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

Part II. BOY PHYSICS

Chapter 9. 'A LATE EROTIC OUTBURST'

'I do not even know what a matrix is', Heisenberg had lamented when told of the origins of the strange multiplication rule that lay at the heart of his new physics. It was a reaction widely shared among physicists when they were presented with his matrix mechanics. Within a matter of months, however, Erwin Schrödinger offered them an alternative that they eagerly embraced. His friend, the great German mathematician Hermann Weyl, later described Schrödinger's astonishing achievement as the product of 'a late erotic outburst'.¹ A serial womaniser, the 38-year-old Austrian discovered wave mechanics while enjoying a secret tryst during Christmas 1925 at the Swiss ski resort of Arosa. Later, after fleeing Nazi Germany, he first scandalised Oxford and then Dublin when he set up home with his wife and yet another mistress under the same roof.

'His private life seemed strange to bourgeois people like ourselves', Born wrote some years after Schrödinger's death in 1961. 'But all this does not matter. He was a most lovable person, independent, amusing, temperamental, kind and generous, and he had a most perfect and efficient brain.'²

Erwin Rudolf Josef Alexander Schrödinger was born in Vienna on 12August 1887. His mother wanted to name him Wolfgang, after Goethe, but allowed her husband to honour an older brother of his who had died in childhood. This brother was the reason why Schrödinger's father inherited the thriving family business manufacturing linoleum and oilcloth, ending his hopes of being a scientist after studying chemistry at Vienna University. Schrödinger knew that the comfortable and carefree life he enjoyed before the First World War was possible only because his father had sacrificed his personal desires on the altar of duty.

Even before he could read or write, Schrödinger kept a record of the day's activities by dictating it to a willing adult. Precocious, he was educated at home by private tutors until the age of eleven when he began attending the Akademisches Gymnasium. Almost from the very first day until he left eight years later, Schrödinger excelled at the school. He was always first in his class without appearing to make much of an effort. A classmate recalled that 'especially in physics and mathematics, Schrödinger had a gift for understanding that allowed him, without any homework, immediately and directly to comprehend all the material during the class hours and to apply it'.³ In truth, he was a dedicated student who worked hard in the privacy of his own study at home.

Schrödinger, like Einstein, had an intense dislike of rote learning and being forced to memorise useless facts. Nevertheless, he enjoyed the strict logic that underpinned the grammar of Greek and Latin. With a maternal grandmother who was English, he began learning the language early and spoke it almost as fluently as German. Later he learnt French and Spanish and was able to lecture in these languages whenever the occasion demanded. Well versed in literature and philosophy, he also loved the theatre, poetry and art. Schrödinger was just the sort of person to leave Werner Heisenberg feeling inadequate. Paul Dirac, when asked once if he played an instrument, replied that he did not know. He had never tried. Nor had Schrödinger, who shared his father's dislike of music.

After graduating from the Gymnasium in 1906, Schrödinger looked forward to studying physics at Vienna University under Ludwig Boltzmann. Tragically, the legendary theoretician committed suicide weeks before Schrödinger started his course. With his grey-blue eyes and shock of swept-back hair, Schrödinger made quite an impression despite being only 5ft 6in. Having shown himself to be an exceptional student at the Gymnasium, much was now expected from him. He did not disappoint, coming top of the class in one exam after another. Surprisingly, given his interest in theoretical physics, Schrödinger gained his doctorate in May 1910 with a dissertation entitled 'On the conduction of electricity on the surface of insulators in moist air'. It was an experimental investigation, showing that Schrödinger was, unlike Pauli and Heisenberg, perfectly at ease in the laboratory. Twenty-three-year-old Dr Schrödinger had a summer of freedom before reporting for military service on 1 October 1910.

All able-bodied young men in Austria-Hungary were required to do three years of military service. But as a university graduate he was able to choose a year's officer training, leading to a commission in the reserve ranks. When he returned to civilian life in 1911, Schrödinger secured a position as an assistant to the professor of experimental physics at his old university. He knew he was not cut out to be an experimenter, but never regretted the experience. 'I belong to those theoreticians who know by direct observation what it means to make a measurement', he later wrote.⁴ 'Methinks it were better if there were more of them.'

In January 1914, Schrödinger, aged 26, became a privatdozent. Like everywhere else, opportunities in theoretical physics in Austria were few. The road to the professorship he desired seemed a long and difficult one. So he toyed with the idea of abandoning physics. Then in August that year the First World War began and he was called up to fight. He had luck on his side from the very beginning. As an artillery officer, he served in fortified positions high on the Italian front. The only real danger he faced during his various postings was boredom. Then he began receiving books and scientific journals that helped to relieve the tedium. 'Is this a life: to sleep, to eat, and to play cards?' he wrote in his diary before the first consignment arrived.⁵ Philosophy and physics were the only things that kept Schrödinger from total despair: 'I no longer ask when will the war be over? But: will it be over?'⁶

Relief came when he was transferred back to Vienna in the spring of 1917 to teach physics at the university and meteorology at an anti-aircraft school. Schrödinger ended the war, as he wrote later, 'without getting wounded and without illness and with little distinction'.⁷ As for most others, the early post-war years were difficult for Schrödinger and his parents, with the family business ruined. As the Habsburg Empire fell apart, the situation was made worse as the victorious allies maintained a blockade that cut off food supplies. As thousands starved and froze during the winter of 1918-19 in Vienna, with little money to buy food on the black market, the Schrödingers were often forced to eat at a local soup kitchen. Things began to improve slowly after March 1919 when the blockade was lifted and the emperor went into exile. Salvation for Schrödinger arrived early the following year with the offer of a job at the University of Jena. The salary was just enough for him to marry 23-year-old Annemarie Bertel.

