am, of course, absolutely enchanted with the days I am spending here', he wrote to Pauli.__ ^{21}__ He had never before met a man like Bohr with whom he could discuss just about anything. Despite his genuine concern for the welfare of everyone at his institute, Sommerfeld upheld the traditional German role of professor, one step removed from his subordinates. In Göttingen, Heisenberg would not have dared to broach with Born the range of subjects he and Bohr had discussed so freely. Unknown to him, it was Pauli, in whose footsteps he always seemed to be following, who was behind Bohr's warm reception.

Pauli always took a keen interest in what Heisenberg was doing, as the pair kept each other informed about their latest ideas. Pauli had returned to Hamburg University when he learnt that Heisenberg was going to spend a few weeks in Copenhagen, and he wrote to Bohr. For a man already notorious for his scathing wit, the fact that he described Heisenberg as a 'gifted genius' who would 'one day advance science greatly' made a deep impression on Bohr.__ ^{22}__But before that day arrived, Pauli was sure that Heisenberg's physics had to be underpinned by a more coherent philosophical approach.

Pauli believed that to overcome the problems besetting atomic physics it was necessary to stop making arbitrary ad hoc assumptions whenever experiments yielded data in conflict with existing theory. Such an approach could only paper over the problems without ever leading to their solution. Given his deep understanding of relativity, Pauli was an ardent admirer of Einstein and the way in which he had constructed the theory using a few guiding principles and assumptions. Believing that it was the correct approach to adopt in atomic physics too, Pauli wanted to emulate Einstein by setting up the underlying philosophical and physical principles before moving on to develop the necessary formal mathematical nuts and bolts that held the theory together. By 1923 it was an approach that had left Pauli in despair. Having avoided introducing assumptions that could not be justified, he nevertheless failed to find a consistent and logical account of the anomalous Zeeman effect.

'Hopefully you will then take atomic theory forward in good measure and solve several of the problems with which I have tormented myself in vain and which are too difficult for me', Pauli wrote to Bohr.__ ^{23}__ 'I hope also that Heisenberg will then bring back home a philosophical attitude in his thinking.' By the time the young German arrived, Bohr had been well briefed. Throughout the two-week visit, the principles of physics rather than any particular problem was the focus of their discussions as Bohr and Heisenberg strolled through Faelledpark next to the institute or chatted over a bottle of wine in the evenings. Many years later, Heisenberg described his time in Copenhagen in March 1924 as a 'gift from heaven'.

^{24}'I shall, of course, miss him (he is a charming, worthy, very bright man, who has become very dear to my heart), but his interest precedes mine, and your wish is decisive for me', Born wrote to Bohr after Heisenberg received an invitation for an extended stay in Copenhagen.__ ^{25}__ Due to spend the forthcoming winter semester teaching in America, Born would not need the services of his assistant until May the following year. At the end of July 1924, having successfully completed his

*habilitation*thesis and gained the right to teach at German universities, Heisenberg left for a three-week hiking tour around Bavaria.

When he returned to Bohr's institute on 17 September 1924, Heisenberg was still only 22 years old, but had already written or co-written an impressive dozen papers on quantum physics. He still had much to learn and knew that Bohr was the man to teach him. 'From Sommerfeld I learned optimism, in Göttingen mathematics, from Bohr physics', he said later.__ ^{26}__ For the next seven months, Heisenberg was exposed to Bohr's approach to overcoming the problems that plagued quantum theory. While Sommerfeld and Born were also troubled by the same inconsistencies and difficulties, neither man was haunted like Bohr by them. He could hardly bring himself to talk of anything else.

From these intense discussions, Heisenberg 'realized how difficult it was to reconcile the results of one experiment with those of another'.__ ^{27}__ Among these experiments was Compton's scattering of X-rays by electrons that supported Einstein's light-quanta. The difficulties just seemed to multiply with de Broglie's extension of wave-particle duality to encompass all matter. Bohr, having taught Heisenberg all that he could, had great hopes for his young protégé: 'Now everything is in Heisenberg's hands – to find a way out of the difficulties.'

