The Quantum Universe_ Everything That Can Happen Does Happen - Brian Cox [48]
We are now going to explore the link between standing waves and particles of definite energy in a different way, without appealing to the special case of the square well. We’ll do this using those little quantum clocks.
Figure 6.6: Four snapshots of a standing wave at successively later times. The arrows represent the clock hands and the dotted line is the projection onto the ‘12 o’clock’ direction. The clocks all turn around in unison.
First, notice that, if an electron is described by a standing wave at some instant in time, then it will be described by the same standing wave at some later time. By ‘the same’, we mean that the shape of the wave is unchanged, as was the case for the standing water wave in Figure 6.1. We don’t, of course, mean that the wave does not change at all; the water height does change, but crucially the positions of the peaks and nodes do not. This allows us to figure out what the quantum clock description of a standing wave must look like, and it is illustrated in Figure 6.6 for the case of the fundamental standing wave. The clock sizes along the wave reflect the position of the peaks and nodes, and the clock hands sweep around together at the same rate. We hope you can see why we’ve drawn this particular pattern of clocks. The nodes must always be nodes, the peaks must always be peaks and they must always stay in the same place. This means that the clocks sitting in the vicinity of the nodes must always be very small, and the clocks representing the peaks must always have the longest hands. The only freedom we have, therefore, is to allow the clocks to sit where we put them and rotate in sync.
If we were following the methodology of the earlier chapters, we would now start from the configuration of clocks shown in the top row of Figure 6.6 and use the shrinking and turning rules to generate the bottom three rows at later times. This exercise in clock hopping is a hop too far for this book, but it can be done, and there is a nice twist because to do it correctly it is necessary to include the possibility that the particle ‘bounces off the walls of the box’ before hopping to its destination. Incidentally, because the clocks are bigger in the centre, we can immediately conclude that an electron described by this array of clocks is more likely to be found in the middle of the box than at the edges.
So, we have found that the trapped electron is described by an array of clocks that all whizz around at the same rate. Physicists don’t usually talk like this, and musicians certainly don’t; they both say that standing waves are waves of definite frequency.5 High-frequency waves correspond to clocks that whizz around faster than the clocks of low-frequency waves. You can see this because, if a clock whizzes around faster, then the time it takes a peak to turn into a trough and then rise back again (represented by a single rotation of the clock hand) decreases. In terms of water waves, the high-frequency standing waves move up and down faster than the low-frequency ones. In music, a middle C is said to have a frequency of 262 Hz, which means that, on a guitar, the string vibrates up and down 262 times every second. The A above middle C has a frequency of 440 Hz, so it vibrates more rapidly (this is the agreed tuning standard for most orchestras and musical instruments across the world). As we’ve noted, however, it is only for pure sine waves that these waves of definite frequency also have definite wavelength. Generally speaking, frequency is the fundamental quantity that describes standing waves, and this sentence is probably a pun.
The million-dollar question, then, is ‘What does it mean