Lecture 04. Wavelengths
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OberflächenbehandlungDeltaVorlesung/Konferenz
Transkript: Englisch(automatisch erzeugt)
00:07
So, last class we talked about how light can be considered a particle, and the experiments that kind of show us a lot of that, that really kind of epitomizes both the wave-particle duality. But there's this other aspect that shows it as much more of a wave than just the
00:23
photoelectric effect really does. So the photoelectric effect does a great job of showing both at once. This stuff that we're going to talk about today does a little bit more of a job of showing it as a wave in a really distinct way that we could actually see if we did. So in order to talk about this experiment effectively, I need to give you a little bit
00:42
more background on what waves are, and specifically what wave interference is. So wave interference is something that you can see. You can see it in waves in a pond or wherever. If you drop a rock into a pond, you make waves, right? You can see the rippling effect going around. And then if you drop two rocks into a lake or a pond, something with relatively still
01:04
water, you can see these weird interference effects. You can do the same thing if you're out on jet skis on a relatively calm day and you can watch the two waves kind of hit each other. This is wave interference. So this is the kind of, you can see it, and you can see you get these interesting patterns that come up that aren't just simple superpositions of the two.
01:26
So what do these look like when you do superimpose the two on each other? So there's two different types of interferences. There's constructive and destructive. And if you have waves of a very particular wavelength, and they're interfering with each other, you can overlap them right on top of each other or exactly, you know,
01:44
like cross on each other. And if you overlap them right on top of each other, they add together. If you overtop them exactly opposite each other, they're going to subtract from each other. Now of course you could also do some sort of, you know, slight off on it too, but these are the two different ways.
02:01
So think back to your algebra classes, whenever you had those, and you may have been forced to go through and, you know, these had little points on a graph and you had to add them together, and if you added this wave together, you had to add this point and this point. So if this was one and this was two, you'd add them together to be three.
02:20
Well this is zero plus zero, and so it adds together to be zero. This is negative one and negative two, so it would add together to be negative three. Same thing over here. Now here only though, you have a negative one and let's say a plus two, giving these really arbitrary units. So if you were to add a negative one and a plus two, well that gives you about a plus one, and so you would get that, and they would subtract from each other.
02:43
You might have been forced to go through this in painstaking detail in your algebra classes. We're not going to do that here, but you need to kind of understand how it works. You need to see that okay, well if they're over top of each other, they're going to subtract from each other. If they're opposite each other, they're going to subtract. And that these are called constructive and destructive interference. Now this is going to be important to understand when we look at this next experiment,
03:04
which is called the double split experiment, because you're going to get patterns of light. And these patterns of light are going to be really bright in some places, and really dim in others. And that's because you get constructive interference, making it really bright, and destructive interference making it very dim. And so that's how this is going to work on this next experiment.
03:23
Now the best way to show you this next experiment is to have a little video of it. So this video does a great job, and you can watch it as much as you want on YouTube. So I'm going to go through and explain it.
03:40
He does do the same thing online with words and such. So the way that the double split experiment works is you have this point source, and you have a place where you can let light through. And so this is sort of showing you how the light would go through, and light expands out like a wave.
04:04
Now okay, we're going to start over. Now the other thing that you're going to be able to do is you can make the exact same thing on the next one over. So instead of just having one point source, you can have two point sources.
04:23
So you have one point source here, and that's going to let light in at a certain amount. Now you put another light source in another place, or just a hole. One of the easiest ways to do this is to actually just have a light source behind the cardboard or board or whatever, and put a hole in it.
04:41
Now if you put these together, what happens is that they interact, and you get these interference patterns. And when you come over here, wherever the light interferes in a constructive way, you're going to get really bright, shiny parts. Wherever it interferes in a destructive way, you're not.
05:03
You're going to get very dim. And you can actually see this. This is a pretty standard freshman physics lab, if you end up taking any physics. And it's really, I mean it looks exactly like this. So this is actually a picture of one from one of these experiments that was done, and it really looks almost identical, using the right lasers and things like this.
05:25
You can actually do this at home if, in a slightly different manner, if you take a piece of cardboard and you cut out a little slat and you put two pieces of hair and you tape it down and you shine flashlights on it, you can get the same sort of situation where you can get little interference patterns.
05:42
I used to do it in class, but it was honestly too hard to see in a big lecture hall, so I quit. So you know, go home and do that. It's fun, you just take two pieces of hair, tape it down onto a piece of cardboard and take a laser pointer and shine it around, and you can see interference patterns too. Now let's look at this a little bit more laid out.
06:06
So here's the experiment that Young did. He had a, something that was light blocking here, and he cut one little hole in it to make sure that the light was coherent. And then he shined a light over here, so that it came through here.
