Plotting Feynman’s Amplitude Spiral after 50 years

I’m tickled pink to have the honor to be (so far as I know) the first person to publish a 3D rendering of one of Feynman’s favourite experiments, and hopefully make a key concept in quantum physics much more tangible to people.

First, here’s the shapes, and then I’ll explain what they actually mean. I’ll also do a follow up post with various ways for making your own 3D models of these:

Feynman Amplitude Spirals – The double slit experiment in 3D

A bit of background. In the early 1960s the kickass physicist Richard Feynman (just a couple of years before he was awarded the Nobel prize) was given the task of teaching undergraduate physics at Caltech. Many years later and the lectures are still remarkably up to date, as well as now being available freely online. (http://www.feynmanlectures.caltech.edu/)

When the class got up to the subject of Quantum physics, in the very first lecture, Feynman wisely decided to teach it in the reverse order to how it was previously done.

The double slit experiment was supposed to be an advanced topic not covered till later, but he felt that it laid a bedrock that was useful and understandable to someone with no previous background in quantum physics.

Feynman introduced it as as:

“…we shall tackle immediately the basic element of the mysterious behavior in its most strange form. We choose to examine a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery. We cannot make the mystery go away by “explaining” how it works. We will just tell you how it works. In telling you how it works we will have told you about the basic peculiarities of all quantum mechanics.”

(emphasis mine)

Feynman went on to describe the double slit experiment, and explain it in a hugely vivid and memorable way. First by comparing what would happen if electrons were ordinary particles (simulated by firing a machine gun randomly at the wall), and what would happen if they were waves, etc. His lecture is well worth a read. (http://www.feynmanlectures.caltech.edu/III_01.html)

Side note: Any good explanation should involve noticing a feeling of confusion about something, examining all parts of it closely, and then not only finding ways to resolve that confusion, but also seeing what other things get explained at the same time. If you’re not confused by something, you don’t need an explanation. If you’re still confused at the end, that’s your brain giving you a hint that there’s stuff left unresolved, and you haven’t found the actual answer.

Anyway. I’m not planning to re-explain Feynman’s awesome intro. Instead, what I want to focus on his diagram below. This blog post should still hopefully be understandable for anyone that hasn’t read his lecture, but be a nice add-on for anyone that has.

I’m also redrawing his experiment just a little (to use photons rather than electrons), but the same principles work equally well for both.

Here’s the setup. It’s super simple, and can be made with about $5 in parts nowadays:

  • A “laser”,
  • two slits in a wall,
  • and a screen.

Light goes from the laser to the wall, and through the slits to the screen.

feynman double slit layout v01.png

By opening and closing the slits, you change the pattern that appears on the screen. Here’s the results you get:

  • If slit 1 is open and slit 2 is closed, you get a smooth curve (P1)
  • If slit 1 is closed and slit 2 is open, you get another smooth curve (P2)
  • If both slits are open, then you get a really lumpy curve (P12)

Notice how some of the valleys of P12 are lower than any point on P1 or P2. What. the. hell.? If one slit lets some amount of light through, then how does opening a second slit, which presumably can only let more light through, cause some parts of the screen to become darker?

I mean, we’d expect that if both slits are open then the amount detected would be the sum of P1 and P2. Or, to put it as bluntly as possible:  How the hell can two smooths add together to make a lumpy?

In other words, we’d expect the system to behave like this:

OK, let’s double check our intuition in another situation. I’m walking through the woods at night, finding my way with a torch. If I ask my friend to turn on a second torch, do we see any parts of the path suddenly get darker? Err.. hell no, everything just seems brighter. Hmmm… OK, I notice I’m confused, so I’m going to ‘bookmark’ this confusion to think about later.

Oh, and there’s one last bit of confusion to throw into the mixture. Back in the lab, we can add more slits in the screen and see the pattern change yet again. We’d want whatever explanation we come up with for the 2 slit case to also be able to explain situations like these:

feynman multislit mosaic v01.png

Now, back to the lecture. Feynman goes on to explain how the flat curves (made of real numbers) make no sense and to get the right answer we must let the contributions from each slit be composed of complex numbers (e.g. be a vector with a both a direction and a length, rather than a bar graph that only has a length.)

Or, to put it graphically, we can explain our graphs by making a bunch of vectors. So, if at each point the green is the sum of the orange and blue:

feynman vector annotation v01.png

OK, well that sounds like it has the potential to explain some things. But how exactly do we figure out the phase at each location?

Here’s the simple version of the equations for calculating it. It neglects phase changes over time, etc. but it’s good enough to give us the shape we need.

Complex Amplitude of a Single Slit

For those that are mathematically inclined, you can see the length is given by the sinc function, and the phase is just a complex rotation based on the ‘flight time’ it takes for the photon to get from the slit to the screen.

So when we graph these functions, here’s what appears on the screen. There’s a contribution due to slit 1, a contribution due to slit 2, and the green is the sum of the two:

Feynman Amplitude Spirals

It looks like we’re pretty close to understanding the lumpiness of our 2D graphs. There’s just one last trick we need to use before we start plotting everything. We need a way to ‘smush’ our 3D graphs back down into 2D. To do that we use the rule that the height of the 2D graph at any point is the absolute square of the vector length. This is known as the Born Rule and is a keystone of quantum physics. It’s written like this:

Now we have all the tricks needed to calculate the whole thing, and see if it matches the flat graphs that started us off.

