 So even though we normally think of light as behaving like a wave, we've seen that it's actually made of particles. Sometimes it behaves like a particle. When we think of individual photons of light, individual particles of light, we're required to think of it as particles instead of a wave. Turns out matter also behaves a little bit schizophrenically like this. Normally we think of matter, atoms, molecules, baseballs as being composed of particles, individual particles. But it turns out sometimes it's also necessary to think of matter as behaving like a wave. And those are the two different aspects of what's called wave particle duality. Everything, every quantum mechanical object, whether it's a photon of light or whether it's an atom or a molecule or an electron, has what's called wave particle duality. Sometimes it behaves like a particle. In other ways, it behaves like a wave. And in particular, when we think of matter having wave-like properties, that begins to sound a little bit weird, so let's talk more about what that means. The different properties of matter that show that it is in some respects like a wave is matter is delocalized. Again, we're used to being able to make statements about matter that describe its particle behavior. It has a position. It has a momentum. We can describe where that particle is and where it's going. But at other times, like a wave, if I ask you where is a wave in an ocean, the wave is delocalized. It's spread out over some region of space. And the same thing turns out to be true about small quantum mechanical objects as well. Two other things that particles can do, matter can do, that we normally think of as being wave-type behaviors, they can diffract and they can interfere with one another. Take two waves and I cross them across each other. They can either interfere constructively or destructively depending on whether two troughs or peaks hit at the same time or whether a peak hits at the same time as a trough and they destructively interfere with that type of interference and diffraction when a wave bends as it passes around an object or through different media. Those are all properties that we think of as describing how waves behave, but they also describe how matter behaves under certain circumstances. So to delve into that a little more deeply for small quantum mechanical particles, let's talk about something called the de Broglie wavelength. So it's not surprising when we talk about the wavelength of light or the wavelength of an ocean wave or something like that, the wavelength being the distance between two troughs or two peaks of a wave, the length of a wave. Turns out particles, matter, have wavelengths as well. In this particular form we call a de Broglie wavelength. I can calculate the wavelength of a quantum mechanical particle as Planck's constant divided by the momentum of the particle as mass times its velocity. So that's a strange thing to think about what is the wavelength of a particle mean if I say what's the wavelength of this marker or what's the wavelength of a baseball or some physical object, that's hard to even understand what that means. So let's do a few calculations to see what these wavelengths are like. So let's say a particle is moving and we'll do this for several different particles. Particle is moving with a velocity of 100 meters per second and we want to know what is the wavelength. I've given you the velocity we know Planck's constant. The only other thing we need to know is the mass. So we'll do this a few different times. First, for the smallest particle we can think of, at least in a chemistry class, that would be an electron. So an electron has a mass, the mass of the electron is 1.9, I believe, I'd better double check, no, 9.1 times 10 to the minus 31 kilograms. So that's a tiny, tiny mass, 9.1 times 10 to the minus 31 kilograms. If we do the full calculation, we can calculate the wavelength. So the way we do this, we'll run through it exactly once. The wavelength is Planck's constant over mass over velocity. So Planck's constant is a number of joule seconds divided by the mass, 9.1 times 10 to the minus 31 kilograms divided by the velocity. We've assumed the particle has a velocity of 100 meters per second. So this electron is flying around at 100 meters per second. Units, to make the units work out, we have to recognize that the joule in Planck's constant is a composite unit. One joule is one kilogram meters squared per second squared. So once we have remembered that joule is a kilogram meters squared per second squared, if I multiply it by seconds, that gets rid of one of these seconds dividing by kilograms in the denominator, it gets rid of kilograms dividing by meters per second, gets rid of one of these meters and the other seconds. And all I'm left with is meters. So this problem is going to work out with units of meters. And if I plug those numbers into a calculator, what I find is that I get 7 times 10 to the minus 6 meters or 7 micrometers. So an electron, according to this equation, the de Broglie wavelength of an electron that's flying around at 100 meters per second has a wavelength of 7 micrometers. So 7 micrometers is small, it's micrometers 10 to the minus 6 meters, but 7 micrometers is certainly a lot larger than an