by Brian Cox
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 pair to which we now turn our attention.
Figure 8.1. The double-well potential at the top and, below it, four interesting wavefunctions describing an electron in the potential. Only the bottom two correspond to an electron of definite energy.
We can safely anticipate that the top two wavefunctions drawn in Figure 8.1 correspond to those for a single electron when it is located either in the left well or the right well (remember we use ‘well’ and ‘atom’ interchangeably). The waves are approximately sine waves, with a wavelength equal to twice the width of the well. Because the wavefunctions are identical in shape we might say that they should correspond to particles with equal energies. But this can’t be right because, as we have already said, there must be a tiny probability that, no matter how deep the wells or how widely separated they are, the electron can hop from one to the other. We have hinted at this by sketching the sine waves as ‘leaking’ slightly through the walls of the well, representing the fact that there is a very small probability to find non-zero clocks in the adjacent well.
The fact that the electron always has a finite probability of leaping from one well to the other means that the top two wavefunctions in Figure 8.1 cannot possibly correspond to an electron of definite energy, because we know from Chapter 6 that such an electron is described by a standing wave whose shape does not change with time or, equivalently, a bunch of clocks whose sizes never change with time. If, as time advances, new clocks are spawned in the originally empty well then the shape of the wavefunction will most certainly change with time. What then, does a state of definite energy look like for a double well? The answer is that the states must be more democratic, and express an equal preference to find the electron in either well. This is the only way to make a standing wave and stop the wavefunction sloshing back and forth from one well to the other.
The lower two wavefunctions we’ve sketched in Figure 8.1 have this property. These are what the lowest-lying energy states actually look like. They are the only possible stationary states we can build that look like the ‘single-well’ wavefunctions in each individual well, and also describe an electron that is equally likely to be found in either well. They are in fact the two energy states that we deduced must be present if we are to put two electrons into orbit around two distant protons to make two almost identical hydrogen atoms in a way consistent with the Pauli principle. If one electron is described by one of these two wavefunctions, then the other electron must be described by the other – this is what the Pauli principle demands.3 For deep enough wells, or if the atoms are far enough apart, the two energies will be almost equal, and almost equal to the lowest energy of a particle trapped in a single isolated well. We should not worry that one of the wavefunctions looks partly upside-down – remember it is only the size of a clock that matters when determining the probability to find the particle at some place. In other words, we could invert all the wavefunctions we’ve drawn in this book and not change the physical content of anything at all. The ‘partly upside-down’ wavefunction (labelled ‘anti-symmetric energy state’ in the figure) therefore still describes an equal superposition of an electron trapped in the left-hand well and an electron trapped in the right-hand well. Crucially though, the symmetric and anti-symmetric wavefunctions are not exactly the same (they could not be, otherwise Pauli would be upset). To see this, we need to look at the behaviour of these two lowest-energy wavefunctions in the region between the wells.
One wavefunction is symmetric about the centre of the two wells, and the other is anti-symmetric (they are labelled as such in the figure). By ‘symmetric’ we mean that the wave on the left is the mirror image of the wave on the right. For the ‘antisymmetric’ wave, the wave on the left is the mirror image of the wave on the right only after it has been turned upside down. The terminology is not very important, but what does matter is that the two waves are different in the region between the two wells. It is this small difference that means that they describe states with very slightly different energies. In fact, the symmetric wave is the one with the lower energy. So turning one of the waves upside down does in fact matter, but only a tiny amount if the wells are deep enough or far enough apart.
It can certainly be confusing to think in terms of particles with definite energy because, as we have just seen, they are described by wavefunctions that are of equal size in either well. This genuinely does mean that there is an equal chance of finding the electron in either well when we look for it, even if the wells are separated by an entire Universe.
Figure 8.2. Upper: an electron localized in the left well can be understood as the sum of the two lowest energy states. Lower: likewise, an electron located in the right well can be understood as the difference between the two lowest energy states.
How should we picture what is going on if we actually place an electron into one well and a second electron into the other well? We said before that we expect the initially empty well to fill with clocks in order to represent the fact that the particle can hop from one side to the other. We even hinted at the
answer when we said that the wavefunction ‘sloshes’ back and forth. To see how this works out, we need to notice that we can express a state localized on one of the protons as the sum of the two lowest-energy wavefunctions. We’ve illustrated this in Figure 8.2 but what does it mean? If the electron is sat in a particular well at some time, then this implies that it does not actually have a unique energy. Specifically, a measurement of its energy will return a value equal to one of the two possible energies corresponding to the two states of definite energy that build up the wavefunction. The electron is therefore in two energy states at once. We hope that, by this stage in the book, this is not a novel concept.
