BB #92: Actually Using Trig

Today (3/14) is Pi Day. People everywhere (or at least math geeks everywhere) are baking decorated pies (or cakes or cookies) to celebrate. And while this is yet another math-y post, it’s not about pi. I’m more of a tau guy, anyway, so I celebrate Tau Day (6/28), because I get twice the (pizza) pie.

Today is also Albert Einstein’s birthday, which I’ve always thought was a cool coincidence. He’s 145 now (and still being widely misquoted).

But this post isn’t about him either.

What it is about is documenting a simple bit of math I did a while back and which has been residing in one of my many notebooks ever since. But I’ve been trying to eliminate all that paper, and rather than just toss it out, I thought I’d toss it into a (short? we’ll see about that!) post.

Its genesis was wanting to make a better diagram than this one:

Figure 1. Schematic representation of the double-slit experiment.

Which was intended to illustrate the famous double-slit experiment (where light shining through two small slits in a mask creates an interference pattern on the wall behind it).

When looked at as a wave phenomenon, there’s nothing at all mysterious about the interference pattern.  Sound waves and water waves would interfere in exactly the same way. But light is also a particle, a photon, and when the double-slit experiment is looked at that way, it’s not just mysterious, it’s downright weird.

Famous physicist (and bongo player) Richard Feynman is widely believed to have said of the double-slit experiment that:

It has in it the heart of quantum mechanics. In reality, it contains the only mystery.

But I can’t find the source of the quote. It may be from a lecture that was never transcribed. (I did find some references to YouTube videos, so I may try to track it down. Feynman was a wonderful lecturer, so they’d be fun to watch anyway.)

Along the way I did find one of his most famous quotes:

I think I can safely say that nobody understands quantum mechanics.

~The Character of Physical Law (1965), chapter six, “Probability and Uncertainty”.

And this one that really speaks to me (because I, too, am a hard-core realist):

The test of all knowledge is experiment. Experiment is the sole judge of scientific “truth”.

~The Feynman Lectures on Physics (1964), volume I, lecture 1, “Atoms in Motion”.

I just started reading Mind, Language and Society: Philosophy in the Real World (1998), by the (somewhat socially controversial) John Searle, another hard-core realist. I’m loving it, because he nicely debunks the rejection of causality by David Hume (1711-1776) as well as the rejection of matter by George Berkeley (1685-1753). With all respect to my philosopher friends, I’ve long thought both those old philosophers were just dead wrong.

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But I digress. The thing about the double-slit experiment is that it has been done with objects we think mostly of as “particles” (a term that I think does a disservice to quantum physics — there are no “particles”, just point-like interactions between waves).

Famously, the experiment was done with electrons (by Davisson & Germer as well as independently by George Thomson). It has also been done with larger objects, for instance buckyball molecules (which have 60 carbon atoms and are big enough — about 0.7 nanometers — to be seen with an electron microscope).

So, Feynman had a good point. If you really understand the consequences of the double-slit experiment, you do run smack into the weirdness of quantum mechanics.

Which is all still a digression, but at least it sets the stage. The diagram above is not bad as a schematic representation, but it lacks a certain je ne sais quoi. I used it back in 2020 when I wrote the Wave-Function Collapse post.

[It took me forever to find that post. My WordPress media page (correctly!) told me it was uploaded for the Wave-Function Collapse post, but I misread it as the Wavefunction Collapse post from 2022, which doesn’t get into the double-slit experiment at all. Very confused was I!]

Even back then I wanted something that looked more like this:

Figure 2. Actual depiction of the double-slit experiment.

Which far better illustrates what’s going on but wasn’t something I could whip up in my image and graphics editor (PaintShop Pro, version 7 — an old but still very useful friend from the last century).

No, it required some (simple) math implemented in software capable of generating an image. So, Python (wonderful programming language) and Pillow (capable image generation) to the rescue.

And some basic trigonometry — the math so many hated in high school and were convinced they’d never use.

Basic trig is actually pretty simple [see Trig Is Easy!], but it’s usually taught abominably. Math in general is often taught very poorly, which is why so many fear and hate it. If it were taught better, I suspect more people would be excited about it (because it’s filled with wonderous things).

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Which brings us to the math-y part (and, fine, you can stop reading now… assuming you even got this far).

We treat the light as an electromagnetic wave, specifically, a sine wave (which is where the trigonometry comes in):

Figure 3. Electromagnetic light wave with wavelength λ and moving in the Z direction. The E axis is the electric field, and the B axis is the magnetic field. For our purposes, we’ll only bother with the E field.