06:20
At which point it spread out like this. Now what that did is it allowed him to have two light sources, as what they showed in the other video, but have it come through it exactly the same way. And then at this point, now we had this double slit, and it comes through here, it
06:41
comes through here, and now you have the exact same wave pattern for each slit. But now because they're close to each other, they're going to be interacting with each other. Now if you draw, if you then have something over here that detects it, human eye works just fine to detect it, but you could also actually detect it with something a little
07:03
bit fancier, you can see these patterns. So where these interfere completely, you get this really bright peak, and then you get these sort of peaks that get more or less intense as you go through, but you still get these interference patterns. So this is how Young's double slit experiment, or split experiment worked.
07:27
So again, now that we've kind of looked at both of these experiments, we've seen how they have properties of both. They can act, light can act like both a wave and a particle, and it does it at the exact same time, this is called wave-particle duality.
07:41
We saw that with the photoelectric effect, you actually did see a little bit of both, right? But you definitely saw the sort of particle-like ways that it worked, where the double split experiment most certainly amplified the wave-like effects. You can't look at that and say, okay, it's not acting like a wave after you've seen that, and you've seen those interference patterns, and you know, you can relate that to real
08:02
life where you see ripples in a pond, and you see those interact. And it's very much the same idea. So, now this is very true of small particles. So the last time that I had looked, they were able to measure these wave-like properties in things up to 1610 amu, which is actually pretty big, if you think about it.
08:24
1000 grams per mole, you can think of it. So when we say that this is true of very small matter, that's about the size that it goes up to. Now, that's just because that's exactly when we can measure it.
08:42
So this kind of bridges us into the idea of, well, does everything have wave-particle duality? I mean, we see it in the small things, but why would an electron have wave-particle duality, but I don't? Why can't I act like a wave? So that's what de Broglie said, in not quite so colloquial of terms.
09:05
So he took wave-particle duality from light and said, well, that's true of matter too. And that's true of larger matter to some extent, but we'll see why it doesn't quite work. So what de Broglie says is, well, wavelength is equal to h, Planck's constant, over p.
09:23
So depending on how much physics you have, you may or may not know what that is. That's momentum, otherwise known as m times v. This is where Font gets us into trouble again, because most of the equation editors like to make it, the v, kind of like that, which looks too much like a nu. But that is a v, that is velocity.
09:42
So p is equal to mv, where that's being v, so just said one more time that that's quite important to make sure you notice. And you're also, this is more just sort of a general problem solving thing that you're going to run into a lot. You're a lot of times going to be given kinetic energy and asked to find the de Broglie
10:03
wavelength. Well, you can do that by finding velocity from the kinetic energy and then plugging that into here. So this takes the wave-particle duality and says, okay, well, that's great, we have it for light, why can't matter have it too? And it gives us an equation that we can actually use for it.
10:24
So quick reminder there on your equation for kinetic energy. Okay, so best way to go through and do some de Broglie wavelengths is to do some problems on it. So for the first one, I say let's find the de Broglie wavelength of an electron.
10:43
So something small. And see, I give you the kinetic energy. So since I give you the kinetic energy, we can't just directly fill it into the de Broglie wavelength because the de Broglie wavelength is h over nv. So we need to find v. So how do we find v?
11:01
Well, we know that kinetic energy is equal to one-half mv squared. Okay, so when we go through and we do this, we solve for velocity, which is two times the kinetic energy, so multiply that up, divide by the mass, take the square
11:26
root, so we can fill in from the problem.
11:45
Now we get to mass and we say, well, I don't know the mass of an electron, luckily this is a homework type problem so you can look it up. On an exam, it would just be given to you on the tables.
12:05
So we get this and we get our velocity. Now something to be kind of careful of here with the mass, it doesn't come up here or in my other example actually, but it'll come up in your homework and it'll come
12:21
up on exams. If I have a mass of an electron or a proton or a neutron or anything like that, I'm going to have to give you that mass. If I ask you for the de Broglie wavelength of a nitrogen atom, do I have to give you the mass? Well, no, it's on the periodic table, right? You have the mass. So if you think about nitrogen, are you just going to go through and fill in the
12:45
mass of, let's say, you know, just an atom, are you just going to go fill in 14? Well, that's in grams per mole. And if you look at this unit, we have joules and we need to get meters per second. So remember what a joule is equal to, right?
13:02
A joule, little sidebar here, a joule is equal to kilogram meters squared over seconds squared. So we can't have something that just, we can't put grams per mole in there.
13:20
So you have to go through and you would have to convert anything from the periodic table into kilograms. So, you know, make a huge note of that in your notes because it comes up and people tend to mess it up. That if you're looking for an atom, you're looking for the de Broglie wavelength of an atom, you're going to be tempted to fill in that grams per mole from the periodic table. You can't do that.
