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That is a very satisfying conclusion to reach. We have learnt that, for hydrogen atoms that are far apart, the tiny difference between the two lowest-lying energy states was only of academic interest, although it did lead us to conclude that every electron in the Universe knows about every other, which is certainly fascinating. On the other hand, the two states get increasingly separated as the protons get closer together, and the lower of the two eventually becomes the state that describes the hydrogen molecule, and that is very far from being of mere academic interest, because covalent bonding is the reason that you are not a bunch of atoms sloshing around in a featureless blob.

Now we can keep pulling on this intellectual thread and start to think about what happens when we bring more than two atoms together. Three is bigger than two, so let’s start there and consider a triple-well potential, as illustrated in Figure 8.5. As ever, we are to imagine that each well is at the site of an atom. There should be three lowest energy states, but looking at the figure you might be tempted to think that there are now four energy states for every state of the single well. The four states we have in mind are illustrated in the figure and they correspond to wavefunctions that are variously symmetric or anti-symmetric about the centre of the two potential barriers.6 This counting must be wrong, because if it were correct then one could put four identical fermions into these four states and the Pauli principle would be violated. To get the Pauli principle to work out we need just three energy states and this, of course, is what happens. To see this, we need merely spot that we can always write any one of the four wavefunctions sketched in the figure as a combination of the other three. At the bottom of the figure, we have illustrated how that works out in one particular case; we have shown how the last wavefunction can be obtained by a combination of adding and subtracting the other three.

Figure 8.5. The triple well, which is our model for three atoms in a row, and the possible lowest-energy wavefunctions. At the bottom we illustrate how the bottom of the four waves can be obtained from the other three.

Figure 8.6. The energy bands in a chunk of solid matter and how they vary with the distance between the atoms.

Having identified the three lowest energy states for a particle sitting in the triple-well potential, we can ask what Figure 8.4 looks like in this case, and it should come as no surprise at all to find that it looks rather similar, except that what was a pair of allowed energy states becomes a triplet of allowed states.

Enough of three atoms – we shall now swiftly move our attention to a chain of many. This is going to be particularly interesting because it contains the key ideas that will allow us to explain a lot about what is happening inside solid matter. If there are N wells (to model a chain of N atoms) then for each energy in the single well there will now be N energies. If N is something like 1023, which is typical of the number of atoms in a small chunk of solid material, that is an awful lot of splitting. The result is that Figure 8.4 now looks something like Figure 8.6. The vertical dotted line illustrates that, for atoms that are separated by the corresponding distance, the electrons can only have certain allowed energies. That should be no big surprise (if it is, then you’d better start reading the book again from the beginning), but what is interesting is that the allowed energies come in ‘bands’. The energies from A to B are allowed, but no other energies are allowed until we get to C, whence energies from C to D are allowed, and so on. The fact that there are many atoms in the chain means that there are very many allowed energies crammed into each band. So many in fact, that for a typical solid we can just as well suppose that the allowed energies form a smooth continuum in each band.