[BLANK_AUDIO]. Hi there. So today we're going to move onto the Total Wavefunctions for Hydrogenic Atoms and how you interpret, interpret them. You recall from our, our previous presentations that we define the total wavefunction, we had the three quantum numbers: n, l, and n sub l. And to define it in, as vertical, polar coordinate system, r theta and phi. We showed that, in order to solve the Schrรถdinger equation, we could separate that into radial part, which we denoted R n l. R and we said that, that was simply total wavefunctions the product of the radial and the angular part. So you had Y l, n l and Y is a function of theta and phi. So say let's look at the first solution that comes out, and that's the Y sorry to, psi 1 0 0. That's r, theta, and phi. And this of course we usually call our, our 1s orbital, but that's now going to be the product of R 1 0 r times Y 0 0, theta to, theta, phi. So if you look back we looked at the solutions for the radial and the angular parts already. So as for the radial part, we got 2 times zed over a 0. All to the power of 3 over 2, e to the minus zed r, over a 0. Now that's the, the radial bit here. And then we times that by the angular of it. And for a, a y 0 0, the solution we mentioned before is 1 over square root of 4 pi. So, this is the, is the radial part here and this is the angular part, and the total wavefunction for the, what we usually describe as a one s orbital is this deposit of these two as given, as given here. So likewise we can, move on to our, our, our next solution and that was the psi. I'll use a slightly different pen, psi 200 or theta, phi and that's equal to the product of R 2 0 or r times Your angular component is the same as above because your angular component is an s orbital again. So again we can look back and we did work on the radial solution which R 2 0 R which of course is not [UNKNOWN] 2 R, 2 S orbital and if you look back that was twice, 2 times zed over 2 a 0 all to the power of 3 over 2. And then you had a polynomial term. 1 minus zed r all over 2 a 0. And then that was mult, there the exponential term, which is e to the minus zed r, all over, 2 a 0. And then of course you multiply that by the angular component. And the angular component is the same. It's an s orbital. So it'll be the same as up there. 1 over the square root of 4 pi. So we can move on then, to the next one which was. Psi 210, r theta phi. And that of course is equal to the radial 2 1 r. Multiplied by Y 1 0 in this case theta, theta phi. So again, we've looked at the radial part for this. And if you look back, it's going to be 1 over the square root of 3. Zed over 2a0. That's also power of 3 over 2. Then you have another component here, zed r over a0. Remember, this r is important, because this is a this is actually the, a 2p, p zed orbital, and p. P orbitals, at a 0 aptitude at the nucleus. We mentioned that already. So then we have e to the minus zed r over two a 0. So that's your radial component that's your r 2 r. And now you need to multiply that by the angular component for this orbital which was for y 1 0 which is 3 over square root of 4 pi cosine of theta. So [UNKNOWN] would be just over 3 over 4 pi. This is your total wavefunction for what you commonly call the two p zed orbital. Okay, so we could carry on like this down if you want, to. We're going, we're going to stop here. More importantly now is to consider well, we've got this total wavefunctions for the hydrogenic atom systems, [SOUND]. How do you interpret, and of course we interpret them in exactly the same way as we interpreted the wave functions for the particle-in-the-box. The way we interpret these is we use Born's, Born's Interpretation. [BLANK_AUDIO]. As you are familiar, that Born's Interpretation, what he said was that the way you interpret it is you find the probability density. And remember, we said that, well said Bora, put the particle in a box, so that the square of the wavefunction, he defined that as the probability density. So we do exactly the same thing for the hydrogenic atom system. The only thing that you might come across that's slightly different, it won't be relevant to our course here. But because you have complex wave functions coming up, what is usually done is, you say it's the magnitude of the wavefunction squared. Because when you have complex wavefunctions. You have to get to the probability you need to multiply by the wavefunction by what is known as it's complex conjugate. This will give you a real number and that then refers to the probability of finding the, the electron in this case. So like wise, so, we'll going to, let's stick to our, our size squared. So we also will define size squared d v or d d what we call a small volume element. So if sized squared is the probability density