Quantum
Page 27
‘Heisenberg is now here and we are all very much occupied with discussions about the new development of the quantum theory and the great prospects it holds out’, Bohr wrote to Rutherford in the middle of May 1926.16 Heisenberg lived at the institute in a ‘cosy little attic flat with slanting walls’ and a view of Faelled Park.17 Bohr and his family had moved into the plush and spacious director’s villa next door. Heisenberg was such a regular visitor that he soon felt ‘half at home with the Bohrs’.18 The enlargement and renovation of the institute had taken far longer than expected and Bohr was exhausted. Sapped of energy, he suffered a severe case of flu. As Bohr spent the next two months recovering, Heisenberg successfully used wave mechanics to account for the line spectrum of helium.
Once Bohr was back to his old self, living next door to him was something of a mixed blessing. ‘After 8 or 9 o’clock in the evening Bohr, all of a sudden, would come up to my room and say, “Heisenberg, what do you think about this problem?” And then we would start talking and talking and quite frequently we went on till twelve or one o’clock at night.’19 Or he would invite Heisenberg over to the villa for a chat that lasted long into the evening, fuelled by glasses of wine.
As well as working with Bohr, Heisenberg gave two lectures a week on theoretical physics at the university in Danish. He was not much older than his students, and one of them could barely believe ‘he was so clever since he looked like a bright carpenter’s apprentice just returned from technical school’.20 Heisenberg quickly adapted to the rhythm of life at the institute and with his new colleagues enjoyed sailing, horse riding, and walking tours at the weekends. But there was less and less time for such activities after Schrödinger’s visit at the beginning of October 1926.
Schrödinger and Bohr had failed to reach any sort of accord over the physical interpretation of either matrix or wave mechanics. Heisenberg saw how ‘terribly anxious’ Bohr was ‘to get to the bottom of things’.21 In the months that followed, the interpretation of quantum mechanics was all that Bohr and his young apprentice talked about as they tried to reconcile theory and experiment. ‘Bohr often came up to my room late at night to talk to me of the difficulties in quantum theory which tortured both of us’, Heisenberg said later.22 Nothing caused them more pain than wave-particle duality. As Einstein told Ehrenfest: ‘On the one hand waves, on the other quanta! The reality of both is firm as a rock. But the devil makes a verse out of this (which really rhymes).’23
In classical physics something can be either a particle or a wave; it cannot be both. Heisenberg had used particles and Schrödinger waves as they discovered their respective versions of quantum mechanics. Even the demonstration that both matrix and wave mechanics were mathematically equivalent had not yielded any deeper understanding of wave-particle duality. The crux of the whole problem, Heisenberg said, was that no one could answer the questions: ‘Is an electron now a wave or is it a particle, and how does it behave if I do this and that and so on?’24 The harder Bohr and Heisenberg thought about wave-particle duality, the worse things seemed to become. ‘Like a chemist who tries to concentrate his poison more and more from some kind of solution,’ remembered Heisenberg, ‘we tried to concentrate the poison of the paradox.’25 As they did so there was an increasing tension between the two men, as each adopted a different approach in an attempt to resolve the difficulties.
In the search for a physical interpretation of quantum mechanics, what the theory revealed about the nature of reality at the atomic level, Heisenberg was totally committed to particles, quantum jumps, and discontinuity. For him the particle aspect was dominant in wave-particle duality. He was not prepared to make room to accommodate anything remotely linked to Schrödinger’s interpretation. To Heisenberg’s horror, Bohr wanted to ‘play with both schemes’.26 Unlike the young German, he was not wedded to matrix mechanics and had never been enthralled by any mathematical formalism. While Heisenberg’s first port of call was always the mathematics, Bohr weighed anchor and sought to understand the physics behind the mathematics. In probing quantum concepts such as wave-particle duality, he was more interested in grasping the physical content of an idea rather than the mathematics it came wrapped in. Bohr believed that a way had to be found to allow for the simultaneous existence of both particles and waves in any complete description of atomic processes. Reconciling these two contradictory concepts was for him the key that would open the door leading to a coherent physical interpretation of quantum mechanics.
