A Brief Guide to the Great Equations (33 page)
Refashioning John Wheeler’s famous remark about Einstein’s theory of general relativity, Stony Brook physicist Alfred S. Gold-haber says the following about Schrödinger’s equation: ‘The wave
tells the particle where to go, the particle tells the wave where to start and stop.’
It is a great irony that this approach, devised to be intuitive, often suffers thanks to a false picture.
The Schrödinger equation implied a radical change in the events on the world-stage. No longer could it be assumed that, eventually, you could plug in numbers, turn the crank, and get predictions. Instead, you plugged in numbers, turned the crank, and got…probabilities. You got the likelihood of an event occurring in a particular place. And nothing could help you more than that: not more information, not adding more pieces to the machinery. No flip book could be constructed, for if you replayed an event over and over, a particle would wind up in different spots on the page, its location a question of averages.
Schrödinger himself never warmed to this interpretation, calling it a ‘resignation.’
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It is ‘convenient’, he wrote, but we cannot allow ourselves to ‘get off so easily.’ We should keep trying to work out the interpretation, the causal mechanism, he insisted. He pointed to some seeming implications that he found ‘quite ridiculous’, including the image of the now-famous cat locked in a box along with a diabolical device that kills the cat upon the decay of a radioactive nucleus. Since the decay is defined by ψ, it appears that whether the cat is alive or dead is so defined as well, seemingly leading to the conclusion that the cat’s existence, too, is superimposed, and is half-alive and half-dead. While the conclusion is false, the image is a brilliant demonstration of the flaws of extending the theories of the microworld to those of the macroworld.
But while Schrödinger’s wave methods are used by all the workers in the field, his premises about the wave structure of reality have been ignored; the interpretation worked out by Heisenberg’s Göttingen allies came to be the favored one. ‘Schrödinger’s methods proved indispensable, writes historian Mara Beller. ‘His philosophy did not.’
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Born’s interpretation of Schrödinger’s equation changes the notion
of what a complete theory of the world consists of. Conventionally, we expect a complete theory to tell us something about reality. Most theories that physicists teach and use fall short by deliberately cutting corners. These theories provide not a complete picture, but a model that is an idealized version of any real situation. For example, the ideal-gas law ignores well understood things like van der Waals forces and hard-core repulsion, but we do not mind because talking about an ideal, rather than a real, object buys us something: a tremendous ease of application. This is what one of my colleagues calls a ‘harmlessly fudging theory.’ Such theories are harmless because these limitations do not threaten normal assumptions about the world. We are cutting corners, know that we are cutting corners, and know that the absence of the corners we are cutting does not affect our impression of the world.
Yet Schrödinger’s equation, as interpreted by Born, is different. It makes us aware that how we interact with the world affects what we bring into it. It makes us explicitly aware, that is, of ourselves as interacting with the world in doing something like taking a measurement. The idea that we are agents comes explicitly into view. We are not watching something happen on a stage and then measuring it, we are making the stage even as we do so. The prevailing interpretation of Schrödinger’s equation brings this to the fore in a way classical physics does not. It is our own expectations that determine whether we find this a sacrifice or an advance. The giant leap forward of quantum mechanics does require a substantial rethinking of what it means to ‘understand’ nature, and how to characterize ‘reality’, in a way that many scientists at the time experienced as what Heisenberg called a sacrifice – so painful a one, in fact, that they struggled mightily against having to make it. Many still do.
THE DOUBLE CONSCIOUSNESS OF SCIENTISTS
…a world which yields him no true self-consciousness, but only lets him see himself through the revelation of the other world. It is a peculiar sensation, this double-consciousness, this sense of always looking at one’s self through the eyes of others, of measuring one’s soul by the tape of a world that looks on in amused contempt and pity.
Schrödinger’s work was greeted with not just skepticism but hostility from Heisenberg and other supporters of matrix mechanics, who revealed what Mara Beller called ‘a dogmatic preference for older conceptions rather than a dispassionate objectivity’, and who were all the more annoyed because of the beauty and simplicity of Schrödinger’s approach compared to the ungainly and complex matrix methods.
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Schrödinger, meanwhile, made no secret of his scorn for matrix mechanics. The formation of quantum mechanics, indeed, was a remarkable episode in the history of science in the way hostile emotions flared in private, and even formal published papers are marked by an ‘unusually emotional undertone.’
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Envy, rivalry, anger, disbelief, conviction, stress, hope, despair, dejection – all can be found in the documents. Yet the existence of this emotional undercurrent to scientific research is omitted from most historical accounts.
Most histories of science run something like this: An unexpected discovery is made. Explanations are applied, but none work well. New equipment is built to make new measurements, but the explanation is still incomplete. The phenomenon is looked at from yet another side, with other instruments and measurements. And so on.
This is what might be called the ‘standard model’ of how science works. It emphasizes the collective and impersonal dimension, and downplays the experiences of specific individuals. The principal structural ingredients are discoveries, instruments, measurements, and theories. It is allied with a conception of science as a way of eradicating mysteries and controlling nature. Its familiarity often leads scientists to tell their own stories in ways that emphasize these ingredients, reinforcing the standard model. In this model, it’s as if the emotional life and experiences, personal successes and disappointments, and so forth, of the person comprised a train running down one track, and the scientific career – the research program, discoveries, events, and so forth – running down another track. And the two tracks are entirely separate, driven by different kinds of locomotives – two sides of one person.
