Continuity, Discretion, and the Perils of Popularization

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.

11 thoughts on “Continuity, Discretion, and the Perils of Popularization


    But this is just because they’re bound states. A free electron can have any energy it likes (above its rest mass). And one can cook up classical systems with discrete solutions as well; the existence of things like discrete bound states is just a property of differential equations. (Think of, say, frequencies of a violin string.) So it’s not clear to me that a distinction between discrete and continuous is a very useful way of conveying what’s different about classical and quantum mechanics.

  2. What got me in QM was that apparent disconnect of an electron having those nice discrete energy values, and that is juxtaposed with a finite probability that it will have tunneled to Moscow while I was out to lunch.
    Not being able to follow the mathematical concepts makes a lot of QM rather counter-intuitive.

  3. Are you sure your editor wouldn’t have balked at my second word? My attempt was an unpracticed nonsense, though obviously an attempt to point to what I felt was left out of yours that I think matters. I agree with your “Popularization is necessarily about making choices about which features are really essential, and which can safely be left out” wholeheartedly. I think of popularization as a very respectable research activity, however, with no fixed endpoint and no absolute best way of doing things. Most popularization drags behind the Physics, but I believe it should be done as if it’s going to change the field. There should be a creative attempt to find a way of thinking that will excite an expert as much as it will excite Einstein’s grandmother. Advances in ways to teach and in ways to popularize are always possible and can feed the edge of research. Indeed, changing the goalposts significantly in teaching and popularization is one way to introduce real change in the research direction of a field (a current example that I approve of, because it meshes fairly well with my own research, is Art Hobson’s attempt to reinvent the teaching of quantum theory, “Teaching Quantum Physics Without Paradoxes”. The Physics Teacher 45, 96-99 (2007), available at ). Making things simple is harder than making things complicated.

    No, I’m a Math degree theorist. I’ve never been in a serious lab. A choice of childhood that one could regret. I’m not, however, a theorist who wants to escape outright from electronics and oil.

    I like your gentle way of dealing with this discussion. Thanks.


    I think the difference is the relative importance of those situations. Waves on strings seem disproportionately important to physicists because they set up ideas that are used in quantum physics, but standing-wave modes of classical systems are not something that most people think about every day.

    And even in classical systems with bound states, there is a continuity that you don’t get in quantum systems. A wave on a string can only have a substantial amplitude at certain frequencies, but what determines the energy is not the frequency but the amplitude. And the amplitude can be anything you want. Likewise a classical pendulum, or a mass on a spring, or any other resonant system.

    A quantum harmonic oscillator, though, has a limited set of discrete allowed energies. As a physicist, you become very accustomed to this sort of thing, but that’s really weird and counterintuitive for someone who isn’t a physicist. If I’m pushing SteelyKid on a swing, I don’t notice any discrete character to her motion. It’s not like she can swing through and angle of 0.1 radians or an angle of 0.2 radians, but never 0.15– her swinging can have any amplitude at all, depending only on how hard she’s pushed. But that’s what you would see with a quantum analogue of a pendulum, and that’s weird and different and critically important to the way the world works.

  5. It may be “popularizations” that got me confused (if I was) about big bad “D”. This is not meant as fault of e.g. the OP in previous descriptions. Indeed, OP once wrote (Many-Worlds and Decoherence: There Are No Other Universes):
    So, photons that have interacted with a big environment become like classical particles and don’t interfere any more, right?
    Wrong. The photons always behave like waves, and they always interfere.
    The cumulative effect of the interactions is to make the photons look as if they were classical particles, taking one definite path or the other. They’re always quantum objects, though, and they always interfere. Decoherence just keeps you from seeing the pattern.

    Well, I agree with that now and always did. Following Chad’s quote, then sure extra processing wouldn’t reveal an actual distinction in final interference results. It’s important to say since I don’t like “feuds” to simmer along any worse than need be. It’s other writers who say things similar to “decoherence converts a superposition into a mixture” etc. Real quote, from The Quantum Challenge (Greenstein, Zajonc) “How does the incessant fluctuation in an object’s environment convert its state from a superposition into a mixture?” (paperback p. 166) I am hoping that even strong DI advocates mean this in a sort of FAPP-ish way, but can you see how such brazen statements instill confusion and resistance. Some of them say, we don’t have to “worry about” the Cat anymore etc. because the issue is taken care of. That professed literalism (?) is what concerns me. FAPP issues wouldn’t resolve the “model problem” of the co-existence of states. So I don’t really understand what they’re saying at that level of middle-brow reading.

