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positions that a particle can have. There are, however, many ways of representing the wavefunction mathematically, and the little clocks in space version is only one of them. We touched on this when we said it is possible to think of the particle as also being represented by a sum over sine waves. If you ponder this point for a moment, you should realize that specifying the complete list of sine waves actually provides a complete description of the particle (because by adding together these waves we can obtain the clocks associated with the position space wavefunction). In other words, if we specify exactly which sine waves are needed to build a wave packet, and exactly how much of each sine wave we need to add in to get the shape just right, then we will have a different but entirely equivalent description of the wave packet. The neat thing is that any sine wave can itself be described by a single imaginary clock: the size of the clock encodes the maximum height of the wave and the phase of the wave at some point can be represented by the time that the clock reads. This means that we can choose to represent a particle not by clocks in space but by an alternative list of clocks, one for each possible value of the particle’s momentum. This description is just as economical as the ‘clocks in space’ description, and instead of making explicit where the particle is likely to be found we are instead making explicit what values of momentum the particle is likely to have. This alternative array of clocks is known as the momentum space wavefunction and it contains exactly the same information as the position space wavefunction.6

This might sound very abstract, but you may well use technology based on Fourier’s ideas every day, because the decomposition of a wave into its component sine waves is the foundation of audio and video compression technology. Think about the sound waves that make up your favourite tune. This complicated wave can, as we have just learnt, be broken down into a series of numbers that give the relative contributions of each of a large number of pure sine waves to the sound. It turns out that, although you may need a vast number of individual sine waves to reproduce the original sound wave exactly, you can in fact throw a lot of them away without compromising the perceived audio quality at all. In particular, the sine waves that contribute to sound waves that humans can’t hear are not kept. This vastly reduces the amount of data needed to store an audio file – hence your mp3 player doesn’t need to be too large.

We might also ask what possible use could this different and even more abstract version of the wavefunction be? Well, think of a particle represented, in position space, by a single clock. This describes a particle located at a certain place in the Universe; the single point where the clock sits. Now think of a particle represented by a single clock, but this time in momentum space. This represents a particle with a single, definite momentum. Describing such a particle using the position space wavefunction would, in contrast, require an infinite number of equally sized clocks, because according to the Uncertainty Principle, a particle with a definite momentum can be found anywhere. As a result, it is sometimes simpler to perform calculations directly in terms of the momentum space wavefunction.

In this chapter, we have learnt that the description of a particle in terms of clocks is capable of capturing what we ordinarily call ‘movement’. We have learnt that our perception that objects move smoothly from point to point is, from the perspective of quantum theory, an illusion. It is closer to the truth to suppose that particles move from A to B via all possible paths. Only when we add together all of the possibilities does motion as we perceive it emerge. We have also seen explicitly how the clock description manages to encode the physics of waves, even though we only ever deal with point-like particles. It is time now to really exploit the similarity with the physics of waves as we tackle the important