by Brian Cox
This is a very important result. If you had put the numbers in for yourself then you’d already have a feel for why this is the case; it’s the smallness of Planck’s constant. Written out in full, it has a value 0.0000000000000000000000000000000066260695729 kg m2/s. Dividing pretty much any everyday number by that will result in a lot of clock winding and a lot of interference, with the result that the exotic journeys of our sand grain across the Universe all cancel each other out, and we perceive this voyager through infinite space as a boring little speck of dust sitting motionless on a beach.
Our particular interest of course is in those circumstances where clocks do not cancel each other out, and, as we have seen, this occurs if the clocks do not turn by more than a single wind. In that case, the orgy of interference will not happen. Let’s see what this means quantitatively.
Figure 4.4. The same as Figure 4.3 except that we are now not committing to a specific value of the size of the clock cluster or the distance to the point X.
We are going to return to the clock cluster, which we’ve redrawn in Figure 4.4, but we’ll be more abstract in our analysis this time instead of committing to definite numbers. We will suppose that the cluster has a size equal to Δx, and the distance of the closest point in the cluster to point X is x. In this case, the cluster size Δx refers to the uncertainty in our knowledge of the initial position of the particle; it started out somewhere in a region of size Δx. Starting with point 1, the point in the cluster closest to point X, we should wind the clock corresponding to a hop from this point to X by an amount
Now let’s go to the farthest point, point 3. When we transport the clock from this point to X, it will be wound around by a greater amount, i.e.
We can now be precise and state the condition for the clocks propagated from all points in the cluster not to cancel out at X: there should be less than one full wind of difference between the clocks from points 1 and 3, i.e.
Writing this out in full, we have
We’re now going to consider the specific case for which the cluster size, Δx, is much smaller than the distance x. This means we are asking for the prospects that our particle will make a leap far outside of its initial domain. In this case, the condition for no clock cancellation, derived directly from the previous equation, is
If you know a little maths, you’ll be able to get this by multiplying out the bracketed term and neglecting all the terms that involve (Δx)2. This is a valid thing to do because we’ve said that Δx is very small compared to x, and a small quantity squared is a very small quantity.
This equation is the condition for there to be no cancellation of the clocks at point X. We know that if the clocks don’t cancel out at a particular point, then there is a good chance that we will find the particle there. So we have discovered that if the particle is initially located within a cluster of size Δx, then at a time t later there is a good chance to find it a long distance x away from the cluster if the above equation is satisfied. Furthermore, this distance increases with time, because we are dividing by the time t in our formula. In other words, as more time passes, the chances of finding the particle further away from its initial position increases. This is beginning to look suspiciously like a particle that is moving. Notice also that the chance of finding the particle a long way away also increases as Δx gets smaller – i.e. as the uncertainty in the initial position of the particle gets smaller. In other words, the more accurately we pin down the particle, the faster it moves away from its initial position. This now looks a lot like Heisenberg’s Uncertainty Principle.
To make final contact, let us rearrange the equation a little bit. Notice that for a particle to make its way from anywhere in the cluster to point X in time t, it must leap a distance x. If you actually measured the particle at X then you would naturally conclude that the particle had travelled at a speed equal to x/t. Also, remember that the mass multiplied by the speed of a particle is its momentum, so the quantity mx/t is the measured momentum of the particle. We can now go ahead and simplify our equation some more, and write
where p is the momentum. This equation can be rearranged to read
and this really is important enough to merit more discussion, because it looks very much like Heisenberg’s Uncertainty Principle.
This is the end of the maths for the time being, and if you haven’t followed it too carefully you should be able to pick the thread up from here.
If we start out with a particle localized within a blob of size Δx, we have just discovered that, after some time has passed, it could be found anywhere in a larger blob of size x. The situation is illustrated in Figure 4.5. To be precise, this means that if we had looked for the particle initially, then the chances are that we would have found it somewhere inside the inner blob. If we didn’t measure it but instead waited a while, then there would be a good chance of finding it later on anywhere within the larger blob. This means that the particle could have moved from a position within the small initial blob to a position within the larger one. It doesn’t have to have moved, and there is still a probability that it will be within the smaller region Δx. But it is quite possible that a measurement will reveal that the particle has moved as far out as the edge of the bigger blob.8 If this extreme case were realized in a measurement then we would conclude that the particle is moving with a momentum given by the equation we just derived (and if you have not followed the maths then you will just have to take this on trust), i.e. p = h/Δx.
Figure 4.5. A small cluster grows with time, corresponding to a particle that is initially localized becoming delocalized as time advances.
Now, we could start from the beginning again and set everything up exactly as before, so that the particle is once again initially located in the smaller blob of size Δx. Upon measuring the particle, we would probably find it somewhere else inside the larger blob, other than the extreme edge, and would therefore conclude that its momentum is smaller than the extreme value.
