Heisenberg’s language was at least as sharp; he described wave mechanics as ‘disgusting’ and as ‘garbage.’ To the extent that wave mechanics was visualizable it was false, he claimed; physicists who use matrix mechanics are less deluded and thus see deeper into nature.32 As Born once said, ‘Mathematics knows better than our intuition.’33
The conflict was soon fought in face-to-face encounters. In July 1926, Schrödinger and Heisenberg met for the first time at a conference in Munich, where Schrödinger had many supporters. Schrödinger gave two talks about wave mechanics, and Heisenberg stood up at the end to object that no theory relying on continuous processes could possibly explain the discontinuities of quantum phenomena, such as Planck’s radiation law and the Compton effect. The audience appeared to be on Schrödinger’s side. Heisenberg seems to have influenced no one, and left feeling defeated. He went to Copenhagen, where he stayed for several months working with Bohr. The two disagreed – Bohr argued that we must use classical concepts to describe experiments, with Heisenberg disagreeing – but they honed their arguments why quantum discontinuities implied that space and time could not be defined, meaning that the quantum realm could neither be represented by theories involving continuous functions nor imagined by the human mind whose picturing ability requires a space-time container.
The next bout between Schrödinger and the matrix allies took place 3 months later, in October 1926. Bohr invited Schrödinger to visit Copenhagen, which was largely matrix territory (though the Copenhagen group had already begun to use some wave mechanics as a tool); Schrödinger, intellectually honest and seemingly riding the popular side, was happy to visit the opposition’s headquarters. But he was utterly unprepared for what followed. Bohr met Schrödinger at the train station, almost immediately began pressing his case, and continued arguing day and night for several days. Bohr had arranged for Schrödinger to stay at his house, so that every possible minute could be used. As Heisenberg recalled:
Bohr was an unusually considerate and obliging person, but in this kind of discussion, which concerned epistemological problems which he thought were of vital importance, he was capable of insisting – with a fanatic terrifying relentlessness – on complete clarity in all argument. Despite hours of struggle, he refused to give up until Schrödinger had admitted his interpretation was not enough, and could not even explain Planck’s law. Perhaps from the strain, Schrödinger got sick after a few days and had to stay in bed in Bohr’s home. Even here it was hard to push Bohr away from Schrödinger’s bedside: again and again, he would say, ‘But Schrödinger, you’ve got to at least admit that…’ Once Schrödinger exploded in a kind of desperation, ‘If you have to have these damn quantum jumps then I wish I’d never started working on atomic theory!’34
With Heisenberg at his side, Bohr persuaded Schrödinger into making a (temporary) retraction. But it did not last, and the originator of wave mechanics was soon back writing papers about it. In November 1926, indeed, he assembled his six seminal papers on wave mechanics – the four part series ‘Quantization as a Problem of Proper Values’ that appeared in the Annalen der Physik, plus his papers on the boundary problem and on the identity between wave and matrix mechanics – and had them published as a book.
By this time, Born had contributed his novel interpretation of wave mechanics. Trying to understand collisions between an electron and an atom, Born had carefully examined Schrödinger’s claim that the ψ-function referred to the electron’s charge density, found it did not make sense, and concluded that it does not tell us about the state of an event but rather about its probability. Pauli then wrote his letter to Heisenberg in which he proposed that ψ2 represented the probability, not of states, but of particles at particular positions. This amounted to a partial restoration of the space-time container and of visualizability. It did not entail that the orbits or paths of electrons from one place to another could be visualized, but that, however they got there, they did have positions.35 The classical properties do exist, and can be measured precisely. Still, it involved the bizarre notion that the strange function that Schrödinger said flowed through space was not a real thing but the probability that a real thing could be found at that spot. At the time, the philosophical novelty of this was not noticed. ‘We were so accustomed to making statistical considerations’, Born remarked later, that ‘to shift it one layer deeper seemed to us not so very important.’36
In the same letter of October 19 in which Pauli made his proposal about the interpretation of the wave function, he also noted implications for the vexing pq – qp issue. Heisenberg had been arguing that neither of the conjugate variables – the noncommuting terms – referred to classical variables such as positions or momenta that could be measured with precision together. Pauli was now saying that one of the pair could be – but if so, the other was only known as a probability. This made the noncommutativity even stranger. ‘The physics of this is unclear to me from top to bottom’, Pauli told Heisenberg. ‘My first question is: why can only the p’s, and not simultaneously both the p’s and the q’s, be described with any degree of precision?’ He was baffled. ‘You can look at the world with p-eyes or with q-eyes, but open both eyes together and you go wrong.’37 What could this mean?