^{28}By the end of April 1925, Heisenberg was back in Göttingen, thanking Bohr for his hospitality and 'sad about the fact that I must carry on wretchedly alone by myself in the future'.__ ^{29}__ Nevertheless, he had learned a valuable lesson from discussions with Bohr and in his ongoing dialogue with Pauli: something fundamental had to give. Heisenberg believed he knew what that might be as he tried to solve a long-standing problem: the intensitie of the spectral lines of hydrogen. The Bohr-Sommerfeld quantum atom could account for the frequency of hydrogen's spectral lines, but not how bright or dim they were. Heisenberg's idea was to separate what was observable and what was not. The orbit of an electron around the nucleus of a hydrogen atom was not observable. So Heisenberg decided to abandon the idea of electrons orbiting the nucleus of an atom. It was a bold step, but one he was now ready to take, having long detested attempts at pictorial representations of the unobservable.

As a teenager in Munich, Heisenberg 'was enthralled by the idea that the smallest particles of matter might reduce to some mathematical form'.__ ^{30}__ At about the same time he came across an illustration in one of his textbooks that he found appalling. To explain how one atom of carbon and two atoms of oxygen formed a carbon dioxide molecule, the atoms were drawn with hooks and eyes by which they could hang together. Heisenberg found the idea of orbiting electrons inside the quantum atom similarly far-fetched. He now abandoned any attempt to visualise what was going on inside an atom. Anything that was unobservable he decided to ignore, focusing his attention only on those quantities that could be measured in the laboratory: the frequencies and intensities of the spectral lines associated with the light emitted or absorbed as an electron jumped from one energy level to another.

Even before Heisenberg adopted this new strategy, Pauli had already expressed his doubts about the usefulness of electron orbits more than a year earlier. 'The most important question seems to me to be this: *to what extent may definite orbits of electrons in stationary states be spoken of at all*', he had written in italics to Bohr in February 1924.__ ^{31}__ Even though he was well on the road that led to the exclusion principle, and concerned about the closure of electron shells, Pauli nevertheless answered his own question in another letter to Bohr in December: 'We must not bind atoms in the chains of our prejudices – to which, in my opinion, also belongs the assumption that electron orbits exist in the sense of ordinary mechanics – but we must, on the contrary, adapt our concepts to experience.'

__They had to stop making compromises and cease trying to accommodate quantum concepts within the comfortable and familiar framework of classical physics. Physicists had to break free. The first to do so was Heisenberg when he pragmatically adopted the positivist credo that science should be based on observable facts, and attempted to construct a theory based solely on the observable quantities.__

^{32}In June 1925, a little more than a month after returning from Copenhagen, Heisenberg was miserable in Göttingen. He was struggling to make headway in calculating the intensities of the spectral lines of hydrogen and admitted as much in a letter to his parents. He complained that 'everyone here is doing something different and no one anything worthwhile'.__ ^{33}__ A very severe attack of hay fever contributed to his low spirits. 'I couldn't see from my eyes, I just was in a terrible state', Heisenberg said later.

__Unable to cope, he had to get away and a sympathetic Born granted him a two-week holiday. On Sunday, 7 June, Heisenberg caught the night train to the port of Cuxhaven on the coast. Arriving early in the morning, tired and hungry, Heisenberg went in search of breakfast at an inn and then boarded a ferry to the island of Helgoland, an isolated barren rock in the North Sea. Originally owned by the British until it was traded for Zanzibar in 1890, Helgoland was 30 miles from the German mainland and less than a square mile in size. It was here that Heisenberg hoped to find relief amid the bracing pollen-free sea air.__

^{34}'On my arrival, I must have looked quite a sight with my swollen face; in any case, my landlady took one look at me, concluded that I had been in a fight and promised to nurse me through the after effects', Heisenberg recalled when he was 70.__ ^{35}__ The guesthouse was high on the southern edge of the distinctive island carved out of red sandstone rock. From the balcony of his second-floor room Heisenberg had a wonderful view of the village below, the beach, and the dark brooding sea beyond. In the days that followed he had time to think about 'Bohr's remark that part of infinity seems to lie within the grasp of those who look across the sea'.

