Applications of Quantum Mechanics

Over at the theoretical physics beach party, Moshe is talking about teaching quantum mechanics, specifically an elective course for upper-level undergraduates. He’s looking for some suggestions of special topics:

The course it titled “Applications of quantum mechanics”, and is covering the second half of the text by David Griffiths, whose textbooks I find to be uniformly excellent. A more accurate description of the material would be approximation methods for solving the Schrodinger equation. Not uncommonly in the physics curriculum, when the math becomes more demanding the physics tends to take a back seat, so we are going to spend quite a bit of the time on what is essentially a course in differential equations, using WKB approximations and perturbation theory and what not. To counter that, I am looking for short and sweet applications of quantum mechanics. Short topics which can be taught in an hour or less, and involve some cool concepts in addition to practicing the new mathematical techniques.

I’m hampered in this by not knowing what’s in the second half of Griffiths (the analogous class at Williams was taught out of Park’s book, because he’s there; I used to have a copy of Griffiths in my office, but it seems to have wandered off). I’m currently teaching a much lower-level version of a similar course, though, so I can suggest a few things:

The “Applications” portion of the class I’m currently teaching is really a mad sprint through whatever QM-related topics I can fit into the last three weeks or so. A couple of these, scaled up appropriately, might work.

One obvious application is solid state physics. It’s relatively easy to sketch out the basic ideas that lead to band structure in solids. The full solution is a bit beyond an undergrad course, but you can do the Kronig-Penney model pretty easily. That works well to show how periodic arrays of potential wells gives you bands of allowed states, with gaps between them. The basic idea of band structure is enough to explain a bunch of useful technology– diodes, transistors, LED’s, etc.

Another area is nuclear physics. I don’t do it in the sophomore-level class that I teach, but you can do a remarkably good job of calculating half-lives of radioactive elements using alpha particle tunneling as a model. Somewhere, I have a Mathematica notebook with code to numerically solve the Schrödinger equation for a bunch of different nuclei, which does a great job of getting the decay rates, and the trend with atomic number.

Those two might very well be in Griffiths already, though. A couple other things come to mind as possible topics, though:

If you’re talking about perturbation theory and approximations, you ought to be able to do the Fermi Golden Rule for transitions between atomic states driven by an oscillating electromagnetic field. From there, you can go for the “lies your teachers taught you” topic of demonstrating that you don’t need photons to explain the photoelectric effect. The model is spelled out in a paper by Mandel in the 60’s (I don’t have the cite here, but I can find it if people want to see it), and doesn’t require anything beyond basic perturbation theory.

If the class includes state-vector notation, you can do the No-Cloning Theorem pretty easily (it’s remarkably simple). That’s a good way of getting into all sorts of fun quantum information topics: teleportation, quantum cryptography, some basic quantum computing, etc.

Several commenters to the original post suggested the Quantum Zeno Effect, which is another good one if you’ve done state vectors. Projective measurement is fun stuff.

It’s also relatively easy to get into a lot of cool quantum optics material– the Hanbury Brown and Twiss experiment can be explained in a very straightforward way, and you can actually calculate the relevant correlation functions for a bunch of different cases. And that gets you to the basic techniques that are used for everything in quantum optics.

That’s what I come up with off the top of my head, without knowing the textbook in question. What did I miss?

Comments

  1. #1 fizzchick
    January 8, 2009

    I love Griffiths’ textbooks. You did pretty well: The second half also includes notes on the Zeeman effect, tunneling, Berry phase and adiabaticity, and scattering. The latter was used as an excuse by my instructor to describe his research involving the formation of ultracold molecules. It was nice to see QM applied to actual current research problems, even if they were (IMO) somewhat esoteric ones. Of course, if you want more math instead of less, Griffiths also introduces Green’s function and Cauchy integrals in an offhand manner in Chapter 11 (along the way to the Born approximation).

  2. #2 Moshe
    January 8, 2009

    Thanks Chad, looking forward also to all the comments here. Incidentally, you have no idea how appropriate “theoretical beach party” is as a description of David’s work place. Less so in mine (there is a beach nearby, but the snow tends to ruin the party).

  3. #3 Asad
    January 8, 2009

    I know and like Griffiths’ books as well, having used three of them in my coursework (QM, EM, and Particles). We used Park for our second-semester QM course, and I found that Griffiths borrows quite heavily from Park for his own QM book.

    If only Griffiths would write a stat mech textbook, I might actually be able to learn it.

  4. #4 Asad
    January 8, 2009

    I know and like Griffiths’ books as well, having used three of them in my coursework (QM, EM, and Particles). We used Park for our second-semester QM course, and I found that Griffiths borrows quite heavily from Park for his own QM book.

    If only Griffiths would write a stat mech textbook, I might actually be able to learn it.

  5. #5 Jonathan Vos Post
    January 8, 2009

    Griffiths is good. I’m reviewing this stuff, for some esoteric research of mine on what the solutions to 4-D of space + 1-D of time equivalent of Schrodinger equation for hyper-atoms of hyper-electrons in orbitals around hyper-nuclei. Hyperspherical polynomials. I’m interested in hyper-Linus Pauling: the Nature of the Chemical Bond for “artificial chemistry” in 4+1 dimensional space. It’s fairly nuts what a science fiction author will do to make his hand-waving look plausible. The standard proofs that there are no possible 4-D or 5-D atoms are fatally flawed, because they neglect the negative energy solutions standard in Quantum Field Theory.

    There have been a flurry of papers since the two co-authored by Hockney, correcting Foppel, on the problem originally posed by J. J. Thomson about arranging unit-charge electrons in a minimum energy configuration on the surface of a unit sphere. This was before the Bohr atom was invented. Some of the new papers look at hyperelectrons in hyperspheres up to 64 dimensions.Weird and lovely stuff, not what “normal” physics looks at. But then Caltech did hire a postdoc whose PhD was on QM if there were an infinite number of space dimensional, and then he went off and wrote for Star Trek, and “Feynman’s Rainbow”, so who knows.

    And wouldn’t it be nice to know the theoretical spectrum of hyperhydrogen? One could look for it, and of course expect not to see it. But you never know until you do the measurements…

  6. #6 bcooper
    January 8, 2009

    If only Griffiths would write a stat mech textbook, I might actually be able to learn it.

    Have you tried Reif? I’m a big Griffiths fan as well and I’ve generally found Reif to be a book in the same spirit.

    In terms of applied quantum mechanics and physical ideas I tend to think quantum information stuff might be a good way to go. Maybe some things like Bell’s/CHSH inequality, teleportation, BB84. I guess these are sort of grab bag when thrown into what the rest of the course sounds like but I bet students would find them interesting. Maybe a lecture on how lasers work?

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