Matter and interactions in quantum mechanics

Here’s another post on how quantum mechanics is weird.

Let’s consider the operation of swapping around different quantum particles. If the coordinates of the first particle are denoted x₁ and the second denoted x₂, we’ll be interested in the transformation:

Ψ(x₁, x₂) → Ψ(x₂, x₁)

Let’s just give this transformation a name. We’ll call it S, for “swap”. S is an operator that is defined as follows:

S Ψ(x₁, x₂) = Ψ(x₂, x₁)

Now, clearly if you swap the particles’ coordinates two times, you get back the same thing you started with. That is:

S² Ψ(x₁, x₂) = Ψ(x₁, x₂)

This implies that S² is an operator with eigenvalue +1. What does this tell us about the spectrum of eigenvalues of S?

Suppose S has an eigenvalue λ. Then, from the above, we see:

S² Ψ = Ψ
S SΨ = Ψ
S λΨ = Ψ
λ SΨ = Ψ
λ² Ψ = Ψ

So λ = ±1.

In other words, the only possible eigenvalues of S are +1 and -1.

This tells us that the wave functions of particles must obey one of the following two equations:

SΨ = +Ψ
or
SΨ = -Ψ

Said another way…

Ψ(x₁, x₂) = Ψ(x₂, x₁)
or
Ψ(x₁, x₂) = -Ψ(x₂, x₁)

This is pretty important! It says that by the nature of what it means to swap particles around and the structure of quantum mechanics, it must be the case that the wave function obtained by switching around coordinates is either the same as the original wave function, or its negative.

This is a huge constraint on the space of functions we’re considering; most functions don’t have this type of symmetry (e.g. x² + y ≠ y² + x).

So we have two possible types of wave functions: those that flip signs upon coordinate swaps, and those that don’t. It turns out that these two types of wave functions describe fundamentally different kinds of particles. The first type of wave functions describes fermions, and the second type describes bosons.

All particles in the standard model of particle physics are either fermions or bosons (as expected by the above argument, since there are only two possible eigenvalues of S). Fermions include electrons, quarks, and neutrinos. Bosons include photons, gravitons, and the Higgs boson. This sampling of particles from each category should give you the intuitive sense that fermions are “matter-like” and bosons “force-like.”

Of course, the definition of fermions and bosons we gave above doesn’t reference matter-like or force-like behavior. Somehow the qualitative difference between fermions and bosons must arise from this simple sign difference. How?

We’ll start exploring this by examining the concept of non-separability of wave functions.

A wave function is separable if it is possible to write it as a product of two individual wave functions, one for each coordinate:

Ψ(x₁, x₂) = Ψ₁(x₁) Ψ₂(x₂)

This is a really nice property. For one thing, it allows us to solve the Schrodinger equation extremely easily by using the differential equations method of variable separation. For another, it tells us that the wave function is simple in a way. If we’re interested in only one set of coordinates, say x₁, then we can easily disregard the whole rest of the wave function and just look at Ψ₁.

But what does it mean? Well, if a wave function is separable, then it is possible to sensibly ask questions about the properties of individual particles, independent of each other. Why? Because you can just look at the component of the wave function corresponding to the particle you’re interested in, and the rest behaves just like a constant for all you care.

If a wave function is non-separable (i.e. if there aren’t any two functions Ψ₁ and Ψ₂ for which Ψ(x₁, x₂) = Ψ₁(x₁) Ψ₂(x₂)), then the story is trickier. For all intents and purposes, it loses meaning to talk about one particle independent of the other.

This is hard to wrap our classical brains around. If the wave function cannot be separated, then the particles just don’t have positions, momentums, and so on that can be described independently.

Now, it turns out that most of matter is composed of non-separable wave functions. Pretty much any time you have particles interacting, their wave function cannot be written as a product of independent particles. A lot of the time, wave functions are very approximately separable (consider the wave function describing two electrons light years away). But in such cases, when we talk about the two electrons as two distinct entities, we’re really using an approximation that is not fundamentally correct.

Now, this all relates back to the fermion/boson distinction in the following way. Suppose we had a system that was described by a separable wave function.

Ψ(x₁, x₂) = Ψ₁(x₁) Ψ₂(x₂)

Now what happens when we apply the swap operator to Ψ?

S Ψ(x₁, x₂) = Ψ(x₂, x₁) = Ψ₁(x₂) Ψ₂(x₁)

As we’d expect, we get the particle described by x₂ in the wave function initially occupied by the particle described by x₁, and vice versa. But now, our system must obey one of two constraints:

Since SΨ = ±Ψ,

Ψ₁(x₂) Ψ₂(x₁) = Ψ₁(x₁) Ψ₂(x₂)
or
Ψ₁(x₂) Ψ₂(x₁) = – Ψ₁(x₁) Ψ₂(x₂)

Let’s take the second possibility for fermions first. It turns out that this constraint can never be satisfied. Why? Look:

Suppose f(x) g(y) = – f(y) g(x)
Then we have:  f(x)/g(x) = -f(y)/g(y)

Since the left is a pure function of x, and the right of y, this implies that they are both equal to a constant.
I.e. f(x)/g(x) = k, for some constant k.
But then f(y)/g(y) = k as well.

Thus k = -k,
which can only be true if k = 0.

