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So welcome back to this lecture about the harmonic oscillator in the previous lecture, we have determined

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the first ion energy and the first ion function.

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And also, we have realized that the methods that we are using to so-called shooting methods is quite

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unstable in terms of the starting conditions.

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So here we have to find some value a which we use as the starting point.

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We will start from minus three in this case, and we have seen that when we go to even even smaller

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values or even larger values in absolute value, let's say eight.

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We can see that the wave function just explodes due to this exponential behavior here.

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So that's something we definitely have to avoid.

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So we make we have to make a sufficiently small.

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But then there is another problem because when our energy is very large, then our parabola for the

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potential will be larger and x direction as well.

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So then maybe it is a is equal to three will be way too low or for the higher energies.

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So what I found out, what I just want to tell you here is that we want to use a certain conditions

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that we want to make this value of a energy dependent.

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So we want to say that our parabola f potential is equal to be times x squared is supposed to be the

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or is equal to the energy.

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But what we want is actually that the energy is much, much smaller than this value.

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So let's say, yeah, let's say you buy such a factor of 10 in terms of the size of this B.

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So now if we solve this with respect to A, we get to square root piece square root 10 plus he will

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v OK.

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So this is now our energy dependent value for.

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So of course, we need to

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do it like this.

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OK, so this I tested before that it works.

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OK, so we still get our same value.

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You can see for the energy of zero point five, the value is just slightly larger than this three that

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we had previously.

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But if we do something like this, you can see now the value is quite different.

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So, yeah, OK, and actually, it turns out that we have found already here another I function.

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OK, but what do you want to do next?

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Like we did previously is we want to find this eigenvalues and eigen functions automatically.

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So again, we are going to use a loop as we had before, and this time since it's basically exactly

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the same thing as we had before.

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I will just copy some code.

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So this is exactly the same thing we had in the previous example where we had the particle inside of

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the box.

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But here with this modified potential and also with this modified a border, and then we have to increase

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here.

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Our energy by D, E and D and D is equal to zero point zero five.

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So as it turns out, this solution here is much slower because we are integrating starting from much

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larger values.

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So here we have to integrate four minus three to three, and previously in the box, we only had to

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integrate from zero to one.

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So I will make the energy steps a bit larger this time so that it doesn't take forever.

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So let's see what comes out.

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The energy value is basically zero point five and we get the same wave function as here.

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So it works very well.

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So it's the same algorithm as before.

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The only thing that is changed is the potential this value for a and of course, here this condition,

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um, we had to make this this critical value here a bit larger because actually our wave function will

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never go back to zero, even in the ideal analytical case.

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But since we are inside of the potential here, it will be quite small.

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But I gave it here some tolerance and made it a bit larger.

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0.02.

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OK, now and the next step, we want to go one step further and want to determine the some more eigenvalues

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and ion energies.

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So I will copy again the code from before.

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And so this is again the same principle procedure as we had in the end, the example for the particle

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in the box.

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So we have here three while loops.

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So basically what was added here is this while loop where we add where we loop over a counter and then

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whenever we found an eigen system, we will add we will save it here and this list ion functions and

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eigen energies and in the X values list as before.

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And we also use here this trick, which avoids double counting of the same eigenvalue so we can now

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run this.

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I was really fast, actually.

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And now we can look at the eye and energies.

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OK, so that's actually really interesting and you can see here the iron values, the iron energies

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of 0.5 1.5 2.5.

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So we have equidistant energy levels and this is in fact true.

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We will see this and one of the following sections where we will dissolve the system analytically.

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Maybe we will have spent quite some time achieving this result that the energy level is actually one

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half plus and where end starts from zero to one, two, three to four and so on.

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And of course, we will have done the prefecture, but because we will determine in the analytical case.

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So now let's look at the eigen functions.

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So let us loop over the counter and plot every single X value list versus the EIGEN function list.

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So this is what it looks like here.

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Actually, I have used and max equal to 10.

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Maybe that's a bit extreme.

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Let's use five so that we can see something.

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OK, so we have here our lowest energy level and blue, which does not have a zero.

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Then the next one will be the orange one with one zero.

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Then there will be the green one and the red one and then the purple one with one two three four zeros.

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That's the fifth eigenvalue.

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So this is, as I told you in general statement, that's always gonna be true.

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Now we will normalize our way function as we did previously.

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So let's test again.

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OK.

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Different from different from one.

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So this is because basically, yeah, we have just assumed some value for the wave function at some

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position here at the very left.

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And of course, this value is not correct.

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So we have to find the correct amplitude in order to make the statement true that the absolute screw

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after wave function represents the probability density for the electron and it has to integrate to one

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because the electron has to be somewhere.

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So what we are going to do is we are going to loop over a counter of all of our away functions.

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So now we determine norm for each of the wave functions and we will divide divide each value in the

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list of design functions by the square root of the norm.

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And then we can check again.

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Yes, no, it's normalized to one.

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And we can plot everything again, and you can see now they will have different amplitudes.

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And this is not released.

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Or it could be surprising for you because in the previous example, all of the way functions had the

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same amplitude.

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But as I told you already there, this is not a general statement.

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So in general, the amplitudes for the wave functions can be different.

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And as you see, here, they are, in fact different.

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So now another thing that we want to do is we want to look at the absolute square plotted over X.

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So this is the probability density.

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So we will plot now the X values of the wave function versus the absolute square, which we have used

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some non pi and p square squares every single element in this list so we can plot this now.