But here is the interesting thing. Because these two states are not of exactly the same energy, their clocks rotate at slightly different rates (as discussed on page 105). This has the effect that a particle initially described by a wavefunction localized around one proton will, after a long enough time, be described by a wavefunction which is peaked around the other proton. We don’t intend to go into details, but suffice to say that this phenomenon is quite analogous to the way that two sound waves of almost the same frequency add together to produce a resultant wave that is at first loud (the two waves are in phase) and then, some time later, quiet (as the two waves are out of phase). This phenomenon is known as ‘beats’. As the frequency of the waves gets closer and closer, so the time interval between loud and quiet increases until, when the waves are of exactly the same frequency, they combine to produce a pure tone. This will be completely familiar to any musician who, perhaps unknowingly, exploits this piece of wave physics when they make use of a tuning fork. The story runs in exactly the same way for the second electron sat in the second well. It too tends to migrate from one well to the other in a fashion that exactly mirrors the behaviour of the first electron. Although we might start with one electron in one well and a second electron in the other, after waiting long enough the electrons will swap positions.
We are now going to exploit what we have just learnt. The really interesting physics happens when we start to move the atoms closer together. In our model, moving the atoms together corresponds to reducing the width of the barrier separating the two wells. As the barrier gets thinner, the wavefunctions begin to merge together and the electron is increasingly likely to be found in the region between the two protons. Figure 8.3 illustrates what the four lowest-energy wavefunctions look like when the barrier is thin. It is interesting that the lowest-energy wavefunction is starting to look like the lowest-energy sine wave we would get if we had a single electron in a single, wide well, i.e. the two peaks merge together to produce a single peak (with a dimple in it). Meanwhile, the second-lowest-energy wavefunction looks rather like the sine wave corresponding to the next-to-lowest energy for a single, wide well. This is what we should expect, because, as the barrier between the wells gets thinner, its effect diminishes and, eventually, when it has no thickness at all, it has no effect at all and so our electron should behave exactly as if it is in a single well.
Having looked at what happens at the two extremes – the wells widely separated and the wells close together – we can complete the picture by considering how the allowed electron energies vary as we decrease the distance between the wells. We’ve sketched the results for the lowest four energy levels in Figure 8.4. Each of the four lines represents one of the four lowest energy levels, and we’ve sketched the corresponding wavefunctions next to them. The right-hand edge of the picture shows the wavefunctions when the wells are widely separated (see also Figure 8.1). As we expect, the difference between the energy levels of the electrons in each well are virtually indistinguishable. As the wells get closer together, however, the energy levels begin to separate (compare the wavefunctions on the left with those in Figure 8.3). Interestingly, the energy level corresponding to the anti-symmetric wavefunction increases, whilst that corresponding to the symmetric wavefunction decreases.
This has a profound consequence for a real system of two protons and two electrons – that is, two hydrogen atoms. Remember that in reality two electrons can actually fit into the same energy level because they can have opposite spins. This means that they can both fit into the lowest (symmetric) energy level and, crucially, this level decreases in energy as the atoms get closer together. This means that it is energetically favourable for two distant atoms to move closer together. And this is what actually happens in Nature:4 the symmetric wavefunction describes a system in which the electrons are shared more evenly between the two protons than one might anticipate from the ‘far apart’ wavefunction, and because this ‘sharing’ configuration is of lower energy, the atoms are drawn towards each other. This attraction is eventually halted because the two protons are positively charged and as such they repel each other (there is also repulsion due to the fact that the electrons have equal charges), but this repulsion only beats the inter-atomic attraction at distances smaller than around 0.1 nanometres (at room temperature). The result is that a pair of hydrogen atoms at rest will eventually nestle together. This pair of nestled hydrogen atoms has a name: it is a hydrogen molecule.
Figure 8.3: Like Figure 8.1 except that the wells are closer together. The ‘leakage’ into the region in between the wells increases. Unlike Figure 8.1, we also show the wavefunctions corresponding to the pair of next-to-lowest energies.
Figure 8.4: The variation of the allowed electron energies as we change the distance between the wells.
This preference for two atoms to stick together as a result of sharing their electrons between them is known as a covalent bond. If you look back at the top wavefunction in Figure 8.3, then this is roughly what the covalent bond in a hydrogen molecule looks like. Remember that the height of the wave corresponds to the probability that an electron will be found at that point.5 There is a peak above each well, i.e. around each proton, which informs us that each electron is still most likely to be in the vicinity of one or other of the protons. But there is also a significant chance that the electrons will spend time between the protons. Chemists speak of the atoms ‘sharing’ electrons in a covalent bond, and this is what we are seeing, even in our toy model with two square wells. Beyond the hydrogen molecule, this tendency for atoms to share electrons is what we invoked when we were discussing chemical reactions on pages 123–4.
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 o
ne 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. This feature of our toy model is preserved in real solid matter – the electrons there really do have energies that come grouped in bands like this, and that has important implications for what kind of solid we are talking about. In particular, these bands explain why some materials (metals) conduct electricity whilst others (insulators) do not.