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OK.

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So let us begin with programming, the wave function and the ion energies of an electron living in the

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box using Python three.

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So first of all, make sure you have these two import commands at the very beginning because we are

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going to plot some data and we are going to use NumPy for some calculations.

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Now next, we will define the potential and to Schrodinger equation and initiating a equation.

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We have the constants each bar and we have two constant m for the mass of the electron.

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And actually, these are very difficult numbers, and we want to make it more easy for us.

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So we just set them to one.

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And later on, we could scale our solution to fit the actual values of these constants.

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But this is not so important here.

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So typically people just set them to one.

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Now next, we will define our potential.

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And here we have our box, which goes from X equal zero to x equal.

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And for a we set the value, which will also be one in this case.

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Now the potential inside of the box, we we will set to zero.

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And outside of the box, the potential will be infinitely large.

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And this makes it a lot easier for us, as we have learned in the analytical case, because in fact,

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we only have to consider the area inside of the box because outside of the box, the potential is infinitely

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large and our wave function will be zero.

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So we don't have to actually define it here.

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The only thing that we need to define is the wave function.

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At the starting position, size is equal to zero.

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So actually, this is just the X dependent solution, so the stationary part.

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But still, I will call it SCI here, but in the energy killer case, we called it PHI.

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Now another thing at the very left of the box, we will have to also define a first derivative.

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So we call this deep sigh.

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And this is equal to a value which is different from zero.

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So far, it doesn't really matter how large this value is.

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We will just use some value and later on we will scale our wave function so that we will get the correct

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value for deep sigh.

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The only thing that's important to you is that it's different from zero.

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OK, so now we have to find some starting conditions, which means the value of science and the first

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derivative of psi at the left part of the box.

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So this means four x equal to zero.

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And now we have to propagate this differential equation, which is the stationary creating equation,

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so that we can calculate these two values for every other position inside of the box.

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So as I told you, we start at X equal to zero.

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And also we have to define some increment.

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So because we cannot solve to the way function continuously from zero to a, we have to define some

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value or some some increments, some steps.

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So I use here eight times.

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Let's let's use something like this.

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So we will have to do 1000 steps until we are at the other end of the box.

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Now what we also need is something to store our data.

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So we have here a list for our X will use, which will be empty in the beginning and also for the values

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of the way function.

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We could also use another list for for deep sigh.

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But actually, we are not really interested in this quantity.

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This is because even the wave function itself doesn't really have a meaning, but we can use the wave

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function to calculate observables and the probability density CI squared.

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OK.

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And now what we will use is a loop, or we use here such a wild statement.

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So what we are going to do is we will increase our value of X by D X until X reaches the value of A.

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So this means in the first loop X will be zero.

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The second loop X will be zero plus the X and then and so on.

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It will keep on increasing in increments of the X until and the last loop.

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It will be equal to a.

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Good.

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Now, what are we going to solve?

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We are going to solve the shooting equation and this means the second derivative of psi, so that's

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called it dead.

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So for a second derivative is equal to two times and over eight bar square.

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And then we can actually combine the potential, which is zero in this case.

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Time is the energy times psi.

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So if you would, yeah, bring this to the other side, you would see that is exactly minus h bar square.

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Over two m d deep psi is equal to E times.

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So this is exactly the stationary shooting equation for this case.

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OK, now we have some statement on how we can calculate deep psi.

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However, what we as so we cannot really use or we cannot calculate this derivative analytically, we

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could, but we don't want to do this here.

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We want to do it numerically.

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And there's actually the second derivative, and you can only calculate a second derivative from the

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first derivative.

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So before we can calculate the second derivative, we need to calculate the first derivative.

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And here we use deep CI is equal to DB ci.

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Plus the derivative of deep CI, which is the deep CI times v x.

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OK, now we have something for the first derivative of CI.

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This we can calculate from from CI itself.

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So CI is equal to CI plus deep CI times d x.

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So we have now and a new value of CI, which is calculated from the derivative times the increment.

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Then we have the first derivative, which is calculated from the second derivative times.

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The increment and the second derivative is related to the wave function itself by just creating an equation.

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OK, so now this shouldn't actually work already.

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What we need now is we we want to store our wax values, so we use here pens so that this value of X

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for each loop is stored in our X list.

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It will add a new entry and the same for sign.

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So again, pens sign.

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Now, what we want to also do is we want to plot.

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So this is the command for plots, so we plot X list versus scientists.

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And another thing that we also have to consider here, the only thing that is left that is not defined

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yet is the energy.

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And actually, we want to determine psi, but also we want to determine the energy.

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So just means so far we don't know what the energy is.

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But this is the whole trick of this method.

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We will just set some value for the energy.

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Let's say one, and we will just test what the wave function looks like for this energy that we have

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to find here.

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And then we will see if it looks reasonable in a physical way.

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So let me run this notebook and see what it looks like.

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So restart, run off.

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OK.

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OK, so here we have now our expel you and remember our box starts from zero.

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Where the wave functions should be zero because we have defined it here.

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And also we have some first derivative, which is different from zero, because here we have to find

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it's actually one.

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So this makes total sense.

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And then the wave function increases in, I would say, something like assign function, which also

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makes sense.

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So that's already good.

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But here at the other end of the box, which is X equal to a which we have set to one here, our wave

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function does not go back to zero.

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So it's in conflict with the other boundary condition.

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And this is, of course, the case because we have not told the code that there is a boundary condition.

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So let's try a different energy.

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OK, so two looks already a bit better, it starts to go down three four.

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OK, we are getting closer.

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Five.

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Oh, OK.

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This looks already pretty good.

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I mean, it's a bit lower than than one.

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Maybe it's something like four point nine.

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OK, yeah, that is actually pretty good.

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So now we could also look at the cyclist and go all the way down.

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So you see, we start here from zero point zero zero one and it goes back down to zero zero zero three

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five.

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OK, that's pretty good, actually.

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And that's the whole point of this method, which is called the shooting method.

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We give it an energy and then we just let the system take a shot and propagate the wave function.

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And if it looks reasonable, then everything is correct and we can say, OK, this is our first eigenvalue

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and this is our first I can function.