Arriving in Jena in April, the couple stayed just six months before Schrödinger was appointed to an extraordinary professorship in October at the Technische Hochschule in Stuttgart. The money was better, and after the experiences of the past few years that mattered to him. By spring 1921 the universities of Kiel, Hamburg, Breslau and Vienna were all looking to appoint theoretical physicists. Schrödinger, who had by then earned a solid reputation, was being seriously considered by all of them. He accepted the offer of a professorship at Breslau.

At the age of 34, Schrödinger might have achieved the ambition of every academic; however, in Breslau he had the title but not the salary to go with it, and he left when the University of Zurich came calling. Not long after arriving in Switzerland in October 1921, Schrödinger was diagnosed with bronchitis and possibly tuberculosis. Negotiations surrounding his future, and the deaths of his parents during the previous two years, had taken their toll. 'I was actually so kaput that I could no longer get any sensible ideas', he later told Wolfgang Pauli.⁸ On doctor's orders, Schrödinger went to a sanatorium in Arosa. It was in this high-altitude Alpine resort not far from Davos that he spent the next nine months recuperating. He was not idle during this time, but found the energy and enthusiasm to publish several papers.

As the years passed, Schrödinger began to wonder if he would ever make a major contribution that would establish him among the first rank of contemporary physicists. At the beginning of 1925 he was 37, long having celebrated the 30th birthday that was said to be the watershed in the creative life of a theorist. Doubts over his worth as a physicist were compounded by a marriage in trouble because of affairs on both sides. By the end of the year Schrödinger's marriage was shakier than ever, but he made the breakthrough that would ensure his place in the pantheon of physics.

Schrödinger was taking an ever more active interest in the latest developments in atomic and quantum physics. In October 1925, he read a paper that Einstein had written earlier in the year. A footnote that flagged up Louis de Broglie's thesis on wave-particle duality caught his eye. As with most footnotes, virtually everyone ignored it. Intrigued by Einstein's stamp of approval, Schrödinger set about acquiring a copy of the thesis, unaware that papers by the French prince had been in print for nearly two years. A couple of weeks later, on 3 November, he wrote to Einstein: 'A few days ago I read with the greatest interest the ingenious thesis of de Broglie, which I finally got hold of.'⁹

Others were also beginning to take note, but in the absence of any experimental support, few were as receptive to de Broglie's ideas as Einstein and Schrödinger. In Zurich, every fortnight, physicists from the university got together with those from the Eidgenossische Technische Hochschule (ETH), for a joint colloquium. Pieter Debye, the ETH professor of physics, ran the meetings and asked Schrödinger to give a talk on de Broglie's work. In the eyes of his colleagues, Schrödinger was an accomplished and versatile theoretician who had made solid but unremarkable contributions in his 40-odd papers that spanned areas as diverse as radioactivity, statistical physics, general relativity and colour theory. Among these were a number of well-received review articles that demonstrated his ability to absorb, analyse and organise the work of others.

On 23 November Felix Bloch, a 21-year-old student, was present when 'Schrödinger gave a beautifully clear account of how de Broglie associated a wave with a particle and how he could obtain the quantization rules of Niels Bohr and Sommerfeld by demanding that an integer number of waves should be fitted along a stationary orbit'.¹⁰ With no experimental confirmation of wave-particle duality, which would come in 1927, Debye found it all far-fetched and 'rather childish'.¹¹ The physics of a wave - any wave, from sound to electromagnetic, even a wave travelling along a violin string - has an equation that describes it. In what Schrödinger had outlined there was no 'wave equation'; de Broglie had never tried to derive one for his matter waves. Nor had Einstein after he read the French prince's thesis. Debye's point 'sounded quite trivial and did not seem to make a great impression', Bloch still remembered 50 years later.¹²

Schrödinger knew that Debye was right: 'You cannot have waves without a wave equation.'¹³ Almost at once he decided to find the missing equation for de Broglie's matter waves. After returning from his Christmas holiday, Schrödinger was able to announce at the next colloquium held early in the New Year: 'My colleague Debye suggested that one should have a wave equation; well, I have found one!'¹⁴ Between one meeting and the next, Schrödinger had taken de Broglie's nascent ideas and developed them into a fully-blown theory of quantum mechanics.

Schrödinger knew exactly where to start and what he had to do. De Broglie had tested his idea of wave-particle duality by reproducing the allowed electron orbits in the Bohr atom as those in which only a whole number of standing electron wavelengths could fit. Schrödinger knew that the elusive wave equation he sought would have to reproduce the three-dimensional model of the hydrogen atom with three-dimensional standing waves. The hydrogen atom would be the litmus test for the wave equation he needed to find.

