 Hi, I'm Professor Stephen Nesheba and I want to tell you a little bit about this idea. Electrons go through nodes. So I laid out here the path of an electron moving from left to right here. We just imagine it's moving along that x-coordinate. And because electrons have wave-like properties, then we can imagine that it has, that wave has positive and negative amplitudes, just like a water wave has positive and negative amplitudes. And that's what I kind of depicted as shown here. It's not at all to indicate that this electron is moving up and down. That's not what this graph is meant to indicate. It's meant to indicate just that as it moves along that path, it's got positive phase for a little while, negative phase, positive phase due to the wave-like nature of the electron. Here's just another way to think about it. I've just drawn along that path, the parts of that path where the electron had positive phase, and here's where it had negative phase, positive phase again. Okay, so as it turns out, quantum mechanics teaches us that the wavelength of that wave, which I've just drawn here, it's the distance from that peak to that peak. It's called the de Broglie wavelength. It depends on how fast the electron is moving. So I've drawn here the situation for a slow-moving electron, and if I wanted to speed up that electron, as it's going from left to right, what you would find is that this oscillation from peaks to troughs to peaks to troughs gets kind of compressed like this, and therefore what happens is that the wavelength, the de Broglie wavelength, got smaller. And that's generally true. It would continue as we made that go even faster. One more bit of terminology here goes something like this. I'm just going to call these zero crossings here. That is, when the wave went from positive to negative and negative to positive, we're going to call that a node. Okay, so you can see that the nodes are packed in a little bit more densely. So the idea is electrons go through nodes. Well, it has to be because the electron is moving from left to right, and there's the nodes. And in fact, the faster the electron is moving, the more nodes are associated with any given interval in its path. So this idea was actually made quantitative and introduced really by de Broglie, Lewis de Broglie in 1924, and he had an equation that described to that, and it goes like this. It's pretty simple. That lambda de Broglie, that wavelength, is equal to, you can calculate it, it's this constant, that's called Planck's constant, that quantity of the denominator is the mass of the electron. But the thing I want to focus on is it also has the speed that V is the speed of the electron in the denominator, so you can see how this must have all played out because as we make the speed bigger, that must mean that the wavelength, the de Broglie wavelength, is smaller, and that's exactly what we saw here in going from this slow electron to this fast electron. Okay.