Recursion relations

Well the QM course is moving along, and we’ve solved the Schrodinger equation (SE) for the harmonic oscillator (and its parabolic potential -kx^2). This involves the creation of 3 new variables from composites of old ones, then transforming the SE to something simpler, which is then solved using an exponential multiplied by a polynomial. To prevent the polynomial from blowing up as q (what we’ve transformed x into) gets large, all powers of the q in the polynomial past a certain point must be zero. Another way of looking at this is that the wavefunction must be zero at infinity (the particle has to be somewhere). This simple fact leads to a recursion relationship among the terms of the polynomial. Very cute, but it didn’t have the impact it did when I first saw recursion relationships used in QM years and years ago (48.5 years ago to be exact).

Back then, budding chemists started out by learning that electrons like to be in filled shells. The first shell has 2 elements, the next 2 + 6 elements etc. etc. It allows the neophyte to make some sense of the periodic table (as long as they deal with low atomic numbers — why the 4s electrons are of lower energy than the 3d electons still seems quite ad hoc to me). Later on we were told that this is because of quantum numbers n, l, m and s. Then we learn that atomic orbitals have shapes, in some wierd way determined by the quantum numbers, etc. etc.

Recursion relations are no stranger to the differential equations course, where you learn to (tediously) find them for a polynomial series solution for the differential equation at hand. I never really understood them, but I could use them (like far too much math that I took back then).

So it wasn’t a shock when the QM instructor back then got to them in the course of solving the hydrogen atom (with it’s radially symmetric potential). First the equation had to be expressed in spherical coordinates (r, theta and phi) which made the Laplacian look rather fierce. Then the equation was split into 3, each involving one of r, theta or phi. The easiest to solve was the one involving phi which involved only a complex exponential. But periodic nature of the solution made the magnetic quantum number fall out. Pretty good, but nothing earthshaking.

Recursion relations made their appearance with the solution of the radial and the theta equations. So it was plug and chug time with series solutions and recursion relations so things wouldn’t blow up (or as Dr. Gouterman put it, the electron has to be somewhere, so the wavefunction must be zero at infinity). MEGO (My Eyes Glazed Over) until all of a sudden there were the main quantum number (n) and the azimuthal quantum number (l) coming directly out of the recursions.

When I first realized what was going on, it really hit me. I can still see the room and the people in it (just as people can remember exactly where they were and what they were doing when they heard about 9/11 or (for the oldsters among you) when Kennedy was shot — I was cutting a physiology class in med school). The realization that what I had considered mathematical diddle, in some way was giving us the quantum numbers and the periodic table, and the shape of orbitals, was a glimpse of incredible and unseen power. For me it was like seeing the face of God.

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  • carlbrannen  On October 17, 2009 at 3:19 pm

    For a cool introductory textbook to quantum mechanics from a more elementary particle point of view, you could pick up Quantum Kinematics and Dynamics by Julian Schwinger, or look for his paper which founded that method “The Algebra of Microscopic Measurement” to get an idea on how it works. The basic concept is to analyze quantum mechanics in terms of simple operations performed on a beam of electrons.

    And for an easy introduction to quantum field theory, without the math, see Feynman’s “QED: The Strange Theory of Light and Matter”.

  • luysii  On October 17, 2009 at 4:46 pm

    Carlbrannan: again, thanks for commenting.

    You’ve got to crawl before you walk and walk before you run. I wrote the following on the 22 September post:

    I am now the proud (if poorer) owner of “Physics for Scientists & Engineers” by Giancoli. Why that book rather than Halliday? Both of them look like comic books with pretty colored diagrams on every page (photos too). Well, it’s what Harvard tells their Freshmen to use. Presumably they know something about physics books. Hopefully I’ll get through some of it before the course ends. Hamlitonians are all well and good, but I’ve forgotten Newton’s laws of motion. Time to crack it open.

  • Anoria  On December 9, 2009 at 12:28 am

    “The realization that what I had considered mathematical diddle, in some way was giving us the quantum numbers and the periodic table, and the shape of orbitals, was a glimpse of incredible and unseen power. For me it was like seeing the face of God.”
    This happened to me last week.
    I just found this post by googling “hydrogen recursion relation” – I’m in my first-and-a-half quantum mechanics course right now and the diff-eq hasn’t stuck with me quite well enough to do the homework without help – but yes, last week we derived n, l, m, and s. My flatmate, sitting next to me in lecture, was subjected to my constant monologue of “ohgod” and “arg no! too much sense!” and “…well, that chalkboard pretty much sums up my entire degree right there” (I took chemistry before adding physics). The elegance in all of it is stunning.

    So I just wanted to leave a little note confirming that even kids today are affected by this revelation as strongly as you were. Thanks for writing this and leaving it up where people like me can stumble across it 🙂

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