Quantum Mechanics for Scientists and Engineers

One aspect of quantum mechanics that is very different from the classical world is that particles can be absolutely identical – so identical that it is meaningless to say which is which. This “identicality” has substantial consequences for what states are allowed, quantum mechanically, and in the counting of possible states. Here, we examine this identicality, introducing the concepts of fermions and bosons and the Pauli exclusion principle that lies behind so much of the physics of materials.

Scattering of identical particles

Suppose we have two electrons in the same spin state, electrons that, for the moment, we imagine we can label as electron 1 and electron 2. We write the spatial coordinates of electron 1 as r1 and those of electron 2 as r2. As far as we know, there is absolutely no difference between one electron and another. They are absolutely interchangeable. We might think, because of something we know about the history of these electrons, that it is more likely that we are looking at electron 1 rather than electron 2, but there is no way by making a measurement so that we can actually know for sure at which one we are looking.

We could imagine that the two electrons were traveling through space, each in some kind of wavepacket. The wavepackets might each be quite localized in space at any given time.