As You Like It

12 July 2013

The quantum equation Brian Koberlein
The quantum equation

Suppose you lose your baseball in the woods, and you and your friend decide to look for it. You know that either you will find it, or your friend will (or it will remain lost). Assuming the ball hasn’t been damaged, it won’t be the case that you and your friend each find half of the ball, or that you both find the ball in different locations. There is only one ball, and it has an exact location, even if you don’t know where it is. It can only be found once, and only by one of you.

A wave behaves differently, however. Suppose the ball is dropped into the center of a pond, and you and your friend stand on the shore to look for the wave. You can be on one side of the pond, your friend on the other side, and you can both see the wave wash upon the shore. You can even both see the wave wash ashore at the same time, because part of the wave is near you, while another part is near your friend.

This is a fundamental difference between particles and waves. Particles are local, with a specific location. Particles are either found in their entirety, or not at all. Waves on the other hand are nonlocal. They are spread over a region, and their influence can be seen in multiple locations at once. You will never detect the entire wave in a single spot. In our everyday experience, the two are completely different. Particles don’t act like waves, and waves don’t act like particles. But last time we saw that things like electrons and light do behave both as particles and as waves. This doesn’t make sense, so let’s look at an experiment that might clear things up.

It is known as the double slit experiment, and it goes basically as follows. Suppose you have a wall on the far end of a room. Against this wall you could place a screen or an array of detectors or whatever you need to detect light, or electrons, or whatever. This is where you observe the outcome of your experiment. Between you and the far wall we place a barrier with two small openings in it. These two openings are typically closely spaced slits, which is why this is called the double slit experiment. The idea is that the barrier will block anything heading toward the far wall unless it passes through the two openings. Next to you is device that can create a beam of light, or a beam of electrons, or whatever we want to examine.

To do the experiment we aim the device at the barrier. The light or electrons hit the barrier. Whatever passes through the openings continues on to the far wall, where it is observed by the screen or array of detectors. What I’ve described might seem like an imaginary experiment, but it is really just a way to visualize real experiments. The real experiments can be much more complicated, but the principle is the same.

The key aspect of this experiment is that it can distinguish particles from waves. If particles are sent toward the barrier, then the detector should observe particles in two small spots, due to the particles passing straight through the two slits. If a wave is passed to the barrier, then a portion of the wave will pass through each slit, and continue spreading from there. The two spreading waves will overlap each other, and the detector will observe an interference pattern of overlapping waves. This is similar to what you might see if you dropped two stones into a pond. The ripples from the stones overlap to create an interference pattern.

So let’s see what happens when we do the experiment. If we shine a beam of light at the barrier, then the detector sees a wave interference pattern, thus light is a wave. But we also know that light is made of photons. Perhaps the wave is due to all those photons interacting with each other, and the wave effect is just an illusion. So let’s do the experiment again, but this time only do a single photon at a time. Fire a photon, detect a photon, fire the next photon, etc. This time our detector array observes a single photon each time. Each is observed as a particle. But as we observe more and more photons, we notice that they aren’t isolated to two spots. Instead, the photons are distributed in a pattern that looks like an interference pattern. In other words, each individual photon is a particle, but as a collective they are distributed in a wave-like pattern.

For a wave, the interference pattern occurred because a part of the wave went through one slit, and a part went through the other. The pattern occurs because of the interference between parts. Our experiment didn’t bother looking at which slit the photon went through, so let’s add another detector to observe which slit each photon goes through. We run the experiment again, one photon at a time, and this time we detect a photon as it goes through the left slit, the next one through the right, then left, right, right, left, etc. Now we know which slit each photon went through. At the far wall, we again detect one photon at a time, but this time the photons are only observed in two small spots, rather than being distributed in an interference pattern. In other words, their behavior has changed to act completely like particles.

If you do the same experiment with electrons, you get exactly the same result. It seems then that photons and electrons are always detected as particles (all or nothing), but they also have a wave behavior. This wave behavior is not due to their collective interactions (the way water molecules collectively act as a water wave). Instead each individual particle has a wave behavior that only appears when compared with other particles. Individually it just looks random. Furthermore, if you try to figure out how the particles use their wave properties (does a photon go through both slits?) then the wave aspect of the particles goes away. Put another way, the way you do your experiment changes the outcome of your experiment.

This is still really strange, but the result of these experiments can be described by the equation below. Here H is known as the operator, and is basically a mathematical description of what you are trying to observe. E is the outcome you observe, and psi (what looks like a trident) is the object you are observing. Psi is known as the wavefunction of your object. Using it you can make all the right predictions for the above experiments, but what it actually is depends on how you interpret it.

One common interpretation is known as the Copenhagen interpretation. In this view, the wavefunction describes the probability of finding a particle in a particular location. The object is in an indefinite, probabilistic state described by the wavefunction until it is observed. When it is observed, the wavefunction collapses, and the object becomes a definite particle with a definite location. This explains why the double slit experiment produces an interference pattern unless we observe which slit the particle goes through. If we don’t observe the object at the slits, it remains wavy, and produces an interference pattern. If we observe it at the slits, we collapse the wavefunction, and the object acts like a particle.

While it is somewhat easy to imagine photons and electrons as fuzzy wavefunctions that collapse into particles when you observe them, there are problems with the Copenhagen interpretation. It presumes a physical wavefunction that can’t be directly observed. It requires the wavefunction to collapse when “observed”, but is never clear on what an observation actually is. If the detector collapses the photon’s wavefunction, what collapses the detector’s. If the answer is that you do (since it’s your experiment), then what collapses you? This doesn’t really make sense.

There are several other interpretations for the equation below that I won’t get into. They each have their advantages and disadvantages. You can also simply not worry about the interpretation, and simply use the equation to predict your outcome. It is a perfectly accurate approach, even if it feels a bit unsatisfying.

But one thing quantum theory is not is magic. The double slit experiment is often proposed as a demonstration of the strange particle-wave duality that exists in nature. This is fine, but too often it is interpreted as being something mystical. You’ll often hear statements that the electron “knows” when it is being observed, or that it “decides” what to do based on how the experiment is done. The implication is that quantum particles are conscious of their surroundings, or have mystical properties that regular objects don’t. This is simply not true. Quantum particles simply do what they do, and it only seems strange to us because they are unfamiliar to our daily lives. To imply more than that is as ridiculous as saying the fall of a baseball means a tossed ball is thinking “I say! I do believe there is a large planet below me! I suppose I should start moving in a downward direction!”

The equation below is known as Schrodinger’s equation, and it is one of the first equations of modern quantum theory. Over the past 100 years we have expanded that theory into the most accurate scientific model ever devised. Quantum theory works. It is true. But it remains extremely strange and counterintuitive.

You can interpret that as you like.

Up Next: Maybe all this quantum weirdness is just due to a limitation of our knowledge. Maybe if we just devise better experiments we can discover some hidden variable that drives it all. Prepare to be disappointed in the final part of the series.