14.4 deBroglie Wavelengths

Okay, our last topic for this chapter in Unit is called the de Broglie wavelengths. de Broglie was a guy who sort of threw something out there and ended up being true. His logic was, "Well, if we said that anything that was a wave can be thought of as a particle, right? So anything that was light or whatever that we knew was waves but now can act like a particle due to Einstein and Planck and the photoelectric effect and Compton effect, and all that. de Broglie said, "Well, the reverse must be true. Two things that we know are actually particles, are they actually traveling as a wave?" And he published that. Nobody really took him seriously, but Einstein read it and kind of agreed with his ideas. He's saying basically, if it works one way, it should work the other way. So what de Broglie did was he combined the momentum of a particle with the momentum of a photon, put those two together, and he said, "Well, what if instead of light just acting like a wave, particles can also act like waves?" And that's what we call the de Broglie wavelength. We can actually calculate the wavelength of a particle by going h over mv. So, what it does is give us the ability to calculate the wavelength of any kind of particle that's moving. So, let's do our first example. Let's actually figure out what the Broglie wavelength of a baseball is. If you throw a baseball, what that means is we have a particle that's moving. It's got mass, it's got velocity. But we can actually calculate its wavelength. So, if we plug these numbers in, we get h (6.63 x 10^-34) divided by the mass of the baseball (0.2) times its velocity (15). If we actually do that calculation, we get a wavelength of 2.2 x 10^-34 meters. So, even though according to this, our baseball is actually traveling as a wave as we're throwing it, but the wavelength is so tiny (2.2 x 10^-34). We'd never be able to see it. It's gonna look like it's just traveling in a straight line. So that was the growing belief's theory. Einstein agreed with it, but the hard part was to prove it. So, remember with diffraction, we basically had to get a diffraction grating small enough that would match the wavelength of light, and then we could actually see the interference patterns. So, in Kong, it's sort of the same thing. We'd have to get a diffraction pattern to the negative 34. Well, that's too hard to create, so that's not going to work. That's when de Broglie came up with the idea, saying, "Well, what happens if we use a smaller particle?" We know an electron is a particle, so let's actually try to figure out the wavelength of an electron that's moving. So, if we go for this question, we've got 100 volts. The first thing we're going to do is figure out what the velocity is going to be when it goes through a potential difference. So here's a good review question: Remember, we have 100 volts. Our electron is going to Move across the plate. So, what we did was, we said our energy equals QV. And we know the energy of the electron is going to be converted to one-half MV squared. So, our first step is to figure out our energy, which is 1.6 times 10 to the negative 19, multiplied by 100 volts. That gives us 1.6 times 10 to the negative 17. Then, we want to take that energy and convert it to one-half MV squared. So, we plug that in and solve it for velocity. I won't bother doing this step; you can do that on your own. Just put that energy divided by one-half times the mass of an electron. We get a velocity of 5.93 times 10 to the 6. Now that we have that, we can basically calculate our wavelength. So, our wavelength would be H over MB. We just go H divided by the mass of an electron times the velocity of the electron. Electrons, and we end up getting a wavelength of 1.23 x 10^-10, so still very small. But these are actually somewhat doable. So there are a couple of guys that said, well, they just by doing some measurements of the size of a crystal. So like, a clear metal ends up having gaps of about that size. So what they did was they would basically shoot electrons through this crystal, and they actually saw the fractured pattern. So by getting a diffraction pattern, then we knew that these particles, these electrons that are being shot, had to be going like beads just going through a diffraction grating. And we ended up getting a pattern that you see in this picture. It ended up giving us just like a Poisson spot or just like wave interference. So what these guys proved and what de Broglie predicted ended up being exactly the same. So indeed, de Broglie was right, basically saying that... Anything that's a particle that's traveling is actually going to be traveling in a wave pattern. So, this is sort of a final tie to these two topics: that all things that are waves are actually particles that are traveling in the wave, and all things that are particles are actually traveling in a wave pattern instead of in a straight line. And that's it for this unit and topic.