 Hi, I'm Professor Steven Nasserband. I want to talk to you a little bit about this claim electrons go through nodes So that as a context for this I'm just imagining first that we have an electron. That's just what we call a free electron It's just moving from left to right in a straight line there and Along this coordinate X and if quantum mechanics tells us that because Electrons have waves like properties all waves have an amplitude That amplitude varies from positive amplitude or what we call positive phase to a negative amplitude negative phase So that's what's happening here positive phase negative phase positive phase Not to say that the electron wiggles up and down like this not at all. It's just that along its path positive phase negative phase positive phase, okay, and and I've drawn two instances of this and the second one here is Electron is moving faster and quantum mechanics tells us that the oscillation between phases is a little bit is a little bit Packed in more densely. I should add one more thing here Which is that what we're graphing here is what we call the real part of the amplitude because in this situation The the the amplitude of the wave function is actually a complex variable So we're just focusing on the real part. So I can talk about because Because I'm looking at this distance from peak to peak here. We can talk about the de Broglie wavelength Okay, I'm just gonna give the symbol Lambda and a subscript db that stands for de Broglie And you can see what's happened here is that the de Broglie wavelength has got from a larger value to a smaller value when the Electron sped up. Okay, that is encapsulated in formula that de Broglie presented And it goes like this the de Broglie wavelength this distance from from peak to peak Goes as it's equal to this constant, which is Planck's constant divided by the mass of the electron The key point here is that it's also divided by how fast the electron is going be for the speed of the electron So you can see that the bigger the speed the smaller the de Broglie wavelength Which is just what we've seen here as we went from slow to fast the de Broglie wavelength got smaller another circumstances to think about this in would be a conjugated polyene as I as I've drawn here and what I've depicted here are the What's called the the highest occupied molecular orbital? this one would have two electrons in it and then I've also drawn the lowest unoccupied molecular orbital and which wouldn't have any electrons in it and if you look at these solutions do an electronic structure program what you will see is that is it has this What we call this nodal structure so One thing to notice about this is that well there must be a little bit of motion from the top to the bottom of this This of this molecule just because I see a node here remember electrons go through nodes But the thing I want you to focus on is the transverse nodes the nodes that get laid out and left to right and You know I can see that these electrons must be able to go left or right because look at all that Probability density from here to here, but in which case is the electron moving faster Well, I can see that the distance from here to here that must be the de Broglie wavelength about Right from here to here that would be lambda de Broglie for the Homo and I can see that in this case lambda de Broglie Is smaller? Okay, and according to our formula when I have a smaller lambda de Broglie It must mean that the speed is faster. So the way we understand this is that sure electrons are zinging back and forth along the You know the width of that the length of that molecule, but they're moving faster in the LUMO Slow a little bit slower in the Homo and how do we know that because of these transverse nodes? There's only four of them here. There are five transverse nodes there And that's what allowed us to identify the de Broglie wavelength. Okay, good