Hi there. So we had previously shown that Albert Einstein had interpreted the photoelectric effect as indicating that electromagnetic radiation behaved or had postulate like behavior. So this was an addition to the obviously wavelike behavior that had previously been exhibited by electromagnetic radiation, and previous to Einstein's theories, had then considered to be just wavelike. Louis de Broglie, who we show on the right here, was a PhD student and he came up with the amazing idea that not only did electromagnetic radiation exhibit wave-like and particle-like characteristics, but he also proposed that moving particles which have obvious particle-like behavior, also exhibited wave-like characteristics. Now to try to understand his reasoning, it's best to go back to the equation we had left off with on the photoelectric effect presentation, where we had shown that you have this momentum, if you like, for a photon as being equal to Planck's constant divided by Lunda. Now, see here, is momentum of the electromagnetic radiation. And we know from Planck and Einstein, that this electromagnetic radiation, that this is the velocity, you can consider this as the velocity of the actual photons of energy, h nu. So what de Broglie's fairly, well easy now but obviously a great insight, showed that if you now have any moving particle with velocity V, you can write a very similar equation in the sense that mv is equal to h over lambda. So these are usually, our usual symbol is small p for momentum. So p, momentum, of any moving particle is Planck's constant divided by its wavelength or lambda is equal to h over p. So essentially, de Broglie here extended this wave-particle duality, that Einstein had proposed for energy or electromagnetic radiation, to all moving particles. And at this time, this was just a postulate, as I indicate here. De Broglie's Postulate. But, it was shown and it's quite not that difficult to show that electrons have indeed particle like activities. And the classic behavior exhibited by waves, of course, is that fraction and If we look down here, here we can show on the left a difaction pattern made by a beam of x rays passing through an atom foil. And you can see there's a diffraction pattern given, but on the right the diffraction pattern here you get a nice pattern is made by a beam of electrons passing through the exact same foil. So here, electromagnetic radiation in the X-ray region is exhibiting diffraction properties, exhibiting wave-like properties. But you can see here on the right, that the electron beam is exhibiting the exact same wave-like behavior. So this is, if you like, experimental proof of the de Broglie theory. So let's move on now to just do a few calculations or examples on this de Broglie theory. If we take a, let's say we have a typical particle object that we are familiar with, so we have a tennis ball. And let's suppose the mass of the tennis ball is, let's say it's 57 grams. And it's velocity, Say it's 80 kilometers per hour, so you could be asked the question what is the. What is lambda? So according to the lambda is equal to h over the momentum, mass times velocity. So we put in our value for Planck's cost which is 6.626 times 10 to the minus 34. So we divide that by the mass of the tennis ball which is 57 times. We like to keep our units in SI units, so I'm gonna convert this to kilograms, so you multiply by 10 to the -3, that gives it in kilograms. And then you multiply [INAUDIBLE] and we need to keep it in meters per second, so it's going, it's 80 kilometers per hour, so it's 80 by tenths to three, meters, and then you divide it by 60 to bring it to minutes, and and then you divide it by another 60 to bring it per second. So that'll give it to you, your velocity in meters per second, which is the SI unit you want. And if you plug the values in you should get a value 5.2 x 10 to the -34 meters. Now what you can see immediately is that this value here, 10 to the minus 34, is negligible compared with, say, the size of a tennis ball. What that means is that every moving particle does have wave-like characteristics. But for everyday objects like a tennis ball, a moving tennis ball, the wavelength is so small that it's negligible and it can be completely neglected. And, sorry to say, it doesn't have any effect on how well you serve. Let's take a look and this would take the typical example of an electron. And now let's say a typical velocity for the electron of say 1.0 by 10 to the 7 meter 2nd minus 1 and you could again ask the question. What is the wavelength of that? So we again use the de Broglie relationship. We said that lambda is equal to h over ms, by the velocity. So it's again Planck's constant, 6.626 by 10 to the minus 34 joule-second divided by, now we have the mass of the electron, 9.11 by 10 to the minus 31, that's kilograms. We can find that from tables. Now we have to multiply that below the line by 1.0 by 10 to the 7th. And again that's [INAUDIBLE] meters per second. If we plug that in we get 7.3 by 10 to the minus 11. Sorry. Just put that there. 10 to the minus 11, and that's in meters. So I'm gonna express that 7.3 by 10 to minus 11 meters. That's equal to 0.73 by ten to the minus ten meters. And 10 to the minus 10 we know it better as 0.73 angstrums. So what you can see here now is that this, the wavelength of the electron, is in the region of the the kind of distance that an electron is moving in, in an atom. So now the wave length of this moving object is about the same. If you like of the region in which it's moving. So now, the wave light properties are going to become significant. And it's going to exhibit properties like diffraction and so forth. So, for the larger objects. Practical objects that we meet in everyday life. They've all got this wave-like characteristics, but because the wavelength is so small, it's not detectable. But for something like an electron which exists inside atoms and molecules. Then the region in which it's confined to is similar to its wavelength. So therefore the wavelength properties become extremely important.