Heisenberg’s response was delayed because he had a hard time retrieving Pauli’s letter from his excited Copenhagen colleagues who were sharing it. Heisenberg finally sent a reply on October 28. He still did not buy the implied restoration of visualizability and classical variables, dismissing Born’s ‘rather dogmatic’ view as ‘only one of several possible interpretations.’ The pq – qp = Ih/2πi relation, he continued to insist, meant that individual ps and qs were meaningless. ‘Above all, I hope there will eventually be a solution of the following type (but don’t spread this around): That time and space are really only statistical concepts, something like, for instance, temperature, pressure, and so on, in a gas. It’s my opinion that spatial and temporal concepts are meaningless when speaking of a single particle, and that the more particles there are, the more meaning these concepts acquire. I often try to push this further, but so far with no success.’
And a few weeks later, on November 15, Heisenberg presented to Pauli what seemed a conclusive argument why the discontinuities of the quantum world made the very concept of individual ps and qs meaningless.38 Let’s say an object such as an electron is at a specific point. Its velocity is defined in terms of the rate at which it moves continuously through points vanishingly close to it – but if space-time is discontinuous, and electrons flit from one state to another, it must lack velocity by definition! A week later, Heisenberg returned obsessively to the issue.39 Because the world is discontinuous, the ‘c-numbers’ (classical numbers) imply that we know way too much about what is happening. ‘What the word ‘wave’ or ‘corpuscle’ mean, one does not know any more.’
Pascual Jordan now stepped in to challenge Heisenberg. In effect, Jordan played contrarian to Heisenberg’s postulate-of-impotence claim that single electrons could not have positions and momenta. What was stopping experimenters from measuring them? Observing equipment is made of atoms, and atoms rattle about at room temperature due to their thermal motion, imposing a practical limit on accuracy. So what if we somehow set up the equipment to make a measurement at absolute zero where thermal motion stops; or, in what amounts to the same thing, what if we use highly energetic probes such as α particles, whose rattling is negligible and whose paths can be tracked?
Born and Pauli had considered the theoretical possibility of fixing one conjugate variable, and noted that the other could only be said to have a certain probability. Jordan was now pointing out experimental conditions in which physicists could indeed measure what was supposedly forbidden: the ‘probability of finding an electron in a certain place.’ It’s not unobservable theoretically, just difficult experimentally.
Jordan’s article troubled Heisenberg.40 The day after it appeared, on February 5, 1927, Heisenberg wrote to Pauli that he found Jor
dan’s paper ‘nice enough but not very exact in places’, because he still thought phrases like ‘probability of finding an electron in a certain place’ were conceptually meaningless. But if things such as the time and position of individual electrons made experimental sense, they had to make theoretical sense. If they made theoretical sense, his approach had to be wrong.
In all these discussions, there was never any question that the mathematics was correct. It was the interpretation that was at issue, and even the nature of interpretation. Bohr demanded more of interpretation than Heisenberg, and both demanded more than Schrödinger.
Heisenberg was still in Copenhagen, working at Bohr’s institute and living in the garret apartment of Bohr’s brother Harald. After supper, Bohr would come by, pipe in hand, and the two would argue about the state of quantum mechanics until the morning hours. The demanding conversation was beginning to wear on their relationship, and the two grew testy. Realizing this, Bohr left to go skiing. One evening while Bohr was gone, Heisenberg took a walk in Faelled Park, behind Bohr’s institute. He pondered ps and qs in theory and in experiment. He thought about Jordan’s microscope. He was as convinced as ever that something had to be wrong with Jordan’s example. Jordan had brought Heisenberg down to earth, derailing his single-minded focus on theoretical meaning, for it forced him to stop philosophizing about concepts and think operationally about what experimenters did. Suppose you looked at a particle at absolute zero; this would mean bouncing a photon off it and capturing the photon in the instrument’s lens. But that would disturb the electron’s position. If you wanted to avoid this, you would have to use a less energetic photon. But the longer the wavelength of the photon, the less precisely you knew its position! The problem might occur, Heisenberg excitedly realized, because of the interaction between the instrument and what you are seeking to measure – between the tools you were using to observe and the system observed.
