A Brief Guide to the Great Equations (37 page)
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
=
I
h
/2π
i
, 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
=
I
h
/2π
i
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
q
1
be the precision with which the value
q
is known (
q
1
is, say, the mean error of
q
), therefore here the wavelength of the light. Let
p
1
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
p
1
and
q
1
stand in the relation
1
q
1
~
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.
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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.’
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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.’
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And Pauli was thrilled. ‘He said something like, ‘Morgenröte einer Neuzeit’ ‘ – the dawn of a new era.
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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.’
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Late in 1927, however, Bohr was scheduled to take a trip to the U.S., and he and Heisenberg were anxious to finalize an interpretation before his departure, so the two agreed to a truce, mainly on Bohr’s terms. The truce was made public that September, at a celebration in Lake Como of the hundredth anniversary of Alessandro Volta’s death. Bohr gave a speech proposing an awkward accommodation of wave and matrix mechanics, and Heisenberg stood up at the end to signal his approval. Waves and particles, Bohr said in effect, are ways we speak about events in the atomic realm. Neither way is entirely accurate, but the two ways have overlapping but restricted spheres of application. They are, Bohr declared, complementary ways of speaking about something of which we can have no direct knowledge. As he once put it, ‘There is no quantum world. There is only an abstract physical description. It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature.’
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Thus the origin of what has become known as the Copenhagen interpretation of quantum mechanics. It was not universally appreciated. Einstein called it ‘shaky’, adding that ‘The Heisenberg-Bohr tranquilizing philosophy – or religion? – is so delicately contrived that, for the time being it provides a gentle pillow for the true believer from which he cannot very easily be aroused.’
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In complete theory, he wrote years later in 1935, with Boris Podolsky and Nathan Rosen, ‘every element of the physical reality must have a counterpart in the physical theory.’ Einstein tried to argue, unsuccessfully, that the incompleteness of quantum mechanics was a flaw revealing that there had to be more to be discovered, so-called hidden variables, the discovery of
which will make its formulations refer directly to the real world. He pressed the argument for years, with Bohr countering that position and momentum were inherently classical concepts, inapplicable to events in the microworld except in loose and, strictly speaking, inaccurate ways.
The Copenhagen interpretation – that somewhere beyond or beneath the macroscopic world lurks something that we cannot visualize, and that is made visualizable by an ensemble or arrangement of things whose behaviour is macroscopic – amounts to a clear, logical interpretation, and appears to be the simplest one consistent with all experimental and theoretical constraints. It is an interpretation that makes us all uncomfortable, but that is a psychological phenomenon, not an argument for or against the interpretation.
THE YOGI AND THE QUANTUM
The idea of intermediate kinds of reality was just the price one had to pay.
In 1929, 2 years after the appearance of the uncertainty principle, a physicist at Harvard University named Percy Bridgman – a future Nobel laureate – published an article in
Harper’s Magazine
about the meaning of the uncertainty principle. The implications are far-reaching, he said, even for the public. ‘The immediate effect will be to let loose a veritable intellectual spree of licentious and debauched thinking.’ For, Bridgman continued, the nonscientist is apt to conclude from the uncertainty principle, not that it stated ‘the end of meaning’, but rather that ‘there is something beyond the ken of the scientist.’ In a remarkably prophetic passage, Bridgman wrote:
This imagined beyond, which the scientist has proved he cannot penetrate, will become the playground of the imagination of every mystic and dreamer. The existence of such a domain will be made the basis of an orgy of rationalizing. It will be made the substance of the soul; the spirits of the dead will populate it; God will lurk in its shadows; the principle of vital processes will have its seat here; and it will be the medium of telepathic communication. One group will find in the failure of the physical law of cause and effect the solution of the age-long problem
of the freedom of the will; and on the other hand the atheist will find the justification of his contention that chance rules the universe.
1
Eighty years later, we see that Bridgman was correct: each of these views has indeed been advanced. Bridgman went on to point to a positive side, saying that eventually, we can develop the ‘new methods of education’ to inculcate into people the ‘habits of thought’ required to reshape the thinking we use in ‘the limited situations of everyday life.’ The end result, Bridgman concluded, will be salutary:
[S]ince thought will conform to reality, understanding and conquest of the world about us will proceed at an accelerated pace. I venture to think that there will also eventually be a favorable effect on man’s character; the mean man will react with pessimism, but a certain courageous nobility is needed to look a situation like this in the face. And in the end, when man has fully partaken of the fruit of the tree of knowledge, there will be this difference between the first Eden and the last, that man will not become as a god, but will remain forever humble.
Eighty years later, we are still working on acquiring courageous nobility, and remaining humble. But we are also still working on how to talk about the physical interpretation of quantum mechanics, on how it connects with other, more familiar and visualizable features of the world.
