The Quantum Universe_ Everything That Can Happen Does Happen - Brian Cox [40]
Of course not – we have forgotten to take something into account. The particle is described by a wave packet that spreads out as time passes. Given enough time, the spreading out will completely wash out the wave packet and that means the particle could be anywhere. This will increase the range of values we get in our measurement of L and spoil our ability to make an arbitrarily accurate measurement of its speed.
For a particle described by a wave packet, we are ultimately still bound by the Uncertainty Principle. Because the particle is initially confined in a region of size d, Heisenberg informs us that the particle’s momentum is correspondingly blurred out by an amount equal to h/d.
There is therefore only one way we can build a configuration of clocks to represent a particle that travels with a definite momentum – we must make d, the size of the wave packet, very large. And the larger we make it, the smaller the uncertainty in its momentum will be. The lesson is clear: a particle of well-known momentum is described by a large cluster of clocks.3 To be precise, a particle of absolutely definite momentum will be described by an infinitely long cluster of clocks, which means an infinitely long wave packet.
We have just argued that a finite-size wave packet does not correspond to a particle with a definite momentum. This means that if we measured the momentum of very many particles, all described by exactly the same initial wave packet, then we would not get the same answer each time. Instead we would get a spread of answers and it does not matter how brilliant we are at experimental physics, that spread cannot be made smaller than h/d.
We can therefore say that a wave packet describes a particle that is travelling with a range of momenta. But the de Broglie equation implies that we can just substitute the word ‘wavelengths’ for ‘momenta’ in the last sentence, because a particle’s momentum is associated with a wave of definite wavelength. This in turn means that a wave packet must be made up of many different wavelengths. Likewise, if a particle is described by a wave with a definite wavelength then that wave must necessarily be infinitely long. It sounds like we are being pushed to conclude that a small wave packet is made up of many infinitely long waves of different wavelengths. We are indeed being pushed down this route, and what we are describing is very familiar to mathematicians, physicists and engineers alike. This is an area of mathematics known as Fourier analysis, named after the French mathematical physicist Joseph Fourier.
Fourier was a colourful man. Amongst his many notable achievements, he was Napoleon’s governor of Lower Egypt and the discoverer of the greenhouse effect. He apparently enjoyed wrapping himself up in blankets, which led to his untimely demise one day in 1830 when, tightly wrapped, he fell down his own stairs. His key paper on Fourier analysis addressed the subject of heat transfer in solids and was published in 1807, although the basic idea can be traced back much earlier.
Fourier showed that any wave at all, of arbitrarily complex shape and extent, can be synthesized by adding together a number of sine waves of different wavelengths. The point is best illustrated through pictures. In Figure 5.4 the dotted curve is made by adding together the first two sine waves in the lower graphs. You can almost do the addition in your head – the two waves are both at maximum height in the centre, and so they add together there, whilst