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The Quantum World


It is difficult to overstate the importance of quantum theory. It is the most successful scientific theory ever, able to predict to many decimal places of accuracy; and it underpins so much of our technology of today - and increasingly it looks like opening up some exciting technology of tomorrow. But it often comes across as difficult to understand - and if a new generation of quantum physicists is to develop successfully, the introductory theory has to be put across to them clearly. We are going to focus on various aspects of the quantum world to highlight some of the way that are being found to make it understandable. And we start by looking at why it is that this particular area of science is apparently fraught with such conceptual problems. 

Being difficult 

One of the disturbing moments for anyone trying to understand quantum theory is when they discover that the founding fathers of the subject apparently didn’t understand it either – and indeed seemed to revel in their difficulty. 

There are a range of oft-quoted statements that reappear time and again, such as Richard Feynman’s remark that: 

‘I think I can safely say that no-one understands quantum mechanics…. Do not keep asking yourself, if you can possibly avoid it, “but how can it be like that?”… Nobody knows how it can be like that.’

Niels Bohr said that:  If someone says that he can think about quantum physics without becoming dizzy, that shows only that he has not understood anything whatever about it.’ 

However, this is not what normally happens in the onward progress of science. New theories explain phenomena, rather than become sources of intrinsic mystery. For example, the reason why planetary orbits were elliptical was strange until Newton developed his theory of gravity. And within a few years, there was not only an acceptance and understanding of elliptical orbits but also of a new model of the universe, running under laws like a machine, and laws that humans could home in on and identify. And the whole worldview of society changed as people adapted to this new picture of the universe – this was the foundation of the Enlightenment. 

Nature and nature’s laws lay hid in night:
God said, let Newton be! And all was light.

  ­was the epitaph that Alexander Pope proposed for Newton. 

The same process of familiarity, as people adapt their perceptions to the new format, as happens in art. ‘A picture may seem extraordinarily strange to you,’ wrote Gertrude Stein, ‘and after some time not only does it not seem strange but it is impossible to find what there was in it that was strange.’ 

But for quantum theory there has been no process of increasing familiarity, and no beam of Newtonian light. Eighty years on from its foundation, it remains in shadows and mist. 

Why should this be? Well first of all, the way that quantum theory came into being was in some ways unusual. Some of the major pieces came into place fast, and almost out of thin air. Schrödinger’s famous equation, for example, comes out of a few lines of elegant mathematics. Suddenly the equation is there – but then comes the equation as to what it may mean. The debate focused first on the symbol ψ, the quantity that was doing the waving? But what was it, and what was it waving in? 

Schrödinger himself did not know. After he had developed the equation, he tried to picture ψ as a kind of electron essence, that could flow as a wave and then concentrate together into a particle, but that only worked for a single electron.  

For two electrons the ψ was a feature of the combined system, depending on the locations of them both, and so could only be plotted in a space of six dimensions, three for the location of the first electron and three for the second. Each of the electrons needs the whole of our three-dimensional space for its waves, and so a mathematical ‘configuration space’ has to be imagined for calculating the combination of the two sets of waves. This space does not have any physical interpretation; it is purely a kind of filing-system, for the purposes of calculation. 

Max Born then came up with the interpretation that ψ was in fact related to probability. ψ is a complex number, and so it has a complex conjugate ψ*. When we multiply the two, we get |ψ|², which turns out to have a physical interpretation – its value at any particular point of space is the probability that the electron will be found there. ‘If ψ is mainly concentrated in one small stormy area, it is practically certain that the electron is there,’ noted Sir Arthur Eddington. So ψ is interpreted as a kind of probability wave. But how can probability, which is a rather abstract and human-centred concept, come in physical waves that travel through space? And again, in what are the waves waving in, and how? 

In Eddington’s words: ‘Something unknown is doing we don’t know what.’ 

The sheer power of Schrödinger’s approach is awesome. How could he handle mathematically something whose essence was so ill-defined? Possible answers come from the story of his life, and particularly his biography by Walter Moore. He had come through traumatic times in service in the First World War and the privations of post-War Vienna, blockaded by the victorious Allies. He had a deep love of philosophy and a strong interest in Eastern texts. And the idea of something unnameable, that we cannot apprehend directly but which underlies all the material world, was a concept that he felt at home with – sufficiently at home to build it up into a mathematical structure. 

So the original uncertainties about ψ should not make us feel uneasy about the equation. It is true that there is something mysterious about it, but it’s a known type of unknown – or at least it was known enough to Schrödinger for him to be able to cast it into a mathematical mould. 

Howie Firth
7 December 2007

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