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the centre of the atom. The figure on the top left is the ground state wavefunction, and it tells us that the electron is, in this case, typically to be found around 1 × 10−10 m from the proton. The energies of the standing waves increase from the top left to the bottom right. The scale also changes by a factor of eight from the top left to the bottom right – in fact the bright region covering most of the top-left picture is approximately the same size as the small bright spots in the centre of the two pictures on the right. This means the electron is likely to be farther away from the proton when it is in the higher energy levels (and hence that it is more weakly bound to it). It is clear that these waves are not sine waves, which means they do not correspond to states of definite momentum. But, as we have been at pains to emphasize, they do correspond to states of definite energy.

Figure 6.9. Four of the lowest energy quantum waves describing the electron in a hydrogen atom. The light regions are where the electron is most likely to be found and the proton is in the centre. The top-right and bottom-left pictures are zoomed out by a factor of 4 relative to the first and the bottom-right picture is zoomed out by a factor of 8 relative to the first. The first picture is around 3 × 10−10 m across.

The distinctive shape of the standing waves is due to the shape of the well and some features are worth discussing in a little more detail. The most obvious feature of the well around a proton is that it is spherically symmetric. This means that it looks the same no matter which angle you view it from. To picture this, think of a basketball with no markings on it: it’s a perfect sphere and it will look exactly the same no matter how you rotate it around. Perhaps we might dare to think of an electron inside a hydrogen atom as if it were trapped inside a tiny basketball? This is certainly more plausible than saying the electron is trapped in a square well and, remarkably, there is a similarity. Figure 6.10 shows, on the left, two of the lowest-energy standing sound waves that can be produced within a basketball. Again we have taken a slice through the ball, and the air pressure within the ball varies from black to white as the pressure increases. On the right are two possible electron standing waves in a hydrogen atom. The pictures are not identical, but they are very similar. So, it is not entirely stupid to imagine that the electron within a hydrogen atom is being trapped within something akin to a tiny basketball. This picture really serves to illustrate the wavelike behaviour of quantum particles, and it hopefully takes some of the mystery out of things: understanding the electron in a hydrogen atom is not more complicated than understanding how the air vibrates inside a basketball.

Figure 6.10: Two of the simplest standing sound waves inside a basketball (left) compared to the corresponding electron waves in a hydrogen atom (right). They are very similar. The top picture for hydrogen is a close-up of the central region in the bottom left picture in Figure 6.9.

Before we leave the hydrogen atom, we would like to say a little more about the potential created by the proton and how it is that the electron can leap from a higher energy level to a lower one with the emission of a photon. We avoided any discussion of how the proton and the electron communicate with each other, quite legitimately, by introducing the idea of a potential. This simplification allowed us to understand the quantization of energy for trapped particles. But if we want a serious understanding of what’s going on, we should try to explain the underlying mechanism for trapping particles. In the case of a particle moving in an actual box, we might imagine some impenetrable wall that is presumably made up of atoms, and the particle is prevented from passing through the wall by interacting with the atoms within it. A proper understanding of ‘impenetrability’ comes from understanding how the particles interact with each other. Likewise, we said that the proton