Dawn of a New Era
Heisenberg then did what he often did when excited: he wrote a letter to Pauli. This one, dated February 23, was unusually long – fourteen pages. The shift in his thinking inaugurated by Jordan’s article is evident at the outset, for he describes several thought experiments involving measurements of ps and qs. Then he writes, ‘One will always find that all thought experiments have this property: When a quantity p is pinned down to within an accuracy characterized by the average error p, then…q can only be given at the same time to within an accuracy characterized by the average error q1 ≈ h/p1.’
This is the uncertainty principle. Like many other equations, its first appearance was not in the form in which it is now famous. Today it is usually written as an inequality:
The uncertainty principle was a conceptual breakthrough. While Born, Pauli, and Jordan had considered cases where one conjugate variable was exactly determined and the other a probability, Heisenberg now showed these are limiting cases, and in between is a spectrum of other cases where neither value is exact. A margin of uncertainty is unavoidable. If the uncertainty (Δx) in the position of, say, an electron is small, then the uncertainty in the momentum (Δp) must be large enough to keep the product, Δx × Δp, on the order of h. If the position of an electron is measured with such precision that the uncertainty is very small, the corresponding uncertainty in the momentum becomes very large. And Heisenberg told Pauli that this was a direct consequence of pq – qp = Ih/2πi, whose interpretation finally seemed clear. Heisenberg put particles back in a space-time stage, at least for the moment, but gave them decidedly unclassical properties.
Heisenberg quickly wrote a paper bearing his thoughts, ‘The Visualizable [anschaulich] Content of Quantum Kinematics and Mechanics.’ It set out to explain to classically trained physicists how quantum mechanics might be visualized in classical terms, and to do so redefines the word in the first sentence: ‘We believe we understand the visualizable (anschaulich) content of a theory when we can see its qualitative experimental consequences in all simple cases and when at the same time we have checked that the application of the theory never contains inner contradictions.’ This definition is too quick and convenient, designed so that Heisenberg eventually can make his theory fit it. But never mind; Heisenberg then says that it might seem difficult for quantum mechanics to fit this definition, for whenever pq – qp = Ih/2πi holds, it is unclear what we mean by things like position and velocity, and we need to clarify matters by specifying experimental conditions. So let’s say we observe an electron under a microscope that illuminates it with light. Because it is very small, we have to use energetic light: γ-rays. But if we use energetic light on tiny things, the Compton effect comes into play; the photon collides with our little electron, and abruptly and discontinuously shoves it away. Heisenberg wrote:
This change is the greater the smaller the wavelength of the light employed – that is, the more exact the determination of the position. At the instant at which the position of the electron is known, its momentum therefore can be known [only] up to magnitudes which correspond to that discontinuous change. Thus, the more precisely the position is determined, the less precisely the momentum is known, and conversely. In this circumstance we see a direct physical interpretation of the equation pq – qp = –ih.
