Quantum One

Michael Nielsen’s recent essay “Why the world needs quantum mechanics,” about the quintessential weirdness of quantum phenomena, provoked Dave Bacon to ask if there’s a better way to teach introductory courses in quantum physics. This question strikes a chord with me, since my first semester of college quantum — the class known as “8.04” — was rather remarkably dreadful.

It began with some fluff about early models of the atom, leaving out most of the ideas actually proposed in favor of a “textbook cardboard” version of the discoveries made in early TwenCen. If we can’t teach history well, why teach it at all? We’re certainly not promoting a genuine understanding of how science works if we only present a caricature of it. I doubt one could even instil an appreciation for the problems which Bohr, Heisenberg, Schrödinger and company solved in the years leading up to 1927: sophomore physics students can’t follow in their footsteps, because sophomore physics students don’t know as much classical physics as professional physicists of the 1920s did. To understand their starting point and the steps they took requires, oddly enough, subject material which even MIT undergrads don’t learn until later.

After the textbook-cardboard version of history, we proceeded to solve the Schrödinger Equation in a great many different circumstances. This was supposed to “build intuition about quantum mechanics,” but I think all it built in most people was a loathing of differential equations. The course was capped off with a cursory treatment of the Bell Inequality.

Looking back, the only thing I learned in 8.04 which I didn’t see derived again, more cleanly and more memorably, in a later course was how a delta-function potential acts in the Schrödinger Equation — and that takes, what, half the back of an envelope to work out? (OK, given small handwriting.)

I have occasionally wondered whether it would be possible to build a first-term quantum physics course out of Feynman’s QED: The Strange Theory of Light and Matter (1985). If you’re teaching to university students, they probably already have experience with complex numbers, trigonometry and some amount of calculus, so you’ll be able to write actual exercises based on the book’s conceptual material.

Either before or after the chapter on the Standard Model’s particle content, expand the book with two-state systems, the Bell Inequality and entanglement stuff. (I might do it after the Standard Model chapter, so I could lead into it with a little kaon physics.) After entanglement, introduce decoherence. Follow with the model of electrons hopping from atom to atom expounded in the Lectures on Physics, chapter III-13, and use it to motivate band-gap behavior and, by shrinking the lattice spacing, the Schrödinger Equation (chapter III-16). Solve the particle-in-a-box and the harmonic oscillator, and then bid the kids a happy summer vacation.

I don’t pretend to know how well this programme would work, and I’ve certainly never tried it on a live audience, but I do think the way I was introduced to QM was wasted effort which fails in its stated aim of “building intuition.” The time is nigh for reform — come, let us tear down the old edifice of incomprehensible authority and build a better one anew! (Singing “Dem Bones” is optional.)

18 thoughts on “Quantum One”

  1. I just took an intro course in quantum mechanics, and I liked it. We used the Griffiths textbook, and we covered a bunch of different potential functions, and then Dirac notation. I don’t have the retrospective viewpoint that you do, so tell me what precisely is wrong with this. I mean, besides the history, which is pretty much a given.

  2. If it’s “pretty much a given” that the history will be taught wrong, then we really shouldn’t be teaching it!

    Not knowing any more about your introductory class than what you’ve just told me, I can really only speak about the one I survived. Instead of “building intuition” about how quantum phenomena differ from classical, it wasted time on laborious solutions of problems which can be solved much more readily if one takes a moderate amount of time to learn some extra math which isn’t particularly difficult anyway. We didn’t even use Dirac notation to any substantial degree (and it certainly shouldn’t be held off until the end of a class).

  3. My undergrad quantum was much like miller’s. We used Griffiths, never substantially deviating from his treatment. We got up to time independent perturbation theory over our two semesters. I went on to grad quantum and felt undergrad quantum had given me a pretty good basis from which to build some real knowledge.

  4. Sorry if that was tangential to your post, but I say that to say I can’t really comment on your suggestions of how to fix undergrad quantum, because mine was fine.

  5. Which makes you what I believe a specialist would call “one lucky bastard.”

    I should add that one of the most frustrating aspects of the whole business was the contrast between 8.04, which was dreadful, and the next two terms of quantum, 8.05 and 8.06, which were superb.