Not long after starting the hunt, Schrödinger thought he had bagged just such an equation. However, when he applied it to the hydrogen atom, the equation churned out the wrong answers. The root of the failure lay in the fact that de Broglie had developed and presented wave-particle duality in a manner consistent with Einstein's theory of special relativity. Following de Broglie's lead, Schrödinger started out by looking for a wave equation that was 'relativistic' in form, and found one. In the meantime, Uhlenbeck and Goudsmit had discovered the concept of electron spin, but their paper did not appear in print until the end of November 1925. Schrödinger had found a relativistic wave equation, but unsurprisingly it did not include spin and therefore failed to agree with experiments.¹⁵

With the Christmas vacation fast approaching, Schrödinger began to concentrate his efforts on finding a wave equation without worrying about relativity. He knew that such an equation would fail for electrons travelling at speeds close to that of light where relativity could not be ignored. But for his purposes such a wave equation would do. Soon, however, there was more than just physics on his mind. He and his wife Anny were having another of their sustained bouts of marital turbulence, one that was lasting longer than most. Despite the affairs and talk of divorce, each seemed incapable and unwilling to permanently part from the other. Schrödinger wanted to escape for a couple of weeks. Whatever excuse he gave his wife, he left Zurich for the winter wonderland of his favourite Alpine resort, Arosa, and a rendezvous with an ex-lover.

Schrödinger was delighted to be back in the familiar and comfortable surroundings of the Villa Herwig. It was here that he and Anny had spent the previous two Christmas holidays, but there was hardly time enough over the next two weeks to feel guilty as Schrödinger spent his passion with his mysterious lady. However distracted he may have been, Schrödinger made time to continue the search for his wave equation. 'At the moment I am struggling with a new atomic theory', he wrote on 27 December.¹⁶ 'If only I knew more mathematics! I am very optimistic about this thing and expect that if I can only … solve it, it will be very beautiful.' Six months of sustained creativity were to follow during this 'late erotic outburst' in his life.¹⁷ Inspired by his unnamed Muse, Schrödinger had discovered a wave equation, but was it the wave equation he was seeking?

Schrödinger did not 'derive' his wave equation; there was just no way to do it from classical physics that was logically rigorous. Instead he constructed it out of de Broglie's wave-particle formula that linked the wavelength associated with a particle to its momentum, and from well-established equations of classical physics. As simple as it sounds, it required all of Schrödinger's skill and experience to be the first to write it down. It was the foundation on which he built the edifice of wave mechanics in the months ahead. But first he had to prove that it was the wave equation. When applied to the hydrogen atom, would it generate the correct values for the energy levels?

After returning to Zurich in January, Schrödinger found that his wave equation did reproduce the series of energy levels of the Bohr-Sommerfeld hydrogen atom. More complicated than de Broglie's one-dimensional standing electron waves fitted into circular orbits, Schrödinger's theory obtained their three-dimensional analogues - electron orbitals. Their associated energies were generated as part and parcel of the acceptable solutions of Schrödinger's wave equation. Banished once and for all were the ad hoc additions required by the Bohr-Sommerfeld quantum atom - all the previous tinkering and tweaking that sat uneasily now emerged naturally from within the framework of Schrödinger's wave mechanics. Even the mysterious quantum jumping between orbits by an electron appeared to be eliminated by the smooth and continuous transitions from one permitted three-dimensional electron standing wave to another. 'Quantization as an Eigenvalue Problem' was received by the Annalen der Physik on 27 January 1926.¹⁸ Published on 13 March, it presented Schrödinger's version of quantum mechanics and its application to the hydrogen atom.

In a career that spanned some 50 years, Schrödinger's average annual output of research papers amounted to 40 printed pages. In 1926 he published 256 pages in which he demonstrated how wave mechanics could successfully solve a range of problems in atomic physics. He also came up with a time-dependent version of his wave equation that could tackle 'systems' that changed with time. Among them were processes involving the absorption and emission of radiation and the scattering of radiation by atoms.

On 20 February, as the first paper was being readied for the printers, Schrödinger used the name Wellenmechanik, wave mechanics, for the first time to describe his new theory. In stark contrast to the cold and austere matrix mechanics that proscribed even the hint of visualisability, Schrödinger offered physicists a familiar and reassuring alternative that offered to explain the quantum world in terms closer to those of nineteenth-century physics than Heisenberg's highly abstract formulation. In place of the mysterious matrices, Schrödinger came bearing differential equations, an essential part of every physicist's mathematical toolbox. Heisenberg's matrix mechanics gave them quantum jumps and discontinuity, and nothing to picture in their mind's eye as they sought to glimpse the inner workings of the atom. Schrödinger told physicists they no longer needed to 'suppress intuition and to operate only with abstract concepts such as transition probabilities, energy levels, and the like'.¹⁹ It was hardly surprising that they greeted wave mechanics with enthusiasm and quickly rushed to embrace it.

As soon as he received complimentary copies of his paper, Schrödinger sent them out to colleagues whose opinions mattered most to him. Planck wrote back on 2 April that he had read the paper 'like an eager child hearing the solution to a riddle that had plagued him for a long time'.²⁰ Two weeks later, Schrödinger received a letter from Einstein, who told him 'the idea of your work springs from true genius'.²¹ 'Your approval and Planck's mean more to me than that of half the world', Schrödinger wrote back.²² Einstein was convinced that Schrödinger had made a decisive advance, 'just as I am convinced that the Heisenberg-Born method is misleading'.²³