__It was in such reflective mood that he relaxed by reading Goethe, taking daily walks around the small resort, and swimming. Soon he was feeling much better. With little to distract him, Heisenberg's thoughts turned once more to problems of atomic physics. But here on Helgoland he felt none of the anxiety that had recently plagued him. Relaxed and carefree, he quickly jettisoned the mathematical ballast he had brought from Göttingen as he tried to solve the riddle of the intensities of the spectral lines.__

^{36}

^{37}In his quest for a new mechanics for the quantised world of the atom, Heisenberg concentrated on the frequencies and relative intensities of the spectral lines produced when an electron instantaneously jumped from one energy level to another. He had no other choice; it was the only available data about what was happening inside an atom. Despite the imagery conjured up by all the talk of quantum jumps and leaps, an electron did not 'jump' through space as it moved between energy levels like a boy jumping off a wall onto the pavement below. It was simply in one place and an instant later it popped up in another without being anywhere in between. Heisenberg accepted that all observables, or anything connected with them, were associated with the mystery and magic of the quantum jump of an electron between two energy levels. Lost forever was the picturesque miniature solar system in which each electron orbited a nuclear sun.

On the pollen-free haven of Helgoland, Heisenberg devised a method of book-keeping to track all possible electron jumps, or transitions, that could occur between the different energy levels of hydrogen. The only way he could think of recording each observable quantity, associated with a unique pair of energy levels, was to use an array:

This was the array for the entire set of possible frequencies of the spectral lines that could theoretically be emitted by an electron when it jumps between two different energy levels. If an electron quantum jumps from the energy level E_{2} to the lower energy level E_{1}, a spectral line is emitted with a frequency designated by v_{21} in the array. The spectral line of frequency v_{12} would only be found in the absorption spectrum, since it is associated with an electron in energy level E_{1} absorbing a quantum of energy sufficient to jump to energy level E_{2}. A spectral line of frequency v_{mn} would be emitted when an electron jumps between any two levels whose energies are E_{m} and E_{n}, where m is greater than n. Not all the frequencies v_{mn} are exactly observed. For example, measurement of v_{11} is impossible, since it would be the frequency of the spectral line emitted in a 'transition' from energy level E_{1} to energy level E_{1} – a physical impossibility. Hence v_{11} is zero, as are all potential frequencies when m=n. The collection of all non-zero frequencies, v_{mn}, would be the lines actually present in the emission spectrum of a particular element.

Another array could be formed from the calculation of transition rates between the various energy levels. If the probability for a particular transition, a_{mn}, from energy level E_{m} to E_{n}, is high, then the transition is more likely than one with a lower probability. The resulting spectral line with frequency v_{mn} would be more intense than for the less probable transition. Heisenberg realised that the transition probabilities a_{mn} and the frequencies v_{mn} could, after some deft theoretical manipulation, lead to a quantum counterpart for each observable quantity known in Newtonian mechanics such as position and momentum.

Of all things, Heisenberg began by thinking about electrons' orbits. He imagined an atom in which an electron was orbiting the nucleus at a great distance – more like Pluto orbiting the sun rather than Mercury. It was to prevent an electron spiralling into the nucleus at it radiated away energy that Bohr had introduced the concept of stationary orbits. However, in accordance with classical physics, the orbital frequency of an electron in such an exaggerated orbit, the number of complete orbits it makes per second, is equal to the frequency of the radiation it emits.

This was no flight of fancy, but a skilful use of the correspondence principle – Bohr's conceptual bridge between the quantum and classical realms. Heisenberg's hypothetical electron orbit was so large that it was on the border that divided the kingdoms of the quantum and the classical. Here in this borderland, the electron's orbital frequency was equal to the frequency of the radiation it emitted. Heisenberg knew that such an electron in an atom was akin to a hypothetical oscillator that could produce all the frequencies of the spectrum. Max Planck had adopted a similar approach a quarter of a century earlier. However, while Planck had used brute force and ad hoc assumptions to generate a formula that he already knew to be correct, Heisenberg was being guided by the correspondence principle onto the familiar landscape of classical physics. Once it was set into motion, he could calculate properties of the oscillator such as its momentum p, the displacement from its equilibrium position q, and its frequency of oscillation. The spectral line with a frequency v_{mn} would be emitted by one of a range of individual oscillators. Heisenberg knew that once he worked out the physics in this territory where the quantum and the classical met, he could extrapolate to explore the unknown interior of the atom.

