Fermions
Fermions are particles with half-integer spin (1/2 ħ, 3/2 ħ, 5/2 ħ, etc.). They serve as the fundamental building blocks of matter. Fermions can be divided into two groups – leptons and quarks. The electron, for instance, is a lepton. The proton and the neutron, however, do not belong to either of the two groups, as they are not elementary particles – both of them are made up of three quarks. Nevertheless, they are still considered fermions. In fact, all composite particles that consist of an odd number of fermions also belong to fermions.
As we have learned in the previous chapter, all fermions have an antisymmetric wave function. This may seem irrelevant, but the opposite is true. Antisymmetric wave functions bring far-reaching consequences in the form of the Pauli exclusion principle.
If we take a look at the structure of an atom, we find out that each atomic orbital is occupied by two electrons at most. This is somewhat peculiar, since everything in the universe has a tendency to stay on the lowest possible energy level. We may notice this when observing the behavior of object in the gravitational field of the Earth – objects fall down to decrease the value of their potential energy. But electrons seem to just ignore this rule entirely – otherwise they would all gather in the orbital with the lowest energy. What prevents them from doing so?
The Pauli exclusion principle states that no two fermions can be in the same quantum state, which means that each fermion must have at least one property (spin, momentum, etc.) different from all the other fermions. Why? The Pauli exclusion principle is associated with antisymmetric wave functions of fermions.
Let us consider two electrons that are described by a combined wave function ψ(1,2). Recall that when swapping the electrons, the sign of the wave function is changed due to its antisymmetric nature: ψ(1,2) = −ψ(2,1) . But also recall that any particle can be in all possible states at once due to the superposition principle, which means that if the given electrons can be described by the wave function ψ(1,2) as well as the wave function ψ(2,1), they are in a superposition of both of these wave functions. This superposition looks as follows:
𝛙 = 𝛙 (1,2) − 𝛙 (2,1)
But what does it have to do with electrons inside of an atom? Electrons belong to fermions; the Pauli exclusion principle therefore applies to them. If all electrons gathered in the orbital with the lowest energy, they would violate this crucial principle, as they would all be in the same quantum state. But there is one more crucial fact to be explained – why are there at most two electrons in each orbital and not just one?
This phenomenon can be explained using spin. The spin of an electron can take two different values: 1/2 or − 1/2. When electrons are in the same orbital, they have the same amount of energy, but they still have different spins – one of them has a spin of 1/2, the other one has a spin of − 1/2. This way, they do not violate the Pauli exclusion principle, as different spins mean different quantum states. However, no other electron can be found in this orbital, because it would inevitably be in the same quantum state with one of the original electrons.
The existence of the Pauli exclusion principle is crucial for stable structures to form. If it did not exist, the universe would be a widely different place. Molecules, for instance, would not form, since atoms would simply not be able to bind to each other.