Here’s my animation showing the process. I’d recommend watching it a couple of times to get the full  effect:

Does it explain our data? Yes, it’s a perfect match!

I also notice I’m no longer confused about how the ‘lumps’ in our graph came about. They’re simply the result of the two spirals being added together, mostly cancelling out at some points, and strongly reinforcing at others points.

Let’s go back to the multiple slit case. Do we still get the right result for the 4-slit case?

feynman multislit mosaic 4slit v01.png

4 slit case & the resulting pattern on the screen

Yep, another win. So once we found the right representation for what each slit provided (a complex number), we’re able to tackle situations with any number of slits (or possible paths the photon could take) just by using ordinary addition. 

So, thinking of things as having complex probability amplitudes rather than (real) probabilities was the key to making it work. I want to quickly cover the rules for doing this, since it’s the source of a lot of confusion about quantum physics. Here are Feynman’s rules for probability amplitudes:

1. The probability of an event is the absolute square of a complex quantity called probability amplitude.

2. When an event can occur in several alternative ways, the probability amplitude is the sum of each probability amplitude for each way considered separately.

3. If an experiment capable of determining which alternative is actually taken is performed, the interference is lost and the prob becomes the sum of the prob for each alternative.

So in other words, we have to be super careful where we apply the Born rule. Here’s a summary:

Guide to the Born Rule

So we’d use the first method  in any situation where we have two sources interfering, and where there is no way, even in principle, to tell which path the photon took. The interference is clear as day, and you’ll see a lumpy curve.

And we use the second method in any situation where there’s some way to ‘know’ which slit the photon went through. For example by putting different polarizing filters on each slits (which is sometimes described as ‘labelling’ the photons with their path information), we could then put another filter on our detectors, and tell which slit the photon came through. If we do that, there’s no interference pattern, and we only see a smooth curve.

Big, important point here. It doesn’t matter if you actually bother to measure the polarization of the photon. If your detector was switched off, or broken, or you forgot to look at the dial or something,  it still doesn’t matter. As long as there’s any way, even just in principle, to tell the which path the photon took, then the complex amplitudes don’t flow to the same final configuration, and the interference pattern disappears.

(Side note: Those two scenarios are the two extremes of something which is actually continuous.  If we had a sensor that performed a ‘shitty labelling’ of the photons, we would see a pattern somewhere in between the smooth and lumpy curves. Feynman explains this better in the lecture, but here’s my graph:)

feynman path detection v01.png

I apologise for my drawing skills

 

There was one final piece of confusion we haven’t tackled yet. Remember the night walk in the woods we talked about before? We noticed that turning on second torch didn’t make any parts of the walking track darker. So what’s up with that? Why did that situation show no apparent interference, but the one in the lab did?

The answer is that double slit experiment only shows the dramatic interference effects when the light sources are coherent, i.e. they have the same frequency and a fixed phase relationship. If we (for some reason) tried to walk along the dark path using only a handheld laser pointer, we’d immediately see the light was speckly, and that the beam had lots of lumpiness to it. And if our friend turned on their laser, we actually would see that some parts of the dirt track did indeed get darker! (as well as some other parts getting lighter, so that the average brightness increased)

(OK, side note for anyone as pedantic as me. While you can see laser speckle really easily with a single laser, strictly speaking any two handheld lasers are almost certainly not reliable enough frequency sources to see that kind of interference effect. It’s still doable in theory, but kinda tricky in practice. Rather than a $3 keychain laser pointer, you’d have to be hiking through the woods with something more like, say, an atomic clock strapped to your back, with a special mechanism that used the clock reference to tune a lab grade laser cavity, which had been modified to take advantage of Zeeman splitting of electron levels in a magnetic field, and has its cavity length controlled through servo feedback in order to guarantee a fixed phase relationship in the output beam. Easy, no?)

So we’re pretty much done. Hopefully the double slit seems a bit less mysterious now, and I hope my plots made it a bit clearer.

Oh yeah, and since (so far as I can see from checking a bunch of textbooks, talking to a couple of quantum physicists, and google image searching various keyword combinations) I’m the first person to plot this 3D shape, I’m going to take the opportunity to name it. I hereby dub it:

“The Feynman Amplitude Spiral”

If anyone knows of somewhere it’s been properly plotted before, please drop me an email? It’s obviously related to the Cornu spiral, but Cornu/Euler spirals are usually plotted as the sum of hundreds of vectors, where this is the sum of only two, and also we’re sweeping this curve in 3D as we vary the screen height.

I still can’t quite believe it took 50+ years for someone to get around to plotting it, advances in computer & display technology notwithstanding. I’m never going to be able to think about the double-slit experiment without seeing this shape.

Edit: There’s now a part 2, on making your own spiral

This entry was posted in physics, Uncategorized. Bookmark the permalink.

3 Responses to Plotting Feynman’s Amplitude Spiral after 50 years

  1. Pingback: Feynman Spirals Part 2 – Making Models | Tinkerings

  2. Pingback: Feynman Spirals Part 3 – FAQ | Tinkerings

  3. Pingback: Why the Bragg Condition Sucks | Tinkerings

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