Of all the founders of quantum mechanics, Niels Bohr was the most insistent that we should try to fully express the quantum world in the framework of ordinary language and classical concepts. ‘[I]n the end’, as Michael Frayn has Bohr’s character say in the play
Copenhagen
, ‘we have to be able to explain it all to Margrethe’, his wife and amanuensis who
serves as the onstage stand-in for the ordinary (that is, ‘classically thinking’) person.
Many physicists, finding this task irrelevant or impossible, were satisfied with partial explanations – and Heisenberg argued that the mathematics works: that’s enough! Bohr rejected such dodges, and rubbed physicists’ noses in what they did not understand or tried to hide. He did not have an answer himself – and knew it – but he had no reason to think one could not be found. His closest answer was the doctrine of complementarity, an ordinary-language way of saying that quantum phenomena behave, apparently inconsistently, as waves or particles depending on how the instruments are set up, and that you need both concepts to fully grasp the phenomena. While this provoked debate among physicists on the ‘meaning’ of quantum mechanics, the doctrine – and discussion – soon all but vanished.
Why? A large part of the answer is that, by 1930, physicists found a perfectly adequate way to represent classical concepts within the quantum framework involving a special abstract mathematical language called Hilbert (infinite-dimensional) space. In this space, the concepts of position and momentum are associated with different sets of coordinate axes that do not line up with each other, resulting in the situation captured in ordinary language terms by
complementarity
.
2
While Bohr used the notion of complementarity to say that quantum phenomena were both particles and waves – somewhat confusingly, and in ordinary language terms – the notion of Hilbert space provided an alternate and much more precise framework in which to say that they are neither. But it was not a language that Margrethe understood; for her, quantum mechanics would have to remain esoteric and she would have to cope with understanding it as best she could. This is what has left the door open for the kinds of fantastic interpretations of meaning for human life mentioned at the beginning of this chapter, and by Bridgman.
What makes the interpretation of quantum mechanics difficult to talk about? It is that we expect a complete theory to fall short of fully describing nature, but in a particular and well-defined way, for it provides a model that is an ideal limit of measurement, with any gaps or discrepancies between it and what we encounter in the laboratory arising from errors and imperfections in the measuring equipment. Many other theories and equations that physicists teach and use have other kinds of gaps and discrepancies, if they omit aspects of nature in the interests of a good approximation. An example is
F
=
ma
, which leaves out mass-energy conversion in the cause of buying ease of application. These are what we might call ‘harmlessly fudging or incomplete descriptive theories.’ Any gaps between the theory and the world are epistemological; that is, they have to do with our knowledge of the world, or the gap between our representations of the world and what it represents.
The uncertainty principle is incomplete in a different sense. It is a mathematical relation, and a feature of the statistical interpretation of the wave function in quantum mechanics. It makes no reference to any underlying physical picture; there are no references to waves or particles, nor to physical experiments. It is not obvious what it refers to, except possibly the clicks of a detector. Yet it is
about
gaps
in
the world itself. These gaps are not epistemological but ontological; having to do not with our knowledge but with the world.
This is strange, but why? It is important to see what the strangeness is
not
due to. The strangeness of the uncertainty principle is not due to the measurement process disturbing the object measured, which would be a feature of any Newtonian theory involving exchange of particles. Nor is it due to the presence of statistics. Rather, the strangeness of quantum mechanics is that quantum formulations are not ‘about’ a real or ideal object in the conventional sense.
In classical physics, deviations of measured quantities from
ideal norms are treated independently in a statistically based theory of errors. But the variations – statistical distributions – of quantum measurements are systematically linked in a single formalism. It tells you that all you can know precisely is the width of a distribution, and that you cannot make individual predictions. The superpositions are possibilities in the world, possibilities of observations. The wave formulas of quantum mechanics are thus neither about an ideal object, nor about a real object, but about a special kind of semiabstract object that admits numerous potential experimental realizations in becoming a real object. This special kind of semiabstract object is incomplete if we try to think of it the way we do other more familiar elements of the real world. One needs to add something to the abstract object to bring it into the world, and the choices that one makes regarding what to measure and how to measure it affect what one is measuring. This abstract object can appear wavelike or particle-like, for instance, depending on the kind of situation we put it in, waves and particles being models in visualizable space-time.
This, then, is the ‘intermediate kind of reality’ that Heisenberg said was the price one had to pay to have quantum phenomena. It has the funny kind of incomplete, semiabstract reality that scripts or scores have – they are programs, as it were, for real things in the world (the produced play, the performed music) that require adding a context, and decisions about that context affect the whole of the abstract object. It brings back the role of human purposes and decisions that Newton left out.
The challenge in explaining the meaning of the uncertainty principle to nonscientists lies in trying to explain this new kind of semiabstract object. And it is important to try, for otherwise there will continue to be information loss and distortion in the public understanding of the uncertainty principle.
Heisenberg proved that. Just not mathematically.