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Your Daily Equation
Your Daily Equation #09: De Broglie Wavelength

#YourDailyEquation with Brian Greene offers brief and breezy discussions of the most pivotal equations of the ages. Even if your math is a bit rusty, these accessible and exciting stories of nature and numbers will allow you to see the universe in a new way. The series includes live Q+As that explore many of the big questions that have occupied some of the greatest thinkers of our age and yielded some of the deepest insights into the nature of reality. Particles and waves collide in quantum physics. In today's episode of Your Daily Equation, Brian discusses the famous double slit experiment and explains the formula that connects particles and waves: the DeBroglie wavelength equation.Learn More

Speaker 1:

Hey everyone. Welcome to today’s Your Daily Equation, and when I asked a little while ago for your favorite equations, equations that people really wanted me to say a few words about, the overwhelming winner of that little informal survey were the equations of quantum mechanics. Schrödinger’s equation, de Broglie equation, things of that sort. So indeed, I’m going to focus on one of those equations today. In fact, it is the de Broglie formula for the wavelength of a particle, and I’m going to jump into the subject in a somewhat ahistorical manner. Then I’ll backtrack and fill in a little bit of the history.

I’m going to begin with a modern day version of an experiment. It’s called the double slit experiment, and I’ll describe the original version of that experiment as it was carried out back in the 1920s but let’s first begin with this very pedagogical experiment. In fact, the famous physicist Richard Feynman once famously said that in the double slit experiment, you kind of see all of quantum mechanics right there, and that’s kind of rare for one single experiment to embody an entire deep and profound subject, but that’s what this experiment is able to do.

Okay, so with that buildup, I think many of you are familiar with it. What is the double slit experiment? Well, let me just bring up a version of it here. Imagine you begin with some kind of gun firing pellets or BBs at a barrier that has two slits, the double slits. What do you think the data will look like in terms of the pellets that make it through? Well as you see on the screen, it’s just what you would think, right? You’ve got some of the pellets that land in a band on the left aligned with that opening, that slit, and you got other pellets that land in a band on the right, aligned with that opening in the barrier, that slit. So this is the double slit experiment carried out with macroscopic objects, pellets, BBs, bullets, whatever. I’m now going to make one little change, and that change is I’m simply going to dial down the size of the projectiles. I’m going to take those metallic pellets, those BBs, dial them down so that they are the size of little particles, little electrons.

Now naively, you would think that simply changing the size of the projectiles should not in any way have some kind of profound impact on the results. So you would think, let me bring it up here, that this is what would happen. Again, those electrons that make it through the left opening land in a band on the left. Those that make through the right should land in a band on the right. Straightforward intuitive logical reasoning leads you to that conclusion.

However, when you actually do the experiment, here is what actually happens. You don’t just get two bands aligned with the two slits in the double slit experiment. Instead, as you see, you get a whole series of bands. In fact, you get a series that goes from dark, to bright, to dark, to bright, dark, bright, and so forth. And the fact that you don’t just get two bands, the fact that you get a whole series of bands, is one of the greatest revolutionary moments in physics because this portends a completely different way of thinking about matter, which itself yields a completely different way of thinking about reality. So where do we get that kind of a deep insight from?

So we look at this data. This bright, dark, bright, dark data. We stare at it and we’re like, “What in the world is going on?” And then a bell kind of rings if you’ve had any training or any study of the motion of waves, right? Because you no doubt are familiar that if you for instance have a pebble and you throw it into a pond, it sets off a nice series of concentric waves that spread outward. These circular waves. If you throw a second pebble into that pond nearby, then those two sets of concentric waves will crisscross, and when they crisscross something very interesting happens.

At those locations where the peak of one wave crosses the peak of another, the wave is high. Those locations where the trough, the bottom of one wave crosses the trough of another, very deep. But there are also locations where the peak of one wave crosses the trough of another and they cancel each other out, and at that location, the water won’t be agitated at all. So for those of you who had a deprived upbringing and never threw pebbles into a pond, let me just show you that. I have a little visual of that one here.

