The Quantum Universe_ Everything That Can Happen Does Happen - Brian Cox [41]
Figure 5.4. Upper graph: Adding together several sine waves to synthesize a sharply peaked wave packet. The dotted curve contains fewer waves than the dashed one, which in turn contains fewer than the solid one. Lower graphs: The first four waves used to build up the wave packets in the upper graph.
The de Broglie equation informs us that each of the waves in the lower graphs of Figure 5.4 corresponds to a particle with a definite momentum, and the momentum increases as the wavelength decreases. We are beginning to see why it is that if a particle is described by a localized cluster of clocks then it must necessarily be made up of a range of momenta.
To be more explicit, let’s suppose that a particle is described by the cluster of clocks represented by the solid curve in the upper graph in Figure 5.4.4 We have just learnt that this particle can also be described by a series of much longer clusters of clocks: the first wave in the lower graphs plus the second wave in the lower graphs, plus the third wave in the lower graphs, and so on. In this way of thinking, there are several clocks at each point (one from each long cluster), which we should add together to produce the single clock cluster represented in upper graph of Figure 5.4. The choice of how to think about the particle is really ‘up to you’. You can think of it as being described by one clock at each point, in which case the size of the clock immediately lets you know where the particle is likely to be found, i.e. in the vicinity of the peak in the upper graph of Figure 5.4. Alternatively, you can think of it as being described by a number of clocks at each point, one for each possible value of the momentum of the particle. In this way we are reminding ourselves that the particle localized in a small region does not have a definite momentum. The impossibility of building a compact wave packet from a single wavelength is an evident feature of Fourier’s mathematics.
This way of thinking provides us with a new perspective on Heisenberg’s Uncertainty Principle. It says that we cannot describe a particle in terms of a localized cluster of clocks using clocks corresponding to waves of a single wavelength. Instead, to get the clocks to cancel outside the region of the cluster, we must necessarily mix in different wavelengths and hence different momenta. So, the price we pay for localizing the particle to some region in space is to admit we do not know what its momentum is. Moreover, the more we restrict the particle, the more waves we need to add in and the less well we know its momentum. This is exactly the content of the Uncertainty Principle, and it is very satisfying to have found a different way of reaching the same conclusion.5
To close this chapter we want to spend a little more time with Fourier. There is a very powerful way of picturing quantum theory that is intimately linked to the ideas we have just been discussing. The important point is that any quantum particle, whatever it is doing, is described by a wavefunction. As we’ve presented it so far, the wavefunction is simply the array of little clocks, one for each point in space, and the size of the clock determines the probability that the particle will be found at that point. This way of representing a particle is called the ‘position space wavefunction’ because it deals directly with the possible