Quantum Mechanics for Dummies #1: Wave Nature of Matter

I got into an involved explanation of Quantum Mechanics a few days ago over at Bad Astronomy, and it reminded me of an idea for a series of posts I've had on the backburner for a while. In this series, my goal is to make Quantum Mechanics somewhat intelligible to those who haven't studied it and clear up some common misconceptions about it.

In this first post in the series, I'm going to discuss what Quantum Physicists call the wave nature of matter, and how a particle can act like both a particle and a wave. But before we go into that, I should explain by what we normally mean by "particle" and "wave."

First, in a classical sense, what is a particle? Particles are generally very small, and possibly infinitely small, being just a point in space. They can have numerous properties such as mass, electric charge, and magnetic moment, which determine how they act and interact in the universe. When they travel from place to place, they do so with a definite, linear path.

Second, what is a wave? Waves are a bit harder to explain, so I'll go with what Wikipedia says on it:

A wave is a disturbance that propagates through space or spacetime, often transferring energy. While a mechanical wave exists in a medium (which on deformation is capable of producing elastic restoring forces), waves of electromagnetic radiation (and probably gravitational radiation) can travel through vacuum, that is, without a medium. Waves travel and transfer energy from one point to another, with little or no permanent displacement of the particles of the medium (there is little or no associated mass transport); instead there are oscillations around fixed positions.

Waves also have properties associated with them, but they're generally different from particles. First, there's amplitude, which is, in a simple sense, the height of the wave. There's also frequency, which is how many oscillations a wave makes per second. Frequency and amplitude together determine the energy of the wave (with the energy being proportional to the frequency and the square of the amplitude). Classically, the amplitude and frequency of a wave are both continuous, so a wave with a given frequency can have any value of energy.

Waves also translate in space differently from particles. Instead of simply following straight lines, waves spread out. If a wave passes through a long, narrow corridor, it will spread out at wide angles when it leaves, unlike how a bunch of particles going through the corridor would act. In fact, waves act in the opposite manner to particles in this case as the thinner the corridor is, the more the wave spreads out upon leaving. This phenomenon is known as diffraction.

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Classically, light was always seen as a wave. It showed all the expected properties of waves, having a measurable frequency, diffracting, etc. But eventually a problem was found: Light of a given frequency could only have quantized energy. What this means is that essentially if you have light that's all of one frequency, there are only certain set values of its total energy that it can have. For instance, it might be allowed to have energy of 1.1 eV, 2.2 eV, 3.3 eV, and so on, but it could never have an energy of 1.5 eV or 0.5 eV unless some of it is at a different frequency.

This also meant that there was a minimum energy that light could have. If light were made out of particles (what we now call photons), this could be explained quite easily: Each particle would have energy equal to a constant times its "frequency," and they added together to form the total energy of the light. The problem was that particles classically couldn't have frequency.

So we were left with a contradiction, and had to form a new theory. Light had properties of both particles and waves. But was it particles that traveled like waves, or maybe waves that just happened to be quantized somehow? Further experiments were necessary to determine what exactly was going on.

The most famous of these experiments was the Double-Slit Experiment. In this experiment, light first diffracts out of one slit, allowing it to spread out and hit two more slits. The light that passes through each of these slits diffracts again, and the wave then hits a detector.

If a conventional wave goes through this, we see a strange interference pattern on the detector. This is cause by the waves coming from each slits being at different points along their wavefunctions. If one is at a peak and the other at a valley, the amplitudes cancel out and no light will appear at that point. If both are at peaks or valleys, then the amplitudes at together, and since we then square the amplitude to get the energy, we get four times the energy at this point as we'd get from a single wave, or twice what we'd get from two waves. Particles, on the other hand, show no interference patterns. This means that we can use a double-slit experiment to determine whether light is acting like a particle or a wave.

