Quantum Physics 2: Multi-Particle Waves

Quantum Physics:
Fields, Waves, and Particles
Multi-Particle Waves

Welcome to the second part in our series about quantum physics! If you haven’t read the first part yet, I highly recommend it, as we will build upon the concepts we learned there. This discussion also takes a non-reductionist view, so you may want to take some time to digest last week’s discussion of object metaphysics before reading this one.

Image found here. Cropped.

To recap, last time we discussed how the universe contains a number of overlapping quantum fields, each one of which is present throughout all space. These fields contain quantities like momentum, energy, and electric charge, which travel around the fields in waves. The fields can trade this information through interactions, and the probability of interaction is correlated with the amplitude of the wave. And finally, there is a smallest possible amount a wave in a quantum field can interact by, and this amount must interact all at once and all in the same place. That “smallest amount of interaction” is what we call a particle.

In this post, when we say “particle,” it is understood that we are talking about collections of information within a wave, not little balls bouncing around.

Quantum Superposition

When the waves of two particles overlap, they add together in superposition. When in superposition, two particles are not two separate waves that happen to be in the same place; they are one wave with two particles’ worth of information.

For example, let’s look at the simplest kind of particle, photons. They are simple because they do not interact with other photons. Imagine two photons on a collision course. Before the intersection, they are flying along as normal. After they have crossed paths, they continue to fly along as if nothing happened.

Remember, photons are waves in the electromagnetic field, and there is only one electromagnetic field. At the moment of intersection, when the two photons are in the same place, their wave amplitudes add together to form a superposition wave that contains the information of the two photons, but is not itself a photon. This information causes the wave to split once again as if they had never joined in the first place.

A superposition wave. The final wave contains the information of the first two, yet it is still a single wave. Image found here. Cropped for better framing.

The key concept here is that when two or more particles are in a superposition wave, they aren’t really two particles, they are one wave with two particles’ worth of information. This is where the reductionist view causes problems, and it is the key point of today’s entire discussion. So if you don’t feel like you understand it well, I would strongly recommend lingering on this section, and perhaps rereading last week’s sections on reductionism, holism, and associative equivalence, until you do.

Bosons and Fermions

There is no limit to the number of photons that can superimpose in the same place, but there is for electrons. Electrons cannot superimpose over one another in the same total state. I do not mean that they repel each other due to their negative electric charges. It’s deeper than that.

In the lingo, a particle’s “state” is the total of its information about its properties: its position, momentum, energy, etc. Two electrons can have the same values in some of these, for instance, position and energy, but there must be at least one property that is different between the two. If you try to plug into the Schrodinger equation a superposition wave that contains two electrons with all of the same properties, in the exact same state, you end up with a contradiction, like 0 = 1. This is called the Pauli exclusion principle because electrons exclude other electrons from being in the same state.

Electrons and photons illustrate two categories all particles fall into. Bosons can exist in the same state in the same place. Fermions cannot. Photons are a type of boson, and electrons are a type of fermion. If this is confusing, don’t worry, it will make more sense after we have looked at some examples.

Lasers

Image found on Wikipedia.

We all know lasers, beams of single-color light packed so tightly that the bright spot where it lands is about the same diameter as the aperture it emerges from. A laser is a very strong coherent electromagnetic wave traveling in the same direction. This can be thought of as a vast number of photons all packed into the same place, superimposing on one another, and giving the wave a very high amplitude.

The higher intensity of the laser, the higher the number of photons superimposing into the wave. Can you guess what the limit is for how many photons can be packed into a single laser beam? Because photons are bosons, they never crowd each other out. Thus, we can keep increasing the power until the concentration of the light is so high that its energy creates a black hole! Don’t worry, though, that would take over a billion times more power than the entire world puts out in a year, all concentrated into one laser beam. That’s a lot of superposition.

Atoms

Each element on the periodic table has a smallest unit, an atom. Atoms are formed when electrons bind to positively charged nuclei. The possible states an electron can have within an atom are quantized; there are only certain specific states allowed, anything else gives a contradiction when put into the Schrodinger equation. If this is confusing to you, I recommend reviewing the pixel analogy from Quantum Physics part 1.

Because electrons are fermions, all electrons in an atom must be in different states. The three things contributing to an electron’s state in an atom are energy, angular momentum, and spin. If you have taken chemistry classes, you have probably heard of the electron states by another name: orbitals. The states with 0 angular momentum are called s orbitals, the states with the smallest non-0 angular momentum are called p orbitals, and then come the d orbitals and the f orbitals. Most of the time, the electrons are in the lowest available energy states.

This image shows the wave modes in the electron field. In real atoms, these get added together in a superposition wave containing all the electrons’ worth of information.

Each orbital letter type has specific energy levels allowed to it. The lowest possible energy for an electron in an atom is the 1s orbital, and the second is the 2s orbital. Then come the three 2p orbitals. There are no 1p orbitals, because trying to put them into the math gives us contradictions. There are one of each s orbital, three of each p orbital, five of each d orbital, and seven of each f orbital. And because electrons have two possible spins, there can be two electrons in each orbital.

