The Quantum Universe_ Everything That Can Happen Does Happen - Brian Cox [39]
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 simply 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