by Manjit Kumar
With the track left behind by an electron passing through a cloud chamber firmly on his mind, Heisenberg examined the concept of the ‘path of the electron’. A path is an unbroken, continuous series of positions taken up by the moving electron in space and time. Under his new criteria, to observe the path involves measuring the electron’s position at each successive point. However, hitting the electron with a gamma ray photon in the act of measuring its position disturbs it, therefore its future trajectory cannot be predicted with certainty. In the case of an atomic electron ‘orbiting’ a nucleus, a gamma ray photon is energetic enough to knock it out of the atom, and only one point in its ‘orbit’ is measured and therefore known. Since the uncertainty principle forbids an exact measurement of both the position and velocity that define the path of an electron or its orbit in an atom, there simply is no path or orbit. The only thing that is known for certain, says Heisenberg, is one point along the path, and ‘therefore here the word “path” has no definable meaning’.47 It is measurement that defines what is being measured.
There is no way of knowing, argued Heisenberg, what happens between two consecutive measurements: ‘It is of course tempting to say that the electron must have been somewhere between the two observations and that therefore the electron must have described some kind of path or orbit even if it may be impossible to know which path.’48 Tempting or not, he maintained that the classical notion of an electron’s trajectory being a continuous, unbroken path through space is unjustified. An electron track observed in a cloud chamber only ‘looks’ like a path, but is really nothing more than a series of water droplets left in its wake.
Heisenberg was desperately trying to understand the sort of questions that it was possible to answer experimentally after his discovery of the uncertainty principle. It was an unspoken basic tenet of classical physics that a moving object possessed both a precise location in space at a given time and a precise momentum, irrespective of whether it was measured or not. From the fact that the position and momentum of an electron cannot be measured with absolute accuracy at the same time, Heisenberg asserted that the electron does not possesses precise values of ‘position’ and ‘momentum’ simultaneously. To talk as if it did, or that it has a ‘trajectory’, is meaningless. To speculate about the nature of reality that lies beyond the realm of observation and measurement is pointless.
In later years, Heisenberg repeatedly chose to highlight the moment he remembered his talk with Einstein in Berlin as the crucial juncture on his journey to the uncertainty principle. Yet as he travelled the road to discovery that ended in the depths of a winter’s night in Copenhagen, others had walked parts of the route with him. His most influential and valued companion was not Bohr, but Wolfgang Pauli.
As Schrödinger, Bohr and Heisenberg were locked in debate in Copenhagen in October 1926, Pauli was in Hamburg quietly analysing the collision of two electrons. He discovered, aided by Born’s probabilistic interpretation, what he described in a letter to Heisenberg as a ‘dark point’. Pauli had found that when electrons collide their respective momenta ‘must be taken as controlled’ and their positions ‘uncontrolled’.49 A probable change in momentum was accompanied by a simultaneous but indeterminable change in position. He had found that one could not ‘ask simultaneously’ about momentum (q) and position (p).50 ‘One can see the world with the p-eye and one can view it with the q-eye,’ Pauli stressed, ‘but if one opens both eyes together, then one goes astray.’51 Pauli took it no further, but his ‘dark point’ lurked in the back of Heisenberg’s mind as he and Bohr grappled with the problem of interpretation and wave-particle duality in the months before the discovery of the uncertainty principle.
On 23 February 1927, Heisenberg wrote a fourteen-page letter to Pauli summarising his work on the uncertainty principle. He relied on the critical judgement of the Viennese ‘Wrath of God’ more than most. ‘Day is dawning in quantum theory’, replied Pauli.52 Any lingering doubts vanished and, by 9 March, Heisenberg had turned the contents of his letter into a paper for publication. It was only then that he wrote to Bohr in Norway: ‘I believe that I have succeeded in treating the case where both [the momentum] p and [the position] q are given to a certain accuracy…I have written a draft of a paper about these problems which yesterday I sent Pauli.’53
Heisenberg chose not to send Bohr either a copy of the paper or the details of what he had done. It was a sign of how strained their relationship had become. ‘I wanted to get Pauli’s reactions before Bohr was back because I felt again that when Bohr comes back he will be angry about my interpretation’, he explained later.54 ‘So I first wanted to have some support, and see whether somebody else liked it.’ Five days after Heisenberg posted his letter, Bohr was back in Copenhagen.
