Quantum
Page 31
Bohr, Heisenberg, Pauli and Born were not entirely sure what Einstein was driving at. He had not clearly stated his aim: to show that quantum mechanics was inconsistent and therefore an incomplete theory. Sure, the wave function collapses instantaneously, they thought, but it was an abstract wave of probability, not a real wave travelling in ordinary three-dimensional space. Nor was it possible to choose between the two viewpoints Einstein outlined on the basis of observing what happens to an individual electron. In both cases an electron passes through the slit and hits the plate at some point.
‘I feel myself in a very difficult position because I don’t understand what precisely is the point which Einstein wants to [make]’, said Bohr.42 ‘No doubt it is my fault.’ Remarkably, he then said: ‘I do not know what quantum mechanics is. I think we are dealing with some mathematical methods which are adequate for [a] description of our experiments.’43 Instead of responding to Einstein’s analysis, Bohr simply went on to restate his own views. But in this game of quantum chess, the Danish grandmaster later recounted in a paper, written in 1949 to celebrate his opponent’s 70th birthday, the reply he gave that evening and on the last day of the conference in 1927.44
According to Bohr, Einstein’s analysis of his thought experiment tacitly assumed that the screen and photographic plate both had a well-defined position in space and time. However, maintained Bohr, this implied that both had an infinite mass, for only then would there be no uncertainty in either position or time as the electron emerged from the slit. As a result, the exact momentum and energy of the electron is unknown. This was the only possible scenario, argued Bohr, given that the uncertainty principle implies that the more precisely the electron’s position is known, the more inexact any concurrent measurement of its momentum must be. The infinitely heavy screen in Einstein’s imaginary experiment left no room for uncertainty in the space and time location of the electron at the slit. However, such precision came at a price: its momentum and energy were completely indeterminate.
It was more realistic, Bohr suggested, to assume that the screen did not have an infinite mass. Although still much heavier, the screen would now move when the electron passed through the slit. While any such movement would be so small as to be impossible to detect in the laboratory, its measurement presented no problem in the abstract world of the idealised thought experiment furnished, as it was, with measuring devices capable of perfect accuracy. Because the screen moves, the position of the electron in space and time is uncertain during the process of diffraction, resulting in a corresponding uncertainty in both its momentum and energy. However, compared to the case of an infinitely massive screen, it would lead to an improved prediction of where the diffracted electron will hit the photographic plate. Within the limits imposed by the uncertainty principle, argued Bohr, quantum mechanics was as complete a description of individual events as was possible.
Unimpressed by Bohr’s reply, Einstein asked him to consider the possibility of controlling and measuring the transfer of momentum and energy between the screen and the particle, be it an electron or a photon, as it passed through the slit. Then, he argued, the state of the particle immediately afterwards could be determined with an accuracy greater than that allowed by the uncertainty principle. As the particle passes through the slit, said Einstein, it would be deflected and its trajectory towards the photographic plate would be determined by the law of conservation of momentum, which requires the sum total of the momenta of two bodies (particle and screen) that interact to remain constant. If the particle is deflected upwards, then the screen must be pushed downwards and vice versa.
Having used the moveable screen introduced by Bohr for his own ends, Einstein modified the imaginary experiment further by inserting a two-slit screen between the moveable screen and the photographic plate.
Figure 15: Einstein’s two-slits thought experiment. At far right, the resulting interference pattern on the screen is shown
Einstein reduced the intensity of a beam until only one particle at a time passed through the slit in the first screen, S1, and one of the two slits of the second screen, S2, before hitting the photographic plate. As each particle left an indelible mark where it hit the plate, something remarkable would happen. What initially appeared to be a random sprinkling of specks was slowly transformed, as more and more particles left their imprint, by the laws of statistics into the characteristic interference pattern of light and dark bands. While each particle was responsible for only a single mark, it nevertheless contributed decisively through some statistical imperative to the overall interference pattern.