13:40
You have to convert to kilograms. And not just kilogram, that doesn't just mean you divide by a thousand either, right? Kilograms per mole isn't what you're looking for. You're looking for kilograms per atom. So you have to convert from kilograms per mole to kilograms per atom or molecule depending on whichever. So make sure you make a note of that, you know, by your mass.
14:01
Okay, continuing on though. So now we have our velocity and we can fill that in.
14:27
You'll notice I sort of changed my units for Planck's constant around. That's just so I can watch all the units cancel.
14:47
And you get this and you end up with that for your answer. Now I always get a few people who ask, well, can't I just take this equation and fill
15:04
it into here and plug it all in at once? That is perfectly fine. Go for it. Probably it's how I would do it if I were doing the problem. But I think probably 80% of you or so prefer this way where you do it out separately. So whichever way you prefer you can do. It doesn't really make too much difference either way.
15:22
So if you prefer to just fill this equation into here, you are welcome to do that. Okay, so that's the equation for an electron. Now, if we move on a little bit, that's, you know, small, right?
15:41
An electron, small. You can definitely assume, okay, it's pretty tiny considering, you know, it's an electron, subatomic particle and all. But what's to say we can't do this for a baseball? So let's do it. Let's calculate the de Broglie wavelength for something big, something that we can
16:02
see in real life. Because, I'm telling you that an electron has a wavelength, which by that same token, your chair should have a wavelength, I should have a wavelength, my water bottle should have a wavelength. So why don't we see that? Why can't we, why can't we look at that and see something there? Well, the easiest way to figure out why we can't see it is to, you know, calculate
16:20
it. So we'll do that. So I give you the mass of a baseball and I give you an approximate meters per second. Obviously, this is going to change a lot if I throw a baseball or if some famous pitcher throws a baseball. So same rules apply. We're going to fill into here.
16:47
Again, just for the sake of watching my units, I'm going to just fill in my Planck's constant units in a slightly different way.
17:04
So we go through, we do the exact same thing we did before. It's a little bit easier this time because I gave you all the values out, right? And we get that. So see, baseball does have a wavelength. Now why don't we see it?
17:21
Well, think of what a baseball looks like, right? It's, you know, a decent size. You can see that size thing. Now look at its wavelength. Okay, to the negative 34th meters. That's not something we can see. That's not something that we would calculate. It's not something that we would, you know, be able to look at. And so, sure, maybe by this definition, macroscopic matter does have a wavelength, but it's
17:43
so tiny that it doesn't matter. And if we look at this equation again, this means that mass is in the numerator, right? Or, excuse me, denominator. So the bigger that the mass gets, what's going to happen to the wavelength? This gets bigger and bigger and bigger, it makes this whole thing smaller and smaller
18:01
and smaller. So the bigger that something gets, the smaller that its wavelength gets. So you're making something larger, you're making its wavelength smaller, and proportionally it ends up just not mattering anymore. When you had an electron, we had a wavelength of this for something that was very, very
18:22
tiny. Now we have a wavelength of this for something that's relatively large. And so the wavelengths just start to not matter. So that's what I meant when I said the largest thing that has sort of shown, that we've been able to measure wave-particle duality in. It's that point where our measurements aren't really good enough and it doesn't really
18:41
make any difference anymore. Something like an electron or a proton or a neutron, it definitely matters for. Something like a baseball, not so much. Okay, so now we get to move on to something called the Heisenberg uncertainty principle.
19:07
So what the Heisenberg uncertainty principle says on one level is that it's impossible to simultaneously know both the momentum and the position. So with absolute certainty.
19:20
So it puts a limit on how well we're able to know this. It says that if you take the uncertainty in x, so notice that's not x, it's the uncertainty in x. The part that we don't know about where the position is. So in really crude terms, maybe if I take a guess and I say I'm three meters away
19:40
from that door, I'm not very good with estimations so my delta x might be a meter. Maybe I measure it now and I say well, now my uncertainty I measure it and it's really close and I'm actually 3.5 meters away. And I can say okay well, I'm pretty sure of that within plus or minus four centimeters because my measuring isn't very good.
20:01
I use a cheap measuring tape. Now maybe I measure it in a standard one place and I use calipers and I'm really careful and I know it within a millimeter where I am. That's what an uncertainty would be. It's how well you know where something is. This comes into play much more, this is really more talked about with small things, you know, electrons, protons, things of that sort, but that's how you want to think of it.
20:22
Momentum is going to be the same way. Now assuming that, keeping in mind that momentum is mass times velocity, a lot of times you'll just see this as mv because you assume you know what the mass of the particle is. You assume you can look that up and you can weigh that and that's pretty perfect.
