My recent hacks at trying to find the scattering and bound states of a semi-infinite line attached to a single weighted edge (the simplest graph imaginable) hadn’t been too successful until today. I had not really been sure about how to approach the problem and had only written down a Hamiltonian and postulated some qualitative behavior of the eigenstates (scattering states should scatter, bound states should be bound, etc). Fortunately, I had a very helpful phone call with Andrew this morning.
I told him what I had been able to work out and where I was stuck and he guided me towards the right approach. The key lessons were that (1) we can assume that scattering states look like plane waves where the graph edges are unweighted, (2) while the reflection coefficient for this baby graph does have amplitude one, it crucially also as a nontrivial phase, and (3) the basic strategy for these types of problems is to investigate the “boring” parts of the graph (unweighted semi-infinite lines), postulate solutions that work there with some undetermined constants, and then use the “interesting” regions (single weighted edge in this case) to pin down the constants.
I’m really looking forward to my time in Waterloo because so far, Andrew is proving to be a great advisor. He was patient enough on the phone to offer hints and then wait for me to think about them and calculate in real time. (I’ve seen far too many professors begin twitching and intervening in the presence of a student deviating slightly from the method of calculation that professor prefers.) His suggestions were just helpful enough so that I understood how to approach a calculation, but vague enough that there was still plenty of the puzzle to figure out.
After the phone call, I spent the next couple of hours jiggling around some equations (anyone who’s ever calculated a reflection coefficient knows scattering problems are an algebraic hellstorm) and was able to find the scattering states and their energies, the bound states and their energies, and some nifty “alternating” bound states and their energies (these states are “alternating” because they are similar to the regular bound states except there’s a sign flip in the amplitude at every other vertex).
For the bound states, I found that:
- the spectrum is discrete
- there is only one bound state
- the transcendental equation yielding that bound states only has solutions for certain minimum strengths of the weight on the weighted edge
Stumbling across the transcendental equation was a nice surprise because last night, my friend and web dev/computational physics guru Casey Stark gave me a crash course in using Mathematica (check out his tips here), and transcendental equations are a great excuse for exploiting Mathematica’s plotting tools.
I’ve still got some algebraic housekeeping to do to clean up the expressions I obtained today, but given today’s success, I should be ready to tackle big boy graphs in no time!
Sounds super interesting. I wish I could see what you are doing instead of just hearing you narrate. Funnily enough Mathematica is a perfect medium: it handles arbitrary mathematical typesetting, inline graphics, dynamic user interface elements (so you can drag a slider around to see the various bound states, for example), hectically powerful automated theorem proving, ODE and PDE solving, 3D plotting, and on and on and on and on.
The only problem is that its not webby, you would have to offer a .nb notebook file to download. But, Coming Soon is a browser plugin. Then, as Stephen Wolfram would put it, “blogs will become computable”, which is a very exciting prospect indeed.
Sorry, I got carried away with Mathematica evangelism there… Forgive me.
Your Mathematica evangelism is well-founded in this case. The vast majority of my work so far has involved solving intractable eigenvalue equations, plotting transcendental equations, and abusing the “Manipulate” function to explore this connection between bound states and phase shifts. Lately though, I’ve become enamored with a complex analysis approach to the problem, so my Mathematica engine has cooled down a bit.
I had a laptop stolen which has derailed my progress reports, but I’ll start ‘em up again soon… complete with glorious Mathematica notebooks full of 3D plots, manipulable data, and (hopefully) clear documentation to boot.
That’s a pity. Sometimes I think data loss is the only kind of material loss that matters — everything else can be replaced, but not those intangible ones and zeros, ironically. I remember when PartitionMagic killed an entire childhood’s worth of QBasic creations, it made me so sad.
For your next machine, you might want to try DropBox. I’m a big fan of its automagical syncing-in-the-cloud across multiple OSs and machines. Plus, by referring you we both get an extra 250 megs free, on top of the gratis 3 gigs
.
Great minds… or at least sufficiently techie blokes.
I’ve been using Dropbox for several months now. Only this week though did I start saving all my research there as well (I was previously just using it as an easy way to get pdfs and other files on to my iPhone). Its probably my favorite app I’ve discovered in the last year.