Fermion Dynamics from a Classical Hamiltonian (for normal people)

Below is a "note" I wrote on Facebook to describe my research project on Fermion dynamics from a classical Hamiltonian. Those of you with a solid background in molecular electronics (or even quantum mechanics generally) may recognize that I oversimplified a few points. But the purpose of this was more to explain why I'd been excited by this work than to teach people subtleties of the field of molecular electronics.

I've been very excited about research lately, and some of you have seen that in the form of me going "hey look, here's the paper I wrote!" (and for the truly unfortunate among you, "hey, will you proofread my paper for me?" -- sorry Dad).

I realize that what I do is pretty esoteric (even my scientist friends will probably scratch their heads at my paper) so let me explain what it's all about, and why I'm so excited.

The physical problem we've been looking at is what is known as "molecular electronics." One part of all this "nanotechnology" business you may have heard about is that people (well, some scientists at least) are excited about the idea of using molecules as wires. Imagine a battery looped around so that one end was hooked up to one side of a molecule, and the other end was hooked to the other side.

Why would molecular wires be interesting? Well, first there's the whole "nano" thing. Molecular wires are about as small as you can make a wire, so as electronics get smaller we may need small wires. Also, the wide variety of possible molecules means that there's a wide variety of possible behaviors for molecular wires. An engineer could tailor the choice of molecule to fit a specific problem (whereas bulk metal has a much narrower range of behaviors.)

So that's why you might want molecular wires, but there are a couple reasons (aside from the engineering challenges) that you don't already have molecular electronics everywhere. It turns out that the equations that describe the behavior of the kinds of wires we've used since we tamed electricity don't quite describe the nanoscale. (More accurately, the physics that are important at the nanoscale aren't important in large systems.) One key effect is that with a single molecule as a wire, quantum mechanics starts to be important.

That's where I come in. My research has been focused on making quantum mechanics easier to do. The short version of the story is the quantum mechanics is correct (as far as chemistry is concerned) but really hard to do. Classical mechanics can be totally wrong for molecular systems, but it's relatively easy to do. The work I do is semiclassical; it straddles the divide between quantum and classical by using classical mechanics (easy to do) to get quantum results (correct answer).

What I (with my boss and collaborators) have done is to find equations that describe the way the quantum molecular wire conducts electricity, but based on classical dynamics. Since electrons are really light, the flow of an electron usually requires quantum mechanics to describe. The fact that our results are phenomenally accurate means that we've found equations that (at least under the circumstances we're using) are really good!

Going back to that picture of the molecule connected to either end of a battery, you might imagine that how that molecule wiggles around could have a big influence on how well it conducts electricity. If it isn't connected tightly on both ends, then it is literally a loose wire! And if you really think about it, remember that the electrodes are made of atoms too, and they're moving around. All this motion might affect how well the molecule conducts electricity.

And that's why we're really excited. It's easy to describe that kind of motion with classical mechanics, and hard to describe it with quantum mechanics. By taking the quantum dynamics of the electrons and bringing that into the realm of classical physics, we're able to treat those complicated motions in a relatively easy way, without sacrificing the important quantum physics that underlies the electron motion.

We hope that we'll be able to describe the effect that molecular motion has on conductivity, which (in the general case) is beyond the realm of other methods for studying molecular electronics.

But when we let ourselves dream, it doesn't stop there. Our methodology for taking quantum mechanics into classical mechanics is based on a mathematical formalism called "second quantization," which can be used to describe many, many problems outside the realm of molecular electronics. There's a chance that our technique we be applicable to those problems as well: and there's almost no limit to those problems. This could become a big deal indeed. Not Nobel-Prize-big-deal, but big enough to lead to a lot of discussion in the scientific community, and that's really the best one can hope for.

So I'm very excited about this.

Oh, and if you want to read the paper, the preprint is publicly available: http://arxiv.org/abs/1103.4405. Academic types can get the final version here: http://link.aip.org/link/doi/10.1063/1.3583366