by Manjit Kumar
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 intensities 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.’32 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.
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.34 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.
‘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’.36 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.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 E2 to the lower energy level E1, a spectral line is emitted with a frequency designated by v21 in the array. The spectral line of frequency v12 would only be found in the absorption spectrum, since it is associated with an electron in energy level E1 absorbing a quantum of energy sufficient to jump to energy level E2. A spectral line of frequency vmn would be emitted when an electron jumps between any two levels whose energies are Em and En, where m is greater than n. Not all the frequencies vmn are exactly observed. For example, measurement of v11 is impossible, since it would be the frequency of the spectral line emitted in a ‘transition’ from energy level E1 to energy level E1 – a physical impossibility. Hence v11 is zero, as are all potential frequencies when m=n. The collection of all non-zero frequencies, vmn, 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, amn, from energy level Em to En, is high, then the transition is more likely than one with a lower probability. The resulting spectral line with frequency vmn would be more intense than for the less probable transition. Heisenberg realise
d that the transition probabilities amn and the frequencies vmn 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 vmn 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’.44 It offered, he wrote to a colleague, ‘a new hope, and a renewed enjoyment of life’.45 ‘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.
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’.48 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.
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