Late one evening on Helgoland, all the pieces began falling into place. The theory built completely out of observables appeared to reproduce everything, but did it contravene the law of the conservation of energy? If it did, then it would collapse like a house of cards. Excited and nervous as he edged ever closer to proving that his theory was both physically and mathematically consistent, the 24-year-old physicist began making simple errors of arithmetic as he checked his calculations. It was almost three in the morning before Heisenberg could put down his pen, satisfied that the theory did not violate one of the most fundamental laws of physics. He was elated, but troubled. 'At first, I was deeply alarmed', Heisenberg recalled later.__ ^{38}__ 'I had the feeling that, through the surface of atomic phenomena, I was looking at a strangely beautiful interior, and felt almost giddy at the thought that I now had to probe this wealth of mathematical structures nature had so generously spread out before me.' Sleep was impossible – he was too excited. So as a new day dawned, Heisenberg walked to the southern tip of the island, where for days he had been longing to climb a rock jutting out into the sea. Fuelled by the adrenaline of discovery, he climbed it 'without too much trouble and waited for the Sun to rise'.

^{39}In the cold light of day, Heisenberg's initial euphoria and optimism faded. His new physics appeared to work only with the help of a strange kind of multiplication where X times Y did not equal Y times X. With ordinary numbers it did not matter in which order they were multiplied: 4×5 gives exactly the same answer as 5×4, 20. Mathematicians called this property, where the ordering in multiplication is unimportant, commutation. Numbers obey the commutative law of multiplication, so (4×5)-(5×4) is always zero. It was a rule of mathematics that every child learned and Heisenberg was deeply troubled by the discovery that when he multiplied two arrays together, the answer was dependent on the order in which they were multiplied. (A×B)-(B×A) was not always zero.^{40}

As the meaning of the peculiar multiplication he had been forced to use continued to elude him, on Friday, 19 June, Heisenberg travelled back to the mainland and headed straight to Hamburg and Wolfgang Pauli. A few hours later, having received words of encouragement from his severest critic, Heisenberg left for Göttingen and the task of refining and writing up what he had discovered. Only two days later, expecting to make quick progress, he wrote to Pauli that 'attempts to fabricate a quantum mechanics advance only slowly'.__ ^{41}__ As the days passed, his frustration grew as he failed to apply his new approach to the hydrogen atom.

Whatever doubts he harboured, there was one thing Heisenberg was certain about. In any calculation, only relationships between 'observable' quantities, or those that could be measured in principle if not in reality, were permissible. He had given the observability of all quantities in his equations the status of a postulate and devoted his 'entire meagre efforts' to 'killing off and suitably replacing the concept of the orbital paths that one cannot observe'.^{42}

'My own works are at the moment not going especially well', Heisenberg wrote to his father at the end of June. A little more than a week later, he had finished the paper that ushered in a new era in quantum physics. Still uncertain about what he had done and its true significance, Heisenberg sent a copy to Pauli. Apologising, he asked him to read and return the paper within two or three days. The reason for the haste was that Heisenberg was due to give a lecture at Cambridge University on 28 July. With other commitments he was unlikely to return to Göttingen until late September and wanted 'either to complete it in the last days of my presence here or to burn it'.__ ^{43}__ Pauli greeted the paper 'with jubilation'.

__It offered, he wrote to a colleague, 'a new hope, and a renewed enjoyment of life'.__

^{44}__'Although it is not the solution to the riddle,' Pauli added, 'I believe that it is now once again possible to move forward.' The man who took those steps in the right direction was Max Born.__

^{45}He had little inkling of what Heisenberg had been doing since returning from the little island in the North Sea. Born was therefore surprised when Heisenberg gave him the paper and requested that he decide whether it was worth publishing or not. Tired by his own exertions, Born put the paper to one side. When a couple of days later he sat down to read it and pass judgement on what Heisenberg had described as a 'crazy paper', Born was immediately captivated. He realised that Heisenberg was being uncharacteristically hesitant in what he was putting forward. Was it a consequence of having to employ a strange multiplication rule? Heisenberg was still groping even at the conclusion of the paper: 'Whether a method to determine quantum-mechanical data using relations between observable quantities, such as that proposed here, can be regarded as satisfactory in principle, or whether this method after all represents far too rough an approach to the physical problem of constructing a theoretical quantum mechanics, an obviously very involved problem at the moment, can be decided only by a more intensive mathematical investigation of the method which has been very superficially employed here.'^{46}