20:41
So what happens if your momentum in uncertainty decreases? Let's say you're using really, really good equipment because do notice that this is greater than or equal to, meaning that it's got to be greater than that. That's the lower limit, but it might be more. It might be that you're in certainty and both are huge because you aren't using very good equipment. But according to this, even the best equipment in the world, which arguably there may be
21:03
some people who have published things saying that they may have broke it, we'll see how that works, but according to this you can't. And if this happens, so let's say best instruments, you're right at that Heisenberg uncertainty principle limit, if your momentum uncertainty decreases, well what does that mean for your
21:24
position? Well it's the uncertainty in your momentum decreases. If you measure that better and better and better, that means that your uncertainty in your position will increase. So every time you improve one, you lose the other.
21:40
So you can improve this, you can know where that particle is at with a little bit more certainty, but you're not going to know its momentum very well. And you can get better and better and better knowing its momentum, but you're not going to know its position very well then. So what happens if your uncertainty in position decreases? Well then your position, your other one is going to increase.
22:02
So it's always this back and forth. You decrease one, you increase the other. You decrease that one, you increase the other. Okay. So this problem is based off a commonly told and really bad chemistry joke. So the joke goes that Heisenberg is driving along and he's pulled over by a police officer
22:24
and the police officer comes up to him and he says, you know, do you know how fast you were going? And he looks at the police officer and says, nope, but I know exactly where I am. So the joke there being that you only know one or the other. So I want to try to prove whether or not we can use this if, you know, I'm driving
22:42
to work arguably probably too fast and I get pulled over, can I argue with the cop on the thing that, well, he knows where I am so he can't know how fast I was going. Obviously there's, you know, some sort of quantum system here that he can't write me a ticket. I don't think I would suggest this argument anyways, but let's see if it's actually valid.
23:02
Let's see if it holds up. Okay. So we're starting from this equation on the slide, which is to say delta x times delta p has to be greater than h over 4 pi or equal to.
23:22
So that's our limit. Now of course it can be more, but it's got to be equal to that. So let's rewrite this a little bit differently so that we have velocity there. We're going to assume I weighed my car and I know exactly how much it weighs since
23:44
this is a little bit of a back of an envelope calculation. Okay. So I tell you that, you know, I think that the officer should be able to measure where my car is by about half a meter. And I know the mass of my car to be 1300 kilograms.
24:06
So we need to solve for delta v. So we can see whether I can argue myself out of a ticket this way. So let's first solve for delta v and then fill in everything that we need.
24:47
Okay. So if we do this, we get a delta v. Our delta v has to be less than or greater to, according to the quantum mechanics, this.
25:03
Okay. So now can I argue to the cop that I shouldn't get a ticket? No right? My delta v here is tiny. So according to quantum mechanics, he knows how fast I was going times 10 to the negative 38 meters per second. I promise you I was speeding by more than that.
25:21
So this quantum mechanics limit isn't an issue here. So this, you know, it's a silly example, but it does show how you would go through and do it for any other particle. You would just have a different mass. So same rules apply here, has to be kilograms, right? So grams per mole isn't going to cut it. You need to change it to kilograms per atom.
25:42
If I give you the mass in grams, you need to make sure you convert it to kilograms. So same rules apply here for any sort of subatomic particle. But it's a little more fun to try to calculate this way. So this is not a good way to get out of a ticket when you get pulled over. Even if your particular police officer does have a strong background in quantum mechanics.
26:16
All right. So next thing that we're going to talk about then. This seems to be every freshman's favorite topic in chemistry.
26:24
So wave functions, energy levels, and particle in the box. So this sort of sets up this next big section of this chapter. We've sort of talked about all the history and the basics to get us to this point where we can start building up a molecule. We can start building up a hydrogen atom and then we can move on to multi-electron
26:43
systems and we can move on to diatomics. And at that point we'll kind of, you know, skip some steps and just have molecules. So what a wave function is, it's represented by this psi, which is another Greek letter. And it's going to describe the movement of a particle, okay?
27:04
So it's just a symbol for a mathematical function. So I'm going to show you those functions in a little bit. We're never really going to have to work with them. I'm going to give you the solutions that come out of them. I'm going to give you the pictures that they pull out of them. You need to know the general way in which these work, though.
