Multiparticle Quantum Mechanics (Jaunary 12, 2000)

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 Idea

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.

Slides

 I gave a talk at the Lawrence-Livermore National Laboratory in October 1999 about my research. Here are the slides for that presentation, in QuickTime format: (2.5 MB)

Examples

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.

 The initial condition here is a Gaussian wavefunction with zero velocity in free space. It widens, yes, but what interesting is that the total probability and the total energy of this wavefunction remains constant, as consistent with theory, and the precise width as a function of time is consistent with theory to five decimal places: (2.0 MB)

 This physical situation is a simple harmonic oscillator (SHO) potential. The initial condition here is n=0 eigenstate. Consistent with theory, the probability density is unchanged while the phase cycles at a precise rate. This is a rigorous test of the code to see if it can duplicate the energy eigenstates: (3.6 MB)

 The same SHO potential, but with the n=7 eigenstate. Again this matches theory very well. I've done the same with other eigenstates. The phase cycles much faster, at a rate proportional to the eigenstate's energy. This cycling can be used as a "fingerprint": (8.7 MB)

 This has the same SHO potential, but the initial condition was a superposition of the n=0, n=1, n=5, and n=7 eigenstates. Because each eigenstate cycles around at a different rate, the size of each eigenstate can be extracted from the data, yielding an energy spectrum. It's like listening to many people each beating drums with different patterns and identifying which is which. This has implications for multiparticle systems, because we could just run a code with arbitrary initial conditions, then extract the energies: (8.0 MB)

 A well with very high walls containing one quantum particle in the n=7 eigenstate. The period with which this state oscillates matches theory to the level I am able to measure. The sound of this movie oscillates between the left and right channels, in sync with the rate of oscillation of this eigenstate: (3.2 MB)

 A Gaussian moving to the right, slams into a wall and bounces back. Listen to how the wavefunction sounds before it hits, then, as soon as is starts interacting with the wall, hear how the timbre of the sound changes. Most of the change in the sound is in the higher frequencies. After the first bounce, it sounds much like what it was before, but soon the wavefunction spreads out enough to interact with the walls continuously: (6.7 MB)

 A Gaussian wavefunction, moving from left to right, runs into a stair step barrier (not drawn) in the middle of the well. A piece tunnels through, and a piece reflects back. The sound of the wavefunction changes as soon as it hits the barrier. After a little practice, you should be able to perceive the impact just by the sound alone. After the two pieces of the wavefunction leave the barrier, it mostly returns to the original sound, except that there are two sources that oscillate in and out of phase with each other: (1.7 MB)

 Two spin-zero particles in a one-dimensional atom potential, with one centered on the atom, and the other offset. As they evolve, their mutual repulsion causes both wavefunctions to distort in complicated ways. This produces a very interesting energy spectrum. The sound you hear is the sum of the two wavefunctions. Most of the frequency range is fairly low in this one, so you may need to use speakers that are bigger than the normal computer speaker. If you're good, you can pick out a ringing sound that occurs towards the end of the simulation before you can see it in the graphics. That ringing came from a problem in the code that I've since fixed: (11.4 MB)

 A well with very high walls around sixteen spin-zero quantum particles. They are initially distributed in position and momentum, then interact in complicated ways: (40.1 MB)

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