## Monday, 31 March 2014

### And nature does it in real-time

A paper of mine appeared today in Physical Review E.  Called, "Extension of the continuum time-dependent Hartree-Fock method to proton states1."  As the name suggests, it is a paper about methodology - so it's not going to feature in any press-releases about exciting new physics results.  It's even an extension to an existing method (from our previous paper), which might make it seem all the less exciting.  I think it's still a good paper, and a useful one, hence the blog post.

The method we developed overcomes a problem inherent in many attempts to solve time-dependent equations in physics problems.  From a mathematical point of view, the problem is that the equations that nature seems to have written itself in are differential equations.  By their very nature, such equations have solutions which combine some functional form, along with boundary conditions.  The functional form gives a kind of general prescription of how to solve the equations for absolutely any case at all, and the boundary conditions then shape the details to fit the exact physical situation at hand.  For example, the general solutions to Maxwell's equations describe all (classical) electromagnetic phenomena, but the boundary conditions pin down whether the particular solutions is for a light wave, the electric field round a charge, or the induction in a generator.

In the case of our paper, we were concerned with the time-dependent Schrödinger equation - the basic equation of quantum mechanics2.  In particular, we are interested in solving it for the case of atomic nuclei undergoing some kind of dynamic process.  In mind we have excited wobbling states, or fusion or fission, or some combination of such things.  More or less any case of interest involves the nucleus being excited in such a way that it can decay by breaking up, either into two or more fragments, or by emitting protons or neutrons.  One long-standing problem with solving the time-dependent Schrödinger equation in such cases is that the only simple way of working with boundary conditions is to assume that we can consider the tiny nucleus to be in a little box a bit less tiny than the nucleus, but a box which either reflects back anything emitted from the excited nucleus, or which lets things pass through but then reappear at the other side. This is kinda bad:  Nature doesn't do it that way.  It lets things that decay off of the nucleus travel far away without some artificial box getting in the way.  The reason that these strange unphysical solutions are the easy ones to implement is that they involve pretending that the inside of the box is everything that there is.  If we have stuff in our system, it has to be somewhere in the box.  It's hard to start having by having a bunch of stuff (nucleons in a nucleus) in our calculation, and then to keep calculating how it changes in time, but to stop keeping track of some of it because it's left our system.  It doesn't sound like a hard problem, or even that it should be a problem at all, but it is, on both counts.

So, our paper is about how to deal with this "open quantum system" (search for that phrase, and there are a whole load of hits - it's a field in itself).  Our method is reasonably general, but we've applied it just to vibrational states of nuclei so far.  That was quite a job in itself.  It is the final work from my PhD student Chris Pardi's thesis, from last year.  It took the paper a while to get into print.  We tried first in Physical Review C, where it was felt to concentrate a bit too much on the technical aspects of the algorithm, and not enough on the nuclear physics - a fair enough comment - and so we asked Physical Review E to take a look.  One referee it was sent to was nice enough to write a long report, ending with "In conclusion, I believe that this paper is excellent and very well written." So - thank you anonymous referee.

The title of this post is a reflection of the fact that we have worked hard to solve some equations, along with their proper boundary conditions, using some computational calculations that take a certain time to run.  Not so very long on the scale of things, but still a few tens of seconds.  They describe a process that happens in nature over a few zeptoseconds.  Nature works out what to do so quickly...

1 C. I. Pardi, P. D. Stevenson and K. Xu (2014). Extension of the continuum time-dependent Hartree-Fock method to proton states Physical Review E, 89 : 10.1103/PhysRevE.89.033312
2 Okay, we could debate what the most basic equation of quantum mechanics is, but calling the Schrödinger equation the basic one is not outrageous.