27:21
You need to know that it describes the movement of a particle. You need to know that it is a function that you can actually graph, that they've solved these functions and they can pull them into a computer and they can graph them. And that's what gives us our orbitals. Different particles are going to have different wave functions. So a particle in a 1s orbital isn't going to have the same wave function as a particle
27:43
in a 2s orbital or a 2p orbital or all of those. Now we have something called psi squared or the probability density. Now what this is, is this is going to give us the probability of finding a particle in a particular region. So if we say we want to find the particle from here to here, what is the probability
28:02
that it's going to be there? And that's what the probability density, what we can gain from it. Now keep in mind if wave function or psi is a mathematical function, what is psi squared? Well it's that function squared. So that's why it's called psi squared. So technically the probability density is the probability of finding a particle divided
28:24
by the volume, so that's just sort of a technical thing to make the numbers work that I wanted to add here, but don't worry about that too much. So let's go through and talk this out with like a super, super simple example, not something that would actually be a wave function, but it gives us a little bit more of a clear
28:43
way in which how these bigger functions that I'm going to show you work. So this is more understanding purposes and being able to replicate it purposes. So let's take an example where psi is a number. I do this to make it simple, right? How can you get a simpler function than a number? So we're going to say that psi is just 0.44 centimeters inverse.
29:02
So your book walks you through this sort of idea too, but I wanted to go over it again. So we graph this function, the 0.44 centimeters inverse, we get this. So what's psi squared going to be? Well it's just 0.44 times 0.44. So we can graph that too.
29:22
Again, really simple examples, these wouldn't be wave functions of something we'd actually calculate, but it gets our point across. So this would describe the movement of the particle, this would describe the probability of us finding it in any particular place. So the wave function, this is sort of nomenclature how we would say this, the wave function
29:41
is equal to 0.44 centimeters inverse. The probability density is equal to 0.2 centimeters cubed. So if I wanted to know, what is the probability of a particle being within 0 to 3 centimeters
30:01
cubed? I say, what is the probability of me finding this particular particle between 0 and 3 centimeters cubed? In other words, between here and then 3 centimeters out from it in a circle. Well the probability of that, you would just multiply this together, right? It's effectively taking an integral, but it's a simple integral because it's a box.
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It's just the 3 times by the 2. So our y-axis times by our x-axis since it's a box. And that would give us our probability density. Now this is a very simple example with a very simple function. When we do this with bigger functions, we're going to get much more complicated probability densities and different shapes and things of that sort.
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But this is how they do it, they just do it with a computer instead of by hand. So let's look at some things about this. So some questions for you. Psi is a function. So at any point can we determine its sign?
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Meaning positive or minus. Does it have to be positive, does it have to be negative? Can it be both? Well it's just a function, right? So sure the function I showed you in the last one was always positive, we agreed that was sort of a simple example. Wave function, it's just a function, so it can be both. It doesn't matter, the sign doesn't matter. So if I show you a graph like the one on the other slide, like this, and I say well
31:24
which one's the wave function, you would know that it's the one that can be negative. This one, now if we move on to Psi squared, and I say well the square of that function, right? It's just taking the function and squaring it. So if you square something can that ever be negative, assuming that we're in the realm
31:41
of real numbers? Well no, that's got to always be positive. So if that's always positive, if I were to give you a graph like this one, and I say I have both the probability, or I have the wave function and Psi squared graphed, you could pick out which one was which. You could say well this one's negative so it's got to be Psi, this one's always
32:03
positive so that one will be Psi squared. Okay, next thing to talk about then is nodes. So nodes occur when Psi is equal to zero. Now if Psi is equal to zero, what does that mean about Psi squared?
32:23
That was going to mean that Psi squared also equals zero, right? Zero times zero is most certainly zero. So wherever wave function is equal to zero, Psi squared is equal to zero as well. Now think about what Psi squared is though, it's a probability density. So it's where you're likely to find something.
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So at a node, what's the likelihood that the particle will be there if Psi squared is zero? It's going to be zero, right? Because if the Psi is zero, that means Psi squared has to be zero, and Psi squared is probability density, so if the probability of finding a particle someplace is zero,
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it's not going to be there. So wave functions have n minus one nodes. That may not mean a lot to you right now, but when we start talking about energy levels, that'll mean a little bit more to you. So if we have a wave function at a particular level for a particle in a box, it's going to have n minus one nodes. So here, we don't really have a node, technically this is asymptotic, so it just approaches
33:26
zero and never quite hits it, and you talk about nodes just being in the center like right here. So this is asymptotically going to zero, it's not a node, this is a node. Okay, so now, Schrodinger equation, this is where we really get into all of your
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guys' favorite things. So what the Schrodinger equation does is it allows us to take the wave function and come up with an energy level. Now we've talked about energy levels already, right? We've talked about the Bohr model, and we've talked about having an n equals one, an n equals two, and an n equals three, as it might be an oversimplification of energy
34:03
levels, but it does work to help you kind of think about it. So the Schrodinger equation gives us a way to actually go through and find these energy levels. Now you don't have to worry about this equation so much, I put it there mostly because your book does, and it's good to have seen once, if you go on in physical chemistry you'll see it a lot.