Others took longer to fully appreciate the product of Schrödinger's 'late erotic outburst'. Sommerfeld initially believed that wave mechanics was 'totally crazy', before changing his mind and declaring: 'although the truth of matrix mechanics is indubitable, its handling is extremely intricate and frighteningly abstract. Schrödinger has now come to our rescue.'²⁴ Many others also breathed easier as they learnt and began using the more famil-iar ideas embodied in wave mechanics rather than having to struggle with the abstract and alien formulation of Heisenberg and his Göttingen colleagues. 'The Schrödinger equation came as a great relief,' wrote the young spin doctor George Uhlenbeck, 'now we did not any longer have to learn the strange mathematics of matrices.'²⁵ Instead Ehrenfest, Uhlenbeck and the others in Leiden spent weeks 'standing for hours at a time in front of the blackboard' in order to learn all the splendid ramifications of wave mechanics.²⁶

Pauli may have been close to the Göttingen physicists, but he recognised the significance of what Schrödinger had done and was deeply impressed. Pauli had strained every ounce of grey matter he possessed as he successfully applied matrix mechanics to the hydrogen atom. Everyone was later amazed by the speed and virtuosity with which he had done so. Pauli sent his paper to the Zeitschrift für Physik on 17 January, only ten days before Schrödinger posted his first paper. When he saw the relative ease with which wave mechanics allowed Schrödinger to tackle the hydrogen atom, Pauli was astonished. 'I believe that the work counts among the most significant recently written', he told Pascual Jordan. 'Read it carefully and with devotion.'²⁷ Not long afterwards, in June, Born described wave mechanics 'as the deepest form of the quantum laws'.²⁸

Heisenberg was 'not very pleased', he told Jordan, by Born's apparent defection to wave mechanics.²⁹ Although he acknowledged that Schrödinger's paper was 'incredibly interesting' with its use of more familiar mathematics, Heisenberg firmly believed that when it came to physics, his matrix mechanics was a better description of the way things were at the atomic level.³⁰ 'Heisenberg from the very beginning did not share my opinion that your wave mechanics is physically more significant than our quantum mechanics', Born confided to Schrödinger in May 1927.³¹ By then it was hardly a secret. Nor did Heisenberg want it to be. There was too much at stake.

As spring had given way to summer in 1925 there was still no quantum mechanics, a theory that would do for atomic physics what Newtonian mechanics did for classical physics. A year later there were two competing theories that were as different as particles and waves. They both gave identical answers when applied to the same problems. What, if any, was the connection between matrix and wave mechanics? It was a question that Schrödinger began to ponder almost as soon as he finished his first ground-breaking paper. After two weeks of searching he found no link. 'Consequently,' Schrödinger wrote to Wilhelm Wien, 'I have given up looking any further myself.'³² He was hardly disappointed, as he confessed that 'matrix calculus was already unbearable to me long before I even distantly thought of my theory'.³³ But he was unable to stop digging until he unearthed the connection at the beginning of March.

The two theories that appeared to be so different in form and content, one employing wave equations and the other matrix algebra, one describing waves and the other particles, were mathematically equivalent.³⁴ No wonder they both gave exactly the same answers. The advantages of having two different but equivalent formalisms of quantum mechanics quickly became apparent. For most problems physicists encountered, Schrödinger's wave mechanics provided the easiest route to the solution. Yet for others, such as those involving spin, it was Heisenberg's matrix approach that proved its worth.

With any possible arguments about which of the two theories was correct smothered even before they could begin, attention turned from the mathematical formalism to the physical interpretation. The two theories might technically be equivalent, but the nature of physical reality that lay beyond the mathematics was altogether different: Schrödinger's waves and continuity versus Heisenberg's particles and discontinuity. Each man was convinced that his theory captured the true nature of physical reality. Both could not be right.

At the beginning there was no personal animosity between Schrödinger and Heisenberg as they began to question each other's interpretation of quantum mechanics. But soon emotions began to run high. In public and in their papers both managed, on the whole, to rein in their true feelings. In their letters, however, there was no need for tact and restraint. When he initially tried but failed to prove the equivalence of wave and matrix mechanics, Schrödinger was somewhat relieved that there might be none, since 'the mere thought makes me shudder, if I later had to present the matrix calculus to a young student as describing the true nature of the atom'.³⁵ In his paper, 'On the Relation Between Heisenberg-Born-Jordan Quantum Mechanics and My Own', Schrödinger was at pains to distance wave mechanics from matrix mechanics. 'My theory was inspired by L. de Broglie and by brief but infinitely far-seeing remarks of A. Einstein', he explained. 'I was absolutely unaware of any genetic relationship with Heisenberg.'³⁶ Schrödinger concluded that, 'because of the lack of visualization' in matrix mechanics, 'I felt deterred by it, if not to say repelled'.³⁷

Heisenberg was even less diplomatic about the continuity that Schrödinger was trying to restore to the atomic realm where, as far as he was concerned, discontinuity ruled. 'The more I think about the physical portion of the Schrödinger theory, the more repulsive I find it', he told Pauli in June.³⁸ 'What Schrödinger writes about the visualizability of his theory "is probably not quite right", in other words it's crap.' Two months earlier, Heisenberg had appeared more conciliatory when he described wave mechanics as 'incredibly interesting'.³⁹ But those who knew Bohr recognised that Heisenberg was employing exactly the sort of language favoured by the Dane, who always called an idea or an argument 'interesting' when in fact he disagreed with it. Increasingly frustrated as more of his colleagues abandoned matrix mechanics for the easier-to-use wave mechanics, Heisenberg finally snapped. He could hardly believe it when Born, of all people, started using Schrödinger's wave equation. In a fit of anger, Heisenberg called him a 'traitor'.

