Parallel Particle In Cell Modeling of Semiclassical Quantum Models
In my research at UCLA, I am constructing a multiparticle quantum mechanical modeller using semiclassical approximation of Feynman path integrals and implemented on parallel computers using techniques (e.g., Particle In Cell) created and used for plasma codes. For more information on quantum mechanics, see my Quantum Orbitals page.
The Feynman path integral for wavefunctions ...
The Feynman path integral is a formulation that, in quantum mechanics, can be used to evolve a wavefunction forward in time. Imagine a wavefunction at one time and at a later time. In this formulation, the information at each time and space are linked to each other via paths, which originate at the earlier time and end at the later time and can be any possible, kinked, straight, whatever, path. Ideally, there are an infinite number of these paths.
In the context of Quantum Field Theory, virtual particles are said to follow these paths. Each virtual particle has a phase attached to them. One particle starts with the phase it is given by the wavefunction at the earlier time. The phase it gains as it follows its path is proportional to the action, the integral of the Lagrangian, along that path.
... reduced using the semiclassical method ...
That's the Feynman path integral for a wavefunction. However, it is possible to reduce the number of paths. Using the semiclassical method, the paths can be reduced to just the classical paths with certain other mathematical details that I don't have room for here. The basic idea is that, instead of many virtual particles, a smaller, but still substantial, number of virtual classical particles can be used to evolve a wavefunction.
... evaluated using plasma code techniques ...
But plasma codes have been moving large numbers of classical particles through electromagnetic fields for decades, and they work very well on parallel computers. Some of the larger plasma runs this group has done has hundreds of millions of classical particles, so if one quantum particle is equivalent in computational cost to ten thousand classical particles, then one could probably do quantum problems with hundreds to thousands of quantum particles.
... to do very difficult Quantum problems.
That's the idea of my research, because if one could do quantum systems like that,
then one could simulate multielectron atoms, molecules, chemical reactions,
semiconductor transistors, electron gases, metals, or other phenomenon
where quantum mechanical effects are significant.
Details, details, details ...
I've left out loads of details because I didn't think I'd have room here, and those details are not trivial, because if it was easy, it would have already been done, which I think is an indication that it makes for an interesting thesis. Email me (see below) if you think I should add more detail here, and, if so, how much.
The examples I have currently are one-dimensional. At the very least, these are meant to be a demonstration of a proof of principle. Early on, it was easy to have the wavefunction to simply fall apart. There is a lot of cancellation going on essential to maintaining the wavefunction, so if the "balance" wasn't quite right, the wavefunction could have torn itself apart in non-physical ways. So I'm only showing more recent examples that behave in good, physical ways.
In the plot, the horizontal axis is position and the vertical is probability density. The phase of the wavefunction is shown in color, where positive real is red, positive imaginary is yellow-green, negative real is cyan, and negative imaginary is purple. The color bars at the top have brightness proportional to probability density.
Also note that some of these movies have sound, denoted with this icon: For each time step, I take the real and imaginary parts of the wavefunction and play them in the left and right channels, respectively. By sonifying my wavefunctions, I'm able to hear, in addition to see, the quantum mechanics going on. This technique has helped me pinpoint problems in the code in the past, and gives me another avenue to analyze the physics hidden in my data.
You can download QuickTime movies of the data by clicking on the animated pictures shown below. The file size of the movie is in the caption.
These and other simulations for this research were computed on two Power Macintosh clusters. Many thanks go to the people at the AppleSeed site for using their cluster and J. Louis Hales-Garcia for his help in using the gSCAD cluster.
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