Ever since Schrödinger’s discovery of wave mechanics it was understood that there was one quantum theory too many. What was needed was a single formulation, especially given that the two were mathematically the same. It was Paul Dirac and Pascual Jordan, independently of each other, who came up with just such a formalism that autumn. Dirac, who had arrived in Copenhagen in September 1926 for a six-month stay, showed that matrix and wave mechanics were just special cases of an even more abstract formulation of quantum mechanics called transformation theory. All that was missing was a physical interpretation of the theory, and the search for it was beginning to take its toll.
‘Since our talks often continued till long after midnight and did not produce a satisfactory conclusion despite protracted efforts over several months,’ recalled Heisenberg, ‘both of us became utterly exhausted and rather tense.’27 Bohr decided that enough was enough and went on a four-week skiing holiday in Guldbrandsdalen, Norway in February 1927. Heisenberg was glad to see him go, so that he ‘could think about these hopelessly complicated problems undisturbed’.28 None was more pressing than the trajectory of an electron in a cloud chamber.
When Bohr met Rutherford at the research students’ Christmas party in Cambridge in 1911, he was struck by the New Zealander’s generous praise for the recent invention of the cloud chamber by C.T.R. Wilson. The Scotsman had managed to create clouds in a small glass chamber that contained air saturated with water vapour. Cooling the air by allowing it to expand caused the vapour to condense into minuscule water droplets on particles of dust, producing a cloud. Before long, Wilson was able to create a ‘cloud’ even after removing all traces of dust from the chamber. The only explanation he could offer was that the cloud was formed by condensation on ions present in the air within the chamber. However, there was another possibility. Radiation passing through the chamber could rip electrons from atoms in the air, forming ions, thereby leaving a trail of tiny water droplets in its wake. It was soon discovered that radiation did exactly that. Wilson appeared to have given physicists a tool for observing the trajectories of alpha and beta particles emitted from radioactive substances.
Particles followed well-defined paths, while waves, because they spread out, did not. However, quantum mechanics did not allow for the existence of the particle trajectories that were clearly visible for all to see in a cloud chamber. The problem seemed insurmountable. But it ought to be possible, Heisenberg was convinced, to establish a connection between what was observed in the cloud chamber and quantum theory, ‘hard though it appeared to be’.29
Working late one evening in his small attic flat at the institute, Heisenberg’s mind began to wander as he pondered the riddle of electron tracks in a cloud chamber where matrix mechanics said there should be none. All of a sudden he heard the echo of Einstein’s rebuke that ‘it is the theory that decides what we can observe’.30 Convinced that he was on to something, Heisenberg needed to clear his head. Although it was well past midnight, he went for a walk in the neighbouring park.
Barely feeling the chill, he began to focus on the precise nature of the electron track left behind in a cloud chamber. ‘We had always said so glibly that the path of the electron in the cloud chamber could be observed’, he wrote later.31 ‘But perhaps what we really observed was something much less. Perhaps we merely saw a series of discrete and ill-defined spots through which the electron had passed. In fact, all we do see in the cloud chamber are individual water droplets which must certainly be much larger than the electron.’32 There was no
continuous, unbroken path, Heisenberg believed. He and Bohr had been asking the wrong questions. The one to answer was: ‘Can quantum mechanics represent the fact that an electron finds itself approximately in a given place and that it moves approximately with a given velocity?’
Hurrying back to his desk, Heisenberg began manipulating the equations he knew so well. quantum mechanics apparently placed restrictions on what could be known and observed. But how did the theory decide what can and cannot be observed? The answer was the uncertainty principle.