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But the way that this works, the way that this equation works, this H is called an operator. And what that means is that you're taking something and you're doing something to this function. An operator operates on a function. So for those of you who have had some calculus, which is most of you, that means that for
34:42
instance if you take a derivative of something, if you take the derivative of sine, you take the derivative of cosine, that taking a derivative of is an operator. Maybe it's something simpler, maybe it's plus two. So you take whatever function you have and you add two, and that's your operator. It just means you're doing something to that function.
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So this happens to be the H, the operator, this is what the hat means, this happens to be what we call the Hamiltonian, and this is the Hamiltonian for this. Now if you care, this is one of those for your info sorts of things, this is the double
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derivative, right? So that would mean you take the wave function and you take the double derivative of it. You multiply it by this. This is just a number, right? It's a big equation but this part is just a number. It's H bar, which is H over two pi, it's just a constant squared, two M which is just a constant. So you have that and then taking the double derivative of it.
35:43
So again, you don't need to worry about this too much, it's just sort of for your info. Part you do need to worry about is knowing that this is a Schrodinger equation, that this means you have to do something to the wave function and then you get your energy back. So one of the easiest ways to think about how this would work is if you take the
36:01
double derivative of a sine or a cosine. If you take the double derivative of a sine, you end up back with sine but now it has a negative one in front of it. This would be like this, you get the same equation back but you have some number in front of it. So taking a double derivative of a sine would give you negative sine and you would get
36:22
the negative one back. If you don't have any calculus, it's a little bit harder to come up with the way this would work, so we'll just kind of leave it at that. But you have to know that you have an operator, it operates on a function, and you get an energy level. That's sort of the condensed version of what I want you to know for this class.
36:41
So the way that we're going to use this is we can use this to find the wave function in E. Now for this class, we're just going to use the results of this. We aren't going to go through and do this, we aren't going to calculate these, we're just going to say okay and here are the energy levels, here's the results. Okay, so we're going to start with a system called a particle in a box.
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Now this got started with, because it's a little bit simpler than something like a hydrogen atom. Hydrogen atom has sort of three dimensional space, a particle in a box is simpler. So the way that this works is you have a box, you can think of it as a box, and the electron can't go through the box, and it's in the middle, it has to be inside
37:21
the box and it's going back and forth, which is a little bit, not quite the way you want to word it, but it's in the box. So kind of the real life way you can think about this, since it's hard to picture an electron being delocalized in a box, is if I took a string and I stretched it from room to room, and then made it like a guitar string and plucked it, and you have
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this wave going back and forth. So you know, go home, steal someone's guitar and look at, you know, pluck the string, see what happens. You'll be able to see the wave kind of oscillating back and forth. Now you're allowed certain wavelengths. This is relatively easy to do with something like a jump rope, right? You can take a jump rope and you can very slowly go like that, and you can get this
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pattern. Or you can speed it up. When you speed it up, you can get a sort of pattern like this. And the faster you go, the more times you can get it to oscillate in one. Eventually your arm won't be able to do it anymore, but that's not an issue with the particle in the box system. So that's how you can kind of think about this working.
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Now, you'll notice only certain wavelengths are allowed. The same is true when you go through and try to do this with a jump rope, right? You're allowed the certain wavelength where you get this, the certain wavelength where you get this, but not in between. And when you try to switch from going the slowly way to a little bit faster, there's this weird point where you get this, it's not a wave anymore, it's not a normal
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wave, it's a transition point. So it kind of works there too. Now, here's where we're going to take the Schrodinger equation and I'm just going to tell you what the result is. So that H psi equals E psi, this, the H psi equals E psi, and I told you that we're
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going to just solve for E and I'm just going to tell you what it is. Okay, well here's our results. So this is psi, this describes the different equations that can relate this electron in the box. And then this is our solution to the Schrodinger equation, this is our energy. Now don't let this equation bother you, there's a lot of things in it, but none
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of them are that big of a deal. N, that's just one, two, three, four, you're thinking about your jump rope, that's which level of energy it's at, how hard you had to move your arm to get that. Now H, Planck's constant, it's not any different than what we've been talking about. Eight, I think you're okay with the number eight, hopefully.
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And we have M, that's the mass, so just whatever particle we're talking about. And then L is the length of the box, so that's not a big deal either, that's just a number that we're going to decide on for any particular system, it's just from here to here. Okay, so what we can do is we can kind of use this to model a particle in the box.
40:07
Example where we're given a wavelength of an electron, or we want to find the wavelength of an electron of a theoretical atom given an approximate diameter. If we say that the diameter is about 100 picometers, and we can find the wavelength of that theoretical atom.