This argument tells us that the only function that satisfies (1) Ψ describes a fermion and (2) Ψ is separable, is Ψ(x₁, x₂) = 0. Of course, this is not a wave function, since it is not normalizable. In other words, fermion wave functions are always non-separable!

What about boson wave functions? The equation

Ψ₁(x₂) Ψ₂(x₁) = Ψ₁(x₁) Ψ₂(x₂)

does have some solutions, but they are highly constrained. Essentially, for this to be true for all x₁ and x₂, Ψ₁ and Ψ₂ must be the same function.

In other words, bosons are separable only in the case that their independent wave functions are completely identical.

So. If fermions cannot be described by a separable wave function, how can we describe them? We can be clever and notice the following:

While Ψ(x₁, x₂) = Ψ₁(x₁) Ψ₂(x₂) does not obey the requirement that SΨ = -Ψ,
Ψ(x₁, x₂) = Ψ₁(x₁) Ψ₂(x₂) – Ψ₂(x₁) Ψ₁(x₂) does!

Let’s check and see that this is right:

S Ψ(x₁, x₂)
= S [Ψ₁(x₁) Ψ₂(x₂) – Ψ₂(x₁) Ψ₁(x₂)]
= Ψ₁(x₂) Ψ₂(x₁) – Ψ₂(x₂) Ψ₁(x₁)
= -Ψ(x₁, x₂).

Aha! It works.

Now we have a general description for fermion wave functions that does not violate the swap constraints. We can do something similar for bosons, giving us:

Fermions
Ψ(x₁, x₂) = Ψ₁(x₁) Ψ₂(x₂) – Ψ₂(x₁) Ψ₁(x₂)

Bosons
Ψ(x₁, x₂) = Ψ₁(x₁) Ψ₂(x₂) + Ψ₂(x₁) Ψ₁(x₂)

(Note: These are not all possible fermion/boson wave functions. They only capture one set of allowable wave functions.)

Now, let’s notice a peculiar feature of the fermion wave function. What happens if we ask for the probability amplitude of the two particles being at the same place at the same time? We find out by taking the fermion equation and setting x₁ = x₂ = x:

Ψ(x, x) = Ψ₁(x) Ψ₂(x) – Ψ₂(x) Ψ₁(x)
= 0

More generally, we can prove that any fermion wave function must have this same property.

SΨ(x₁, x₂) = -Ψ(x₁, x₂)
Ψ(x₂, x₁) = -Ψ(x₁, x₂)
Ψ(x, x) = -Ψ(x, x)
Ψ(x, x) = 0

Crazy! Apparently, two fermions cannot be at the same position. And since wave functions are smooth, fermions will generally have a small probability of being close together.

This is the case even if the fermions are interacting by a strong attractive force. It’s almost as if there is an intrinsic repulsive force keeping fermions away from each other. But this would be the wrong way to think about it. This feature of the natural world is not discovered by exploring different possible forces and potentials. Instead we got here by reasoning purely abstractly about the nature of swapping particles. We are a priori guaranteed this unusual feature: that if fermions exist, they must have zero probability of being located at the same point in space.

The name for this property is the Pauli exclusion principle. It’s the reason why electrons in atoms spread out in ever-larger orbitals around nuclei instead of all settling down to the nucleus. It is responsible for the entire structure of the periodic table, and without it we couldn’t have chemistry and biology.

(Quick side note: You might have thought of something that seems to throw a wrench in this – namely, that the ground state of atoms can hold two electrons. In general, electron orbitals in atoms are populated by more than one electron. This is possible because the wave function has extra degrees of freedom such as spin, the details of which determine how many electrons can fit in the same spatial orbital. I.e. two fermions can have the same spatial amplitude distribution, so far as they have some other distinct property.)

Mathematically, this arises because of interference effects. As the two fermions get closer and closer to each other, their wave functions interfere with one another more and more, turning their joint probability to zero. Fermions destructively interfere.

And bosons? They constructively interfere! At x₁ = x₂ = x, for bosons, we have:

Ψ(x, x) = Ψ₁(x) Ψ₂(x) + Ψ₂(x) Ψ₁(x)
= 2 Ψ₁(x) Ψ₂(x)

In other words, while fermions disperse, bosons cluster! Just like how fermions seemed to be pulled away from each other with an imaginary force, bosons seem to be pulled together! (Once again, this is only a poetic description, not to be taken literally. There is no extra force concentrating bosons, as can be evidenced by the straight trajectory of parallel beams of light).

I’m not aware of a name for this principle to complement the Pauli exclusion principle. But it explains phenomena like lasers, where enormous numbers of photons concentrate together to form a powerful beam of light. By contrast, an “electron laser” that concentrates enormous numbers of electrons into a single beam would be enormously difficult to create.

The intrinsic tendency for fermions to repel each other leads them to form complicated structures that spread out through space, and give them the qualitative material resistance to objects passing through them. You just can’t pack them arbitrarily close together – eventually you’ll run out of degrees of freedom and the particles will push each other away.

Bosons, on the other hand, are like ghosts – they can vanish into each others wave functions, congregate together in large numbers and behave as a single entity, and so on.

These differences between fermions and bosons pop straight out of their definitions, and lead us to the qualitative differences between matter and forces!