But listen carefully to the scientists speaking about their work – as some popular biographies do – and you can hear another story as well, highlighting human experience. In it, motivating forces include excitement at the discovery, puzzlement at why the explanations do not work, curiosity about what might explain it, growing perplexity as more explanations do not work, imagination at devising new instruments, and wonder as the explanations shed a different kind of light than thought at the outset, even awe at learning something fundamental. As these more fully integrated pictures of science in process show, the standard model has limits, there is something beyond it, and it is ultimately destined to be superceded. It is due to be succeeded by a program of grand unification, in
which these two tracks are seen to be merged, in which science is done by individuals, not dividuals, whose life and work are part and parcel of the same person.
To see what I mean, look at the collection of Richard Feynman’s letters published a few years ago. In them, you see Feynman’s character, poses and all, as inextricably intertwined with his craft. You see that his curiosity, presumption, haranguing, and desire to set people straight were seamlessly interwoven – that the physicist and educator and his character cannot be disentangled. The same force fueled both his scientific inquiries and his interactions with others. ‘The real fun of life’, he wrote, ‘is this perpetual testing to realize how far out you can go with any potentialities.’ And you see this testing in the way he dealt with cranks, editors, ordinary people, and with nature. This provides a taste, I think, of what lies beyond the standard model.
We can also see this grand unification in Einstein. Here science historian Gerald Holton has written a good article called ‘Einstein’s Third Paradise.’ Einstein’s ‘first paradise’ refers to the intensely religious phase Einstein went through in childhood – a period of ‘the religious paradise of youth’, he called it. This phase is well attested to by Einstein himself, and by his sister Maja. This paradise ended when he was about twelve, after reading popular science books that revealed to him that not all the biblical stories could be true. He also discovered the joys of Euclidean plane geometry after being given a little book on it – he called the book ‘holy’, and a ‘wonder.’ He became acquainted with other works of science that presented nature as ‘a great, eternal riddle’, contemplation of which could give him ‘inner freedom and security.’ He called this, too, a ‘Paradise.’ Breaking away from the first paradise to enter the second paradise, he wrote, was an attempt to ‘free myself from the chains of the ‘merely personal’, from an existence dominated by wishes, hopes, and primitive feelings.’
Biographers have tended to contrast these two paradises, deeming them two separate, disconnected phases of his life, from a religious to a nonreligious phase. But Holton disagrees. At the heart of Einstein’s mature identity Holton sees a fusion of the first and second paradises, ‘where the meaning of a life of brilliant scientific activity drew on the remnants of his fervent first feelings of youthful religiosity.’
In this third paradise, Einstein seems to exemplify someone who had feelings that we can call religious and that were essential to his work but who did not credit the existence of a Master Mechanic. As he says in one letter, he was a ‘deeply religious unbeliever.’ This third paradise, then, is the kind of thing that would be described in what I called the grand unification. Consider Einstein’s speech honoring the sixtieth birthday of Max Planck. In it, Einstein said the search for a simplified, lucid image of the world was not only a scientific goal, but corresponded to a deep psychological need. A scientist could make the effort to pursue this goal the ‘centre of gravity of his emotional life.’ And, Einstein added, pursuing the most difficult scientific problems requires ‘a state of feeling similar to that of a religious person or a lover.’ Holton then mentions instances by Einstein and others in which they were brought to great despair, or great joy, by developments in science, in which the psychological commitment of these people cannot be treated as separate from the tasks they set for themselves. The science and the personal commitment are bound up with each other. Holton treats Einstein’s drive to unify apparently different phenomena as an example of this interpenetration of emotional life and career. He points to a letter to Grossmann in 1901, referring to his very first paper, on capillarity, which unifies opposing behaviours of bodies. ‘It is a wonderful feeling’, Einstein wrote (echoing Kant), ‘to recognize the unity of a complex of appearances which, in direct sense experiences, appear to be quite separate things.’ And in another letter,
15 years later, Einstein says that he is ‘driven by my need to generalize.’ Holton points out that, practically, too, ‘Einstein lived under the compulsion to unify.’ He loathed nationalisms, and dreamed of a unified world government. Holton sums up: ‘No boundaries, no barriers: none in life, as there are none in nature. Einstein’s life and his work were so mutually resonant that we recognize both to have been carried on together in the service of one grand project – the fusion into one coherency.’ Likewise, Holton says, ‘there were no boundaries or barriers between Einstein’s scientific and religious feelings.’ In his writings on science and religion late in life, Einstein often uses the same phrases to refer to the aims of science and religion. ‘I maintain that the cosmic religious feeling is the strongest and noblest motive for scientific research… A contemporary has said not unjustly that in this materialistic age of ours the serious scientific workers are the only profoundly religious people.’ And again, ‘The most beautiful experience we can have is the mysterious. A knowledge of the existence of something we cannot penetrate, our perceptions of the profoundest reason and the most radiant beauty, which only in their most primitive forms are accessible to our minds – it is this knowledge and this emotion that constitute true religiosity; in this sense, and in this alone, I am a deeply religious man.’ Thus in Einstein, too, we can see glimpses of what lies beyond the standard model: an account of science in which character and personal feeling are not marginal to the scientific process, not a prelude to a person’s scientific labours, but what sustains them and carries them forward.
Living with Uncertainty:
THE HEISENBERG UNCERTAINTY PRINCIPLE
DESCRIPTION:
Establishing the position of a particle in a small region of space makes its momentum uncertain, and vice versa, and the overall uncertainty is greater than or equal to a certain amount.