Refreshed after his month-long vacation, Bohr dealt with pressing institute business before carefully reading the uncertainty paper. When they met to discuss it, he told a stunned Heisenberg that it was ‘not quite right’.55 Bohr not only disagreed with Heisenberg’s interpretation, but he had also spotted an error in the analysis of the gamma-ray microscope thought experiment. The workings of the microscope had nearly proved to be Heisenberg’s undoing as a student in Munich. Only the intervention of Sommerfeld had secured his doctorate. Afterwards, a contrite Heisenberg had read up on microscopes, but he was about to discover that he still had some more to learn.
Bohr told Heisenberg it was wrong to place the origin of the uncertainty in the momentum of the electron in the discontinuous recoil it suffers due to the collision with the gamma-ray photon. What prohibits the precise measurement of the momentum of the electron is not the discontinuous and uncontrollable nature of the momentum change, Bohr argued, but the impossibility of measuring that change exactly. The Compton effect, he explained, allows the change in momentum to be calculated with pinpoint accuracy as long as the angle by which the photon is scattered after the collision through the aperture of the microscope is known. However, it is impossible to fix the point where the photon enters the microscope. Bohr identified this as the source of the uncertainty in the momentum of the electron. The electron’s position when it collides with the photon is uncertain, since the finite aperture of any microscope limits its resolving power and therefore its ability to locate any microphysical object exactly. Heisenberg had failed to take all this into account, and there was worse to come.
Bohr maintained that a wave interpretation of the scattered light-quantum was indispensable for the correct analysis of the thought experiment. It was the wave-particle duality of radiation and matter that was at the heart of quantum uncertainty for Bohr as he linked Schrödinger’s wave packets with Heisenberg’s new principle. If the electron is viewed as a wave packet, then for it to have a precise, well-defined position requires it to be localised and not spread out. Such a wave packet is formed from the superposition of a group of waves. The more tightly localised or confined the wave packet is, the greater the variety of waves needed, the greater the range of frequencies and wavelengths involved. A single wave has a precise momentum, but it was an established fact that a group of superimposed waves of differing wavelengths cannot have a well-defined momentum. Equally, the more precisely defined the momentum of a wave packet, the fewer component waves it has and the more spread out it is, thereby increasing the uncertainty in its position. The simultaneously precise measurement of position and momentum is impossible, as Bohr showed that the uncertainty relations could be derived from the wave model of the electron.
Figure 12: (a) Position of the wave can be precisely determined but not the wavelength (and hence momentum); (b) wavelength can be measured accurately but not the position, since the wave is spread out
What troubled Bohr was that Heisenberg had adopted an approach based exclusively on particles and discontinuity. The wave interpretation, Bohr believed, could not be ignored. He regarded Heisenberg’s failure to accommodate wave-particle duality as a deep conceptual flaw. ‘I did not know exactly what to say to Bohr’s argument,
’ Heisenberg said later, ‘so the discussion ended with the general impression that now Bohr has again shown that my interpretation is not correct.’56 He was furious and Bohr upset at the reaction of his young protégé.
Living next to door to each other and with their offices on the ground floor of the institute separated only by a staircase, Bohr and Heisenberg did well to avoid one another for a few days before meeting again to discuss the uncertainty paper. Bohr hoped that, having had time to cool down, Heisenberg would see reason and rewrite it. He refused. ‘Bohr tried to explain that it was not right and I shouldn’t publish the paper’, Heisenberg said later.57 ‘I remember that it ended by my breaking out in tears because I just couldn’t stand this pressure from Bohr.’58 There was too much at stake for him to simply make the changes being demanded.