What was the meaning of the mysterious multiplication law? It was a question that so obsessed Born, he could think of little else during the days and nights that followed. He was troubled by the fact that there was something vaguely familiar about it, but he could not pinpoint exactly what. 'Heisenberg's latest paper, soon to be published, appears rather mystifying, but is certainly true and profound', Born wrote to Einstein, even though he was still unable to explain the origin of the strange multiplication.__ ^{47}__ Praising the young physicists at his institute, especially Heisenberg, Born admitted 'that merely to keep up with their thoughts demands at times considerable effort on my part'.

__After days of considering nothing else, the effort on this occasion was rewarded. One morning, Born suddenly recalled a long-forgotten lecture he had attended as a student and realised that Heisenberg had accidentally stumbled across matrix multiplication in which X times Y does not always equal Y times X.__

^{48}On being told that the mystery of his strange multiplication rule had been solved, Heisenberg complained that 'I do not even know what a matrix is'.__ ^{49}__ A matrix is nothing more than an array of numbers placed in a series of rows and columns, just like the arrays that Heisenberg constructed in Helgoland. In the mid-nineteenth century the British mathematician Arthur Cayley had worked out how to add, subtract, and multiply matrices. If A and B are both matrices, then A×B can yield a different answer from B×A. Just like Heisenberg's array of numbers, matrices do not necessarily commute. Although they were established features of the mathematical landscape, matrices were unfamiliar territory for the theoretical physicists of Heisenberg's generation.

Once Born had correctly identified the roots of the strange multiplication, he knew that he needed help to turn Heisenberg's original scheme into a coherent theoretical framework that embraced all the multifarious aspects of atomic physics. He knew the perfect man for the job, one well versed in the intricacies of both quantum physics and mathematics. As luck would have it, he too would be in Hanover, where Born was due to attend a meeting of the German Physical Society. Once there, he immediately sought out Wolfgang Pauli. Born asked his former assistant to collaborate with him. 'Yes, I know you are fond of tedious and complicated formalisms', came the reply as Pauli refused. He wanted no part in Born's plans: 'You are only going to spoil Heisenberg's physical ideas by your futile mathematics.'__ ^{50}__ Feeling unable to make progress alone, he turned in desperation to one of his students for help.

In choosing 22-year-old Pascual Jordan, Born had unwittingly found the perfect collaborator for the task ahead. Entering the Technische Hochschule in Hanover in 1921 with the intention of studying physics, Jordan found the lectures rather poor and turned instead to mathematics. A year later he transferred to Göttingen to study physics. However, he rarely attended the lectures because they were too early in the morning, starting at either 7am or 8am. Then he met Born. Under his supervision, Jordan began to study physics seriously for the first time. 'He was not only my teacher, who in my student days introduced me to the wide world of physics – his lectures were a wonderful combination of intellectual clarity and horizon widening overview', Jordan later said of Born. 'But he was also, I want to assert, the person, who next to my parents, exerted the deepest, longest lasting influence on my life.'^{51}

With Born as his guide, Jordan soon began concentrating on problems of atomic structure. Somewhat insecure and with a stutter, he appreciated Born's patience whenever they discussed the latest papers touching on atomic theory. Fortuitously, he had moved to Göttingen in time to attend the Bohr Festspiele and, like Heisenberg, was inspired by the lectures and the discussions that followed. After his doctoral dissertation in 1924, Jordan worked briefly with others before being asked by Born to collaborate with him on an attempt to explain the width of spectral lines. Jordan is 'exceptionally intelligent and astute and can think far more swiftly and confidently than I', Born wrote to Einstein in July 1925.^{52}

