The Quantum Universe_ Everything That Can Happen Does Happen - Brian Cox [34]
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