that's the probability at like a point in it's space. Where [UNKNOWN] d v is the actual probability of finding the electron particle in a small volume element d v. So let's write that down,. So that's the probability. [BLANK_AUDIO]. N d v volume, volume element. So let's look at a few examples of this interpretation. From the wavefunctions that we've just, obtained. So again, again it's useful to break it down into the radio part and the angular part at least initially. So let's take our simplest radio function, the radio part for the 1s orbital, so you have R 1 0. Now, again, we've written this down before. I'm just going to put a site called the factory, values are the numbers before the exponential factor with constant k. So this is a constant k, which corresponds to these. And, it's e to the minus zedor, over, over a 0. So let's define that, k is just a, a constant. We're not going to write it out in full for this, for this [UNKNOWN], because this will suit our, our purposes. Now, what we said is we want to get the probability density based on Born's interpretation. As we said, this is the wave function, the radio wave function. You just square that. So you have r 1 0 R, and you square it. So you should be up to square this function up here, that's going to give you k squared. And it's going to be e to the minus square exponential you are going to use minus two z r all over, all over a 0. So as we did when we worked, looked at the wave function initially itself. Let's just, let's just get a very quick sketch of what that might, that might look at. So here we have as we did before just a rough sketch here. So here we had, we had zator over a 0 as we had before and then on this axis, we had our r 1 0 r. Now you should be able to recall that there was a nice, just plotting this function here, constant times. A nice exponential function. So you had a finite value of constant K here. Start off with that and then it's moving the k's down exponentially and tending towards, towards 0. As [INAUDIBLE]. Tend to tends to vanish here the radius tends tends to infinity now lets look at this function here. Here you have r one zero squared function, so you square the initial value here. Lets not worry about that but this part here you have now e to the minus two. So, again, based on your mathematics, you should know that this is going to increase the slope. The larger value here, the exponent, is going to increase the. The slope of the rate of decrease. So what you'll get is something let's put in a different color. So you would get a sharper let's see if we can reproduce that, a sharper decrease. And then again, again it tends towards, towards 0. So the square of the wave function in this case is similar, but it's as I say, the slope's a little bit, a little bit steeper. Now let's move on to to the next radial part, wavefunction we've looked at. And that was let's do it in a, another color here, so that was our, our 2 0 R. And again that was some constant. You can look back on this where the constant factor here. And then you had this polynomial function zed r all over 2 a 0. And it was e to the minus zed r. All over two over 2a zero. So when we ,well let's, yeah let's do the squares, square that because we want to get the probability density so that's going to be r zero r. All squared. So what we're going to get here now is we're going to get k squared and let's write it out like this. We're going to get one minus zr over two a 0 and that's going to be multiplied by [UNKNOWN] so one minus z over. All over 2 a 0, and then we are going to get e to the minus 2 zed over 2 a 0 or e to the minus zed over a 0. So this is the function [UNKNOWN] equation and this is the square and this gives us the probably density for the radial part of that function. So that's, now let's plot these two as we did before. So let's leave the functions there. So on one side, I'm going to plot the wave function itself. And again, this is Z r. Over a0, on your x-axis if you like. And here we have R20 r axis. Now we've done this before, but let's just show it again. Sketch it out. And remember if you went back to your notes on the video parts, this polynomial here causes this function, as are increasing, causes it to go to zero, pass through, pass through zero. So it comes down this wave function. Comes down something like that, and then the exponential brings it up. So then if you like you, as we mentioned before, you have a positive and we'll call this a, the negative, negative part of the, of the wave function. So what does the squared wave function look like? So we're going to plot that over here. So here we have, similar type of axis system. Again, we're Zr over a0 on our x if you like. And then we have R2 or. That's all, that's all squared. So, what's going to