Imagine you throw a little pebble on the right and there you’ve got the concentric water waves going around. Now throw another pebble in on the left, and here you have the crisscrossing of the water waves, and again, there are locations where the wave comes together from the left and the right sources and make the water highly agitated. But there are places in between, and you can sort of see them here. I’ve highlighted them in these dark regions, where the waves are canceling each other out. So in essence you have a lot of agitation, a little agitation, no agitation, a lot of agitation, little agitation, no agitation. And that is very similar to the bright, dark, bright, dark pattern that we saw in the double slit experiment with electrons.

Let me even make that connection a little bit more obvious. Here is a version of throwing a pebble into a pond. One single pebble. I have one single opening, and as you see, the water’s going through that opening, causing these concentric waves. And on that back screen, I’m using the brightness of the screen to indicate the agitation of the water, so it’s highly agitated where it’s very bright and as you head off left and right, the waves die out and it becomes darker on the screen. So that’s like throwing one pebble into a pond.

Now we throw two in. How do we do that? I have two openings. The two concentric waves are now crisscrossing and as I described, you get regions of high agitation, little agitation, and no agitation. That’s the surprising thing where the waves crisscross, the top of one crosses the bottom of another, and now if on the screen I also use brightness to indicate the agitation of the water, look at what we have. Highly agitated, bright, dark when there’s no agitation, bright, dark, bright, dark. Look at that back screen. It’s just like the data that we had when we fired particles at the barrier with the two openings, the barrier with the two slits in it.

So we seem to be coming upon some weird unexpected wavelike phenomenon, but in an experiment when we are firing particles at the barrier with the two openings. Now I could imagine at first sight you say, “Hey, you know, sounds surprising, but if you think it through it’s actually not that surprising because we all know that water is itself composed of particles, composed of molecules, H2O molecules and so even in the experiment with water, we are actually throwing particles at the barrier with the two openings, and so maybe the fact that we get this interference pattern which is the technical term for the bright, dark, bright, dark, the interference of the two waves from the two openings, maybe it isn’t surprising that we also get that when we fire a different species of particles, not H20 molecules, but little electrons,” and that’s a reasonable assumption, or assumption’s the wrong word. That’s a reasonable suggestion, a better word, but let’s follow it through and see whether that could be the answer.

Could it be that in the experiment with the electrons, it’s just that the electrons are acting like water waves, which are highly choreographed motion of H2O molecules? Maybe the electrons are following a highly choreographed motion and you’re sort of getting a wave from the large collection of electrons just like you can get a wave from a large collection of H2O molecules. How would you test that?

Well, go back and do the double slit experiment firing one electron at a time. If you fire one at a time, then there’s no opportunity for a large collection to somehow bang into each other and yield this pattern from their choreographed motion. So let’s do that. So fire one electron at a time, and at first it’s just a random sequence of dots that are forming on that detector screen, but if you wait long enough and allowed the dots to build up over time, holy smokes. Look what happens. You recreate the interference pattern, and in this case, it’s not as though a wave from one opening is crossing literally a wave from the other if you’re thinking purely in the particle language.

Here the individual particles dot by single dot are building up that interference pattern, which tells us that even individual particles have some intrinsic wavelike quality because they’re yielding wavelike data on that detector screen being built up single particle by single particle, so your suggestion was a good one that maybe it was a large number of electrons, like a large number of water molecules somehow banging off each other, but those individual electrons are not banging off each other. They’re hitting that screen individually one at a time, yet we’re still getting the interference pattern.

Now look, I could also imagine you say to me, “Hey, nice animation, but does this actually happen in the real world?” I mean, a good question is how many of you have actually ever seen the double slit experiment really done? I mean, not just done in pictures in a textbook or animation, seen it really done. And I have to tell you, until I don’t know, about eight or ten years ago, I’d never actually seen it done.