If we imagine repeating this experiment again and again, measuring the momentum of a particle that starts out inside a small cluster of size Δx, then we will typically measure a range of values of p anywhere between zero and the extreme value h/Δx. Saying that ‘if you do this experiment many times then I predict you will measure the momentum to be somewhere between zero and h/Δx’ means that ‘the momentum of the particle is uncertain by an amount h/Δx’. Just as for the case of the uncertainty in position, physicists assign the symbol Δp to this uncertainty, and write ΔpΔx ∼ h. The ‘∼’ sign indicates that the product of the uncertainties in position and momentum is roughly equal to Planck’s constant – it might be a little bigger or it might be a little smaller. With a little more care in the mathematics we could get this equation exactly right. The result would depend upon the details of the initial clock cluster, but it is not worth the extra effort to spend time doing that because what we have done is sufficient to capture the key ideas.
The statement that the uncertainty in a particle’s position multiplied by the uncertainty in its momentum is (approximately) equal to Planck’s constant is perhaps the most familiar form of Heisenberg’s Uncertainty Principle. It is telling us that, starting from the knowledge that the particle is located within some region at some initial time, a measurement of the particle’s position at some time later will reveal that the particle is moving with a momentum whose value cannot be predicted more accurately than ‘somewhere between zero and h/Δx’. In other words, if we start out by confining a particle to be in a smaller and smaller region, then it has a tendency to want to jump further and further away from that region. This is so important, it is worth restating a third time: the more precisely you know the position of a particle at some instant, the less well you know how fast it is moving and therefore where it will be sometime later.
This is exactly Heisenberg’s statement of the Uncertainty Principle. It lies at the heart of quantum theory, but we should be quite clear that it is not in itself a vague statement. It is a statem
ent about our inability to track particles around with precision, and there is no more scope for quantum magic here than there is for Newtonian magic. What we have done in the last few pages is to derive Heisenberg’s Uncertainty Principle from the fundamental rules of quantum physics as embodied in the rules for winding, shrinking and adding clocks. Indeed, its origin lies in our proposition that a particle can be anywhere in the Universe an instant after we measure its position. Our initial wild proposal that the particle can be anywhere and everywhere in the Universe has been tamed by the orgy of quantum interference, and the Uncertainty Principle is in a sense all that remains of the original anarchy.
There is something very important that we should say about how to interpret the Uncertainty Principle before we move on. We must not make the mistake of thinking that the particle is actually at some single specific place and that the spread in initial clocks really reflects some limitation in our understanding. If we thought that then we would not have been able to compute the Uncertainty Principle correctly, because we would not admit that we must take clocks from every possible point inside the initial cluster, transport them in turn to a distant point X and then add them all up. It was the act of doing this that gave us our result, i.e. we had to suppose that the particle arrives at X via a superposition of many possible routes. We will make use of Heisenberg’s principle in some real-world examples later on. For now, it is satisfying that we have derived one of the key results of quantum theory using nothing more than some simple manipulations with imaginary clocks.
Let’s stick a few numbers into the equations to get a better feel for things. How long will we have to wait for there to be a reasonable probability that a sand grain will hop outside a matchbox? Let’s assume that the matchbox has sides of length 3 cm and that the sand grain weighs 1 microgram. Recall that the condition for there to be a reasonable probability of the sand grain hopping a given distance is given by
where Δx is the size of the matchbox. Let’s calculate what t should be if we want the sand grain to jump a distance x = 4 cm, which would comfortably exceed the size of the matchbox. Doing a very simple bit of algebra, we find that
and sticking the numbers in tells us that t must be greater than approximately 1021 seconds. That is around 6 × 1013 years, which is over a thousand times the current age of the Universe. So it won’t happen. Quantum mechanics is weird, but not weird enough to allow a grain of sand to hop unaided out of a matchbox.
To conclude this chapter, and launch ourselves into the next one, we will make one final observation. Our derivation of the Uncertainty Principle was based upon the configuration of clocks illustrated in Figure 4.4. In particular, we set up the initial cluster of clocks so that they all had hands of the same size and were all reading the same time. This specific arrangement corresponds to a particle initially at rest within a certain region of space – a sand grain in a matchbox, for example. Although we discovered that the particle will most likely not remain at rest, we also discovered that for large objects – and a grain of sand is very large indeed in quantum terms – this motion is completely undetectable. So there is some motion in our theory, but it is motion that is imperceptible for big enough objects. Obviously we are missing something rather important, because big things do actually move around, and remember that quantum theory is a theory of all things big and small. We must now address this problem: how can we explain motion?
