This is the third in a series attempting to introduce the most basic elements of modern physics in a condensed form. The goal is to share some of the strangest knowledge we have, and in the process to immunize people against some of the bullshit that flies around in this field, e.g. on Huffpo. Our first story described a little about relativity, the bottom line being that Einstein did not claim to show that everything is relative, although he did show that the things that aren't relative aren't the ones you'd guess, and that the geometry of spacetime is a lot different from what our intuitions say. Our second story, preparing the ground for quantum mechanics, described some of the experimental proof that our world cannot be described by any local realist theory.
Today I'll try to introduce some of the content of quantum mechanics. The story will be a bit complex, and, unlike the previous installments, it will not be possible to stick strictly to points on which there's virtually unanimous agreement. Although there's no doubt about how to use quantum mechanics as a nuts-and-bolts working theory, the interpretation remains open. Any description uses some vocabulary with arguable implications for the interpretation, but I'll try to label any disputed points. I'll leave many open questions, in the hope that a few of you will struggle through far enough to raise objections and questions yourselves. As I wrote before, you may feel some discomfort.
In classical mechanics (including the relativistic kind) there are precise mathematical rules describing how the state of the world changes as a function of what its current state is. Many of you have seen an approximate piece of that, F=ma, in which the velocity of a particle of mass m changes (via acceleration a) as a function of the known features of its environment, the nearby things responsible for the force F. If you put together all the pieces, the collection of these rules determines the future based on the current state of things. There are two types of ingredients in that state. One is a collection of particles, things that have positions and velocities as well as other properties. The other is a collection of fields, like the electrical field, things which permeate all space, although with varying strengths. The particles give rise to fields, and the fields exert forces on the particles, in a big self-consistent system. We've now seen that no such picture can describe our actual universe.
In quantum mechanics, things are in some ways simpler. Despite what you may have heard, there is only one type of ingredient, space-permeating fields. Things like electrons, which you may have seen pictured as little dots, never behave like anything except spread-out fields, although the degree of spreading can vary. One can write the current state of things as a field (sometimes wave-like) in space representing not just one type of particle but all sorts of different interacting types. One odd feature, however, is that this field doesn't have definite values for variables that you think ought to have definite values. Objects are always smeared out in space. They also always have a smeared out range of velocities. Except in the most boring cases (cases where nothing is happening) they have a range of energies. They typically have a range of particle numbers.
Despite that unfamiliar feature, in quantum mechanics we still have a rule giving exactly how the state is changing in time as a function of what the state currently is. The rule is:
ihd|Ψ>/dt=2πH|Ψ> .
Here |Ψ> is just the name of the state, d|Ψ>/dt is the rate at which it is changing in time, and H stands for a linear function of the state. "i" is the square root of -1, and the rest of the terms are boring constants. Don't worry if that looks confusing, because we're only going to use one simple feature of it.
So far, things look clear enough. You've heard about the random side of quantum mechanics, but we have a plain old equation with no randomness. You've heard about the dreaded Uncertainty Principle, but so far we just have a little spread in some fields. There's nothing more uncertain about that than the way a ripple in a pond certainly spreads out. Our equation is even local, meaning that the changes in |Ψ> can be calculated just from the part of |Ψ> nearby in space, so there's nothing spooky about it. So what's the big problem, other than that things are a little more spread out than you might have expected?
We know that trouble is coming, because that was a local theory, and we had found experimentally that no local theory can represent what we see. The first problem comes from the simplest feature of our equation, its strict linearity. The important part about linear equations (and this is really just simple school math) is that they obey superposition, meaning that if you have two different solutions you can add up some of one and some of the other and still get a solution to the same equation.
Why is that nice feature a problem here? Let's say you have a quantum thingy, say a single blip of light, which either makes it through a polarizer or not, depending on which way it's polarized. If it's horizontal (H), it gets blocked and nothing happens. If it's vertical (V) it gets through, triggers a detector, and starts a mechanism that kills a cat. (sorry for the traditional nasty example.) What's fishy about that? It's easy to prepare that blip polarized somewhere between H and V. The quantum state representing that is a combination of some H and some V. The H part was the start of a solution in which the cat lived. The V part was the start of a solution in which the cat died. So our exact superposition says what this solution looks like: the output is part dead cat and part live cat.