I’m a theorist. I sort of stay away from equipment, but I had a show which basically it’s going to go on the road at some point I think in the next year or two called Spooky Action, a stage show, and in that stage show I brought out the double slit experiment, did it in real time in front of the audience, and I have to tell you the audiences were often well versed in the ideas of quantum mechanics, but when the curtain went up and the actual double slit experiment was undertaken on stage, there were oohs and aahs. It’s sort of like people were shocked to actually see this thing in real life, so let me show you some real data.

I’d love to set up the double slit experiment behind me right now and do it in real time, but it can’t actually do that. I don’t have the equipment here, but let me show you some real data that was taken I think first actually in the 1980s. Even though these are old ideas, this version of the experiment was really only first done in the 1980s. A group at Hitachi gathered together this data, and let me just show you it. This is these dots here now, not animation. These dots are actual landing locations of actual electrons in an actual version of the double slit experiment, and look. You sort of have a constellation of random dots at first.

I’m going to speed this up because we’d be sitting here for a long time, and look, the real world is never as clean and pristine as a theoretical world, the animated world that is based on equations, but even though the data is a little bit noisy and messy compared to the clean pictures that I showed you, look. You can see the interference pattern here, right? You can clearly see that there are brighter regions and darker regions and brighter regions. You see that band-like interference pattern, even in real world data of the double slit experiment.

So as I mentioned, this was I think first done in the 1980s. The experiment itself of course was carried out first in the 1920s. In fact, I can show you one of the early papers of Davisson and Germer who carried out this experiment at Bell Labs and point out to you, I don’t know if you can see it on the screen but you can google this paper online of course, but if you look at the first paragraph after the abstract, it says, I’ll just read it to you. “The investigation reported in this paper was begun as the result of an accident which occurred in the laboratory in April, 1925.” An accident, right?

Now, most physicists today, if there was an accident that yielded an interesting result, they kind of wouldn’t tell you it was an accident. They’d say, “Hey, I was sitting in my office and I had this inspired idea and I just carried it out here.” We typically don’t give you the behind the scenes look any longer, so this is kind of beautiful back in the 1920s that they were willing to tell the story of what actually happened, and the story is they were firing electrons at a nickel crystal, okay? But they turned up the intensity so high that the evacuated chamber, which is surrounded by glass, exploded. And when it exploded, the nickel sample got tarnished by the air that rushed in, so they clean up the nickel sample by heating it to get off that tarnished surface.

Unwittingly by heating up the sample, they changed it. They cause the scattering centers to blend, to meld together into larger and more separated scattering centers, which as it turns out gives you a version of the double slit experiment. Not completely obvious, but there’s a direct link between these two ideas of scattering electrons off of this nickel crystal that’s been modified in the manner that they did by heating it and the double slit that I showed you, the more intuitively obvious version, but in the end they’re actually the same. And the data that they acquired, let me just show you to it on the next page here. You can see in the graphs that you have this up and down, this bright, dark, bright, dark pattern in the data.

So this is this paper written in 1927 and what Davisson and Germer had unwittingly done is actually confirmed an idea that had been put forward by somebody else, Prince Louis de Broglie, a very brilliant physicist who was thinking about Einstein’s ideas and the ideas that were emerging from conversations of people like Niels Bohr, one of the founding fathers of quantum mechanics, and came up with the following idea which finally is our equation of the day, our daily equation, so let me just show you this over here. So what was de Broglie doing? De Broglie said, “Hey, there’s something very interesting that happened in the photoelectric effect with Einstein.” What happened there was Einstein said, “Hey, you normally think about light as a wave, but now I’m asking you to think about light as a particle, as photons to explain the data in the photoelectric effect.” So we kind of went from waves to particles. De Broglie says, “Hey, maybe the reverse is also true. Maybe things that we normally think about as particles may also be waves,” and he kind of reasons in the following way.

