[BLANK_AUDIO]. Okay, so what we move on to now in this presentation is how we actually interpret the wave function. We've seen that if you solve the Schrรถdinger equation. For a simple system like a particle in a box, then you predict a number of different wave functions for the system having different energies. But what, what do these wave functions actually mean? And the person that gives us the interpretation of these wave functions is this person shown here and he was called Max, Max Born. He was a German, a German scientist, and what he decided he would interpret or you can interpret the wave function laws. He decided that if you squared the wave function, so you have a wave function psi and you square that. What he said was that this represented the probability. [BLANK_AUDIO]. Density of finding or the value of that wave function squared represents the probability density of finding a particular at that particular point, point in space. Now this idea came from the idea that if you have electromagnetic radiation as a wave, in waveform. Then the square of the amplitude of the wave is referred to as the, as the intensity. So what Born suggested was that here you have a particle wave, and therefore the square of the wave function. Will give you the probability of finding the particle at that particular point in space. So what you interpret, what Bones interpretation, is that you interpret the wave functions that come out from the solutions of the shunju equation as probability amplitude functions. So if you again take a very simple system, where you're moving along one direction, the x direction. And here's your point, a point x, and now you go a small region of space, an infinitesimally. Infinitesimally small region, Dx, then another way of stating this is that the square of the wave function times Dx, and that now will correspond to the probability. [BLANK_AUDIO]. Of finding that particle in a region x plus dx. So if you'd like it's trying to find out an infinitesimally small region here. What's the probability of finding the particle in that, in that particular region? So if we move down, then we can look at our solutions for particle in a box step we did in a previous presentation. And let's take the the first solution we've found, we've found that we had psi 1 of x. And we had worked out that that was equal to some constant a sin of pi x, all over L. Now we had represented that function when we, when we drew it. We were going from zero to L. And we said that that represented like half a, half a wavelength, and we give it a positive, positive amplitude. Now what Born interpretation of that, to find out what it means and how to interpret it, and to interpret it in terms of a probability distribution, you would say that psi 1 of x, and I'll square it. And if you square A function you going to get A squared, and you going to get sine squared, pi x all over L. And now if we roughly draw what that function would look like again its going from zero to L over here on the right. And what you'd find is, at is narrows up a little bit, you'll find much more accurate, representations of these if you go to your text book. So your online resource and I've, that I've recommended. And again squared is time, and it will be positive again. And what you're saying really is if you go along, see here your particles going from zero to L. What you can do is if you pick off regions. Along here, so say you pick up this region here. Again Bone's interpretation says that the probability is the square of the wave function times this little area Dx here. And then if you compare it thus here to here in the middle of the box. Another small region here. Again Dx, And let's again go to the same region here at the end of the box. Now what you note is that this magnitude here, this amplitude, is larger than these pieces. So what the interpretations is telling us is if you have a partial moving back and forth between zero and L in a classical description of that particle. The probability of finding a particle at any point along here is going to be the same as the constant velocity. However, what the wave mechanical interpretation is telling us is that the probability of finding the particle is highest here at the center apox. And then it goes down gradually on each side. So that was for the first wave function that we, for the project in the box. Now if we move on to the second one, and again from our project in a box presentation we show that sign 2 of x is equal to A sign. And this time it's two pi x, over L. So again we plot a dot from zero to L. And what we found for this was that it occupied one wavelength, so you had positive and negative region, something I like that, and that's positive and that's negative. Now if we go over here and we square that function, so that's two of X squared that's equal to A squared, sign squared, 2 pi x, all over L. So now we plot that function. [BLANK_AUDIO]. What we get when we square that function is we now get 2 positive. And so the, positive bits if you like from the, on square function stays the same, but narrows a bit. And now you have two positive values. [BLANK_AUDIO]. So what that's telling us now, for this, is that the maximum probability of find the particle is now at these two peaks here if you like a quarter way out the box quarter way along the line or three quarters of the way along the line. And now what is telling us here is in this region here at that point there you can see it goes to zero. What we've been told is the probability of finding the particle here, the probability density at that specific point is zero. So, in this small region here, it's, it's, it's, it's very low. So it's different to what we saw for the, for the first wave function. But again it contrasts completely with what we talk about classical behavior where again if this particle is moving back and forth between the two, these two. Points then you would expect it to have a constant, the probability of finding that part to go with 22 points at any point between them should be the same. And likewise then we can look at the third way function, that we solve for the particle in the box, so we call this psi 3. And again we've found the last day or the last part, presentation on the part is in a box so now it's equal to 3 pi x, all over L. And now first r square wave function is equal to a squared sin squared 3 pi x, all over L. So again, if we drew out our, our wA well we did draw it out the last A, zero to L. We now, have three half wave lengths. So we go up like that, that, and something like that. So its plus minus for the actual wave function. For the square wave function what we now get is we get. Lets draw it out here to make it quite reasonably accurate. So we got three humps like that, if you like. So this is zero to L. And this is zero to L here. And we'll just call them plus and plus again. So now you see if we look at our probability here, we have highest probability at this point, this point and this point. And if we draw in regions, we have our highest probability of regions here at the top of these peaks and now we have two regions where the actual probability of finding the particle is predicted. It's predicted to be zero. [BLANK_AUDIO]