electron. An electron itself is no bigger than 10 to the minus 10 meters or so. So that is considerably bigger, many orders of magnitude bigger than the electron itself, the size of the electron, the radius of the electron. That'll be important in just a second after we consider the next few examples. Let's consider another small quantum mechanical object, but not quite as small, a single atom, maybe a hydrogen atom or let's take a somewhat larger atom. The mass of a carbon atom, for example, we know is 12 grams per mole. So if I convert that to kilograms, convert grams to kilograms and use Avogadro's number to get rid of grams per mole, we won't run through that calculation. But that turns out to be about 10 to the minus 29 kilograms or so. So it's maybe 10 to the minus 28 kilograms. That's several thousand times heavier than an electron. So when I use that mass in the de Broglie wavelength calculation, it turns out that the de Broglie wavelength of a carbon atom flying around at 100 meters per second, that works out to be 3 times 10 to the minus 10th meters for this particular example. And in SI units, that would be, well, that is SI units, but 3 times 10 to the minus 10 meters is about 3 angstroms. And a hydrogen atom itself is about a half an angstrom large. So that's larger than or at least on the same order of magnitude as the atom itself. It's comparable to the size of the atom itself. So it's not much greater than anymore. If I continue with an example that's a little bit larger, let's combine several atoms together and make a molecule. And it's not really important what molecule we're talking about, maybe a benzene molecule or maybe a caffeine molecule or some molecule that's made up of a handful of atoms. Let's just say, roughly speaking, I've combined a bunch of carbons and hydrogens and other atoms together, that molecule might have a mass of somewhere around 100 grams per mole. If we calculate the de Broglie wavelength of that atom, we find, again, because the mass is in the denominator and this mass is getting larger, the de Broglie wavelength is going to get shorter. So this de Broglie wavelength has gotten a little bit shorter. Mass has gotten bigger by almost a factor of 10, so the de Broglie wavelength got shorter by almost a factor of 10. It's only now a fraction of an angstrom. The molecule itself is now several angstroms large because I've tacked together several molecules, each of which might be about an angstrom in size. So this size is now smaller than the original object. So the de Broglie wavelength of molecule is smaller than the molecule itself when the velocity is about 100 meters per second. And as a last example, let's compare those small molecule atom electron-sized quantum mechanical objects with something large, something that we can see in the real world, like a baseball or a pen or a marker or something large. And just as a rough estimate, let's say we're talking about something with a mass of a kilogram. So 2.2 pounds or so, a kilogram, if I plug that number into the equation for de Broglie's wavelength, this is now many orders of magnitude heavier than an individual molecule. And what I find is the de Broglie wavelength has gotten quite a bit shorter. If I use a kilogram, moving at 100 meters per second, I find that the de Broglie wavelength of that object is 10 to the minus 36 meters. So ridiculously small. An atom, remember, is only about 10 to the minus 10 meters in size. So this is 26 or so orders of magnitude smaller than an individual atom. So the wavelength of this macroscopic object is so small that we can't even measure it, see it, detect it in any way. It's not only smaller than, it's much, much smaller than the original object. So that last statement is why it seems a little bit nonsensical for us to talk about the wavelength of a marker or the wavelength of a baseball or something like that, is because the wavelength itself is so small that we can't measure it. Effectively, what this wavelength means is the same thing it would mean for a water wave or a pressure wave or a sound wave or something like that. It's the length of the wave that determines how that thing diffracts or interferes with itself. It's sort of nonsensical to say if I throw two baseballs at each other and they collide, they can interfere constructively or destructively. Of course, that's not true. They just collide the way particles do. And that's because the delocalization or the interference of those baseballs happens on a size that's only a tiny little fraction of the molecule itself. So only at undetectable levels do they interfere or diffract off of one another. Whereas quantum mechanical objects like molecules or atoms or electrons, their wavelengths is in fact comparable to or sometimes much larger than the object itself. So they can actually behave more like a wave than a particle if their de Broglie wavelengths are much larger than the size of the particles themselves. So it certainly seems weird at the macroscopic level because we're not used to seeing objects behave that way. But at the quantum mechanical small atomic level, things definitely do behave, matter does behave like a wave. And let me illustrate for you the way we know this is true or one of the first pieces of evidence we