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is perhaps the most familiar form of Heisenberg’s Uncertainty Principle. It is telling us that, starting from the knowledge that the particle is located within some region at some initial time, a measurement of the particle’s position at some time later will reveal that the particle is moving with a momentum whose value cannot be predicted more accurately than ‘somewhere between zero and h/Δx’. In other words, if we start out by confining a particle to be in a smaller and smaller region, then it has a tendency to want to jump further and further away from that region. This is so important, it is worth restating a third time: the more precisely you know the position of a particle at some instant, the less well you know how fast it is moving and therefore where it will be sometime later.

This is exactly Heisenberg’s statement of the Uncertainty Principle. It lies at the heart of quantum theory, but we should be quite clear that it is not in itself a vague statement. It is a statement about our inability to track particles around with precision, and there is no more scope for quantum magic here than there is for Newtonian magic. What we have done in the last few pages is to derive Heisenberg’s Uncertainty Principle from the fundamental rules of quantum physics as embodied in the rules for winding, shrinking and adding clocks. Indeed, its origin lies in our proposition that a particle can be anywhere in the Universe an instant after we measure its position. Our initial wild proposal that the particle can be anywhere and everywhere in the Universe has been tamed by the orgy of quantum interference, and the Uncertainty Principle is in a sense all that remains of the original anarchy.

There is something very important that we should say about how to interpret the Uncertainty Principle before we move on. We must not make the mistake of thinking that the particle is actually at some single specific place and that the spread in initial clocks really reflects some limitation in our understanding. If we thought that then we would not have been able to compute the Uncertainty Principle correctly, because we would not admit that we must take clocks from every possible point inside the initial cluster, transport them in turn to a distant point X and then add them all up. It was the act of doing this that gave us our result, i.e. we had to suppose that the particle arrives at X via a superposition of many possible routes. We will make use of Heisenberg’s principle in some real-world examples later on. For now, it is satisfying that we have derived one of the key results of quantum theory using nothing more than some simple manipulations with imaginary clocks.

Let’s stick a few numbers into the equations to get a better feel for things. How long will we have to wait for there to be a reasonable probability that a sand grain will hop outside a matchbox? Let’s assume that the matchbox has sides of length 3 cm and that the sand grain weighs 1 microgram. Recall that the condition for there to be a reasonable probability of the sand grain hopping a given distance is given by

where Δx is the size of the matchbox. Let’s calculate what t should be if we want the sand grain to jump a distance x = 4 cm, which would comfortably exceed the size of the matchbox. Doing a very simple bit of algebra, we find that

and sticking the numbers in tells us that t must be greater than approximately 1021 seconds. That is around 6 × 1013 years, which is over a thousand times the current age of the Universe. So it won’t happen. Quantum mechanics is weird, but not weird enough to allow a grain of sand to hop unaided out of a matchbox.

To conclude this chapter, and launch ourselves into the next one, we will make one final observation. Our derivation of the Uncertainty Principle was based upon the configuration of clocks illustrated in Figure 4.4. In particular, we set up the initial cluster of clocks so that they all had hands of the same size and were all reading the same time. This specific arrangement corresponds to a particle initially at rest within