But so much was not at all clear at the time. Heisenberg’s path unfolded in fits and starts over the first few months of 1925. He co-wrote a paper with Hendrik Kramers in Göttingen with equations containing no classical variables but only frequencies and amplitudes. Kramers’s contribution was an important clue, for he showed that only when these frequencies and amplitudes are associated with pairs of states does one get the correct matrices. Then Heisenberg injured himself skiing and spent several weeks in Munich recovering. He visited Copenhagen and Göttingen, took another trip to the mountains, and by the end of April returned to Born’s institute in Göttingen to prepare to teach a summer session. All of these visits prepared Heisenberg to try to rewrite Bohr’s quantum descriptions of electron momentum (p) and position (q) in purely mathematical terms. He did not tell his supervisor what he was up to, keeping the idea, as Born once put it, ‘dark and mysterious.’8
Then, Heisenberg once recalled, ‘My work along these lines was advanced rather than retarded by an unfortunate personal setback.’9 In May, he was hit by an attack of hayfever so severe that he asked Born for 2 weeks off. Born agreed, and Heisenberg headed for Helgoland, an isolated, rocky island in the North Sea that is inhospitable to grass, weeds, and other allergen producers. The evening before his departure, the landlady of the Gasthaus who showed him his room was so horrified by his swollen face that she assumed he had been in a fight. On the island, once able to take up his work again, he tried to see if his ideas were consistent with the conservation of energy. When they were, he grew excited. He made mathematical errors, owing to his condition and fatigue, but caught them and continued working late into the night, finally sorting out everything by about 3 A.M.
At first I was deeply alarmed. 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. I was far too excited to sleep, and so, as a new day dawned, I made for the southern tip of the island, where I had been longing to climb a rock jutting out into the sea. I now did so without too much trouble, and waited for the sun to rise.10
Heisenberg returned to Göttingen late in June, and was soon scheduled to leave to lecture in Cambridge. In a few days he dashed off a paper, ‘On the Quantum-Mechanical Reinterpretation of Kinematic and Mechanical Relations.’11 The word ‘reinterpretation’ (Umdeutung) reveals Heisenberg’s audacity: it was the manifesto of a new approach to atomic physics. The abstract boldly declared that the aim was ‘to establish a basis for theoretical quantum mechanics founded exclusively upon relationships between quantities which in principle are observable.’ We have not been able to ‘associate an electron with a point in space’ based on the experimental information, he continued, and ‘in this situation it seems sensible to discard all hope of observing hitherto unobservable quantities, such as the position and period of the electron.’ Heisenberg was groping in the dark here; it would turn out that quantum mechanics would contain the possibility of measuring position and momentum to any degree of accuracy, just not simultaneously. The paper showed how to compile tables of amplitudes and frequencies associated with transitions between states – he called such tables ‘quantum-theoretical quantities’ – and how the tables could be related by a new kind of calculus, which he called ‘quantum-mechanical relations.’
The paper associated physical quantities with tables whose rows and columns were labeled with the ‘allowed’ quantum states that Bohr had postulated in his groundbreaking paper on the hydrogen spectrum. This had been done before (for example, Einstein’s A and B coefficients are ‘tables’ labeled with two states), but Heisenberg applied the idea to a more fundamental set of quantities and went a step further and found a rule for ‘multiplying’ two such tables to make formulas similar to those used in classical mechanics. This was new, and it opened the way to do quantum calculations far beyond the limited capacity of the previous attempts, such as Born’s, at a ‘quantum mechanics.’