Electron Spin

We mentioned something mysterious in the previous section: electron spin. What is it? It’s not angular momentum, as that is a different property. Spin is the property which determines how a particle interacts with magnetic fields. The term is confusing, because nothing is actually spinning; the electron is a spread-out wave with no central point for an axis. It is called spin for historical reasons.

In this section, we will talk about magnetic fields as if they are separate objects from one another, because that makes spin much simpler to explain. Keep in mind, however, that it is more true to say there is one magnetic field with different strengths throughout the universe, and that it is a part of the electromagnetic field.

When an electron interacts with a magnetic field, there are two possible outcomes: the electron’s own magnetic field could be aligned with the external magnetic field, or it could be aligned opposite. These are called “spin up” and “spin down.” Other alignments are impossible, because they give mathematical contradictions.

If you pick an axis, you might think an electron has a set spin, up or down, along that axis. However, it does not. Remember how we talked about electron waves, and the probability of interaction being proportional to the wave’s amplitude? The same thing is true for its spin. If the electron has not yet interacted, then it has an amplitude for spin up and an amplitude for spin down. Just like the electron does not have a single definite position before it interacts, it also does not have a definite spin.

Non-Spatial Probability Amplitudes

Let’s not let what I just said slip by. In an electron wave, there is a position component and a spin component. The position has an amplitude, and the spin also has an amplitude. If the electron interacts in such away that its spin is not involved, the position component of its wave collapses, but the spin component does not. This means an electron can move around and interact with other objects, but keep its spin in a non-determined state.

If I am not mistaken, the wave function of a particle has components for all properties that have more than one value, whether it feels intuitive to conceptualize them as waves or not. Polarization of light is the other well-known example. In a particle interaction, only the components of the wave function for the properties involved in the interaction collapse. The rest remain in superposition.

This means that if something interacts with an electron by its electric field, and not by its magnetic field, its wave will collapse to the position of the interaction, but its spin will still be undetermined.

Quantum Entanglement

Now that we have looked at superposition and the fact that each of a quantum wave’s qualities has its own probability amplitude, we are primed for one of the coolest and most famous aspects of quantum physics: entanglement.

Suppose two electrons are in a helium atom. What we have is a single wave with two electrons’-worth of information. Two units of interaction ability, and two opposite spins. Now we remove the wave from the atom and separate it into two, each with one unit of interaction ability; we can comfortably say that electron A is over there and electron B is over here.

However, if we choose our interactions with the electrons such that their spins are not involved, then the spins are still in superposition. As far as the spin is concerned, the two electrons are still part of the same wave. The only spin information this wave has is that there are two spins, and they are opposite. Which electron has which spin has not been determined, and their probability functions have not collapsed!

This phenomenon, when the position components of a multi-particle wave have separated, but the components of one or more of their other properties has not, is quantum entanglement.

What does this mean? What effects does entanglement have on human experiences? When we measure—cause an interaction with—the entangled property of one of the particles, we know what the other one will be when we measure it too. If we measure the spin of electron A, then we know before measuring the spin of electron B that it will be the opposite.

Most people wonder how the measurement of one electron can affect the properties of another electron instantaneously, ignoring the speed of light. Even Einstein was uncomfortable with it, calling it “spooky action at a distance.” But here’s the catch: the measurement of an entangled property does not affect the other particle. There is no causation between quantum entangled properties, only correlation. If we view entanglement in terms of superposition waves, we find it is not spooky, and it is not action at a distance.

We humans feel that if two things are guaranteed to correlate, then there must be some common cause. Either information is being transferred instantaneously from one particle to the other, or there is some kind of unmeasurable information that determined the outcome when the two particles were together. However, this is nothing more than a human assumption. There is a reason why the spins correlate, but not a cause. The reason entangled properties correlate is because if they didn’t, there would be a contradiction in the math. That is sufficient to make it true. Causation is found almost everywhere within reality, but non-contradiction is absolute.

If you aren’t sold on the connection between math and reality, check out the four-point argument I make at the end of last year’s post on the subject.

A common question people ask is whether quantum entanglement can be used for faster-than-light communication. There has been a lot of discussion about this in the literature, but the bottom line is that no, it cannot. We discussed one of the most compelling reasons to me in a previous post about the relationship between faster-than-light travel and time travel, the paradox that there is no objective way to determine whether the message would go from A to B or B to A.

So there you have it. Between this post and the one that came before, quantum physics explained in 4,500 words. If you understand it, then congratulations! You understand the basics of quantum physics as well as the experts, and perhaps even better than some. You also have a little insight into how the world of our experience emerges from it. If you have questions, feel free to ask them in the comments. There are still things we have not talked about, such as quantum computers, which are interesting enough to get their own discussion. Also, if you take what is said in these two posts at face value, it is known as the Copenhagen Interpretation. There is an alternative view called the Many-Worlds interpretation, which I go back and forth on, as you can see in my argument for it and subsequent argument against it.

So yeah. Quantum physics explained from scratch so that non-experts can understand it. Take that, Feynman!