Fermions and Bosons

All fundamental particles in nature can be categorized as either a fermion or a boson.

Bosons have integral spin and can occupy the same "state". All of the force carriers (Photons, Gluons, etc.) and mesons (Pions, Kaons, etc.) are bosons.

Fermions have half integral spin and cannot occupy the same state (Pauli Exclusion Principle). Quarks (up, down, strange, etc.) and Leptons (Electrons, Muons, etc.), as well baryons (Protons, Neutrons, etc.) are all Fermions.

Proton Example:

A proton is a Baryon. Baryons by definition are made of three quarks, which are fermions. A proton is made up of two up quarks and a down quark (uud). Each quark is a fermion, so a combination of three quarks is a fermion as well. So a proton is a fermion.

So where do Fermions and Bosons come from? Why are there two types of fundamental particles? Is there a quantum mechanical explanation?

The Importance of Being Identical

In probability, things that are identical lead to different results than things that aren't. For instance, say you have a bag with two marbles. If the marbles are different colors, say red and blue, then the odds of pulling out a specific color, say blue, is 50%. However if both marbles are blue, in other words identical (in terms of color), then the probability of pulling out a blue marble is 100%.

I know that seems pretty obvious, but it's important to see that distinction because it's about to become much less clear. Classically if you have two identical things, you can tell them apart by their position. You could say, blue marble A is to the left of blue marble B. You could also tell them apart if you knew their momentum. You could say blue marble A has larger momentum than blue marble B.

So here's the problem, in Quantum Mechanics you don't know where a particle is exactly, nor its momentum exactly, so you're suddenly out of tricks to tell identical particles apart.

The Solution

In quantum mechanics, we don't talk about individual particles, we talk about the state of a system which could be 1 particle or many. This is because of the uncertainty principle (Here is an earlier blog entry where I talk about this). For a two particle system, the state can be defined as:

where r₁²=x₁²+y₁²+z₁² and r₂²=x₂²+y₂²+z₂² with r₁ corresponding to particle number 1 and r₂ particle number 2.

Thus to calculate the probability of the wavefunction we write:

|ψ(r₁,r₂)|²

But since the particles are identical, we can switch the particles and we get the same probability as before which means:

|ψ(r₁,r₂)|² = |ψ(r₂,r₁)|²

Solving the equation above we get, we get

ψ(r₁,r₂) = ± ψ(r₂,r₁)

So what does it mean? Well it turns out it means something very, very important. It says that there are two different types of two identical particles. One in which switching the two particles results in no change in the wavefunction (symmetric) and one in which switching the particles adds a negative sign to the wavefunction (antisymmetric). So how can we represent these two different types of wavefunctions for identical particles? Lets try:

ψ_S = ¹/_√2(ψ(r₁,r₂) + ψ(r₂,r₁))
ψ_A = ¹/_√2(ψ(r₁,r₂) - ψ(r₂,r₁))

We can see that ψ_S = 0 when ψ(r₁,r₂) = - ψ(r₂,r₁), but ψ_S exists when ψ(r₁,r₂) = ψ(r₂,r₁). Also you can see for yourself that ψ_A = 0 when ψ(r₁,r₂) = ψ(r₂,r₁), and exists only when ψ(r₁,r₂) = - ψ(r₂,r₁) . Thus ψ_S is representative of the symmetric solution for identical particles and ψ_A is representative of the antisymmetric solution for identical particles.

Bosons and Fermions - What's in a name?

So now we know that if you have two identical particles, there are two types of wavefunctions which can describe the system. One is symmetric and the other is antisymmetric, but aside from being an interesting mathematical result, aren't we taking it a little too seriously? You'll calculate the same probability for both wavefunctions, which is what is what is measurable anyway. Wavefunctions are just abstractions, who cares.

Well, it turns out that it matters, a lot. Lets take a closer look at the symmetric and antisymmetric wavefunctions and consider a special case to understand why:

Here are the wavefunctions for symmetric and antisymmetric wavefunctions:

ψ_S = ¹/_√2(ψ(r₁,r₂) + ψ(r₂,r₁))
ψ_A = ¹/_√2(ψ(r₁,r₂) - ψ(r₂,r₁))

So what happens if r₁=r₂, in other words, the particles are in the same position (for those of you who object, relax, this is just a particular example of the more general "state").

ψ_S = ¹/_√2(ψ(r₁,r₁) + ψ(r₁,r₁)) = ²/_√2 ψ(r₁,r₁) = 2 ψ(r₁,r₁)
ψ_A = ¹/_√2(ψ(r₁,r₁) - ψ(r₁,r₁)) = 0

So what does this mean?

Well, lets remember that the modulus squared (|ψ²|) of a wavefunction is related to the probability. Lets also remember that |0²| = 0. Lastly lets remember that there is a law called the Conservation of Probability which says that probability cannot be created or destroyed.

So lets say you've got two identical particles in the universe, described by a nonzero state. In the symmetric case:

ψ_S = ¹/_√2(ψ(r₁,r₂) + ψ(r₂,r₁))

At r₁=r₂ (when the two particles are at the same spot), the probability is nonzero and conserved, everythings ok.

However,

In the antisymmetric case:

ψ_A = ¹/_√2(ψ(r₁,r₂) - ψ(r₂,r₁))

At r₁=r_2, (when the two particles are in the same spot), the probability goes to zero, which means that probability is not conserved. Since conservation of probability is a law, that means only one thing, identical antisymmetric particles can't ever exist at the same location (or more generally "in the same state") at the same time. This is called the Pauli exclusion principle.

Location, Location, Location - How the statistics are effected

Because of this unique difference between the ψ_A and ψ_S , there are dire consequences to the statistics that arise when the number of particles increase. ψ_S follow Bose-Einstein statistics which is why they are called Bosons. ψ_A follow Fermi-Dirac statistics and thus are called Fermions. Fermions follow the Pauli Exclusion Principle which says that two identical fermions cannot occupy the same state.

Certain Uncertainty

So there you have it. The uncertainty principle creates a situation that can only be solved by two types of fundamental particles, fermions and bosons. Those particle types follow specific statistics that gives rise to matter and forces. If there wasn't uncertainty, then we would be able to distinguish between fundamental particles and the antisymmetric two particle state, fermions, wouldn't exist. No electrons, protons, neutrons, and countless other particles. It's also worth mentioning that if there was only one particle, none of these issues come up, since the properties of Fermions and Bosons are only relevant when there are more than one particle. The addition of a second particle has led to symmetry breaking.

That's all for now. Remember if you'd like something derived or would like to have a subject covered in this blog, please email me your suggestions.

Roger

Special thanks to Wikipedia and Quantum Mechanics by B.K. Agarwal and Hari Prakash.