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By Root 753 0

the cluster conspire to read the same time. But point X is not the only special point – all points to the left of X for a distance equal to the length of the original cluster also share the same property that the clocks add together constructively. To see this, notice that we could take clock 2 and transport it to a point a distance d to the left of X. This would correspond to moving it a distance x, which is exactly the same distance that we moved clock 1 when we moved it to X. We could then transport clock 3 to this new point through a distance x + d, which is exactly the same distance that we previously moved clock 2. These two clocks should therefore read the same time when they arrive and add together. We can keep on doing this for all the clocks in the cluster, but only until we reach a distance to the left of X equal to the original cluster size. Outside of this special region, the clocks largely cancel out because they are no longer protected from the usual orgy of quantum interference.1 The interpretation is clear: the cluster of clocks moves, as illustrated in Figure 5.2.

Figure 5.2. The cluster of clocks moves at constant speed to the right. This is because the original cluster had its clocks wound relative to each other as described in the text.

This is a fascinating result. By setting up the initial cluster using offset clocks rather than clocks all pointing in the same direction, we have arrived at the description of a moving particle. Intriguingly, we can also make a very important connection between the offset clocks and the behaviour of waves.

Remember that we were motivated to introduce the clocks back in Chapter 2 in order to explain the wave-like behaviour of particles in the double-slit experiment. Look back at Figure 3.3 on page 35, where we sketched an arrangement of clocks that describes a wave. It is just like the arrangement of the clocks in our moving cluster. We’ve sketched the corresponding wave below the cluster in Figure 5.1 using exactly the same methodology as before: 12 o’clock represents the peak of the wave, 6 o’clock represents the trough and 3 o’clock and 9 o’clock represent the places where the wave height is zero.

As we might have anticipated, it appears that the representation of a moving particle has something to do with a wave. The wave has a wavelength, and this corresponds to the distance between clocks showing identical times in the cluster. We’ve also drawn this on the figure, and labelled it λ.

We can now work out how far the point X should be away from the cluster in order for adjacent clocks to add constructively. This will lead us to another very important result in quantum mechanics, and make the connection between quantum particles and waves much clearer. Time for a bit more mathematics.

First, we need to write down the extra amount by which clock 2 is wound relative to clock 1 because it has further to travel to point X. Using the results on page 75, this is

Again, you may be able to work this out for yourself by multiplying out the brackets and throwing away the d2 bits because d, the distance between the clocks, is very small compared to x, the distance to point X a long way away from the original cluster.

It is also straightforward to write down the criterion for the clocks to read the same time; we want the extra amount of winding due to the propagation of clock 2 to be exactly cancelled by the extra forward wind we gave it initially. For the example shown in Figure 5.1, the extra wind for clock 2 is ¼, because we’ve wound the clock forward by a quarter of a turn. Similarly, clock 3 has a wind of ½, because we’ve wound it around ½ a turn. In symbols, we can express the fraction of one full wind between two clocks quite generally as d/λ, where d is the distance between the clocks and λ is the wavelength. If you can’t quite see this, just think of the case for which the distance between two clocks is equal to the wavelength. Then d = λ, and therefore d/λ = 1, which is one full wind, and both clocks will read the same time.

Bringing this all together, we