De Broglie's Matter Waves
The de Broglie wavelength is the wavelength, \(\lambda\), De Broglie first used Einstein's famous equation relating matter and energy. The de Broglie relation is taken to define the wavelength of a matter wave. It can only be derived for massless particles, but using the same. The de Broglie wavelength is inversely proportional to the particle .. and 3) runaway of the electron matter from the nucleus due to rotation.
It must be jittering around a bit, but there are few instruments capable of detecting this jitter. As we increase the mass of the object further, the uncertainty in the speed of the object becomes completely negligible.
So while the uncertainty principle also holds for massive, macroscopic particles, it has no practical consequences. Dispersion Photons always move with the speed of light c.
Deriving the de Broglie Wavelength
If we build a wave packet from plane waves of the above form, the speed of the wave packet is c. Particles can move with any speed. This relationship is called the dispersion relation for free particles. This may seem surprising. Our classical intuition suggests that a wave describing a particle moving with speed v should move with speed v itself.
de Broglie concept of matter waves: dual nature of matter
How else can it track the particle? It is impossible to track a particle whose momentum is exactly known. The dispersion relation for free particles implies that the plane matter waves of particles do not all have the same speed. The wave function for a particle for which we have some position information is a wave packet.
We have to superimpose plane waves describing particles with slightly different momenta and energies, or with slightly different wavelengths and frequencies. But these component waves now all move with slightly different speeds.
So what happens to the shape of a wave packet as time goes on?
Let us investigate using a spreadsheet. A moving square pulse The spreadsheets shows the moving wave packet and the dominant component wave. We observe the wave packet changing shape as time progresses. We also observe that the wave packet travels with a higher speed than the component wave. The most likely speed of the particle is the speed of the center of the wave packet.
This speed is approximately twice the speed of the component waves. Electron Diffraction The first experimental verification of de Broglie's hypothesis came from two physicists working at Bell Laboratories in the USA in They scattered electrons off Nickel crystals and noticed that the electrons were more likely to appear at certain angles than others.
The work was carried out by Clinton Davisson and Lester Germer.
- Matter wave
- De Broglie wavelength
The Davisson-Germer apparatus is a vacuum glass tube which has in its interior an accelerator of electrons, a known crystal structured substance as a target and an electron detector. The figure on the right shows a simplified sketch of the experimental setup.
This simple apparatus send an electron beam with an adjustable energy to a crystal surface, and then measures the current of electrons detected at a particular scattering angle theta. The electron beam struck a crystal target and was diffracted. They don't necessarily have to be at a particular point at a particular time. This is a little misleading, this picture here, I'm just not sure how else to represent this idea in a picture that they only deposit their energies in bunches.
So this is a very loose drawing, don't take this too seriously here. But people had already discovered this relationship for photons. And that might bother you, you might be like, "Wait a minute, how in the world can photons have momentum?
Because parallel to all these discoveries in quantum physics, Einstein realized that this was actually not true when things traveled near the speed of light. The actual relationship, I'll just show you, it looks like this.
The actual relationship is that the energy squared, is gonna equal the rest mass squared, times the speed of light to the fourth, plus the momentum of the particles squared, times the speed of light squared.
This is the better relationship that shows you how to relate momentum and energy. This is true in special relativity, and using this, you can get this formula for the wavelength of light in terms of its momentum. It's not even that hard. In fact, I'll show you here, it only takes a second. Light has no rest mass, we know that, light has no rest mass, so this term is zero.
We've got a formula for the energy of light, it's just h times f. So e squared is just gonna be h squared times f squared, the frequency of the light squared, so that equals the momentum of the light squared, times the speed of light squared, I could take the square root of both sides now and get rid of all these squares, and I get hf equals momentum times c, if I rearrange this, and get h over p on the left hand side, if I divide both sides by momentum, and then divide both sides by frequency, I get h over the momentum is equal to the speed of light over the frequency, but the speed of light over the frequency is just the wavelength.
And we know that, because the speed of a wave is wavelength times frequency, so if you solve for the wavelength, you get the speed of the wave over the frequency, and for light, the speed of the wave is the speed of light.
So c over frequency is just wavelength. That is just this relationship right here. So people knew about this. And de Broglie suggested, hypothesized, that maybe the same relationship works for these matter particles like electrons, or protons, or neutrons, or things that we thought were particles, maybe they also can have a wavelength. And you still might not be satisfied, you might be like, "What, what does that even mean, "that a particle can have a wavelength?
How would you even test that? Well, you'd test it the same way you test whether photons and light can have a wavelength. You subject them to an experiment that would expose the wave-like properties, i.
Deriving the de Broglie Wavelength - Chemistry LibreTexts
So, if light can exhibit wave-like behavior when we shoot it through a double slit, then the electrons, if they also have a wavelength and wave-like behavior, they should also demonstrate wave-like behavior when we shoot them through the double slit. And that's what people did. There was an experiment by Davisson and Germer, they took electrons, they shot them through a double slit.
If the electrons just created two bright electron splotches right behind the holes, you would've known that, "Okay, that's not wave-like. Davisson and Germer did this experiment, and it's a little harder, the wavelength of these electrons are really small.
So you've gotta use atomic structure to create this double slit.
It's difficult, you should look it up, it's interesting. People still use this, it's called electron diffraction. But long story short, they did the experiment. They shot electrons through here, guess what they got? They got wave-like behavior. They got this diffraction pattern on the other side.
And when they discovered that, de Broglie won his Nobel Prize, 'cause it showed that he was right. Matter particles can have wavelength, and they can exhibit wave-like behavior, just like light can, which was a beautiful synthesis between two separate realms of physics, matter and light.
Turns out they weren't so different after all. Now, sometimes, de Broglie is given sort of a bum rap. People say, "Wait a minute, all he did "was take this equation that people already knew about, "and just restate it for matter particles? If you go back and look at his paper, I suggest you do, he did a lot more than that. The paper's impressive, it's an impressive paper, and it's written beautifully.