Heisenberg’s reputation as the wunderkind of physics rested on his discovery of matrix mechanics aged just 24. The growing popularity of Schrödinger’s wave mechanics threatened to overshadow, even undermine, that astonishing achievement. Before long he was complaining about the number of papers being written that simply reworked into the language of wave mechanics results first obtained using matrix methods. Although he too had employed the alternative to matrix mechanics as a handy set of mathematical tools with which to calculate the spectrum of helium, Heisenberg harboured hopes of slamming the door on Schrödinger’s wave mechanics and the Austrian’s claims at having restored continuity. With the discovery of the uncertainty principle, and his interpretation of it based on particles and discontinuity, Heisenberg thought he had closed the door and locked it. He wept tears of frustration as he tried to prevent Bohr from opening it again.
Heisenberg believed that his future was intimately bound to whether it was particles or waves, discontinuity or continuity that ruled in the atomic domain. He wanted to publish as quickly as possible and challenge Schrödinger’s claim that matrix mechanics was unanschaulich, unvisualisable, and therefore untenable. Schrödinger disliked discontinuity and a particle-based physics as much as Heisenberg loathed a physics of continuity and waves. Armed with the uncertainty principle and what he deemed to be the correct interpretation of quantum mechanics, Heisenberg went on the attack as he consigned his rival to a footnote in his paper: ‘Schrödinger describes quantum mechanics as a formal theory of frightening, indeed repulsive, abstractness and lack of visualizability. Certainly one cannot overestimate the value of the mathematical (and to that extent physical) mastery of the quantum-mechanical laws that Schrödinger’s theory has made possible. However, as regards questions of physical interpretation and principle, the popular view of wave mechanics, as I see it, has actually deflected us from exactly those roads which were pointed out by the papers of Einstein and de Broglie on the one hand and by the papers of Bohr and by quantum mechanics [i.e. matrix mechanics] on the other hand.’59
On 22 March 1927, Heisenberg posted his paper, ‘On the perceptual content of quantum theoretical kinematics and mechanics’, to the Zeitschrift für Physik, the quantum theorist’s journal of choice.60 ‘I quarrel with Bohr’, he wrote to Pauli two weeks later.61 ‘By exaggerating one side or the other,’ protested Heisenberg, ‘one can discuss a lot without saying anything new.’ Believing that he had dealt with Schrödinger and his wave mechanics once and for all, Heisenberg now faced a far more tenacious opponent.
While Heisenberg was busy exploring the consequences of the uncertainty principle in Copenhagen, on the ski slopes in Norway, Bohr came up with complementarity. It was for him no mere theory or a principle, but the necessary conceptual framework hitherto missing for describing the strange nature of the quantum world. Complementarity, Bohr believed, could accommodate the paradoxical nature of wave-particle duality. The wave and particle properties of electrons and photons, matter and radiation, were mutually exclusive yet complementary aspects of the same phenomenon. Waves and particles were two sides of the same coin.
Complementarity neatly sidestepped the difficulties that arose from having to use two disparate classical descriptions, waves and particles, to describe a non-classical world. Both particles and waves were, according to Bohr, indispensable for a complete description of quantum reality. Either description by itself is only partially true. Photons paint one picture of light, waves another. Both hang side by side. But to avoid contradictions, there were limitations. The observer can look at only one of them at any given time. No experiment would ever reveal a particle and a wave at the same time. Bohr argued that ‘evidence obtained under different conditions cannot be comprehended within a single picture, but must be regarded as complementary in the sense that only the totality of the phenomena exhausts the possible information about the objects’.62
Bohr found support for his emerging ideas when he saw something in the uncertainty relations, pqh/2 and Eth/2, that Heisenberg, blinded by his intense dislike of waves and continuity, did not. The Planck-Einstein equation E=h and de Broglie’s formula p=h/ embodied wave-particle duality. Energy and momentum are properties commonly associated with particles, whereas frequency and wavelength are both characteristics of waves. Each equation contained one particle-like and one wave-like variable. The meaning of this combination of particle and wave characteristics in the same equation was something that niggled Bohr. After all, a particle and a wave are two wholly distinct physical entities.