    Here’s a cute way to tease about “real mixtures” in these Mach-Zehnder problems: First pretend the stream of photon waves from BS1 was literally converted into a mixture before reaching BS2 (as indicated by their inability to show interference patterns there.) Then, BS2 would re-split and convert them back into superpositions anyway! What help would that be, for explaining in any way why we get specific (“collapsed”) detector outcomes on past BS2? But if the output from BS2 is a literal mixture instead – don’t take my word for it, check and see if you can tell the difference. I’ll just watch and learn unless prompted.

  6. onymous pointed out not only that classical systems can have discrete frequencies — to which Chad correctly responded that they are still “continuous” in a way that quantum mechanical systems are not — but also that the discreteness of quantum systems is a consequence of their being in bound states.

    A “particle in a box” or a simple harmonic oscillator have discrete energy eigenvalues because their wavefunctions go to zero as one gets far from the middle of the box or the equilibrium position of the harmonic oscillator. But they can only do this because their potential energy functions *diverge* as one gets far away. For a hydrogen atom, where the potential approaches zero as the electron gets far from the proton, the *negative* energy eigenstates are discrete, as you say, but the *positive* energy eigenstates — what classically would be the orbits for which the electron escapes from the atom — are continuous.

    Can we leave out the less essential details? Sure… But leaving someone with the idea that there are no continuous spectra in quantum mechanics doesn’t seem appropriate to me. Why not qualify this with the criterion that it applies to bound states — or if that’s too technical, states that are restricted to some region?

  7. I agree with onymous that the distinction between continuous and discrete is not uniquely quantum-mechanical, and with Alex that it’s as much a consequence of boundary conditions and (general) wave mechanics as anything else.

    Revising my previous suggestion, how about combining #2 and #3 to say that
    “the outcome of measuring a quantum system is often random, and the possible outcomes are often discrete”?

    Admittedly, “often” makes things MUCH less punchy than the catchier “states are discrete” and “probability is all we ever know”.

    I’d argue the above revision is “correcter” and equally nontechnical. The price paid, as you point out, is probably one of emotional impact.

  8. Well, to get back to my nit-picking about interpretations, I am probably oversensitive about popular accounts of quantum theory that are not interpretation neutral, i.e. accounts that do not distinguish between what you *must* believe if you accept quantum theory and what different groups of physicists believe is the reality behind those facts. Therefore, I tend to put any account through the filter of all the reasonable interpretations that I know to see if it stands up to scrutiny. I am not terribly attached to the main realist interpretations myself, so it is not really an issue with realism per se, but I do generally find that the realist approaches are usually the sources of counter-examples to statements in popular accounts, and hence I do end up harping on about them.

    In any case, I believe that interpretation neutrality is a good thing to strive for in popular accounts because I think we need to be honest with the public when there is a legitimate controversy in science. If we don’t then they will become confused when they hear contradictory statements of “fact” from different experts and this gives the quantum flapdoodle merchants the space they need to make their crackpot claims seem legitimate. I accept that not everyone would agree with my insistence on complete neutrality, since it involves getting into the sort of subtleties that would have spilled a lot of red editors ink if they had appeared in your book. Nevertheless, I hope we can agree that what you definitely want to avoid in popularizations of quantum physics is inconsistent mixing of ideas from different interpretations. For example, if one were to say that quantum theory entails parallel universes in a matter-of-fact way (as some people do) then you shouldn’t also be saying that nothing exists in the world until we measure it. This is because a large part of the reason why anyone would accept the former is because it gets rid of the latter.

    Unfortunately, I think that current popularizations of quantum theory give the impression that all sorts of weird and crazy things (e.g. parallel universes, collapse being a real physical process, consciousness causing collapse, nonlocal influences, particles being spread over all of space, no reality until we look, etc.) are true simultaneously, whereas in fact some of them are posited precisely to avoid having to admit the truth of others. In reality, quantum weirdness is like a tennis ball stuck under the carpet that you can move from one room to the other but you cannot get rid of completely. Sadly, the public perception is that we are walking on a surface made entirely from tennis balls.

  9. If you had to explain the American culture to some one from a far different culture, you might say to them, “Don’t insult someone’s mother. In our society, we don’t like that.” Some of you would be the guys who would protest and try to explain that, “Insulting your mother is a high form of social interaction. Many people tell ‘Yo Momma is fat’ jokes. It is highly prized in some subset of our culture and it is imperative that you know that as a basic fact of our culture. So don’t tell that person not to insult someone’s mother because in some situations it might be ok. It is wrong to not tell them about ‘Yo momma’ jokes and when to use them.”


    I do the same when explaining to clients that ‘a printer driver is a piece of software that allows a computer to talk to the printer’.

    They understand what it does, without becoming mired in how it does it.

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