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So I've changed this example around a little bit from what was on the slides. So this is what was on the worksheets, and I say we're going to go ahead and we're
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going to solve for this. So actually I've just decided to skip this step, we decided to skip that and do that at
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a later time once we know a little bit more about the way quantum mechanics work. I think it'll be a little bit easier to understand then. So in the meantime, let's now take and go to the next step. So the particle in the box is a nice model, it kind of lets us take, if you actually
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go through and do the quantum mechanics, it's a little bit easier to work with than something like a hydrogen atom, the equations are simpler. So that was kind of why it was developed. But now we also have hydrogen atoms, and we can actually go through and we can find the equation for that. Now I'm going to talk about two different ways about thinking of all of this with the hydrogen atom.
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We're going to talk about the Rydberg equation, and then I'm going to go through and do it with the Schrodinger equation as well. Now the results of these are exactly the same. So what you're going to find is regardless of which way we do it, you're going to have the same answer and we're going to have the same equation. So you can sort of pick the easier of the equations when you're trying to solve for
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an actual number here. So in my opinion, I think the Rydberg equation is probably easier to work with. Now both of these were used to describe the hydrogen atom, but the difference is in how they were discovered. Rydberg did it experimentally. He looked at a whole bunch of spectra coming in and said, notice patterns and notice the
42:44
way numbers work together and came up with an equation that described it. And it had a constant in there that we're calling the Rydberg constant, and it had all of the energy levels and everything in it. Now what Schrodinger did is he said, well, I'm going to start from, you know, the wave function and I'm going to put the Hamiltonian on it and then I'm going to solve
43:03
for the energy and then that's going to give me all of the energy levels. So one was done experimentally, one was done theoretically. But either way you do it, you get the exact same results. So Rydberg was done first and then Schrodinger's agreed, which is always great. In science, that's what you want.
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You want someone to do it experimentally and then someone and or someone to do it theoretically and have the two results agree because that means that your theory is relatively sound. If your theory doesn't agree with your experiment or your experiment doesn't agree with your theory, you're obviously not understanding something. So this showed that, you know, they both kind of had it right.
43:41
So this is going to be true for the hydrogen-like atoms. And what I mean by hydrogen-like atoms is one-electron systems. So if you take some sort of atom and you take away all but one of its electrons, you can use this solution set to do it with one little change. Now I'm going to start backwards from reality and start with Schrodinger because I
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think having the theory base first and then working into the experimental is a little bit easier to understand. So with Schrodinger, this is going to walk through and do the exact same thing that we did with particle in the box. So now remember I said I was just going to give you the results. So this is the results when you come out.
44:21
This is the energy level. Now Z is equal to the atomic number. So Z is one of those letters that is kind of universal in chemistry. So you want to recognize that as atomic number. H planks constant. We'll talk about R in just a minute. We'll leave that alone for a sec.
44:40
And then N is the energy level. So now what R is is this whole series of equation or this whole series of constants. None of these are a big deal. They're all just constants that you can look up. It just gets a little tedious to write them all in. So you just have a ton of constants here and then the Z squared H over N squared.
45:03
So really H is a constant too, right? So you really just have Z squared over N squared times a whole bunch of constants. So keep that in mind when we go to do Rydberg's equation. It's really just Z squared over N squared times a whole bunch of constants.
45:22
Now let's look at the Bohr model of the atom one more time. So remember we have our nucleus with all of our positive charges and we have our rings of electrons going around the outside. Now this arose because they knew that we had a proton filled nucleus
45:40
and we had the electrons going around them. Now the interesting thing here is that if you look at the laws of physics, it says that the electrons should spiral into the nucleus. That it should be going around and around and around and then just spiraling in and everything collides in together. Well after Planck discovered quantization and Planck said,
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hey these particles are emitting at all different sorts of wavelengths and they're very particular wavelengths and not in between, Bohr said oh well that's because the energy levels are going to be quantized. That you can have this energy and you can have this energy but you can't have an energy in between the two. Now this particular model is only going to work for one electron systems.
46:21
It does turn out that they're quantized just in a slightly different way than what Bohr thought. Now the energy difference is given by negative rh z squared over n squared and that's what Rydberg went through and said. So before we talk about exactly what Rydberg said, I think it's kind of worth noting the stair and ball analogy
46:42
for quantization and energy levels. The idea here is always that if you set a ball down on, you know, this energy level or this energy level or this energy level, it can be at any stair step energy level. It's never going to be in between. It can transfer between the energy levels. It can go from here to here but it cannot be like held in the middle somewhere.