Wait! That's crazy. That's not what you see. So let's go beyond the cat to ask what the equation says you should see. Just write out the quantum state for the blip, the apparatus, the cat, and you. The output is a combination of a you seeing the live cat and a you seeing the dead cat. It doesn't equal your perceived experience.
Just to avoid misunderstanding, you should realize that such "quantum measurement" situations are not just contrived lab artifacts. We're bathed in them constantly- UV light blips interacting with your DNA, leading to a superposition of a you with a dead skin cell and a you dead from melanoma, etc. That also doesn't correspond to your perception.
What happens in our experience? We don't experience that whole output state. We experience one piece of it, a piece where big things (cats, people) have nearly definite properties. So this is where probability and uncertainty come in. We don't know which piece of the output state will describe our experience. It turns out we can correctly calculate the probabilities of each of the possible outcomes from a measure of how much of the quantum state headed toward that outcome.
If you have a spread-out electron wave hitting a tv screen, the quantum state is initially certainly spread out. When you see a blip of light from that wave hitting the screen, it comes from some particular region. The wave just gave you probabilities for what you would see. What had been a simple spread is converted to an uncertainty by whatever process causes us not to experience the whole output state predicted by our equation.
At this point, anybody in their right mind (if any are still reading) would be thinking that there must be something else, some little hidden clue to tell the blip where to show up, to make the cat live or die. This is where our last story comes in. Any such "hidden variable" hypothesis has implications, and those implications are experimentally false.
Then you might think, ok somehow nature throws some secret dice to tell that electron where to show up, but here's where it gets stranger. When little objects interact, their quantum states become "entangled". Then when one of the possibilities for one of the objects becomes the actual outcome you see, that tells you what actual outcome you'll see for the other object, no matter how far away it is. As we saw before, there's no way to put little quantum dice at each spot and reproduce the observed behavior, because there's no way for the remote places to coordinate their results, and the actual results are coordinated.
Now may be a good point to remind you that this whole crazy story provides the tools for doing the calculations at the heart of modern physics and chemistry. The calculations are often extraordinarily precise. We know how to calculate the possibilities and what probabilities to assign them. For all practical purposes, we don't really need to know more about how this works.
Nevertheless, we would like some more understanding, some feel for what's up even if it's weird. At least a little start is provided by an analysis of something called "decoherence", beyond the scope of this note, which explains how interactions with the outside world cause a quantum state of some region to break up into parts which quit showing wave-like effects with each other. Decoherence is already implicit in our basic story, and in many cases it can be explicitly calculated. (It's done a lot, since it's decoherence that makes it hard to build quantum computers.) One decoherent part would have a live cat and a you seeing a live cat, another decoherent part would have a dead cat and a you seeing a dead cat.
That still leaves some very big open questions, including:
1. Is there some sort of collapse of the quantum state to just one outcome? We usually speak as if this were how things happened.
2. Do all the parts still exist, just losing contact with each other? That's called the Many Worlds interpretation.
3. Was there some sort of "pre-collapse" in which there really was only one actual value of the coordinates? That's called the Bohm interpretation.
4. Is the traditional Copenhagen Interpretation correct?
There are other questions and other interpretations beyond the ones I discuss. This should give you some of the flavor. You won't find the answers here.
(1) implicitly invokes some sort of process outside the known equation. Attempts to make an explicit version have not yet succeeded. One central problem is the one we've mentioned- how do remote entangled objects get their random stories straight?
(2) is in some ways the most obvious interpretation. It has the advantage of not having to explain how nature's choices at remote locations get coordinated, because nature isn't making any choices. Everything happens. The beauty of not adding any extra processes has a drawback, however. There's no extra step where you can sneak in the rule for what the probabilities are. In my opinion no one has successfully explained how it leads to the correct values for the observed probabilities.
(3) used to look promising before the non-locality results were in. Now instead of having a little coordinate dot for each particle, we need one dot in some high dimensional space representing every particle in the universe. That doesn't seem to clear anything up.
(4) is still what many textbooks advocate, but I'm not sure what it means. One common version of it is "shut up and calculate", which may be good advice. Perhaps it's time for me to take it.