5. Movement as an Illusion
In the previous chapter we derived Heisenberg’s Uncertainty Principle by considering a particular initial arrangement of clocks – a small cluster of them, each with hands of the same size and pointing in the same direction. We discovered that this represents a particle that is approximately stationary, although the quantum rules imply that it jiggles around a little. We shall now set up a different initial configuration; we want to describe a particle in motion. In Figure 5.1, we’ve drawn a new configuration of clocks. Again it is a cluster of clocks, corresponding to a particle that is initially located in the vicinity of the clocks. The clock at position 1 reads 12 o’clock, as before, but the other clocks in the cluster are now all wound forwards by different amounts. We’ve drawn five clocks this time simply because it will help make the reasoning more transparent, although as before we are to imagine clocks in between the ones we have drawn – one for each point in space in the cluster. Let’s apply the quantum rule as before and move these clocks to point X, a long way outside the cluster, to once again describe the many ways that the particle can hop from the cluster to X.
In a procedure that we hope is becoming more routine, let’s take the clock from point 1 and propagate it to point X, winding it around as we go. It will wind around by an amount
Now let’s take the clock from point 2 and propagate it to point X. It’s a little bit further away, let’s say a distance d further, so it will wind a bit more
Figure 5.1: The initial cluster (illustrated by the clocks marked 1 to 5) is made up of clocks that all read different times – they are all shifted by three hours relative to their neighbours. The lower part of the figure illustrates how the time on the clocks varies through the cluster.
This is exactly what we did in the previous chapter, but perhaps you can already see that something different will happen for this new initial configuration of clocks. We set things up such that clock 2 was initially wound forwards by three hours relative to clock 1 – from 12 o’clock to 3 o’clock. But in carrying clock 2 to point X, we have to wind it backwards by a little more than clock 1, corresponding to the extra distance d that it has to travel. If we arrange things so that the initial forward wind of clock 2 is exactly the same as the extra backward wind it gets when travelling to X, then it will arrive at X showing exactly the same time as clock 1. This will mean that, far from cancelling out, it will add to clock 1 to make a larger clock, which in turn means that there will be a high probability that the particle will be found at X. This is a completely different situation from the orgy of quantum interference that occurred when we began with all the clocks reading the same time. Let’s now consider clock 3, which we have wound forwards six hours relative to clock 1. This clock has to travel an extra distance 2d to make it to point X and again, because of the offset in time, this clock will arrive pointing to 12 o’clock. If we set all the offsets in the same manner, then this will happen right across the cluster and all of the clocks will add together constructively at X.
This means that there will be a high probability that the particle will be found at the point X at some later time. Clearly point X is special because it is that particular point where all the clocks from the cluster conspire to read the same time. But point X is not the only special point – all points to the left of X for a distance equal to the length of the original cluster also share the same property that the clocks add together constructively. To see this, notice that we could take clock 2 and transport it to a point a distance d to the left of X. This would correspond to moving it a distance x, which is exactly the same distance that we moved clock 1 when we moved it to X. We could then transport clock 3 to this new point through a distance x + d, which is exactly the same distance that we previously moved clock 2. These two clocks should therefore read the same time when they arrive and add together. We can keep on doing this for all the clocks in the cluster, but only until we reach a distance to the left of X equal to the original cluster size. Outside of this special region, the clocks largely cancel out because they are no longer protected from the usual orgy of quantum interference.1 The interpretation is clear: the cluster of clocks moves, as illustrated in Figure 5.2.
Figure 5.2. The cluster of clocks moves at constant speed to the right. This is because the original cluster had its clocks wound relative to each other as described in the text.
This is a fascinating result. By setting up the initial cluster using offset clocks rather than clocks all pointing in the same direction, we have arrived at the description of a moving particle. Intriguingly, we can also make
a very important connection between the offset clocks and the behaviour of waves.
Remember that we were motivated to introduce the clocks back in Chapter 2 in order to explain the wave-like behaviour of particles in the double-slit experiment. Look back at Figure 3.3 on page 35, where we sketched an arrangement of clocks that describes a wave. It is just like the arrangement of the clocks in our moving cluster. We’ve sketched the corresponding wave below the cluster in Figure 5.1 using exactly the same methodology as before: 12 o’clock represents the peak of the wave, 6 o’clock represents the trough and 3 o’clock and 9 o’clock represent the places where the wave height is zero.
As we might have anticipated, it appears that the representation of a moving particle has something to do with a wave. The wave has a wavelength, and this corresponds to the distance between clocks showing identical times in the cluster. We’ve also drawn this on the figure, and labelled it λ.
We can now work out how far the point X should be away from the cluster in order for adjacent clocks to add constructively. This will lead us to another very important result in quantum mechanics, and make the connection between quantum particles and waves much clearer. Time for a bit more mathematics.