By then Jordan had already heard of Heisenberg's latest ideas. Before he left Göttingen at the end of July, Heisenberg gave a talk to a small circle of students and friends about his attempt to construct a quantum mechanics based solely on the relations between observable properties. When Born asked him to collaborate, Jordan jumped at the chance to recast and extend Heisenberg's original ideas into a systematic theory of quantum mechanics. Unknown to Born, as he sent Heisenberg's paper to the journal *Zeitschrift für Physik*, Jordan was well versed in matrix theory through his background in mathematics. Applying these methods to quantum physics, in two months Born and Jordan laid the foundations for a new quantum mechanics that others would call matrix mechanics.^{53}

Once Born identified Heisenberg's multiplication rule as a rediscovery of matrix multiplication, he quickly found a matrix formula that connected position q and momentum p using an expression that included Planck's constant: pq–qp=(ih/2)I, where I is what mathematicians call a unit matrix. It allowed the right-hand side of the equation to be written as a matrix. It was from this fundamental equation using the methods of matrix mathematics that all of quantum mechanics was constructed in the months that followed. Born was proud to be 'the first person to write a physical law in terms of non-commuting symbols'.__ ^{54}__ But it 'was only a guess, and my attempts to prove it failed', he recalled later.

__Within days of being shown the formula, Jordan came up with the rigorous mathematical derivation. No wonder Born was soon telling Bohr that, aside from Heisenberg and Pauli, he considered Jordan 'to be the most gifted of the younger colleagues'.__

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^{56}In August, Born went on his summer holiday to Switzerland with his family while Jordan stayed in Göttingen to write up a paper by the end of September for publication. Before it appeared in print they sent a copy to Heisenberg, who was in Copenhagen at the time. 'Here, I got a paper from Born, which I cannot understand at all', Heisenberg said to Bohr as he handed him the paper.__ ^{57}__ 'It is full of matrices, and I hardly know what they are.'

Heisenberg was hardly alone in not being familiar with matrices, but he set about learning the new mathematics with gusto and mastered enough to begin collaborating with Born and Jordan while still in Copenhagen. Heisenberg returned to Göttingen in the middle of October in time to help write the final version of what became known as the *Drei-Männer-Arbeit*, the 'three-man paper' in which he, Born and Jordan presented the first logically consistent formulation of quantum mechanics – the long-sought-after new physics of the atom. However, there were already reservations being expressed about Heisenberg's initial work. Einstein wrote to Paul Ehrenfest: 'In Göttingen they believe it (I don't).'__ ^{58}__ Bohr believed it was 'a step probably of fundamental importance' but 'it has not yet been possible to apply [the] theory to questions of atomic structure'.

__While Heisenberg, Born and Jordan had been concentrating on developing the theory, Pauli had been busy using the new mechanics to do just that. By early November, while the 'three-man paper' was still being written, he had successfully applied matrix mechanics in a stunning tour de force. Pauli had done for the new physics what Bohr had done for the old quantum theory – reproduced the line spectrum of the hydrogen atom. For Heisenberg, to add insult to injury, Pauli had also calculated the Stark effect – the influence of an external electric field on the spectrum. 'I myself had been a bit unhappy that I could not succeed in deriving the hydrogen spectrum from the new theory', Heisenberg recalled.__

^{59}__Pauli had provided the first concrete vindication of the new quantum mechanics.__

^{60}'The Fundamental Equations of Quantum Mechanics' read the title. Born had been in Boston for a nearly a month, as part of a five-month lecture tour of the United States, when one December morning he opened his post and received 'one of the greatest surprises' of his scientific life.__ ^{61}__ As he read the paper by one P.A.M. Dirac, a senior research student at Cambridge University, Born realised that 'everything was perfect in its way'.

__Even more remarkably, Born soon discovered that Dirac had sent his paper to the__

^{62}*Proceedings of the Royal Society*containing the nuts and bolts of quantum mechanics a whole nine days before the 'three-man paper' was finished. Who was Dirac and how had he done it, wondered Born?

Paul Adrien Maurice Dirac was 23 years old in 1925. The son of a Swiss, French-speaking father, Charles, and an English mother, Florence, he was the second of three children. His father was such an overbearing and dominant figure that when he died in 1935, Dirac wrote: 'I feel much freer now.'__ ^{63}__ It was the trauma of having to remain silent in the presence of his father, a teacher of French, as he grew up that made Dirac a man of few words. 'My father made the rule that I should only talk to him in French. He thought it would be good for me to learn French in that way. Since I found that I couldn't express myself in French, it was better for me to stay silent than to talk in English.'

