Werner Heisenberg is driving down a long, deserted country road, when suddenly a police car appears behind him, lights flashing. Heisenberg pulls his automobile onto the shoulder and stops. The policeman parks his car behind him, gets out, and walks over. As Heisenberg rolls down his window the policeman leans down and says, "Sir, did you realize that you were travelling at 97 miles per hour?" Heisenberg groans loudly, covers his face with his hands, and says "No, but now that I do, I no longer know where I am!"

This is one of a handful of jokes told by physics students that riff on the bizarre truth embodied by Heisenberg's uncertainty principle — namely, that at the subatomic level, certain pieces of information about particles, such as position and momentum, are not isolated facts, but merge fuzzily together so that the precision of each is like a too-small blanket that can be assigned to one number only at the expense of the other. If Heisenberg's car was a subatomic particle, then it would actually be the case that the act of measuring its speed (and therefore its momentum) to enough precision would cause its location to become unclear. And "unclear" doesn't just mean less well-known, but actually less well-defined. And this is only one of the first steps into the oddness of quantum mechanics.

What is perhaps even odder, though, is that there really aren't very many examples of jokes like this. Every branch of science has its tome of in-jokes, the appreciation of which requires specialized knowledge. And physicists are no exception to this. So you might expect that quantum physics, famously the source of so much counter-intuitive phenomena, would be ripe with material for jokes like the one above. But it isn't. Why is that? It's hard not to suspect that the oddity of quantum physics is too extreme, making it difficult to find anything even remotely familiar with which to leverage a concept up into the everyday world. Heisenberg's uncertainty principle is strange, but its ingredients are familiar (position and momentum, which can be mapped to location and speed well enough). So much of quantum physics, however, deals with nothing recognizable at all.

Quantum physics is extremely confusing: this is almost a truism. Many people know almost nothing about quantum physics except that it is hard to understand. Not hard like rocket science, which is problematic mainly due to involving high-energy reactions that are to difficult to study in a controlled fashion, or brain surgery, which gets its difficulty by the delicacy of its application and the rather large amount of the subject matter that is still poorly understood: Quantum physics is difficult mainly because it violates so many of our intuitive knowledge of basic physics without giving us new ones to replace them with.

One can argue that Newton did the same thing when he unseated classical Aristotelian physics, but the analogy doesn't really hold. Developing an intuition for Newtonian physics isn't actually too hard (especially nowadays when we have space stations and you can watch videos of tea and flames in microgravity). Furthermore, Newtonian physics has that elegance that is the hallmark of all the most beautiful scientific theories, a simplicity that Aristotle's world utterly lacks, down to its very foundation. But perhaps most significantly, Newtonian physics still deals comfortably with everyday objects, objects that we can see and touch directly, and I suspect that this is a crucial factor. Nobody has ever taken a picture of a single electron, and for that matter I'm not sure that that's even a meaningful concept.

It's bad enough that quantum particles behave so irregularly, with this "uncertainty" built right into their behavioral structure and all. But on top of that, we don't even know what they should look like. Most of us start out imagining them as tiny billiard balls, or maybe as little specks of dust. Then we learn that this visual can be completely misleading. Because at the same time as they are pointlike particles, they are also waves, and therefore sort of smeared out over an unspecified volume, and this volume varies as they are acted upon by various forces and/or bombarded by other particles that may or may not find their ways to some detector later on, perhaps causing them to get noticed by some lumbering macroscopic beast like ourselves.

Compared with macroscopic objects and all that we think we know about them, an electron hardly seem deserving of the term "matter". Eventually most people give up and go back to visualizing electrons as billiard balls again, only now they feel vaguely ashamed for doing so.

Physicists of course are no less disappointed by the lack of intuitive imagery for quantum physics than anyone else. But that doesn't stop them from using it, and so on they go. There's work to be done. Some of them may vaguely hope that some future Grand Unified Theory might eventually point the way towards some good analogies, one of which will work with human intuition instead of against it, all without doing disservice to reality. But given such facts as the absence of "hidden variables", one may be forgiven for being skeptical about such possibilities. At this time, the evidence is in favor of the Standard Model looking a lot like a finished product, and if there is a underlying deeper structure, it has to be at a completely new level.

And so we're left with what we have, a theory that is both remarkably predictive and utterly opaque, and another generation of would-be physicists struggle through their university courses, cursing at the pitfalls set in place by their own brains, optimized more for tree-swinging and rock-throwing than for particles that can generate interference patterns with themselves.

I wish I could say, here is an interpretation of quantum mechanics that will give you an intuitive and visual grasp of the true nature of electrons. I don't have that. It might be that no such analogy exists to begin with. On the other hand, I have found it useful to consider analogies, even if they are wrong, that help you see how to stretch your idea of what an elementary particle even is. All of which is a roundabout way of saying that the following visual analogy is not at all close to the truth, but it may still be worth your time.

Imagine for the moment, an electron. Not the squishy, smeared-out skull-busting electron of reality, but the old-fashioned, sub-sub-microscopic billiard ball electron that you probably already carry around in your head. (You can even imagine it as being white, if you like, as long as you remember that objects smaller than a single wavelength of visible light can't really have a color. But one can't really imagine a billiard-ball-shaped object without any color at all.)

Now, imagine these electrons scattered more or less evenly throughout all of space. Not so dense enough as to be touching, but dense enough to feel the effect of each other. Each electron carries a negative electromagnetic charge, so all the electrons repel each other. And as a result they will tend to settle into a universe-sized lattice, each electron being roughly equidistant from all of its immediate neighbors in an attempt to stay as far away from each other as possible.