As he corrected Heisenberg’s analysis of the microscope thought experiment, Bohr spotted that the same was true for the uncertainty relations. It was a finding that led him to interpret the uncertainty principle as revealing the extent to which two complementary but mutually exclusive classical concepts, either particles and waves or momentum and position, could be applied simultaneously without contradiction in the quantum world.63
The uncertainty relations also implied that a choice has to be made between what Bohr called a ‘causal’ description based on the conservation laws of energy and momentum (E and p in the uncertainty relations), and a ‘space-time’ description in which events are followed in space and time (q and t). The two descriptions were mutually exclusive but complementary so as to account for the results of all possible experiments. To Heisenberg’s dismay, Bohr had reduced the uncertainty principle to a special rule exposing the limits inherent in nature on any simultaneous measurements of complementary pairs of observables such as position and momentum or on the simultaneous use of two complementary descriptions.
There was another difference of opinion. Whereas the uncertainty principle led Heisenberg to question the extent to which classical concepts such as ‘particle’, ‘wave’, ‘position’, ‘momentum’ and ‘trajectory’ were applicable in the atomic realm, Bohr argued that the ‘interpretation of the experimental material rests essentially upon the classical concepts’.64 While Heisenberg insisted upon an operational definition of these concepts, a sort of meaning through measurement, Bohr argued that their meanings were already fixed by how they were used in classical physics. ‘Every description of natural processes,’ he had written in 1923, ‘must be based on ideas which have been introduced and defined by the classical theory.’65 Regardless of any limitations imposed by the uncertainty principle, they could not be replaced for the simple reason that all experimental data, its discussion and interpretation, by which theories are put to the test in the laboratory, is of necessity expressed in the language and concepts of classical physics.
Heisenberg suggested that since classical physics was found wanting at the atomic level, why should these concepts be retained? ‘Why should we not simply say that we cannot use these concepts with a very high precision, therefore the uncertainty relations, and therefore we have to abandon these concepts to a certain extent’, he argued in the spring of 1927.66 When it comes to the quantum, ‘we must realize that our words don’t fit’. If words fail, then the only sensible option for Heisenberg was to retreat into the formalism of quantum mechanics. After all, he maintained, ‘a new mathematical scheme is just as good as anything because the new mathematical sc
heme then tells what may be there and what may not be there’.67
Bohr was unconvinced. The gathering of every piece of information about the quantum world, he pointed out, involves performing an experiment the results of which are recorded as fleeting flashes of light on a screen, or as clicks of a Geiger counter, or registered by the movement of needles on voltmeters and the like. Such instruments belong to the everyday world of the physics laboratory, but they are the only means by which an event at the quantum level can be magnified, measured, and recorded. It is the interaction between a piece of laboratory equipment and a microphysical object, an alpha particle or an electron, which triggers the click of a Geiger counter or causes the needle of a voltmeter to move.
Any such interaction involves the exchange of at least one quantum of energy. The consequence of this, Bohr said, is the ‘impossibility of any sharp distinction between the behaviour of atomic objects and the interactions with the measuring instruments which serve to define the conditions under which the phenomena appear’.68 In other words, it was no longer possible to make the separation that existed in classical physics between the observer and the observed, between the equipment used to make a measurement and what was being measured.
Bohr was adamant that it was the specific experiment being performed that revealed either the particle or wave aspects of an electron or a beam of light, of matter or radiation. Since particle and wave were complementary but mutually exclusive facets of one underlying phenomenon, in no actual or imaginary experiment could both be revealed. When equipment was set up to investigate the interference of light, as in Young’s famous two-slits experiment, it was the wave nature of light that was manifest. If it was an experiment to study the photoelectric effect by shining a beam of light onto a metal surface, then it was light as a particle that would be observed. To ask whether light is either a wave or a particle is meaningless. In quantum mechanics, said Bohr, there is no way of knowing what light ‘really is’. The only question worth asking is: Does the light ‘behave’ like a particle or a wave? The answer is that sometimes it behaves like a particle and at others like a wave, depending upon the choice of experiment.