be the difference on this occasion? Okay, your exponential factor, let's move it up a little bit, your exponential factor here, again. Is, is, is bigger for the square function, so that means the initial slope, slope is going to be is going to be greater like the, like it was for the, for the 1 [UNKNOWN] orbital. But there's also a difference here because when we can, when we, plot the, wave function here we come down and this term 1 z r over 2 a zero is equal to one, this goes to zero and then as it increases it goes negative. You get a negative number in here. So there for when negative [UNKNOWN] is pulled up eventually twords 0 by the exponential term. So what happens here, again when Z R over 2 a 0 is equal to 1 goes to 0 both of these go to 0. At that point, so lets draw it in. So the canvas is a very qualitative diagram. So lets say its going down like that, very lovely. But now what happens here, this. This goes to 0. But now as this gets bigger it just goes negative. And this goes negative. So a negative by negative is positive. So what happens is, is instead of going negative, it goes, it goes positive. And again, it's pull down. So, if we compare them, so what you can see now is you've got two positive switch is what you want really because we're talking about the probability of finding the electron as you go out. A, a, an [UNKNOWN] distance from, from the nucleus, and of course a probability has to be, has to be positive. Okay, so similar to the radial part, we've also got to, the top wave function is the, the radial times the, the angular part. But the angular part as well we've got to, we've got to square, so we're now going to move on to the angular. Are we going to move on to one or two examples of the angular part. And again we start off at our let's start off at our Y, one zero. Now we know that's theta, theta phi dependence, and again. You should know this is equal to one over square root of four pi. So our Y one zero theta phi. Squared is equal to one over four pi. So that's fairly simple, but this is the, the, the y, sorry that is wrong here. So this should be y zero zero. And this should be Y 0 here. [UNKNOWN]. So this is our our [UNKNOWN] orbital, so S orbital is at equal to zero and [UNKNOWN] zero. So it'S Y0 is 0 [UNKNOWN] here. So it's 1 over the Y function is 1 over square root 4 pi, so the [UNKNOWN] function then. Is 1 over 4 pi. So this is going to be a smaller number than this. And remember this is what gave the, the s-orbital its spherical shape, so this squared function, for the other, one that corresponds to probability density, is going to correspond to a sphere, spherical shape again. It's going to be. Spherical, that is going to be a slightly smaller. Smaller value if you like than the than, than the actual way function itself. So we can now move on to the next angular Y function,. And here we had our Y one zero which I was trying to like, wrote it down incorrectly the last time. Now again if you look back in your notes, you'll find that this is equal to the square root of three over four pi and this is cosine of theta. So the square of that, so we have y one zero, t to [INAUDIBLE], square that and of course you're going to get three over four pi, cosine squared. So what effect will that have? Well you can do this more accurately but as we showed before for the angular, this actually corresponds with what we had before, P Z, the P Z orbital. So when you plot that, so you have let's say this is defined, this is our, our y axis here. Remember the p orbital, the peel away functions are, we called two dumbbell shapes. So you have two different phases. So. Plus and minus. So that corresponds to, roughly to that, that here. And of course if it's a three dimensional in three dimensions as well. And this is your this is your zed, zed axis. Now, when you square it out. So, we're going to square it that way function. And we're going to get this one. [BLANK_AUDIO]. Going to get this one here. Now, what you'll find and it's mainly because of this you're going to have cosine theta. The cosine squared theta, remember again for the angular parts we do a, we do a polar plot. What you find is if you represent it the same way. So here's are y axis. You find is a more compressed looking about something like that. Actual it's the kind of loop you see mainly written. In textbooks for p orbitals. And, and, and this, use, should be taken to correspond to the squared angular part. This one, this one here and the same thing will happen for, for the p x and p y we're not going to go into that they're there kind of a signed squared dependence, but again because the signed squared opposed to sign. You get a more compressed looking function of course then it's the same psi, sign on, on, on both [UNKNOWN]. [BLANK_AUDIO]