had that illustrated that the small objects do in fact diffract and interfere the way waves do. And that's an experiment called the two slit experiment. And to start with, I'll describe what a one slit experiment would look like when we're talking about not particles, but waves. So let's imagine a wave like a wave in the ocean. So imagine here's the line of waves, top view of a line of waves that are moving in the ocean. They're about to break on the shore. But before they get to shore, they get to a wall that someone has built near the shore that has a gap in it. So what's going to happen if the wall were solid, the waves would just crash into the wall and they wouldn't move past it. But if we put a gap in the wall or a slit in the wall, what happens when the waves reach the slit is they won't pass through regions where the wall is. They will pass through here, but they'll diffract around that opening and they'll move out the ripples. If you can imagine the waves of the ripples, they'll move in this same pattern that you'd see if you threw a rock into a pond. The ripples would move out in this circular pattern. That's how we understand waves behave in a two slit experiment. If our wall has two gaps in it, and again we have waves that are approaching the wall with the slits in it, what will happen is the waves will diffract out from these openings, but the ripples that get formed when the waves diffract out of these openings, those ripples will begin to overlap themselves at some point. So the place where I've just drawn the troughs or let's say the peaks of these waves both occurring at the same place, that's going to be a site of constructive interference. The wave will be twice as high here as it would if there were only one wave from a single slit. So there will be various places where there's constructive interference, and there's various other places where there's destructive interferences where the peak of one wave overlaps with the trough of another wave. So we'll see constructive and destructive interference the way we'd expect from normal waves to behave. Now things get a little bit weird when I imagine what should happen and what actually does happen for particles. So these are waves moving towards a wall with some gaps in it. I can also show when particles arrive at a wall with gaps in it. So let's imagine we're throwing baseballs or shooting bullets at a wall with a gap in it. So I've got a lot of, instead of a lot of waves, I've got a lot of particles that I'm sending towards this gap. And certainly what we'd expect to happen is that a particle that hits the wall won't pass through. A particle that arrives at the gap will pass straight through. And for one slit, things aren't too different than what you'd expect. If I then ask, if I draw another wall back here and I ask where these particles hit on the wall, you'd expect them to hit only in places that allowed them, where they were allowed to pass through the slit. And that's, for the most part, what you see. Occasionally you'll get one hitting a little further away because it gets bent or diffracted as it passes through the slit. But that's not too weird yet. Where it gets really strange is in the two slit experiment. If I have particles being sent toward this wall with some gaps in it, in particular two gaps, and I ask back here where do I find that the particles land after having passed through these slits? Again, what you'd expect is maybe particles can pass through here and can pass through here. So they ought to land here and they ought to land here. But that's in fact not what we see. We see perhaps some particles landing there, but we might see some particles landing here, maybe even more particles landing here. So I'm drawing a little histogram or diagram of where we see these particles. And what we see is a series of peaks and valleys. There's certain places where a lot of particles hit. There's other places where very few particles hit, or relatively few particles hit, and there's some places like here where no particles hit at all. And the only way to understand how that can happen is if these particles as they pass through the wall don't fly through like bullets, instead they diffract through the slit, constructively and destructively interfere with one another. The places where we find them a lot on this wall is where there's a lot of constructive interference. The places where we never find them is where there's complete destructive interference. So again, if the particles were purely particles, macroscopic bullets or baseballs don't behave that way, but small atoms and electrons do in fact behave that way, and the only way we can understand that they do is the fact that these electrons and atoms behave as much like waves as they do like particles and can constructively and destructively interfere with each other. So that was one of the first experiments that let us know for sure that small objects do in fact behave like waves. And that completes our tour of the quantum mechanical properties at the qualitative level that you already perhaps know something about. Our next step will be to get a little more quantitative and to understand quantum mechanics at the level of equations and numbers so that we can begin to make concrete predictions.