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By Root 804 0

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