The basic task is to determine the intensity (height) of the wave as it travels along a straight line. In Figure 2, to the left of the mask (white vertical line on the left side), a beam of light is coming from some out-of-sight source far to the left. That beam is mostly blocked by the mask except for two slits that allow the light to pass through.

From the slits the light spreads out (as light often does) and eventually hits the screen on the right edge of the image. What’s important here is that the two beams of light interfere with each other (which is just a fancy way to say they combine).

Light always travels in a straight line, so the intensity of the wave for any given pixel of the image is just a matter of figuring where that pixel is in the sine wave. The formula is pretty simple:

Where I is the intensity (or height) of the wave. Note that the height can be positive or negative as the wave oscillates up and down. The sin (sine) function gives us the y (height) value for any given x along the horizontal axis. The simplest expression of that is:

Here we calculate x as the product of the frequency of the light times the distance from some starting point. The frequency is important because it tells us how rapidly the wave goes up and down in a given span of distance.

But we’re not calculating the sine wave along a (single) horizontal axis. We need to figure out the intensity for any pixel, and those pixels are everywhere in the two-dimensional space. That means we need to work with a diagonal line from the source to the pixel.

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Enter good old Pythagoras and his famous theorem (which I referenced recently in The Irrational Square). In English, the square of the hypotenuse of a right triangle is the sum of the squares of the other two sides.

In our case, the hypotenuse is the diagonal line from the light source to the pixel in question. The sides of the right triangle are the horizontal distance from the light source and the vertical distance above or below it.

Figure 4. Two light paths.

Here, I’ll focus on doing that to the right of the two-slit mask, but the same idea obtains on the left. It’s just simpler there because there’s only one light source. With two light sources, we must calculate the intensity from both and add them together.

Since the wave can be positive or negative (or zero or anywhere in between), the sum of two waves can reenforce or cancel each other (or anywhere in between).

Figure 4 illustrates how we approach this. On the left are the two slits in the mask, labeled S1 and S2. We want to know the intensity at some point to the right. I’ve shown the light paths for two such points, labeled P1 and P2.

In fact, we need to calculate the intensity for every pixel along that vertical axis, and we need to do it for every vertical column of pixels in the final image.

The distance from the slits to a given vertical column is L, which give us one of the legs of the triangle. L is the same for both light sources. The other leg is the distance above or below a given slit (not labeled in the diagram, but we’ll call them y1 and y2). The hypotenuses (d1 and d2) are the actual distances from the light source to the pixel.

Put it all together, and we have:

Easy peasy, lemon squeezy! The f parameter is the frequency of the light wave. The square roots are because, while the Pythagorean theorem is:

We want the hypotenuse, not its square, so, taking the square root of both sides:

Which, by the way is how we’d calculate the intensity from a single light source:

Where x is the horizontal distance from the light source and y is the vertical distance (and f is the frequency).

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The only other wrinkle is that the light coming from the slits drops off as the angle increases, so my code also calculates the angle of the line from the slit to the pixel. There’s a term in the final determination of intensity that looks roughly like this:

Where x and y are the horizontal and vertical distances. The arctan function gives the angle, and the cos (cosine) function turns that into a value from 1.0 (when the angle is straight ahead from the slit) to 0.0 (when the angle is 90° from the slit). I squared that to make the falloff sharper.

Lastly, there’s also a term that considers the distance from the light sources such that the intensity decreases with distance. I found that the final image looked wrong without that.

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Once nice consequence of using software to draw the image is that, because the software is parameterized (meaning its variables are all things I can easily change), it’s easy to create different versions of the image. For instance, the frequency can be low:

Figure 5. Low frequency light.

Or high:

Figure 6. High frequency light.

And I can move the slits close together:

Figure 7. Slits close together.

Or put them further apart:

Figure 8. Slits far apart.

And, obviously, I can mix and match those. Which let me create an animated series showing the effect of changing the distance or frequency in action. I made those images into animated GIF (“jiff” damnit, not “giff”) files. I find webpages with animated GIFs distracting and annoying (plus, big ones take time to load), so if you’re curious:

• Frequency animation (3 megabytes)
• Slit separation animation (2 megabytes)

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Now I can toss that page from my notebook — one less piece of paper!

More to the point of this post, despite the adage that “I’ll never use trig in real life!”, I’ve found myself using it a lot in my hobbies and even at work. It’s certainly helpful if you ever want to know exactly how tall your flagpole or tree is.

Stay trigonometric, my friends! Go forth and spread beauty and light.

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