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pool, the string will vibrate in some combination of the different possible standing waves, depending on how it is plucked. The actual shape of the string can always be obtained by adding together the standing waves corresponding to each of the harmonics present. The harmonics and their relative sizes give the sound its characteristic tone. Different guitars will have different distributions of harmonics and therefore sound different, but a middle C (a pure harmonic) on one guitar is always the same as a middle C on another. For the guitar, the shape of the standing waves is very simple: they are pure sine waves whose wavelengths are fixed by the length of the string. For the swimming pool, the standing waves are more complicated, as shown in Figure 6.1, but the idea is exactly the same.

You may be wondering why these special waves are called ‘standing waves’. It is because the waves do not change their shape. If we take two snapshots of a guitar string vibrating in a standing wave, then the two pictures will only differ in the overall size of the wave. The peaks will always be in the same place, and the nodes will always be in the same place because they are fixed by the ends of the string or, in the case of the swimming pool, by the sides of the pool. Mathematically, we could say that the waves in the two snapshots differ only by an overall multiplicative factor. This factor varies periodically with time, and expresses the rhythmical vibration of the string. The same is true for the swimming pool in Figure 6.1, where each snapshot is related to the others by an overall multiplicative factor. For example, the last snapshot can be obtained from the first by multiplying the wave height at every point by minus one.

In summary, waves that are confined in some way can always be expressed in terms of standing waves (waves that do not change their shape) and, as we have said, there are very good reasons for devoting so much time to understanding them. At the top of the list is the fact that standing waves are quantized. This is very clear for the standing waves on a guitar string: the fundamental has a wavelength of twice the length of the string, and the next longest allowed wavelength is equal to the length of the string. There is no standing wave with a wavelength in between these two and so we can say that the allowed wavelengths on a guitar string are quantized.

Standing waves therefore make manifest the fact that something gets quantized when we trap waves. In the case of a guitar string, it is clearly the wavelength. For the case of an electron inside a box, the quantum waves corresponding to the electron will also be trapped, and by analogy we should expect that only certain standing waves will be present in the box, and therefore that something will be quantized. Other waves simply cannot exist, just as a guitar string doesn’t play all the notes in an octave at the same time no matter how it is plucked. And just as for the sound of a guitar, the general state of the electron will be described by a blend of standing waves. These quantum standing waves are starting to look very interesting, and, encouraged by this, let’s start our analysis proper.

To make progress, we must be specific about the shape of the box inside which we place our electron. To keep things simple, we’ll suppose that the electron is free to hop around inside a region of size L, but that it is totally forbidden from wandering outside this region. We do not need to say how we intend to forbid the electron from wandering – but if this is supposed to be a simplified model of an atom then we should imagine that the force exerted by the positively charged nucleus is responsible for its confinement. In the jargon, this is known as a ‘square well potential’. We’ve sketched the situation in Figure 6.3, and the reason for the name should be obvious.

Figure 6.3. An electron trapped in a square well potential.

The idea of confining a particle in a potential is a very important one that we’ll use again, so it will be useful to make sure we understand exactly