Heisenberg was cavalier about his use of the unit matrix I in this equation, and it is frequently omitted in the literature as well. He continued by quantifying this interpretation:
Let q1 be the precision with which the value q is known (q1 is, say, the mean error of q), therefore here the wavelength of the light. Let p1 be the precision with which the value p is determinable; that is, here, the discontinuous change of p in the Compton effect. Then, according to the elementary laws of the Compton effect p1 and q1 stand in the relation
p1q1 ~ h
Now comes an odd thing whose significance has not been noted until recently, by John H. Marburger, III. Heisenberg proceeded to say that this equation is ‘a straightforward mathematical consequence of the rule equation pq – qp = –ih’, but he does not show it. There is no derivation of the uncertainty relation in Heisenberg’s paper! While it was accepted by Heisenberg and Bohr, and it was clearly a good conjecture, neither bothered to prove it, and the first proof of the principle to which Bohr refers is flawed.41
This ‘visualizable’ paper was less radical than the ‘reinterpretation’ paper of 2 years before. It did not argue that an electron lacked position or velocity, only that it had no simultaneous definite position and velocity, leaving the door open for one or the other to have a precise value. Heisenberg restored enough visualizability to claim that ‘quantum mechanics should no longer be considered as abstract and non-visualizable.’ In a kind of coup de grâce, he quoted Schrödinger’s remark about how ‘disgusting and frightening’ matrix mechanics is, to set up a retort that the real enemy is Schrödinger’s misconceived understanding of visualizability. The atomic world is visualizable, but what one could visualize was clearly not classical. A careful reading leaves one unsure whether Heisenberg was really committed to visualization at all. As Beller writes, ‘Heisenberg assumed the classical picture of the world in order to refute it.’42
After finishing the paper, Heisenberg wrote to Jordan that he felt ‘very, very happy’ that after a year of being continuously suspended, he now felt the ‘discontinuous ground under my feet.’43 And Pauli was thrilled. ‘He said something like, ‘Morgenröte einer Neuzeit’ ‘ – the dawn of a new era.44
But the new era got off to a rocky start. When Bohr returned and Heisenberg showed him the paper, Bohr spotted several blatant errors. Even in the atomic world, Bohr pointed out, energy and momentum are conserved, and if you disturb an electron by knocking it with a photon you can still figure out its momentum by catching the photon, eradicating the uncertainty. Yet, Bohr continued, Heisenberg’s idea was still correct, but because of the wave nature of particles. You cannot determine the momentum of recoiling particles precisely – not even if you use electrons instead of photons – because they all spread
out in a wavelike manner just as Schrödinger’s equation described, which is why you use a microscope lens to focus them. But this meant acknowledging that Schrödinger’s waves played an essential role in the theory. The conversation quickly deteriorated, and neither Bohr nor Heisenberg budged from his deeply entrenched position: Bohr said you needed waves, Heisenberg that you could do without them. Bohr told Heisenberg not to publish the paper, and the latter eventually burst into tears with frustration.45 But as Beller points out, these tears are as much due to Bohr’s ruthlessness as to Heisenberg’s stubbornness.
Heisenberg ignored Bohr’s advice and refused to withdraw or even fix the paper; he merely appended a brief note, entitled ‘Addition in Proof’, which stated that ‘Bohr has brought to my attention that I have overlooked essential points in the course of several discussions in this paper.’ But he did not fix the overlooked points.
For months, Bohr and Heisenberg continued to disagree about the interpretation of quantum mechanics. Both agreed that the mathematics was right and, as Einstein noted, had to guide the interpretation. But Bohr had a better idea of how to go about it. Heisenberg argued that you could use either matrix or wave language, Bohr that you needed both. Heisenberg’s position was essentially Platonist: he wanted to say that the mathematics alone describes what exists in the atomic realm. Bohr’s position was Kantian: nature forces human beings to experience and imagine according to certain (classical) categories and schemata structured by a space-time stage; as Marburger puts it, reality is a macroscopic phenomenon. These categories and schemata are adequate for macroscopic events, and appropriate for the classical physics which sought to provide the theory for such events. But these categories and schemata do not apply to microscopic events – to apply them and assume they are valid is to make what might be called the macroscopic fallacy. Still, we cannot get around these classical schemata in our thinking and imagining. Therefore, Bohr concluded, in our thinking about the microscopic world we are forced to depend on classical categories and schemata – such as position and momentum – but these categories are to be used in overlapping, nonclassical ways, as in ‘complementary’ pairs. We have to abandon the notion that the concepts and schemata adequate for sensible phenomena in the macroscopic world correspond to what is real in the microworld. Bohr’s Kantian approach therefore severed an ontological connection between the quantum theory and the world of ‘real’ phenomena. Down there, it’s stranger than we can say. ‘[A]n independent reality in the ordinary physical sense can neither be ascribed to the phenomena nor to the agencies of observations.’46
A Brief Guide to the Great Equations Page 25