Heisenberg then hit a snag. The tables and the multiplication rule he invented for them obeyed a new kind of algebra that mathematicians had discovered long before, but was unfamiliar to most physicists, himself included. Most strikingly, the rule did not follow the ‘commutative law’, the mathematical principle according to which the order in which one multiplies two numbers does not affect the result: ab = ba. When Heisenberg used his new calculus to multiply one quantum-theoretical table (let’s call it A) by another (B), the result depended on the order: AB ≠ BA. The feature ‘was very disagreeable to me’, he said later, and try as he might he could not rid his theory of it.12 ‘I felt this was the only point of difficulty in the whole scheme, otherwise I would be perfectly happy.’ Heisenberg then did what many people do when a nuisance threatens to spoil a promising new invention: he swept it under the rug. He waved at it in a single sentence – ‘Whereas in classical theory [AB] is always equal to [BA], this is not necessarily the case in quantum theory’ – mentioned circumstances in which the difficulty does not arise, and then dropped the subject. Heisenberg concluded his paper with a disclaimer of the sort that is often seen in early papers in a field, wondering whether it was ‘satisfactory’ or ‘still too crude an approach’ to quantum mechanics. The answer, he declared, would have to await ‘a penetrating mathematical investigation.’13
After he finished, around July 9, Heisenberg gave Born a copy of the paper, asking his supervisor to see whether it was worth publishing, and to see whether he could investigate the basic idea, which Heisenberg knew seemed awkward and even bizarre. Born so promised – but set the paper aside for a few days, exhausted after a semester teaching and from research he was doing with his other assistant, Pascual Jordan.
Born read the paper only after Heisenberg’s departure. Impressed, he sent it on to the Zeitschrift für Physik, and on July 15 wrote to Einstein that Heisenberg’s work appeared ‘very mysterious, but is surely correct and profound.’14 But Born also had a nagging feeling about Heisenberg’s tables and the strange mathematical rules used to multiply them. It looked so familiar! After a restless week in which he could hardly sleep, it hit Born that he had seen this peculiar structure in his high school maths classes. His intrepid young assistant had reinvented the wheel. The tables were what mathematicians called matrices, arrangements of numbers (or variables) in rows and columns – though Heisenberg’s tables had infinite numbers of elements. And Heisenberg’s funny quantum-mechanical relations were actually the most natural way that mathematicians had discovered to ‘multiply’ matrices.
Born was overjoyed. The mathematics of matrices gave him a framework in which to investigate and systematize Heisenberg’s work. He knew that matrices can be noncommutative – the order in which one multiplied them mattered. This explained Heisenberg’s embarrassing difficulty that, for instance, the matrix p associated with momentum and q with position did not commute; the matrix pq was not the same as qp (by convention, physicists often indicate matrices with bold symbols). But there was more. This pair of variables – known as canonically conjugate variables – was not commutative, but in a special way. Though Born could not prove it, the difference between pq and qp seemed to be a specific matrix proportional to Planck’s constant: pq – qp = Ih/2πi, where I is the unit matrix – ‘ones’ along the diagonal entries and zeros everywhere else. Born wrote later, ‘I was as excited by this result as a sailor would be who, after a long voyage, sees from afar, the longed-for land, and I felt regret that Heisenberg was not there.’15
A few days later, on July 19, Born ran into Pauli on a train, excitedly explained how Heisenberg’s paper could be translated into matrix language, and asked his former assistant if he wanted to collaborate in investigating the topic. Pauli was dismissive, and sarcastically accused Born of trying to ‘spoil Heisenberg’s physical ideas’ with ‘futile mathematics’ and ‘tedious and complicated formalism.’ (Historians find this remark
humorous, for Heisenberg’s ideas in this case were formal and even more tedious than conventional matrix analysis.) The next day, July 20, Born approached Jordan, who happened to be knowledgeable in the mathematics of matrices. Within a few days the two were able to show how to derive the relation pq – qp = Ih/2πi from Heisenberg’s work, and again Born was awestruck: ‘I shall never forget the thrill I experienced when I succeeded in condensing Heisenberg’s ideas on quantum conditions in the mysterious equation pq – qp = Ih/2πi, which is the centre of the new mechanics and was later found to imply the uncertainty relations.’16
By the end of September they sent off a paper, ‘On Quantum Mechanics.’ It carried out the ‘penetrating mathematical investigation’ that Heisenberg had hoped for, and was the first formulation of what became known as matrix mechanics. The maths was unfamiliar – many physicists had to bone up on matrices to understand the paper – and its methods were unwieldy, but it worked on the limited number of problems for which calculations could be carried through to completion. The authors sent a copy to Heisenberg, who by then had left Cambridge and was in Copenhagen. He showed the paper to Bohr, saying, ‘Here, I got a paper from Born, which I cannot understand at all. It is full of matrices, and I hardly know what they are.’17 But after Heisenberg brushed up on matrices, he too, shared their excitement, and on September 18 wrote to Pauli that Born’s bright idea, pq – qp = Ih/2πi, was the foundation of the new mechanics. Heisenberg, Born, and Jordan began a feverish discussion by letter, and Heisenberg interrupted his stay in Copenhagen and returned to Göttingen so that the three of them could finish work on another paper generalizing the results of the Born-Jordan paper before Born departed for a long-scheduled trip to the U.S. in October.18
The result was a paper written by Born, Heisenberg, and Jordan entitled ‘On Quantum Mechanics II’, known to historians of physics as ‘the three-man paper.’ Its central feature is what they called the ‘fundamental quantum-mechanical relation’, the strange equation pq – qp = Ih/2πi. The paper is a landmark in the history of physics, for it is the first map of the quantum domain. Around the same time, Pauli published a paper in which he had successfully – though with considerable difficulty – applied matrix mechanics to the test case of the hydrogen atom.
