The Quantum Universe_ Everything That Can Happen Does Happen - Brian Cox [60]
Now comes a new quantum rule – it says that we are to associate a single clock with the process as a whole, i.e. there is a clock whose size squared is equal to the probability to find electron 1 at A and electron 2 at B. In other words, there is a single clock associated with the upper picture in Figure 7.3. We can see that this clock must have a size equal to the square root of 9%, because that is the probability for the process to happen. But what time does it read? Answering this question is the domain of Chapter 10 and it involves the idea of clock multiplication. As far as this chapter is concerned, we don’t need to know the time, we only need the important new rule that we have just stated, but which is worth repeating because it is a very general statement in quantum theory: we should associate a single clock with each possible way that an entire process can happen. The clock we associate with finding a single particle at a single location is the simplest illustration of this rule, and we have managed to get this far in the book with it. But it is a special case, and as soon as we start to think about more than one particle we need to extend the rule.
This means that there is a clock of size equal to 0.3 associated with the upper picture in the figure. Likewise, there is a second clock of size equal to 0.1 (because 0.1 squared is 0.01 = 1%) associated with the lower picture in the figure. We therefore have two clocks and we want a way to use them to determine the probability to find an electron at A and another at B. If the two electrons were distinguishable then the answer would be simple – we would just add together the probabilities (and not the clocks) associated with each possibility. We would then obtain the answer of 10%.
But if there is absolutely no way of telling which of the diagrams actually happened, which is the case if the electrons are indistinguishable from each other, then following the logic we’ve developed for a single particle as it hops from place to place, we should seek to combine the clocks. What we are after is a generalization of the rule which states that, for one particle, we should add together the clocks associated with all of the different ways that the particle can reach a particular point in order to determine the probability to find the particle at that point. For a system of many identical particles, we should combine together all the clocks associated with all of the different ways that the particles can reach a set of locations in order to determine the probability that particles will be found at those locations. This is important enough to merit reading a few times – it should be clear that this new law for combining clocks is a direct generalization of the rule we have been using for a single particle. You may have noticed that we have been very careful with our wording, however. We did not say that the clocks should necessarily be added together – we said that they should be combined together. There is a good reason for our caution.
The obvious thing to do would be to add the clocks together. But before leaping in we should ask whether there is a good reason why this is correct. This is a nice example of not taking things for granted in physics – exploring our assumptions often leads to new insights, as it will do in this instance. Let’s take a step back, and think of the most general thing we could imagine. This would be to allow for the possibility of giving one of the clocks a turn or a shrink (or expansion) before we add them. Let’s explore this possibility in more detail.
What we are doing is saying, ‘I have two clocks and I want to combine them to make a single clock, so that I can use that to tell me what the probability is for the two electrons to be found at A and B. How should I combine them?’ We are not pre-empting the answer, because we want to understand if adding clocks together