By Hugh Lippincott

I want to try and explain some of the math behind the double-slit experiment. The goal here is not to explain the weird nature of light mathematically, which is beyond the scope of a blog. I do want to show how the double-slit experiment proves light behaves as a wave quantitatively and give an example of how math can be used to explain the results of an experiment.

After a brief discussion with my mom, I realize that I will have to start by explaining what the sine function is (hence the “Part 1″ in the title. For the reader who knows what the sine function is, I apologize. Hopefully you will enjoy this post anyway [I always like reading about something I know, it's egotistically gratifying and maybe there will be some interest to be found here from a pedagogical standpoint]). From looking at some of the comments, I fear that just the mention of something called a sine function will cause eyes to glaze over, so let me try and explain why I think it’s cool. Math is a language, and each additional element in the language expands the scope of what you can talk about. For example, English with just nouns and verbs would be a boring language (“I wrote”). This is like math with just multiplication. But when you include adverbs and adjectives, all of a sudden you can say something interesting (“I wrote a fascinating post on math, and everyone unanimously agreed that I was the best blogger around who specializes in explaining physics to his mother.”). In math, it’s the same way; the sine function is a tool that enables a discussion of a whole host of things that were previously unavailable, and in particular, waves.

Part 1
Let’s start with a circle like the one shown and draw a line from the center of the circle to the edge. I’m going to trace out the circumference of the circle with this line. At any one time, the angle between that line and the horizontal axis is θ (my mom will ask about the variable names again, for whatever reason angles are always given Greek letters, and θ is always the first one given), and the projection of that line on the horizontal and vertical axes are x and y respectively. I’m particularly interested in the vertical projection, y (hence the color). Initially, when θ=0, the line is entirely horizontal, and y=0. As θ increases, then so does y, until reaching its maximum value when the line is entirely in the vertical axis. Then y decreases before reaching 0 again, and then goes negative, before finally returning to where it started. We can imagine going around the circle again and getting the exact same thing.

Now plot y as a function of θ, and we get a wave.
This is the sine function – or more technically, the ratio of y to the radius of the circle (we could have performed a similar exercise for the x coordinate and obtained the cosine function, which is [clearly] a very close relative of the sine). It describes a wave as well as circular motion. It also represents a relationship between the sides of a triangle (the alert reader will have noticed that x, y and the radius created a triangle for each angle, suggesting that the sine of an angle relates the length of the sides of a triangle to the hypotenuse). Among many other things. All in all, it’s really useful.

Part 2

(click on the picture for a larger view)

Now that we know what a sine wave is, we can understand the double-slit experiment. I need to start with a few definitions that I probably should have put in the last post: the wavelength is the length between two successive peaks in the wave (often represented by λ) and the amplitude is the height of the wave (we’ll call it A). There is a symmetry property of the wave; if you shifted the wave to the right or left by its wavelength, it would look exactly the same. In fact, you could shift the wave by any integer times the wavelength, and you wouldn’t be able to tell the difference. This will be important later on.

To understand the double slit experiment, we need to ask what happens when two waves overlap. The answer depends on their “phase,” or where each wave is in its oscillation relative to the other. For example, suppose two waves are perfectly “in phase,” so that when one wave is peaking, so is the other. When you add these two waves together, you’ll get a wave that is twice as big in amplitude.