PhD., University of Maryland
Jonathan is a published author and recently completed a book on physics and applied mathematics.
De Broglie wavelength is the wavelength associated with a matter wave. Matter, though it can behave like particles, also behaves like a wave. Both light and matter behave like a wave on a large scale and like a particle on a small scale. To calculate the matter wave, we use the formula de broglie wavelength = planck's constant / momentum.
So let's talk about De Broglie wavelength. The De Broglie wavelength is named after Louis De Broglie who wrote about it in 1924, actually in his doctoral dissertation and he actually won a Nobel Prize for it in Physics making him the first guy to win a Nobel Prize in Physics for something that he did in his doctoral dissertation.
But anyway, what did he do? Well, what he said was that matter behaves like a wave. So the De Broglie wavelength is the wavelength that's associated with matter. Now this is kind of strange because this was in 1924 and in 1905 Einstein had told us that light which is supposed to be a wave behaves like a particle. So we've got light behaving like a particle, now we got matter behaving like a wave? What's going on? Well, it turns out that everything's kind of mixed together at the fundamental microscopic level. So what we have is a momentum for a photon, the particle associated with light, which is given by h, Planck's constant divided by the wavelength of the light. So what De Broglie did, was he just said, "well jeez. If that's the relationship between momentum and wavelength, then what's the relationship between momentum and wavelength?"
So if I've got matter, then I can easily calculate the momentum of just by doing mass times velocity. Then I can determine a wavelength by h over the momentum. So this is the De Broglie wavelength associated with the matter, alright? Now it all seems very strange everything's just kind of flowing together, but the idea is that both light and matter will behave like a wave if the wavelength is much much much bigger than the scale at which you're probing the system. And both light and matter will behave like a particle if the wavelength is much much much smaller than the scale associated with the probe that you're using.
So the idea is that matter because the momentum is so so so large compared to the momentum of photons, we'll have an extremely short wavelength. So except in the most bizarre situations, we'll always have a wavelength that's much much much smaller than the scale that you're probing the system at and that means that it will behave like a particle and that's why we're used to matter behaving like a particle not behaving like a wave. Alright. So let's just do an example real quick.
What is the De Broglie wavelength of an electron that's moving at 2.2 times 10 to the 6 meters per second. Now, this is really really really fast. I mean, jeez. 2.2 million meters per second. But it's not relativistic, alright? It's still slow compared to the speed of light so we can still do everything fairly classically.
So the first thing that we'll do is we'll calculate the momentum. p=mv alright? the mass of an electron is 9.11 times 10 to the -31 in SI units and then we'll multiply by 10.2 times 10 to the 6 meters per second. Alright. Again, numbers first and then the tens. So when we multiply this up, we end up with 2 times 10 to the -24 newton seconds. I know the unit because it's a momentum and that means it has to be force times time. Alright. So there's the momentum. So how do we get the De Broglie wavelength? Well, we say lambda equals h over p. So that's 6.626 times 10 to the -34 over 2 times 10 to the -24. And what this turns out to be is 3.33 times 10 to the -10. What's the unit? It's a wavelength so it's got to be a length, so it's meters. Alright. 3.33 times 10 to the -10 meters.
Now this is an important number because this speed is the speed the kind of average speed of an electron in the ground state of hydrogen. So, what this says is that an electron in the ground state of hydrogen actually has a wavelength of 3.33 times 10 to the -10 meters. So that means that I can't squish it smaller than that. It's on it's own. It's just going to be 3.33 times 10 to the -10 meters. So this gives me a measure of the size of the hydrogen atom. Its size is determined by the De Broglie wavelength of the electron. It's not determined by the size of the nucleus or the size of the electron. Size of the electron instead of using that term, we say De Broglie wavelength of the electron. And so what this means is that if I probe a hydrogen atom is its ground state with a probe that has a scale that's much much much bigger than this, then I' going to probe the whole atom at once. It will look to me like the atom itself is a particle. This size. Alright? But if I probe it much much much smaller than this size, then I'm going to go inside of it and I'll be able to tell that it's a wave.
And that's De Broglie wavelength.