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Our goal in the remainder of this chapter is to explain why it is that electrons behave in the strange way articulated by the Exclusion Principle. As ever, we shall make good use of those quantum clocks.

Figure 7.3. Two electrons scattering.

We can attack the question by thinking about what happens when two electrons ‘bounce’ off each other. Figure 7.3 illustrates a particular scenario where two electrons, labelled ‘1’ and ‘2’, start out somewhere and end up somewhere else. We have labelled the final locations A and B. The shaded blobs are there to remind us that we have not yet thought about just what happens when two electrons interact with each other (the details are irrelevant for the purposes of this discussion). All we need to imagine is that electron 1 hops from its starting place and ends up at the point labelled A. Likewise, electron 2 ends up at the point labelled B. This is what is illustrated in the top of the two pictures in the figure. In fact, the argument we are about to present works fine even if we ignore the possibility that the electrons might interact. In that case, electron 1 hops to A oblivious to the meanderings of electron 2 and the probability of finding electron 1 at A and electron 2 at B would be simply a product of two independent probabilities.

For example, suppose the probability of electron 1 hopping to point A is 45% and the probability of electron 2 hopping to point B is 20%. The probability of finding electron 1 at A and electron 2 at B is 0.45 × 0.2 = 0.09 = 9%. All we are doing here is using the logic that says that the chances of tossing a coin and getting ‘tails’ and rolling a dice and getting a ‘six’ at the same time is one-half multiplied by one-sixth, which is equal to (i.e. just over 8%).2

As the figure illustrates, there is a second way that the two electrons can end up at A and B. It is possible for electron 1 to hop to B whilst electron 2 ends up at A. Suppose that the chance of finding electron 1 at B is 5% and the chance of finding electron 2 at A is 20%. Then the probability of finding electron 1 at B and electron 2 at A is 0.05 × 0.2 = 0.01 = 1%.

We therefore have two ways of getting our two electrons to A and B – one with a probability of 9% and one with a probability of 1%. The probability of getting one electron at A and one at B, if we don’t care which is which, should therefore be 9% + 1% = 10%. Simple; but wrong.

The error is in supposing that it is possible to say which electron arrives at A and which one arrives at B. What if the electrons are identical to each other in every way? This might sound like an irrelevant question, but it isn’t. Incidentally, the suggestion that quantum particles might be strictly identical was first made in relation to Planck’s black body radiation law. A little-known physicist called Ladislas Natanson had pointed out, as far back as 1911, that Planck’s law was incompatible with the assumption that photons could be treated as identifiable particles. In other words, if you could tag a photon and track its movements, then you wouldn’t get Planck’s law.

If electrons 1 and 2 are absolutely identical then we must describe the scattering process as follows: initially there are two electrons, and a little later there are still two electrons located in different places. As we’ve learnt, quantum particles do not travel along well-defined trajectories, and this means that there really is no way of tracking them, even in principle. It therefore makes no sense to claim electron 1 appeared at A and electron 2 at B. We simply can’t tell, and it is therefore meaningless to label them. This is what it means for two particles to be ‘identical’ in quantum theory. Where does this line of reasoning take us?

Look at the figure again. For this particular process, the two probabilities we associated with the two diagrams (9% and 1%) are not wrong. They are, however, not the whole story. We know that quantum particles are described by clocks, so we should associate a clock with electron