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a field. The temperature field is simply an array of numbers, one for every point. In the case of a quantum particle, the field is more complicated because it requires a clock face at each point rather than a single number. This field is usually called the wavefunction of the particle. The fact that we need an array of clocks for the wavefunction, whilst a single number would suffice for the temperature field or for water waves, is an important difference. In physics jargon, the clocks are there because the wavefunction is a ‘complex’ field, whilst the temperature or water wave heights are both ‘real’ fields. We shall not need any of this language, because we can work with the clock faces.1

We should not worry that we have no direct way to sense a wave-function, in contrast to a temperature field. The fact that it is not something we can touch, smell or see directly is irrelevant. Indeed, we would not get very far in physics if we decided to restrict our description of the Universe to things we can directly sense.

In our discussion of the double-slit experiment for electrons, we said that the electron wave is largest where the electron is most likely to be. This interpretation allowed us to appreciate how the stripy interference pattern can be built up dot by dot as the electrons arrive. But this is not a precise enough statement for our purposes now. We want to know what the probability is to find an electron at a particular point – we want to put a number on it. This is where the clocks become necessary, because the probability that we want is not simply the wave height. The correct thing to do is to interpret the square of the length of the clock hand as the probability to find the particle at the site of the clock. This is why we need the extra flexibility that the clocks give us over simple numbers. That interpretation is not meant to be at all obvious, and we cannot offer any good explanation for why it is correct. In the end, we know that it is correct because it leads to predictions that agree with experimental data. This interpretation of the wavefunction was one of the thorny issues facing the early pioneers of quantum theory.

The wavefunction (that is our cluster of clocks) was introduced into quantum theory in a series of papers published in 1926 by the Austrian physicist Erwin Schrödinger. His paper of 21 June contains an equation that should be etched into the mind of every undergraduate physics student. It is known, naturally enough, as the Schrödinger equation:

The Greek symbol Ψ (pronounced ‘psi’) represents the wavefunction, and the Schrödinger equation describes how it changes as time passes. The details of the equation are irrelevant for our purposes because we are not going to follow the Schrödinger approach in this book. What is interesting, though, is that, although Schrödinger wrote down the correct equation for the wavefunction, he initially got the interpretation wrong. It was Max Born, one of the oldest of the physicists working on the quantum theory in 1926, who, at the grand old age of forty-three, gave the correct interpretation in a paper submitted just four days after Schrödinger’s. We mention his age because quantum theory during the mid 1920s gained the nickname ‘Knabenphysik’ – ‘boy physics’ – because so many of the key protagonists were young. In 1925 Heisenberg was twenty-three, Wolfgang Pauli, whose famous Exclusion Principle we shall meet later on, was twenty-two, as was Paul Dirac, the British physicist who first wrote down the correct equation describing the electron. It is often claimed that their youth freed them from the old ways of thinking and allowed them fully to embrace the radical new picture of the world represented by quantum theory. Schrödinger, at thirty-eight, was an old man in this company and it is true that he was never completely at ease with the theory he played such a key role in developing.

Born’s radical interpretation of the wavefunction, for which he received the Nobel Prize for physics in 1954, was that the square of the length of the clock hand at