 So when we solved the one-dimensional particle in a box problem, getting these wave functions with these particular energies, our real motivation for solving that particular problem was just that it was the easiest problem that it was possible to solve Schrodinger's equation for. We chose the simplest possible potential energy and the simplest possible one-dimensional problem rather than a three-dimensional problem. But it turns out that problem can actually be used to discuss some real problems. So let's remind ourselves what the assumptions were that we used when constructing the 1D particle in a box problem. They were, first of all, we chose potential energy equal to zero, again, because it made the problem easier to solve. We also required that the particle be confined to a box. Our particle lives in a box of size A. The position is only between x equals zero and x equals A. So the particle is confined to some region of space. And it was only a one-dimensional problem. The particle moves in one dimension along an x-axis. It doesn't have a y or z-axis. It only moves in one dimension. So if we can think of real-world problems that at least approximately obey this set of constraints, then it turns out the one-dimensional particle in a box is a good model for studying that problem as well. And one such problem is linear hydrocarbons, the pi electrons in particular in linear hydrocarbons. So let's take a linear hydrocarbon like I'll just draw a linear hydrocarbon with four double bonds. So there's eight carbons in that molecule. That's an octane molecule with four double bonds. So that makes it octatetrene. So that particular molecule has some pi electrons in it. So each of these double bonds consists of two pi electrons. And of course, there's also some sigma electrons in the single bonds. And half of the electrons in each double bond are also sigma electrons. But many two, four, six, eight of the electrons in this molecule are pi electrons. And what you know about pi electrons in conjugated molecules is they're conjugated. They are delocalized. They can spread out and move throughout the whole backbone of this molecule. So those eight pi electrons are spread out and occupy this whole backbone. So that's maybe not exactly a one-dimensional problem, but they mostly move along an x-axis, a linear axis in this direction. They don't move very much in the y or z directions. They're confined to only a certain region of space. So I can think of these adoelectrons as being confined to a one-dimensional box. And it's not quite true that the potential energy is zero, but what's important is that the potential energy is relatively unchanging as I move around to different positions in my one-dimensional space. So the reason these electrons are delocalized is because there's not much of an energy barrier preventing their flowing from one carbon atom to the other. So the energy barriers between different carbon atoms are relatively low. The electrons are spread out over the whole backbone as if they had no potential energy, at least when they're within this box. So at least plausibly, linear molecules, linear conjugated molecules like octetetrine have our systems that could be modeled by these wave functions. So the claim would be then that these wave functions and these energies describe these pi electrons, the ones that can float back and forth in this molecule's backbone. So what that means is based on this prediction of the energies, the energies of the pi electrons should have this form. The eight pi electrons in this molecule, there's going to be a pretty low E sub 1, E sub 2. The energies are going up as E squared, so they're getting a little bit further apart as if they climb the ladder here. But there's this whole hierarchy of energy levels that these pi electrons can occupy. So the electrons live on this particular ladder of energy levels. The question is how could we test whether that's true? How would we know whether the pi electrons have this energy or this energy or some different energy? And we typically measure the energies of electrons in molecules by shining light on them. So I can take some light, photons of light, H times nu, or Hc over lambda, describes the energy of those photons. And if those photons get absorbed by, let's say, a molecule whose pi electrons live in this energy level, if I make them jump up to this energy level, then I have to provide an amount of energy, delta E, that's exactly equal to the energy of the photon has to exactly match the difference in energy between these two states. If I want to lift this molecule to here, I should provide that much energy. If I were to want to lift this molecule up to here or up to here, I would provide a photon with a different amount of energy. So we can check what the energy levels are by shining light on the molecules and seeing what colors of light get absorbed. We can also ask ourselves how this model would make predictions about things like, let's say we take a linear conjugated hydrocarbon of a couple different lengths. So one of these might be the octatetrine that we've been talking about so far, but let's also imagine a shorter one. This is a six carbon atom molecule, so this would be hexatriene. So how would the light absorbed by these two different molecules differ? The only difference between these two molecules, at least as far as the particle in a box model is concerned, is that the length of the box is different for octatetrine than it is for hexatrine. The eight carbon atom molecule is longer. It has a longer bond length. So without putting numbers on things just yet, when the box gets longer for octatetrine compared to hexatrine, when the box gets longer, that's in the denominator of the energy. So the energies are going to decrease. When A goes up, the energies go down. So what that means on this energy ladder, if I've drawn this letter for one molecule like hexatrine, if I were to draw it again for, so this would be for hexatrine. If I draw the ladder again for the longer molecule, the energy levels are a little bit lower, so they've decreased. Each one of these energy levels has decreased for the longer molecule. I'm really not interested as much in the energy levels when I shine light on them as I am in the difference between the energy levels. But if each of these energy levels has decreased, the gaps between the energy levels have also decreased. And what that means is, let's say I'm going to decide what wavelength of light to shine on these molecules so that it will be absorbed and cause a transition between these levels. If the difference in energy goes down, then the wavelength, because wavelength is inversely proportional to energy, the wavelength is going to go up. So my prediction would be that the longer molecules absorb light of a longer wavelength than do the shorter molecules. And we can actually test that prediction. So what I'll do now is I'll pull up a table here of the structures of several of these linear conjugated hydrocarbons, changing in length as I go down the table. So from short molecules like ethylene, so here's our hexatrine and octatetrine. So the two we've considered in this example are this one and this one. But in general, as I go down this table, the length of the molecules is getting longer. And I should predict that the wavelength that they absorb light is also getting longer. And that's in fact what we see. So the smallest of these molecules, ethylene, absorbs light in the ultraviolet at 207 nanometers. But as the molecules get longer, the light absorbed by them gets longer in wavelength until they approach the visible end of the spectrum. So 200, 250, 300, 350 wavelength. So the longer I make these linear conjugated hydrocarbons, the longer the wavelength of light they absorb. And that's because, purely because of this particle in a box effect. The box we're confining the electrons to has gotten longer. And particle in a box model tells us that the wavelength should get longer in response. So the way we would describe this phenomenon when a wavelength is getting longer, perhaps because it's a particle in a box and a box size is getting longer. But in any circumstance where a wavelength is getting longer, we call that a redshift because at least in the visible portion of the spectrum, red is on the long wavelength side of the spectrum, violet is on the short wavelength side of the spectrum. So if it's getting longer in wavelength, we say it's been redshifted and the opposite. So whenever a wavelength gets longer, we say that wavelength has been redshifted. The opposite you might expect, especially based on what I said a minute ago, would be if the wavelength is getting shorter, we would call that a violet shift. But in fact, what we actually call that is a blue shift. So you'll find that terminology used pretty often. If someone tells you that some light has been redshifted or blueshifted, all they're really trying to tell you is the wavelength of light has increased or decreased or equivalently that the frequency of the light has decreased or increased. They're just telling you the direction in which the energy or the wavelength or the frequency of that photon has changed. So we've seen perhaps a little bit surprisingly that the one-dimensional particle in a box isn't purely theoretical. It can be used to describe real-world molecules like these conjugated hydrocarbons. So we might wonder next, maybe it's good for more than just qualitative predictions, predicting not just the direction of a redshift or a blueshift, but being able to make quantitative predictions about these specific absorption wavelengths. So that's what we'll tackle in the next video lecture.