The Quantum Universe_ Everything That Can Happen Does Happen - Brian Cox [44]
Figure 6.1. Six successive snapshots of a standing wave in a tank of water. The time advances from the top left to the bottom right.
Generally speaking, waves are complicated things. Imagine jumping into a swimming pool full of water. The water will slosh around all over the place, and it would seem to be futile to try to describe what is going on in any simple fashion. Underlying the complexity, however, there is hidden simplicity. The key point is that the water in a swimming pool is confined, which means that all the waves are trapped inside the pool. This gives rise to a phenomenon known as ‘standing waves’. The standing waves are hidden away in the mess when we disturb the pool by jumping into it, but there is a way to make the water move so that it oscillates in the regular, repeating patterns of the standing waves. Figure 6.1 shows how the water surface looks when it is undergoing one such oscillation. The peaks and troughs rise and fall, but most importantly they rise and fall in exactly the same place. There are other standing waves too, including one where the water in the middle of the tank rises and falls rhythmically. We do not usually see these special waves because they are hard to produce, but the key point is that any disturbance of the water at all – even the one we caused by our inelegant dive and subsequent thrashing around – can be expressed as some combination or other of the different standing waves. We’ve seen this type of behaviour before; it is a direct generalization of Fourier’s ideas that we encountered in the last chapter. There, we saw that any wave packet can be built up out of a combination of waves each of definite wavelength. These special waves, representing particle states of definite momentum, are sine waves. In the case of confined water waves, the idea generalizes so that any disturbance can always be described using some combination of standing waves. We’ll see later in this chapter that standing waves have an important interpretation in quantum theory, and in fact they hold the key to understanding the structure of atoms. With this in mind, let’s explore them in a little more detail.
Figure 6.2. The three longest wavelength waves that can fit on a guitar string. The longest wavelength (at the top) corresponds to the lowest harmonic (fundamental) and the others correspond to the higher harmonics (overtones).
Figure 6.2 shows another example of standing waves in Nature: three of the possible standing waves on a guitar string. On plucking a guitar string, the note we hear is usually dominated by the standing wave with the largest wavelength – the first of the three waves shown in the figure. This is known in both physics and music as the ‘lowest harmonic’ or ‘fundamental’. Other wavelengths are usually present too, and they are known as overtones or higher harmonics. The other waves in the figure are the two longest-wavelength overtones. The guitar is a nice example because it’s simple enough to see why a guitar string can only vibrate at these special wavelengths. It is because it is held fixed at both ends – by the guitar bridge at one end and your finger pressing against a fret at the other. This means that the string cannot move at these two points, and this determines the allowed wavelengths. If you play the guitar, you’ll know this physics instinctively; as you move your fingers up the fret board towards the bridge, you decrease the length of the string and therefore force it to vibrate with shorter and shorter wavelengths, corresponding to higher-pitched notes.
The lowest harmonic is the wave that has only two stationary points, or ‘nodes’; it moves everywhere except at the two fixed ends. As you can see from the figure, this note has a wavelength of twice the length of the string. The next smallest wavelength is equal to the length of the string, because we can fit another node in the centre. Next, we can get a wave with wavelength equal to ⅔ times the length of the string, and so on.
In general, just as in the case of the water confined in a swimming