So, this experiment is then performed. A lot of light is shot out, and it does indeed show the interference pattern. So, if light is a bunch of particles, they can somehow interfere with each other, it would seem. We had the technology to decrease the emission rate of light low enough that only one photon was being sent out at a time, so this was the logical next step.

When we performed this experiment, the results were extremely surprising. When you plotted the frequency with which the photon would strike different points of the screen, it matched up with the interference pattern! Even single photons were acting like waves. This is something that just wasn't possible if you treated them like particles. The first problem was that particles wouldn't diffract like waves, but these photons were doing this. The second problem is that, even if particles could diffract, you would expect them to go through one of the two slits, and then diffract onto the detector. The pattern that appears should then be the some of the diffraction patterns from the two slits, but it was instead the interference pattern.

Then things got stranger. We tried firing things that we were pretty sure were particles through a double-slit experiment, such as electrons. They, too, showed a diffraction pattern. We went bigger and shot atoms through it. Same deal. Our record so far has been shooting Bucky Balls (spherical molecules of 60 carbon atoms) through it, and even they act like waves.

It was becoming cliché at this point, but there was an even stranger development still to come. We figured that if these particles were acting like particles, they had to be going through just one of the slits. We then set up detectors at both slits that would tell us if a particle was passing through it. We did so, and we got results from it: a 50/50 spread of particles between the two slits. But there was a problem. When the detector was on, the interference pattern went away! If we turned off the detector, the interference pattern appeared again. Things were seriously screwed up.

Take a few minutes to ponder this. It's taken scientists many decades, and most of them still don't have a good picture of what's actually going on that could cause this. There are a few theories out there, but none are very well accepted. I'm going to go into my personal interpretation of how this works, but remember that there are others.

When any particle is traveling, it does so as a probability wave, that is, a wave that represents the probability of the particle being at a certain point if we measure it. This isn't a theoretical wave (in my picture, at least), and the probabilities aren't simply an oversimplification like in Statistical Mechanics. Instead, the wave is an actual object, and the probability is a fundamental law of the universe.

This probability wave has properties like normal waves, even if it represents a "particle" like an electron. These properties include frequency and the value of the wavefunction at a point. Squaring the value of the wavefunction is what gives us the probability of it showing up in a certain area. Like other waves, if its path is split up, it can interfere with itself, causing an interference pattern in the probability it will show up in an area.

Now, from this description, the more statistically inclined of you might be wondering about one possible problem. Take a simplified case where a probability wave has a 50% chance of resolving into a particle in the right half of an area, and a 50% chance of resolving into a particle in the left half. Also, let's assume that the wave hits the entire area at the same instant of time. Shouldn't the probability distribution look something like the following?

Shows up in right only: 25%
Shows up in left only: 25%
Shows up in both: 25%
Shows up in neither: 25%

Well, the above distribution is only valid if the probabilities are independent. We know from experiments that this isn't the case; a particle will always resolve in exactly one spot. But the wave hits the entire surface at one instant, and Relativity tells us that the speed of light is an upper limit on data transfer. How does the left half of the wave know whether the particle has resolved in the right half, if information can't get there fast enough?

What I've described here is part of what's known as the EPR paradox. Somehow, for quantum mechanics to work the way it does, there must be some form of information transfer from one part of a probability wave to another so that particles don't randomly disappear or split into two. It would seem at first glance that this would violate Relativity, but this isn't quite so. The information you get from it resolving or not resolving in one area is no more than logical inference, and this is all the universe is doing as well. In addition to a basic law of randomness, the universe also seems to have a basic law of inference on the resolution of these waves, so that we end up with conservation of energy.

Well, that's enough for today. If there's anything in there that's still confusing, please leave a comment and ask for clarification on it; I'll be glad to explain.

Note to anyone who already knows some QM: Yes, I'm aware I came very close to talking about entanglement in the last couple of paragraphs, but I chose not to go into detail about it. This is simply because entanglement is getting its own entire post in this series.

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Other posts in this series:

Quantum Mechanics for Dummies #2: Observation