by Manjit Kumar
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 mechanics.’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.73 Schrödinger 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.
Chapter 10
UNCERTAINTY IN COPENHAGEN
As Werner Heisenberg stood in front of the blackboard, with his notes spread out on the table before him, he was nervous. The brilliant 25-year-old physicist had every reason to be. It was Wednesday, 28 April 1926, and he was about to deliver a lecture on matrix mechanics to the famed physics colloquium at Berlin University. Whatever the merits of Munich or Göttingen, it was Berlin that Heisenberg rightly called ‘the stronghold of physics in Germany’.1 His eyes scanned the faces in the audience and settled on four men sitting in the front row, each with a Nobel Prize to his name: Max von Laue, Walter Nernst, Max Planck, and Albert Einstein.
Any nerves at this ‘first chance to meet so many famous men’ quickly subsided as Heisenberg, by his own reckoning, presented ‘a clear account of the concepts and mathematical foundations of what was then a most unconventional theory’.2 As the audience drifted away after the lecture, Einstein invited Heisenberg back to his apartment. During the half-hour stroll to Haberlandstrasse, Einstein asked Heisenberg about his family, education and early research. It was only when they were comfortably seated in his apartment that the real conversation began, recalled Heisenberg, as Einstein probed ‘the philosophical background of my recent work’.3 ‘You assume the existence of electrons inside the atom, and you are probably right to do so’, said Einstein. ‘But you refuse to consider their orbits, even though we can observe electron tracks in a cloud chamber. I should very much like to hear more about your reasons for making such strange assumptions.’4 This was just what he had hoped for, a chance to win over the 47-year-old quantum master.
‘We cannot observe electron orbits inside the atom,’ replied Heisenberg, ‘but the radiation which an atom emits during discharges enables us to deduce the frequencies and corresponding amplitudes of its electrons.’5 Warming to his theme, he explained that ‘since a good theory must be based on directly observable magnitudes, I thought it more fitting to restrict myself to these, treating them, as it were, as representatives of the electron orbits’.6 ‘But you don’t seriously believe,’ Einstein protested, ‘that none but observable magnitudes must go into a physical theory?’7 It was a question that struck at the very foundations on which Heisenberg had constructed his new mechanics. ‘Isn’t that precisely what you have done with relativity?’ he countered.
A ‘good trick should not be tried twice’, smiled Einstein.8 ‘Possibly I did use this kind of reasoning,’ he conceded, ‘but it is nonsense all the same.’ Although it might be heuristically useful to bear in mind what one has actually observed, in principle, he argued, ‘it is quite wrong to try founding a theory on observable magnitudes alone’. ‘In reality the very opposite happens. It is the theory which decides what we can observe.’9 What did Einstein mean?
Almost a century before, in 1830, the French philosopher Auguste Comte had argued that, while every theory has to be based on observation, the mind also needs a theory in order to make observations. Einstein tried to explain that observation was a complex process, involving assumptions about phenomena that are used in theories. ‘The phenomenon under observation produces certain events in our measuring apparatus’, said Einstein.10 ‘As a result, further processes take place in the apparatus, which eventually and by complicated paths produce sense impressions and help fix the effects in our consciousness.’ These effects, Einstein maintained, depend on our theories. ‘And in your theory,’ he told Heisenberg, ‘you quite obviously assume that the whole mechanism of light transmission from the vibrating atom to the spectroscope or to the eye works just as one has always supposed it does, that is, essentially according to Maxwell’
s law. If that were no longer the case, you could not possibly observe any of the magnitudes you call observable.’11 Einstein continued to press: ‘Your claim that you are introducing none but observable magnitudes is therefore an assumption about a property of the theory that you are trying to formulate.’12 ‘I was completely taken aback by Einstein’s attitude, though I found his arguments convincing’, Heisenberg later admitted.13
While Einstein was still a patent clerk he had studied the work of the Austrian physicist Ernst Mach, for whom the goal of science was not to discern the nature of reality, but to describe experimental data, the ‘facts’, as economically as possible. Every scientific concept was to be understood in terms of its operational definition – a specification of how it could be measured. It was while under the influence of this philosophy that Einstein had challenged the established concepts of absolute space and time. But he had long since abandoned Mach’s approach because, as he told Heisenberg, it ‘rather neglects the fact that the world really exists, that our sense impressions are based on something objective’.14
As he left the apartment disappointed at his failure to persuade Einstein, Heisenberg needed to make a decision. In three days’ time, on 1 May, he was due in Copenhagen to begin his dual appointment as Bohr’s assistant and as a lecturer at the university. However, he had just been offered an ordinary professorship at Leipzig University. Heisenberg knew it was a tremendous honour for one so young, but should he accept? Heisenberg told Einstein of the difficult choice he had to make. Go and work with Bohr, was his advice. The next day, Heisenberg wrote to his parents that he was turning down the Leipzig offer. ‘If I continue to produce good papers,’ he reassured himself and them, ‘I will always receive another call; otherwise I don’t deserve it.’15