By controlling and measuring the transfer of momentum between the particle and the first screen it was possible, said Einstein, to determine if the particle was deflected towards the upper or lower slit in the second screen. From where it hit the photographic plate and the movement of the first screen, it was possible to trace through which of the two slits the particle had passed. It appeared that Einstein had devised an experiment in which it was possible to simultaneously determine the position and momentum of a particle with a greater precision than the uncertainty principle allowed. In the process he also seemed to have contradicted another fundamental tenet of the Copenhagen interpretation. Bohr’s framework of complementarity posited that either particle-like or wave-like properties of an electron or a photon could be manifest in any given experiment.
There had to be a flaw in Einstein’s argument, and Bohr set out to find it by sketching the sort of equipment needed to conduct the experiment. The apparatus he focused on was the first screen. Bohr realised that the control and measurement of the transfer of momentum between the particle and screen hinged on the screen’s ability to move vertically. It is the observation of the screen moving either up or down as the particle passes through the slit that allows the determination of whether it passes through either the upper or lower slit in the second screen, after it strikes the photographic plate.
Einstein, despite his years at the Swiss Patent Office, had not considered the details of the experiment. Bohr knew that the quantum devil lay in the details. He replaced the first screen with one hanging by a pair of springs fixed to a supporting frame so that its vertical motion due to the transfer of momentum from a particle passing through the slit could be measured. The measuring device was simple: a pointer attached to the supporting frame and a scale engraved on the screen itself. It was crude, but sensitive enough to allow the observation of any individual interaction between screen and particle in an imaginary experiment.
Figure 16: Bohr’s design of a moveable first screen
Bohr argued that if the screen was already moving with an unknown velocity greater than any due to an interaction with a particle as it passed through the slit, then it would be impossible to ascertain the degree of momentum transfer and with it the trajectory of the particle. On the other hand, if it was possible to control and measure the transfer of momentum from particle to screen, the uncertainty principle implied a simultaneous uncertainty in the position of the screen and slit. However precise the measurement of the screen’s vertical momentum, it was strictly matched, in accordance with the uncertainty principle, by a corresponding imprecision in the measurement of its vertical position.
Bohr went on to argue that the uncertainty in the position of the first screen destroys the interference pattern. For example, D on the photographic plate is a point of destructive interference, a dark spot in the interference pattern. A vertical displacement of the first screen would result in a change in the length of the two paths ABD and ACD. If the new lengths differed by half a wavelength, then instead of destructive interference there would be constructive interference and a bright spot at D.
To accommodate uncertainty in the vertical displacement of the first screen, S1, requires an ‘averaging’ over all of its possible positions. This leads to interference somewhere between the extremes of total constructive and total destructive interference, resulting in a washed-out pattern on the photographic pla
te. Controlling the transfer of momentum from the particle to the first screen allows the trajectory of the particle through a slit in the second screen to be tracked; however, it destroys the interference pattern, argued Bohr. He concluded that Einstein’s ‘suggested control of momentum transfer would involve a latitude in the knowledge of the position of the diaphragm [S1] which would exclude the appearance of the interference phenomena in question’.45 Bohr had not only defended the uncertainty principle but also the belief that the wave and particle aspects of a microphysical object cannot both appear in a single experiment, imaginary or not.
Bohr’s rebuttal rested on the assumption that controlling and measuring the momentum transferred to S1 accurately enough to determine the particle’s direction afterwards results in an uncertainty in the position of S1. The reason for this, Bohr explained, lay in reading the scale on S1. To do so, it has to be illuminated, and that requires the scattering of photons from the screen and results in an uncontrollable transfer of momentum. This impedes the precise measurement of the momentum transferred from the particle to the screen as it passes through the slit. The only way to eliminate the impact of the photon is by not illuminating the scale at all, making it impossible to read. Bohr had resorted to employing the same concept of ‘disturbance’ that he had earlier criticised Heisenberg for using as an explanation of the origin of uncertainty in the microscope thought experiment.
