Wave Functions

This section deals with the basics of quantum mechanics. Don’t worry too much about it, but this is some good general information.

Definition of an Atomic Orbital

Concept: The probability of finding electrons in a given place.

7m
Video Transcript

Why is quantum mechanics important? Why do we have to talk about it for orgo? Because maybe you guys remember that the smaller particles get, the more that they're going to function as both particles and as waves. What that means is that in regular physics, if you have a ball and it hits something and it collides, that's what we can Newtonian physics. But as these particles get very, very small, for example, electrons, they're not just going to behave as particles anymore. They're not just going to have collisions. They're also going to behave as waves and interfere with each other.
There are these mathematical equations that make this – the math is very confusing. We don't need to know all that, but we do need to know a few things. For example, the Heisenberg Uncertainty Principle, what does that mean? Well, what that says is that because these are acting as waves, we can't simultaneously know an electron's speed and it's position. We can know one or the other. We can know where it is, but not how fast it's going or we can know how fast it's going, but not where it is.
What that means is that instead of focusing so much on where the electron is at a given moment, what we're going to do is we're going to focus more on probabilities. We're going to say, “Okay, what are the chances that an electron is in this space?” That's actually going to be a lot more important for this course.
Remember that I was just talking about these mathematical equations. Well, the reason that these equations are complicated is because these aren't just particles; these are acting as waves. There's a lot more complexity to it. The way that we describe these is through wave functions. So you can think of a function in math. It's just an equation that's going to describe the energy state of an electron at a given time.
There is a Greek letter that we use to symbolize the wave function, so go ahead and circle that. It's the Greek letter psi, so I kind of drew a bad psi, but whatever. That's a psi. The psi is my letter to substitute for this wave function that I'm not going to teach you because it would be way too long and way too tedious.
But it turns out that if I take a derivation of that equation and in fact, if I square it, so the psi squared, what that's going to give me is the relative probability of finding an electron in a certain space. That's really important because now if I can take that equation and square it, that's going to tell me what are the chances that an electron is in a certain place and that's what's important to me as an organic chemist.
Finally, where does this all go? Finally, if you take the 3D plot of this psi squared, because remember this is just an equation, so if I take a 3D representation of it, so just right here of the psi squared, what I'm going to get is a region of space called an atomic orbital. That's where this all comes together.
Basically, I'm using these really fancy equations to describe where our electron is going to be. That is what we call the atomic orbital. So the atomic orbital is just a mathematical representation of where these electrons might be at a given time. That's a place where the chances of finding an electron are high.
So now let's go ahead and talk a little bit about what these orbitals look like. I know you guys might remember this from gen chem, but let's just go over it really quick. The simplest type of orbital is the 1s orbital. The 1s orbital is just a sphere. It's kind of small. That's it. Remember that the first shell can hold two electrons and the 1s orbital – each orbital can hold how many electrons? Two. So that's it. There's really nothing more to know. That first shell only has one orbital and it's 1s.
Then once we get into the second shell, the second shell we're going to learn can actually hold eight electrons. I'll just put e's. Eight electrons. What that means is that it consists of four different orbitals that can hold two electrons each. The first one and the lower energy one is going to be the 2s. Now the 2s looks a lot like the 1s, it's just bigger because these electrons are now a little bit further away from the nucleus, so they have a little bit more energy. So that one can also hold two.
Then finally, we have these three orbitals that are of a little higher energy state and they're all the p orbitals. The way that I think of it is they kind of look like peanuts. Your professor might have said dumbbells or whatever. Whatever the case is, I just think they look like peanuts. So there's three of these. They go in different directions. The important part to know is that they all have the same energy and they can hold two electrons. Good.
What if we were, for example, trying to draw the atomic orbital diagram for carbon. Let's say we're doing carbon right here and trying to figure out where the electrons are going to go. So first of all, you guys have to tell me how many electrons would a carbon atom have. It would have six. So let's go ahead and write six electrons. That's because carbon has an atomic number of six, so it would have the same amount of electrons.
So where do we put these electrons? Well, the first two should definitely go in the 1s. So now I have my two first electrons at the lowest energy state possible. Now, that energy state is full. That shell is full, so now I have to jump up to the next higher energy and that would actually be the 2s. Remember that I said 2s is a little bit more. I would then put two more electrons in the 2s. Now I have four electrons that are filled in orbitals, but I still have two left. Where do you guys think I should put those? This one's a little trickier. I should put one in one p orbital and one in another p orbital.
Now just so you know, all three of these p orbitals are mathematically equivalent, but it's just normal convention to use x and y first. If you really wanted to be a rebel, you could use z first, but your professor would just think what are you doing. Also, notice that I drew them with an up spin. If you drew them with a down spin, that would also be fine as long as you're consistent. Don't draw one up and one down. That would be bad. But if you draw them both down, that's fine.
Cool. Just so you know this little meme that I drew, I made this because it's like Schrodinger's cat is a thought experiment to talk about that a cat could be half alive or half dead at the same time. It's used to talk about how electrons could be in one place or they could not be in one place. It's interesting. Whatever. This meme did not go viral, so I should probably just stick to organic chemistry because I guess it wasn't that funny.