He may have been envious of the growing popularity of Schrödinger's alternative, but after its discovery it was Heisenberg who was responsible for the next great triumph of wave mechanics. He might have been annoyed at Born, but Heisenberg had also been seduced by the mathematical ease with which Schrödinger's approach could be applied to atomic problems. In July 1926 he used wave mechanics to account for the line spectra of helium.⁴⁰ Just in case anyone read too much into his adoption of the rival formulation, Heisenberg pointed out that it was nothing more than expediency. The fact that the two theories were mathematically equivalent meant he could use wave mechanics while ignoring the 'intuitive pictures' Schrödinger painted with it. However, even before Heisenberg posted his paper, Born had used Schrödinger's palette to paint an entirely different picture on the same canvas when he discovered that probability lay at the heart of wave mechanics and quantum reality.

Schrödinger was not trying to paint a new picture, but attempting to restore an old one. For him there were no quantum jumps between different energy levels in an atom, but only smooth, continuous transitions from one standing wave into another, with the emission of radiation being the product of some exotic resonance phenomenon. He believed that wave mechanics allowed the restoration of a classical, 'intuitive' picture of physical reality, one of continuity, causality and determinism. Born disagreed. 'Schrödinger's achievement reduces itself to something purely mathematical,' he told Einstein, 'his physics is wretched.'⁴¹ Born used wave mechanics to paint a surreal picture of a reality with discontinuity, acausality and probability, instead of Schrödinger's attempt at a Newtonian-inspired old master. These two pictures of reality hang on different interpretations of the so-called wave function, symbolised by the Greek letter psi, , in Schrödinger's wave equation.

Schrödinger had known from the very beginning that there was a problem with his version of quantum mechanics. According to Newton's laws of motion, if the position of an electron is known at a certain time together with its velocity, then it is theoretically possible to determine exactly where it will be at some later time. However, waves are much more difficult to pin down than a particle. Dropping a stone into a pond sends ripples of waves across its surface. Exactly where is the wave? Unlike a particle, a wave is not localised at a single place, but is a disturbance that carries energy through a medium. Like people taking part in a 'Mexican wave', a water wave is just individual water molecules bobbing up and down.

All waves, whatever their size and shape, can be described by an equation that mathematically maps their motion, just as Newton's equations do for a particle. The wave function, , represents the wave itself and describes its shape at a given time. The wave function of a wave rippling across the surface of a pond specifies the size of the disturbance, the so-called amplitude, of the water at any point x at time t. When Schrödinger discovered the wave equation for de Broglie's matter waves, the wave function was the unknown part. Solving the equation for a particular physical situation, such as the hydrogen atom, would yield the wave function. However, there was a question that Schrödinger was finding difficult to answer: what was doing the waving?

In the case of water or sound waves, it was obvious: water or air molecules. Light had perplexed physicists in the nineteenth century. They had been forced to invoke the mysterious 'ether' as the necessary medium through which light travelled, until it was discovered that light was an electromagnetic wave with interlocked electric and magnetic fields doing the waving. Schrödinger believed that matter waves were as real as any of these more familiar types of waves. However, what was the medium through which an electron wave travelled? The question was akin to asking what does the wave function in Schrödinger's wave equation represent? In the summer of 1926 a witty little ditty summed up the situation that confronted Schrödinger and his colleagues:

Erwin with his psi can do

Calculations quite a few.

But one thing has not been seen:

Just what does psi really mean?⁴²

Schrödinger finally proposed that the wave function of an electron, for example, was intimately connected to the cloud-like distribution of its electric charge as it travelled through space. In wave mechanics the wave function was not a quantity that could be directly measured because it was what mathematicians call a complex number. 4+3i is one example of such a number, and it consists of two parts: one 'real' and the other 'imaginary'. 4 is an ordinary number and is the 'real' part of the complex number 4+3i. The 'imaginary' part, 3i, has no physical meaning because i is the square root of -1. The square root of a number is just another number that multiplied by itself will give the original number. The square root of 4 is 2 since 2×2 equals 4. There is no number that multiplied by itself equals -1. While 1×1=1, -1×-1 is also equal to 1, since by the laws of algebra, a minus times a minus generates a plus.

The wave function was unobservable; it was something intangible that could not be measured. However, the square of a complex number gives a real number that is associated with something that can actually be measured in the laboratory.⁴³ The square of 4+3i is 25.⁴⁴ Schrödinger believed that the square of the wave function of an electron, |(x,t)|², was a measure of the smeared-out density of electric charge at location x at time t.

As part of his interpretation of the wave function, Schrödinger introduced the concept of a 'wave packet' to represent the electron as he challenged the very idea that particles existed. He argued that an electron only 'appeared' to be particle-like but was not actually a particle, despite the overwhelming experimental evidence in favour of it being so. Schrödinger believed that a particle-like electron was an illusion. In reality there were only waves. Any manifestation of a particle electron was due to a group of matter waves being superimposed into a wave packet. An electron in motion would then be nothing more than a wave packet that moved like a pulse sent, with a flick of the wrist, travelling down the length of a taut rope tied at one end and held at the other. A wave packet that gave the appearance of a particle required a collection of waves of different wavelengths that interfered with one another in such a way that they cancelled each other out beyond the wave packet.

