The Quantum Universe_ Everything That Can Happen Does Happen - Brian Cox [38]
We can simplify this, as we’ve done before, by noticing that mx/t is the momentum of the particle, p. So with a little bit of rearrangement, we get
This result is important enough to warrant a name, and it is called the de Broglie equation because it was first proposed in September 1923 by the French physicist Louis de Broglie. It is important because it associates a wavelength with a particle of a known momentum. In other words it expresses an intimate link between a property usually associated with particles – momentum – and a property usually associated with waves – wavelength. In this way, the wave-particle duality of quantum mechanics has emerged from our manipulations with clocks.
The de Broglie equation constituted a huge conceptual leap. In his original paper, he wrote that a ‘fictitious associated wave’ should be assigned to all particles, including electrons, and that a stream of electrons passing through a slit ‘should show diffraction phenomena’.2 In 1923, this was theoretical speculation, because Davisson and Germer did not observe an interference pattern using beams of electrons until 1927. Einstein made a similar proposal to de Broglie’s, using different reasoning, at around the same time, and these two theoretical results were the catalyst for Schrödinger to develop his wave mechanics. In the last paper before he introduced his eponymous equation, Schrödinger wrote: ‘That means nothing else but taking seriously the de Broglie–Einstein wave theory of moving particles.’
We can gain a little more insight into the de Broglie equation by looking at what happens if we decrease the wavelength, which would correspond to increasing the amount of winding between adjacent clocks. In other words, we will reduce the distance between clocks reading the same time. This means that we would then have to increase the distance x to compensate for the decrease in λ. In other words, point X needs to be further away in order for the extra winding to be ‘undone’. That corresponds to a faster-moving particle: smaller wavelength corresponds to larger momentum, which is exactly what the de Broglie equation says. It is a lovely result that we have managed to ‘derive’ ordinary motion (because the cluster of clocks moves smoothly in time) starting from a static array of clocks.
Wave Packets
We would now like to return to an important issue that we skipped over earlier in the chapter. We said that the initial cluster moves in its entirety to the vicinity of point X, but only roughly maintains its original configuration. What did we mean by that rather imprecise statement? The answer provides a link back to the Heisenberg Uncertainty Principle, and delivers further insight.
We have been describing what happens to a cluster of clocks, which represents a particle that can be found somewhere within a small region of space. That’s the region spanned by our five clocks in Figure 5.1. A cluster like this is referred to as a wave packet. But we have already seen that confining a particle to some region in space has consequences. We cannot prevent a localized particle from getting a Heisenberg kick (i.e. its momentum is uncertain because it is localized), and as time passes this will lead to the particle ‘leaking out’ of the region within which it was initially located. This effect was present for the case where the clocks all read the same time and it is present in the case of the moving cluster too. It will tend to spread the wave packet out as it travels, just as a stationary particle spreads out over time.
If we wait long enough, the wave packet corresponding to the moving cluster of clocks will have totally disintegrated and we’ll lose any ability to predict where the particle actually is. This will obviously have implications for any attempt we might make to measure the speed of our particle. Let’s see how