Yet few besides its creators fully recognized the importance of matrix mechanics, for the appreciation of its value was hampered by several obstacles. One was its complexity: while matrix mathematics was not intrinsically that difficult, Heisenberg’s application of it appeared to be horrendously complicated, and most physicists had to take matrix mechanics on faith while they struggled to master it. Typical was the reaction of George Uhlenbeck, then a student at the University of Leiden, who remarked much later, ‘Everything became these infinite numbers of equations that you had then to solve, and so nobody knew exactly how to do it.’19 Others were put off by the unanschaulichkeit – by the fact that matrix mechanics deliberately refrained from providing a picture of the mechanics of the atomic domain, and that its fundamental terms, the matrices, were strictly speaking meaningless, just formal symbolic artifacts.20 Still others were bothered by the failure to explain the transition from the microworld to the macroworld – between the unvisualizable world without space and time to the familiar space-time container that we and our imaginations live in. Many scientists therefore took, as historian Mara Beller once wrote, a ‘wait and see’ attitude, and even its originators viewed it as but a first and imperfect step toward an adequate theory.21
But shortly after the three-man paper appeared in February 1926, its authors had unwelcome company.
Matrix vs. Wave Mechanics
Schrödinger’s first and second papers on wave mechanics appeared in the Annalen der Physik in March and April 1926. Wave mechanics mapped the same terrain as matrix mechanics, but physicists found the map much easier to read. It lacked the obstacles of matrix mechanics. First, the maths was part of the bread-and-butter training of classical physicists, who had been using and solving wave equations since their high school days. Second, wave equations were visualizable. Physicists saw water, sound, and light waves, and their properties – frequency, amplitude, and wavelength – smoothly and continuously propagating around them daily. They had trained themselves to see other wave properties such as nodes and interference. There was the small matter of the ψ-function, which existed in multidimensional ‘configuration space’ – three dimensions for each particle in the system – but even that seemed somewhat visualizable, as something that traveled through space or stayed ‘perched’ in a standing-wave-like way when bound inside an atom. Third, wave mechanics provided a natural way of describing the transition from the microworld to the macroworld, as Schrödinger’s third paper that year showed, as particle-like groupings or packets of waves moved along the classical paths, which were the rays perpendicular to the phase fronts of the ψ-function.22
Small wonder most physicists took to wave mechanics. Planck was awestruck, Einstein ecstatic. U.S. physicist Karl Darrow reported that wave mechanics ‘captivated the world of physics’ as it promised ‘a fulfillment of that long-baffled and insuppressible desire’ to return to classical physics, its comfortable and continuously propagating functions.23 A flood of papers used Schrödinger’s approach to tackle atomic issues. Allies of the Göttingen physicists reacted badly: Heisenberg called it ‘too good to be true’, Dirac reacted with ‘hostility’, Pauli called it ‘crazy.’24 But many of them soon fell under its spell. Pauli, who had just laboriously worked out the theory of the hydrogen spectrum using matrix mechanics, found wave mechanics much easier to use for the same purpose. Born wrote to Schrödinger that he grew so excited upon reading the first wave mechanics paper that he wanted ‘to defect…to continuum physics…[to] the crisp, clear conceptual foundations of classical physics’,25 though his ardor soon cooled.