There was another curious phenomenon associated with the two-slit experiment. If one of the two slits has a shutter that is closed, then the interference pattern disappears. Interference occurs only when both slits are open at the same time. But how was that possible? A particle can go through only one slit. How did the particle ‘know’ that the other slit was open or closed?
Figure 17: Two-slit experiment (a) with both slits open; (b) with one slit closed
Bohr had a ready answer. There was no such thing as a particle with a well-defined path. It was this lack of a definite trajectory that was behind the appearance of an interference pattern, even though it was particles, one at a time, which had passed through the two-slit set-up, and not waves. This quantum fuzziness enables a particle to ‘sample’ a variety of possible paths and so it ‘knows’ if one of the slits is open or closed. Whether it is open or not affects the particle’s future path.
If detectors are placed in front of the two slits to sneak a look at which slit a particle is going to pass through, then it seems possible to close the other slit without affecting the particle’s trajectory. When such a ‘delayed-choice’ experiment was later actually conducted, instead of an interference pattern there was an enlarged image of the slit. In trying to measure the position of the particle to establish through which slit it would pass, it is disturbed from its original course and the interference pattern fails to materialise.
The physicist has to choose, says Bohr, between ‘either tracing the path of a particle or observing interference effects’.46 If one of the two slits of S2 is closed, then the physicist knows through which slit the particle passed before hitting the photographic plate, but there will be no interference pattern. Bohr argues that this choice allows an ‘escape from the paradoxical necessity of concluding that the behaviour of an electron or a photon should depend on the presence of a slit in the diaphragm [S2] through which it could be proved not to pass’.47
The two-slit experiment was for Bohr ‘a typical example’ of the appearance of complementary phenomena under mutually exclusive experimental conditions.48 Given the quantum mechanical nature of reality, he argued, light was neither a particle nor a wave. It was both, and sometimes it behaved like a particle and sometimes like a wave. On any given occasion, nature’s answer to whether it was a particle or a wave simply depended on the question asked – on the type of experiment performed. An experiment to determine through which slit in S2 a photon passed was a question that solicited a ‘particle’ answer and therefore no interference pattern. It was the loss of an independent, objective reality and not probability, God playing dice, that Einstein found unacceptable. quantum mechanics, therefore, could not be the fundamental theory of nature that Bohr claimed it to be.
‘Einstein’s concern and criticism provided a most valuable incentive for us all to re-examine the various aspects of the situation as regards the description of atomic phenomena’, recalled Bohr.49 A major point of contention, he stressed, was ‘the distinction between the objects under investigation and the measuring instruments which serve to define, in classical terms, the conditions under which the phenomena appear’.50 In the Copenhagen interpretation the measuring instruments were inextricably linked with the object under investigation: no separation is possible.
While a microphysical object such as an electron was subject to the laws of quantum mechanics, the apparatus obeyed the laws of classical physics. Yet Bohr had to retreat in the face of Einstein’s challenge as he applied the uncertainty principle to a macroscopic object, the first screen S1. By doing so, Bohr had imperiously consigned an element of the large-scale world of the everyday to the realm of the quantum as he failed to establish where is ‘the cut’ between the classical and the quantum worlds, the border between the macro and micro. It would not be the last time that Bohr played a questionable move in his game of quantum chess with Einstein. The spoils for the victor were just too high.
Einstein spoke only once more during the general discussion, when he asked a question. De Broglie recalled later that ‘Einstein said hardly anything beyond presenting a very simple objection to the probability interpretation’ and then ‘he fell back into silence’.51 However, with all the participants staying at the Hotel Metropole, it was in its elegant art deco dining room that the keenest arguments took place, not in the conference room at the Institute of Physiology. ‘Bohr and Einstein,’ said Heisenberg, ‘were in the thick of it all.’52
Surprisingly for an aristocrat, de Broglie spoke only French. He must have seen Einstein and Bohr deep in conversation in the dining room, with the likes of Heisenberg and Pauli listening closely. As they spoke in German, de Broglie did not realise that they were engaged in what Heisenberg called a ‘duel’.53 The acknowledged master of the thought experiment, Einstein would arrive at breakfast armed with a new proposal that challenged the uncertainty principle and with it the much-lauded consistency of the Copenhagen interpretation.