These funny orbital shapes represent the 3-D plots of the equations that describe the probability of finding electrons at any given place as their energy states increase.

Atomic Orbital Interference

Concept: Constructive vs. destructive interference.

4m
Video Transcript

Let's keep going so now let's talk about one more thing in terms of quantum mechanics and that is what I was hinting to earlier with the fact that these particles don't just act as particles they don't just bump into each other they act more like waves into a sort of certain extent so as waves they have the ability to interfere with each other so let's write in that word interfere, and what that means is that instead of colliding they actually may interfere constructively or destructively so what does that mean? Well think about it like this instead of thinking about these two orbitals are like balls and like they will hit each other and bounce off each other no that's not how they act at all instead think of these orbitals as waves as regions of waves and once these orbital overlap with each other they can either amplify each other or they can basically cancel each other out, you could also think about it for example a pool of water I make two different splashes in two different places some of the winds are going to come together and make bigger waves, right? But also, some other waves are going to come together and actually cancel each other out, maybe you've noticed that that there will be some regions that are really big waves and some regions that have like not that many waves, OK? That's the difference between constructive and destructive interference so I have constructive interference the waves come together and they make waves of greater amplitude, alright? With destructive interference that means that these Orbitals come together but they wind up canceling each other out in some way and actually making no net wave at the end now what do these waves represent? I know you like thinking where is Johnny going with this? Well when you increase the amplitude of your wave what you're doing is you're increasing the chances of finding an electron in a certain place, OK? So, notice that I have this region of overlap right here, OK? That is between the two. When I have constructive interference, what that means is that now the chances of finding electrons in this space right here are higher, OK? When I have destructive interference then what that means is that the chances of finding electrons in the middle are lowered, OK? And that's what I've drawn here at the top and the bottom when they constructively interfere that's what we call a bonding molecular orbital and that's the whole idea behind the chemical bond, a chemical bond isn't just some random stick floating out in space what it is is that there's a higher mathematical probability of finding electrons in this region right here, OK? And that's what we call a bond so anytime you think of bond now think that's where electrons are at a higher likelihood of them being, OK? Now the opposite is true when they destructively interfere, when they destructively interfere that's called an anti-bonding orbital because what happens is that the chances of finding electrons in the middle actually goes straight to 0, remember that I said that psi squared is the orbital or the relative probability so basically what we're saying is that there is no relative probability of finding electrons right in between there, by the way just so you guys know any area that has a probability of 0 is called a node, OK? And that's something random that you might need to know just some.... Whatever you get what I'm saying it's some terminology that you guys need to know.

Instead of colliding into each other, wave functions have the ability to interfere with each other upon meeting.

The type of interference determines if a new bond will be created between the two orbitals.