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a particular point represents the probability of finding a particle there. For example, if the hour-hand on the clock located at some place has a length of 0.1 then squaring this gives 0.01. This means that the probability to find the particle at this place is 0.01, i.e. one in a hundred. You might ask why Born didn’t just square the clocks up in the first place, so that in the last example the clock hand would itself have a length of 0.01. That will not work, because to account for interference we are going to want to add clocks together and adding 0.01 to 0.01 say (which gives 0.02) is not the same as adding 0.1 to 0.1 and then squaring (which gives 0.04).

We can illustrate this key idea in quantum theory with another example. Imagine doing something to a particle such that it is described by a specific array of clocks. Also imagine we have a device that can measure the location of particles. A simple-to-imagine-butnot-so-simple-to-build device might be a little box that we can rapidly erect around any region of space. If the theory says that the chance of finding a particle at some point is 0.01 (because the clock hand at that point has length 0.1), then when we erect the box around that point we have a one in a hundred chance of finding the particle inside the box afterwards. This means that it is unlikely that we’ll find anything in the box. However, if we are able to reset the experiment by setting everything up such that the particle is once again described by the same initial set of clocks, then we could redo the experiment as many times as we wish. Now, for every 100 times we look in the little box we should, on average, discover that there is a particle inside it once – it will be empty the remaining ninety-nine times.

The interpretation of the squared length of the clock hand as the probability to find a particle at a particular place is not particularly difficult to grasp, but it does seem as if we (or to be more precise, Max Born) plucked it out of the blue. And indeed, from a historical perspective, it proved very difficult for some great scientists, Einstein and Schrödinger among them, to accept. Looking back on the summer of 1926, fifty years later, Dirac wrote: ‘This problem of getting the interpretation proved to be rather more difficult than just working out the equations.’ Despite this difficulty, it is noteworthy that by the end of 1926 the spectrum of light emitted from the hydrogen atom, one of the great puzzles of nineteenth-century physics, had already been computed using both Heisenberg’s and Schrödinger’s equations (Dirac eventually proved that their two approaches were in all cases entirely equivalent).

Einstein famously expressed his objection to the probabilistic nature of quantum mechanics in a letter to Born in December 1926. ‘The theory says a lot but does not really bring us any closer to the secret of the “old one”. I, at any rate, am convinced that He is not playing at dice.’ The issue was that, until then, it had been assumed that physics was completely deterministic. Of course, the idea of probability is not exclusive to quantum theory. It is regularly used in a variety of situations, from gambling on horse races to the science of thermodynamics, upon which whole swathes of Victorian engineering rested. But the reason for this is a lack of knowledge about the part of the world in question, rather than something fundamental. Think about tossing a coin – the archetypal game of chance. We are all familiar with probability in this context. If we toss the coin 100 times, we expect, on average, that fifty times it will land heads and fifty times tails. Pre-quantum theory, we were obliged to say that, if we knew everything there is to know about the coin – the precise way we tossed it into the air, the pull of gravity, the details of little air currents that swish through the room, the temperature of the air, etc. – then we could, in principle, work out whether the coin would land heads or tails. The emergence of probabilities in this context is therefore a reflection of our lack of knowledge