If giving up particles and reducing everything to waves rid physics of discontinuity and quantum jumps, then for Schrödinger it was a price worth paying. However, his interpretation soon ran into difficulties as it failed to make physical sense. Firstly, the wave packet representation of the electron began to unravel when it was discovered that the constituent

Figure 11: A wave packet formed from the superposition of a group of waves

waves would spread out across space to such a degree that they would have to travel faster than the speed of light if they were to be connected with the detection of a particle-like electron in an experiment.

Try as he might, there was no way for Schrödinger to prevent this dispersal of the wave packet. Since it was made up of waves that varied in wavelength and frequency, as the wave packet travelled through space it would soon begin to spread out as individual waves moved at different velocities. An almost instantaneous coming together, a localisation at one point in space, would have to take place every time an electron was detected as a particle. Secondly, when attempts were made to apply the wave equation to helium and other atoms, Schrödinger's vision of the reality that lay beneath his mathematics disappeared into an abstract, multi-dimensional space that was impossible to visualise.

The wave function of an electron encodes everything there is to know about its single three-dimensional wave. Yet the wave function for the two electrons of the helium atom could not be interpreted as two three-dimensional waves existing in ordinary three-dimensional space. Instead the mathematics pointed to a single wave inhabiting a strange six-dimensional space. In each move across the periodic table from one element to the next, the number of electrons increased by one and an additional three dimensions were required. If lithium, third in the table, required a nine-dimensional space, then uranium had to be accommodated in a space with 276 dimensions. The waves that occupied these abstract multi-dimensional spaces could not be the real, physical waves that Schrödinger hoped would restore continuity and eliminate the quantum jump.

Nor could Schrödinger's interpretation account for the photoelectric and Compton effects. There were unanswered questions: how could a wave packet possess electric charge? Could wave mechanics incorporate quantum spin? If Schrödinger's wave function did not represent real waves in everyday three-dimensional space, then what were they? It was Max Born who provided the answer.

Born was nearing the end of his five-month stay in America when Schrödinger's first paper on wave mechanics appeared in March 1926. Reading it on his return to Göttingen in April, he was taken completely 'by surprise' as others had been.⁴⁵ The terrain of quantum physics had dramatically changed during his absence. Almost out of nowhere, Born immediately recognised, Schrödinger had constructed a theory of 'fascinating power and elegance'.⁴⁶ He was quick to acknowledge the 'superiority of wave mechanics as a mathematical tool', as demonstrated by the relative ease with which it solved 'the fundamental atomic problem' - the hydrogen atom.⁴⁷ After all, it had taken someone of Pauli's prodigious talent to apply matrix mechanics to the hydrogen atom. Born might have been taken by surprise but he was already familiar with the idea of matter waves long before Schrödinger's paper was published.

'A letter from Einstein directed my attention to de Broglie's thesis shortly after its publication, but I was too much involved in our speculations to study it carefully', Born admitted more than half a century later.⁴⁸ By July 1925 he had made time to study de Broglie's work and wrote to Einstein that 'the wave theory of matter could be of very great importance'.⁴⁹ Enthused, he had already begun 'speculating a little about de Broglie's waves', Born told Einstein.⁵⁰ But just then he shoved de Broglie's ideas aside to make sense of the strange multiplication rule in a paper given to him by Heisenberg. Now, almost a year later, Born solved some of the problems encountered by wave mechanics, but at a price far higher than Schrödinger demanded with his sacrifice of particles.

The rejection of particles and quantum jumps that Schrödinger advocated was too much for Born. He witnessed regularly in Göttingen what he called 'the fertility of the particle concept' in experiments on atomic collisions.⁵¹ Born accepted the richness of Schrödinger's formalism but rejected the Austrian's interpretation. 'It is necessary,' Born wrote late in 1926, 'to drop completely the physical pictures of Schrödinger which aim at a revitalization of the classical continuum theory, to retain only the formalism and to fill that with a new physical content.'⁵² Already convinced 'that particles could not simply be abolished', Born found a way to weave them together with waves using probability as he came up with a new interpretation of the wave function.⁵³

Born had been working on applying matrix mechanics to atomic collisions while in America. Back in Germany with Schrödinger's wave mechanics suddenly at his disposal, he returned to the subject and produced two seminal papers bearing the same title, 'Quantum mechanics of collision phenomena'. The first, only four pages long, was published on 10 July in Zeitschrift für Physik. Ten days later the second paper, more polished and refined than the first, was finished and in the post.⁵⁴ While Schrödinger renounced the existence of particles, Born in his attempt to save them put forward an interpretation of the wave function that challenged a fundamental tenet of physics - determinism.

The Newtonian universe is purely deterministic with no room for chance. In it, a particle has a definite momentum and position at any given time. The forces that act on the particle determine the way its momentum and position vary in time. The only way that physicists such as James Clerk Maxwell and Ludwig Boltzmann could account for the properties of a gas that consists of many such particles was to use probability and settle for a statistical description. The forced retreat into a statistical analysis was due to the difficulties in tracking the motion of such an enormous number of particles. Probability was a consequence of human ignorance in a deterministic universe where everything unfolded according to the laws of nature. If the present state of any system and the forces acting upon it are known, then what happens to it in the future is already determined. In classical physics, determinism is bound by an umbilical cord to causality - the notion that every effect has a cause.