47:02
It needs to be on one of these particular levels. So that's sort of how you want to think of these is you can be here, you can be here, but you can never be in between. Okay so now moving on to what Rydberg said. Rydberg looked at all these sorts of spectra that we're going to talk about in a minute and said well this is what I think the energy levels are at.
47:22
I think that it's at some constant that I'm going to call the Rydberg constant because my name is Rydberg and so I like the way that sounds. Times z squared over n squared. Now keep in mind what did we have for Schrodinger's equation? We had a whole bunch of random constants that all have numbers associated with them times z squared over n squared.
47:43
So the two aren't any different. The only difference is is that when Bohr did it he just knew it was a number. He just knew because experimentally this had to be a number and he just called it one constant. Where Schrodinger because he did it theoretically was able to say well I know what all those constants are. I know that they're you know it's all of these combined together.
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If you multiply all of those together it comes out to be the Rydberg constant. It comes out to be the same number and so that's how these two are related. I think that this one's the easier one to work with because we only have to worry about one constant as opposed to those like five or six different constants.
48:22
Okay so that's our energy level. Don't forget z is atomic number. So that's we're still always going to be talking about one electron systems but because we can have ions z can change. We can have a helium one plus or a lithium two plus or you know things of that sort that we're calculating. So this is what the Rydberg constant turns out to be.
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There are some other versions of the Rydberg constant that you'll see a lot of times you'll see it written in terms of Hertz and you'll see the Rydberg equation that we're going to talk about in a minute written in a different way is one over lambda. The way I'm doing it on the slides is the way that I've found the least amount of people to make mistakes on actually doing the problem.
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I would suggest doing it this way. Your homework suggests doing it a different way. I don't remember what your particular book says. There's like four or five different ways to do all these problems. I think this is the best way to not make mistakes at your guys's level. So z is again here atomic number drive that point home.
49:21
Okay so what is our energy difference? So we're going to care a lot about these transitions. How much energy it takes to go from this level to this level. How much energy we get back if instead now it drops down to this level. And we were going to want to be able to calculate all of these. You can do this with both Schrodinger's and Rydberg's equation.
49:41
But again Schrodinger's is a little bit complicated because of all these different constants where Rydberg only has R. So it's a little bit easier to work with. So we're going to go through and we're going to work with Rydberg's. So if we want to find the difference in energy levels, the difference that it takes for an electron to go from here to here or from here to here or so on, it's a difference.
50:02
So it's always final minus initial. So we're going to go through and we're going to derive this as sort of our last thing of the day. So for a given energy level it's going to be defined by this, right? This is just from the last slide.
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So we know that we can find the difference in energy by taking the final and subtracting the initial. That's just the definition of change. Now we also know that we can define both states. We can define the initial and we can define the final. So all I did was rewrite this equation two times.
50:41
One time calling it arbitrarily initial and one time calling it final. Now we can fill these in to this equation. So we're going to take the final and fill it in. We're going to take the initial and fill it in. So change is equal to final minus initial. So we fill everything in. So again, taking the energy levels, filling it in.
51:02
So we're trying to find the difference in energy between two different levels. Now we have RH which doesn't change and Z which isn't going to change either, right? It's not switching atoms, it's just switching within one atom. So we can just factor those out of the equation, set them aside for the sake of algebra.
51:21
So when we do that, we get these. Where we take and we have the RH and the Z squared. You can factor the Z squared out too. Some books will leave it in, some books won't. You end up with this equation. Now something that gets talked about a lot and students have issues with is the fact that this is written two different ways depending on where you happen to find it.
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Sometimes you have a negative sign and sometimes you don't. So the difference here is that here you have a final minus initial and you factored out the negative. Here you basically put the negative inside but when you did that you ended up with initial minus final.
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It doesn't really matter which one you use. On the final I'm going to give you one of them so just use that one. But when you're doing your homework, be careful if you're doing this off memory, not to mess this up. The one with a negative is final minus initial, the one without the negative is initial minus final. And it's not that one of them is wrong or one of them is right, it's just different ways of writing the exact same thing.
52:22
So keep that in mind. So next time we'll go through and we'll do some examples using this and we're going to do a lot of talking about sign convention because the signs for all of these are really important. And honestly is one of the most commonly missed things on the entire exam. If you mess up the sign you're going to have problems.
52:41
Again going back to what I said before there's different versions of this equation. You'll see it with one over lambda. Sapling does that, some books do that. I think it's easier to do it this way and I'll show you why when we go to worry about sign conventions and we do some examples. You have to be really careful about which direction these transitions are going.
53:02
Delta E is going to be allowed to be positive or negative and you're going to run into problems if you mess that up. If you mess up whether the delta E is positive or negative. So we'll talk about all that next time. All the sign conventions, examples, things of that sort and really kind of finish up the hydrogen atom next time.