So far, we have only been applying quantum mechanics to one particle at a time. Interesting things happen when you add even one more particle to the mix, especially if it is the same kind of particle.
All electrons, for example, are identical in every way. If you have two electrons to play with, the only way to tell them apart would be to keep them widely separated. If you kept one in the living room, and the other in the bedroom at the other end of the house, then you could call one of them Romulus and the other Remus, and you could keep track of which was which.
If Romulus and Remus ever get close together, however, all bets are off. Remember, the position of an electron has some unavoidable uncertainty. So if the wave packets representing your two electrons ever overlap, it will no longer be possible to determine which one was Remus and which one was Romulus!
Why does this matter? Well, because quantum physics can only make predictions about what can possibly be known. If it is impossible to tell the difference between two electrons in a system, quantum mechanics must make the same prediction whether Remus is here and Romulus there, or vice versa. You have to use a two-electron wave function that gives you the right answer if you were to swap any label used to identify the component states. Any two-particle wave function that will be subject to the same potential energy field has to give the same results when the two particles are switched. Otherwise, it can’t represent reality.
We’ll spare you all the math, but it turns out that there are exactly two ways to write wave functions that will have this property. We call these two symmetric and anti-symmetric wave functions.
DEFINITION
A symmetric two-particle wave function does not change at all when the identities of the two particles are switched.
An anti-symmetric two-particle wave function changes its sign (e.g., from positive to negative) when the two particles are switched, but nothing else about it changes. Since only the square of the wave function is observable, probabilities and expectation values will not be changed.
What does this have to do with particle spin, the alleged topic? The quantum math for particles with spin reveals a surprising relationship. Given the complexity of the math involved, part of the surprise is that the result is so simple to state: identical particles with whole number spins must have symmetric wave functions, while identical particles with half-integer spins (1⁄2, 3⁄2, 5⁄2, etc.) must have anti-symmetric wave functions.
This simple result has far-reaching consequences for all sorts of many-particle quantum systems. The two types of particles behave in very different ways when they get together, so we give them handy names. The particles with whole-number (i.e., integer) spin are called bosons, and all particles with half-integer spin we call fermions. These are not new subatomic particles that we are talking about, but rather categories of particles. Electrons, which have a spin of 1⁄2, fall into the fermion category while photons turn out to be bosons.
Remember that this thing we call spin in quantum mechanics is an intrinsic property of all fundamental particles. There are a lot of possible values for the amount of spin, but they are all either whole numbers (including 0, 1, 2, and so on) or numbers halfway between the whole numbers. As far as we know, all fundamental particles in nature fall into one category or the other.
DEFINITION
Any particle with whole number spin is considered a boson. A wave function representing two bosons must be symmetric when the particles switch places.
Any particle with half-integer spin is considered a fermion. A wave function which is a combination of two fermions must be anti-symmetric when the two particles switch places.
This categorization and the corresponding behaviors are predicted by quantum physics and have now been verified in numerous experiments. It reflects something truly fundamental about the microscopic world.
Leave a Reply