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a certain region of space – a sand grain in a matchbox, for example. Although we discovered that the particle will most likely not remain at rest, we also discovered that for large objects – and a grain of sand is very large indeed in quantum terms – this motion is completely undetectable. So there is some motion in our theory, but it is motion that is imperceptible for big enough objects. Obviously we are missing something rather important, because big things do actually move around, and remember that quantum theory is a theory of all things big and small. We must now address this problem: how can we explain motion?

5. Movement as an Illusion

In the previous chapter we derived Heisenberg’s Uncertainty Principle by considering a particular initial arrangement of clocks – a small cluster of them, each with hands of the same size and pointing in the same direction. We discovered that this represents a particle that is approximately stationary, although the quantum rules imply that it jiggles around a little. We shall now set up a different initial configuration; we want to describe a particle in motion. In Figure 5.1, we’ve drawn a new configuration of clocks. Again it is a cluster of clocks, corresponding to a particle that is initially located in the vicinity of the clocks. The clock at position 1 reads 12 o’clock, as before, but the other clocks in the cluster are now all wound forwards by different amounts. We’ve drawn five clocks this time simply because it will help make the reasoning more transparent, although as before we are to imagine clocks in between the ones we have drawn – one for each point in space in the cluster. Let’s apply the quantum rule as before and move these clocks to point X, a long way outside the cluster, to once again describe the many ways that the particle can hop from the cluster to X.

In a procedure that we hope is becoming more routine, let’s take the clock from point 1 and propagate it to point X, winding it around as we go. It will wind around by an amount

Now let’s take the clock from point 2 and propagate it to point X. It’s a little bit further away, let’s say a distance d further, so it will wind a bit more

Figure 5.1: The initial cluster (illustrated by the clocks marked 1 to 5) is made up of clocks that all read different times – they are all shifted by three hours relative to their neighbours. The lower part of the figure illustrates how the time on the clocks varies through the cluster.

This is exactly what we did in the previous chapter, but perhaps you can already see that something different will happen for this new initial configuration of clocks. We set things up such that clock 2 was initially wound forwards by three hours relative to clock 1 – from 12 o’clock to 3 o’clock. But in carrying clock 2 to point X, we have to wind it backwards by a little more than clock 1, corresponding to the extra distance d that it has to travel. If we arrange things so that the initial forward wind of clock 2 is exactly the same as the extra backward wind it gets when travelling to X, then it will arrive at X showing exactly the same time as clock 1. This will mean that, far from cancelling out, it will add to clock 1 to make a larger clock, which in turn means that there will be a high probability that the particle will be found at X. This is a completely different situation from the orgy of quantum interference that occurred when we began with all the clocks reading the same time. Let’s now consider clock 3, which we have wound forwards six hours relative to clock 1. This clock has to travel an extra distance 2d to make it to point X and again, because of the offset in time, this clock will arrive pointing to 12 o’clock. If we set all the offsets in the same manner, then this will happen right across the cluster and all of the clocks will add together constructively at X.

This means that there will be a high probability that the particle will be found at the point X at some later time. Clearly point X is special because it is that particular point where all the clocks from