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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