Heisenberg had discovered that quantum mechanics forbids, at any given moment, the precise determination of both the position and the momentum of a particle. It is possible to measure exactly either where an electron is or how fast it is moving, but not both simultaneously. It was nature’s price for knowing one of the two exactly. In a quantum dance of give-and-take, the more accurately one is measured the less accurately the other can be known or predicted. If he was right, then Heisenberg knew that it meant no experiment probing the atomic realm would ever succeed in overcoming the limits imposed by the uncertainty principle. It was, of course, impossible to ‘prove’ such a claim, but Heisenberg was certain it must be so, given that all processes involved in any such experiment ‘had necessarily to satisfy the laws of quantum mechanics’.33
In the days that followed he tested the uncertainty principle, or as he preferred to call it, the indeterminacy principle. In the laboratory of the mind, he conducted one imaginary ‘thought experiment’ after another in which it might be possible to measure position and momentum simultaneously with an accuracy that the uncertainty principle said was impossible. As calculation after calculation revealed that the uncertainty principle had not been violated, one particular thought experiment convinced Heisenberg that he had successfully demonstrated that ‘It is the theory which decides what we can and cannot observe’.
Heisenberg had once discussed with a friend the difficulties surrounding the concept of electron orbits. His friend had argued that it should, in principle, be possible to construct a microscope that allowed electron paths inside the atom to be observed. However, such an experiment was now ruled out because, according to Heisenberg, ‘not even the best microscope could cross the limits set by the uncertainty principle’.34 All he had to do was prove it theoretically by trying to determine the exact position of a moving electron.
To ‘see’ an electron required a special kind of microscope. Ordinary microscopes use visible light to illuminate an object and then focus the reflected light into an image. The wavelengths of visible light are much larger than an electron and therefore could not be used to determine its exact position as they washed over it like waves over a pebble. What was required was a microscope that used gamma rays, ‘light’ of extremely short wavelength and high frequency, to pinpoint its position. Arthur Compton, in 1923, had investigated X-rays striking electrons and found conclusive evidence for the existence of Einstein’s light-quanta. Heisenberg imagined that, like two billiard balls colliding, when a gamma ray photon hits the electron, it is scattered into the microscope as the electron recoils.
There is, however, a discontinuous shove rather than a smooth transition in the electron’s momentum due to the impact of the gamma ray photon. Since the momentum that an object possesses is its mass multiplied by its velocity, any change in its velocity causes a corresponding change in its momentum.35 When the photon hits the electron it jolts its velocity. The only way to minimise the discontinuous change in the electron’s momentum is by reducing the energy of the photon, thereby lessening the impact of the collision. To do so entails using light of a longer wavelength and lower frequency. However, such a switch in wavelength means that it is no longer possible to pin down the exact position of the electron. The more precisely the electron’s position is measured, the more uncertain or imprecise any measurement of its momentum and vice versa.36
Heisenberg showed that if p and q (where is the Greek letter delta) are the ‘imprecision’ or ‘uncertainty’ with which the momentum and the position are known, then p multiplied by q is always greater than or equal to h/2: pqh/2, where h is Planck’s constant.37 This was the mathematical form of the uncertainty principle or the ‘imprecision in knowledge of simultaneous measurements’ of position and momentum. Heisenberg also discovered another ‘uncertainty relation’ involving a different pair of so-called conjugate variables, energy and time. If E and t are the uncertainties with which the energy E of a system can be determined and the time t at which E is observed, then Eth/2.
At first there were some who thought that the uncertainty principle was the result of the technological shortcomings of the equipment used in an experiment. If the equipment could be improved, they believed, then the uncertainty would disappear. This misunderstanding arose because of Heisenberg’s use of thought experiments to draw out the significance of the uncertainty principle. However, thought experiments are imaginary experiments employing perfect equipment under ideal conditions. The uncertainty discovered by Heisenberg is an intrinsic feature of reality. There could be no improvement, he argued, on the limits set by the size of Planck’s constant and enforced by the uncertainty relations on the precision of what is observable in the atomic world. Rather than ‘uncertain’ or ‘indeterminate’, ‘unknowable’ may have been a more apt description of his remarkable discovery.