__Dirac's preference for silence, the legacy of a deeply unhappy childhood and adolescence, would become legendary.__

^{64}Although interested in science, in 1918, Dirac acted on his father's advice and enrolled to study electrical engineering at the University of Bristol. Three years later, despite graduating with a first-class honours degree, he could not find a job as an engineer. With his employment prospects looking bleak as Britain's post-war depression continued, Dirac accepted the offer of free tuition for two years to study mathematics back at his old university. He would rather have gone to Cambridge, but the scholarship he had won did not cover all the expenses of studying at the university. However, in 1923, after gaining his mathematics degree and receiving a government grant, he finally arrived in Cambridge as a PhD student. His supervisor was Ralph Fowler, Rutherford's son-in-law.

Dirac had a thorough grasp of Einstein's theory of relativity, which had generated a firestorm of publicity around the world in 1919 while he was still an engineering student, but he knew very little about Bohr's decade-old quantum atom. Until his arrival in Cambridge, Dirac always considered atoms 'as very hypothetical things', hardly worth bothering about.__ ^{65}__ He soon changed his mind and set about making up for lost time.

The quiet, secluded life of a budding Cambridge theoretical physicist was tailor-made for the shy and introverted Dirac. Research students were largely left to work alone in either their college rooms or in the library. While others might have struggled with a lack of human contact day after day, Dirac was perfectly happy to be left alone in his room to think. Even on a Sunday as he relaxed by walking in the Cambridgeshire countryside, Dirac preferred to do it alone.

Like Bohr, whom he met for the first time in June 1925, Dirac chose his words, written or spoken, very carefully. If he gave a lecture and was asked to explain a point that had not been understood, Dirac would often repeat word for word what he had said before. Bohr had gone to Cambridge to lecture on the problems of quantum theory and Dirac had been impressed by the man, but not by his arguments. 'What I wanted was statements which could be expressed in terms of equations,' he said later, 'and Bohr's work very seldom provided such statements.'__ ^{66}__ Heisenberg, on the other hand, arrived from Göttingen to give a lecture having spent months doing just the sort of physics that Dirac would have found stimulating. But he did not hear about it from Heisenberg, who chose not to mention it as he spoke about atomic spectroscopy.

It was Ralph Fowler who alerted Dirac to Heisenberg's work by giving him a proof copy of the German's soon-to-be-published paper. Heisenberg had been Fowler's house-guest during his brief visit and had discussed his latest ideas with his host, who asked for a copy of the paper. When it arrived, Fowler had little time to study it thoroughly and so passed it on to Dirac, asking him for his opinion. When he first read it in early September, he found it difficult to follow and failed to appreciate what a breakthrough it represented. Then, as one week turned into two, Dirac suddenly realised that the fact that A×B did not equal B×A lay at the very heart of Heisenberg's new approach and 'provided the key to the whole mystery'.^{67}

Dirac developed a mathematical theory that also led him to the formula pq–qp=(ih/2)I by distinguishing between what he called q-numbers and c-numbers, between those quantities that do not commute (AB does not equal BA) and those that do (AB=BA). Dirac showed that quantum mechanics differs from classical mechanics in that the variables, q and p, representing the position and momentum of a particle, do not commute with one another but obey the formula that he had found independently of Born, Jordan and Heisenberg. In May 1926, he received his PhD with the first-ever thesis on the subject of 'quantum mechanics'. By then physicists were beginning to breathe a little easier after being confronted by matrix mechanics, which was difficult to use and impossible to visualise, even though it generated the right answers.

'The Heisenberg-Born concepts leave us all breathless, and have made a deep impression on all theoretically orientated people', Einstein wrote in March 1926. 'Instead of dull resignation, there is now a singular tension in us sluggish people.'__ ^{68}__ They were roused out of their stupor by an Austrian physicist who found time while conducting an affair to produce an entirely different version of quantum mechanics that avoided what Einstein called Heisenberg's 'veritable calculation by magic'.

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