Last week’s Seven Essential Elements of Quantum Physics post sparked a fair bit of discussion, though most of it was at the expert level, well above the level of the intended audience. such is life in the physics blogosphere.

I think it’s worth a little time to unpack some of the disagreement, though, as it sheds a little light on the process of writing this sort of thing for a general audience, and the eternal conflict between broad explanation and “dumbing down.” And, if nothing else, it lets me put off grading the exams from last night for a little while longer.

So, what’s the issue? The strongest single objection probably comes from Peter Morgan, who didn’t like my element 2):

2) Quantum states are discrete.The “quantum” in quantum physics refers to the fact that everything in quantum physics comes in discrete amounts. A beam of light can only contain integer numbers of photons– 1, 2, 3, 137, but never 1.5 or 22.7. An electron in an atom can only have certain discrete energy values– -13.6 electron volts, or -3.4 electron volts in hydrogen, but never -7.5 electron volts. No matter what you do, you will only ever detect a quantum system in one of these special allowed states.

He commented:

NOOOO!!!!! You need to talk about measurement operators, not about states, if you want to say “discrete”.

Perhaps: Measurement operators that have discrete spectra are used to represent measurement apparatus/procedures that produce discrete measurement results. Measurement operators that have continuous spectra are idealizations that do not correspond to real experimental data that is written in lab books or in computer memory.

The state space is usually taken to be vectors in a Hilbert space over the complex field, or density operators (arguably always one of these, by quantum physicists?), which are pretty much continuous linear spaces.

Leaving aside the intimidating language (my editor wouldn’t've gotten five words into the suggested alternative), there’s a real objection here, which is something I’ve glossed over. I would argue (obviously) that glossing over that was the right thing to do given my intended audience and goal for the piece.

The objection is, to paraphrase it a bit, that the mathematical descriptions we use to describe quantum objects are not in themselves discrete– that is, when we write a wavefunction to describe, say, the position of an electron, that wavefunction is a continuous mathematical object, with a value at every point in space. There are no gaps, no places where the wavefunction is not defined.

When we *measure* the position of the electron, we get discrete values, but Peter is arguing (as I understand his point, which he elaborated on in email) that that is a function of our measuring apparatus– we can only detect the position of our electron as being at one of the pixels of our measurement apparatus, say. So the outcomes are discrete, mathematically– there’s no possibility of measuring the position to be between two pixels– but the wavefunction describing the electron is not. If you upgraded your measuring apparatus to have more pixels, you would find that there is a probability of finding it between two of the original pixels.

This is a real objection, and if you look at it this way, there’s even a certain amount of tension between the first two points on my list. The wave phenomena described in the first point are fundamentally about the behavior of continuous idealized wave functions, which are not discrete in the sense Peter objects to. This is largely a problem with language, though.

What I was trying to do in that post was to set down in simple and non-mathematical language a list of the distinguishing features of quantum mechanics, that separate it from other theories of physics and other branches of science. “Discrete” was not intended to be a formal statement about the mathematical structure of the theory, but rather a reference to the way the allowed states of quantum objects are different from those we use in classical physics.

In classical physics, all of the properties you might measure for an object are continuous in the mathematical sense. If I throw a tennis ball for Emmy to chase, I can throw it at 15 meters per second, or 20, or 25, but also at 15.1376439 m/s, 0r 21.9876 m/s, or any arbitrary number you like in that range. Classical objects like tennis balls can have any velocity you like.

This is not the case for a quantum object like an electron in a hydrogen atom. We tend to express the result in terms of energy, rather than velocity, but there are certain values of that energy that are allowed, and you will only ever find the electron having one of those energies, as I said in the post. The total energy of that electron can be -13.6 eV, or -3.4 eV, but never anything in between those. That’s the sense in which I mean “discrete”– an electron inside an atom will always be found in one of a discrete set of states, the energy eigenstates of that atom.

This is the essential feature of quantum theory that sets it apart from classical mechanics. It’s what surprises undergraduates, and surprised even the people who came up with the theory. If you want a bullet-point list of Things You Need to Know About Quantum Physics that will fit on a card in your wallet (or in a blog post), that’s one of them.

Now, is this a simplification? Absolutely. The electron wavefunction is still continuous, and if you did some sort of idealized measurement of the precise location of the electron near the atom, you would find (after zillions of repeated measurements) that it has a probability of being at any point you like. And you would even find some spread in the energy, due to a variety of small effects and fluctuations.

But as a high-level statement about the way the theory works, suitable for readers who are not now and will not become physicists, “Electrons occupy discrete states” is a perfectly good statement of the predictions of the theory. It’s a simplification, yes, but not one that makes any important difference. People who go on to become physicists will need to learn more than that, but for everyone else, the simpler statement is just fine.

To put it in physics terms, it’s a little like the different levels you can choose for looking an atom. If you’re doing thermodynamics or fluid dynamics, an atom is a discrete and effectively indivisible particle with minimal structure. If you’re doing atomic or molecular physics, an atom is a collection of electrons orbiting a nucleus, which is a very small positively charged particle with minimal internal structure. If you’re doing nuclear physics, the nucleus is a collection of protons and neutrons bound together by the strong force, and so on into particle physics and string theory or M-theory of whatever Ultimate Theory you prefer. At each step up from the theory of everything, you’re making some simplification, and obscuring some underlying structure, but the details you lose don’t make any significant difference. Someone studying the behavior of a macroscopic gas in a box doesn’t need to worry about vacuum fluctuations and the Lamb shift, let alone the quark structure of nucleons.

The goal of popularizations, like my book and this blog, is to give people as much correct information as they need to understand the important features of a given branch of science, *and no more*. Things like Hilbert spaces and the distinction between wavefunctions and measurement operators just confuse the issue. If you want to be a professional quantum mechanic, you need all that stuff, but if you just want somebody to get the big picture, it’s better to leave all that out.

In a later comment, Peter attempts to draw a negative comparison between my posts on experimental details and my glossing-over of theory. The flippant response to that is “if you think I’m giving *all* the details, you’ve never worked in an experimental lab.” I gloss over a lot of experimental stuff, too, but it’s not as obvious.

A more serious response would be that there are details, and there are details, and some are more intimidating than others when you’re writing for a general audience. Details about plumbing and wiring are things that everyone can relate to, because they’re ordinary, comprehensible, physical tasks. Those tend to add some flavor to a description of an expeiment (though you’ll note that when I write up something like the single-photon cooling paper, I don’t simultaneously talk about all the plumbing details– too much detail is deadly).

Details involving math, on the other hand, are bad news. I wish it weren’t the case, but it is. One of my beta readers, terrifically smart person on literary matters, reported becoming physically angry (in the sense of “throw the book aside with great force”) when I used a couple of equations in one explanation. You just can’t get away with the same level of detail with regard to theoretical and mathematical matters that you can with plumbing. Expecting readers to parse more mathematically correct descriptions of quantum physics just isn’t realistic.

Popularization is necessarily about making choices about which features are really essential, and which can safely be left out– as I called it in the original post, a selection of the things “that everyone ought to know, at least in broad outlines.” This will always entail saying things that are only high-level approximations of deeper theories. This will inevitably leave some people unhappy. The trick is to have more happy people than unhappy ones at the end of the post/book/day.

The other serious objection raised were Matt Leifer’s comments on realist interpretations, probability, and measurement. There’s a long discussion that could be had about those points, but this post is already positively Zivkovician in length, and I doubt I’d have any readers left by the end of that. Another time, maybe; for now, I will bow to Matt’s vastly greater knowledge of realist theories.