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The probabilities in quantum theory are not like this at all; they are fundamental. It is not the case that we can only predict the probability of a particle being in one place or another because we are ignorant. We can’t, even in principle, predict what the position of a particle will be. What we can predict, with absolute precision, is the probability that a particle will be found in a particular place if we look for it. More than that, we can predict with absolute precision how this probability changes with time. Born expressed this beautifully in 1926: ‘The motion of particles follows probability laws but the probability itself propagates according to the law of causality.’ This is exactly what Schrödinger’s equation does: it is an equation that allows us to calculate exactly what the wavefunction will look like in the future, given what it looks like in the past. In that sense, it is analogous to Newton’s laws. The difference is that, whilst Newton’s laws allow us to calculate the position and speed of particles at any particular time in the future, quantum mechanics allows us to calculate only the probability that they will be found at a particular place.

This loss of predictive power was what bothered Einstein and many of his colleagues. With the benefit of over eighty years of hindsight and a great deal of hard work, the debate now seems somewhat redundant, and it is easy to dismiss it with the statement that Born, Heisenberg, Pauli, Dirac and others were correct and Einstein, Schrödinger and the old guard were wrong. But it was certainly possible back then to believe that quantum theory was incomplete in some way, and that the probabilities appear, just as in thermodynamics or coin tossing, because there is some information about the particles that we are missing. Today that idea gains little purchase – theoretical and experimental progress indicate that Nature really does use random numbers, and the loss of certainty in predicting the positions of particles is an intrinsic property of the physical world: probabilities are the best we can do.

4. Everything That Can Happen Does Happen

We’ve now set up a framework within which we can explore quantum theory in detail. The key ideas are very simple in their technical content, but tricky in the way they challenge us to confront our prejudices about the world. We have said that a particle is to be represented by lots of little clocks dotted around and that the length of the clock hand at a particular place (squared) represents the probability that the particle will be found at that place. The clocks are not the main point – they are a mathematical device we’ll use to keep track of the odds on finding a particle somewhere. We also gave a rule for adding clocks together, which is necessary to describe the phenomenon of interference. We now need to tie up the final loose end, and look for the rule that tells us how the clocks change from one moment to the next. This rule will be the replacement of Newton’s first law, in the sense that it will allow us to predict what a particle will do if we leave it alone. Let’s begin at the beginning and imagine placing a single particle at a point.

Figure 4.1. The single clock representing a particle that is definitely located at a particular point in space.

We know how to represent a particle at a point, and this is shown in Figure 4.1. There will be a single clock at that point, with a hand of length 1 (because 1 squared is 1 and that means the probability to find the particle there is equal to 1, i.e. 100 per cent). Let’s suppose that the clock reads 12 o’clock, although this choice is completely arbitrary. As far as the probability is concerned, the clock hand can point in any direction, but we have to choose something to start with, so 12 o’clock will do. The question we want to answer is the following: what is the chance that the particle will be located somewhere else at a later time? In other words, how many clocks do we have to draw, and where