He says, “Look, in the photoelectric effect, we learned that E equals h nu for the energy of a photon where the light has frequency nu.” Now there’s also some other good equations to record, so the speed of light is equal to the wavelength of light times its frequency, and there’s also another little equation that’s good to bear in mind which is that energy can be written as momentum times c when you’re talking about light and look, there’s an interesting whole other equation that’s worth thinking about. Let me just record it over here, which is E squared minus p squared c squared equals m naught squared c to the fourth.

This is another version of basically E equals m c squared, and the reason it’s good to have this is people often say, “Hey, if light is composed of photons and photons don’t have any mass and E equals m c squared, if mass is equal to zero, then we should also have energy is equal to zero.” But that’s actually not true. The rest mass of photons is equal to zero, but that then just leads the equation E squared equals p squared c squared, or E equals pc as I wrote down over there.

So that’s another equation. Maybe at some point I’ll spend a little bit of time on that, but the bottom line that I’m driving at right here is de Broglie says, “Hey, if you look at all these equations, then we can write E as pc equals h nu or we can write p equals h nu over c, but nu over c from this equation over here is just equal to one upon lambda, one divided by the wavelength. So that means p is equal to h over lambda, or lambda is equal to h over p,” and that is the equation that is de Broglie’s formula.

Now he says, “Look, I may have derived this in the context of photons, but let me imagine, let me conjecture, let me hypothesize, let me propose that this is a more general formula that might work for particles of matter.” So a particle of matter, like an electron, has a certain momentum, right? Non-relativistically, you may recall from high school p is equal to m times v. That v is not a nu. That’s the velocity of the particle. So non-relativistically p equals mv, and you could stick that p in this formula and from that you’d have a wavelength associated with a particle, and there you have a nice little formula that says lambda, the wavelength of a wave associated with a particle, is given by this number h. Oh, remember, this is Planck’s constant. Perhaps I should have said that. It’s just a number which is about 6.6 times 10 to the -34 joule seconds. Just a number that comes out of experiment, and the beautiful thing is if you use this formula, clearly I have an upcoming appointment that you just saw on that little reminder so I better finish this up.

If you use this formula in trying to understand the data that comes out of say the Davisson Germer experiment, then the separation between the peaks and troughs, the dark and light bands in the interference pattern, is dependent upon the wavelength of the waves that are going through the two slits. And indeed, this formula allows you to explain the data that Davisson and Germer had found unwittingly in that experiment that emerged from the accident.

So you see that things are really coming together now in the 1920s. You have de Broglie suggesting that not only can waves look like particles, but particles can look like waves and actually gives a formula for the wavelength associated with a particle of a given momentum, and then in this experiment by Davisson and Germer, they actually find that electron, through what amounts to a double slit, behave like particles whose wavelength is indeed given by this number h divided the momentum of the particles, confirming the idea that de Broglie had put forward.

This is a giant step forward in the trajectory toward the quantum laws of physics. There are a few vital steps still to go that we will cover in subsequent episodes, but this profound formula that comes from de Broglie that the wavelength comes from h over p is a key step in that progression and we’ll carry on in subsequent episodes with the Schrödinger equation and the probabilistic interpretation of quantum mechanics. All that is coming up, but for today that’s all I wanted to say, so until next time, take care. This is Your Daily Equation.

Your Daily Equation
Your Daily Equation #09: De Broglie Wavelength

#YourDailyEquation with Brian Greene offers brief and breezy discussions of the most pivotal equations of the ages. Even if your math is a bit rusty, these accessible and exciting stories of nature and numbers will allow you to see the universe in a new way. The series includes live Q+As that explore many of the big questions that have occupied some of the greatest thinkers of our age and yielded some of the deepest insights into the nature of reality. Particles and waves collide in quantum physics. In today's episode of Your Daily Equation, Brian discusses the famous double slit experiment and explains the formula that connects particles and waves: the DeBroglie wavelength equation.Learn More

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