Like two billiard balls colliding, when an electron slams into an atom it can be scattered in almost any direction. However, that is where the similarity ends, argued Born as he made a startling claim. When it comes to atomic collisions, physics could not answer the question 'What is the state after collision?', but only 'How probable is a given effect of the collision?'⁵⁵ 'Here the whole problem of determinism arises', admitted Born.⁵⁶ It was impossible to determine exactly where the electron was after the collision. The best that physics could do, he said, was to calculate the probability that the electron would be scattered through a certain angle. This was Born's 'new physical content', and it all hinged on his interpretation of the wave function.

The wave function itself has no physical reality; it exists in the mysterious, ghost-like realm of the possible. It deals with abstract possibilities, like all the angles by which an electron could be scattered following a collision with an atom. There is a real world of difference between the possible and the probable. Born argued that the square of the wave function, a real rather than a complex number, inhabits the world of the probable. Squaring the wave function, for example, does not give the actual position of an electron, only the probability, the odds that it will found here rather than there.⁵⁷ For example, if the value of the wave function of an electron at X is double its value at Y, then the probability of it being found at X is four times greater than the probability of finding it at Y. The electron could be found at X, Y or somewhere else.

Niels Bohr would soon argue that until an observation or measurement is made, a microphysical object like an electron does not exist anywhere. Between one measurement and the next it has no existence outside the abstract possibilities of the wave function. It is only when an observation or measurement is made that the 'wave function collapses' as one of the 'possible' states of the electron becomes the 'actual' state and the probability of all the other possibilities becomes zero.

For Born, Schrödinger's equation described a probability wave. There were no real electron waves, only abstract waves of probability. 'From the point of view of our quantum mechanics there exists no quantity which in an individual case causally determines the effect of a collision', wrote Born.⁵⁸ And he confessed, 'I myself tend to give up determinism in the atomic world.'⁵⁹ Yet while the 'motion of particles follows probability rules', he pointed out, 'probability itself propagates according to the law of causality'.⁶⁰

It took Born the time between his two papers to fully grasp that he had introduced a new kind of probability into physics. 'Quantum probability', for want of a better term, was not the classical probability of ignorance that could in theory be eliminated. It was an inherent feature of atomic reality. For example, the fact that it was impossible to predict when an individual atom would decay in a radioactive sample, amid the certainty that one would do so, was not due to a lack of knowledge but was the result of the probabilistic nature of the quantum rules that dictate radioactive decay.

Schrödinger dismissed Born's probability interpretation. He did not accept that a collision of an electron or an alpha particle with an atom is 'absolutely accidental', i.e. 'completely undetermined'.⁶¹ Otherwise, if Born was right, then there was no way to avoid quantum jumps and causality was once again threatened. In November 1926, he wrote to Born: 'I have, however, the impression that you and others, who essentially share your opinion, are too deeply under the spell of those concepts (like stationary states, quantum jumps, etc.), which have obtained civic rights in our thinking in the last dozen years; hence, you cannot do full justice to an attempt to break away from this scheme of thought.'⁶² Schrödinger never relinquished his interpretation of wave mechanics and the attempt at a visualisability of atomic phenomena. 'I can't imagine that an electron hops about like a flea', he once memorably said.⁶³

Zurich lay well outside the golden quantum triangle of Copenhagen, Göttingen and Munich. As the new physics of wave mechanics spread like wildfire through Europe's physics community in the spring and summer of 1926, many were eager to hear Schrödinger discuss his theory in person. When the invitation arrived from Arnold Sommerfeld and Wilhelm Wien to give two lectures in Munich, Schrödinger readily accepted. The first, on 21 July, to Sommerfeld's 'Wednesday Colloquium', was routine and well-received. The second, on 23 July, to the Bavarian section of the German Physical Society, was not. Heisenberg, who at the time was based in Copenhagen as Bohr's assistant, had returned to Munich in time to hear both of Schrödinger's lectures before going on a hiking tour.

As he sat in the packed lecture theatre for a second time, Heisenberg listened quietly until the end of Schrödinger's talk, entitled 'New results of wave mechanics'. During the question-and-answer session that followed, he became increasingly agitated until he could no longer remain silent. As he rose to speak, all eyes were on him. Schrödinger's theory, he pointed out, could not explain Planck's radiation law, the Frank-Hertz experiment, the Compton effect, or the photoelectric effect. None could be explained without discontinuity and quantum jumps - the very concepts that Schrödinger wanted to eliminate.

Before Schrödinger could reply, with some in the audience already expressing their disapproval at the remarks of the 24-year-old, an annoyed Wien stood up and intervened. The old physicist, Heisenberg told Pauli later, 'almost threw me out of the room'.⁶⁴ The pair had a history going back to Heisenberg's days as a student in Munich and his poor showing during the oral examination for his doctorate on anything connected to experimental physics. 'Young man, Professor Schrödinger will certainly take care of all these questions in due time', Wien told Heisenberg as he motioned for him to sit down.⁶⁵ 'You must understand that we are now finished with all that nonsense about quantum jumps.' Schrödinger, unfazed, replied that he was confident that all remaining problems would be overcome.

Heisenberg could not stop himself from lamenting later that Sommerfeld, who had witnessed the whole incident, had 'succumbed to the persuasive force of Schrödinger's mathematics'.⁶⁶ Shaken and dejected at being forced to retire from the arena vanquished before battle had been properly joined, Heisenberg needed to regroup. 'A few days ago I heard two lectures here by Schrödinger,' he wrote to Jordan, 'and I am rock-solid convinced of the incorrectness of the physical interpretation of QM presented by Schrödinger.'⁶⁷ He already knew that conviction alone was not enough, given that 'Schrödinger's mathematics signifies a great progress'.⁶⁸ After his disastrous intervention, Heisenberg had sent a dispatch to Bohr from the front line of quantum physics.