  6. I wonder if the problem isn’t the whole semester system. I mean, shoehorning an expansive topic into three months or so doesn’t seem quite fair. But then, maybe this is actually a case of trying to expand what’s actually rather limited introductory material into a whole semester course, when really it could be covered just as well in a quarter.

    I remember very little of my Modern Physics class at BU, but to the extent that I do it seems similar to the one you took. We did a few interesting things with Schroedinger’s Equation like deriving quantum tunneling, and in all it wasn’t horrible (we didn’t get the botched attempt at a history lesson that you did, which freed us up to spend a little bit more time discussing electron orbitals as they’re currently understood that in your class was spent re-covering the same few broken models that physics classes always discuss), but the SE was basically the entirety of the course. I never took any QM beyond that, so I can’t say how appropriate that is for an introductory course.

    I remember the lab stuff being kind of cool, though. Lots of stuff with optics and spectra.

  7. I think the problem with the science history lessons wasn’t so much that it was wrong, but that it was so half-hearted. They spent only the first lecture, mentioning various problems with classical physics, what so and so scientist did whenever, and so forth. I remember hardly any of it. From experience, I’ve never really expected to get any good history from my science classes, if I get it at all.

  8. I can safely say that I didn’t ‘get’ much of QM when I took the one semester course. But of course that was back before I’d decided whether to pursue physics or not, so it was a very busy semester.

    It certainly never had the time to ‘gel’ for me, and I think that I’d benefit from studying it again now. As a way of getting my head back in order, I guess. I don’t recall much analysis or algebra off the top of my head, but I’ve had it and I think there’s a token understanding of sorts lurking inside me that might help.

    I’m pretty sure we go all the way to timedependent perturbationtheory (which I could never spell), but the only thing I do recall is getting spherical integrals wrong because our calculusbook had θ and φ swapped (to better reflect polar coördinates).

  9. History: The Making of the Atomic Bomb is an incredible book. The first third or so of it — 300 pages — is all set in the time before we knew that a U-235 chain reaction might be possible. Later, the few pages of Hiroshima survivors describing the bomb’s aftermath are enough to justify the book’s existence on their own.

    I’m about to post a lot of senseless blather about classes and such, and did not want the goodness of The Making of the Atomic Bomb to get lost in it.

  10. So, politics major here. Last math class was at age 16. Physics-blog-commenting license was permanently voided shortly thereafter, was a bummer, so don’t tell anyone I’m here. But I think I see something folks are getting at here, because it’s largely about education and not about physics.

    What made some of my best classes successful is that they left me with a few simple, salient, intuitively gripping ideas as hooks on which the details hung — tasty, shiny high-degree nodes in the network of ideas.

    One of the very last math classes I took used Serge Lang’s Linear Algebra book, and I think it mostly failed at structure; there was some flow in that lesson n’s exercises would nudge you towards the ideas you’d actually prove in lesson n+2, but it still felt like a train of small, unremarkable proofs about matrices and such; it never jumped out with a flourish to show why some result was remarkable or a useful tool or whatever.

    A couple of my economics courses more nearly succeded: a game theory intro hit some memorable high points (how it can make sense to include randomness in a rational strategy; just how a duopoly works; what helps determine whether self-interested folks will cooperate), and an introductory policy class introduced me to the weird and distinctive geoclassical approach to land prices (and got me into occasionally gratifying arguments with the prof).

    It seems like 8.04 is treating learning QM like physical exercise or a martial art: take a basic tool of QM, the Schrödinger equation, and have students do variation on variation of it to build up that muscle. If profs instead tried to lead students to a few high points, perhaps they’d instead have students derive varied, interesting, counterintuitive QM results. There’d be time to work out other, less interesting cases later, but it’d give everyone a few salient mental anchors for the often non-intuitive math.

    And, again, politics major, but a book in the QM canon subtitled The Strange Theory of Light and Matter sure sounds like a great place to start with that.

  11. For those of you who have recommended things to read, rest assured, I’ll get around to them, ahem, soon.

    :) Fair enough.

  12. The algorithm: if Blake doesn’t mind my shameless self-promotion, maybe you’d like to check out my recent (and continuing) posts about linear algebra and let me know what you think about my approach?

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