### The Aharonov-Bohm Effect, or, How is a Coffee Cup Like a Donut?

Into this Universe, and Why not knowing

Nor Whence, like Water willy-nilly flowing;

And out of it, as Wind along the Waste,

I know not Whither, willy-nilly blowing.

(from the Rubaiyat of Omar Khayyam)

(Edited: my first version made it sound like I just make stuff up as I go along while teaching, which certainly is not true!)

Some of my colleagues prepare lecture notes for their entire semester before the term even begins; others teach directly out of books or out of their own memory. In introductory physics, it’s been several years before I’ve really needed notes since I’ve taught it so many times, but for advanced topics I haven’t taught, crutches are often needed. I’m capable of working in advance—don’t get me wrong—but I find when I review my notes I often want to put new stuff in that has occurred to me in the interim. Sometimes it works; often it doesn’t; sometimes the success depends on the particular students in the room. Today was one of those days where I felt the improvisational approach worked, since I was able to go beyond the material in the text to connect with some really cool physics, the kind of stuff I love in my research.

Specifically, we were talking today about a topic in magnetism: if you have a solenoid (a long coil of wire with current running around it), the magnetic field will be constant inside it and zero outside—something pretty useful for a lot of experiments. The inside is where most people focus their interest, but if you look at the region outside, you see something interesting and a little surprising.

Here’s the experimental setup: take a beam of electrons and split it so that half the electrons travel along one side of the solenoid and half travel along the other, then recombine them. From quantum mechanics, we know that electrons behave like waves as well as like particles, so when you recombine the beams, they interfere just like light does. If the current in the solenoid is turned off, the electrons will be perfectly in phase–the interference will be constructive. If the solenoid is then turned on, the electron interference pattern changes—even though the electrons never pass through the magnetic field! The electrons don’t experience a force, but they still somehow are still influenced by the presence of the magnetic field. This is known as the Aharonov-Bohm effect (for Yakir Aharonov and David Bohm, who described it in 1959, though they weren’t the first to discover it).

Showing how this effect works mathematically is beyond what I want to do on this blog, but we can understand how electrons can feel an effect without a force by examining a system that on its surface looks very different. Look at the coffee mug-donut transformation in the image to the right: you can continuously transform the coffee cup into the donut and back again without tearing because both shapes have a single hole. In fancy math terms, we say the donut and coffee cup are topologically equivalent (which is coincidental to how wonderfully coffee and donuts go together); neither is topologically equivalent to a sphere, which has no holes, or a pair of scissors, which has two holes.

When the solenoid is off, it’s like the sphere: there’s no hole, so the two electron paths could be merged into each other without bumping into anything. With the solenoid on, it’s like the donut: the two paths can’t be deformed into each other because the “hole”–the magnetic field–is in the way. Even though the electron paths aren’t influenced by the magnetic field, the environment is different—the shape of the space is not the same, so the electrons are affected. This is analogous to what we see in general relativity: there is no “force of gravity”, but the presence of a mass like our Sun affects the shape of spacetime. At its heart, physics is geometrical, and when I can touch on that kind of deep connection in class, I feel I have shown my students something significant.

Now I want a donut.