Or rather, that's what they would do in a calm, quiescent space-time continuum. Our universe is anything but, however, and so to this static picture we need to add motion and change. The continuum is full of radiation in various forms, from the microwave background to the gamma rays of hot stars, pushing and pulling all matter about. The result is that the electrons are always moving around, sometimes quite speedily, jostling each other in the process. They will never get to settle down anywhere near long enough for the pertubations to smooth out into that perfect universal lattice of minimal energy. At large scales the lattice will be smooth on average, but down at the subatomic level, there will always be inconsistencies. More electrons in some places, less in others.

Most people's immediate objection to this model is that we don't see the entire universe filled with a negative electromagnetic field. But if this field of negative charge is smooth on average, then how could we detect it as such? That field would become our zero point, the baseline from which we measure everything else. It would be what we would call "the vacuum of space." And so, in this hyperabundant model, the object that we referred to as "an electron" would actually be any position in space that contains one more electron (our billiard-ball kind of electron, that is) than "the vacuum".

If we have a volume of space in which the electrons are more or less evenly dispersed, and then a new electron arrives, travelling at a high rate of speed, it's easy to see that it will cause a ripple through this space. Each electron will get pushed out of the way as the intruder approaches, only to quickly move back again after it passes. Thus we get the classic bow shock and wake phenomena, like a speedboat cutting the surface of a lake. Voila: waves.

And these are not just waves in name alone; they mimic the waves associated with quantum particles. From a distance an observer only sees that this space contains an electron (that is, it contains exactly one more billiard-ball electron than it does normally). But if the observer then tries to pin down the electron's exact location, it becomes, well, fuzzy. Because the intruding electron has pushed away its closest neighbors slightly, the immediate area around the newcomer actually has slightly less of the full negative charge that an electron carries. Meanwhile all the pushed-aside electrons are crowding into the surrounding space slightly more closely than they would otherwise, making the immediate neighborhood slightly more negative than the vacuum state. To any observer that treats the vacuum state as being empty and uncharged, it appears that the electron has been smeared out.

Not just appears: it has been smeared out. It is inextricably mixed with the vacuum of space that surrounds it, for it is made of exactly the same stuff. In this model, an electron is no longer simply an object: it is a state of one part of the space-time continuum. Consider the case of electrons orbiting an atomic nucleus. In this situation, there is no longer a simple trajectory: the electrons swarm around in their orbits, remaining largely within various shapes (depending on the orbit), but following no directional path that we know of. Here, it is easy to imagine a captured electron simply merging in with the crowd, making the whole area of the orbit vaguely more full but without maintaining any sense of apartness from the other billiard balls. If and when the electron is kicked out of its orbit and leaves the atom for adventures elsewhere, there's no telling if the billiard ball that actually left was the same one or not.

Some of the electron oribts in the atom have strange discontinuities in their shapes. The s2 orbit contains a sphere with an infinitely thin surface, upon which there is a flat zero percent chance of the electron ever being located there — even though the electron has a non-zero chance of being inside the sphere and outside the sphere. They are a bit like the gaps between Saturn's rings, places where the resonant frequencies of the bodies in question line up perfectly enough to push away, either inwards or outwards, anything that tries to stay there. Now, if you try to maintain a naive visual model of electrons, one that assumes that in order to get from point A to point B a particles has to pass through every point inbetween, this is a very unhappy fact to have to contend with. (I can remember my own vague distate with in back in high school.) Our billiard-ball model can cope with this situation, however: electrons can cross the zero-percent barrier as long as the density of electrons remains constant. Here an electron disappearing from one side of the sphere and reappearing on the other is a little like the bump in the carpet that when you push it down causes another to pop up a few inches away. The electron density can shift around in ways that are independent of the motion of any particular electron.

Perhaps you find this model too extreme to be anything more than a weird curiosity, demanding as it does a universe crammed full of electrons. If so then it may surprise you to learn that none other than Paul Dirac seriously suggested a model very much like this. Dirac's equation, which he formulated in the 1920s, predicted the existence of antimatter before it had ever been observed. Attempting to explain what appeared to him (and everyone) as a rather far-fetched idea, Dirac suggested that an antimatter electron could be explained as an "electron hole" if one presupposed a universe full of electrons. In our model, we imagine a gamma ray of the necessary energy level striking one of our billiard-ball electrons directly. The energy imparted causes it to accelerate so rapidly that it escapes its immediate neighborhood before the surrounding electrons can move away from it, thus becoming a "new" visible electron in the vacuum, and leaving behind a relatively positive charge in the area where it used to be, which manifests itself as a positron: an antimatter electron. Likewise, if an electron and a positron happen to find each other in the vacuum, the billiard-ball electron can easily fit into the hole, causing both to effectively disappear into the vacuum, and releasing pent-up energy as the billiard-ball lattice gets rid of not just one but two density imbalances.

One noteworthy consequence of this model is that antimatter should fall up, not down, in a gravitational field, exactly like a bubble of air rising in a glass of carbonated water. So this model is not just a whimsical exercise in silliness: it actually makes predictions that can be tested. Sadly, this particular test is a very difficult one to do, due to the effort involved to make (and preserve) enough antimatter to make the gravitational effect on it measurable, and at this point in time no one has succeeded. However, nearly every physicist currently believes that antimatter will fall down in a gravity field, for a variety of reasons that are largely unrelated.

So there's not much hope for our hyperabundant billiard-ball model. And in fact there are actually a number of features that fail to conform to observed reality, so (as I warned you) it was purely a whimsical exercise after all. But hopefully it has helped you see, however dimly, one way in which the things we call a quantum particle may not actually be anything like particles, and yet still permit the universe to make sense. If nothing else, thinking about the behavior of particles in this model can function as a sort of warm-up exercise, before tackling the real thing.

I still hold out hope one day that someone will craft a model for quantum mechanics that appeals to our intuitions. Or at least to mine.