Quantum Mechanics/Fermions and Boson

The fermions are the particles that obey Pauli's exclusion principle, which states that two particles can not have the same quantic numbers. They also have non integer spin. The fermions are divided in two groups (by the standard model) : leptons and quarks. Some well known fermions are: the electron, muon, tau. They are described by the statistical laws stated by Fermi and Dirac.

The bosons do not obey Pauli's exclusion principle. They have integer spin (or zero). Some common bosons are: the photon, the graviton and the He 4 nuclei; mesons are included here. Bosons are described by statistical laws stated by Bose and Einstein.

Identical Particles

Fermions and bosons arise from the theory of identical particles. Consider two cars. Even if they are the same make and model, you can be fairly sure that there are tiny differences that make each car unique. If even that fails, you know which car is which based on which car is where. But electrons have no similar identifying marks. They only have simple properties, such as intrinsic spin, intrinsic parity, electric charge, and the like. To make it worse, they also may lack well-defined positions. If the wave functions of two electrons mix, when you force those functions to collapse through direct observation, which electron is which?

For a two particle system, if the two particles are not identical (i.e., are of different types), and their Hamiltonian is separable, we can write down their wave function simply. Let us assume that we have one particle in state a, and the other particle in state b. Since the particles are distinguishable, we can tell which particle is which.

$\Psi (\mathbf {r} _{1},\mathbf {r} _{2})=\Psi _{a}(\mathbf {r} _{1})\Psi _{b}(\mathbf {r} _{2})$

This is a simple enough exercise. But what if we have the same situation with identical particles? Then we can't tell the difference between having particle 1 in state a, or having it in state b. All we know is that there are two particles, one is in state a and one is in state b. To account for both of those states, we have to write the total state as a superposition of those two states.

The easiest way to write this, of course would be to write:

$\Psi _{+}(\mathbf {r} _{1},\mathbf {r} _{2})={\frac {1}{\sqrt {2}}}[\Psi _{a}(\mathbf {r} _{1})\Psi _{b}(\mathbf {r} _{2}){+}\Psi _{a}(\mathbf {r} _{2})\Psi _{b}(\mathbf {r} _{1})]$

But as we should know from our study of Quantum Mechanics so far, things are not quite this easy. There are actually two different final states that we can end up in, and this is only the first one. The second one is actually:

$\Psi _{-}(\mathbf {r} _{1},\mathbf {r} _{2})={\frac {1}{\sqrt {2}}}[\Psi _{a}(\mathbf {r} _{1})\Psi _{b}(\mathbf {r} _{2})-\Psi _{a}(\mathbf {r} _{2})\Psi _{b}(\mathbf {r} _{1})]$

The first expression is the symmetric expression, and the second is the anti-symmetric expression. That difference between + and - looks fairly insignificant, but it is perhaps the fundamental difference between elementary particles. To illustrate this, let us examine the peculiar case when we have two particles in the same state. If a and b are identical, then the symmetric expression is written as:

$\Psi _{+}(\mathbf {r} _{1},\mathbf {r} _{2})={\frac {1}{\sqrt {2}}}[\Psi _{a}(\mathbf {r} _{1})\Psi _{a}(\mathbf {r} _{2})+\Psi _{a}(\mathbf {r} _{2})\Psi _{a}(\mathbf {r} _{1})]$

an expression that is fairly easy to simplify. However, the anti-symmetric expression simplifies to something completely different:

$\Psi _{-}(\mathbf {r} _{1},\mathbf {r} _{2})={\frac {1}{\sqrt {2}}}[\Psi _{a}(\mathbf {r} _{1})\Psi _{a}(\mathbf {r} _{2})-\Psi _{a}(\mathbf {r} _{2})\Psi _{a}(\mathbf {r} _{1})]=0$

Because those two wave function expressions cancel out, the anti-symmetric wave function obeys a fundamental rule, no two identical, anti-symmetric particles can be in the same state. If they were, their wave functions would cancel out to zero, a physical impossibility! This is a theoretical derivation of the famed Pauli Exclusion Principle, whose effect on the world of Chemistry and Particle Physics has been tremendous.

Particles that are symmetric are referred to as bosons. If you examine Fermi and Bose statistics, you will find that bosons tend to congregate in the ground state, because there are no barriers to this accumulations. Anti-symmetric particles are known as fermions, and their inability to stay in the same state as their neighbors has lead to the intricate structure of electron valence orbitals.

Practically speaking, the difference between fermions and bosons is greatest when dealing with elementary particles. There bosons are the force carriers, particles like the graviton and the photon that are the mechanism by which particles interact. Fermions become the building blocks, out of which all matter is made. This distinction is one of the foundations of the Standard Model, a major topic which will be covered later in this book.