The point.

Okay folks, this is a long one.

As most of you know, I am working on a quantum mechanics research project this summer. Since 1) I think it is really cool and 2) I'm not very good at explaining it on the fly, I thought I'd post what I'm doing here so if anyone is interested you can see what I'm doing. Plus there is a movie at the end...that was one of my major accomplishments (don't laugh, my adviser thought it was really cool too!).

In the largest sense, I am applying an alternate interpretation of quantum mechanics to a basic concept in physics, namely, the harmonic oscillator. The harmonic oscillator is used as a model in almost all subdivisions of physics. Photons are modeled as harmonic oscillators in theories of light, pendulums in classical mechanics can be modeled as harmonic oscillators, etc. The quantum harmonic oscillator is one of the few systems that can by solved analytically in quantum mechanics (that is, the answer is exact equations instead of making approximations and using a computer to hack through). But even though an analytical solution is possible with standard quantum mechanics, the results are not necessarily conceptually fulfilling.

Standard quantum mechanics works on the assumption that a particle’s positions or velocity or any other feature is inherently probabilistic. Whereas on a human scale I can tell you the exact position I am sitting or exactly how fast I am moving, on the scale of an electron the best I can tell you is a range where the electron will be found and a position where it will most likely be.

This is a pretty unsatisfying view of the microscopic world. And since the macroscopic world is built from these probabilistic particles, quantum physics has profound implications for how we view the world. Einstein’s dissatisfaction with these ideas prompted the quotable line, “I am convinced that God does not play dice with the universe”.

As standard quantum mechanics was being developed (well before it was referred to as “standard”), other theories to explain experimental results were being developed. Why one theory becomes accepted and the others get relegated to footnotes, I am not enough of historian to explain. But so it happens. One of these “alternate” theories is now known as Bohmian mechanics, after its conceiver David Bohm.

Bohmian mechanics works on the principle that a probabilistic wave-function can be seen as a collection particles that can be treated just like particles in classical mechanics. We envision that each of these particles has a probability of existing until a measurement is made at which point only one particle actually exists. The true value of this interpretation comes from the fact that the behavior of wave-functions can be understood in terms of classical forces. All of those Physics I concepts like Force = Mass * Acceleration become useful tools in Bohmian mechanics. In standard quantum mechanics many problems can’t be understood conceptually, one just has to “shut up and calculate” as theoretical physicist Richard Feynman is quoted as saying. Bohmian mechanics gives us a way to picture the quantum world in terms of graspable classical concepts.

My specific project this summer is to apply Bohmian mechanics to the time-dependent harmonic oscillator. The simplest example of a harmonic oscillator is a weight on a spring sliding along a frictionless surface. The weight is pulled back from its equilibrium point and let go. It oscillates back and forth with a certain period and maximum distance from the equilibrium point depending on how far it is pulled back and the strength of the spring. This is a time-independent harmonic oscillator since the strength of the spring is constant in time. A time-dependent harmonic oscillator would be one where the strength of the spring changes over time. This isn’t a very meaningful model in the spring/mass example, but time-dependent harmonic oscillators are very useful models in quantum mechanics.

What we do is start with a specific wave-function whose shape is a Gaussian. A Gaussian is a distribution useful in statistics, you may know it as a normal distribution or a bell curve. A Gaussian is useful for our purposes because when the forces of a harmonic oscillator are applied to it, it stays a Gaussian. Its width and height changes, but it never loses the properties of a Gaussian. This makes it exceptionally nice to follow through time since we only have to worry about how one function changes in a harmonic oscillator.

Bohmian mechanics works by adding the force from the harmonic oscillator with the so called quantum force that is derived from the Schrödinger equation (the equation that governs standard quantum mechanics). By looking at how these forces cause the Gaussian’s width to grow or shrink we can predict how the probability distribution will change with time and where we are most likely to find the electron that this distribution models. Although there are still probabilities and uncertainties in this treatment, why they behave like they do is much more clear than in standard quantum mechanics. And since we come to the same conclusions as people who use standard quantum mechanics and as are achieved experimentally, our interpretation has validity.

My end results are best viewed as movies. The blue curve is the Gaussian probability distribution which represents where the particle is likely to exist. The green line is the quantum force that depends on the width of the probability distribution and the red line is the classical force which I can make change as any arbitrary time-dependent function. Sometimes physics is watching squiggly lines move on a screen!