At first the conflict played out in arguments about the scientific merit of the two approaches: Which approach did the better job? The hydrogen atom – which Pauli solved with both methods – was a first key test case. It was the drosophila fly or lab rat of atomic physicists, the problem that any model had to tackle first – for the hydrogen atom had been successfully analysed with the old quantum theory, in close agreement with experiment, meaning that formula could be compared. Another test case was to account for the transition between the quantum and the classical world, or how to get from its rungs to ours. Schrödinger had shown that wave mechanics had an answer to this, but none was apparent yet for matrix mechanics. Yet another key problem was how to handle collisions between things in the atomic world, which would require showing how a system evolved over time.
The answer to the question of which approach had more scientific merit was soon resolved: In May, in his fourth paper of 1926, Schrödinger proved that, mathematically speaking, the two approaches were identical.26 Pauli reached the same conclusion. It was not yet clear how to handle all the test cases, but the demonstration of mathematical equivalence showed that neither approach had more or less mathematical merit than the other. Scientifically, though, wave mechanics could do more than matrix mechanics, for it was essential for analyzing the continuous part of the spectrum.
As Heisenberg’s biographer David Cassidy notes, however, this conclusion only restructured the conflict so it could begin in earnest. With the issue of mathematical equivalence settled, the partisans now were liberated to argue about the physical interpretations of the theories. These were dramatically opposed: wave mechanics – at least as interpreted somewhat hopefully by Schrödinger – portrayed the atomic world as woven out of continuous processes that are causally responsible for what seem to be discontinuous events, and as unfolding in space and time, while matrix mechanics portrayed the atomic realm as lacking continuous processes and causal relations, as not related to space and time, and as not being a world at all in any way humans can imagine. This conflict was less decidable and more emotional than the one about scientific merit, for it reflected the adversaries’
sense of what physics was all about, of what the world was, and of the most fundamental relationship between human beings and the world.
Yet, as Beller notes, ‘in the initial stages of the controversy over interpretation nobody had a clearly articulated position, let alone a handle on the ‘truth.’ ‘27 The restructured clash now forced the partisans to argue for the physical interpretations of their theories. Schrödinger had to argue that, at its most fundamental level, the world was full of continuities, and that to describe it one did not require Heisenberg’s awkward formal methods. Schrödinger also had to explain how wave packets could hold together, elaborate the meaning of the ψ-function, and demonstrate how the discontinuities of quantum phenomena arise from continuous wave processes. Schrödinger, finally, had to admit the problem of a wave that existed in multidimensional configuration space. Heisenberg and his allies had to argue that the world was full of discontinuities, and that it was misleading to present it otherwise. They had to provide some way of connecting the formal symbolic terms of matrix mechanics to familiar properties, and show why, to the extent that wave mechanics was visualizable, it was false. As partisans are wont, they were not always consistent; Beller has shown how each side ‘cheated’ a little, incorporating aspects of the other in order to make the theories work. But this new and ferocious clash now set the stage for the emergence of both the uncertainty principle and the Copenhagen interpretation of its meaning.28
Schrödinger began sniping already in his paper proving the identity of the two approaches. While they were indeed identical, he said, he was ‘discouraged, if not repelled’ by the ‘very difficult’ mathematical methods of matrix mechanics and by its lack of visualizability.29 Later he said that it was ‘extraordinarily difficult’ to attack atomic issues such as the transition problem so long as one has to ‘repress intuition’ and ‘operate only with such abstract ideas as transition probabilities, energy levels, etc.’30 He wrote to Wien that the avowals about the necessity of restricting physics to observables ‘only glosses over our inability to guess the right pictures.’31
A Brief Guide to the Great Equations Page 24