Thursday, August 11, 2005

Least action and non-classical paths

Here's a funky piece of physics discovered in condensed matter research that deserves broader attention because it is so weird.

So, the Feynmann-Hibbs path integral approach to quantum mechanics says that the way to calculate the probability of a quantum system starting at configuration {a} and ending at configuration {b} is to add up the complex amplitudes for all possible paths between {a} and {b}. For each path, you can compute the classical action, S, by integrating the Lagrangian along the path, and the amplitude for such a path is given by exp^(i S / \hbar). This is particularly nice because paths that extremize S end up constructively interfering (having very similar phases). So, when one passes to the classical limit (\hbar -> 0), one finds that the dominant classical trajectory for the system is the one that extremizes the action. Unsurprisingly, the trajectory that dominates is something smooth that looks like a sensible classical path (e.g. for a particle propagating in free space, it's a straight line from point a to point b).

Here's the weirdness, though. There is at least one system (quantum tunneling of spin in molecular magnets) for which the action-extremizing paths are discontinuous. In our free particle discussion, these would be analogous to trajectories where the particle's path in space looks like _____----____--__ , complete with discontinuous changes in coordinates. Such trajectories are always in the calculations (in fact, in quantum field theory you need to include paths with space-like discontinuities (!) in order to get the calculations to come out correctly), but this is the only case I've ever seen where they can be the dominant trajectories.

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