The analysis would begin over coffee and croissants. It continued as Einstein and Bohr headed to the Institute of Physiology, usually with Heisenberg, Pauli and Ehrenfest trailing alongside. As they walked and talked, assumptions were probed and clarified before the start of the morning session. ‘During the meeting and particularly in the pauses we younger people, mostly Pauli and I, tried to analyse Einstein’s experiment,’ Heisenberg said later, ‘and at lunch time the discussions continued between Bohr and the others from Copenhagen.’54 Late in the afternoon, following further consultations among themselves, the collaborative effort would yield a rebuttal. During dinner back at the Metropole, Bohr would explain to Einstein why his latest thought experiment had failed to break the limits imposed by the uncertainty principle. Each time Einstein could find no fault with the Copenhagen response, but they knew, said Heisenberg, ‘in his heart he was not convinced’.55
After several days, Heisenberg later recalled, ‘Bohr, Pauli and I – knew that we could now be sure of our ground, and Einstein understood that the new interpretation of quantum mechanics cannot be refuted so simply’.56 But Einstein refused to yield. Even if it failed to capture the essence of his rejection of the Copenhagen interpretation, he would say, ‘God does not play dice’. ‘But still, it cannot be for us to tell God, how he is to run the world’, replied Bohr on one occasion.57 ‘Einstein, I am ashamed of you,’ said Paul Ehrenfest only half-joking, ‘you are arguing against the new quantum theory just as your opponents argue about relativity theory.’58
The only impartial witness to the private encounters between Einstein and Bohr at Solvay 1927 was Ehrenfest. ‘Einstein’
s attitude gave rise to ardent discussions within a small circle, in which Ehrenfest, who through the years had been a close friend of us both,’ recalled Bohr, ‘took part in a most active and helpful way.’59 A few days after the conference ended, Ehrenfest wrote a letter to his students at Leiden University vividly describing the goings-on in Brussels: ‘Bohr towering completely over everybody. At first not understood at all (Born was also there), then step by step defeating everybody. Naturally once again the awful Bohr incantation terminology. (Poor Lorentz as interpreter between the British and the French who were absolutely unable to understand each other. Summarizing Bohr. And Bohr responding with polite despair.) Every night at 1 a.m. Bohr came into my room just to say ONE SINGLE WORD to me, until 3 a.m. It was delightful for me to be present during the conversations between Bohr and Einstein. Like a game of chess. Einstein all the time with new examples…. to break the UNCERTAINTY RELATION. Bohr from out of the philosophical smoke clouds constantly searching for the tools to crush one example after the other. Einstein like a jack-in-the-box, jumping out fresh every morning. Oh, that was priceless. But I am almost without reservation pro Bohr and contra Einstein.’60 However, Ehrenfest admitted ‘that he would not be able to find relief in his own mind before concord with Einstein was reached’.61
At Solvay 1927 the discussions with Einstein were conducted, Bohr said later, in ‘a most humorous spirit’.62 Yet he noted wistfully, ‘a certain difference in attitude and outlook remained, since with his mastery for coordinating apparently contrasting experiences without abandoning continuity and causality, Einstein was perhaps more reluctant to renounce such ideals than someone for whom renunciation in this respect appeared to be the only way to proceed with the immediate task of coordinating the multifarious evidence regarding atomic phenomena, which accumulated from day to day in the exploration of this new field of knowledge.’63 It was Einstein’s very successes, implied Bohr, that kept him anchored in the past.