Deriving the de Broglie Wavelength - Chemistry LibreTexts
Louis de Broglie, in full Louis-Victor-Pierre-Raymond, 7e duc de Broglie, (born August 15, , Dieppe, France—died March 19, , Louveciennes), French . The de Broglie equation relates a moving particle's wavelength with its momentum. Louis de Broglie() put forward an idea which is known as de Broglie. hypothesis of Louis de Broglie (): particles may have wave-like properties. • note: it The wavelength of this matter wave is given by the de Broglie relation.
The de Broglie wavelength, he figured it out, and he realized it was this. So, he actually postulated it. He didn't really prove this.
De Broglie wavelength
He motivated the idea, and then it was up to experimentalists to try this out. So he said that the wavelength associated with things that we thought were matter, so sometimes these were called matter waves, but the wavelength of, say, an electron, is gonna be equal to Planck's constant, divided by the momentum of that electron.
And so, why did he say this? Why did he pick Planck's constant, which, by the way, if you're not familiar with Planck's constant, it is like the name suggests, just a constant, and it's always the same value, it's 6.
This was a constant discovered in other experiments, like this photoelectric effect, and the original blackbody experiments that Planck was dealing with. It's called Planck's constant, it shows up all around modern physics and quantum mechanics.
So how did Louis de Broglie even come up with this? Why Planck's constant over the momentum? Well, people already knew for light, that the wavelength of a light ray is gonna also equal Planck's constant, divided by the momentum of the photons in that light ray. So the name for these particles of light are called photons. I'm drawing them localized in space here, but don't necessarily think about it that way.
Think about it just in terms of, they only deposit their energy in bunches. 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.
De Broglie wavelength (video) | Khan Academy
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.
- Louis de Broglie
- Deriving the de Broglie Wavelength
- Matter wave
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. 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. During this same period de Broglie's brother Maurice was studying experimental physics, and he was particularly interested in x rays.
The brothers frequently discussed x rays, and their dual nature both wavelike and particle-like behavior suggested to Louis that this same particle-wave duality might also apply to particles such as electrons. The following year American physicists Charles J. Davisson and Lester H.
Germer and others demonstrated experimentally that electrons can be diffracted just like light. That is, as electrons pass through a narrow slit, they spread out in a wavelike pattern similar to that of diffracted light.
De Broglie accomplished his most important work in physics while still a young man, receiving the Nobel Prize in physics in In he also became professor of theoretical physics at the Sorbonne, retiring from that post in Throughout his long life, de Broglie remained active in the development and interpretation of quantum mechanics and wrote more than twenty-five books on various topics related to this field of study.
As a member of the French Commission on Atomic Energy, he was a long-time advocate for the peaceful use of atomic power. He was the recipient of many awards and honors for his work in quantum mechanics. As a young scientist de Broglie had believed that the statistical nature of modern physics masks our ignorance of the underlying reality of the physical world, but for much of his life he also believed that this statistical nature is all that we can know.