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And so, yeah, this is what it is.

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What you can see is the yeah, the probability density.

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So basically, you can estimate here for this blue curve, it goes up to zero point five or something,

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and here it goes from minus two to two.

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So it really has the area of one.

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Now, before we use this alternative way of plotting the wave functions, I want to show you something

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pretty cool that I really liked.

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So first of all, let's calculate a few more energy levels.

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So let's go to 10.

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OK, now we have your 10 eigen functions.

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Um, let's normalize them again.

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OK, so now we have here.

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Ten functions, and we can also plot this one.

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OK.

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Good.

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Now what I want to show you is that we can actually compare this.

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This probability density, let's say off the level, the highest level that we have calculated so and

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maximum minus one which looks like this, we can actually compare this to the to the classical case.

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So this is quantum mechanics and the probability here.

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And we can compare this to the classical probability.

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OK, so classically when you have a harmonic oscillator, for example, you have a pendulum, then you

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will have as a solution something like this.

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You will have, for example, a cosine or a sign function of T times some frequency band that say,

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we have cosine of two.

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You

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know, the classical probability is actually one over the velocity, because if you think about it,

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when our pendulum spins or when your pendulum has the highest velocity, it will spend the least amount

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of time at a certain position.

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So when we look at the pendulum at a random point of time, it will be what the other, the possibility

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of finding the pendulum in a certain state will be largest when the velocity is zero.

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So I think to extrema and in the middle, where the velocity is highest, the probability is smallest

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for finding the pendulum in this state.

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So we can now use this equation to determine the the velocity.

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So this is actually the of T is equal to minus sine of T.

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And also, we want to express this velocity not in terms of T, but in terms of X.

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So actually, we want to have this one here.

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So what we use here is T of X is the inverse function, of course.

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So we have our cosine of X.

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So we can just write minus sine or we can can do it like this or the of X is equal to minus sign of

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Orcus cosine of X.

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So this is now the classical solution for the velocity.

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And now we calculate this probability.

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So P of X is one over the velocity, so it's basically a minus one.

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We have X is minus one over the sign of the R because cosine of X.

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And actually, when you use some, some basic trigonometry, you can find that this is actually equal

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to.

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Two minus one over the square root of one minus x squared.

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And now for the probability since it's always the absolute value, we can just do it like this.

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What's happening here?

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OK?

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Um, so we will get this one here.

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So this is the equation for the classical probability that we are now going to plot.

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So let us do this.

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No, let's plot this classical equation on top of this quantum mechanical result.

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First of all, we need some X values for this plot.

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So we have it and p range.

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Um, let's say, you know, let's let's let's we scale it later minus one to one, and we use 100 points.

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Now let's let's just plot it pretty plots.

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And we start with this x one.

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That's for a moment not care about the scaling on the x axis.

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We just write down what we had here.

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So this is one minus x one squared.

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OK.

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OK, this looks pretty large, let's.

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OK.

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Still pretty large, 20.

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Mm hmm.

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OK.

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And now we can scale the X coordinate properly, probably.

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So he we need to square roots of the energy that we are using and then G's.

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And yes, this is the eigenen energy that we're using, and we just divide it by the value of the potential

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because then we get our proper X value.

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OK, so we have no wind blew the probability of finding the particle in the quantum mechanical case

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and in orange, we have to probability for finding the classical particle.

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Actually, I would have to normalize this function here because yeah, if just arbitrarily divided by

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20.

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But that's that's not so important, I would say for the moment.

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Let's see.

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You can you can really see how the classical distribution resembles the quantum mechanical distribution

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here.

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So you can see how similar quantum mechanics is to classical mechanics in a way.

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So these maxima really follow the same curve, but additionally, you will have these quantum oscillations.

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And this is again very typical, very typical observation on how classical physics and quantum physics

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relate to each other.

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And especially, you will notice that this will become more and more similar when you go to larger length

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scales so that quantum effects are not that relevant anymore.

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OK, so this was just a short course that we will also discuss later again when we will solve the harmonic

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oscillator analytically.

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But now the only thing that's left to do is we use our alternative plotting scheme.

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So as before we used wave function, it is already normalized and now we add on top our energy will

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use the ion energies.

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So this will just shift these functions a year or better, say these functions here away from each other.

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So let's do this and then let's just re plot these functions.

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OK.

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So that looks pretty cool.

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Now, additionally, what I want to do is I want to plot the potential

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just to show you some things.

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So we have again our X1 range and this time we plot

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this is X range over just about potential.

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So that's just three times x squared.

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OK.

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Yeah.

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Of course, we have to plot from minus eight to a because otherwise it will just give this.

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OK, so here you can really see how use you get such an oscillating behavior inside of the potential.

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So basically, strictly speaking, in a physical sense, this means that when the energy is larger than

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your potential, then our electrons behave a bit like free electrons, so they start to oscillate.

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So we will get some cosine like function.

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We will see later in the analytical case that it's not actually a sign or co-sign, but it's much,

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much more difficult solution.

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But here, especially in the middle, it looks like just a sign function.

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And then when we come close to the case where the energy is equal to the potential, or even when the

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energy is lowered and the potential, then we will again see our decaying exponential function.

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So this is because, classically speaking, the particle would be forbidden to be in such a state.

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But quantum mechanically, the particle can tunnel or it can penetrate into this potential barrier,

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and therefore we will have a very small probability density inside of two potential.