After reading Heisenberg's version of events in Munich, Bohr invited Schrödinger to Copenhagen to give a lecture and participate in 'some discussions for the narrower circle of those who work here at the Institute, in which we can deal more deeply with the open questions of atomic theory'.⁶⁹ When Schrödinger stepped off the train on 1 October 1926, Bohr was waiting for him at the station. Remarkably, it was the first time they had ever met.

After the exchange of pleasantries, battle began almost at once, and according to Heisenberg, 'continued daily from early morning until late at night'.⁷⁰ There was to be little respite for Schrödinger from Bohr's continual probing in the days ahead. He installed Schrödinger in the guest room at his home to maximise their time together. Although usually the most kind and considerate of hosts, in his desire to convince Schrödinger that he was in error, Bohr appeared even to Heisenberg to act as a 'remorseless fanatic, one who was not prepared to make the least concession or grant that he could ever be mistaken'.⁷¹ Each man passionately defended his deeply-rooted convictions concerning the physical interpretation of the new physics. Neither was prepared to concede a single point without putting up a fight. Each pounced on any weakness or lack of precision in the argument of the other.

During one discussion Schrödinger called 'the whole idea of quantum jumps a sheer fantasy'. 'But it does not prove that there are no quantum jumps', Bohr countered. All it proved, he continued, was that 'we cannot imagine them'. Emotions soon ran high. 'You can't seriously be trying to cast doubt on the whole basis of quantum theory!' asked Bohr. Schrödinger conceded there was much that still needed to be fully explained, but that Bohr had also 'failed to discover a satisfactory physical interpretation of quantum mechanics'. As Bohr continued to press, Schrödinger finally snapped. 'If all this damned quantum jumping were really here to stay, I should be sorry I ever got involved with quantum theory.' 'But the rest of us are extremely grateful that you did,' Bohr replied, 'your wave mechanics has contributed so much to mathematical clarity and simplicity that it represents a gigantic advance over all previous forms of quantum mechanic.'⁷²

After a few days of these relentless discussions, Schrödinger fell ill and took to his bed. Even as his wife did all she could to nurse their house-guest, Bohr sat on the edge of the bed and continued the argument. 'But surely Schrödinger, you must see …' He did see, but only through the glasses that he had long worn, and he was not about to change them for ones prescribed by Bohr. There had been little, if any, chance of the two men ever reaching a concord. Each remained unconvinced by the other. 'No real understanding could be expected since, at the time, neither side was able to offer a complete and coherent interpretation of quantum mechanics', Heisenberg later wrote.⁷³Schrödinger did not accept that quantum theory represented a complete break with classical reality. As far as Bohr was concerned, there was no going back to the familiar ideas of orbits and continuous paths in the atomic realm. The quantum jump was here to stay whether Schrödinger liked it or not.

As soon as he arrived back in Zurich, Schrödinger recounted Bohr's 'really remarkable' approach to atomic problems in a letter to Wilhelm Wien. 'He is completely convinced that any understanding in the usual sense of the word is impossible', he told Wien. 'Therefore the conversation is almost immediately driven into philosophical questions, and soon you no longer know whether you really take the position he is attacking, or whether you really must attack the position that he is defending.'⁷⁴ Yet despite their theoretical differences, Bohr and 'especially' Heisenberg had behaved 'in a touchingly kind, nice, caring and attentive manner', and all 'was totally, cloudlessly amiable and cordial'.⁷⁵ Distance and a few weeks had made it seem less of an ordeal.

A week before Christmas 1926, Schrödinger and his wife travelled to America, where he had accepted an invitation from the University of Wisconsin to give a series of lectures for which he would receive the princely sum of $2,500. Afterwards he criss-crossed the country, giving nearly 50 lectures. By the time he arrived back in Zurich in April 1927, Schrödinger had turned down several job offers. He had his eye on a far greater prize, Planck's chair in Berlin.

Having been appointed in 1892, Planck was due to retire on 1 October 1927 to an emeritus professorship. Heisenberg, 24, was too young for such an elevated position. Arnold Sommerfeld had been first choice, but at 59, he decided to stay in Munich. It was now either Schrödinger or Born. Schrödinger was appointed as Planck's successor and it was the discovery of wave mechanics that had clinched it. In August 1927, Schrödinger moved to Berlin and found someone there who was just as unhappy with Born's probabilistic interpretation of the wave function as he was - Einstein.

Einstein had been the first to introduce probability into quantum physics in 1916 when he provided the explanation for the spontaneous emission of light-quanta as an electron jumped from one atomic energy level to another. Ten years later, Born had put forward an interpretation of the wave function and wave mechanics that could account for the probabilistic character of quantum jumps. It came with a price tag that Einstein did not want to pay - the renunciation of causality.

In December 1926, Einstein had expressed his growing disquiet at the rejection of causality and determinism in a letter to Born: 'Quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but does not really bring us any closer to the secret of the "old one". I, at any rate, am convinced that He is not playing at dice.'⁷⁶ As the battle lines were being drawn, Einstein was unwittingly the inspiration for a stunning breakthrough, one of the greatest and profoundest achievements in the history of the quantum - the uncertainty principle.