Heisenberg believed it was the act of measuring the position of the electron that made the precise determination of its momentum at the same time impossible. The reason appeared, as far as he was concerned, to be straightforward. The electron is disturbed unpredictably when struck by the photon used to ‘see it’ in order to locate its position. It was this unavoidable disturbance during the act of measurement that Heisenberg identified as the origin of uncertainty.38
It was an explanation that he believed was supported by the fundamental equation of quantum mechanics: pq–qp=–ih/2, where p and q are the momentum and position of a particle. It was the inherent uncertainty of nature that lay behind non-commutativity – the fact that p×q does not equal q×p. If an experiment to locate an electron were followed by one measuring its velocity (and therefore its momentum) they would give two precise values. Multiplying the two values together yields an answer A. However, repeating the experiments in reverse order, measuring the velocity first and then the position, would lead to a completely different result, B. In each case the first measurement caused a disturbance that affected the outcome of the second. If there had been no disturbance, which was different in each experiment, then p×q would be the same as q×p. As pq–qp would then equal zero, there would be no uncertainty and no quantum world.
Heisenberg was delighted as he saw the pieces fit neatly together. His version of quantum mechanics was built out of matrices representing observables such as position and momentum that do not commute. Ever since he discovered the strange rule that made the order in which two arrays of numbers were multiplied an essential component of the mathematical scheme of his new mechanics, the physical reason why this was so had been shrouded in mystery. Now he had lifted the veil. It was, according to Heisenberg, ‘only the uncertainty specified by pqh/2’, that ‘creates room for the validity of the relations’ in pq–qp=–ih/2.39 It was uncertainty, he claimed, that ‘makes possible this equation without requiring that the physical meaning of the quantities p and q be changed’.40
The uncertainty principle had exposed a deep fundamental difference between quantum and classical mechanics. In classical physics both the position and momentum of an object can in principle be simultaneously determined to any degree of accuracy. If the position and velocity were known precisely at any given moment, then the path of an object, past, present and future, could also be exactly mapped out. These long-established concepts of everyday physics ‘can also be defined exactly for the atomic processes’, said Heisenberg.41 However, the limitations of these concepts are laid bare when attempts are made to measure simultaneously a pair of conjugate variables: position a
nd momentum or energy and time.
For Heisenberg the uncertainty principle was the bridge between the observation of what appeared to be electron tracks in a cloud chamber and quantum mechanics. As he built that bridge between theory and experiment, he assumed that ‘only such experimental situations can arise in nature as can be expressed in the mathematical formalism’ of quantum mechanics.42 He was convinced that if quantum mechanics said it could not happen, then it did not. ‘The physical interpretation of quantum mechanics is still full of internal discrepancies,’ Heisenberg wrote in his uncertainty paper, ‘which show themselves in arguments about continuity versus discontinuity and particle versus wave.’43
It was a sorry state of affairs that arose because concepts that had been the foundation of classical physics ever since Newton ‘fit nature only inaccurately’ at the atomic level.44 He believed that with a more precise analysis of concepts such as position, momentum, velocity, and the path of an electron or atom it might be possible to eliminate ‘the contradictions evident up to now in the physical interpretations of quantum mechanics’.45
What is meant by ‘position’ in the quantum realm? Nothing more or less, Heisenberg answered, than the result of a specific experiment designed to measure, say, the ‘position of the electron’ in space at a given moment, ‘otherwise this word has no meaning’.46 For him there simply is no electron with a well-defined position or a well-defined momentum in the absence of an experiment to measure its position or momentum. A measurement of an electron’s position creates an electron-with-a-position, while a measurement of its momentum creates an electron-with-a-momentum. The very idea of an electron with a definite ‘position’ or ‘momentum’ is meaningless prior to an experiment that measures it. Heisenberg had adopted an approach to defining concepts through their measurement that harked back to Ernst Mach and what philosophers called operationalism. But it was more than just a redefinition of old concepts.