Quantum Views

By Angela DePace

How can we “see” the behavior of objects at the atomic scale? Atoms are smaller than a nanometer - one thousand-millionth of a meter - challenging our best microscopes and usually only seen indirectly by diffraction. Facts taken for granted in our daily world - like the ability to measure the position and velocity of a particle at a given instant - are incompatible with the behavior of particles at this submicrometer scale. This is the world described by quantum mechanics, the famously counterintuitive physical theory that predicts a wide range of spectacular and yet experimentally verified phenomena, such as supersolids and quantum teleportation.

We spoke with physicist Michael Trott about how he uses visualization of Bohmian mechanics, an interpretation of quantum mechanics, to gain insight into the microworld.

What is the theory of Bohm mechanics?

Bohm’s theory makes one of the most mysterious puzzles of traditional quantum mechanics — the so-called measurement postulate — evaporate, allowing one to calculate deterministic paths for microparticles. However, this comes at a price of incorporating unobservable additional parameters. The theory, called today Bohmian mechanics or de Broglie-Bohm theory, starts by using a traditional description of a microparticle as a wave function (density matrix), and then builds trajectory descriptions of microparticles on top of it. This is achieved by calculating a ‘quantum potential’, the shape of which dictates a ‘quantum force’ to determine a fictitious path of a quantum particle. But because we cannot get access to the to the ‘secret’ initial conditions used in the Bohm theory, the predictive power of this theory is exactly the same as quantum mechanics. We have to average over an ensemble of initial conditions.

How does visualization of Bohm mechanics aid in understanding the theory?

First of all, the Bohm theory is interesting from a mathematical point of view. For example, what kind of trajectories can occur? Under which conditions are the trajectories periodic or chaotic? How chaotic can the trajectories became as a function of time and a functional of the outer applied forces? Visualizing the equations makes the answers to these questions more apparent.

Second, while the Bohm theory doesn’t provide more predictive power than other interpretations of quantum mechanics, it can give us some kind of intuitive picture about the world of microparticles. This is critical if we are to propose new experiments and nanodevices that operate on this scale.

Finally, calculations and visualization based on Bohm’s quantum mechanics, along with the decoherence program, are beginning to address the measurement problem in quantum mechanics. How do we derive rules for how macroscopic objects that follow the rules of classical mechanics are built from microscopic objects that follow the rules of quantum mechanics? Advanced versions of these sorts of images can add some intuitive understanding of how phenomena like “einselection” and “pointer states” arise dynamically.

Where are these sorts of images used?

The first images of actual Bohm trajectories were produced about 25 years ago and used to visualize how the interference pattern of the archetypical quantum mechanics experiment - the interference pattern on a double slit diffraction - arises. More recently, images such as those shown here are used in various parts of physics that operate on a submicrometer scale. In a few decades, transistors on microchips will be of a size that quantum effects will not only be important, but even dominating. “Seeing” how a group of electrons will behave in such semiconductor micro- and nanostructures allows more easily to optimize their design.

What are the steps involved in making such images?

Three steps are typically involved:

1. Write down the governing physical principles of the problem and derive the actual equations of motion.
2. Solve the equations of motion, taking into account concrete initial and boundary conditions. Sometimes, it is possible to find closed-form solutions, but in most cases, one has to resort to numerical solutions.
3. Visualizations of the solutions as 2D and/or 3D plots.

In the first step, usually there are few choices to make (from time to time, you can introduce a parameter here and there). In the second step, you have to choose from a large number of possible system parameter and initial conditions. For visualization, you can choose from an enormous variety of strategies; you can use surfaces, lines, tubes, wireframes, 2D or 3D graphics, colors, stills or animated images to emphasize certain features.

How difficult is it to make such images?

I do all of my work exclusively with Mathematica. It is an integrated system that is strong at symbolic computing, powerful numerical capabilities, and flexible customizable graphics. Ten years ago, calculations such as the ones shown here were only possible with high-end workstations and using various different specialized software systems. Today the visualizations shown here are doable in Mathematica with a small amount of code and all graphics can be calculated and rendered, and exported to a format of your choice within minutes on a Mac laptop.

What is the most important thing to see in these images?

The whole series of images considers one of the simplest nontrivial quantum systems - the time-dependence of a particle in a 2D box. The horizontal plane is the x,y-plane and time runs vertically upward. In the probability pictures, you see that even a simple possible quantum system exhibits quite complicated behavior. In the trajectory pictures you see two things. First, how the time-dependent density of the trajectories mimics the probability density of the wave function. Second, that the Bohm trajectories of this simple system can be quite intricate, showing a seemingly sporadic mix of loops, straight lines, and helixes.

I think the 2D version of the trajectories has an aesthetic appeal because of its similarity to a hand-drawn representation - do you agree?

Yes. Sometimes showing less is actually showing more. Even in 3D things can get ‘crowded’. And in 3D graphics, one typically needs trajectories to have a finite thickness (although mathematically they are infinitely thin) to obtain a proper depth impression of the resulting picture. This means instead of lines, one uses thin tubes made from polygons such that their normal direction changes with the angle and one gets a depth perception. In 2D one does not have to worry about depth and one can use much more trajectories, each being represented by a thin line. This graphic shows nicely how initially particles starting from a small neighborhood stay together but then slowly diverge, eventually behaving qualitatively quite differently from one another.

Ultimately, the aesthetics of these images reflect the fact that nature itself is beautiful. And the equations of mathematics are beautiful in their own way. And (fortunately) the real world can be described through mathematics (while this might seem obvious, it is far from nontrivial and the reason is only poorly understood). And the mathematical description of nature, even of simple systems can exhibit an extremely rich and fascinating complexity of shapes, figures, and forms.