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By Root 812 0

about in Chapter 6. The width of the well determines the allowed spectrum of energy levels, and as the size of the well is changed, the energy level spectrum changes. The same is true here in that the shape of the well inside which our electrons are sitting, and therefore the energy levels they are allowed to occupy, is determined by the positions of the protons. If there are two protons, the energy spectrum is determined by the position of both of them. And if there are 1080 protons forming a universe, then the position of every one of them affects the shape of the well within which 1080 electrons are sitting. There is only ever one set of energy levels and when anything changes (e.g. an electron changes from one energy level to another) then everything else must instantaneously adjust itself such that no two fermions are ever in the same energy level.

The idea that the electrons ‘know’ about each other instantaneously sounds like it has the potential to violate Einstein’s Theory of Relativity. Perhaps we can build some sort of signalling apparatus that exploits this instantaneous communication to transmit information at faster-than-light speeds. This apparently paradoxical feature of quantum theory was first appreciated in 1935, by Einstein in collaboration with Boris Podolsky and Nathan Rosen; Einstein called it ‘spooky action at a distance’ and did not like it. It took some time before people realized that, despite its spookiness, it is impossible to exploit these long-range correlations to transfer information faster than the speed of light and that means the law of cause and effect can rest safe.

This decadent multiplicity of energy levels is not just an esoteric device to evade the constraints of the Exclusion Principle. In fact, it is anything but esoteric because this is the physics behind chemical bonding. It is also the key idea in explaining why some materials conduct electricity whilst others do not and, without it, we would not understand how a transistor works. To begin our journey to the transistor, we are going to go back to the simplified ‘atom’ we met in Chapter 6, when we trapped an electron inside a potential well. To be sure, this simple model didn’t allow us to compute the correct spectrum of energies in a hydrogen atom, but it did teach us about the behaviour of a single atom and it will serve us well here too. We are going to use two square wells joined together to make a toy model of two adjacent hydrogen atoms. We’ll think first about the case where there is a single electron moving in the potential created by two protons. The upper picture in Figure 8.1 illustrates how we’ll do it. The potential is flat except where it dips down to make two wells, which mimic the influence of the two protons in their ability to trap electrons. The step in the middle helps keep the electron trapped either on the left or on the right, provided it is high enough. In the technical parlance, we say that the electron is moving in a double-well potential.

Our first challenge is to use this toy model to understand what happens when we bring two hydrogen atoms together – we will see that when they get close enough they bind together, to make a molecule. After that, we shall contemplate more than two atoms and that will allow us to appreciate what happens inside solid matter.

If the wells are very deep, we can use the results from Chapter 6 to determine what the lowest-lying energy states should correspond to. For a single electron in a single square well, the lowest energy state is described by a sine wave of wavelength equal to twice the size of the box. The next-to-lowest energy state is a sine wave whose wavelength is equal to the size of the box, and so on. If we put an electron into one side of a double-well, and if the well is deep enough, then the allowed energies must be close to those for an electron trapped in a single deep well, and its wavefunction should therefore look quite like a sine wave. It is to the small differences between a perfectly isolated hydrogen atom and a hydrogen atom in a distantly separated