by Brian Cox
First, we need to write down the extra amount by which clock 2 is wound relative to clock 1 because it has further to travel to point X. Using the results on page 75, this is
Again, you may be able to work this out for yourself by multiplying out the brackets and throwing away the d2 bits because d, the distance between the clocks, is very small compared to x, the distance to point X a long way away from the original cluster.
It is also straightforward to write down the criterion for the clocks to read the same time; we want the extra amount of winding due to the propagation of clock 2 to be exactly cancelled by the extra forward wind we gave it initially. For the example shown in Figure 5.1, the extra wind for clock 2 is ¼, because we’ve wound the clock forward by a quarter of a turn. Similarly, clock 3 has a wind of ½, because we’ve wound it around ½ a turn. In symbols, we can express the fraction of one full wind between two clocks quite generally as d/λ, where d is the distance between the clocks and λ is the wavelength. If you can’t quite see this, just think of the case for which the distance between two clocks is equal to the wavelength. Then d = λ, and therefore d/λ = 1, which is one full wind, and both clocks will read the same time.
Bringing this all together, we can say that for two adjacent clocks to read the same time at point X we require the extra amount of wind we put into the initial clock to be equal to the extra amount of wind due to the difference in propagation distance:
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 this works out.
A good way to measure a particle’s speed is to make two measurements of its position at two different times. We can then deduce the speed by dividing the distance the particle travelled by the time between the two measurements. Given what we’ve just said, however, this looks like a dangerous thing to do because if we make a measurement of the position of a particle too precisely then we are in danger of squeezing its wave packet, and that will change its subsequent motion. If we don’t want to give the particle a significant Heisenberg kick (i.e. a significant momentum because we make Δx too small) then we must make sure that our position measurement is sufficiently vague. Vague is, of course, a vague term, so let’s make it less so. If we use a particle detection device that is capable of detecting particles to an accuracy of 1 micrometre and our wave packet has a width of 1 nanometre, then the detector won’t have much impact on the particle at all. An experimenter reading out the detector might be very happy with a resolution of 1 micron but, from the electron’s perspective, all the detector did was report back to the experimenter that the particle is in some huge box, a thousand times bigger than the actual wave packet. In this case, the Heisenberg kick induced by the measurement process will be very small compared to that induced by the finite size of the wave packet itself. That’s what we mean by ‘sufficiently vague’.
We’ve sketched the situation in Figure 5.3 and have labelled the initial width of the wave packet d and the resolution of our detector Δ. We’ve also drawn the wave packet at a later time; it’s a little broader and has a width d′, which is bigger than d. The peak of the wave packet has travelled a distance L over some time interval t at a speed v. Apologies if that particular flourish of formality reminds you of your long-forgotten school days sitting behind a stained and eroded wooden bench listening to a science teacher’s voice fading into the half-light of a late winter’s afternoon as you slide into an inappropriate nap. We are covering ourselves in chalk dust for good reason, and it is our hope that the conclusion of this section will jolt you back to consciousness more effectively than the flying board dusters of your youth.
Figure 5.3. A wave packet at two different times. The packet moves to the right and spreads out as time advances. The packet moves because the clocks that constitute it are wound around relative to each other (de Broglie) and it spreads out because of the Uncertainty Principle. The shape of the packet is not very important but, for completeness, we should say that where the packet is large the clocks are large, and where it is small the clocks are small.
Back in the metaphorical science lab, with renewed vigour, we are trying to measure the speed v of the wave packet by making two measurements of its position at two different times. This will give us the distance L that the wave packet has travelled in a time t. But our detector has a resolution Δ, so we won’t be able to pin down L exactly. In symbols, we can say that the measured speed is
where the combined plus or minus sign is there simp
ly to remind us that, if we actually make the two position measurements, we will generally not always get L but instead ‘L plus a bit’ or ‘L minus a bit’, where the ‘bit’ is due to the fact we agreed not to make a very accurate measurement of the particle’s position. It is important to bear in mind that L is not something we can actually measure: we always measure a value somewhere in the range L ± Δ. Remember also that we need Δ to be much larger than the size of the wave packet otherwise we will squeeze the particle and that will disrupt it.
Let’s rewrite the last equation very slightly so that we can better see what’s going on:
It seems that if we take t to be very large then we will get a measurement of the speed v = L/t with a very tiny spread, because we can choose to wait around for a very long time, making t as large as we like and consequently Δ/t as small as we like whilst still keeping Δ comfortably large. This looks like we have a nice way to make an arbitrarily precise measurement of the particle’s speed without disturbing it at all; just wait for a huge amount of time between the first and the second measurements. This makes perfect intuitive sense. Imagine you are measuring the speed of a car driving along a road. If you measure how far it has travelled in one minute, you will tend to get a much more precise measurement of its speed than if you measure how far it travelled in one second. Have we dodged Heisenberg?
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 they tend to cancel each other out at the ends. The dashed curve is what happens if we add together all four of the waves illustrated in the lower graphs – now the peak in the centre is becoming more pronounced. Finally, the solid curve shows what happens when we add together the first ten waves, i.e. the four shown plus six more of progressively decreasing wavelength. The more waves we add in to the mix, the more detail we can achieve in the final wave. The wave packet in the upper graph could describe a localized particle, rather like the wave packet illustrated in Figure 5.3. In this way it really is possible to synthesize a wave of any shape at all – it is all achieved by adding together simple sine waves.
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, wh
atever 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 positions that a particle can have. There are, however, many ways of representing the wavefunction mathematically, and the little clocks in space version is only one of them. We touched on this when we said it is possible to think of the particle as also being represented by a sum over sine waves. If you ponder this point for a moment, you should realize that specifying the complete list of sine waves actually provides a complete description of the particle (because by adding together these waves we can obtain the clocks associated with the position space wavefunction). In other words, if we specify exactly which sine waves are needed to build a wave packet, and exactly how much of each sine wave we need to add in to get the shape just right, then we will have a different but entirely equivalent description of the wave packet. The neat thing is that any sine wave can itself be described by a single imaginary clock: the size of the clock encodes the maximum height of the wave and the phase of the wave at some point can be represented by the time that the clock reads. This means that we can choose to represent a particle not by clocks in space but by an alternative list of clocks, one for each possible value of the particle’s momentum. This description is just as economical as the ‘clocks in space’ description, and instead of making explicit where the particle is likely to be found we are instead making explicit what values of momentum the particle is likely to have. This alternative array of clocks is known as the momentum space wavefunction and it contains exactly the same information as the position space wavefunction.6